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+Inference for Large Panel Data with Many Covariates∗
+Markus Pelger†
+Jiacheng Zou‡
+December 31, 2022
+Abstract
+This paper proposes a new method for covariate selection in large dimensional panels. We de-
+velop the inferential theory for large dimensional panel data with many covariates by combining
+post-selection inference with a new multiple testing method specifically designed for panel data.
+Our novel data-driven hypotheses are conditional on sparse covariate selections and valid for
+any regularized estimator. Based on our panel localization procedure, we control for family-wise
+error rates for the covariate discovery and can test unordered and nested families of hypothe-
+ses for large cross-sections. As an easy-to-use and practically relevant procedure, we propose
+Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with valid
+post-selection p-values of a generalized LASSO, that allows to incorporate priors. In an empir-
+ical study, we select a small number of asset pricing factors that explain a large cross-section of
+investment strategies. Our method dominates the benchmarks out-of-sample due to its better
+control of false rejections and detections.
+Keywords: panel data, high-dimensional data, LASSO, number of covariates, post-selection
+inference, multiple testing, adaptive hypothesis, step-down procedures, factor model
+JEL classification: C33, C38, C52, C55, G12
+∗We thank conference and seminar participants at Stanford, the California Econometric conference and the NBER-NSF
+SBIES conference for helpful comments. Jiacheng Zou gratefully acknowledges the generous support by the MS&E Departmental
+Fellowship, and Charles & Katherine Lin Fellowship.
+†Stanford University, Department of Management Science & Engineering, Email: mpelger@stanford.edu.
+‡Stanford University, Department of Management Science & Engineering, Email: jiachengzou@stanford.edu
+arXiv:2301.00292v1  [econ.EM]  31 Dec 2022
+
+1
+Introduction
+Our goal is the selection of a parsimonious sparse model from a large set of candidate covariates
+that explains a large dimensional panel. This problem is common in many social science applica-
+tions, where a large number of potential covariates are available to explain the time-series of a large
+cross-section of units or individuals. An example is empirical asset pricing, where the literature has
+produced a “factor zoo” of potential risk factors to explain the large cross-section of stock returns.
+This problem requires a large panel, as a successful asset pricing model should explain the many
+available investment strategies, resulting in a large panel of test assets. At the same time, there is
+no consensus about which are the appropriate factors, which leads to a statistical selection problem
+from a large set of candidate risk factors. So far, the literature has only provided solutions for one
+of the two subproblems, while keeping the dimensionality of the other problem small. Our paper
+closes this gap.
+The inferential theory on a large panel with many covariates is a challenging problem. As a first
+step, we have to select a sparse set of covariates from a large pool of candidates with a regularized
+estimator. The challenge is to provide valid p-values from this estimation that account for the
+post-selection inference. Furthermore, researchers might want to impose economic priors on which
+variables should be more likely to be selected. The second challenge is that the panel cross-section
+results in a large number of p-values. Hence, some of them are inadvertently very small, which if
+left unaddressed leads to “p-hacking”. The multiple testing adjustment conditional on the selected
+subset of covariates from the first step is a novel problem, and requires to redesign what hypotheses
+should be tested jointly. A naive counting of all tests is overly conservative, and the test design
+and simultaneity counts need to be conditional on the covariate selection.
+This paper proposes a new method for covariate selection in large dimensional panels, tackling
+all of the above challenges. We develop the inferential theory for large dimensional panel data with
+many covariates by combining post-selection inference with a new multiple testing method specifi-
+cally designed for panel data. Our novel data-driven hypotheses are conditional on sparse covariate
+selections and valid for any regularized estimator. Based on our panel localization procedure, we
+control for family-wise error rates for the covariate discovery and can test unordered and nested
+families of hypotheses for large cross-sections. As an easy-to-use and practically relevant procedure,
+we propose Panel-PoSI, which combines the data-driven adjustment for panel multiple testing with
+valid post-selection p-values of a generalized LASSO, that allows to incorporate priors.
+Our paper proposes the novel conceptual idea of data-driven hypotheses family for panels. This
+allows us to put forward a unifying framework of valid post-selection inference and multiple test-
+ing. Leveraging our data-driven hypotheses family, we adjust for multiple testing with a localized
+simultaneity count, which increases the power, while maintaining false discovery rate control. An
+essential step for a formal statistical test is to formulate the hypothesis. This turns out to be
+non-trivial for a large panel with a first stage selection step for the covariates. It is a fundamental
+insight of our paper, that the hypothesis of our test has to be conditional on the selected set of
+1
+
+active covariates of the first stage. Once we have defined the appropriate hypothesis, we can deal
+with the multiple testing adjustment, which by construction is also conditional on the selection
+step.
+Our method is a disciplined approach based on formal statistical theory to construct and in-
+terpret a parsimonious model. It goes beyond the selection of a sparse set of covariates as it also
+provides the inferential theory.
+This is important as it allows to rank the covariates based on
+their statistical significance and can also be applied for relatively short time horizons, where cross-
+validation for tuning a regularization parameter might not be reliable. We answer the question
+which covariates are needed to explain the full panel jointly, and can also accommodate “weak”
+covariates or factors that only affect a small subset of the cross-sectional units.
+Our data-driven hypothesis perspective exploits the geometric structure implied by the first
+stage selection step.
+Given valid post-selection p-values of a regularized sparse estimator from
+time-series regressions, we collect them across the large cross-section into a “matrix” of p-values.
+Only active coefficients, that are selected in the first stage, contribute p-value entries, whereas
+covariates that were non-active lead to “holes” in this matrix. We leverage the non-trivial shape
+of this matrix to form our adaptive hypotheses. This allows us to make valid multiple testing
+adjusted inference statements, for which we design a panel modified Bonferroni-type procedure
+that can control for the family-wiser error rate (FWER) in discovery of the covariates. As one
+loosens the FWER requirements, the inferential thresholds admits more and more explanatory
+variables, which suggests that the amount of covariates we expect to admit and the FWER control
+level form an “false-discovery control frontier”. We provide a method that allows us to traverse the
+inferential results and determine the least number of covariates that have to be included given a
+user-specified FWER level. In other words, we provide a statistical significance test for the number
+of factors in a panel.
+We propose the novel procedure Panel-PoSI, which combines the data-driven adjustment for
+panel multiple testing with valid post-selection p-values of a generalized LASSO. While our multiple
+testing procedure is valid for any sparsity constrained model, Panel-PoSI is an easy-to-use and prac-
+tically relevant special case. We propose Weighted-LASSO for the first stage selection regression and
+provide valid p-values through post-selection inference (PoSI), which yields a truncated-Gaussian
+distribution for an adjusted LASSO estimator. This geometric perspective is less common in the
+LASSO literature, but has the advantage that it avoids the use of infeasible quantities, in particu-
+lar the second moment of the large set of potential covariates. The Weighted-LASSO generalizes
+LASSO by allowing to put weights onto prior belief sets. For example, a researcher might have
+economic knowledge that she wants to include in her statistical selection method, and impose an in-
+finite prior weight to include specific covariates in the sparse selection model. Our Weighted-LASSO
+makes several contributions. First, the expression for the truncated conditional distribution with
+weights become much more complex than for the special case of the conventional LASSO. Second,
+we provide a simple, easy-to-use and asymptotically valid conditional distribution in the case of an
+estimated noise variance.
+2
+
+We demonstrate in simulations and empirically that our inferential theory allows us to select
+better models. We compare different estimation approaches to select covariates and show that our
+approach better trades off false discovery and correct selections and hence results in a better out-
+of-sample performance. Our empirical analysis studies the fundamental problem in asset pricing of
+selecting a parsimonious factor model from a large set of candidate factors that can jointly explain
+the asset prices of a large cross-section of investment strategies. We consider a standard data set
+of 114 candidate asset pricing factors to explain 243 double sorted anomaly portfolios. We show
+that Panel PoSI selects 3 factors which form the best model to explain out-of-sample the expected
+returns and the variations of the test assets. The selected factors are economically meaningful and
+we can rank them based on their relative importance. A prior on the Fama-French factors does not
+improve the model. Our findings contributes to the discussion about the number of asset pricing
+factors.
+The rest of the paper is organized as follows. Section 1.1 relates our work to the literature.
+Section 2 introduces the model and the Weighted-LASSO. Section 3 discusses the appropriate
+hypotheses to be considered for inference on the entire panel. Section 4 proposes a joint unordered
+test for the panel using multiple testing adjustment so that we can maintain FWER control, and
+shows how to traverse this procedure to acquire the least factor count associated with each FWER
+target. In section 5 we consider the case of nested hypotheses, where the covariates observe a fixed
+ordering, which is of independent interest, and we propose a step-down procedure for this setting
+that maintains false discovery control. Section 6 provides the results of our simulation and Section
+7 discusses our empirical studies on a large asset pricing panel data set. Section 8 concludes. The
+proofs and more technical details are available in the Online Appendix.
+1.1
+Related Literature
+The problem of multiple testing is an active area of research with a long history. The statistical
+inference community has studied the problem of controlling the classical FWER since Bonferroni
+(1935), and controlling for false-discover rate (FDR) going back to Benjamini and Hochberg (1995)
+and Benjamini and Yekutieli (2001). Bonferroni (1935) allows for arbitrary correlation in the test
+statistics because its validity comes from a simple union bound argument, and is in fact the optimal
+test when statistics are “close to independent” under true sparse non-nulls. FDR control on the
+other hand requires a discussion about the estimated covariance in the test statistics.
+Recent
+developments include a stream of papers led by Barber and Cand´es (2015) and Cand´es, Fan,
+Janson, and Lv (2018), which constructs a generative model to produce fake data and control
+for FDR. Fithian and Lei (2022) is a more recent work that iteratively adjusts the threshold for
+each hypothesis in the family to seek finite sample exact FDR control and dominates Benjamini
+and Hochberg (1995) and Benjamini and Yekutieli (2001) in terms of power. Another notion on
+temporal false discovery control has been revived more recently by Johari, Koomen, Pekelis, and
+Walsh (2021), who consider the industry practice of constantly checking p-values and provide an
+early stopping in line with Siegmund (1985) that adjusts for bias from sequentially picking favorable
+3
+
+evidence, whereas we consider a static panel that is not an on-going experiment.
+There are cases where the covariates warrant a natural order such that the hypothesis family
+possesses a special testing logic. A hierarchical structure in covariates arises when the inclusion of
+the next covariate only make sense if the previous covariates is included. An example is the use of
+principal component (PC) factors, where PCs are included sequentially from the dominating one
+to the least dominating one. We distinguish this from putting weights and assigning importance
+on features because this variant of family of hypotheses warrants a new definition of FWER. We
+propose a step-down procedure that can be considered as a panel extension of G’Sell, Wager,
+Chouldechova, and Tibshirani (2016), relying on an approximation of the R´enyi representation of
+p-values. The step-down control for nested FWER is based on Simes (1986), which along with
+Bonferroni (1935) can be seen as comparing sorted p-values against linear growth. Our framework
+contributes to estimating the number of principal component factors in a panel. There are have been
+many studies that provide consistent estimators for the number of PCs based on the divergence in
+eigenvalues of the covariance matrix, which include Onatski (2010), Ahn and Horenstein (2013) and
+Pelger (2019). Another direction uses sequential testing procedures that presume correct nested
+family of hypotheses, which include Kapetanios (2010) and Choi, Taylor, and Tibshirani (2017).
+In contrast, we characterize the least amount of factors (which can also be based on principal
+components), which should be expected when a FWER rate is provided. The nested version of
+our procedure is close in nature to a panel version of “when-to-stop” problem of a multiple testing
+procedure.
+The problem of post-LASSO statistical testing for small dimensional cross-sections is studied in
+a stream of papers including Meinshausen and B¨uhlmann (2006), Zhang and Zhang (2014), van de
+Geer, B¨uhlmann, Ritov, and Dezeure (2014) and Javanmard and Montanari (2018), which consider
+inference statements by debiasing the LASSO estimator. An alternative stream of post-selection
+or post-machine learning inference literature includes Chernozhukov, Hansen, and Spindler (2015),
+Kuchibhotla, Brown, Buja, George, and Zhao (2018) and Zrnic and Jordan (2020), who provide
+non-parametric post-selection or post-regularization valid confidence intervals and p-values. These
+papers do not make conditional statements and presume that the researcher sets the hypotheses
+before seeing the data, which we will refer to as data agnostic hypothesis family. We follow a dif-
+ferent train of thought that treats LASSO, among a family of conic maximum likelihood estimator,
+as a polyhedral constraint on the support of the response variable. This geometric perspective
+that provides inferential theory post-LASSO is pioneered by the work of Lee, Sun, Sun, and Taylor
+(2016) and followed up by Fithian, Sun, and Taylor (2017) and Tian and Taylor (2018), assum-
+ing Gaussian linear model. Markovic, Xia, and Taylor (2018) extends the results to LASSO with
+cross-validation, Tian, Loftus, and Taylor (2018) discusses a square-root LASSO variant that takes
+unknown covariance into consideration and Tian and Taylor (2017) considers the asymptotic results
+when removing the Gaussian assumption. This body literature is often referred to as PoSI, and
+traverses the Karush-Kuhn-Tucker (KKT) condition of a LASSO optimization problem to show
+that the LASSO fit can be expressed as a polyhedral constraint on the support of the response
+4
+
+variable. We extend this work by allowing to put weights onto prior belief sets, and by bringing it
+to the panel setting with multiple testing adjustment.
