Large Panel Test of Factor Pricing Models

Transcription

1 Large Panel est of Factor Pricing Models Jianqing Fan, Yan Liao and Jiawei Yao Department of Operations Research and Financial Engineering, Princeton University Bendheim Center for Finance, Princeton University Department of Mathematics, University of Maryland Abstract We consider testing the mean-variance efficiency in the context of a highdimensional mlti-factor model, with the nmber of assets mch larger than the timeseries dimension. Most of the existing tests are based on a qadratic form of estimated alphas. Under high dimensionality, however, they all sffer from low powers becase the accmlation of a large amont of estimation errors overrides the signals of the tre nonzero alphas. o resolve this isse, we propose a new test that deals with highdimensional hypothesis testing problems, called power enhancement. A screening statistic is introdced to screen off most of the estimation errors and consistently select stocks with significant alphas. We develop a feasible standardized Wald statistic sing a consistent estimator of the high-dimensional weight matrix based on thresholding. In addition, by attaching the screening statistic to the traditional qadratic-form tests, or proposed test significantly enhances the power of the Wald-type tests nder most of the alternatives, while keeping a correct asymptotic size. Finally, the proposed methods are applied to the secrities in the S&P 500 index as an empirical application. he empirical stdy shows that market inefficiency is primarily cased by a small portion of mispriced stocks, instead of aggregated alphas. Moreover, most of the significant alphas are de to extra retrns (nderpriced). Keywords: high dimensionality, approximate factor model, test for mean-variance efficiency, power enhancement Address: Department of Operations Research and Financial Engineering, Sherrerd Hall, Princeton University, Princeton, J 08544, USA, jqfan@princeton.ed, yanliao@md.ed, jiaweiy@princeton.ed. he research was partially spported by DMS and IH R0GM , IH R0-GM0726.

2 Introdction One of the fndamental assmptions of the Arbitrage Pricing heory (AP) developed by Ross (976) is that asset retrns follow a factor model strctre. Assme the excessive retrn of an asset over the risk-free rate satisfies y it = α i + b if t + it, i =,...,, t =,...,, (.) where f t = (f t,..., f Kt ) are the excessive retrns of K factors, b i = (b i,..., b ik ) are nknown factor loadings, and it represents the idiosyncratic error. he key implication from the asset pricing theory is that all the elements of the intercept vector α = (α,..., α ) shold be zero, known as mean-variance efficiency. esting H 0 : α = 0 in the mlti-factor model (.) is of crcial importance in many practical applications, inclding portfolio selection and fnd evalation. It also incldes the test of the Capital Asset Pricing Model (CAPM) as a special case. Factor models have wide impacts on both economics and finance. In classical factor analysis, the cross-sectional dimension is assmed fixed and the idiosyncratic components are cross-sectionally ncorrelated (that is, the idiosyncratic covariance matrix is diagonal). hese assmptions are no longer sitable for modern financial applications when data are often widely available for a large nmber of assets over a short time span. In addition, a large panel often introdces correlations among idiosyncratic errors, making a strict factor model in the traditional sense very restrictive. In this paper, we consider the problem of testing the mean-variance efficiency when is relatively large compared to the time-series dimension. It is also desirable to allow for an approximate factor strctre in the sense of Chamberlain and Rothschild (983), which admits cross-sectional correlations among the idiosyncratic components. Most of the existing tests are based on the qadratic statistic W = α V α, where α is the OLS estimator for α, and V is some positive definite matrix whose eigenvales are α, where stochastically bonded. he Wald statistic, for instance, takes the form τ α Σ is the estimated inverse of the error covariance, and τ is a positive scalar that depends on Σ the factors only. Another prominent example is the test given by Gibbons, Ross and Shaken (989, GRS test), who showed that the exact distribtion of an adjsted Wald test is F - distribtion nder normal assmptions. hese tests are condcted when does not grow with, and therefore rle ot the possibility of a large panel. When diverges, Pesaran 2

3 and Yamagata (202, PY test) obtained the asymptotic nll distribtion of the standardized qadratic form: (W EW )/ var(w ). oting that the sample residal covariance is no longer invertible when >, they chose V = diag( Σ ), and showed that the standardized W is asymptotically normal. In addition to these work, Bealie et al. (2007, BDK test) developed a likelihood ratio test, and MacKinlay and Richardson (99) stdied a GMM test; both are based on the qadratic form as well. In practice, the nmber of assets nder consideration can be of thosands. Bt to prevent possible strctral changes on the factor loadings and risks, a relatively short time series is mostly appropriate, which contains only hndreds of daily observations, or tens of monthly data. Sch a high-dimension-low-sample-size context cases the tests based on the qadratic statistic to have a very low power and inconsistent against many common alternatives. o see this, we note that the rejection region for the qadratic statistic at significant level q takes the form α V α > c q for some critical vale c q, regardless of the choice of V or whether standardization is applied. When the dimension of α is large, a large critical vale has to be sed in order to correctly control the size. For example, in the ideal case when { t } t= are i.i.d. normally distribted and V = cov( α), the critical vale is proportional to χ 2,q /, which is of order /, and diverges when = o(). Conseqently, sch a qadratic test is only consistent when α 2 is large enogh nder the alternative, and will have a low detection power whenever α 2 is either bonded or growing at a slow rate. his is especially the case when it is only a few significant alphas that arose market inefficiency. herefore, one of the fndamental difficlties of the high-dimensional test arises from the qadratic form α V α: it accmlates a hge amont of estimation errors, and loses power against many alternatives nder which α does not grow so rapidly. Or empirical stdy on the consititents of the S&P 500 index confirms this isse. In this paper, we introdce a new concept for high-dimensional testing problems called power enhancement (PEM), and develop a PEM test as J = J 0 + J. Here J 0 0 is a sre-screening statistic that serves as a power enhancement part, and J can be any qadratic form based test that has a correct asymptotic size (e.g., GRS, PY, BDK). he sre-screening statistic is defined to be J 0 = α Ŝ V Ŝ α Ŝ 3

