A general randomized test for Alpha

A general randomized test for Alpha

\address{\textsuperscript{1}King's Business School, King's College London, UK } \address{\textsuperscript{2}Cambridge Judge Business School and Girton College, University of Cambridge, UK; and Centre for Economic Policy Research (CEPR)} \address{\textsuperscript{3}Universita' di Pavia, Italy} \address{\textsuperscript{4}University of Leicester, UK} \address{\textsuperscript{5}Institute of Finance, Universita' della Svizzera Italiana (USI Lugano), Switzerland}

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Introduction

Testing for \textquotedblleft alpha\textquotedblright\ - i.e. the component of expected returns that cannot be explained by a linear factor model - is central in asset pricing, and yet poses a number of issues. Available testing procedures are often marred by low power or poor size control, hinge on strong assumptions on the data generating process (DGP), and generally involve the estimation or inversion of (often large) covariance matrices giglio2022factor. \newline In this paper, we propose a general methodology to test for \textquotedblleft zero alpha\textquotedblright\ - that is the null that for each test asset, on average, realized excess returns equal those implied by a linear factor pricing model. In order to make the presentation easier to follow, we mainly focus on the following linear factor pricing model with tradable and observable factors

where: $y_{i,t}$ is the excess return on the $i$-th security at time $t$; $ f_{t}$ is a $K$-dimensional vector of pricing factors; $ {\Greekmath 010C} _{i}$ is a $K$-dimensional vector of factor loadings; and $u_{i,t}$ is a zero mean error term. Equation ((ref)) is the workhorse model employed in asset pricing, and it encompasses several popular specifications such as the Capital Asset Pricing Model (CAPM; see sharpe1964capital), and the three- and five-factors models (fama1993common; and fama2015five).\footnote{In Sections (ref) and (ref) in the Supplement, we study the extension to the cases of non-tradable and of latent factors respectively.} In the context of ((ref)), we propose a novel approach to test for the null hypothesis that all the ${\Greekmath 010B} _{i}$s are jointly equal to zero versus the alternative that at least one ${\Greekmath 010B} _{i}$ is nonzero, viz.

We base our analysis on the equivalent version of ((ref))

Testing for ((ref)) is interesting per se. In the context of asset pricing, ((ref)) implies $\mathbb{E} \left( y_{i,t}\right) ={\Greekmath 010B} _{i}+{\Greekmath 010C} _{i}^{\prime }\mathbb{E}\left( f_{t}\right) $; hence, ${\Greekmath 010B} _{i}$ represents the excess return on the $i$-th test asset not explained by the $K$ factors. When no relevant pricing factor is omitted from (ref), a non-zero ${\Greekmath 010B}_i$ can still appear due, for example, to market rigidities that prevent security prices from reaching their theoretical fair values and that are not allowed for in the factor pricing model. As these securities are misspriced, this ${\Greekmath 010B}_{i}$ is often called pricing error. However, a non-zero ${\Greekmath 010B} _{i}$ can originate from multiple other sources of model misspecification, such as the omission of one or more pricing factors and/or non-linear pricing relations. Hence, depending on the nature of model misspecification, ${\Greekmath 010B} _{i}\neq0$ is not necessarily a pure pricing error but rather a signal that the model in (1.1) is misspecified from an asset pricing perspective.\footnote{In the case of testing for alpha in fund returns, rather than security returns, alpha can also capture skill of the fund manager giglio2021thousands.}