+2
+Sparse linear models
+We consider a large dimensional panel data set Y ∈ RT×N which we want explain with a large
+number of potential covariates X ∈ RT×J. The panel data and explanatory variables are both
+observed over T time periods.1 The size of the cross-section N and the dimension of the covariate
+candidate set J are both large in our problem. We assume a linear relationship between Y and X:
+Yt,n =
+J
+�
+j=1
+Xt,jβ(n)
+j
++ ϵt,n
+for n = 1, ..., N,
+which reads in matrix notation as
+Y = Xβ + ϵ
+(1)
+We refer to the coefficients β as loading matrix, where the nth column β(n) corresponds to the nth
+unit and β(n)
+j
+denotes the loading of the nth unit on the jth covariate. The remainder term ϵ is
+unexplained noise.
+We assume that a sparse linear model can explain jointly the full panel. Formally, a sparse
+linear model with s active covariates is
+Y = XSβS + ϵ
+(2)
+where s = |S| is the cardinality of the set of active covariates S = {j : ∃β(j)
+n
+̸= 0, n ∈ {1, ..., N},
+that is, the set of covariates with non-zero loadings. XS is the subset of covariates that belong to
+S. Our goal is to estimate this low dimensional model, that can explain the full panel, from a large
+number of candidate covariates, and provide a valid inferential theory.
+Note that our sparse model formulation allows for two important properties. First, different
+units can be explained by different covariates with different loadings. This means that β(n) ̸= β(m)
+for n ̸= m.
+For example, a subset of the cross-sectional units might be modeled by different
+covariates than the remaining part of the panel. Second, we can accommodate “weak” covariates. A
+covariate is included in S if it is required by at least one cross-sectional unit requires as explanatory
+variable. In other words, a sparse model can include covariates in XS that explain only a very
+small subset of the panel Y .
+The first step is to estimate the sparse models over the time-series for each unit separately due
+to the heterogeneity in the loadings. In a second step, we provide the valid inferential theory for
+the loadings on the full panel. The time-series estimation requires an appropriate regularization to
+1Our setting and multiple testing results can be readily extended to the case of unbalanced panel, although we
+focus on the balanced panel case for now to highlight the core multiple testing insight of our method. We will further
+discuss on this once we introduce our main procedure in Section 4
+5
+
+select a small subset of covariates that contains all the relevant covariates for each unit. We allow
+for a prior belief weight ω ∈ ¯RJ
++, so that different X can have different relative penalizations, and
+a global λ ∈ R+ scalar penalty parameter. For the nth unit, we denote its β(n) estimate as ˆβ(n)
+and the active set M(n) = {j : ˆβ(n)
+j
+̸= 0} as the set of j’s with non-zero loadings ˆβ(n)
+j
+. A general
+regularized linear estimator solves the following optimization problem
+ˆβ(j)(λ, ω) = arg min
+β
+1
+2T ∥Y (j) − Xβ∥2
+2 + λ · f(β, ω)
+(3)
+for a penalty function f and appropriate weights. In this paper, we consider the weighted LASSO
+estimator with the regularization function
+f(β, ω) =
+J
+�
+j=1
+fj(βj, ωj)
+where fj(βj, ωj) =
+�
+�
+�
+|βj|
+ωj
+ωj < ∞
+0
+o.w.
+(4)
+and weights ωj > 0 for all j ∈ {1, ..., J} and ∥ω−1∥1 = J.
+We assume that the penalty λ is
+selected such that the set ∥ˆβ(j)∥0 = |M(j)| is low dimensional. Importantly, we do not need to
+assume that the selected set contains all active covariates. Our goal it is provide a valid inferential
+theory conditional on the selected set. Our estimator generalizes the conventional LASSO with
+the l1 regularization function of Tibshirani (1996) by allowing for different relative weighting in
+the penalty. Importantly, we also allow for an infinite weight, which can be interpreted as a prior
+on a set of covariates.
+This allows researchers to take advantage of prior information and for
+example ensure that a specific set of covariates will always be included. The weighted LASSO
+will be particularly relevant in our empirical study, where we can answer the question which risk
+factors should be added to a given set of economically motivated risk factors. Our weighted LASSO
+formulation can also be interpreted as a Bayesian estimator with the canonical Laplacian prior.
+Conventional regression theory will not provide correct inferential statements on the weighted-
+LASSO estimates. We face two challenges. First, regularized estimation results in a bias, which
+needs to be corrected. Second and more challenging, post-selection inference changes the distribu-
+tion of the estimators. When we observe an active ˆβ(n)
+j
+from (3), it would be incorrect to simply
+calculate its p-value from a conventional t-distribution. This invalidity stems from the fact that
+conditional on observing a LASSO output, β(n)
+j
+must be large enough in magnitude for its ˆβ(n)
+j
+to
+be active. In other words, the probability distribution of the estimators is truncated.
+The correct inference has to be conditional on the covariates being selected by the LASSO
+estimator. Hence, valid p-values have to be the tail probability conditional on being in the selection
+set. The key to quantify such styles of inference is to recognize that a sparsity constrained estimator
+is typically the result of solving Karush-Kuhn-Tucker (KKT) conditions, which can in turn be
+geometrically characterized as polyhedral constraints on the support of response variables. This
+is first established in Lee, Sun, Sun, and Taylor (2016), who provide the stylized results that
+Post-Selection Inference (PoSI) of debiased non-weighted LASSO estimators can be calculated as
+6
+
+polyhedral truncation on Y . This line of research is also referred to as Selective Inference in other
+literature such as Taylor and Tibshirani (2015). We extend this line of literature to allow for the
+Weighted-LASSO. We derive these results with assumptions common in the PoSI LASSO literature,
+detailed in Appendix A, and referred to as conventional regularity conditions for ease of exhibition.
+THEOREM 1. Truncated Gaussian Distribution of Weighted-LASSO
+Under conventional regularity conditions, the debiased estimate ¯βi for the i-th Weighted-LASSO
+active covariate is conditionally distributed as
+¯βi|Weighted-LASSO ∼ T N {η⊤Y :AY ≤b(ω)}
+(5)
+where T N A is truncated-Gaussian with truncation A, and the weights ω only appear in b(ω)
+Theorem 1 has two elements. First, it debiases the LASSO estimate by a shifting argument.
+While we use a geometric argument to remove the bias, the bias adjustment takes the usual form
+in the LASSO literature as for example in Belloni and Chernozhukov (2013). The debiased LASSO
+estimator simply equals a standard OLS estimation on the subset Mn selected by the Weighted-
+Lasso. Second, the distribution of the linear coefficients is not a usual Gaussian distribution, but it
+is truncated due to studying post-selection coefficients. This geometric perspective is less common
+in the LASSO literature, but provides several advantages. One advantage of the geometric approach
+is that it avoids the use of infeasible quantities, in particular the second moment of the large set of
+potential covariates. Furthermore, the distribution result is not asymptotic in T, but also valid in
+finite samples. We can obtain these results because we make the stronger assumption that the data
+is normally distributed. Appendix A provides the detailed information on constructing ¯β and the
+definitions of η, A, b(ω) along with lemmas that lead up to this result. It also discusses extensions
+and the effect of estimating the variance of the noise. The empirical analysis is based on the explicit
+form of Theorem 1 formulated in Theorem A.3.
+Our Weighted-LASSO results make several contributions. First, the expression for the trun-
+cated conditional distribution with weights become much more complex than for the special case
+of the conventional LASSO. Second, we provide a simple, easy-to-use and asymptotically valid
+conditional distribution in the case of an estimated noise variance. Last but not least, we show the
+formal connection with alternative debiased LASSO estimators by showing that debiasing can be
+interpreted as one step in a Newton-Ralphson method of solving a constrained optimization.
+Theorem 1 allows us to obtain valid p-values for Weighted-LASSO coefficients. We obtain these
+p values from the simulated cumulative distribution function of the truncated Gaussian distribution.
+Crucially, all results for multiple testing adjustment in panels that we study in the following sections
+neither require us to use a weighted Lasso estimator nor to use the p-values implied by Theorem
+1. We only require to have a set of valid p-values for sparsity constrained models. These can be
+obtained with any suitable regularized estimator and post-selection inference. The key element is
+the selection of a low dimensional subset with p-values conditional on this selection. We propose
+the weighted LASSO conditional inference results as an example of the type of sparsity constraint
+7
+
+models we are interested in, and demonstrate a machinery with which we can obtain valid p-values
+for sparsity constrained models. In our empirical studies, we use Weighted-LASSO as our sparsity
+constrained model since we want to specify strong prior beliefs on a few covariates and it is common
+practice to use LASSO in the context of our empirical studies. Nonetheless, the testing methods
+in the next sections accommodate any sparse estimator, and can be detached from inference for
+Weighted-LASSO.
+3
+Data-Driven Hypotheses
+Our goal is to provide formal statistical tests that allow us to establish a joint model across a
+large cross-section with potentially weak covariates. This requires us to provide a form of statistical
+significance test with multiple testing adjustment that properly accounts for covariates that only ex-
+plain a small subset of the cross-sectional units. This is important as in many problems in economic
+and finance there is substantial cross-sectional variation in the explanatory power of covariates, and
+a model that simply minimizes an average error metric might neglect weaker covariates.
+An essential step for a formal statistical test is to formulate the hypothesis. This turns out to be
+non-trivial for a large panel with a first stage selection step for the covariates. It is a fundamental
+insight of our paper, that the hypothesis of our test has to be conditional on the selected set of
+active covariates of the first stage. Once we have defined the appropriate hypothesis, we can deal
+with the multiple testing adjustment, which by construction is also conditional on the selection
+step.
+The hypothesis formulation and test construction only requires valid p-values from a first stage
+selection estimator. The results of the next two sections do not depend on a specific model for
+obtaining these p-values and the active set. The results are valid for any model including non-
+linear ones. The input to the analysis is a N ×J matrix, which specifies which covariates are active
+for each unit and the corresponding p-values. The Weighted-LASSO is only one possible model,
+but it can be replaced by any regularized model. We have introduced the sparse linear model as
+it is the horse race model for many problems in economics and finance, and therefore of practical
+relevance.
+We illustrate the concept of a data-driven hypothesis with a simple example, which we will
+use throughout this section. For simplicity we assume that we have J = 4 covariates and want
+to explain N = 6 cross-sectional units. In the first stage, we have estimated a Weighted-LASSO
+and have obtained the post-selection valid p-values for each of the N units. We collect the fitted
+sparse estimator ¯β(n) for the nth unit in the matrix ¯β. Note, that this matrix has “holes” due to
+the sparsity for each ¯β(n). Figure 1(a) illustrates ¯β for this example.
+Similarly, we collect the corresponding p-values in the matrix P . For the nth unit, we only
+have p-values for those covariates that are active in the nth linear sparse model. Thus, Figure
+1(b) also has white boxes showing the same pattern of unavailable p-values due to the conditioning
+on the output of the linear sparse model. These holes can appear at different positions for each
+8
+
+Figure 1: Illustrative example of data-driven selection
+(a) Matrix ¯β
+(b) Matrix P of p-values
+This figure illustrates in a simple example the data-driven selection of a linear sparse model. In a first stage, we have
+estimated a regularized sparse linear model for each of the N = 6 units with J = 4 covariates. Each row represents
+the selected covariates with their estimated coefficients and p-values. The columns represent the J = 4 different
+covariates. The grey shaded boxes represent the active set, while white boxes indicate the inactive covariates. The
+numbers are purely for demonstrative purposes.
+unit, which makes this problem non-trivial. This non-trivial shape of either subplot (a) or (b)
+is completely data-driven and a consequence of linear sparse model selection. We show that the
+hypothesis should be formed around these non trivial shapes as well, which is why we name it the
+data-driven hypothesis family.
+We want to test which covariates are jointly insignificant in the full panel. A data-agnostic
+approach would simply test if all covariates are jointly insignificant, independent of the data-driven
+selection step in the first stage. A data-agnostic hypothesis is unconditional as it does not depend
+on any model output.
+However, as we will show, this perspective is problematic for the high-
+dimensional panel setting with many covariates as it ignores the dimension reduction from the
+selection step. Therefore, an unconditional multiple testing adjustment accounts for “too many”
+tests, which severely reduces the power.
+We propose to form the hypothesis conditional on the first stage selection step. The data-driven
+hypothesis only tests the significance of the covariates that were included in the selection, and hence
+can drastically reduce the number of hypothesis. However, given the non-trivial shape of the active
+set, the multiple testing adjustment for the data-driven hypothesis is more challenging.
+Before formally defining the families of hypothesis, we illustrate them in our running example.
