diff --git "a/59E4T4oBgHgl3EQfBgvS/content/tmp_files/load_file.txt" "b/59E4T4oBgHgl3EQfBgvS/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/59E4T4oBgHgl3EQfBgvS/content/tmp_files/load_file.txt" @@ -0,0 +1,1636 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf,len=1635 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='04853v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='EM] 12 Jan 2023 Testing for Coefficient Randomness in Local-to-Unity Autoregressions Mikihito Nishi∗1 1Graduate School of Economics, Hitotsubashi University January 13, 2023 Abstract In this study, we propose a test for the coefficient randomness in autoregressive models where the autoregressive coefficient is local to unity, which is empirically relevant given the results of earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under this specification, we theoretically analyze the effect of the correlation between the random coefficient and disturbance on tests’ properties, which remains largely unexplored in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Our analysis reveals that the correlation crucially affects the power of tests for coefficient randomness and that tests proposed by earlier studies can perform poorly when the degree of the correlation is moderate to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The test we propose in this paper is designed to have a power function robust to the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because the asymptotic null distribution of our test statistic depends on the correlation ψ between the disturbance and its square as earlier tests do, we also propose a modified version of the test statistic such that its asymptotic null distribution is free from the nuisance parameter ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The modified test is shown to have better power properties than existing ones in large and finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Keywords: random coefficient autoregression, local to unity, Bonferroni test JEL Codes: C12, C22 ∗I am greatly indebted to Eiji Kurozumi, my advisor, for discussions and his help, support and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' I am also grateful to Mototsugu Shintani and Yohei Yamamoto for their helpful com- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' All errors are mine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Address correspondence to: Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' e-mail: ed225007@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='hit-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='jp 1 Introduction In this paper, we consider the random-coefficient autoregressive (RCA) model, yt = (ρ + ωvt)yt−1 + εt, t = 1, 2, · · · , T, (1) where (εt, vt)′ is a random vector with E[εt] = E[vt] = 0 and V[vt] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Model (1) is a gener- alization of the usual AR(1) model, in that the autoregressive coefficient fluctuates over time around its mean ρ with variance ω2, instead of being constant at ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Since Nicholls and Quinn (1982), much attention has been paid to the estimation and inference theory of model (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' see, for example, Hwang and Basawa (2005), Aue and Horv´ath (2011), Horv´ath and Trapani (2019) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One aspect of the inferential theory for (1) that has attracted econometricians and statis- ticians is how to test the hypothesis of ω2 = 0, that is, how to test the null hypothesis of the usual autoregressive specification against the RCA alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nicholls and Quinn (1982) and Lee (1998) proposed test statistics for testing H0 : ω2 = 0 under the stationarity condition ρ2 + ω2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nagakura (2009a) showed the Lee (1998) test statistic has the same asymptotic null distribution when ρ = 1 as when |ρ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 The condition, ρ2 + ω2 ≤ 1, which these tests rest on, however, is restrictive and limits their applicability in practice, in view of the empirical analyses by Hill and Peng (2014) and Horv´ath and Trapani (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' They applied model (1) to several macroeconomic variables and estimated ρ to be near 1 for most of the variables, with ρ estimated to be greater than 1 for some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For instance, the ρ estimates obtained by Horv´ath and Trapani (2019) took values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='9884 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='0021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Their results indicate that nonstationary RCA models where ρ2 + ω2 and ρ may be greater than 1 with ρ near unity are empirically relevant, and thus testing procedures are required that are valid under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Horv´ath and Trapani (2019) proposed a test statistic for H0 : ω2 = 0 valid under the nonstationarity conditions (as well as under the stationarity conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Another insight earlier studies provide is that the variation of the autoregressive root is smaller (if it exists) than assumed in the extant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For example, according to the empirical analysis conducted by Horv´ath and Trapani (2019), 4 variables out of 7 were estimated to have positive variance (ω2 > 0) in their coefficient, which is between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 × 10−5 1Nagakura (2009b) also showed the Lee (1998) test is consistent when ρ2+ω2 > 1 and some other conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, he did not show the test statistic has the same limiting null distribution when ρ > 1 as when |ρ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, Horv´ath and Trapani (2019) studied the finite-sample power of their test only for ω2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5 × 10−1, which are of larger magnitude than observed in practice (as long as macroeconomic variables are concerned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2 Therefore, it is unclear whether their test performs well when ω2 is of small magnitude that seems to be typical in empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In fact, their test has almost no distinguishing power when ω2 is relatively small, as our simulation studies will reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Given these observations, we consider model (1) with ρ near unity and ω2 near zero and employ the following near unit root RCA model: yt = (ρT + ωT vt)yt−1 + εt, (2) where ρT = 1 + a/T and ωT = c/T 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When ω2 T = 0, (2) is the popular (conventional) local-to-unity AR(1) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This specification has been useful in analyzing the power of unit root tests (Elliott, Rothenberg and Stock, 1996) and in developing estimation and inference theory for models involving persistent variables (see, inter alia, Phillips, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Stock, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Campbell and Yogo, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Phillips, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The local-to-zero variance ω2 T = c2/T 3/2 has been employed under ρT = 1 by McCabe and Smith (1998) and Nishi and Kurozumi (2022) to derive and compare the local asymptotic power functions of unit root tests of ω2 = 0 against ω2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Introducing the local-to-zero variance provides us with a convenient framework in which we can evaluate power functions of several tests against the alternatives that are close to the null of ω2 = 0 and seem relevant for empirical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3 Model (2) integrates these two local-to-unit-root parametrizations, thereby rendering itself an empirically relevant random-coefficient model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In the literature, testing procedures for H0 : ω2 = 0 have been studied in the spe- cial case with ρT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Such a model is called stochastic unit root (STUR) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Test statistics for H0 : ω2 = 0 in the STUR model have been proposed by earlier studies, includ- ing McCabe and Tremayne (1995), Leybourne, McCabe and Tremayne (1996) and Distaso (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Moreover, McCabe and Smith (1998) and Nishi and Kurozumi (2022) analyzed the power properties of several tests for H0 : ω2 = 0 under the STUR modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' On the one hand, the STUR model is a generalization of the pure unit root model, which is a main reason authors have paid much attention to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' But on the other hand, the assumption of ρT exactly being unity is a restrictive condition that is unlikely to hold in empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Our model, 2Horv´ath and Trapani (2022) applied the RCA model to a cryptocurrency index and estimated ω2 to be around 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 3For example, when T = 200 and 0 < c2 ≤ 50, the variance ω2 T takes values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='8 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2 (2), generalizes the STUR model by allowing ρT to take a value different from unity (but near it), which is a more realistic assumption given the empirical analyses conducted by earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nishi and Kurozumi (2022) revealed that, under the STUR (ρT = 1) modelling with Corr(εt, vt) = 0, the test for H0 : ω2 = 0 proposed by Lee (1998) and Nagakura (2009a) (hereafter, LN) has a higher local asymptotic power function than other tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We will demonstrate that this is also the case when ρT = 1+ a/T and Corr(εt, vt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In this paper, however, it will be shown that the LN test can perform poorly when Corr(εt, vt) ̸= 0, for both the cases ρT = 1 and ρT = 1 + a/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We will therefore propose several tests for H0 : ω2 = 0 whose power properties are robust to the value of Corr(εt, vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One of those tests turns out to be more powerful than the LN test for moderate to large values of Corr(εt, vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To the best of our knowledge, this study is the first to investigate how the value of Corr(εt, vt) affects the power properties of tests for H0 : ω2 = 0, with the only exception of Su and Roca (2012), who analyzed through simulation this effect in finite samples under ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The other issue that this paper tackles is how to remove the effect of nuisance parameters from the null distributions of the test statistics mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As pointed out by Nagakura (2009a), the limiting null distribution of the LN test statistic depends on the correlation between εt and ε2 t −σ2 ε, Corr(εt, ε2 t −σ2 ε), and so do the test statistics proposed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' If the true value of ρT is known, the nuisance parameter can be removed straightforwardly by a similar way to the modification proposed by Nagakura (2009a), but this is not the case when the true ρT is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This problem is caused by the fact that the localizing parameter a cannot be consistently estimated, as will be made clear later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As a solution for this problem, we propose a testing procedure based on the Bonferroni approach as Campbell and Yogo (2006) and Phillips (2014) did in the context of predictive regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In Section 2, we consider the local- to-unity model (2) with the true value of ρT (or equivalently a) known, to uncover the effect Corr(εt, vt) has on power properties of tests for H0 : ω2 = 0 and propose new tests that have power functions robust to this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We also propose a modification to make these tests independent of Corr(εt, ε2 t − σ2 ε), a nuisance parameter, under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In Section 3, we consider model (2) with unknown ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because the modification proposed in the preceding section cannot be directly applied in this case due to the fact that a is not consistently estimable, we base our tests on the Bonferroni approach by constructing a confidence interval for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Section 4 analyzes the finite-sample properties of our tests and compares them with 3 those of existing tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In Section 5, we apply our tests to real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2 The Case of Known ρT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 The effect of Corr(εt, vt) In this section, we begin our analysis under the assumption that the true value of ρT is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This assumption will be relaxed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Our analysis for model (2) in this and the next section is conducted under the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (εt, vt)′ ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d (0, Ω), where Ω := � σ2 ε σεv σεv 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Also, E[ε4 t] < ∞ and E[v8 t ] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Moreover, y0 = op(T 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Define ηt := ε2 t − σ2 ε and σ2 η := E[η2 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Define also the partial sum process (Wε,T, Wη,T )′ on [0, 1] by Wε,T(r) := T −1/2σ−1 ε �⌊Tr⌋ t=1 εt and Wη,T (r) := T −1/2σ−1 η �⌊Tr⌋ t=1 ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, it follows from the functional central limit theorem (FCLT) that under Assumption 1, as T → ∞ �Wε,T Wη,T � ⇒ �Wε Wη � , in the Skorokhod space D[0, 1], where (Wε, Wη)′ is a vector Brownian motion with the co- variance coefficient �1 ψ ψ 1 � with ψ := E[εtηt]/(σεση).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Note that Wε and Wη are not necessarily independent because of the covariance ψ, and the Brownian motion Wη satisfies the following equality in distribution: Wη d= ψWε + � 1 − ψ2W1, (3) where W1 is a standard Brownian motion independent of Wε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Following the argument by Nishi and Kurozumi (2022), we can show that under Assumption 1, the standardized process T −1/2y⌊T·⌋ on [0, 1] constructed from (2) weakly converges to the Ornstein-Uhlenbeck (OU) process σεJa(·), where Ja solves dJa(r) = aJa(r)dr+dWε(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We can also construct consistent estimators of variances, namely, σ2 ε and σ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Define zt(ρT ) := yt − ρT yt−1(= ωT vtyt−1 + 4 εt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, the estimators ˆσ2 ε,T(ρT ) := T −1 �T t=1 z2 t (ρT ) and ˆσ2 η,T (ρT ) := T −1 �T t=1{z2 t (ρT ) − ˆσ2 ε,T(ρT )}2 are consistent for σ2 ε and σ2 η, respectively, which is proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Several tests of H0 : ω2 T = 0 for model (2) have been proposed by earlier work such as LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The LN test statistic is defined by LNT (ρT ) := �T t=1(y2 t−1 − T −1 �T t=1 y2 t−1)z2 t (ρT ) ˆση,T (ρT ){�T t=1(y2 t−1 − T −1 �T t=1 y2 t−1)2}1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nishi and Kurozumi (2022) found that the LN test has a high power function under the assumption that σεv = 0 and ρT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As pointed out by Nishi and Kurozumi (2022), the LN test, which was originally derived as a locally best invariant test, can also be obtained as the t-test for H0 : ω2 T = 0 under σεv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To see this, note that from model (2), a simple calculation gives z2 t (ρT ) = σ2 ε + ω2 Ty2 t−1 + ξt, (4) where ξt := ω2 Ty2 t−1(v2 t − 1) + 2ωT yt−1εtvt + (ε2 t − σ2 ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because E[ξt] = 0 and E[y2 t−1ξt] = 0 under Assumption 1 with σεv = 0, model (4) may be viewed as the linear regression model with ξt playing the role of the disturbance, and the LN test statistic is obtained as the t-test for H0 : ω2 T = 0 (with the variance estimator under the null used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In view of this observation, the LN test seems to be a natural test for H0 : ω2 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' How- ever, this may not be the case when σεv ̸= 0, because it results in E[y2 t−1ξt] = 2ωT σεvE[y3 t−1] ̸= 0, that is, endogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In fact, as we will shortly show, the power of the LN test is crucially affected by the value of σεv, and the greater the value of σεv (in absolute value), the more poorly the LN test performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because we consider the localized model (2), the setup for analyzing the influence σεv has is also a localized one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To derive relevant local asymptotic distributions, we localize the correlation (rather than the covariance σεv) between εt and vt in the following way: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Corr(εt, vt) = σεv/σε = q/T 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Here, q is interpreted as the correlation coefficient in the limit as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under this localization, the LN test statistic has the following asymptotic distribution: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under Assumptions 1 and 2, we have LNT (ρT ) ⇒ � 1 0 � Ja,2(r)dWη(r) �� 1 0 �� Ja,2 �2(r)dr �1/2 + σ2 ε ση � c2 � 1 0 �� Ja,2 �2(r)dr + 2cq � 1 0 � Ja,2(r)� Ja,1(r)dr �� 1 0 �� Ja,2 �2(r)dr �1/2 � , (5) where � Ja,1(r) := Ja(r) − � 1 0 Ja(s)ds and � Ja,2(r) := J2 a(r) − � 1 0 J2 a(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 5 Note that when q = 0 and ρT = 1 (or a = 0), the limit distribution reduces to the one Nishi and Kurozumi (2022) derived (their equation (19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nishi and Kurozumi (2022) found that the LN test performs better than other tests when q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To see how the value of q alters the LN test’s power properties, we simulate the asymptotic distribution in (5) by 100,000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The replications are based on εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1), so that σ2 ε = 1, σ2 η = 2 and ψ = E[εtηt]/(σεση) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Note that the effect of q on the limit distribution is symmetric when ψ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' that is, q = ¯q and q = −¯q produce the identical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This is because (Wε, Wη)′ d= (−Wε, Wη)′ when ψ = 0, and hence (Ja, Wη)′ d= (−Ja, Wη)′ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus, in this simulation, and in the replications conducted later where ψ = 0 holds, we only consider positive values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Figure 1 shows the power functions of the LN test for q = 0, 1, 2, 3 under a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One noticeable feature is that as q gets larger, the power function gets lower and flatter for c2 ≥ 5, while the power becomes greater for c2 ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Since the power gain over c2 ≤ 5 is obviously outweighed by the power loss over c2 ≥ 5, we conclude that the LN test performs poorly when q is large (in absolute value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4 The LN test’s poor performance for large q could be attributed to the endogeneity E[y2 t−1ξt] = 2cσεqT −1E[y3 t−1] ̸= 0 in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One solution for this endogeneity is to augment model (4) by adding yt−1 as a regressor, z2 t (ρT ) = σ2 ε + δT yt−1 + ω2 T y2 t−1 + ξ∗ t , (6) where δT := 2cσεqT −1 and ξ∗ t := ω2 T y2 t−1(v2 t − 1) + 2ωT yt−1(εtvt − σεv) + (ε2 t − σ2 ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because this augmented model is free from the endogeneity, that is, E[ξ∗ t ] = E[yt−1ξ∗ t ] = E[y2 t−1ξ∗ t ] = 0 under Assumption 1, we propose using the t-test for H0 : ω2 T = 0 in model (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We also propose using the Wald test for (δT , ω2 T ) = (0, 0), because ω2 T = 0 if and only if (δT , ω2 T ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To express the t and Wald test statistics, first regress z2 t (ρT ), yt−1 and y2 t−1 on a constant: �z2 t (ρT ) = δT � yt−1 + ω2 T � y2 t−1 + �ξ∗ t , (7) where �z2 t (ρT ) := z2 t (ρT ) − T −1 �T t=1 z2 t (ρT ), and � yt−1, � y2 t−1 and �ξ∗ t are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Let � Z2(ρT ) := ( �z2 1(ρT ), �z2 2(ρT ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � z2 T (ρT ))′, � X1 := ( �y0, �y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � yT−1)′, � X2 := ( �y2 0, �y2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � y2 T−1)′, and � Ξ∗ := ( �ξ∗ 1, �ξ∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � ξ∗ T )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, model (7) can be expressed in matrix notation as � Z2(ρT ) = � XθT + � Ξ∗, (8) 4In our unreported simulations, we also found a similar tendency