4 where αŝ = ( α j : α j > δ ) is a sbvector of α screened ot by a threshold δ, and VŜ is a corresponding sbmatrix of certain weight V. he threshold δ is chosen sch that nder the nll hypothesis, P (J 0 = 0 H 0 ), and that J 0 diverges when max j α j > O( log ). Hence the asymptotic nll distribtion of the PEM test is completely determined by that of J, while the sre-screening part significantly enhances the power of J nder many mild alternatives in which α 2 may not be large. his incldes the sparse alternative as an example, where most of the α j s are either zero or nearly so, with only a small (compared to / ) portion of α j s standing ot. Since J J, the rejection region of J strictly contains that of J. We also develop an operational Wald statistic even when > and Σ = cov( t ) is not a diagonal matrix. he statistic is based on a consistent sparse estimator Σ for the inverse error covariance. We show that as, and is possibly mch larger than, for some scalar τ > 0, J sq = τ α Σ α d (0, ) 2 nder the nll hypothesis. his test takes into accont the cross-sectional dependence among the idiosyncratic errors. echnically, in order to show that the effect of replacing Σ the sparse estimator Σ is negligible, we need to establish, nder H 0, τ α ( Σ with Σ ) α = o p (). (.2) 2 ote that a simple ineqality α ( Σ Σ ) α α 2 Σ Σ wold not work when > becase the estimation errors in α 2 accmlate in high dimensions. Instead, we have developed a new technical strategy to prove (.2), which wold be also potentially sefl in high-dimensional inference sing GMM methods when one needs to estimate the optimal weight matrix. We frther take J = J sq, and combine it with or sre-screening statistic J 0 to propose a power enhancement test, which is mch more powerfl than sing J sq itself, while maintaining the same asymptotic nll distribtion. As a by-prodct, the sre-screening step also identifies the individal significant alphas which we show to be sefl to detect market inefficiency in a real application. In contrast, most of the existing tests do not possess this featre. he proposed methods are applied to the secrities in the S&P 500 index as an empirical application. he empirical stdy shows that indeed, market inefficiency is primarily cased by a small portion of mispriced stocks instead of aggregated alphas. In addition, most of 4

5 the significant alphas are de to extra retrns (that is, de to a large min αj >0 α j instead of a large α 2 ). his is captred by the proposed PEM test. he remaining of the paper is organized as follows. Section 2 sets p the preliminaries and discss the limitations of traditional tests. Section 3 proposes the power enhancement method, derives the asymptotic behaviors of the sre-screening statistic and analyzes its performances nder different alternatives. Section 4 combines the PEM with the standardized qadratic form. An improved qadratic test based on thresholding is considered in Section 5. Simlation reslts are presented in Section 6, along with an empirical application to the secrities in the S&P 500 index in Section 7. Section 8 concldes. All the proofs are given in the appendix. hroghot the paper, for a sqare matrix A, let λ min (A) and λ max (A) represent its minimm and maximm eigenvales. Let A and A denote its operator norm and l norm respectively, defined by A = λ /2 max(a) and max i j A ij. For two deterministic seqences a and b, we write a b (or eqivalently b a ) if a = o(b ). Also, a b if there are constants C, C 2 > 0 so that C b a C 2 b for all large. Finally, for a finite set S, we denote S 0 as the nmber of elements in S. 2 Factor models and traditional tests Consider the following linear factor pricing model in a matrix form y t = α + Bf t + t, t =,...,, (2.) where y t = (y t,..., y t ) is an vector of observed asset retrns at time t, B = (b,..., b ) with b i being a K vector of loadings, f t = (f t,..., f Kt ) is a vector of common factors, t = ( t,..., t ) denotes the idiosyncratic component, and α = (α,..., α ). We set p the model to have an approximate factor strctre as in Chamberlain and Rothschild (983) where the idiosyncratic components are cross-sectionally correlated over i. Here both y t and the common factors f t are observable. Or goal is to test the hypothesis that those alphas are jointly zero across the panel: H 0 : α = 0. (2.2) We are particlarly interested in a high-dimensional sitation where can be mch larger than. In practice, the nmber of assets nder consideration can reach as many as thosands, bt the observation period might be mch smaller, becase a long observation window is likely to introdce strctral breaks in factor loadings. In addition, endogeneity 5

6 problems arise when sing a testing strategy based on portfolios instead of individal secrities. As a reslt, a reliable test of the above factor model often entails a large panel of high dimensions relative to. Moreover, a large panel natrally imposes a sparsity assmption on the error covariance matrix Σ = cov( t ), that is, many of the off-diagonal elements are either zeros or close to zero. Since the common factors have sbstantially mitigated the co-movement across the whole panel, a particlar asset s idiosyncratic volatility is sally correlated with no more than a few nmber of other assets. For example, some shocks only exert inflences on a particlar indstry, bt are not pervasive for the whole economy [Connor and Korajczyk (993)]. Sch a sparse assmption will be sed to reliably estimate the error covariance later. 2. Wald-type tests If we frther denote y i = (y i,..., y i ) ; as an vector consisting of ones, F = (f,..., f ) and i = ( i,..., i ), then model (2.) can be written as y i = α i + Fb i + i. (2.3) he model is a seemingly nrelated regression model with common factors. We therefore can rn the regression stock by stock according to (2.3). Define K vectors f = t= f t, w = ( t= f tf t) f, and a scalar τ = f w. hen the ordinary least sqares (OLS) estimator yields Frther calclations yield α = ( α,..., α ), α i = τ α i = α i + τ y it ( f tw). (2.4) t= it ( f tw). (2.5) t= Assming no serial correlation among t, the conditional covariance of α is Σ /( τ), given the factors. If Σ is known, a classical way to bild the test statistic wold be W = τ α Σ α. (2.6) When is fixed, Σ can be estimated and replaced by the inverse of the sample residal covariance matrix, and the reslting test statistic is then in line with the well-known test by Gibbons, Ross and Shaken (989, GRS test). With the normality assmption, GRS obtained the exact finite sample distribtion of this test statistic. Asymptotically, the test statistic 6