+9
+
+1
+2
+3
+4 
+1 
+-5.43
+2.15
+2
+-1.10
+4.78
+-0.08
+m
+0.19
+4.59
+4 
+4.44
+2.10
+5
+1.44
+4.53
+2.10
+6
+-0.46
+4.701 
+2
+3
+4 
+1
+0.127
+0.587
+2
+0.005
+0.001
+0.871
+3
+0.526
+0.001
+4 
+:0.001
+≤0.001
+5
+:0.0010.001
+0.001
+6
+0.102
+0.010The data-agnostic hypothesis HA for explaining the full panel takes the following form:
+HA = {HA0,1, HA0,2, HA0,3, HA0,4}
+= {β(1)
+1
+=β(2)
+1
+= β(3)
+1
+= β(4)
+1
+= β(5)
+1
+= β(6)
+1
+= 0,
+β(1)
+2
+=β(2)
+2
+= β(3)
+2
+= β(4)
+2
+= β(5)
+2
+= β(6)
+2
+= 0,
+β(1)
+3
+=β(2)
+3
+= β(3)
+3
+= β(4)
+3
+= β(5)
+3
+= β(6)
+3
+= 0,
+β(1)
+4
+=β(2)
+4
+= β(3)
+4
+= β(4)
+4
+= β(5)
+4
+= β(6)
+4
+= 0}
+(6)
+The data-driven hypothesis HD only includes the active set and hence equals
+HD = {β(2)
+1
+=0,
+β(1)
+2
+=β(3)
+2
+= β(5)
+2
+= β(6)
+2
+= 0,
+β(1)
+3
+=β(2)
+3
+= β(3)
+3
+= β(4)
+3
+= β(5)
+3
+= β(6)
+3
+= 0,
+β(2)
+4
+=β(4)
+4
+= β(5)
+4
+= 0}
+(7)
+Obviously, HA has a larger cardinality of |HA| = 24 > |HD| = 14. This holds in general, unless the
+first stage selects all covariates for each unit, in which case the two hypotheses coincide.
+Formally, the data-agnostic family of hypothesis is defined as follows:
+DEFINITION 1. Data-agnostic family
+The data-agnostic family of hypotheses is
+HA = {HA0,i|i ∈ [d]}
+where HA0,i =
+�
+j∈[N]
+H(j)
+A0,i and H(j)
+A0,i : β(j)
+i
+= 0.
+(8)
+It is evident that HA does not need any model output or exploratory analysis, so it is indeed
+data-agnostic.
+As soon as we use a sparsity constrained model that has censoring capabilities, we no longer
+observe (Y , X) from its data generating process. Consequently, unless our hypotheses depend on
+how we built the model, or equivalently on how the data was censored, the data-agnostic hypotheses
+forgo power without any benefit in false discovery control. Therefore, we formulate the hypothesis
+on the ith covariate H(j)
+0,i only if i ∈ M(j), that is, it is in the active set. Conditional on observing
+the model output, there is no inference statement to be made about H(j)
+0,i if i /∈ M(j), because its
+estimator is censored by the model.
+We denote as Ki the set of units for which the ith covariate is active. We define the cross-
+sectional hypothesis for the ith covariate as:
+H0,i =
+�
+j∈Ki
+H(j)
+0,i
+����M,
+∀i : Ki ̸= ∅
+(9)
+By combining all covariates {i : Ki ̸= ∅} that show up at least once in one of the active sets of our
+sparse linear estimators, we arrive at a data-driven hypothesis associated with our panel. This is
+defined as follows:
+10
+
+DEFINITION 2 (Data-driven family). The data-driven family of hypotheses conditional on M
+is
+HD = {H0,i|i : Ki ̸= ∅}
+(10)
+This demonstrates the non-trivial nature of writing down a hypothesis in high-dimensional
+panel: we can only collect Ki - the set of units for which the ith covariate is active - after seeing
+the sparse selection estimation result.
+4
+Multiple Testing Adjustment for Data-Driven Hypothesis
+4.1
+Simultaneity Counts through Panel Localization
+We show how to adjust for multiple testing of data-driven hypotheses. Given the p-values p(j)
+i
+for i ∈ M and j ∈ Ki, we form the data-driven hypothesis HD. Our goal is to reject members of
+HD while controlling the Type I error, and the common way to measure such error is the family-
+wise error rate. This is the same underlying logic that is used to define confidence intervals and
+determine significance of covariates in a conventional setup. The crucial difference is that we need
+to account for multiple testing given the large number of cross-sectional units. The family-wise
+error rate (FWER) is defined as follows:
+DEFINITION 3. Family-wise error rate
+Let V denote the number of rejections of H(j)
+0,i |M(j) when the null hypothesis is true. The family-
+wise error rate (FWER) is P(V ≥ 1).
+Similar to the conventional definition, we simply count the false rejections V and define FWER
+as the probability of making at least one false rejection.
+Importantly, Definition 3 accounts for the fact that we might repeatedly test on �
+j∈[N] |Mj|
+rather than a single hypothesis test of the form H(j)
+0,i : β(j)
+i
+= 0|M(j). Our contribution to FWER
+control in the panel setting is thus to take into consideration both the multiplicities in units and
+covariates when we deal with the “matrix” of p-values P . To achieve this goal, we propose a new
+simultaneity account for the ith covariate, calculated as
+Ni =
+�
+j∈Ki
+|Mj|
+(11)
+Figure 2 illustrates the simultaneity counting for our running example with N = 6 units and
+J = 4 covariates. The blue boxes represent the active set for a specific covariate. The yellow boxes
+indicate the “co-active” covariates, which have to be accounted for in a multiple testing adjustment.
+In the case of the first covariate j = 1, only the second unit n = 2 has selected this covariate. This
+second unit has also selected covariate j = 3 and j = 4, which are jointly tested with the first
+covariates. Hence, they are “co-active”, and the simultaneity count equals N1 = 3. Intuitively,
+Nj represents all relevant comparisons for the jth covariate because it counts how many covariates
+11
+
+Figure 2: Simultaneity counts Ni in the illustrative example
+(a) N1 = 3
+(b) N2 = 9
+(c) N3 = 14
+(d) N4 = 8
+This figure shows the simultaneity counts Ni in the illustrative example. The subplots represent the simultaneity
+counts for the J = 4 covariates. The blue boxes indicate the active set Kj of the j covariates, while yellow boxes
+indicate the “co-active” covariates of the jth covariate. The simultaneity counts are the sum of yellow and blue
+boxes.
+are active with the jth covariate in the regressions. Hence, Nj quantifies the number of “multiple
+tests” for each covariate.
+In subplot 2(a), we see that K1 = {2} for the 1st covariate, indicated by the blue box, because
+it is only active in the second unit’s regression. The multiple testing adjustment needs to consider
+all yellow boxes, and N1 = 3 is thus the total count of 1 blue and 2 yellow boxes. Similarly, for
+the second covariate, K2 = {1, 3, 5, 6}, so we shade boxes yellow for the 2nd, 3rd and 5th units
+and obtain N2 = 9. We can already see that our design of simultaneity count takes all relevant
+pairwise comparisons into considerations, but avoids counting the white boxes - which would cause
+overcounting and result in over-conservatism.
+Our multiplicity counting is a generalization of the classical Bonferroni adjustment for multiple
+testing. A conventional Bonferroni method for the data-agnostic hypothesis HA has a simultaneity
+count of |HA| = N · J = 24 for testing each covariate. A direct application of a vanilla Bonfer-
+roni method to the panel of all selected units and the data-driven hypothesis HD, would use a
+simultaneity count of |HD| = 14 for testing each covariate. Our proposed multiplicity counting is
+a refinement that leverages the structure of the problem, and takes the heterogeneity of the active
+sets for each covariate into account. Our count has only N1 = 3, N2 = 9 and N4 = 8 for the
+covariates j = 1, 2 and 4. Only for covariate j = 3 is the simultaneity count the same as a vanilla
+Bonferroni count applied to HD, i.e. N3 = 14.
+In addition to the simultaneity count of each covariate, we need an additional “global” metric
+for our testing procedure. We define a panel cohesion coefficient ρ as a scalar that measures how
+12
+
+1
+2
+3
+4 
+2
+4 
+5
+61
+2
+3
+4
+1
+2
+4 
+5
+61
+2
+3
+4
+1
+2
+4 
+5
+62
+3
+4
+1
+2
+3
+4 
+5
+6Figure 3: Illustration of the cohesion coefficient
+(a) ρ = J−1 = 0.25
+(b) ρ = 0.44
+(c) ρ = 1
+This figure illustrate the cohesion coefficient ρ in three separate examples. It shows the smallest, largest and in-
+between cases of ρ. The columns represent the J = 4 different covariates.The blue boxes indicate the active sets for
+each panel.
+sparse or de-centralized the proposed hypotheses family is:
+ρ =
+�
+��
+j
+|Kj|
+Nj
+�
+�
+−1
+(12)
+The panel cohesion coefficient ρ is conditional on the data-driven selection of the overall panel. It is
+straightforward to compute once we observe the sparse selection of the panel. This coefficient takes
+values between J−1 and 1,2 where larger values of ρ imply that the active set is more dependent in
+the cross-section. This can be interpreted as that the panel Y has a stronger dependency due to
+the covariates X. Intuitively, in the extreme case when ρ = J−1, the panel can be separated into
+J smaller problems, each containing a subset of response units explained by only one covariate.
+Thus the panel would be very incohesive, and could be studied with J independent tests. In the
+other extreme, if ρ approaches 1, the first-stage models include all active covariates for all units.
+We consider this as a very cohesive panel. If ρ is between theses bounds, the panel is cohesive in
+a non-trivial way such that some units can be explained by some covariates and there is no clear
+separation of the panel into independent subproblems.
+Figure 3 illustrates the panel cohesion coefficient in three examples. The subplots show three
+active sets that are different from our running example. The left subplot 3(a) shows the extreme
+case of ρ = J−1, where the panel is the least cohesive. The right subplot 3(c) illustrates the other
+extreme for ρ = 1, where the panel is the most cohesive. The middle subplot 3(b) is the complex
+case of a medium cohesion coefficient.
+2We prove this bound in the Appendix, without leveraging sparsity of first-stage models but rather as an algebraic
+result with intuitive interpretations.
+13
+
+1
+2
+3
+4 
+1
+2
+3
+4 
+5
+61
+2
+3
+4 
+1
+2
+4 
+5
+61
+2
+3
+4 
+1
+2
+4 
+5
+6Our novel simultaneity count and cohesiveness measure are the basis for modifying a Bonferroni
+test for FWER-controlled inference. Theorem 2 formally states the FWER control. The proof is
+in the Online Appendix.
+THEOREM 2. FWER control
+The following rejection rule has FWER≤ γ on HD:
+min
+n∈Kj
+�
+p(n)(j)
+�
+≤ ρ γ
+Nj
+⇒ Reject H0,j
+(13)
+where p(n)(j) are valid p-values for each univariate unit n, and ρ is the panel cohesion coefficient.
+This completes the joint testing procedure. First, we calculate p-values after running a sparse
+linear estimator time-series regression. Second, we use the sparse linear estimator output to write
+down a hypothesis and, third, we provide a FWER control inference procedure by combining the
+p-values across the cross-section and test the hypothesis.
+The difference between a naive Bonferroni and our FWER control is particularly pronounced
+for weak covariates that affect only a subset of the cross-sectional units. Given a FWER control
+level of γ, the rejection threshold for a naive Bonferroni test is
+γ
+JN for every covariate. The rejection
+threshold for our FWER control is always higher, and differs in particular when Nj is small and ρ
+is large. This is the case for weak covariates in a cohesive panel.
+As it is common in statistical inference, we focus on Type I error control. Type II error rates
+require the specification of alternatives. While we do not provide formal theoretical results for the
+power of our inference approach, we show comprehensively in the simulation and empirical part,
+that our approach has substantially higher power than conventional approaches.
+We point out that the validity of our procedure holds for unbalanced panels as well.
+This
+is because even when there are different number of observations for the nth and mth units, i.e.
+Tn ̸= Tm for n ̸= m, they can still be estimated separately in the first stage of the regularized
+regression. The hypothesis testing and selection of a parsimonious model only requires the matrix
+P of valid p-values, which can be based on different samples.
+4.2
+Least Number of Covariates: Traversing the Threshold
+The typical logic of statistical inference is to determine which covariates we should admit from
+XM, given a significance level γ. We use K to denote the number of selected covariates. When
+γ is specified as a lower quantity, we expect K to decrease as well, that is, the rejection becomes
+harsher.
+As the number of admitted covariates of our procedure is monotone in γ, we want to ask
+the following converse question: How low do we need to set γ such that we reject K covariates?
+Concretely, we are interested in finding:
+14
+
+γ∗(K) = sup
+�
+�
+�γ|K =
+J
+�
+j=1
+1
+�
+min
+n∈Kj
+�
+p(n)
+j
+�
+≤ ρ γ
+Nj
+��
+�
+� .
+(14)
+Let pj = minn∈Kj{p(n)
+j
+} be the 1st order statistic for j = 1, ..., J. Then (14) is simply the K-th
+order statistics of Njpj/ρ:
+γ∗(K) = min{Nipi/ρ|∃j1, j2, ..., jK ∈ {1, ..., J} : Nipi ≥ Njkpjk}.
+(15)
+Since this minimization scan is monotone, we can determine how many covariates at least
+should be admitted, given a control level, which is similar to the “SimpleStop” procedure described
+in Choi, Taylor, and Tibshirani (2017). The following corollary formalizes this inversion method
+that finds the least number of covariates to admit:
+COROLLARY 1. Least number of covariates
+Given the FWER level γ, there exists a unique number K∗(γ) such that
+K∗(γ) =
+�
+�
+�
+arg max0≤K≤J γ∗(K) ≤ γ
+∃K : γ∗(K) ≤ γ
+d
+o.w.