in the power function of Distaso (2008)’s LM test, which is based on the assumption of εt and vt being independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The results are omitted to save space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 6 where � X := (� X1, � X2) and θT := (δT , ω2 T )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The t and Wald test statistics are then given by tˆω2 T (ρT ) := ˆσ−1 � ξ∗ (ρT )(� X2 ′M1 � X2)−1/2 � X2 ′M1� Z2(ρT ) and WT (ρT ) := ˆθT (ρT )′( � X′ � X)ˆθT (ρT ) ˆσ2 � ξ∗(ρT ) , where M1 := IT − � X1(� X1 ′ � X1)−1 � X1 ′, and ˆσ2 � ξ∗(ρT ) and ˆθT (ρT ) are the OLS variance and coefficient estimators of (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We shall call the t and Wald tests in model (6) augmented t and Wald tests, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The limiting distributions of these augmented test statistics are collected in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under Assumptions 1 and 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' we have tˆω2 T (ρT ) ⇒ � 1 0 Qa(r)dWη(r) �� 1 0 Q2a(r)dr �1/2 + c2σ2 ε ση �� 1 0 Q2 a(r)dr �1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (9) where Qa(r) := � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r) − � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(s)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(s)ds � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(s)ds � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' and WT (ρT ) ⇒ � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) � + σ2 ε ση � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr ��′ × � � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dr � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr �−1 × � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) � + σ2 ε ση � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (10) In particular, the asymptotic null distributions (c2 = 0) are tˆω2 T (ρT ) ⇒ � 1 0 Qa(r)dWη(r) �� 1 0 Q2a(r)dr �1/2 and WT (ρT ) ⇒ �� 1 0 � Ja,1(r)dWη(r) � 1 0 � Ja,2(r)dWη(r) �′ � � 1 0 �� Ja,1 �2(r)dr � 1 0 � Ja,1(r)� Ja,2(r)dr � 1 0 � Ja,2(r)� Ja,1(r)dr � 1 0 �� Ja,2 �2(r)dr �−1 �� 1 0 � Ja,1(r)dWη(r) � 1 0 � Ja,2(r)dWη(r) � , 7 which are standard normal and chi square with 2 degrees of freedom, respectively, when ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' There are two points worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' First, the augmented t test statistic in model (6) has the asymptotic null distribution independent of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This is a direct result of the augmentation, which is intended to remove the endogeneity from the linearized model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When performing the augmented t test, we regress the endogenous regressor � y2 t−1 (and �z2 t (ρT )) on � yt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In the limit, this projection amounts to replacing � Ja,2 in the limiting distribution of LNT with Qa, the residual from the linear projection of � Ja,2 on � Ja,1 in the Hilbert space (see, for example, Phillips and Ouliaris (1990)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The second point is that the limiting null distributions of the augmented t and Wald test statistics are standard normal and chi square with 2 degrees of freedom, respectively, when ψ = E[εtηt]/(σεση) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This will be seen immediately upon noting Ja and Wη are independent when ψ = 0 (due to the independence between Wε and Wη), and the limit distributions under c2 = 0 conditional on Wε are standard normal and chi square with 2 degrees of freedom (and so are they unconditionally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Along the lines of this argument, the asymptotic null distribution of the LN test statistic is seen to be standard normal when ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 Removing ψ from the limiting null distributions Unless ψ = 0, the test statistics discussed thus far have asymptotic null distributions dependent on ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Actually, this dependence stems from the strong persistence of the regressors yt−1 and y2 t−1 and the long run endogeneity present in the linearized models (4) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To illustrate this, consider model (4) and the LN test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under the null of ω2 T = 0, the model reduces to z2 t (ρT ) = σ2 ε + ω2 T y2 t−1 + ηt, (11) where ηt = ε2 t − σ2 ε and yt = ρT yt−1 + εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This model is free from endogeneity because E[ηt] = E[y2 t−1ηt] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, the regressor y2 t−1 is, in a sense, “endogenous” in the limit as T → ∞, because of the correlation ψ between its innovation εt and the disturbance ηt = ε2 t − σ2 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Indeed, the LN test statistic becomes LNT (ρT ) = �T t=1 � y2 t−1ηt ˆση,T (ρT ){�T t=1( � y2 t−1)2}1/2 ⇒ � 1 0 � Ja,2(r)dWη(r) �� 1 0 �� Ja,2 �2(r)dr �1/2 , 8 where � Ja,2(r) (≈ T −1 � y2 ⌊Tr⌋) is correlated with the differential dWη(r) (≈ T −1/2η⌊Tr⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This correlation originates from that between Wε and Wη, which is denoted by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Therefore, the correlation between the regressor’s innovation εt and the regression disturbance ηt affects the test statistic’s behavior in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Interestingly, the situation we are in is analogous to the one that has been considered in the literature on predictive regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In predictive regressions, a main aim is typically to investigate whether stock returns (rt) can be predicted by another economic or financial variable (xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To test the predictability of rt, predictive regressions involve regressing rt on a constant and the lag of xt: rt = µ + βxt−1 + ur,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Here, β represents the predictability of rt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' the stock return is not predictable by xt if β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In the literature on predictive regressions, it has been well known that the usual t test for the hypothesis β = 0 can be misleading when the regressor xt is persistent, or has a generating mechanism of the form xt = ρx,Txt−1 + ux,t with ρx,T = 1 + ax/T, and its innovation ux,t is correlated with the regression disturbance ur,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The problem arising in such a case is that the asymptotic null distribution of the t statistic depends on Corr(ux,t, ur,t) and is not standard normal unless Corr(ux,t, ur,t) = 0 (see, for example, Campbell and Yogo, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Our situation here is essentially the same: the regressor y2 t−1 in (11) is persistent with the mechanism yt = ρT yt−1+εt and its innovation εt is correlated with the regression disturbance ηt, which results in the test statistic having the limiting null distribution dependent on the correlation ψ = Corr(εt, ηt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For the predictive regression model, there is an extensive literature on this problem, and numerous solutions have been proposed (Campbell and Yogo, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Phillips and Lee, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Phillips, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Kostakis, Magdalinos and Stamatogiannis, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For our case, fortunately, one of those solutions can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Specifically, following Campbell and Yogo (2006), we modify the test statistics (LN, augmented t and augmented Wald) so that their asymptotic null distributions are standard normal and chi square with 2 degrees of freedom, irrespective of the value of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To explain the idea, take the LN test as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Note that under the null of c2 = 0, z2 t (ρT ) in the numerator of the LN test statistic may be asymptotically expressed as z2 ⌊Tr⌋(ρT ) = σ2 ε + (ε2 ⌊Tr⌋ − σ2 ε) ≈ σ2 ε + T 1/2 × σηdWη(r) d= σ2 ε + T 1/2 × σηψdWε(r) + T 1/2 × ση � 1 − ψ2dW1(r), 9 given the distributional equivalence (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This observation leads us to propose the following modification to remove the effect of ψ: z2 t (ρT ) → z2∗ t (ρT ) := z2 t (ρT ) − ˆση,T (ρT ) ˆψT (ρT )(zt(ρT )/ˆσε,T (ρT )) � 1 − ˆψ2 T (ρT ) , (12) where ˆψT (ρT ) := T −1 �T t=1 zt(ρT ){z2 t (ρT ) − ˆσ2 ε,T(ρT )}/(ˆσε,T (ρT )ˆση,T (ρT )) is an estimator of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5 In Appendix B, we show that ˆψT (ρT ) is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Note that zt(ρT ) is used as a proxy for T 1/2 × σεdWε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' With the replacement given in (12), the modified LN test statistic is defined by LN∗ T (ρT ) := �T t=1(y2 t−1 − T −1 �T t=1 y2 t−1)z2∗ t (ρT ) ˆση,T (ρT ){�T t=1(y2 t−1 − T −1 �T t=1 y2 t−1)2}1/2 The modified augmented t and Wald tests are based on the following regression model: � Z∗ 2(ρT ) = � XθT + � Ξ∗∗, (13) where � Z∗ 2(ρT ) := ( � z2∗ 1 (ρT ), � z2∗ 2 (ρT ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � z2∗ T (ρT ))′ with � z2∗ t (ρT ) := z2∗ t (ρT )−T −1 �T t=1 z2∗ t (ρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The modified augmented test statistics are t∗ ˆω2 T (ρT ) := ˆσ−1 � ξ∗∗(ρT )(� X2 ′M1 � X2)−1/2 � X2 ′M1� Z∗ 2(ρT ) and W ∗ T (ρT ) := ˆθ∗′ T (ρT )( � X′ � X)ˆθ∗ T (ρT ) ˆσ2 � ξ∗∗(ρT ) , where ˆσ2 � ξ∗∗(ρT ) and ˆθ∗ T (ρT ) are the OLS variance and coefficient estimators of (13), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under Assumptions 1 and 2, we have LN∗ T (ρT ) ⇒ � 1 0 � Ja,2(r)dW1(r) �� 1 0 �� Ja,2 �2(r)dr �1/2 + σ2 ε ση � 1 − ψ2 � c2 � 1 0 �� Ja,2 �2(r)dr + 2cq � 1 0 � Ja,2(r)� Ja,1(r)dr �� 1 0 �� Ja,2 �2(r)dr �1/2 � , (14) t∗ ˆω2 T (ρT ) ⇒ � 1 0 Qa(r)dW1(r) �� 1 0 Q2a(r)dr �1/2 + c2σ2 ε ση � 1 − ψ2 �� 1 0 Q2 a(r)dr �1/2 , (15) 5In fact, Campbell and Yogo (2006) proposed the modification based on the optimality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 10 and W ∗ T (ρT ) ⇒ � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dW1(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dW1(r) � + σ2 ε ση � 1 − ψ2 � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr ��′ × � � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dr � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr �−1 × � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dW1(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dW1(r) � + σ2 ε ση � 1 − ψ2 � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (16) where W1 is a