7 becomes the Wald statistic which follows a χ 2 -distribtion nder the nll hypothesis. When grows, the traditional asymptotics for the nll distribtion of W does not apply. Instead, Pesaran and Yamagata (202, PY test) developed an alternative asymptotics for the Wald statistic, based on the standardized version of W. More specifically, nder some reglarity conditions, they showed J = τ α Σ α d (0, ). (2.7) 2 as. Hence at the significant level q (0, ), P (J > z q H 0 ) q. Regardless of the type of asymptotics, we shall refer to the test based on W (with Σ an estimator of α. Σ possibly replaced with ) as Wald-type statistic, or qadratic test becase W is a qadratic form 2.2 wo main challenges In a data-rich environment, the panel size can be mch larger than the nmber of observations. Sch a high dimensionality brings new challenges to the test statistics based on W, and the alternative asymptotics introdced by Pesaran and Yamagata (202) only partially solves the problem. Specifically, there are two main challenges. he first challenge arises from estimating Σ. It is well known that the sample residal covariance matrix becomes singlar when >. Even if <, replacing Σ in W with the inverse sample covariance can still bring a hge amont of estimation errors when 2 is close to. his can distort the nll distribtion of the test statistic. Another challenge comes from the concern of the power. Even when Σ is known so that W is directly feasible, a test statistic based on W has very low powers against varios alternative hypotheses when is large, still de to the accmlation of estimation errors. his can be illstrated in the following example. Example 2.. Sppose = O(), and we want to test H 0 against finitely many nonzero α s: H a : α i = c i, i r, α i = 0, r < i, where r is fixed, and all the c i s are bonded away from zero. Assme Σ = I to be known, and α is the OLS estimator with niform accracy: max i ˆα i α i = O p ( Under H 0, α is a pre estimation noise, and W = τ α Σ α H 0 = O p ( ) 7 α i 2 = O p (). i= ).

8 Under H a, each of the first r components in α is stochastically close to c i, whereas the estimation noises constitte the remaining r components. Hence W Ha = O p ( r c 2 i + ( r)) = O p (). i= Apparently, the Wald-type test has the same order nder both H 0 and H a, making it hard to distingish H 0 from H a. Let s now derive the critical vale. Sppose { t } t= are i.i.d. normal with a known covariance I. hen conditional on {f t } t=, ( α α) (0, Σ /τ). herefore, W rejects H 0 if W > c q = χ 2,q /( τ). In addition, as, W 2 d (0, ). Hence an alternative critical vale is c q = ( 2z q + )/( τ). Either way, the critical vale c q O(/ ) c q. Bt in this example, α 2 = o(/ ) nder the alternative. So a test based on W hardly has power against H a. In the previos example, there are only a small portion of nonzero alphas in the alternative, whose signals are dominated by the aggregated high-dimensional estimation errors: i>r α2 i. ote that when (2.7) holds, we can reject H 0 as long as J > z q at the significant level q (0, ), where z q is the critical vale for the standard normal distribtion. However, it can be shown that when = o( ), and there are only r = o( ) nonzero alphas in the alternative, this test is not consistent. We formally present this reslt in the following theorem. For simplicity, we assme both Σ and Σ to be bonded. heorem 2.. Sppose = o( ). Consider J that satisfies (2.7). In addition, sppose Assmption 3.2 below holds, and both Σ and Σ are bonded away from infinity. When = O( ), consider the following alternative: H a : there are at most r = o( ) nonzero α j s, which are also bonded away from infinity. hen nder H a, P (J > z q H a ) 2q + o() for any q (0, 0.5). herefore the test based on J is inconsistent. A more sensible approach is to focs on those alternatives that may have only a few nonzero alphas compared to, which are also of interest in practice. In what follows, we develop a new testing procedre that significantly improves the power of the Wald-type test against sch alternatives. 8

9 3 Power Enhancement raditional tests of factor pricing models are especially directed against alternatives regarding portfolios, rather than individal assets. he power of the GRS test relies on the sqared Sharpe ratio for the tangency portfolio, regardless of how individal assets behave nder the alternatives. It trns ot that even if some individal asset are either significantly overpriced or nderpriced, their trivial contribtion to the whole portfolio is not enogh for making inferences. We aim to bild a test statistic that not only takes care of alternatives involving portfolios, bt also deals with sparse alternatives where only a few assets have nonzero alphas. By doing so, we keep track of featres of both portfolios and individal assets. For sch prposes, we propose a class of test statistics that consist of two parts: J = J 0 + J, (3.) where J 0 0 is a proposed sre-screening statistic, designed to detect both sparse alternatives and significant individal alphas, and J is based on some existing Wald-type statistic that mainly controls the size of the test. We reconcile the two parts so that the asymptotic size of the test is that of J nder the nll, and the asymptotic power of the test is mainly driven by J 0. Becase J J always holds, the power of J is enhanced as a reslt. 3. Sre-screening statistic For a given factor pricing model with a large panel, we divide market inefficiency into two forms according to different configrations of alphas.. Average case: he average of alphas deviates from zero. 2. Sparse case: A few alphas are significantly away from zero, while all the others are close to zero. ote that the above two cases are not mtally exclsive. A few alphas with large absolte vales wold drive the average p to some extent, implying market inefficiency in an average sense. In trn, a large enogh average indicates at least a few alphas are as large. he separation, however, gives s an insight on how to select stocks by telling s whether we shold focs on a small portion of them or all of them. And if it is a small portion, it compels or interest to identify those stocks that lead to the market inefficiency. As we shall see, or proposed test J 0 + J atomatically identifies significantly mispriced stocks. 9