+(16)
+The statement simply states that the simplest linear model should have at least K∗(γ) covariates
+for a given γ. Note that it is possible that, for example, γ∗(5) and γ∗(6) are both equal to 0.05,
+while γ∗(7) > 0.05. In this case the minimum number of covariates is K∗(0.05) = 6 because it does
+not hurt FWER-wise to include 6 covariates in the model. Hence, we are making a slightly different
+statement than that there would be exactly K∗(γ) covariates in the true linear model. The number
+of covariates is obviously conditional on the set of candidate covariates X, and we can only make
+statements for this given set.
+In our empirical study we consider candidate asset pricing factors X to explain the investment
+strategies Y . More generally, the linear model that we consider is often referred to as a factor model.
+Therefore, we will also refer to the selected covariates as factors, and use these two expressions as
+synonyms moving forward. This directly links our procedure to the literature on estimating the
+number of factors to explain a panel. A common approach in this literature is to use statistics based
+on the eigenvalues of either Y or X to make statements about the underlying factor structure. Our
+approach is different, as it provides significance levels for the selected factors and FWER control
+for the number of factors.
+Table 1 illustrates the estimation of the number of factors and their ranking with our running
+example introduced in Figure 1. We calculate the simultaneity counts Ni’s as given in (11) and
+demonstrated in Figure 2, and pi as the smallest p-values associated with the ith covariate. Then,
+the rejection rule in Theorem 2 is based on whether a pre-specified level γ satisfies pi < ργ
+Ni , which
+is equivalent to Ni · piρ < γ.
+Thus, the natural ranking of the covariates is to sort all covariates in descending order of the
+15
+
+Table 1: Sorted p-values for the running example
+Factor (j)
+pj
+Simultaneity count for HD
+Conventional Bonferroni for HA
+ρ−1 · Nj
+ρ−1 · Nj · pj
+J · N
+J · N · pj
+3
+< 0.001
+22.1
+< 0.001
+24
+0.002
+4
+< 0.001
+11.1
+0.001
+24
+0.003
+1
+0.005
+4.7
+0.024
+24
+0.120
+2
+0.002
+14.3
+0.028
+24
+0.051
+This table constructs “significance” levels for the running example introduce in Figure 1. We compare the simul-
+taneity count for the data-driven hypotheses HD and a onventional Bonferroni count for data-agnostic hypotheses
+HA. The products Nj · pj, respectively J · N · pj, can be interpreted as the significance levels for the corresponding
+approach. Given a FWER control γ all factors with ρ−1 · Nj · pj (respectively J · N · pj) below this threshold are
+selected.
+Ni · pi/ρ values as shown in Table 1. It is then trivial to determine K∗(γ) for any choice of γ. For
+example, for γ = 1%, we would select factors 3 and 4, but not 1 and 2. On the other hand, for
+γ > 2%, we would include all four factors. Hence, the ranking of Nipi/ρ directly maps into K∗(γ).
+The list of Nipi/ρ encompasses more information than just the number of factors. Naturally, it
+provides an importance ranking of the factors. Furthermore, the number Ni reveals if significant
+factors are “weak”. In our case, factor 1 has N1 = 3, which indicates that it affects only a small
+number of hypothesis. Its p-value p1 is sufficiently small to still imply significance in terms of
+FWER control.
+For comparison, Table 1 also includes the corresponding analysis for the data-agnostic hypoth-
+esis and a conventional Bonferroni correction. The Bonferroni analysis uses the same p-values but
+a different multiple testing adjustment. In our case, the p values would be multiplied by J ·N = 24
+as this corresponds to the total number of hypothesis tests. This will obviously make the inference
+substantially more conservative. Indeed, even for a FWER control of γ = 4%, we would only select
+factors 3 and 4. We would need to raise the FWER control to γ = 12% to include factor 1. Hence,
+weak factors, like factor 1, are more likely to be discarded by the data-agnostic hypothesis with
+conventional multiple testing adjustment.
+We want to emphasize that a data-agnostic hypotheses with conventional Bonferroni correction
+does provide correct FWER control, but it is overly conservative. By construction, the data-agnostic
+Bonferroni approach will test a larger number of hypothesis, which means that the corresponding
+“significance levels” will always be lower or equal to our data-driven simultaneity count. Second,
+the data-agnostic Bonferroni approach does not differentiate the “strength” of the factors, while
+our approach provides a selection-based heterogeneous adjustment of the p-values. This is essential
+for detecting weak factors.
+Having introduced all building blocks of our novel method to detect covariates, we put the entire
+procedure together as “Panel-PoSI”:
+PROCEDURE 1. Panel-PoSI
+The Panel-PoSI procedure consists of the following steps:
+16
+
+1. For each unit n = 1, ..., N unit, we fit a linear sparse model ˆβ(n)
+X,Y (c, ω) given (X, Y , λ, ω). We
+suggest cross-validation to select the LASSO penalty λ. We construct the sparse estimators
+¯β(n) and the corresponding p-values for the active covariates for each unit, and collect them
+in the “matrix” of p-values P .
+2. We collect the panel-level sparse model selection event M and construct the data-driven hy-
+pothesis HD.
+3. Given the FWER control level γ and based on the the simultaneity counts Nj, we make
+inference decision for the sparse model. We can rank covariates in terms of their significance
+and select a parsimonious model that explains the full panel.
+As we have now all results in place, we can summarize the advantages of our procedure. First,
+we want to clarify that our goals and results are different from just some form of optimal shrink-
+age selection. Selecting a shrinkage parameter with some form of cross-validation in a regularized
+estimator like LASSO does not provide the same insights and model that we do.
+A shrinkage
+estimator can either be applied to each unit separately, as we do it in our first step, or to the
+full panel in a LASSO panel regression. The separate covariate selection for each cross-sectional
+unit does not answer the question which covariates are needed to explain the full panel jointly. A
+shrinkage selection on the full panel for some form of panel LASSO can neglect weaker factors,
+as those receive a low weight in the cross-validation objective function. Second, tuning parameter
+selection with cross-validation requires a sufficiently large amount of data. Our approach is attrac-
+tive as we can do the complete analysis on the same data. That means, an initial LASSO is used
+to first reduce the number of covariates, but this set is then further trimmed down using inferential
+theory. Hence, we can construct a parsimonious model even for data with a relatively short time
+horizon, but large cross-sectional dimension. Third, the statements that we can make are much
+richer than a simple variable selection. We can formally assess the relative importance of factors
+in terms of their significance. The model selection is directly linked to a form of significance level,
+which allows us to assess the relevance of including more factors. Last but not least, we can also
+make statements about the strength of factors. In summary, Panel-PoSI is a disciplined approach
+based on formal statistical theory to construct and interpret a parsimonious model.
+5
+Ordered Multiple Testing on Nested Hypothesis Family
+So far, our hypothesis family HD has no hierarchy and consequently, we have not imposed a
+sequential structures on the admission order of covariates of X. However, there are cases where
+the covariates or factors warrant a natural order such that the family possesses a special testing
+logic. A hierarchical structure in covariates arises when the inclusion of the next covariate only
+make sense if the previous covariates is included. One example would be if the next covariates
+refines a property of the previous covariate. Another case is the use of principal component (PC)
+factors.
+The conventional logic is to include PCs sequentially from the dominating one to the
+17
+
+least dominating one. This is similar to the motivation for Choi, Taylor, and Tibshirani (2017),
+but different from them, we treat the PCs as exogenous without taking the estimation of PCs
+explicitly into account. In this section, we will use exogenous PCs as hierarchical covariates, as this
+is the main example in our empirical study. However, all the results hold for any set of exogenous
+hierarchical covariates.
+Without loss of generality, we presume X has the jth column as the jth nested factor. A
+k-order nested model N(k) is of the following form
+N(k) model : Y = X[k]β[k]
+(17)
+where [k] = {1, ..., k} is the set that includes indices up to k. For example, a hierarchical three
+factor model corresponds to X{1,2,3}. When formulating our hypothesis family, we must represent
+the sequential testing structure. This is reflected in our definition of nested families of hypotheses:
+DEFINITION 4. Data-driven nested family
+The data-driven nested family of hypotheses conditional on M is
+HN = {HN,k : k = 0, 1, ..., J},
+HN,k =
+�
+j∈Kk
+H(j)
+N,k
+����M,
+H(j)
+N,k : {i′ : β(j)
+i′
+̸= 0} ≤ k.
+(18)
+HN,0 completes the case when no rejection on any factor is made. Whenever HN,k is true, then
+HN,k′ is also true for k < k′ ≤ J. Moreover, in the cases where Kk = ∅ but Kk′ ̸= ∅ with k < k′,
+the notation ensures that the hypothesis HN,k is included in HN simply because Kk′ is present.
+In other words, if a less dominating hypothesis HN,k′ is suggested by data (that is, its active set is
+non-empty Kk′ ̸= ∅), HN would automatically include all HN,k for k ≤ k′.
+The FWER control property needs to be adapted to the nested nature of this family. Choi,
+Taylor, and Tibshirani (2017) argue that the proper measurement is to control for ordered factor
+count over-estimation with level γ, as follows:
+DEFINITION 5. FWER for nested family
+For a test that rejects HN,k for k = 1, 2, ..., ˆk of HN, the FWER control at the level γ satisfies
+P(ˆk ≥ s) ≤ γ, where s is the true factor count.
+Given the hierarchical belief about the model, we need to add the following additional assump-
+tion:
+ASSUMPTION 1. Tail p-values
+Under H(j)
+N,k, there is p(j)(i′) iid
+∼ Unif [0, 1] if i′ > k.
+Assumption 1 only needs to hold for the tail hierarchical covariates. In the case of PCs, it only
+applies to the lower order tail PC factors that should not be included for a given null hypothesis.
+For example, if the true model is HN,s, we only need p(j)(i) iid
+∼ Unif[0, 1] for i > s, which is a
+usual type of assumption in this literature such as in G’Sell, Wager, Chouldechova, and Tibshirani
+18
+
+Figure 4: Example of hierarchical simultaneity counts Norder
+k
+for HN
+(a) N order
+4
+= 3
+(b) N order
+3
+= 5
+(c) N order
+2
+= 8
+(d) N order
+1
+= 12
+This figure shows the simultaneity counts N order
+i
+in an illustrative example. The subplots represent the simultaneity
+counts for the J = 4 covariates and N = 6 units. The dark blue columns present the active factors, while the light
+blue columns capture factors of higher-order. The sub-plots from left-to-right represent our calculation order from
+the highest-order factor to the 1st factor.
+(2016). Moreover, because the nested nature guarantees that the higher-order PCs are more likely
+to be null, a step-down procedure is expected to increase the power relative to a step-up procedure.
+As our focus is to control for false discoveries, we also need to adjust our simultaneity counts
+to the sequential testing. Concretely, we consider first taking a union to obtain the active unit set
+Korder
+k
+and then calculate conservative simultaneity counts Norder
+k
+:
+Korder
+k
+=
+�
+i∈{k,k+1,...,J}
+Ki,
+Norder
+k
+=
+�
+j∈Korder
+k
+|Mj|.
+(19)
+It is possible for some |Mk| to be 0 (that is, the kth PC could be inactive for all units), but its
+Norder
+k
+would be 0 if and only if higher-order PCs all have |Mk′| = 0 for k′ > k.
+Figure 4 illustrates the process of our step-down simultaneity count. From the left, we start
+with factor k = 4 and move step-wise down to factor k = 1 on the right. The dark blue columns
+present the active factors, while the light blue columns capture factors of higher-order. In the
+left-most sub-figure, we only need to account for the 4th PC, implying Norder
+4
+= 3, whereas in the
+mid-left sub-figure, the 3rd PC has Norder
+3
+= 2 + 3 = 5. Eventually, in the right-most sub-figure,
+we have swept through the entire panel and the 1st PC has a simultaneity count of Norder
+1
+= 12.
+Now we can introduce a step-down procedure adapted to the nested structure of HN:
+PROCEDURE 2. Step-down rejection of nested ordered family HN
+The step-down rejection procedure consists of the following steps:
+1. For each k ∈ {1, ..., J} calculate the ordered simultaneity count Norder
+k
+.
+19
+
+1
+2
+3
+4 
+1 
+2
+3
+4 
+5
+61 
+2
+4 
+1
+2
+4 
+5
+61 
+2
+4
+1
+2
+4 
+5
+61
+2
+4
+1
+2
+3
+4 
+5
+62. For each k ∈ {1, ..., J} calculate the approximated R´enyi representation Zorder
+k
+and its trans-
+formed reversed order statistics qorder
+k
+:
+Zorder
+k
+=
+J
+�
+i=k
+�
+j∈Ki
+ln(p(j)(k))
+Norder
+1
+− Norder
+i+1 1{i ̸= J},
+qorder
+k
+= exp(−Zorder
+k
+)
+(20)
+3. Reject hypothesis 1, 2, ..., ˆk, where ˆk = max{k : qorder
+k
+≤ γNorder
+k
+JN
+}.
+This procedure will have FWER control at level γ as stated in the following theorem:
+THEOREM 3. FWER control for ordered hypothesis
+Under Assumption 1, Procedure 2 has FWER control of γ for the ordered hypothesis HN.
+The proof is deferred to the Online Appendix. This design extends Procedure 2 from G’Sell,
+Wager, Chouldechova, and Tibshirani (2016) and “Rank Estimation” from Choi, Taylor, and Tib-
+shirani (2017), both of which focus on a single sequence of p-values rather than the panel setting.