Arguably, the first contribution to propose a test for \textquotedblleft zero alpha\textquotedblright\ in linear pricing models is the paper by gibbons1989test, where an $F$-test is proposed for the joint null hypothesis that ${\Greekmath 010B} _{1}={\Greekmath 010B} _{2}=...={\Greekmath 010B} _{N}=0$. Such an approach is entirely natural, but - as well as needing several restrictions on the error terms $u_{i,t}$ - it requires the restriction that $N$ is fixed with $ N<T$, because the test uses a (consistent) estimate of the $N\times N$ covariance matrix of the regression residuals, which subsequently needs to be inverted. Hence, the case $N>T$, often encountered in applied work, cannot be addressed with the GRS test. fan2015power, gagliardini2016time and pesaran2023testing propose several solutions towards this technical difficulty by developing average-type alpha tests - i.e.\ tests where individual statistics on each of the ${\Greekmath 010B} _{i}$ s are averaged - for the joint asymptotics case $\min \left\{ N,T\right\} \rightarrow \infty $. All these tests can deal with cross-sectional dependence among the errors (in essence, allowing for cross-sectional dependence as long as a Central Limit Theory holds for cross-sectional averages). However, their implementation still hinges on estimating a large dimensional, $N\times N$ covariance matrix, which requires several restrictions on the covariance structure of the errors, and which can become computationally intensive when certain estimators are considered (this is e.g.\ the case for the threshold estimator of bickel2008regularized, used by gagliardini2016time). Further, all the tests mentioned above are based on the assumption of serially independent errors, which is bound to cause problems in the presence of conditional heteroskedasticity, and the approaches by fan2015power and gagliardini2016time further require Gaussian errors. In a recent contribution, feng2022high propose a max-type test for the null that $\max_{1\leq i\leq N}\left\vert {\Greekmath 010B} _{i}\right\vert =0$; their test is based on using the maximal t-statistic, and therefore its asymptotics hinges on an Extreme-Value-type argument, rather than a \textquotedblleft central\textquotedblright\ one like average-type tests. As we also discuss in greater detail in Section (ref), their asymptotics requires weak cross-sectional dependence (and serial independence). ardia2024robust test the null hypothesis in ((ref)) whether all alphas are zero by combining $p$-values from asset-specific tests using the Cauchy combination approach proposed by liu2020cauchy. While their test can be applied under quite general assumptions, its implementation hinges on several tuning parameters which significantly impact the outcome of the procedure. Finally, chernov2025test directly generalize the GRS test to the high-dimensional case by considering a ridge-regularized estimator of the $ N\times N$ covariance matrix of the residuals, still resting on similar assumptions to those of the average-type tests discussed above.\footnote{ Other relevant references include gungor2016multivariate, ma2020testing, and raponi2020testing. Further, the correct specification of linear factor pricing models can also be assessed by testing whether they imply a pricing kernel with zero Hansen-Jagannathan distance (see e.g., hansen1997assessing, hodrick2001evaluating, and carrasco2022hansen).}