standard Brownian motion independent of Wε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' It should be noticed that the modified test statistics have pivotal asymptotic null distribu- tions (standard normal and chi square with 2 degrees of freedom), thanks to the independence between Ja and W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Also note that the limiting distributions are unaffected by our modifi- cation when ψ = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Theorems 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Figure 2 compares the asymptotic power functions of the modified LN, augmented t and augmented Wald tests under q = 0, 1, 2, 3 and a = 0 with εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When q is small (q = 0, 1), the LN test performs best, and the Wald test’s power function is slightly below the LN test’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for the comparison between the augmented t and Wald tests, the Wald test has better power properties than the t test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When q = 2, 3, the Wald test outperforms the LN test (and the augmented t test) with the greater dominance by the Wald test for larger q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To translate these results into finite-sample ones, consider for example the case of T = 200, in which case −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='76 < q < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Based on the results for the local asymptotic case, it is expected that the augmented Wald test performs more poorly than the LN test if |q| ≤ 1, or |Corr(εt, vt)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='266 in this case, and outperforms the LN test otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To see whether this reasoning gives a good approximation of finite-sample results, we present the size-adjusted power functions for T = 200 in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The calculation of these power functions are based on 20,000 replications where Corr(εt, vt) ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='75} and ω2 T ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='0177 so that c2 ≤ 50 under T = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Figure 3 shows the local asymptotic analysis can well predict the finite-sample results: the LN test performs slightly better than the augmented Wald test when |Corr(εt, vt)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='25, but the latter test outperforms the former otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We also compute the asymptotic power functions under a ∈ {−5, −10}, to investigate the effect of the a value on the power properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The computed power functions are displayed in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When a < 0, the general pattern of the power properties is the same 11 as when a = 0: the LN test performs best for small q, and the Wald test performs best for moderate to large q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, the powers of all the tests we consider get lower as a deviates from 0 (as long as a < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='6 A similar tendency of the power properties of tests for H0 : ω2 T = 0 has been observed through simulation by earlier work such as Nagakura (2009a) and Horv´ath and Trapani (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nonetheless, even when a < 0, the Wald test’s power function is increasing in q while the LN test’s is decreasing in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This renders the Wald test preferable in empirical applications, where in general the degree of the correlation between the random coefficient and disturbance is unknown to practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Although all the above results are based on the assumption that the true ρT is known so that the tests discussed so far are infeasible, they suggest the potential of the augmentation to enhance the ability to detect the nonzero variance of the autoregressive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 3 The Case of Unknown ρT In this section, we consider model (2) with the true ρT unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To deal with the uncertainty about ρT , we use the OLS estimator ˆρT of ρT , which is defined by ˆρT := �T t=1 yt−1yt/ �T t=1 y2 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In Appendix C, we show that ˆρT is T−consistent, and that other estimators defined in the preceding section such as ˆσ2 ε,T(ρT ) remain consistent if they are computed using ˆρT instead of the true ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Given these consistency results, one may expect that the asymptotic behaviors of LN∗ T (ˆρT ), t∗ ˆω2 T (ˆρT ) and W ∗ T (ˆρT ) are the same as those of LN∗ T (ρT ), t∗ ˆω2 T (ρT ) and W ∗ T (ρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Unfortunately, however, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Indeed, it can be shown that the limiting null distributions of the test statistics, if calculated using ˆρT , are no longer normal or chi square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This is essentially because a is not consistently estimable, which has been well known in the literature (Campbell and Yogo, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To deal with this problem, following Cavanagh, Elliott and Stock (1995) and Campbell and Yogo (2006), we base our tests on the Bonferroni approach by using a confidence interval for ρT in- stead of its point estimate ˆρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The Bonferroni-based test consists of two steps: first, construct a confidence interval for ρT , and then repeat either of the tests considered above using all the hypothetical ρT values belonging to the confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Specifically, letting ST (ρT ) denote either the modified LN, augmented t or augmented Wald test statistic (calculated using ρT ), the testing procedure based on the Bonferroni approach is described as follows: 6According to the results of our simulation studies given later, deviations from 0 by positive a seem to lead to higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 12 Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Given the data {yt}T t=0, calculate for each hypothetical ¯ρT = 1 + ¯a/T (on some grid) the t statistic t ¯ ρT := (ˆρT − ¯ρT )(ˆσ−2 T (ˆρT ) �T t=1 y2 t−1)1/2, to construct the (equal-tailed) 100(1 − α1)% confidence interval for ρT , denoted by CI(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' That is, CI(α1) is the collection of all the ¯ρT = 1 + ¯a/T that satisfies cv(¯a)α1/2 ≤ t ¯ ρT ≤ cv(¯a)1−α1/2 with cv(¯a)α1/2 and cv(¯a)1−α1/2 denoting the lower and upper α1/2 quantiles of the asymptotic distribution of t ¯ ρT derived under the assumption that ρT = ¯ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For each ¯ρT ∈ CI(α1), calculate ST (¯ρT ) and compare its value with the critical value cvα2 with significance level α2 (based on the standard normal or chi square distribu- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Reject the null only if all the calculated ST (¯ρT ) (¯ρT ∈ CI(α1)) exceed the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We shall call tests following these two steps Bonferroni tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The asymptotic distribution of t ¯ ρT under ρT = ¯ρT is � 1 0 J¯a(r)dWε(r)/{ � 1 0 J2 ¯a(r)dr}1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The quantiles cv(¯a)α1/2 and cv(¯a)1−α1/2 can be found through simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' By the Bonferroni inequality, the resulted test is (asymptotically) a test with significance level α = α1 + α2: under H0, Pa,ψ � � ¯ρT ∈CI(α1) � ST(¯ρT ) > cvα2 �� = Pa,ψ � � ¯ρT ∈CI(α1) � ST (¯ρT ) > cvα2 � |a ∈ CI(α1) � Pa,ψ � a ∈ CI(α1) � + Pa,ψ � � ¯ρT ∈CI(α1) � ST (¯ρT ) > cvα2 � |a /∈ CI(α1) � Pa,ψ � a /∈ CI(α1) � ≤ α2(1 − α1) + α1 ≤ α, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Campbell and Yogo (2006) constructed the confidence interval for ρT by inverting the Dickey-Fuller type t statistic, which is calculated by centering ˆρT by unity rather than ¯ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, it has been known in the literature on predictive regressions that the use of the Dickey-Fuller type t ratio leads to severe size distortion (Phillips, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We found in our unreported simulations that our Bonferroni tests also suffer from the same problem when it is based on the Dickey-Fuller t ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Following the theoretical analysis by Phillips (2014), we here construct the confidence interval for ρT using the t statistic centered by hypothetical ¯ρT ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 13 Although the Bonferroni test defined above is a valid test with significance level α = α1 +α2, the test’s type 1 errors (dependent on (a, ψ)) can be quite smaller than α, as pointed out by Cavanagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (1995) and Campbell and Yogo (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To make the type 1 errors close to the desired nominal level, ˜α (say), they proposed using a pair (α1, α2) with which the Bonferroni test’s type 1 errors are close to the given ˜α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Theoretically, we would find such numerous pairs (α1, α2) by changing both the values of α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To simplify the searching process, we fix α2 and set α2 = ˜α, following Campbell and Yogo (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In this study, we consider the case ˜α = α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05, giving the Bonferroni test with significance level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, for each ψ ∈ (−1, 1), we numerically find the value of α1(ψ) such that under the null Pa,ψ �� ¯ρT ∈CI(α1(ψ)) � ST (¯ρT ) > cvα2 �� ≤ ˜α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05, and this probability is as close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 as possible, for all a on some grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The simulation process to find the α1(ψ) values is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Table 1 displays the significance level α1(ψ) of the confidence interval for ρT along with corresponding intervals of |ψ| values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='7 Given the good asymptotic performance by the mod- ified Wald test, we report in Table 1 α1 values for the case of ST(ρT ) being the modified augmented Wald test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' When performing the Bonferroni-Wald test, first estimate ψ by its consistent estimator ˆψ(ˆρT ), and then select the value of α1 based on Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For example, if | ˆψ(ˆρT )| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='17, α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='31 is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' With the selected α1 value, one can per- form the Bonferroni-Wald test following the two-step testing procedure outlined before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The finite-sample performances of the Bonferroni-Wald test are investigated through simulation in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 4 Finite-Sample Performance As described in Appendix A, the simulation exercise to determine the values of α1 is based on the asymptotic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus, we need to verify whether the Bonferroni-Wald test we have proposed performs well in finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 Empirical size Following Nagakura (2009a), we employ three data generating mechanism to evaluate empirical sizes: (i) εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1), so that ψ = 0, (ii) εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d (χ2(10) − 10)/ √ 20, so 7We assign α1 values to each interval of ψ instead of each single ψ (on some grid) for computational ease of the Bonferroni-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 14 that ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5, and (iii) εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d (χ2(1) − 1)/ √ 2, so that ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For each case, we set y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The sample size we use is T ∈ {200, 500, 1000}, and the number of replications is 5,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The ρT = ρ values are fixed across T, and we consider ρ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The simulation results are collected in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The general pattern is that the empirical rejection rates under H0 are relatively small when ρ < 1 and tend to be greater than the nominal level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 when ρ is near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for the normal case (Figure 6(a)), rejection rates are stable around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 over ρ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01], with rejection rates approaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for the chi square cases (Figure 6(b) and (c)), rejection rates stay around the nominal level, but they get farther away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 when |ψ| is larger and T is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In particular, when εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d (χ2(1) − 1)/ √ 2 and T = 200, the rejection rates can be as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='09 (around ρ = 1), although they approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' A similar tendency can be observed in the finite-sample behavior of modified LN tests proposed by Nagakura (2009a) according to his simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One possible cause of this phenomenon would be the finite-sample bias involved in the estimation of ψ when the value of ψ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' One may be advised to use a more conservative Bonferroni-Wald test (with smaller α1 values) when the |ψ| estimate is large and the sample size is not so large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 Finite-sample power comparison Next, we investigate tests’ ability to detect the nonzero variance in the autoregressive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The simulation design is as follows: T = 200, εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1) and vt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1) with Corr(εt, vt) ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='75}, ρ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='95}, and ω2 ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01] (corresponding to c2 ∈ (0, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The tests we consider here are the Bonferroni-Wald test, the infeasible modified Wald test (calculated using the true ρ), and one of the modified LN tests proposed by Nagakura (2009a), which is denoted by �GT,1 in his notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The infeasible Wald test is taken as a benchmark, and thereby we can evaluate the power loss originating from using the confidence interval for ρ to perform the Bonferroni-Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Nagakura (2009a)’s modified LN test statistic is designed to converge in distribution under H0 to the standard normal irrespective of the ψ value, under the data generating mechanism with ρ ∈ (−1, 1] fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=', independent of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Because Nagakura (2009a) did not show its asymptotic null distribution remains standard normal when ρ > 1, we do not perform it for the case ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We also consider the test proposed recently by Horv´ath and Trapani (2019) (hereafter HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The HT test is a randomized one, and their test statistic ΘT,R (in their notation) converges in 15 distribution to the chi square with one degree of freedom, for almost all realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='8 The HT test is not originally proposed under the local-to-unity specification, but it will be informative to practitioners to reveal the performance of the test under this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The simulation results are shown in Figures 7 through 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For the case ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01, where all but the LN test are performed, the infeasible and Bonferroni-Wald tests have good power, and the discrepancy between their power functions is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The latter result is due to the refinement on the construction of the confidence interval for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In contrast, the power function of the HT test stays around the nominal level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 over ω2 ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='01], which implies it has almost no distinguishing power for these alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Indeed, HT conjectured (based on some theoretical analysis) that their test would have no power when 1 − ρT = O(T −1) and ω2 T = O(T −1), which is of larger magnitude than our local-to-zero variance ω2 T = O(T −3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='9 Our simulation results corroborate their statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Turning to the cases ρ ≤ 1, it is easily noticeable that for each ρ, the LN test performs well for small values of Corr(εt, vt), but Wald type tests outperform the LN test otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In particular, the greater value of Corr(εt, vt) leads to the greater dominance by the Wald type tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The power function of HT test, again, stays around the nominal level for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Overall, the Bonferroni-Wald test performs better than the LN and HT tests for moderate to large values of Corr(εt, vt) (irrespective of the ρ value) and performs almost as efficiently as its ideal, infeasible counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 5 Empirical Application In this section, we apply the Bonferroni-Wald test along with the LN and HT tests10 to several U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' macroeconomic and financial time series, following Hill and Peng (2014) and HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The dataset includes CPI, real GDP, industrial production, M2, S&P500, the 3 month Treasury bill rate and the unemployment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We take the logarithm of the first 5 series before detrending all the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' All data have been extracted from Federal Reserve Economic Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The data description and testing results are displayed in Tables 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 8The HT test needs a tuning parameter, x ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5), to be performed, and they stated their test’s per- formance is insensitive to the choice of the x value and set x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' However, in our simulation, their test’s performance is somehow affected by the choice of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus, we set x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='38, to obtain simulation results similar to those of HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 9Nishi and Kurozumi (2022) showed that under the STUR specification, the LN and some other tests are consistent when ω2 T = O(T −1), from which they concluded that this case should be regarded as capturing stochastic “moderate” departures from a unit root rather than local departures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 10Based on our simulation results, we set the tuning parameter x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='38 for the HT test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 16 Before discussing main results, it should be recalled from our simulation results that the Bonferroni-Wald test can be oversized when T is not large, ψ is large and ρ is near unity (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus, we need to check whether all the three conditions hold for our series or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' According to Table 3, all the ρ estimates are near unity, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We also have obtained moderate estimates of ψ for the CPI and S&P500 series, but the sample sizes for them are large enough that it is unlikely the Bonferroni-Wald test is oversized for these series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for GDP, the sample size is T = 292 but the ψ estimate is near zero, and hence we expect the Bonferroni-Wald test is not severely oversized for this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Overall, all the three conditions for the potential oversize problem do not seem to jointly hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For GDP and T-bill rate, we have obtained consistent results from all the three tests: the null of H0 : ω2 = 0 is not rejected for these series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Given (great) differences in power between the three tests, this coincidence should be viewed as strong evidence