+In Step 2, we use Assumption 1 to transform p-values into ln(p(j)(k)), which are i.i.d. standard
+exponential random variables. Since the family HN has J members, we need to modify our simul-
+taneity count and in a sense condense the panel into a sequence of statistics associated with the
+ordered covariates. We built a staircase sequence of conservative simultaneity count Norder
+k
+in Step
+1 to accumulate the number of p-values we use up to the kth ordered covariate, starting from the
+end. By the R´enyi representation of R´enyi (1953), the Zorder
+k
+of Step 2 approximate exponential
+order statistics and the qorder
+k
+approximate uniform order statistics. The nature of these approxima-
+tions is to create a more conservative rejection, the technical details of which are examined in the
+proof in our Online Appendix. Finally, we run the order statistics through a step-down procedure
+proposed by Simes (1986) so that we find the ˆk largest number of ordered covariates rejected by
+the data with FWER control. Also note that even if the global null, i.e. HN,0, is true, and every
+linear sparse model active set is empty, that is Norder
+1
+= 0, the procedure in Step 3 is still valid
+because we do not reject HN,1.
+6
+Simulation
+We demonstrate in simulations that our inferential theory allows us to select better models.
+We compare different estimation approaches to select covariates and show that our approach bet-
+ter trades off false discovery and correct selections and hence results in a better out-of-sample
+performance.
+Table 2 summarizes the benchmark models. Our framework contributes among three dimen-
+sions: the selection step for the sparse model, the construction of the hypothesis and the multiple
+testing adjustment. We consider variations for these three dimensions which yields in total six
+estimation methods. By varying the different elements of the estimators, we can understand the
+benefit of each component.
+20
+
+Table 2: Summary of estimation methods
+Name
+Abbreviation
+Selection
+Hypothesis
+Multiple Testing
+Rejection rule
+Naive OLS
+N-OLS
+OLS without LASSO
+Agnostic HA
+No adjustment
+pOLS < γ
+Bonferroni OLS
+B-OLS
+OLS without LASSO
+Agnostic HA
+No adjustment
+pOLS <
+γ
+JN
+Naive LASSO
+N-LASSO
+LASSO without PoSI
+Agnostic HA
+No adjustment
+pLASSO < γ
+Bonferroni Naive LASSO
+B-LASSO
+LASSO without PoSI
+Agnostic HA
+Bonferroni
+pLASSO <
+γ
+JN
+Bonferroni PoSI
+B-PoSI
+LASSO with PoSI
+Agnostic HA
+Bonferroni
+pPoSI <
+γ
+JN
+Panel PoSI
+P-PoSI
+LASSO with PoSI
+Data-driven HD
+Simultaneity count
+pPoSI < ργ
+Ni
+This table compares the different methods to estimate a set of covariates from a large dimensional panel. For each
+method, we list the name and abbreviation. The selection refers to the regression approach for each univariate
+time-series. The hypothesis is either agnostic or data-driven given the selected subset of covariates. The multiple
+testing adjustment includes no adjustment, a conventional Bonferroni adjustment and our novel simultaneity count
+for a data-driven hypothesis. The rejection rules combine the valid p-values and multiple testing adjustment. pOLS
+is the p-value for a conventional t-statistics of an OLS estimator. pLASSO is the p-value without removing the lasso
+bias or adjusting for post-selection inference, that is, it is simply the OLS p-values using the selected subset of
+regressors. pPoSI is the debiased post-selection adjusted p-value based on Theorem 1.
+Our baseline model is Panel PoSI, which uses post-selection inference LASSO, and a simultane-
+ity count for a data driven hypothesis. The first component that we modify is the selection of the
+sparse model. A simple OLS regression without shrinkage does not produce a sparse model. This
+gives us the methods Naive OLS and Bonferroni OLS. A conventional LASSO results in a sparse
+selection, but the p-values are not adjusted for the post-selection inference and the bias adjustment.
+The corresponding models are the Naive LASSO and the Bonferroni Naive LASSO. The second
+component is the hypothesis, which is agnostic for methods besides Panel PoSI. For the comparison
+models, we either consider no multiple testing adjustment or the conventional Bonferroni adjust-
+ment. Under the multiple testing adjustment we obtain the Bonferroni OLS, the Bonferroni Naive
+LASSO and the Bonferroni PoSI. The outcome of all the estimations are adjusted p-values for the
+covariates, which we use to select our model for a given target threshold. For a given value of γ we
+include a covariate if its adjusted p-value is below the critical values summarized in the last column
+of Table 2.
+We simulate a simple and transparent model. Our panel follows the linear model
+Yt,n =
+J
+�
+j=1
+Xt,jβ(n)
+j
++ ϵt,n
+for t = 1, ..., T, n = 1, ..., N and j = 1, .., J.
+The covariates and errors are sampled independently as normally distributed random variables:
+Xt,j
+iid
+∼ N(0, 1),
+ϵt
+iid
+∼ N(0, Σ).
+The noise is either generated as independent noise with covariance matrix Σ = σ2I or as cross-
+sectionally dependent noise with non-zero off-diagonal elements Σij = κ and diagonal elements
+Σii = σ2. Note that our theorems for PoSI assume homogeneous noise, while dependent noise
+violates our assumptions. Hence, the dependent noise allows us to test how robust our method is
+21
+
+Figure 5: Design of loadings β
+This figure demonstrates the setting of our simulations with 10 factors, where loadings are shaded based on whether
+they are active. In this staircase setting, the first factor affects all units, the 2nd factor affects 90%, and so on, and
+lastly the 10th factor affects 10% of all units.
+to misspecification. We set σ2 = 2 and κ = 1, but the results are robust to all these choices.
+We construct the active set based on the staircase structure depicted in Figure 5. Of the J
+covariates in X, we have K = 10 active independent factors. Figure 5 demonstrates the setting
+for the 10 factors, where loadings are shaded based on whether they are active. The first factor
+affects all units, the 2nd factor affects 90%, and so on, and lastly the 10th factor affects 10% of all
+units. This setting is relevant, and also challenging from a multiple testing perspective. It results
+in a large cohesion coefficient ρ, which makes the correct FWER control even more important. The
+loadings are sampled from a uniform distribution, if they are in the active set:
+β(n)
+j
+iid
+∼ Unif
+�
+−1
+2, 1
+2
+�
+for j in the active set,
+β(n)
+j
+= 0
+for j outside the active set.
+We simulate a panel of dimension N = 120, J = 100 and T = 300 with K = 10 active factors.
+The first half of the time-series observations is used for the in-sample estimation and selection, while
+the second half serves for the out-of-sample analysis. All results are averages of 100 simulations.
+We use the covariates selected on the in-sample data for regressions out-of-sample. Our focus is
+on the inferential theory, and not on the bias correction for shrinkage. Hence, we first use the
+inferential theory on the in-sample data to select our set of covariates. Second, we use the selected
+subset of covariates in an OLS regression on the in-sample data to obtain the loadings. Last but
+not least, we apply the estimated loadings of the selected subset to the out-of-sample data to obtain
+the model fit. Note that this procedure helps a Naive LASSO, which in contrast to PoSI LASSO
+does not have a bias correction. The out-of-sample explained variation is measured by R2, which
+is the sum of explained variation normalized by the total variation. The rejection FWER is set to
+γ = 5% or γ = 1%. The LASSO shrinkage penalty λ is selected by 5-fold cross-validation on the
+in-sample data.
+22
+
+."
+.".
+...
+"
+"
+"
+"Table 3: Simulation Comparison between Selection Methods
+Independent noise
+Method
+# Selections
+# False Selections
+# Correct Selections
+OOS R2
+FWER γ = 5%
+Panel PoSI
+10.8
+2.8
+7.9
+10.0%
+Bonferroni PoSI
+4.7
+0.0
+4.7
+8.0%
+Bonferroni Naive LASSO
+0.0
+0.0
+0.0
+0.0%
+Naive LASSO
+0.2
+0.0
+0.2
+0.4%
+Bonferroni OLS
+1.0
+0.0
+1.0
+1.7%
+Naive OLS
+99.2
+89.2
+10.0
+-144.2%
+FWER γ = 1%
+Panel PoSI
+8.6
+1.1
+7.5
+10.6%
+Bonferroni PoSI
+2.7
+0.0
+2.7
+5.2%
+Bonferroni Naive LASSO
+0.0
+0.0
+0.0
+0.0%
+Naive LASSO
+0.1
+0.0
+0.1
+0.3%
+Bonferroni OLS
+0.2
+0.0
+0.2
+0.5%
+Naive OLS
+46.4
+36.5
+9.9
+-19.3%
+Cross-sectionally dependent noise
+Method
+# Selections
+# False Selections
+# Correct Selections
+OOS R2
+FWER γ = 5%
+Panel PoSI
+10.1
+2.2
+7.9
+8.0%
+Bonferroni PoSI
+4.4
+0.0
+4.4
+7.2%
+Bonferroni Naive LASSO
+0.0
+0.0
+0.0
+0.0%
+Naive LASSO
+0.4
+0.0
+0.4
+0.5%
+Bonferroni OLS
+0.9
+0.0
+0.9
+1.3%
+Naive OLS
+83.7
+73.7
+10.0
+-83.8%
+FWER γ = 1%
+Panel PoSI
+7.9
+0.6
+7.3
+10.3%
+Bonferroni PoSI
+2.4
+0.0
+2.4
+3.9%
+Bonferroni Naive LASSO
+0.0
+0.0
+0.0
+0.0%
+Naive LASSO
+0.0
+0.0
+0.0
+0.0%
+Bonferroni OLS
+0.3
+0.0
+0.3
+0.4%
+Naive OLS
+31.0
+21.2
+9.8
+-6.8%
+This table compares the selection results for different methods in a simulation. For each method we report the num-
+ber of selected covariates, the number of falsely selected covariates and the number of correctly selected covariates.
+We also report the out-of-sample R2 of the models that estimated with the selected covariates on the out-of-sample
+data. All results are averages of 100 simulations. The rejection FWER is set to γ = 5% or γ = 1%. We simulate a
+panel of dimension N = 120, J = 100, T = 300. The first half of time-series observations is used for the in-sample
+estimation and selection, while the second half serves for the out-of-sample analysis. The panel is generated by 10
+independent factors. The active set of the factors follows the staircase structure of Figure 5. The first factor affects
+all units, the second 90%, and lastly the 10th factor affects 10%. The unknown error variance is estimated based as
+a homogenous sample variance. The noise is either generated as independent noise with covariance matrix Σ = σ2I
+or as cross-sectionally dependent noise with Σij = κ and Σii = σ2 for σ2 = 2 and κ = 1.
+Table 3 compares the selection results for the different methods. For each method we report the
+number of selected covariates, the number of falsely selected covariates and the number of correctly
+23
+
+selected covariates. We also report the out-of-sample R2. The upper panel shows the results for
+independent noise, while the lower panel collects the results for cross-sectionally dependent noise.
+PanelPoSI clearly dominates all models. It provides the best trade-off between correct and
+false selection, which results in the best out-of-sample performance. In the case of γ = 5% and
+independent noise, Panel PoSI selects 10.8 factors in a model generated by 10 factors.
+7.9 of
+these factors are correct. A simple Bonferroni correction is overly conservative. The Bonferroni
+PoSI selects only 4.7 correct factors. While this overly conservative selection protects against false
+discovery, it omits over half of the relevant factors which lowers the out-of-sample performance.
+Using post-selection inference is important, as a naive lasso provides wrong p-values which makes
+the overly conservative selection even worse. The other extreme is to have neither shrinkage nor
+multiple testing adjustment. As expected the naive OLS has an extreme number of false selections
+with a correspondingly terrible out-of-sample performance.
+As expected, tightening the FWER control to 1% lowers the number of false rejections, but
+also the number of correct selections. It reveals again that Panel PoSI provides the best inferential
+theory among the benchmark models. Panel PoSI selects 7.5 correct covariates, while it controls
+the false rejections at 1.1. The overly conservative Bonferroni methods select even fewer correct
+covariates, which further deteriorates the out-of-sample performance. The gap in OOS R2 between
+Panel PoSI and Bonferroni PoSI widens to 5.4%. All the other approaches cannot be used for a
+meaningful selection.
+Panel PosI performs well, even when some of the underlying assumptions are not satisfied. The
+lower panel of Table 3 shows the results for dependent noise. As the dependence in the noise is
+relatively strong, it can be interpreted as omitting a relevant factor in the set of candidate covariates
+X. Even thought the PoSI theory is developed for homogeneous noise, Panel PoSI continues to
+perform very well. In contrast, the comparison methods perform even worse, and the Bonferroni
+approaches select even less correct covariates.
+7
+Empirical Analysis
+7.1
+Data and Problem
+Our empirical analysis studies a fundamental problem in asset pricing. We select a parsimonious
+factor model from a large set of candidate factors that can jointly explain the asset prices of a large
+cross-section of investment strategies. Our data is standard and obtained from the data libraries
+of Kenneth French and Hou, Xue, and Zhang (2018).
+We consider monthly excess returns from January 1967 to December 2021, which results in a
+time dimension of T = 660. Our test assets are the N = 243 double-sorted portfolios of Kenneth
+French’s data library summarized in Table A.1 in the Appendix. The candidate factors are J = 114
+univariate long-short factors based on the data of Hou, Xue, and Zhang (2018). We include all
+univariate portfolio sorts from their data library that are available for our time period, and construct
+top minus bottom decile factor portfolios. In addition, we include the five Fama-French factors of
+24
+
+Fama and French (2015) from Kenneth French’s data library.