Testing methodology and the contribution of this paper

In this paper, we fill the gaps mentioned above, by proposing a test for ( (ref)) which allows for the errors to have: (i) (weak) serial dependence, including conditional heteroskedasticity such as volatility clustering, or the \textquotedblleft leverage effect\textquotedblright\ ( black1976studies); (ii) non-Gaussianity, such as skewness and excess kurtosis, indeed relaxing some of the moment assumptions in the aforementioned papers; (iii) strong cross-sectional dependence; and also (iv) allowing for $N>T$. Importantly, as we further explain in Section (ref), our test statistic only requires a consistent estimator (and its rate of convergence) for the individual ${\Greekmath 010B} _{i}$s, whereas no second order property (such as asymptotic Gaussianity, or asymptotic efficiency) is needed. This allows to ignore the cross-sectional structure of the data, and therefore the test does not require the estimation of any $N\times N$ covariance matrices, and needs virtually no tuning. \\ The technical details are described in the next sections; here we offer a preview of the main arguments, which are based on the construction of a randomized test statistic. In order to construct the test statistic, we estimate the ${\Greekmath 010B} _{i}$s from $N$ separate time series regressions, unit by unit: at no stage do we require joint estimation. Although any consistent estimator can be employed, here we use OLS, obtaining, say, $\widehat{{\Greekmath 010B} }_{i}$. Under our assumptions, it can be expected that $\widehat{{\Greekmath 010B} }_{i}$ will converge to ${\Greekmath 010B} _{i}$ at a rate $T^{-1/2}$. We then pre-multiply each $\left\vert \widehat{{\Greekmath 010B} } _{i}\right\vert $ by a function of $T$ which diverges as $T\rightarrow \infty $, but at a rate slower than $O\left( T^{1/2}\right)$. Hence, we obtain $N$ statistics which drift to zero when ${\Greekmath 010B} _{i}=0$, and diverge to positive infinity whenever ${\Greekmath 010B} _{i}\neq 0$. We then perturb the resulting $N$ statistics by adding to each of them a $ \mathcal{N}\left( 0,1\right) $ shock, with the $N$ shocks forming an i.i.d. sequence. As a consequence, we obtain an $N$-dimensional sequence which, under $\mathbb{H}_{0}$, is (roughly) \textit{i.i.d.}$\mathcal{N} \left( 0,1\right) $ conditionally on the sample. Finally, we take the largest of these perturbed statistics as a test statistic for $\mathbb{H}_{0} $ in ((ref)): conditionally on the sample, this is distributed as a Gumbel under $\mathbb{H}_{0}$, whereas it diverges under the alternative that at least one asset is mis-priced. This methodology has, at least conceptually, some similarities with the one proposed in fan2015power , where a sequence is added to a test statistic, constructed so as to drift to zero under the null (thus introducing no distortion in the asymptotics under the null), and diverging under the alternative (so as to boost the power). However, unlike fan2015power, we do not need to estimate at any stage the asymptotic covariance matrix of the estimated $\mathbf{{\Greekmath 010B} } =\left( {\Greekmath 010B} _{1},...,{\Greekmath 010B} _{N}\right) ^{\prime }$, or define a high-dimensional weight matrix: only the individual, unit-by-unit, estimates of the ${\Greekmath 010B} _{i}$s are required. Moreover, we only require a rate of convergence for the estimate $\widehat{{\Greekmath 010B} }_{i}$. Hence, the assumptions on serial dependence and moment existence can be relatively mild.

To the best of our knowledge, our contribution is the first application to asset pricing of tests based on randomizing a function of the data, i.e.\ a statistic. While novel to this setting, this type of randomized tests have been used in econometrics and statistics, particularly where a limiting distribution is unavailable or non-pivotal, or where its derivation requires excessively restrictive assumptions; although a comprehensive literature review goes beyond the scope of this paper, we refer to corradi2006 for a first application of these randomized tests in econometrics, and to the paper by he2023one for references.\footnote{We emphasize that this paper is not, in general, the first alpha testing procedure based on a randomization. The literature has considered bootstrap-based tests for alpha; see, for example, sullivan1999data, white2000reality, kosowski2006can and fama2010luck. While bootstrap and randomization are both based on an added source of randomness, their core statistical mechanisms are very different, and we extensively discuss similarities and differences between bootstrap and randomized tests in Section (ref). }

We make at least four contributions to the current literature. First, whilst we focus on the specific case of \textquotedblleft testing for alpha\textquotedblright , we propose a novel methodology to construct tests involving a growing number of parameters with no need for joint estimation and, in essence, no need to take the dimensionality of the problem into account. In a similar spirit, although in ((ref)) we assume that the common factors $f_{t}$ are the same across all units, our approach can be readily extended e.g.\ to the case where pricing factors are heterogeneous across assets. Indeed, the factor structure could be so heterogeneous that no factor influences all the assets, which makes our testing procedure robust to the presence of weak and semi-strong pricing factors, and we do not need any factor to be \textquotedblleft strong\textquotedblright\ in our model.\footnote{ Following chudik2011weak, the $k$-th pricing factor is strong if ${\Greekmath 010C} _{i,k}\neq 0$ for all $i=1,\dots ,N$; and it is weak (resp. semi-strong) when ${\Greekmath 010C} _{i,k}\neq 0$ for $\lfloor N^{{\Greekmath 010D} }\rfloor $ of the assets with $0<{\Greekmath 010D} <1/2$ (resp. $1/2\leq {\Greekmath 010D} <1)$. } Second, we relax several technical conditions required in the extant literature, allowing for serial and cross-sectional dependence, conditional heteroskedasticity, thicker tails and a larger $N/T$ ratio than allowed for in other contributions. Third, we enhance the randomized test by proposing a decision rule which shows excellent control for Type I and Type II errors in simulations. Fourth and last, our methodology is flexible and can be readily extended to other frameworks.