for nonexistence of randomness in the autoregressive root for these series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for CPI, industrial production and S&P500, it is seen that the Bonferroni-Wald and LN tests reject the null while the HT test does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This result will be attributed to the fact that the former two tests are much more powerful than the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Finally, as for M2 and unemployment rate, only the Bonferroni- Wald test rejects the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This outcome will be due to the fact that the Bonferroni-Wald test tends to be the most powerful among the three tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In Table 3, we also present the ω2 estimates proposed by Horv´ath and Trapani (2019) (denoted by ˆω2 HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' They showed this estimator is consistent under the non-local RCA models (with ρ and ω2 fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The values of ˆω2 HT seem to support our testing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For instance, we have a negative ˆω2 HT for T-bill rate, for which none of the tests reject the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Moreover, for the other series, ˆω2 HT takes values around 10−4 to 10−3, magnitudes of the coefficient randomness with which the HT test tends to be powerless under the sample size given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For example, for M2, the sample size is T = 2044, and ˆω2 HT is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='12× 10−4, translating into ˆc2 = ˆω2 HT ×T 3/2 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='83 estimate of the localizing coefficient c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' According to our simulation results, the HT test has almost no power against the alternative of this magnitude, hence the testing result given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Given these findings, our empirical application seems to illustrate the merit in choosing our Bonferroni-Wald test over existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 17 6 Conclusion and Discussion Given the results of empirical analyses conducted by earlier studies, the local-to-unity RCA models, which extend the STUR modelling, are empirically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Under this setting, we can analyze the effect of the correlation between the random coefficient and disturbance on the power properties of tests for coefficient randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Theoretical and simulation analyses reveal that tests proposed by earlier studies can perform poorly when the degree of the correlation is moderate to large and the coefficient randomness is local to zero, while the augmented-Wald test we have proposed performs well even in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Our test is also independent of the nuisance parameter ψ, the correlation between the disturbance and its square, so that it is implementable without the knowledge about the value of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To deal with the uncertainty about the mean ρ of the autoregressive root, we have proposed using a confidence interval for ρ, leading to the Bonferroni-Wald test, where the significance level for the confidence interval is selected according to the value of the ψ estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Embedding this selection process into the Bonferroni-Wald test helps stabilize the test’s size and improve the test’s power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Several directions for future research are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' First, from the similarity in the con- struction of test statistics between our model and predictive regressions, it is expected that the theory developed by numerous studies on predictive regressions can be applied to testing for coefficient randomness in local-to-unity autoregressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For example Phillips and Lee (2013, 2016) proposed the use of the so-called IVX procedure for predictability testing (see also Kostakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This procedure leads to good size and power properties and also requires less computational burden for implementation than the Bonferroni approach em- ployed by Campbell and Yogo (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The use of the IVX approach may facilitate testing for coefficient randomness in local-to-unity autoregressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The analysis of tests’ perfor- mance when ρ is distant from unity will also be of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In such a case, more preferable tests might be available than the Bonferroni-Wald test, which is based on the local-to-unity asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' References Aue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Horv´ath (2011) Quasi-Likelihood Estimation in Stationary and Nonstationary Autoregressive 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Roca (2012) Examining the Power of Stochastic Unit Root Tests without Assuming Independence in the Error Processes of the Underlying Time Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Applied Economics Letters, 19 (4), 373–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 20 Appendix A: Procedure to Determine α1 Values for the Bonferroni- Wald Test In this appendix, we describe the procedure to determine the α1 values for the confidence interval for ρT explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The procedure is based on simulating asymptotic distributions with 5,000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In each replication, we first generate {yt}T t=1 by the mechanism yt = (1 + a/T)yt−1 + εt, t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , T (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) with y0 = 0, T = 2000, a ∈ [−300, 10] and εt ∼ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, with given α1, we conduct the two-step procedure for the Bonferroni-Wald test explained in Section 3 and calculate the frequency of H0 being rejected for the α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Note that εt used in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) satisfies ψ = Corr(εt, ε2 t − σ2 ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To calculate the false rejection frequencies for other ψ values, we artificially produce an environment where the Wald test statistic depends on ψ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Noting that under the null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' z2 t (¯ρT ) = 1 + (ε2 t − 1) + (¯a/T − a/T)2y2 t−1 − 2(¯a/T − a/T)yt−1εt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' and ηt = ε2 t − 1 (combined with εt) determines the value of ψ on which the asymptotic distribution of W ∗ T(¯ρT ) depends,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' we artificially replace ηt = ε2 t − 1 with ηrep t := � 1 − ψ2ηt + ψ √ 2εt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' obtaining z2 t (¯ρT )rep := 1 + ηrep t + (¯a/T − a/T)2y2 t−1 − 2(¯a/T − a/T)yt−1εt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' in view of the distributional equivalence Wη d= � 1 − ψ2W1 + ψWε and the fact that V[ηt] = σ2 η = 2 and Corr(ηt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' εt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' In this replacement, the newly crafted variable ηrep t takes over the role of ηt = ε2 t − 1, satisfying E[ηrep t ] = 0, V[ηrep t ] = 2 = V[ηt] and Corr(ηrep t , εt) = ψ by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This replacement can be justified by the fact that under the null, the Wald test statistic asymptotically depends only on ψ (and a) and is not dependent on any other moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' It follows that the Bonferroni-Wald test statistic using z2 t (¯ρT )rep in place of z2 t (¯ρT ) asymptotically depends on ψ, so that we can calculate the false rejection frequencies for any value of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For each ψ on some grid, we calculate the false rejection rates of the Bonferroni-Wald test with a moving over a grid on [−300, 10] and determine the value of α1 for the given ψ such that the false rejection rate is less than or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 21 Appendix B: Proofs of Results in Section 2 In this appendix, we prove the theorems stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Define the stochastic process YT on [0, 1] by YT (r) := T −1/2y⌊Tr⌋, 0 ≤ r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, YT ⇒ σεJa in the Skorokhod space D[0, 1], where Ja solves dJa(r) = aJa(r)dr + dWε(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The proof is essentially the same as that of Lemma 1(a) of Nishi and Kurozumi (2022) and hence is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, we have (a) ˆσ2 ε,T(ρT ) p→ σ2 ε, (b) ˆσ2 η,T (ρT ) p→ σ2 η, (c) ˆψT (ρT ) p→ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The proofs of parts (a) and (b) are identical to those of Lemma 1(b) and (c) of Nishi and Kurozumi (2022) and hence are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (c), it suffices to show T −1 T � t=1 zt(ρT ) � z2 t (ρT ) − ˆσ2 ε,T(ρT ) � p→ E[ε3 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' A simple calculation gives T −1 T � t=1 zt(ρT ){z2 t (ρT ) − ˆσ2 ε,T (ρT )} = T −1 T � t=1 (cT −3/4yt−1vt + εt){c2T −3/2y2 t−1v2 t + 2cT −3/4yt−1εtvt + ε2 t − ˆσ2 ε,T (ρT )} = T −1 T � t=1 ε3 t + AT , where AT := c3T −13/4 T � t=1 y3 t−1v3 t + 3c2T −5/2 T � t=1 y2 t−1εtv2 t + 3cT −7/4 T � t=1 yt−1ε2 t vt − ˆσ2 ε,T(ρT ) × � cT −7/4 T � t=1 yt−1vt + T −1 T � t=1 εt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 22 The first term of AT satisfies c3T −13/4 T � t=1 y3 t−1v3 t = c3E[v3 t ]T −13/4 T � t=1 y3 t−1 + c3T −13/4 T � t=1 y3 t−1(v3 t − E[v3 t ]) = c3E[v3 t ]T −3/4 � 1 0 Y 3 T (r)dr + c3T −5/4 � 1 0 Y 3 T (r)dWv3−E[v3],T(r) = Op(T −3/4), say, because {v3 t − E[v3 t ]} is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='d with zero mean and finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Similarly, we can prove that the other terms of AT is Op(T −1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus T −1 T � t=1 zt(ρT ){z2 t (ρT ) − ˆσ2 ε,T(ρT )} = T −1 T � t=1 ε3 t + op(1) p→ E[ε3 t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For later reference, we give several results on the weak convergence of components of test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, we have (a) T −3/2 T � t=1 � y2 t−1z2 t (ρT ) ⇒ σησ2 ε � 1 0 � Ja,2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,2)2(r)dr + 2cσ4 εq � 1 0 � Ja,1(r)� Ja,2(r)dr, (b) T −1 T � t=1 � yt−1z2 t (ρT ) ⇒ σησε � 1 0 � Ja,1(r)dWη(r) + c2σ3 ε � 1 0 � Ja,1(r)� Ja,2(r)dr + 2cσ3 