+Our analysis projects out the excess return of the market factor.
+We are interested in the
+question which factors explain the component that is orthogonal to market movements. Hence,
+we regress out the market factor from the test assets and use the residuals as test assets. We also
+do not include a market factor in the set of long-short candidate factors. The original test assets
+have a market component as they are long only portfolios. Our results are essentially the same
+when we include the market component in the test assets, with the only difference that we would
+need to include the market factor as an additional factor in our parsimonious models. The market
+factor would always be selected by all models as significant, but this by itself is neither a novel nor
+interesting result.
+We present in-sample and out-of-sample results.
+The in-sample analysis uses the first 330
+observations (January, 1967 to June, 1994), while the out-of-sample results are based on the second
+330 observations (July, 1994 to December, 2021). As in the simulation, we first use the inferential
+theory on the in-sample data to select our set of covariates. Second, we use the selected subset
+of covariates in an OLS regression on the in-sample data to obtain the loadings. Last but not
+least, we use the estimated loadings on the selected subset of factors for the out-of-sample model.
+The LASSO penalty λ is selected via 5-fold cross-validation on the in-sample data to minimize the
+squared errors.3 Hence, LASSO represents a first-stage dimension reduction tool, and we need the
+inferential theory to select our final sparse model.
+We allow our selection to impose a prior on two of the most widely used asset pricing models.
+More specifically, we estimate models without a prior, and two specific priors that impose an infinite
+weight on the Fama-French 3 factors (FF3) and the Fama-French 5 factors (FF5). This prior as
+part of PoSI LASSO enforces that the FF3 and FF5 factors are included in the active set. Note
+that because we work with data orthogonal to the market return, we do not include the market
+factor in the prior, but only the size and value factors for FF3 and in addition the investment and
+profitability factor for FF5. We denote these weights by ωFF3 and ωFF5. This is an example where
+the researcher has economic knowledge that she wants to include in her statistical selection method.
+We evaluate the models with standard metrics. The root-mean-squared error (RMSE) is based
+on the squared residuals relative to the estimated factor models. Hence, in-sample the models are
+estimated to minimize the RMSE. The pricing error is the economic quantity of interest. It is
+the time-series mean of the residual component of the factor model, and corresponds to the mean
+return that is not explained by the risk premia and exposure to the factors. In summary, we obtain
+the residuals as ˆϵ = Yt,n − XS ˆβS for the selected factors, where the loadings are estimated on the
+in-sample data. The metrics are the RMSE and mean absolute pricing error (MAPE):
+RMSE =
+�
+�
+�
+� 1
+N T
+N
+�
+i=1
+T
+�
+t=1
+ˆϵ2,
+MAPE = 1
+N
+N
+�
+i=1
+�����
+1
+T
+T
+�
+t=1
+ˆϵ
+����� .
+3We select λ from the grid exp(a) · log J/
+√
+T with a = −8, ..., 8. This grid choice satisfies the Assumptions in
+Chatterjee (2014) and hence Assumption A.4.
+25
+
+In addition to Panel PoSI without and with the FF3 and FF5 priors, we consider the benchmark
+methods of Table 2. We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and
+FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni Naive LASSO (B-LASSO), Naive LASSO (N-
+LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS). Our main analysis sets the FWER
+control to the usual γ = 5%.
+7.2
+Asset Pricing Results
+Panel PoSI selects parsimonious factor models with the best out-of-sample performance among
+the benchmarks. For the FWER rate of γ = 5% the number of factors differs substantially among
+the different methods. Panel PoSI selects 3 factors. Imposing infinite priors on FF3 or FF5 results
+in 4 and 5 factors for P-PoSI ωFF3 respectively ωFF5. In contrast, the alternative approaches select
+too many factors. Bonferroni Naive LASSO includes 10, Naive Lasso 70, Bonferroni OLS 107 and
+Naive OLS 114. These over-parametrized models lead to overfitting of the in-sample data.
+Figure 6 shows in-sample and out-of-sample RMSE for each set of double-sorts. The composition
+of the double sorts is summarized in Table A.1 in the Appendix. The in-sample performance in
+the left subfigure has the expected result that more factors mechanically decrease the RMSE.
+The important findings are in the right subfigure with the out-of-sample RMSE. The uniformly
+best performing model is Panel PoSI without any priors. In fact, imposing a prior on the Fama-
+French factors increases the out-of-sample RMSE. The conventional LASSO and OLS estimates
+have substantially higher RMSE, which can be more than twice as large.
+The Panel PoSI models also explain the average returns the best. In Figure 7, we compare
+the mean absolute pricing errors among the benchmarks for each set of double sorts. Importantly,
+the pricing errors are not used as in objective function of the estimation, and hence the fact that
+the models with the smallest RMSE explain expected returns is an economic finding supporting
+arbitrage pricing theory. Our Panel PoSI has the smallest out-of-sample pricing errors, which can
+be up to six times smaller compared to the OLS estimates. Including the Fama-French factors as a
+prior does not improve the models, except for the profitability and investment double sort, which
+uses the same information as two of the Fama-French factors.
+The Panel PoSI models select economically meaningful factors. Table 4 reports the ranking of
+factors based on their FWER bound without prior and infinite prior weights on the Fama-French
+3 and 5 factors. The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds to
+the FWER bound. It allows us to infer the number of factors for different levels of FWER control
+values. Setting γ = 5% leads to 3, 4 and respectively 5 factors, while a γ = 1% results in 2, 4 and
+5 factors, respectively.
+In addition to their significance, we can infer the relative importance of factors. The baseline
+PoSI with γ = 5% selects a size, dollar trading volume and value factor. The size and value factors
+are among the most widely used asset pricing factors. Their selection is in line with their economic
+importance and confirms the Fama-French 3 factor model. The dollar trading volume factor is less
+conventional, but is correlated with many assets in our cross-sections. The size factor is the most
+26
+
+Figure 6: RMSE across cross-sections
+(a) In-sample
+(b) Out-of-sample
+This figure shows the in-sample and out-of-sample root-mean-squared errors (RMSE) for each cross-section of test
+assets for different factor models. The test assets are the N = 243 double-sorted portfolios, and we show the RMSE
+for each set of double-sorts. The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate
+factor portfolios. The time dimension is T = 660. We use the first half for the in-sample estimation and selection,
+while the second half serves for the out-of-sample analysis. We compare Panel PoSI (P-PoSI), Panel PoSI with
+infinite priors on FF3 and FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO
+(N-LASSO), Bonferroni OLS (B-OLS) and Naive OLS (N-OLS).
+important as measured by the FWER bound, that is, the product of the number of relevant assets
+and its minimum p-value are the smallest. The short term reversal factor is less important and
+would require a FWER control of 10% to be included.
+Imposing a prior affects the p-values of PoSI and the simultaneity count. For example, the
+cohesiveness coefficient increases from ρ = 0.16 for no priors to ρ = 0.18 in the case of the two
+priors. Hence, the FWER bounds of all factors can change when we impose a prior. The FF3 prior
+increases the significance of the short-term reversal factor, which is widely used in asset pricing.
+Interestingly, even for a FF5 prior, the profitability and investment factors remain insignificant.
+7.3
+Number of Factors
+Our method contributes to the discussion about the number of asset pricing factors. Many
+popular asset pricing models suggest between three and six factors. Our approach allows a disci-
+27
+
+OP
+1.32
+1.35
+1.49
+1.82
+2.02
+2.01
+2.00
+INV
+ME
+1.16
+1.18
+1.30
+1.58
+1.74
+1.73
+1.62
+Prior60
+ME
+1.13
+1.16
+1.29
+1.88
+1.99
+1.94
+1.93
+Prior12
+ME
+1.08
+1.10
+1.23
+1.49
+1.68
+1.53
+1.53
+Priorl
+ME
+0.95
+0.96
+1.06
+1.32
+1.44
+1.43
+1.42
+OP
+ME
+0.95
+0.97
+1.07
+1.32
+1.43
+1.42
+1.42
+INV
+ME
+0.59
+0.60
+0.68
+0.92
+1.02
+1.01
+1.01
+EP
+ME
+0.62
+0.64
+0.71
+0.98
+1.09
+1.06
+1.06
+DP
+ME
+0.57
+0.58
+0.66
+0.90
+1.00
+1.00
+1.00
+CFP
+BEME
+1.53
+1.57
+1.72
+2.11
+2.29
+2.28
+2.27
+OP
+BEME
+1.37
+1.40
+1.56
+1.91
+2.09
+2.08
+2.07
+INV
+BE
+0.99
+1.01
+1.11
+1.39
+1.48
+1.47
+1.46
+ME
+N-OLS
+B-OLS
+N-LASSO B-LASSO
+P-POSI
+P-POSI
+P-POSI
+WFF3
+WFF5OP
+2.72
+2.62
+2.33
+1.56
+0.91
+0.95
+0.96
+INV
+ME
+3.05
+3.03
+2.80
+2.28
+2.05
+2.10
+2.22
+Prior60
+ME
+3.39
+3.35
+3.16
+2.12
+1.82
+2.04
+2.02
+Prior12
+ME
+3.14
+3.08
+2.82
+2.29
+1.78
+2.23
+2.24
+Priorl
+ME
+2.87
+2.84
+2.65
+2.18
+1.91
+1.93
+1.95
+OP
+ME
+2.83
+2.79
+2.58
+2.10
+1.94
+1.98
+1.99
+INV
+ME
+2.39
+2.39
+2.31
+2.15
+1.99
+2.01
+2.01
+EP
+ME
+2.19
+2.14
+2.05
+1.90
+1.82
+1.88
+1.87
+DP
+ME
+2.35
+2.34
+2.29
+2.16
+1.97
+1.99
+2.00
+CFP
+BEME
+3.42
+3.30
+2.98
+2.39
+1.62
+1.67
+1.67
+OP
+BEME
+2.88
+2.78
+2.45
+1.79
+1.46
+1.50
+1.52
+INV
+BE
+3.04
+3.00
+2.84
+2.39
+2.28
+2.30
+2.31
+ME
+N-OLS
+B-OLS
+N-LASSO B-LASSO
+P-POSI
+P-POSI
+P-POSI
+WFF3
+wFF5Figure 7: MAPE across cross-sections
+(a) In-sample
+(b) Out-of-sample
+This figure shows the mean absolute pricing errors (MAPE) for each cross-section of test assets for different factor
+models. The test assets are the N = 243 double-sorted portfolios, and we show the average |α| for each set of double
+sorts. The rejection FWER is set to γ = 5% The candidate factors are the 114 univariate factor portfolios. The
+time dimension is T = 660. We use the first half for the in-sample estimation and selection, while the second half
+serves for the out-of-sample analysis. We compare Panel PoSI (P-PoSI), Panel PoSI with infinite priors on FF3 and
+FF5 (P-PoSI ωFF3 respectively ωFF5), Bonferroni LASSO (B-LASSO), Naive LASSO (N-LASSO), Bonferroni OLS
+(B-OLS) and Naive OLS (N-OLS).
+plined estimate for the number of factors based on inferential theory. The level of sparsity of a
+linear model also depends on the rotation of the covariates. Therefore, we also study the principal
+components (PCs) of the covariates X as candidate factors. In this case, we use the step-down
+procedure, which we refer to as “Ordered PoSI” or O-POSI for short.