10 o prevent the accmlation of estimation errors in a large panel, we propose a srescreening statistic. For some predetermined threshold vale δ > 0, define a screening set Ŝ = { j : α j σ j } > δ, j =,...,, (3.2) where α j is the OLS estimator and σ j = t= û2 jt/ τ is the sample estimator of the variance of α j. Denote by α I αŝ = ( α j : j Ŝ) = Ŝ. (3.3) α I Ŝ the screened-ot alpha estimators, which can be interpreted as rejecting the efficiency of the corresponding assets individally. Let Σ be a nonsinglar estimator of Σ, to be defined later. Let ΣŜ denote the sbmatrix of Σ formed by the rows and colmns in {j : j Ŝ}, so that ΣŜ/( τ) is the estimated conditional covariance matrix of αŝ given the common factors. With the notations above, we define or sre-screening statistic as J 0 = τ α Σ α Ŝ Ŝ Ŝ. (3.4) he choice of δ mst sppress most of the noises, reslting in an empty set of Ŝ nder the nll hypothesis. On the other hand, δ cannot be too large to filter ot important signals of alphas nder the alternative. For this prpose, noting that the maximm noise level is O p ( log / ), we let With this choice of δ, if we define, for σ j = Σ,jj / τ, S = log δ = (log log ). (3.5) { j : α j σ j } > 2δ, j =,...,, (3.6) then nder mild reglarity conditions, P (S = Ŝ), and α Ŝ mimics α S = (α j : j S). herefore, nder H 0, Ŝ gets rid of most of the estimation noises, and it follows that P (J 0 = 0 H 0 ). Under most of the alternatives, Ŝ preserves the important individal alphas by mimicking the oracle set S. When α S 2, J 0 is stochastically nbonded. Hence the nll and alternative hypotheses are well distingished by the asymptotic behaviors 0

11 of J 0. Σ S he sre-screening statistic depends on a nonsinglar covariance matrix. Derived from an estimate of Σ, Σ Ŝ Σ Ŝ that mimics is sed for standardization in order for J 0 to attain a proper scale. he covariance Σ S can be very large when the tre S is large, and is rather difficlt to estimate. o obtain an operational Σ, we assme Σ Ŝ to be a sparse covariance and estimate it by thresholding. Let û jt be the residal from the OLS estimator. For s ij = t= ûitû jt, let ( Σ s ij, if i = j, ) ij = th(s ij ), if i j, (3.7) log where th( ) is a thresholding fnction, with threshold vale h ij = C(s ii s jj )/2 for some constant C > 0. When the hard-thresholding fnction is sed, this is the estimator proposed by Fan et al. (20). Many other thresholdings also apply, e.g., soft thresholding and SCAD (Fan and Li 200). In general, th( ) shold satisfy: (i) th(z) = 0 if z < h ij ; (ii) th(z) z h ij. (iii) here are constants a > 0 and b > sch that th(z) z ah 2 ij if z > bh ij. We can also replace ΣŜ with DŜ = diag{s jj : j Ŝ}. his is particlarly sefl when Σ is not sparse. he sre-screening statistic is then defined as J 0 = τ αŝ D α Ŝ Ŝ = τ α js 2 jj. j Ŝ 3.2 Power enhancement test he screening statistic J 0 is powerfl in detecting significant alphas. However, nder the nll hypothesis, J 0 = 0 with probability approaching one. Hence J 0 by itself cannot be sed directly. We combine it with other standard test statistics in order to control the size of the test. Sppose J is some test statistic for H 0 : α = 0, and assme that there is Ω R sch that J has power against α Ω, in the sense that the test based on J is consistent against the alternative hypothesis H a : α Ω. Or test statistic is formally defined as: J = J 0 + J. Becase the sre-screening statistic J 0 is zero nder the nll with probability approaching one, the nll distribtion of J is asymptotically determined by that of J. Let c q be the qth

12 qantile of J nder the nll, that is, P (J 2 c q H 0 ) q. Or proposed J-test then reject H 0 if J > c q nder the significant level q. Hence adding J provides s a non-degenerate nll distribtion, which is needed to control the size of the test. On the other hand, as demonstrated by Example 2., traditional tests often sffer from low powers de to the accmlation of estimation errors nder the high dimensionality. Inclding the term J 0 enhances the power of J becase J J always holds. In fact, we will show below that the new test based on J has power against Ω {α R : max j α j > 2δ min j σ j}. As a reslt, adding J 0 significantly enhances the power of J. We shall ths address or test based on J to be power enhancement test (PEM test). o formally see the fact of power enhancement, we impose the following assmptions. First of all, we assme Σ to be a sparse matrix so that one can apply the thresholding method to consistently estimate it. he notion of generalized sparsity in Bickel and Levina (2008) is sed: for some q [0, ), define m = max i (Σ ) ij q. (3.8) j= Under the following sparsity assmption, the thresholded covariance estimator Σ is positive definite, and consistently estimates Σ nder the operator norm. Assmption 3.. here is q [0, ), so that, for log = o( ), ( m = o ( log )( q)/2 A special case of Assmption 3. occrs when m is bonded from above, in the case of block-diagonal matrix with finite block sizes. Sparsity is one of the commonly sed assmptions on high-dimensional covariance matrix estimations. here have been extensive stdies recently in the statistical literatre on estimating a large sparse covariance matrix. We refer to El Karoi (2008), Bickel and Levina (2008), Lam and Fan (2009), Cai and Li (20), and the references therein. Let F 0 and F denote the σ-algebras generated by {(f t, t ) : t 0} and {(f t, t ) : t } respectively. In addition, define the mixing coefficient ). α( ) = sp A F 0,B F P (A)P (B) P (AB). (3.9) 2