The remainder of the paper is organized as follows. We discuss our set-up and assumptions in Section (ref). The hypotheses of interest and the randomized testing approach are discussed in Section (ref); we offer a heuristic description of the core statistical mechanism of our methodology, and its relationship with other randomization methods, in Section (ref); we report the derandomized decision rule in Section (ref); and, in Section (ref), we offer guidelines for the practical implementation of our procedure. Simulations are in Section (ref), while Section (ref) contains an empirical illustration. Conclusions and further lines of research are in Section (ref). Further Monte Carlo and empirical evidence, extensions, technical lemmas and proofs are in the Supplement.

NOTATION. We define the probability space $\left( \Omega ,\boldsymbol{B}, \mathbb{P}\right) $, where $\Omega $ is the sample space with elements $ {\Greekmath 0121} \in \Omega $, $\boldsymbol{B}$ the space of events, and $\mathbb{P} $ the probability function. Given a random variable $X$, $\mathbb{E}\left( X\right) $ is the mean and $\mathcal{V} \left( X\right) $ is the covariance viz. $\mathcal{V}\left( X\right) = \mathbb{E}\left[ \left( X-\mathbb{E}\left( X\right) \right) \left( X-\mathbb{ E}\left( X\right) \right) ^{\prime }\right] $; further, given $r>0$, we denote the $\mathcal{L}_{r}$-norm of $X$ as $\left\vert X\right\vert _{r}=\left( \mathbb{E}\left\vert X\right\vert ^{r}\right) ^{1/r}$. The indicator function of a set $\mathcal{A}$ is denoted as $\mathbb{I}\left( \mathcal{A}\right) $. We use: \textquotedblleft a.s\textquotedblright\ for \textquotedblleft almost sure(ly)\textquotedblright ; \textquotedblleft $ \rightarrow $\textquotedblright\ to denote the ordinary limit; and \textquotedblleft $\overset{a.s.}{\rightarrow }$\textquotedblright\ to denote almost sure convergence. Orders of magnitude for an almost surely convergent sequence with multiparameter index $\Pi $, say $s_{\Pi }$ are denoted as $O_{a.s.}\left( b_{\Pi }\right) $\ and $o_{a.s.}\left( b_{\Pi }\right) $\ when, respectively, $\mathbb{P}\left( \limsup_{\Pi \rightarrow \infty }\left\vert b_{\Pi }^{-1}s_{\Pi }\right\vert <\infty \right) $ $=$ $1$ , and $b_{\Pi }^{-1}s_{\Pi }\overset{a.s.}{\rightarrow }0$, as $\Pi \rightarrow \infty $. Positive, finite constants are denoted as $c_{0}$, $ c_{1}$, ... and their value may change from line to line. Other, relevant notation is introduced later on in the paper.

Model and assumptions

Recall our workhorse model ((ref))

We begin with a definition of weak dependence which we use throughout the paper.

Decomposable Bernoulli shifts (ibragimov1962some) are a convenient way to model stationary, dependent time series. Virtually all the most common DGPs in econometrics and statistics satisfy Definition (ref). liu2009strong provide various theoretical results, and numerous examples including ARMA-GARCH sequences, and other nonlinear time series models (e.g. Random Coefficient AutoRegressive and threshold models).

We are now ready to present our assumptions.