εq � 1 0 (� Ja,1)2(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (c) ˆσ2 � ξ∗(ρT ) p→ σ2 η, (d) ˆσ2 � ξ∗∗(ρT ) p→ σ2 η, where ˆσ2 � ξ∗(ρT ) and ˆσ2 � ξ∗∗(ρT ) are the OLS variance estimators of (8) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' For part (a), a straightforward calculation gives T −3/2 T � t=1 � y2 t−1z2 t (ρT ) = B1,T + B2,T + B3,T , 23 where B1,T := T −3/2 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 � (ε2 t − σ2 ε), B2,T := c2T −3 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 � y2 t−1v2 t , and B3,T := 2cT −9/4 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 � yt−1εtvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' It is straightforward to show B1,T ⇒ σησ2 ε � 1 0 � Ja,2(r)dWη(r), using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 of Hansen (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for B2,T , we have B2,T = c2T −3 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 �2 + c2T −3 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 � y2 t−1(v2 t − 1) = c2 � 1 0 � Y 2 T (r) − � 1 0 Y 2 T (s)ds �2 dr + c2T −1/2 � 1 0 � Y 2 T (r) − � 1 0 Y 2 T (s)ds � Y 2 T (r)dWv2−1,T ⇒ c2σ4 ε � 1 0 (� Ja,2)2(r)dr, where the last convergence follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 and the continuous mapping theorem (CMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' By a similar argument, we obtain B3,T = 2cσεqT −5/2 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 �� yt−1 − T −1 T � t=1 yt−1 � + 2cT −9/4 T � t=1 � y2 t−1 − T −1 T � t=1 y2 t−1 � yt−1(εtvt − σεv) ⇒ 2cσ4 εq � 1 0 � Ja,2(r)� Ja,1(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Therefore, we arrive at T −3/2 T � t=1 � y2 t−1z2 t (ρT ) ⇒ σησ2 ε � 1 0 � Ja,2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,2)2(r)dr + 2cσ4 εq � 1 0 � Ja,2(r)� Ja,1(r)dr, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Part (b) can be proven in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Write T −1 �T t=1 � yt−1z2 t (ρT ) as T −1 T � t=1 � yt−1z2 t (ρT ) = C1,T + C2,T + C3,T , 24 where C1,T := T −1 �T t=1 � yt−1(ε2 t −σ2 ε), C2,T := c2T − 5 2 �T t=1 � yt−1y2 t−1v2 t , and C3,T := 2cT − 7 4 �T t=1 � yt−1yt−1εtvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, it is straightforward to show C1,T ⇒ σησε � 1 0 � Ja,1(r)dWη(r), C2,T ⇒ c2σ3 ε � 1 0 � Ja,1(r)� Ja,2(r)dr, and C3,T ⇒ 2cσ3 εq � 1 0 (� Ja,1)2(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Combining the above results completes the proof of part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (c), we define M := IT − � X( � X′ � X)−1 � X and write ˆσ2 � ξ∗(ρT ) as ˆσ2 � ξ∗(ρT ) = T −1� Ξ∗′M � Ξ∗ = T −1 T � t=1 ( �ξ∗ t )2 − T −1 ��T t=1 � yt−1ξ∗ t �T t=1 � y2 t−1ξ∗ t � × � �T t=1( � yt−1)2 �T t=1 � yt−1 � y2 t−1 �T t=1 � y2 t−1 � yt−1 �T t=1( � y2 t−1)2 �−1 ��T t=1 � yt−1ξ∗ t �T t=1 � y2 t−1ξ∗ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) The first term of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) becomes T −1 T � t=1 ( �ξ∗ t )2 = T −1 T � t=1 (ξ∗ t )2 − � T −1 T � t=1 ξ∗ t �2 = T −1 T � t=1 {c2T −3/2y2 t−1(v2 t − 1) + 2cT −3/4yt−1(εtvt − σεv) + (ε2 t − σ2 ε)}2 − � T −1 T � t=1 {c2T −3/2y2 t−1(v2 t − 1) + 2cT −3/4yt−1(εtvt − σεv) + (ε2 t − σ2 ε)} �2 = T −1 T � t=1 (ε2 t − σ2 ε)2 + op(1) p→ σ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 25 The second term of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) satisfies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −1 ��T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1ξ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1ξ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='� � �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�−1 ��T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1ξ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1ξ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='= T −1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −1 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1(ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t − σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='ε) + op(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −3/2 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1(ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t − σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='ε) + op(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −2 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −5/2 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −5/2 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −3 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='�−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −1 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='yt−1(ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t − σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='ε) + op(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T −3/2 �T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t=1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t−1(ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='t − σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='ε) + op(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='= Op(T −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2) Hence, we obtain ˆσ2 � ξ∗(ρT ) p→ σ2 η, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (d), let � Z1 := ( �z1(ρT ), �z2(ρT ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' , � zT (ρT ))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, � Z∗ 2(ρT ) is expressed as � Z∗ 2(ρT ) = 1 � 1 − ˆψ2 T (ρT ) � � Z2(ρT ) − ˆση,T (ρT ) ˆψT (ρT ) ˆσε,T(ρT ) � Z1(ρT ) � , which yields ˆσ2 � ξ∗∗ = T −1� Z∗ 2(ρT )′M � Z∗ 2(ρT ) = 1 1 − ˆψ2 T (ρT ) � D1,T − 2 ˆση,T (ρT ) ˆψT (ρT ) ˆσε,T(ρT ) D2,T + ˆσ2 η,T (ρT ) ˆψ2 T (ρT ) ˆσ2 ε,T (ρT ) D3,T � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3) where D1,T := T −1� Z2(ρT )′M � Z2(ρT ), D2,T := T −1� Z2(ρT )′M � Z1(ρT ), and D3,T := T −1� Z1(ρT )′M � Z1(ρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Since D1,T is ˆσ2 � ξ∗(ρT ), we have already proven in part (c) that D1,T p→ σ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4) In view of equation (8), D2,T becomes D2,T = T −1� Ξ∗(ρT )′M � Z1(ρT ) = T −1 T � t=1 ξ∗ t �zt(ρT ) − T −1 ��T t=1 � yt−1ξ∗ t �T t=1 � y2 t−1ξ∗ t � × � �T t=1( � yt−1)2 �T t=1 � yt−1 � y2 t−1 �T t=1 � y2 t−1 � yt−1 �T t=1( � y2 t−1)2 �−1 ��T t=1 � yt−1zt(ρT ) �T t=1 � y2 t−1zt(ρT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 26 As for the first term of D2,T , we have T −1 T � t=1 ξ∗ t �zt(ρT ) = T −1 T � t=1 {c2T −3/2y2 t−1(v2 t − 1) + 2cT −3/4yt−1(εtvt − σεv) + (ε2 t − σ2 ε)} × � zt(ρT ) − T −1 T � t=1 zt(ρT ) � = T −1 T � t=1 (ε2 t − σ2 ε)εt + op(1) p→ E[ε3 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' We can also show that the second term of D2,T is Op(T −1) in the same way as we did in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Thus, we get D2,T p→ E[ε3 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5) Lastly, D3,T becomes D3,T = T −1� Z1(ρT )′M � Z1(ρT ) = T −1 T � t=1 �zt2(ρT ) − T −1 ��T t=1 � yt−1zt(ρT ) �T t=1 � y2 t−1zt(ρT ) �′ × � �T t=1( � yt−1)2 �T t=1 � yt−1 � y2 t−1 �T t=1 � y2 t−1 � yt−1 �T t=1( � y2 t−1)2 �−1 ��T t=1 � yt−1zt(ρT ) �T t=1 � y2 t−1zt(ρT ) � = ˆσ2 ε,T(ρT ) − � T −1 T � t=1 zt(ρT ) �2 + Op(T −1) p→ σ2 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='6) Substituting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4) through (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='6) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3) and applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2, we deduce ˆσ2 � ξ∗∗ p→ 1 1 − ψ2 � σ2 η − 2σηψ σε E[ε3 t] + σ2 ηψ2 σ2ε σ2 ε � = 1 1 − ψ2 (σ2 η − 2σ2 ηψ2 + σ2 ηψ2) = σ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' First, note that LNT (ρT ) = T −3/2 �T t=1 � y2 t−1z2 t (ρT ) ˆση,T (ρT ) � T −3 �T t=1( � y2 t−1)2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 27 Then, using Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3 and the CMT, we deduce LNT (ρT ) ⇒ σησ2 ε � 1 0 � Ja,2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,2)2(r)dr + 2cσ4 εq � 1 0 � Ja,1(r)� Ja,2(r)dr ση � σ4ε � 1 0 (� Ja,2)2(r)dr �1/2 = � 1 0 � Ja,2(r)dWη(r) �� 1 0 �� Ja,2 �2(r)dr �1/2 + σ2 ε ση � c2 � 1 0 �� Ja,2 �2(r)dr + 2cq � 1 0 � Ja,2(r)� Ja,1(r)dr �� 1 0 �� Ja,2 �2(r)dr �1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' First, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3(c) and the CMT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' the denominator of tˆω2 T (ρT ) divided by T 3/2 becomes ˆσ � ξ∗(ρT )T −3/2(� X2 ′M1 � X2)1/2 = ˆσ � ξ∗(ρT ){T −3(M1 � X2)′(M1 � X2)}1/2 = ˆσ � ξ∗(ρT ) � T −3 T � t=1 � � y2 t−1 − �T t=1 � yt−1 � y2 t−1 �T t=1( � yt−1)2 � yt−1 �2�1/2 = ˆσ � ξ∗(ρT ) �� 1 0 � � Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (r) − � 1 0 � Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (s) � Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (s)ds � 1 0 ( � Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T )2(s)ds � Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (r) �2�1/2 ⇒ ση � σ4 ε � 1 0 � � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r) − � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(s)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(s)ds � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1)2(s)ds � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r) �2 dr �1/2 = σησ2 ε �� 1 0 Q2 a(r)dr �1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' where � Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T(r) := YT (r) − � 1 0 YT(s)ds and � Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T(r) := Y 2 T (r) − � 1 0 Y 2 T (s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Next, applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' the numerator of tˆω2 T (ρT ) divided by T 3/2 is seen to satisfy T −3/2 � X2 ′M1� Z2(ρT ) = T −3/2 T � t=1 � y2 t−1z2 t (ρT ) − T −5/2 �T t=1 � yt−1 � y2 t−1T −1 �T t=1 � yt−1z2 t (ρT ) T −2 �T t=1( � yt−1)2 ⇒ σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr − σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr × σησε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) + c2σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cσ3 εq � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1)2(r)dr σ2ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1)2(r)dr = σησ2 ε � 1 0 Qa(r)dWη(r) + c2σ4 ε � 1 0 Q2 a(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 28 Combining the above results gives tˆω2 T (ρT ) ⇒ � 1 0 Qa(r)dWη(r) �� 1 0 Q2a(r)dr �1/2 + c2σ2 ε ση �� 1 0 Q2 a(r)dr �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To derive the asymptotic distribution of WT (ρT ), note that WT (ρT ) = ˆσ−2 � ξ∗ ( � X′� Z2(ρT ))′( � X′ � X)−1( � X′� Z2(ρT )) = ˆσ−2 � ξ∗ � T −1 �T t=1 � yt−1z2 t (ρT ) T −3/2 �T t=1 � y2 t−1z2 t (ρT ) �′ � T −2 �T t=1( � yt−1)2 T −5/2 �T t=1 � yt−1 � y2 t−1 T −5/2 �T t=1 � y2 t−1 � yt−1 T −3 �T t=1( � y2 t−1)2 �−1 × � T −1 �T t=1 � yt−1z2 t (ρT ) T −3/2 �T t=1 � y2 t−1z2 t (ρT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3 