13 Assmption 3.2. (i) {f t, t } t is strictly stationary, and E t = 0 and E t f t = 0. In addition, for s t, E t s = 0, and Ef t(ef t f t) Ef t <. (ii) here exit constants c, c 2 > 0 sch that max i b i < c 2, c < λ min (Σ ) λ max (Σ ) < c 2, and c < λ min (cov(f t )) λ max (cov(f t )) < c 2. (iii) Exponential tail: here exist r, r 2 > 0, and b, b 2 > 0, sch that for any s > 0, max i P ( it > s) exp( (s/b ) r ), max i K P ( f it > s) exp( (s/b 2 ) r 2 ). (iv) Strong mixing: here exists r 3 > 0 and C > 0 satisfying: for all Z +, α( ) exp( C r 3 ). hese conditions are standard in the time series literatre. In Condition (i), we reqire the idiosyncratic error t be serially ncorrelated across t. Under this condition, the conditional covariance of α is Σ /( τ). Estimating Σ when > is already challenging. When the serial correlation is present, the atocovariance of t wold also be involved in the covariance of the OLS estimator for alphas, and needs be estimated. We rle ot these atocovariance terms to simplify the technicalities, and or method can be extended to the case of serial correlation, with frther sparsity conditions. On the other hand, we allow the factors to be weakly dependent via the strong mixing condition. Also, it is always tre that Ef t(ef t f t) Ef t. We rle ot the eqality to garantee that the asymptotic variance of αj does not degenerate for each j. he following theorem qantifies the asymptotic behavior of the sre-screening statistic J 0, and provides sfficient conditions for the set consistency in selecting significant alphas. Recall that Ŝ and S are defined in (3.2) and (3.6) respectively. Define = {j : α j δ, j =,..., }. heorem 3.. Sppose log = o( ), Assmption 3.2 hold. As,, (i) P (S Ŝ), P (Ŝ \ S ). (ii) Under the nll hypothesis, P (Ŝ = ). Hence P (J 0 = 0 H 0 ), We are particlarly interested in a type of alternative hypothesis that satisfy the following grey area condition. 3

14 Assmption 3.3 (Grey area). he alternative hypothesis H a satisfies: =. he grey area represents a class of alternatives that have no nonzero α j s on the bondary of the screening set S. his condition is very weak becase the chance of falling exactly at the bondary is very low. Intitively speaking, when an α j is on the bondary of the threshold, it is hard to decide whether to eliminate it from the screening step or not. According to heorem 3., the difference between the set estimator Ŝ and the oracle set S is contained in the grey area with probability approaching one. So the grey area condition sffices to achieve the screening consistency: P (S = Ŝ). Corollary 3.. Sppose Assmption 3.3 holds. Under the assmptions of heorem 3., P (Ŝ = S). ote that S ranks the importance of individal alphas nder the alternative, representing the tre set of significant alphas we wish to identify. Hence the sre-screening consistency as in Corollary 3. enables s to identify these significant alphas. We now formally present the asymptotic behavior of the PEM test. A test is said to have power against a set A R if α A, the probability of rejection converges to. heorem 3.2. Sppose log = o( ), and Assmptions hold. In addition, sppose there is a test J whose rejection region takes the form J > C, and a non-degenerate distribtion F, so that nder the nll hypothesis J d F, and J has power against Ω R. hen the PEM test J = J 0 + J satisfies: (i) Under H 0, J d F, (ii) nder H a sch that α satisfies: α Ω {α R : max j α j > 2δ min j σ j} Ω, the PEM test has power against any sbset of Ω. Let F q denote the qth qantile of F. Part (i) of heorem 3.2 shows that J and J reject the nll if J > F q and J > F q respectively. It follows immediately from J J that P (J > F q ) P (J > F q ), which means J at least has power against Ω. In addition, J 0 indeed plays the role of power enhancement, in the sense that (i) it does not affect the nll 4

15 distribtion, and (ii) it significantly enhances the power of the test by broadening the range of α for which the test is consistent. ote that the introdced restriction for the alternative: {α R : max j α j > 2δ min j σ j} reqires that S be nonempty. his is a very weak restriction for the alternative. For instance as long as there is an alpha standing ot nder H a, the nonemptyness of S is satisfied. Under the nll, J dominates J 0, and gives a correct size, while nder the alternative, J 0 dominates J, and is stochastically nbonded. In addition, the sre-screening statistic atomatically identifies all the significant alphas that are in Ŝ. 4 PEM for the Standardized qadratic test In principle, any existing consistent test can serve as J. In this section, we consider a standardized qadratic test (Wald-type) recently developed by Pesaran and Yamagata (202). heorem 2. shows that this test by itself sffers from low powers nder the high dimensionality. We will see that PEM significantly enhances its power. 4. Standardized qadratic statistic Σ he Wald-type statistic τ α Σ α depends on a high-dimensional inverse covariance. As described in Section 2.2, if Σ is not diagonal, estimating Σ is a challenging problem when >. Alternatively, one can se only the diagonal entries of Σ, and consider the following qadratic form, which takes into accont the error cross-sectional heteroskedasticity only: for s ii = t= û2 it, W = τ α D α, D = diag{s,..., s }. ote that W is a sm of individal sqared t-statistics, the nmber of which grows in or large panel settings. A standardized version is given by: W E(W ) var(w ). (4.) One needs to calclate and estimate both E(W ) and var(w ) nder the nll. As D ignores the off diagonal entries of Σ, when Σ is indeed non-diagonal, W is no longer χ 2 even nder the normal assmption. Hence the calclation is not a trivial task. Pesaran and Yamagata 5

16 (202) showed that a feasible version of (4.) is J sq = W 2( + ξ ) where ξ = ˆρ 2 2 ijiˆρ ij >a, a = Φ ( c/) i j for some c (0, 0.5). Here ˆρ ij denotes the sample correlation between it and jt based on the residals, and Φ ( ) denotes the inverse standard normal cmlative distribtion fnction. Applying a linear-qadratic form central limit theorem (Kelejian and Prcha 200) yields that J sq is standard normal as,. We call J sq to be the standardized qadratic statistic. Pesaran and Yamagata (202) also proposed a slightly different (bt asymptotically eqivalent) statistic that corrects the finite sample bias. 4.2 Combining sre-screening with standardized qadratic test We now formally investigate the power of J sq combined with the sre-screening statistic J 0, namely, the PEM test: J = J 0 + J sq. he size of the test statistic is controlled by J sq, as J 0 is zero with probability approaching one nder the nll hypothesis. o appreciate the power enhancement to J sq, note that J J sq, so the rejection region of J 0 contains that of J sq. In fact, the rejection region is significantly enlarged, as J 0 diverges nder varios interesting alternatives, inclding the sparse alternative case in the sense that there are only a few α i 0. he PEM test J combines the rejection region of both two statistics, achieving a mch enhanced power, withot sacrificing the good size of J sq. Formally, we have the following reslts concerning the size and the power of the PEM test. heorem 4.. Sppose log = o( ) and Assmptions hold. Also, i j (Σ ) 2 ij = O(). As, and / 3 0, we have: (i) nder the nll hypothesis H 0 : α = 0, J d (0, ), 6