Assumptions (ref) and (ref) allow for (weak) serial dependence in $u_{i,t}$ and $f_{t}$, e.g. due to nonlinear phenomena such as conditional heteroskedasticity. In contrast, the tests by fan2015power , gagliardini2016time, feng2022high and pesaran2023testing all assume independence over time of $u_{i,t}$, thus being unable to accommodate idiosyncratic conditional heteroskedasticity in asset returns. The assumptions imply that the unconditional variances of errors and pricing factors are constant over time, similarly to Assumptions A1 and A2 in feng2022high. However, as we also discuss in Section (ref), we can extend our set-up to the case of unconditional heteroskedasticity, allowing for different regimes along similar lines as Assumption 2.2 in horvath2025detecting.\footnote{That is, assuming $ \left\{ f_{t},1\leq t\leq T\right\} =\bigcup_{\ell =1}^{q}\left\{ f_{\ell ,t},t_{\ell -1}<t\leq t_{\ell }\right\} $ with $t_{0}=1$\ and $ t_{q}=T$\, and $\left\{ f_{\ell ,t},-\infty <t<\infty \right\} $\ satisfying Assumption (ref) for each segment $1\leq \ell \leq q$\ (and similarly for $u_{i,t}$).} Our assumptions also require error terms to have four finite moments, as opposed to the Gaussianity assumption in fan2015power. Similarly, Assumption (ref) is substantially milder than the sub-exponential tails constraint on $f_t$ in fan2015power and feng2022high.\\ Assumption (ref) also allows for more general forms of cross-sectional dependence in the errors, compared with the extant literature. For instance, existing max-type tests such as the one by feng2022high require some restrictions on the covariance matrix of the error term.\footnote{In particular, Assumption 3 in feng2022high implies that the covariance matrix of the error term be invertible and that cross-covariances are absolutely summable. In turn, this is required in order for the individual t-statistics associated with the ${\Greekmath 010B}_i$s to be weakly correlated, thus being able to apply standard Extreme Value Theory when deriving the asymptotic distribution of their maximum.} We do not impose any such requirement and, in principle, our methodology can also deal with strong cross-sectional dependence in $u_{i,t}$ as long as the maintained zero-mean assumption is satisfied. However, in the context of asset pricing such a case requires attention. Indeed, a possible cause of cross-sectional dependence could be the omission of strong or semi-strong factors, which then show up in the error term $u_{i,t}$. If these omitted tradable factors are priced - thus having a non-zero risk-premium (mean) - then ${\Greekmath 010B}_i$ captures both the pure pricing error and the risk premium of the omitted factors, regardless of their correlation with the factors included in the model giglio2021thousands. However, ${\Greekmath 010B}_i$ remains a pure pricing error when there are {\textquotedblleft time series factors\textquotedblright} in $u_{i,t}$, that is tradable factors $g_t$ which have zero mean $\mathbb{E}\left( g_{t}\right) =0$ -- and hence do not contribute to the pricing of the assets -- and are orthogonal to $f_t$, $\mathbb{E}\left( f_{t}g_{t}^{\prime }\right) =0$. Omitting $g_t$ results in $u_{i,t}={\Greekmath 010D} _{i}^{\prime }g_{t}+{\Greekmath 0118} _{i,t}$, where ${\Greekmath 0118} _{i,t}$ is a purely idiosyncratic component. This, in turn, renders all the existing procedures invalid, but it is allowed under our Assumptions (ref) - (ref).

Assumption (ref) is a weak exogeneity requirement, less restrictive than the independence assumption in fan2015power, feng2022high, and pesaran2023testing.

The test

Recall the null hypothesis of ((ref)), i.e. $ \mathbb{H}_{0}:\max_{1\leq i\leq N}\left\vert {\Greekmath 010B} _{i}\right\vert =0$. As mentioned in the introduction, any consistent estimator of ${\Greekmath 010B}_i$ could be employed to construct our test statistics. Here, we focus on the unit-by-unit OLS estimator,\footnote{We note that the $1 \leq i \leq N$ equations in ((ref)) share the same set of regressors; hence, the OLS estimator coincides with a system-based (Feasible) GLS estimation.} viz.

for $\overline{y}_{i}=T^{-1}\sum_{t=1}^{T}y_{i,t}$, $\overline{f} =T^{-1}\sum_{t=1}^{T}f_{t} $, and $ \widehat{{\Greekmath 010C} }_{i,T}=\left[ \sum_{t=1}^{T}\left( f_{t}-\overline{f}\right) \left( f_{t}-\overline{f}\right) ^{\prime }\right] ^{-1}\left[ \sum_{t=1}^{T}\left( f_{t}-\overline{f}\right) y_{i,t}\right] $. Recall that the data admit at least ${\Greekmath 0117} \geq 4$ moments. We use the transformation

where the rescaling sequence is based on the squared OLS residuals $\widehat{ u}_{i,t}^{2}$:

As we only work with rates of convergence -- requiring ${\Greekmath 0120} _{i,NT}$ to drift to zero under the null and diverge under the alternative -- in principle any rescaling sequence which removes the measurement unit from $\widehat{{\Greekmath 010B} }_{i,T}$ can be used in ((ref)). For example, one may use an estimate of the variance of the individual $\widehat{{\Greekmath 010B} } _{i,T}$, and indeed such estimate does not even need to be consistent. We propose $\widehat{s}_{NT}$ because the cross-sectional averaging smooths away any potentially problematic behavior such as the presence of spikes in the unit specific variances; this choice also turns out to deliver the best performance in simulations, and to help in the empirics (see the discussion in Section (ref)).

We now explain the construction of the test statistic. Heuristically, under our assumptions it should hold that $\widehat{ {\Greekmath 010B} }_{i,T}-{\Greekmath 010B} _{i}=O_{a.s.}\left( T^{-1/2}\right) $. Hence, under the null of ((ref)), ${\Greekmath 0120} _{i,NT} \overset{a.s.}{\rightarrow }0$ by construction; conversely, under the alternative, ${\Greekmath 0120} _{i,NT}\overset{a.s.}{\rightarrow }\infty $. In order to have a statistic to test for $\mathbb{H}_{0}$, we now perturb the ${\Greekmath 0120} _{i,NT}$s by adding a sequence of i.i.d. Gaussian variables

where ${\Greekmath 0121} _{i}\overset{i.i.d.}{\sim }\mathcal{N}\left( 0,1\right) $, generated independently of the sample $\left\{ \left( u_{i,t},f_{t}^{\prime }\right) ^{\prime },1\leq i\leq N,1\leq t\leq T\right\} $. Thus, under the null $z_{i,NT}$ should be an i.i.d. sequence of standard normals; under the alternative, there should be (at least) one spike due to the fact that ${\Greekmath 010B} _{i}\neq 0$ for some $i$. Hence, we base our test on the maximally selected $z_{i,NT}$:

In order to study the asymptotics of $Z_{N,T}$, let

and consider the following restriction:

Assumption (ref) poses a constraint on the relative rate of divergence of $N$ and $T$ as they pass to infinity: the more moments the data admit, the larger $N$ can be relative to $T$. A comparison with the similar Assumption A1(iii) in feng2022high may shed further light: if, similarly to feng2022high, we assumed independence between $f_{t}$ and $u_{i,t}$, then Assumption (ref) would become $N=O\left( T^{{\Greekmath 0117} /2-1-{\Greekmath 0122} }\right) $ for some (arbitrarily small) ${\Greekmath 0122} >0$, which coincides with Assumption A1(iii) in feng2022high. Similarly, the asymptotics in pesaran2023testing requires $N=o\left( T^{2}\right) $, under the assumptions of deterministic regressors and at least eight finite moments for the errors. In our case, as long as ${\Greekmath 0117} \geq 6$, the condition that $ N=o\left( T^{2}\right) $ is satisfied, and therefore we have either the same asymptotic regime with a milder moment condition, or, with the same moment condition, a larger $N$ relative to $T$. In Section (ref) in the Supplement, we discuss the possibility of relaxing -- given ${\Greekmath 0117} $ -- Assumption (ref) to allow for a broader set of combinations of $ N $ and $T$.

Let $\mathbb{P}^{\ast }$ denote the probability conditional on the sample $\left\{ \left( u_{i,t},f_{t}^{\prime }\right) ^{\prime },1\leq i\leq N,1\leq t\leq T\right\} $.