+Figure 8 shows the number of factors for different FWER rates γ. The factor count is obtained
+by traversing K∗(γ) equal to 0.01, 0.02, 0.05 and 0.1. Panel PoSI without priors selects 2 factors
+for γ = 0.01 and 3 for γ = 0.05. Once, we impose an infinite weight on the Fama-French 3 factors,
+we select 4 factors for all FWER levels, while the prior on the Fama-French 5 factors results in a
+5 factor model for all FWER levels. The Ordered PoSI with PCA rotated factors selects 3 factors
+for all FWER levels. In summary, our results confirm that depending on the desired significance,
+the number of asset pricing factors for a good model seems to be between 2 and 4. Note that our
+analysis is orthogonal to the market factor, which would also be added to the final model. Thus,
+28
+
+OP
+0.04
+0.04
+0.04
+0.08
+0.17
+0.17
+0.17
+INV
+ME
+0.04
+0.04
+0.05
+0.12
+0.10
+0.10
+0.11
+Prior60
+ME
+0.04
+0.04
+0.05
+0.28
+0.36
+0.39
+0.38
+Prior12
+ME
+0.06
+0.06
+0.08
+0.17
+0.21
+0.19
+0.19
+Priorl
+ME
+0.03
+0.03
+0.03
+0.07
+0.11
+0.11
+0.11
+OP
+ME
+0.03
+0.03
+0.04
+0.08
+0.11
+0.10
+0.10
+INV
+ME
+0.02
+0.02
+0.03
+0.05
+0.10
+0.10
+0.10
+EP
+ME
+0.02
+0.02
+0.03
+0.06
+0.09
+0.09
+0.09
+DP
+ME
+0.02
+0.02
+0.04
+0.06
+0.11
+0.11
+0.11
+CFP
+BEME
+0.04
+0.04
+0.05
+0.12
+0.17
+0.17
+0.17
+OP
+BEME
+0.06
+0.05
+0.06
+0.10
+0.13
+0.13
+0.13
+INV
+BE
+0.04
+0.04
+0.04
+0.10
+0.11
+0.10
+0.10
+ME
+N-OLS
+B-OLS
+N-LASSO B-LASSO
+P-POSI
+P-POSI
+P-POSI
+WFF3
+WFF5Op
+0.19
+0.21
+0.17
+0.14
+0.03
+0.03
+0.02
+INV
+ME
+0.10
+0.11
+0.10
+0.10
+0.09
+0.10
+0.09
+Prior60
+ME
+0.18
+0.19
+0.17
+0.11
+0.08
+0.09
+0.08
+Prior12
+ME
+0.14
+0.13
+0.09
+0.10
+0.08
+0.09
+0.09
+Priorl
+ME
+0.18
+0.17
+0.16
+0.14
+0.08
+0.08
+0.08
+OP
+ME
+0.14
+0.14
+0.13
+0.10
+0.09
+0.08
+0.08
+INV
+ME
+0.13
+0.14
+0.13
+0.09
+0.08
+0.08
+0.08
+EP
+ME
+0.12
+0.13
+0.15
+0.11
+0.07
+0.07
+0.08
+DP
+ME
+0.13
+0.12
+0.12
+0.10
+0.07
+0.07
+0.07
+CFP
+BEME
+0.27
+0.27
+0.23
+0.15
+0.05
+0.05
+0.05
+OP
+BEME
+0.17
+0.14
+0.13
+0.09
+0.05
+0.05
+0.05
+INV
+BE
+0.14
+0.12
+0.14
+0.10
+0.09
+0.09
+0.09
+ME
+N-OLS
+B-OLS
+N-LASSO B-LASSO
+P-POSI
+P-POSI
+P-POSI
+WFF3
+wFF5Table 4: Selected factors with Panel PoSI
+Factor
+Nj
+pj
+ρ−1Njpj
+Order
+No prior
+Size (SMB)
+1824
+<0.00001
+<0.0001
+1
+Dollar Trading Volume (dtv 12)
+2099
+<0.00001
+<0.0001
+2
+Value (HML)
+1191
+<0.00001
+0.0280
+3
+Short-Term Reversal (srev)
+1050
+0.00001
+0.0974
+4
+Forecast Revisions (rev 1)
+242
+0.00018
+0.2782
+5
+Investment (CMA)
+998
+0.00112
+>0.9999
+6
+Profitability (RMW)
+797
+0.00123
+>0.9999
+7
+FF3 prior (ωFF3)
+Size (SMB)
+2802
+<0.00001
+<0.0001
+1
+Value (HML)
+2802
+<0.00001
+<0.0001
+2
+Dollar Trading Volume (dtv 12)
+779
+<0.00001
+0.0017
+3
+Short-Term Reversal (srev)
+1106
+<0.00001
+0.0049
+4
+Profitability (RMW)
+819
+0.00006
+0.2527
+5
+Investment (CMA)
+874
+0.00087
+>0.9999
+6
+FF5 prior (ωFF5)
+Size (SMB)
+2911
+<0.00001
+<0.0001
+1
+Value (HML)
+2911
+<0.00001
+<0.0001
+2
+Forecast Revisions (rev 1)
+230
+<0.00001
+0.0005
+3
+Short-Term Reversal (srev)
+1140
+<0.00001
+0.0052
+4
+Dollar Trading Volume (dtv 12)
+661
+<0.00001
+0.0072
+5
+Profitability (RMW)
+2911
+0.00001
+0.1937
+6
+Investment (CMA)
+2911
+0.00001
+0.1996
+7
+Gross profits-to-assets (gpa)
+1151
+0.00013
+0.8382
+8
+This table reports ranking of factors based on their FWER bound for no prior, and infinite weight priors on the
+Fama-French 3 and 5 factors. The test assets are the N = 243 double-sorted portfolios and the candidate factors are
+J = 114 univariate long-short factors. The rows are ordered based on sorted ascending ρ−1Njpj, which corresponds
+to the FWER bound.
+the final model would have between 3 and 5 factors.
+Table 5 further confirms our findings about the number of asset pricing factors. We compare
+the number of factors for γ = 5% selected either from the univariate high-minus-low factors (HL),
+their PCA rotation or the combination of the high-minus-low factors and their PCs. Panel PoSI
+selects consistently 3 factors from the long-short factors and their PCs. When combined, PoSI
+selects 4 factors, which is plausible as the optimal sparse model can be different for this larger set
+of candidate factors. The Bonferroni PoSI is overly conservative and selects only 2 HL factors. The
+models based on Naive LASSO or OLS select excessively many factors independent of the rotation.
+Overall, the findings support that parsimonious asset pricing models can be described by three to
+four factors. Of course, any discussion about the number of asset pricing factors is always subject
+to the choice of test assets and candidate factors.
+29
+
+Figure 8: Number of selected factors for different FWER
+(a) Univariate factors with priors (P-POSI)
+(b) PCA rotated factors (O-POSI)
+This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different
+FWER rates γ. The factor count is obtained by traversing K∗(γ) for γ ranging from 0.01 to 0.1. The left subfigure
+uses univariate high-minus-low factors as candidate factors. We consider the case of no prior, and the cases of an
+infinite weight on the Fama-French 3 factor model (ωFF3) and an infinite weight on the Fama-French 5 factor model
+(ωFF5). The right subfigure uses the PCA rotation as candidate factors with the step-down procedure Ordered PoSI
+(O-POSI).
+Table 5: Number of selected factors for different methods
+HL
+PCs
+HL + PCs
+Panel PoSI
+3
+3
+4
+Bonferroni PoSI
+2
+3
+2
+Bonferroni Naive LASSO
+10
+29
+10
+Naive LASSO
+70
+50
+76
+Bonferroni OLS
+107
+13
+117
+Naive OLS
+114
+50
+164
+This figure shows the number of selected factors to explain the test assets of double-sorted portfolios for different
+methods and different sets of candidate factors. The rejection FWER is set to γ = 5%. The factor count is obtained
+by traversing K∗(γ) for γ. The number of factors is selected on the in-sample data. For PCs, we use the step-down
+method for the nested hypothesis.
+8
+Conclusion
+This paper proposes a new method for covariate selection in large dimensional panels.
+We
+develop the conditional inferential theory for large dimensional panel data with many covariates
+by combining post-selection inference with a new multiple testing method specifically designed for
+panel data. Our novel data-driven hypotheses are conditional on sparse covariate selections and
+valid for any regularized estimator.
+Based on our panel localization procedure, we control for
+family-wise error rates for the covariate discovery and can test unordered and nested families of
+hypotheses for large cross-sections. We provide a method that allows us to traverse the inferential
+30
+
+P-PoSI
+P-PoSL WE3
+6
+P-PoSI WFF5
+Factor count
+5
+5
+5
+5
+5
+4
+4
+4
+44
+4
+3
+3 
+2
+2
+2
+0.01
+0.02
+0.05
+0.17
+6
+PC count
+5
+4.
+3
+3
+3
+3
+3 :
+0.01
+0.02
+0.05
+0.1results and determine the least number of covariates that have to be included given a user-specified
+FWER level.
+As an easy-to-use and practically relevant procedure, we propose Panel-PoSI, which combines
+the data-driven adjustment for panel multiple testing with valid post-selection p-values of a gen-
+eralized LASSO, that allows to incorporate weights for priors. In an empirical study, we select a
+small number of asset pricing factors that explain a large cross-section of investment strategies.
+Our method dominates the benchmarks out-of-sample due to its better control of false rejections
+and detections.
+A
+Post-selection Inference with Weighted-LASSO
+A.1
+Weighted-LASSO: Linear Truncation Results
+This appendix collects the assumptions and formal statements underlying Theorem 1.
+We
+present the results for the Weighted-LASSO, which includes the conventional LASSO as a special
+case. In order to ensure uniqueness of the LASSO solution, we impose the following condition,
+which is standard in the LASSO literature:
+Definition A.1. General position
+The matrix X ∈ RT×J has columns in general position if the affine span of any J0 + 1 points
+(σ1Xi1, ..., σJ0+1XiJ0+1) in RT for arbitrary σ1, ...σd0+1 ∈ {±1} does not contain any element of
+{±Xi : i /∈ {i1, ..., iJ0+1}}, where J0 < J4 and Xi denotes ith column of X.
+This position notion will help us to avoid ambiguity in the LASSO solution. Note that this
+condition is a much weaker requirement than full-rank of X, and states that if one constructs a
+J0-dimensional subspace, it must contain at most J0 + 1 entries of {±X1, ..., ±XJ}. Even though
+this appears to be a complicated and mechanical condition, by a union argument it turns out that
+with probability 1, if the entries of X ∈ RT×J are drawn from a continuous probability distribution
+on RT×J then X is in general position.5 Then, we will be able to discuss the LASSO solution for
+general design with relative ease, thanks to Lemma 3 of Tibshirani (2013). It shows that if X lie
+in general position, it is sufficient to have a unique LASSO solution regardless of the penalty scalar
+λ. This condition will later be used in establishing our Lemma A.2.
+We can now state the formal assumptions:
+Assumption A.1. Unique low dimensional model
+(a) Low dimensional truth:
+The data satisfies Y = XSβS + ϵ where |S| = O(1);
+(b) General position design:
+The covariates X have columns in general position as given by Definition A.1;
+4The original condition needs to hold for J0 < min{T, J} but in the scope of our study, we consider T > J.
+5See Donoho (2006) and §2.2 of Tibshirani (2013) for more discussions on uniqueness and general position.
+31
+
+We start our analysis with the simpler model of known error variance, and later extend it to
+the case of estimated unknown variance.
+Assumption A.2. Gaussian residual with known variance
+The residuals are distributed as ϵ ∼ N(0, Σ) where Σ is known.
+Before formalizing the inferential theory, we need to clarify the quantity for which we want
+to make inference statements. As stated before, we only test the hypothesis on a covariate if its
+LASSO estimate turns out active. This is exactly the approach how researchers in practice conduct
+explorations in high-dimensional datasets. In other words, we focus on ˆβM and quantities associated
+with it, where M denotes the active set of selected covariates.
+We study the inferential theory of the “debiased estimator”, which is a shifted version of the
+LASSO fit as defined below. We show that this debiased estimator is unbiased, consistent and
+follows a truncated Gaussian distribution, with profound connections to the debiased LASSO lit-
+erature such as Javanmard and Montanari (2018), but has different properties by a subtle different
+descent direction. More concretely, given M, clearly ˆY = XM ˆβM is the fitted value since ˆβ−M = 0,
+where −M is the complement of the set M. We let ˆϵM := Y − XM ˆβM be the residual from the
+LASSO estimator. By considering only the partial LASSO loss of ℓ(Y, XM, λ, β) and given we are
+currently at the LASSO estimator ˆβ, the Newton step is X+
+Mˆϵ following (Boyd and Vandenberghe,
+2004, § 9.5.2), where we denote X+
+M = (X⊤
+MXM)−1X⊤
+M as the pseudo-inverse of the active subma-
+trix of X. The invertibility of X⊤
+MXM either is observed when we are in the fixed design regime
+or happens almost surely when we are dealing with continuous quantities, as a consequence of
+Assumption A.1(b) as argued in Tibshirani (2013) and Lee, Sun, Sun, and Taylor (2016). Now we
+can formally define the main object of our inferential theories:
+Definition A.2. Debiased Estimator
+The debiased Weighted-LASSO estimator ¯βM given M is given by
+¯βM = ˆβM + X+
+MˆϵM
+(21)
+It is now evident why some of the literature refers to the debiased estimator also as the one-step
+estimator: given that ˆβM solves the Karush-Kuhn-Tucker (KKT) condition and reaches the optimal
+sub-gradient for the full loss ℓ(Y, X, c, β), our debiased estimator ¯βM is the result of moving one
+more Newton-Ralphson method step after ˆβM, but only taking XM rather than X as a whole into
+the likelihood loss function. Hence, the update step is actually only a partial update from the
+LASSO solution point. Intuitively, ¯βM should still be close to solving the KKT conditions, and
+would exactly solve the KKT conditions if XM happen to be the true covariates (i.e. XM = XS).
+If we were to take a Newton’s method step with gradient and Hessian calculated with the entirety
+of data X, or equivalently taking a full update from the stationary point, we will recover the ˆβd
+M
+proposed in Javanmard and Montanari (2018). The material difference is that the full-update would
+require the J ×J precision matrix Ω = Γ−1, where Γ = X⊤X if X assumed fixed or Γ = E[X⊤X] if
+X assumed to be generated from a stationary process. Using ℓ(Y, XM, λ, β) instead of ℓ(Y, X, λ, β),
+32
+
+our debiased estimator would not need the full Hessian, which is leveraging LASSO’s screening
+property and uses (X⊤
+MXM)−1X⊤
+M (i.e. X+
+M) as a much lower-dimensional alternative of ΩX⊤.
+Without loss of generality, we assume that the covariate indexed i ≤ |M| is part of M, and we can
+always rearrange the columns of X to have the first |M| covariates as active. Let η = (X+
+M)⊤ei ∈ RT
+be a vector where ei ∈ R|M| is a vector with 1 at ith coordinate and 0 otherwise. Hence, the η
+vector is the linear mapping from Y to the ith coordinate of an OLS estimator. In particular, the
+debiased estimator and the response satisfy the following relationship:
+Lemma A.1. Debiased Estimator is OLS-post-LASSO
+The debiased estimator is a linear mapping of Y . Specifically, given η = (X+
+M)⊤ei:
+¯βi = η⊤Y
+(22)
+Moreover, ¯βM is the OLS estimate of regressing XM on Y :
+¯βM = arg min
+β
+1
+2T ∥Y − XMβ∥2
+2.