17 (ii) nder the alternative H a sch that α {α R : α 2 ( log )/ } {α R : max j α j > 2δ min j (Σ ) jj } B, the PEM test has power against any sbset of B. We see that adding J 0 in the PEM test does not lose anything nder the nll asymptotically, and it significantly enhances the power of J sq nder many important alternatives. he set {α R : α 2 ( log )/ } itself is the region of α in the alternative that J sq has power against. his is a very restrictive region when >. For instance, it rles ot the type of alternative in which there are finitely many nonzero alphas. In contrast, set B is mch more enlarged, and contains many interesting alternatives. We investigate a few examples below that fall within this category. In these examples, we denote H a A to mean that the alternative set for α belongs to a given set A. Example 4. (Minimm alpha). Consider an alternative sch that the minimm nonzero alpha is not too small: H a A {min j { α j 0} δ }. In this case, the oracle set S is not empty, and meanwhile there exists no nonzero alphas that are at or below the level δ. his set is a sbset of B in heorem 4., and hence the PEM test J asymptotically has power against this alternative. ote that we allow min{ α j, j =,..., } to vanish in the limit. A special case is the sparse alternative: H a : min j r { α j } δ, α j = 0, when j > r where r is fixed. In this case, there are only finitely many significant alphas. he PEM still has power against it (the power approaches one). However, J sq itself has very low power becase i α2 i = O() nder the alternative. In fact, as long as r grows slowly compared to, the power of any test based on α V α with V being positive definite does not converge to one. Example 4.2 (Empty grey area). Consider an alternative H a {j : α j δ } =, and {j : α j δ }. Under this alternative, the grey area is empty and there exists at least one significant alpha. We allow the existence of very small bt nonzero alphas, that is, { α j 0, α j = o(δ )}, regardless of the nmber of them. he PEM test is still consistent by heorem 4.. In 7

18 contrast, if the nmber of significant alphas is not large enogh, J sq itself still cannot detect H a. An important featre of the above alternatives is that we do not reqire α 2 nder the alternative to be very large for the test consistency. In contrast, we only reqire a few alphas stand ot. In addition, besides the enhanced power, the PEM test is able to identify all the significant alphas via Ŝ if the grey area condition holds (S = Ŝ with probability approaching one). Withot screening, traditional tests sch as GRS test and J sq alone cannot detect these alternatives, especially when the panel is very large. When there are no estimated alphas above the level δ nder the alternative, Ŝ can be empty, leading to J 0 = 0. However, de to the component J sq in J, we may still achieve the consistency. his is illstrated in the following example. Example 4.3. Consider the following alternative: H a max j α j = O(δ ), and α 2 ( log ) c, log for c (0, ). Under sch alternatives, a typical alpha lies in the interval [, log log ], and the nmber of nonzero alphas is not small. In this case, the probability that the srescreening set Ŝ is empty might be positive. We therefore expect P (J 0 = 0) > 0. However, since there are so many small bt nonzero alphas, these aggregated alphas lead to a nottoo-small j= α2 j. As a reslt, J sq stands ot and gains power, making J still has power converging to one. Indeed, or simlated reslts demonstrate that the PEM test has a good power in this case (see Section 6). Finally, when both the maximm magnitde of alphas and α 2 are small, the PEM test statistic has a low power. Bt the market can be considered approximately efficient in this sitation. 5 An improved qadratic test based on thresholding So far we have been sing a diagonal weight matrix D for the standardized qadratic test to combine with J 0, becase the tre Σ is nknown and non-diagonal. On the other hand, this loses powers against certain local alternatives, becase as long as Σ is not diagonal, sing a diagonal weight matrix ignores the cross-sectional correlations among the error components. A better PEM shold be: J 0 + τ α Σ α. (5.) 2 8

19 When Σ is known, Pesaran and Yamagata (202) showed that the above statistic withot J 0 is asymptotically (0, ) nder the nll. Hence, J sq can be improved by replacing D with a consistent estimator for Σ. In this section, we show that this indeed can be done by applying the thresholded covariance estimator nder the sparsity condition of Σ. We reqire log = o( 2 ), bt still allow to be mch larger than. 5. A technical challenge When Σ is a sparse matrix, Fan et al. (20) obtained a thresholded estimator Σ as described in Section 3., which satisfies: Σ Σ log = O p (m ). (5.2) Here m = max i j= (Σ ) ij is assmed to be either growing slowly with or even bonded. For instance, m is bonded if Σ is a block-diagonal matrix with bonded block sizes, which is the case when the idiosyncratic noises are ncorrelated across indstries. ote that the above convergence achieves the minimax optimal rate for sparse covariance estimation, as shown by Cai and Zho (202). However, it comes with a technical challenge when >, which we now explain. When replacing Σ in (5.) with Σ, one needs to show that the effect of sch a replacement is asymptotically negligible, namely, nder H 0, Σ α (Σ ) α = o p (). (5.3) ote that when α = 0, α 2 = O p ((log )/ ) (or O p (/ ) with a more carefl analysis). A simple application of (5.2) yields α (Σ Σ ) α Σ α 2 Σ log = O p (m log ). We see that even if m is bonded, it still reqires log = o( ). Σ However, the above derivation ses a very crde bond α (Σ Σ ) α α 2 Σ, which accmlates the estimation errors in α α 2 nder a large. In fact, α (Σ Σ ) α is a weighted estimation error of Σ Σ, where the weights α help to redce the crse of dimensionality, and shold reslt in an improved rate of convergence. he formalization of this argment reqires frther reglarity conditions, which we shall present in the following sbsection. 9