and the CMT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' we get WT (ρT ) ⇒ σ−2 η � σησε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) + c2σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cσ3 εq � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1)2(r)dr σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr �′ × � σ2 ε � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dr σ4 ε � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr �−1 × � σησε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) + c2σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cσ3 εq � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1)2(r)dr σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � = � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) � + σ2 ε ση � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr ��′ × � � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dr � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr �−1 × � �� 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)dWη(r) � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) � + σ2 ε ση � c2 � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr + 2cq � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 �2(r)dr c2 � 1 0 �� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2 �2(r)dr + 2cq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove Theorem 3, we use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, we have (a) T −3/2 T � t=1 � y2 t−1z2∗ t (ρT ) ⇒σησ2 ε � 1 0 � Ja,2(r)dW1(r) + (1 − ψ2)−1/2� c2σ4 ε � 1 0 (� Ja,2)2(r)dr + 2cσ4 εq � 1 0 � Ja,1(r)� Ja,2(r)dr � , 29 (b) T −1 T � t=1 � yt−1z2∗ t (ρT ) ⇒σησε � 1 0 � Ja,1(r)dW1(r) + (1 − ψ2)−1/2� c2σ3 ε � 1 0 � Ja,1(r)� Ja,2(r)dr + 2cσ3 εq � 1 0 (� Ja,1)2(r)dr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (a), note that T −3/2 T � t=1 � y2 t−1z2∗ t (ρT ) = 1 � 1 − ˆψ2 T (ρT ) � E1,T − ˆση,T (ρT ) ˆψT (ρT ) ˆσε,T(ρT ) E2,T � , where E1,T := T −3/2 �T t=1 � y2 t−1z2 t (ρT ) and E2,T := T −3/2 �T t=1 � y2 t−1zt(ρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3(a), E1,T satisfies E1,T ⇒ σησ2 ε � 1 0 � Ja,2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,2)2(r)dr + 2cσ4 εq � 1 0 � Ja,1(r)� Ja,2(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for E2,T , a straightforward calculation yields E2,T = T −3/2 T � t=1 � y2 t−1(cT −3/4yt−1vt + εt) = T −3/2 T � t=1 � y2 t−1εt + Op(T −1/4) ⇒ σ3 ε � 1 0 � Ja,2(r)dWε(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' we obtain T −3/2 T � t=1 � y2 t−1z2∗ t (ρT ) ⇒ 1 � 1 − ψ2 � σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWη(r) + c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr − σηψ σε σ3 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dWε(r) � = σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)d �Wη(r) − ψWε(r) � 1 − ψ2 � + 1 � 1 − ψ2 � c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � d= σησ2 ε � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dW1(r) + 1 � 1 − ψ2 � c2σ4 ε � 1 0 (� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2)2(r)dr + 2cσ4 εq � 1 0 � Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1(r)� Ja,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(r)dr � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 30 in view of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' This proves part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The proof of part (b) is similar and thus is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The proof for LN∗ T (ρT ) is essentially the same as that of Theorem 1 except that we consider �T t=1 � y2 t−1z2∗ t (ρT ) in the numerator of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Dividing both the numerator and denominator by T 3/2 and applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4(a) leads to the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' The proof for the augmented tests goes along the same lines as those of Theorem 2 if we replace z2 t (ρT ) with z2∗ t (ρT ) and apply Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='3(d) and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Appendix C: Proofs of Results in Section 3 In this appendix, we prove the asymptotic results mentioned in Section 3: namely, the asymptotic distribution of ˆρT and the consistency of ˆσ2 ε,T(ˆρT ), ˆσ2 η,T (ˆρT ) and ˆψT (ˆρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, we have T(ˆρT − ρT ) ⇒ � 1 0 Ja(r)Wε(r) � 1 0 J2a(r)dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' From the definition of ˆρT , we have T(ˆρT − ρT ) = T −1 �T t=1 yt−1(cT −3/4yt−1vt + εt) T −2 �T t=1 y2 t−1 = cT −1/4 � 1 0 Y 2 T (r)dWv,T (r) + σε � 1 0 YT (r)dWε,T(r) � 1 0 Y 2 T (r)dr ⇒ � 1 0 Ja(r)dWε(r) � 1 0 J2a(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Consider model (2) under Assumptions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Then, we have (a) ˆσ2 ε,T(ˆρT ) p→ σ2 ε, (b) ˆσ2 η,T (ˆρT ) p→ σ2 η, (c) ˆψT (ˆρT ) p→ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' As for part (a), ˆσ2 ε,T(ˆρT ) satisfies ˆσ2 ε,T (ˆρT ) = T −1 T � t=1 z2 t (ˆρT ) = T −1 T � t=1 {zt(ρT ) − (ˆρT − ρT )yt−1}2 = T −1 T � t=1 z2 t (ρT ) − 2T(ˆρT − ρT )T −2 T � t=1 yt−1zt(ρT ) + T 2(ˆρT − ρT )2T −3 T � t=1 y2 t−1, for which we have T −1 �T t=1 z2 t (ρT ) = ˆσ2 ε,T(ρT ), T 2(ˆρT − ρT )2T −3 �T t=1 y2 t−1 = Op(T −1) by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1 and the CMT, and T −2 T � t=1 yt−1zt(ρT ) = T −2 T � t=1 yt−1(cT −3/4yt−1vt + εt) = cT −5/4 � 1 0 Y 2 T (r)dWT,v(r) + T −1σε � 1 0 YT (r)dWε,T(r) = Op(T −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Therefore ˆσ2 ε,T(ˆρT ) = ˆσ2 ε,T (ρT ) + op(1) p→ σ2 ε, given Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (b), write ˆσ2 η,T (ˆρT ) as ˆσ2 η,T (ˆρT ) = T −1 T � t=1 {z2 t (ˆρT ) − ˆσ2 ε,T(ˆρT )}2 = T −1 T � t=1 z4 t (ˆρT ) − ˆσ4 ε,T(ˆρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) The first term is T −1 T � t=1 z4 t (ˆρT ) = T −1 T � t=1 {zt(ρT ) − (ˆρT − ρT )yt−1}4 = T −1 T � t=1 z4 t (ρT ) − 4T(ˆρT − ρT )T −2 T � t=1 z3 t (ρT )yt−1 + 6T 2(ˆρT − ρT )2T −3 T � t=1 z2 t (ρT )y2 t−1 − 4T 3(ˆρT − ρT )3T −4 T � t=1 zt(ρT )y3 t−1 + T 4(ˆρT − ρT )4T −5 T � t=1 y4 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 32 Straightforward calculations reveal T −2 T � t=1 z3 t (ρT )yt−1 = Op(T −1/2), T −3 T � t=1 z2 t (ρT )y2 t−1 = Op(T −1), T −4 T � t=1 zt(ρT )y3 t−1 = Op(T −2), T −5 T � t=1 y4 t−1 = Op(T −2), which gives T −1 T � t=1 z4 t (ˆρT ) = T −1 T � t=1 z4 t (ρT ) + Op(T −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Substituting this and ˆσ2 ε,T(ˆρT ) = ˆσ2 ε,T(ρT ) + op(1) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1), we arrive at ˆσ2 η,T (ˆρT ) = T −1 T � t=1 z4 t (ρT ) − ˆσ4 ε,T(ρT ) + op(1) = ˆσ2 η,T (ρT ) + op(1) p→ σ2 η, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To prove part (c), it suffices to show T −1 �T t=1 zt(ˆρT ){z2 t (ˆρT ) − ˆσ2 ε,T(ˆρT )} p→ E[ε3 t ], given that ˆσ2 ε,T(ˆρT ) p→ σ2 ε and ˆσ2 η,T (ˆρT ) p→ σ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' we have T −1 T � t=1 zt(ˆρT ){z2 t (ˆρT ) − ˆσ2 ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (ˆρT )} = T −1 T � t=1 {zt(ρT ) − (ˆρT − ρT )yt−1}{z2 t (ρT ) − 2(ˆρT − ρT )yt−1zt(ρT ) + (ˆρT − ρT )2y2 t−1 − ˆσ2 ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T(ˆρT )} = T −1 T � t=1 zt(ρT ){z2 t (ρT ) − ˆσ2 ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (ˆρT )} − 3T(ˆρT − ρT )T −2 T � t=1 yt−1z2 t (ρT ) + 3T 2(ˆρT − ρT )2T −3 T � t=1 y2 t−1zt(ρT ) − T 3(ˆρT − ρT )3T −4 T � t=1 y3 t−1 − ˆσ2 ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (ˆρT )T(ˆρT − ρT )T −2 T � t=1 yt−1 = T −1 T � t=1 zt(ρT ){z2 t (ρT ) − ˆσ2 ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='T (ρT )} + op(1) p→ E[ε3 t ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' in view of the last line of the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 33 Table 1: Significance levels of the equal-tailed confidence interval for a |ψ| ∈ α1 |ψ| ∈ α1 [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='42 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='65] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='38 [0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='11 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='95, 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='55] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='44 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' Entries in the second and fourth columns are the significance level of the equal-tailed confidence in- terval for ρT when the significance levels of the Bonferroni-Wald test (˜α) and individual modified Wald test (α2) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' To determine the α1 value for the interval (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='95, 1), we actually computed type 1 errors over (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 34 Table 2: Data description of the series used in Section 5 Series Frequency Sample period T CPI (1982-1984=100) Monthly Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 1913-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2019 1284 Real GDP (2012 chained) Quarterly Q1 1947-Q4 2019 292 Industrial production (2017=100) Monthly Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 1919-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2019 1212 M2 Weekly Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 3, 1980-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 30, 2019 2044 S&P 500 Daily Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 3, 2012-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 31, 2019 1782 3 month T-bill rate Daily Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 3, 2012-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 31, 2019 1770 Unemployment rate Monthly Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 1948-Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content=' 2019 864 35 Table 3: Estimation and testing results from the empirical analysis in Section 5 Series T ˆρT ˆψ( ˆ ρT ) ˆω2 HT Bonf-Wald LN ( �GT,1) HT (ΘT,R) CPI 1284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='211 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='12 × 10−4 Yes Yes GDP 292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='065 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='16 × 10−3 Industrial production 1212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='054 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='98 × 10−3 Yes Yes M2 2044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='066 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='12 × 10−4 Yes S&P 500 1782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='258 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='38 × 10−3 Yes Yes T-bill rate 1770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='108 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} +page_content='24 × 10−5 Unemployment rate 864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfBgvS/content/2301.04853v1.pdf'} 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