+(23)
+The proof of Lemma A.1 is deferred to the Online Appendix. Although its proof is simple, this
+lemma reveals that our debiased estimator is the same as the least-square after LASSO estimator
+proposed in Belloni and Chernozhukov (2013).
+Our strategy to obtain a rigorous statistical inferential theory with p-values is as follows. First
+we perform an algebraic manipulation to transform ˆβM into ¯βM in the linear form of (22). Then, we
+follow the strategy in Lee, Sun, Sun, and Taylor (2016) to traverse the KKT subgradient optimal
+equations for general X by writing it equivalently into a truncation in the form of {AY ≤ b}, as
+we will do in Lemma A.2. Finally we will circle back to ˆβM by the linear mapping between ¯βM
+and Y and the distributional results induced by the fact that Y is truncated by {AY ≤ b}.
+For our Weighted-LASSO, the KKT sub-gradient equations are
+X⊤(X ˆβ − Y ) + λ
+�
+s
+v
+�
+⊙ ω−1 = 0
+where
+�
+�
+�
+si = sign(ˆβi)
+if ˆβi ̸= 0, ωi < ∞
+vi ∈ [−1, 1]
+if ˆβi = 0, ωi < ∞
+(24)
+In other words, when ω is specified, the KKT conditions can be identified using the tuple of
+{M, s}, where M is the active covariates set and s is the signs of LASSO fit. This is a consequence
+of how LASSO KKT condition can separate the slacks into s for active variables and v for inactive
+variables. If we have infinite importance weights (J ̸= ∅), we would simply need si < ∞ for i ∈ J
+because λsi/ωi = 0 is guaranteed. We rigorously characterize the KKT sub-gradient conditions as a
+combinations of signs and infinity norm bounds conditions by the following lemma, which parallels
+Lemma 4.1 of Lee, Sun, Sun, and Taylor (2016):
+Lemma A.2. Selection in norm equivalency
+33
+
+Consider the following random variables
+w(M, s, ω) = (X⊤
+MXM)−1(X⊤
+MY − λs ⊙ ω−1
+M )
+u(M, s, ω) = ω−M ⊙
+�
+X⊤
+−M(X+
+M)⊤s ⊙ ω−1
+M + 1
+λX⊤
+−M(I − PM)Y
+�
+(25)
+where PM = XMX+
+M ∈ RT×|M| is the projection matrix. The Weighted-LASSO selection can be
+written equivalently as
+{M, s} = {sign(w(M, s, ω)) = s, ∥u(M, s, ω)∥∞ < 1}
+(26)
+Using this characterization, we are then able to provide the distributional results for the debiased
+estimators. Consider ξ = Ση(η⊤Ση)−1 ∈ RT as a covariance-scaled version of our η, and a mapping
+of Y using residual projection matrix: z = (I − ξη⊤)Y . Note that z can be calculated once we
+observe (X, Y ), so it can be conditioned on were we to do so. We will soon see that the truncation
+set will depend on the variable z, but this does not cause any issues thanks to the following lemma,
+the proof of which is deferred to the Online Appendix:
+Lemma A.3. Ancillarity in truncation
+The projected z and the debiased estimator ¯βi are independently distributed.
+As a result of Lemma A.3, when describing the distribution of ¯βi, we can use z in its truncation
+conditions as long as we condition on z as well. To simplify notation, we can collect all quantities
+we need to condition on into
+˜
+M := ((M, s), z, ω, X). Now we can assemble the consequences of
+Lemmas A.1, A.2, A.3 to arrive at the truncated Gaussian statements for the debiased estimator
+similar to Lee, Sun, Sun, and Taylor (2016), but for weighted-LASSO:
+Theorem A.1. Truncated Gaussian
+Under Assumptions A.1 and A.2 for i ∈ M, ¯βi is conditionally distributed as:
+¯βi| ˜
+M ∼ T N(βi, η⊤Ση; [V −(z), V +(z)])
+(27)
+where T N is a truncated Gaussian with mean βi, variance η⊤Ση and truncation set [V −(z), V +(z)].
+βi denotes the ith entry of the true β. The vector of signs is s = sign(ˆβM) ∈ R|M| and the truncation
+set depends on
+A =
+�
+��
+λ−1X⊤
+−M(I − PM)
+−λ−1X⊤
+−M(I − PM)
+−diag(s)X+
+M
+�
+�� ∈ R(2J−|M|)×T ,
+b =
+�
+��
+ω−1
+−M − X⊤
+−M(X+
+M)⊤s ⊙ ω−1
+M
+ω−1
+−M + X⊤
+−M(X+
+M)⊤s ⊙ ω−1
+M
+−λ · diag(s)(X⊤
+MXM)−1s ⊙ ω−1
+M
+�
+�� ∈ R2J−|M|
+V −(z) =
+max
+j:(Aξ)j<0
+bj − (Az)j
+(Aξ)j
+,
+V +(z) =
+min
+j:(Aξ)j>0
+bj − (Az)j
+(Aξ)j
+.
+Notice that Theorem A.1 is decoupled across M, which is to say we are able to deal with
+1-dimensional statistics. We arrive at this form because the construction of (V −, V +) over the
+34
+
+extreme points of the linear inequality system (or vertices of the polyhedral) has decomposed the
+dimensionality of the truncation. This decoupling is of significant practical value, in that it would
+be otherwise a non-trivial task to calculate a statistic of multivariate (in our case |M|-dimensional)
+truncated Gaussian and then marginalize over |M| − 1 dimensions.
+A.2
+Weighted-LASSO Quasi-Linear Truncation with Estimated Variance
+This section generalizes the distribution results to the practically relevant case when the noise
+variance is unknown and has to be estimated. This becomes a challenging problem for post-selection
+inference. We replace Assumption A.2 by the following assumption:
+Assumption A.3. Gaussian residual with simple unknown variance
+The residuals are distributed as ϵi
+iid
+∼ N(0, σ2) where σ2 is unknown.
+The simple structure of unknown variance of Assumption A.2 is common in the post-selection
+inference literature as for example in Lee, Sun, Sun, and Taylor (2016) and Tian, Loftus, and Taylor
+(2018). A feasible conditional distribution replaces σ2 with an estimate. Under Assumption A.2,
+we can estimate the variance using LASSO residuals and then reiterate the previous truncation
+arguments. The most common standard variance estimator is
+ˆσ2(Y ) = ∥Y − X ˆβ∥2
+2/(T − |M|).
+(28)
+In classical regression analysis, the normally distributed estimated coefficient divided by an
+estimated standard deviation follows a t-statistic. Hence, we would expect that a truncated normal
+debiased estimator divided by a sample standard deviation might yield a truncated t-distribution.
+However, the arguments are substantially more involved. Simply using ˆσ(Y ) of (28) in the expres-
+sion η⊤Ση of Theorem A.1 changes the truncation. Specifically, Y having truncated support means
+ˆσ(Y )2 is not χ2-distributed supported on the entire R+, which makes the support of ¯β/ˆσ(Y ) non-
+trivial. Therefore, in order to correctly assess the truncation of the studentized quantity, we have
+to disentangle how much truncation is implied in ˆσ(Y )−1 and ¯β simultaneously. Geometrically, as
+ˆσ(Y ) is a non-linear function of Y and ¯β, the truncation on Y is in fact no longer of the simple
+linear form {AY ≤ b} such as in Theorem A.1.
+Instead of a polyhedral induced by affine constraints, we have a “quasi-affine constraints” form
+of {CY ≤ ˆσ(Y )b} because LASSO KKT conditions preserve the estimated variance throughout
+the arguments. Thus, both sides of the inequality CY ≤ ˆσ(Y )b have Y , and in right-hand-side
+the ˆσ(Y ) is non-linear in Y . A significantly more complex set of arguments are needed compute
+the exact truncation, which is equivalent to solve for a |M|-system of non-linear inequalities rather
+than linear inequalities that constrain the support of Y for inference on each ¯βi. Theorem A.2
+shows the appropriate truncation based on those arguments:
+Theorem A.2. Truncated t-distribution for estimated variance
+Under Assumptions A.1 and A.3, and the null hypothesis that βi = 0, the studentized quantity
+35
+
+¯βi/∥η∥ˆσ(Y ) follows
+¯βi/∥η∥ˆσ(Y ) ∼ TTd;Ω,
+(29)
+where TT is a truncated t-distribution with d degrees of freedom and truncation set Ω.
+The
+truncation set Ω = �
+i∈M{t : t
+√
+Wνi + ξi
+√
+d + t2 ≤ −θi
+√
+W} is an |M|-intersection of simple
+inequality-induced intervals based on the following quantities.
+The active signs are denoted as
+s = sign(ˆβM) ∈ R|M|. The scaled LASSO equivalent penalty is ˜λ2 =
+λ2
+ˆσ2(Y )·(T−|M|)+∥(X+
+M)⊤s⊙ω−1
+M ∥2
+2λ2 .
+θi = (˜λsi
+�
+T − |M|
+1 − ˜λ2∥(X+
+M)⊤s ⊙ ω−1
+M ∥2
+2
+) · e⊤
+i
+�
+(X⊤
+MXM)−1s ⊙ ω−1�
+for i ∈ M
+C = −diag(s)X+
+M ∈ R|M|×T ,
+ν = Cη ∈ R|M|,
+ξ = C(PM − ηη⊤)Y ∈ R|M|,
+d = tr(I − PM),
+W = ˆσ2(Y ) · d + (η⊤Y )2
+The quantities θ and C describe the quasi-linear constraints, whereas ν and ξ transform them
+into the form of Ω. Note that the Ω set is obtained from solving a low-dimensional set of quadratic
+inequalities that do not necessarily yield a single interval after intersection. We provide a proof of
+this result in the Online Appendix.
+Using Theorem A.2 in practice poses several challenges. First, the computations are much more
+involved, especially as each βi requires calculation of Ω which includes |M| actual constraints, each
+of which involves solving a simple but still non-linear inequality. It is non-trivial to ensure that the
+numerical stability holds at every step of the calculations. Second, since Ω is not necessarily an
+interval, it is harder to interpret the truncation and also calculate the cumulative density function
+through Monte-Carlo simulations when there is a non-trivial truncation structure. Third, in fact,
+the authors in Tian, Loftus, and Taylor (2018) recommend a regularized likelihood minimizing
+variance estimator that deviates from the simple ˆσ(Y ), which would in turn involves more numerical
+integration and optimization steps. Last but not least, this result was proposed initially for studying
+scale-LASSO, which is why there has to be a penalty term transformation of λ to ˜λ. Our goal is to
+provide a set of tools that can be useful for a wide range of applications including the LASSO with
+l2 squared norm loss rather than un-squared norm loss. These implementation difficulties are also
+discussed in more detail in the Online Appendix, which provides the accompanying proofs and the
+exact forms of the truncations.
+We provide a practical solution based on an asymptotic normal argument.
+We impose the
+standard assumption that we have a consistent estimator of the residual variance:
+Assumption A.4. Consistent estimator ˆσ(Y )
+Given λ, the residual variance estimator is consistent ˆσ(Y )
+p→ σ2 as T → ∞.
+This general assumption includes many common scenarios such as the results specified in Corol-
+lary 6.1 of van de Geer and B¨uhlmann (2011), or in Theorem 2 of Chatterjee (2014). For example,
+for diminishing c
+�
+log(J)/T → 0 as J, T grow and our Assumptions A.1 and A.3, we obtain con-
+sistency of ˆσ(Y ) of (28) by Chatterjee (2014).
+36
+
+Theorem A.3. Asymptotic truncated normal distribution
+Suppose Assumptions A.1, A.3 and A.4 hold. Under the null hypothesis that βi = 0 and for T → ∞
+the studentized quantity ¯βi/∥η∥ˆσ(Y ) follows
+¯βi/∥η∥ˆσ(Y ) ∼ TNΩ,
+(30)
+where TN is a truncated normal distribution with truncation Ω = [V −(z)/∥η∥2ˆσ(Y ); V +(z)/∥η∥2ˆσ(Y )],
+where V −(z) and V +(z) are the same as in Theorem A.1.
+The asymptotic distribution result has several advantages. First, it is intuitive since it parallels
+the classical OLS inference with a t-statistic converging to Gaussianity. Secondly, it is computa-
+tionally more tractable than results of Appendix Theorem A.2. With this result, one could obtain
+asymptotically valid post-selection p-values.
+B
+Appendix: Empirics
+Table A.1: Compositions of DS portfolios
+Sorted by
+# portfolios
+Sorted by
+# portfolios
+Sorted by
+# portfolios
+Sorted by
+# portfolios
+BEME, INV
+25
+ME, CFP
+6
+ME, INV
+25
+ME, Prior1
+25
+BEME, OP
+25
+ME, DP
+6
+ME, OP
+25
+ME, Prior12
+25
+ME, BE
+25
+ME, EP
+6
+OP, INV
+25
+ME, Prior60
+25
+This table lists the composition of double sorted portfolios that we use as test assets in our empirical study. All the
+double sorted portfolios are from Kenneth French’s data library.
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