20 5.2 Assmptions In order to show that the effect of replacing Σ with its consistent estimator is asymptotically negligible, especially when >, frther assmptions are needed, bt they are still qite reasonable. First of all, we need to refine the sparsity condition on Σ, which is similar to that of Lam and Fan (2009). Let S L and S U denote two disjoint sets and respectively inclde the indices of small and large entries of Σ, and {(i, j) : i, j } = S L S U. (5.4) We assme that most of the indices (i, j) belong to S L when i j. For the banded matrix as an example, (Σ ) ij 0 if i j k; (Σ ) ij = 0 if i j > k for some fixed k. hen S L = {(i, j) : i j > k} and S U = {(i, j) : i j k}. Formally, we assme: Assmption 5.. here is a partition {(i, j) : i, j } = S L S U i j,(i,j) S U = O() and (i,j) S L (Σ ) ij = O(). In addition, sch that max (Σ ) ij (i,j) S L log min (Σ ) ij. (i,j) S U We reqire the elements in S L and S U be well-separable. For example, if Σ is a block covariance matrix with finite block sizes, this assmption is natrally satisfied as long as the signal is not too-weak (that is, log = o(min (i,j) SU (Σ ) ij )). he partition {(i, j) : i, j } = S L S U may not be niqe. Most importantly, we do not need to know either S L or S U ; hence the block size, the banding length, or the locations of the zero entries can be completely nknown. Or analysis sffices as long as sch a partition exists. o introdce or next assmption, define ξ t = Σ t = (ξ t,..., ξ t ), which is an - dimensional vector with mean zero and covariance Σ, whose entries are stochastically bonded. Since E t f t = 0 and E t = 0, we have t= ξ it = O p () and t= ξ itf t = O p () for each i. We assme the following: Assmption 5.2. Let w = (Ef t f t) Ef t, then (i) E i= ( 2 it E 2 it)( t= ξ is ( f sw)) 2 2 = o() s= 20

21 (ii) E i j,(i,j) S U ( it jt E it jt )[ t= ξ is ( f sw)][ s= ξ jk ( f kw)] 2 = o() k= ote that in the literatre of high-dimensional panels and factor analysis (e.g., Stock and Watson 2002, Bai 2003, 2009), it is sally assmed that the cross-sectional and serial doble sm is bonded: E i= t= (2 it E 2 it) 2 = O(), which is sally garanteed by the central limit theorem across both i and t. Here, Condition (i) of Assmption 5.2 is with respect to the weighted doble sms, where the weight ( s= ξ is( f s w))2 is stochastically bonded becase both t= ξ it and t= ξ itf t are asymptotically normal. Condition (ii) is new in the literatre becase it is related to the sparsity condition. It is however very similar to (i) in that the index set of the cross-sectional sm is changed from {(i, j) : i = j} to {(i, j) : i j, (i, j) S U }, where S U is defined as the set of big entry indices. Recall that Assmption 5. assmes i j,(i,j) S U = O(). So this condition is still reasonable for sparse enogh covariances. Primitive conditions for Assmption 5.2 will be provided in Section PEM test with improved standardized qadratic form With the help of Assmptions 5. and 5.2, we show in the appendix (Proposition A.) that (5.3) indeed holds. As a reslt, the effect of replacing Σ with its consistent thresholded estimator is asymptotically negligible even if >. ow define a new PEM test with an improved standardized qadratic form: J = J 0 + J sq where, with Σ defined in (3.7) as in Fan et al. (20), J sq = τ α Σ α. 2 We have the following theorem. heorem 5.. Sppose m 4 (log )4 = o( 2 ), and Assmptions 3., 3.2, 5., 5.2 hold. hen (i) nder the nll hypothesis H 0 : α = 0, J d (0, ), Jsq d (0, ), 2

22 (ii) nder the alternative H a sch that α {α R : α 2 ( log )/ } {α R : max j α j > 2δ min j (Σ ) jj } B, the PEM test J has power against any sbset of B. ote that when m is bonded, we then only reqire (log ) 4 = o( 2 ). herefore is allowed to be mch larger than. 5.4 Sfficient conditions for Assmption 5.2 As a simple example, Assmption 5.2 is satisfied if it is i.i.d. across both i and t. Example 5.. As an example, Assmption 5.2 can be simply verified if it is i.i.d. across both i and t and = o( 2 ), bt still, can be larger than. Write X = i= t= (2 it E 2 it)( s= ξ is( f sw)) 2. Let e is = ξ is ( f sw). When it is i.i.d. across both i and t, var(x) = var[ ( 2 it E 2 it)( t= e is ) 2 ] E[ s= ( 2 it E 2 it)( t= e is ) 2 ] 2. s= It is bonded by {E[ t= (2 it E 2 it)] 4 } /2 {E[ s= e is] 8 } /2, by the Cachy-Schwarz ineqality, which is O() by the central limit theorem. On the other hand, EX = E[ ( 2 it E 2 it)( e is ) 2 ] = cov( ( 2 it E 2 it), ( e is ) 2 ) t= s= t= s= = ( cov( 2 it, e 2 it) + 2 cov( 2 it E 2 it, e is e it ) + cov( 2 it, e is e ik )). t= t= s t t= s,k t he second term on the right is zero becase when s t, cov( 2 it E 2 it, e is e it ) = E[( 2 it E 2 it)e is e it ] = E[( 2 it E 2 it)e it ]E(e is ) = 0. he third term is also zero becase t is serially independent. hs EX = cov( 2 it, e 2 it). ogether, we have EX2 = var(x)+ (EX)2 = o() + o( 2 ) = o() as long as = o( 2 ). {(i, j) : i j} = so the sms are zero. Condition (ii) is natrally satisfied becase When cross-sectional heteroskedasticity and correlations are present, Assmption 5.2 can still be verified when Σ is sparse enogh. For simplicity, we assme Σ to be strictly sparse, in the sense that all its small entries are zero: (Σ ) ij = 0 for (i, j) S L. In addition, it is assmed that both Σ and Σ as follows. have bonded row sms. hese are stated in the assmption 22

23 Assmption 5.3. (i) (Σ ) ij = 0 for all (i, j) S L, where S L is defined in Assmption 5.. (ii) here is a constant C > 0 sch that Σ + Σ < C. Let m = max i j= I (Σ) ij 0, D = i,j I i j,(σ) ij 0 = i j,(i,j) S U. Here m represents the maximm nmber of nonzeros in each row, corresponding to q = 0 in (3.8), and D represents the nmber of nonzero off-diagonal entries. We consider two kinds of sparse matrices, and verify Assmption 5.2 in both cases. In the first case, Σ is reqired to have no more than O( ) off-diagonal nonzero entries, bt allows a growing m ; in the second case, m shold be bonded, bt Σ can have O() off-diagonal nonzero entries. he latter allows block-diagonal matrices with finite size of blocks. his is particlarly sefl when firms individal shocks are correlated only within indstries bt not across indstries. Formally, we assme: Assmption 5.4. One of the following cases holds: (i) D = O( ); (ii) D = O(), and m = O(). he following lemma shows that Assmption 5.2 can be verified nder the reqired sparsity assmptions when >. For simplicity, we focs on the Gassian errors and serially independent time series. Lemma 5.. Sppose t (0, Σ ), where Σ satisfies Assmptions 5.3 and 5.4. In addition, the seqence { t, f t } t is independent across t, and satisfy Assmption 3.2 (i)- (iii); t and f t are also independent. hen as,, log = o( 2 ), Assmption 5.2 is satisfied. 6 Monte Carlo Experiments We examine the power enhancement via several nmerical examples. Excess retrns are assmed to follow the three-factor model by Fama and French (992): y it = α i + b if t + it. 6. Simlation We simlate {b i } i=, {f t } t= and { t } t= independently from 3 (µ B, Σ B ), 3 (µ f, Σ f ), and (0, Σ ) respectively. he same parameters as in the simlations of Fan et al. (203) 23

24 are sed, which are calibrated sing the data on daily retrns of S&P 500 s top 00 constitents, for the period from Jly st, 2008 to Jne 29 th 202. hese parameters are listed in the following table. µ B Σ B µ f Σ f able : Means and covariances sed to generate b i and f t wo types of Σ are considered. Diagonal Σ () is a diagonal matrix with diagonal entries (Σ ) ii = + v i 2, where v i are generated independently from 3 (0, 0.0I 3 ). In this case no cross-sectional correlations are present. Block-diagonal Σ (2) = diag{a,..., A /5 } is a block-diagonal covariance, where each diagonal block A j is a 5 5 positive definite matrix, generated from a cross-sectional MA(3) process as follows: for each j, generate {a i, b i, c i } i 5 independently from (0, 0.0). Let {e i } i 5 be i.i.d. (0, ), v = e, v 2 = e 2 + a e, v 3 = e 3 + a 2 e 2 + b e, and for i = 3, 4, v i+ = e i+ + a i e i + b i e i + c i 2 e i 2. Set A j be the covariance matrix of (v i : i 5), which only depends on {a i, b i, c i } i 5. In the nmerical stdy, we wold not assme we know the block-diagonal strctre thogh, and apply soft-thresholdings to estimate Σ (2). We stdy two types of alternatives (we set > ): 0.3, i sparse alternative Ha : α i = 0, i > log weak alpha Ha 2 : α i =, i , i > 0.4 Under H a, many components of α are zero bt the nonzero alphas are not weak. Under H 2 a, the nonzero alphas are all very weak. In or simlation setp, log / varies from 0.05 to 0.0. We therefore expect that nder Ha, P (Ŝ = ) is close to zero becase most of the first / estimated alphas shold srvive from the screening step. In contrast, P (Ŝ = ) shold be mch larger nder H 2 a becase the nonzero alphas are too week. 24

25 able 2: Size and Power comparison when Σ = Σ () is diagonal H 0 Ha Ha 2 J sq PEM P (Ŝ = ) J sq PEM P (Ŝ = ) J sq PEM P (Ŝ = ) he freqencies of rejection and Ŝ = ot of 500 replications are calclated. Here J sq is the standardized qadratic test sing diagonal weight matrix as in Pesaran and Yamagata (202); PEM represents the power enhancement test J 0 + J sq. 6.2 Reslts When Σ = Σ (), we assme the diagonal strctre to be known, and compare the performances of the standardized qadratic test J sq (considered by Pesaran and Yamagata 202) with the power enhanced test J 0 + J sq. When Σ = Σ (2), we do not assme we know the block-diagonal strctre. In this case, for tests are carried ot and compared: () the standardized qadratic test with diagonal weight J sq based on α D α, (2) the improved standardized qadratic test J sq with thresholded covariance weight matrix, based on α Σ α, (3) the power enhanced test J 0 + J sq, and (4) the power enhanced test J 0 + J sq. For the thresholded covariance matrix, we se the soft-thresholding fnction and fix the threshold at.5 log /. For each test, we calclate the freqency of rejection nder H 0, H a and H 2 a based on 500 replications, with the 0.05 significance level. We also calclate the freqency of Ŝ being empty, which approximates P (Ŝ = ). Reslts are smmarized in ables 2-4. When Σ is diagonal, we see that nder the nll the PEM test has slightly larger rejection probabilities, and P (Ŝ = ) is close to one, which demonstrates that the screening statistic J 0 indeed manages to screen ot most of the estimation errors nder the nll. On the other hand, nder H a, the PEM test significantly improves the power of the standardized qadratic test. In this case, P (Ŝ = ) is nearly zero becase the estimated nonzero alphas still stay after screening. Under H 2 a, however, the nonzero alphas are very week, which leads to a large probability that Ŝ is an empty set. As a reslt, the PEM test only slightly improves the power of the qadratic test. 25