Theorem (ref) describes the limiting behavior of the test statistic $ Z_{N,T}$ both under the null and under the alternative hypotheses. By ((ref)) the suitably normed version of $Z_{N,T}$ converges (in distribution, a.s.\ conditionally on the sample) to a Gumbel distribution.\footnote{In Section (ref) in the Supplement, we also propose an alternative, fixed $N$ version of the critical values. As discussed in Section (ref), these alternative critical values return better finite sample results when $T$ grows larger for a fixed $N$.} Equation ((ref)) implies that asymptotic critical values at nominal level ${\Greekmath 011C} $ are given by

Similarly, ((ref)) roughly states that under the alternative (the suitably normed version of) $Z_{N,T}$ diverges to positive infinity in probability, a.s.\ conditional on the sample.\footnote{Following the proof of the theorem, it can be readily shown that a sufficient condition to have asymptotic unit power is that, as $\min \{N,T\} \rightarrow \infty$, it holds that $\sqrt{T/\log N} \max_{1 \leq i \leq N} |{\Greekmath 010B}_i| \rightarrow \infty$. This entails that our test has power as long as one alpha is (mildly) larger than zero. The same result can be shown to hold, a fortiori, in the case of a \textquotedblleft small\textquotedblright , pervasive alternative whereby $|{\Greekmath 010B}_i|=|{\Greekmath 010B}|$ for all $i$, with $\sqrt{T/\log N}|{\Greekmath 010B}| \rightarrow \infty$ - a case known in the literature as the \textquotedblleft diffuse alpha\textquotedblright\ case (chernov2025test).}

Discussion

We now discuss how and why randomization works in our case, and what relationship it has with other approaches based on \textquotedblleft adding randomness\textquotedblright\ such as the bootstrap. As zhang2023randomization put it, \textquotedblleft [t]he meaning of randomization tests has become obscure in statistics education and practice over the last century \textquotedblright\ (p.\ 2928). Hence, some clarifications on the core statistical mechanism underpinning our randomized test are in order. As we expound hereafter, the main feature of our approach is that our randomization is based on adding randomness to the test statistic, rather than to the data.

Our approach works as follows. To start, we construct the statistic ${\Greekmath 0120} _{i,NT}$ defined in ((ref)), based on the data, as in any \textquotedblleft traditional\textquotedblright\ testing approach. However, we do not require its second order properties (that is, its limiting distribution and/or asymptotic efficiency), and use only its first order properties (that is, its rate of convergence). Being able to focus only on rates requires simpler arguments and milder assumptions; furthermore, at no stage do we require the estimation of asymptotic variances.\footnote{Indeed, this entails that our approach, by its very nature, places more emphasis on the robustness of the estimator employed.} In the construction of ${\Greekmath 0120}_{i,NT}$, we pre-multiply $\widehat{{\Greekmath 010B} }_{i}$\ by the scaling factor $T^{1/{\Greekmath 0117} }$, which - heuristically - is designed to \textquotedblleft wash out\textquotedblright\ the estimation error $\widehat{{\Greekmath 010B}}_i-{\Greekmath 010B}_i$, and therefore the randomness coming from the data. Our theory uses almost sure rates for the statistic ${\Greekmath 0120} _{i,NT}$; the results in Lemma (ref) in the Supplement yield that, for each $i$

so that, in our proofs, we can work with the premise that (under the null) $\lim_{\min \left\{ N,T\right\} \rightarrow \infty}{\Greekmath 0120} _{i,NT}=0$.\footnote{Similarly, the theory also entails that, for each $i$, $\mathbb{P}\left( {\Greekmath 0121} :\lim_{\min \left\{ N,T\right\} \rightarrow \infty }{\Greekmath 0120} _{i,NT}=\infty \right) =1\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, under }\mathbb{H}_{A}$; hence, in our proofs, we can work with the premise that (under the alternative) $\lim_{\min \left\{ N,T\right\} \rightarrow \infty }{\Greekmath 0120} _{i,NT}=\infty$. } Seeing as the randomness of the data has been washed out by pre-multiplying $\widehat{{\Greekmath 010B} }_{i}$\ by $T^{1/{\Greekmath 0117} }$, in order to construct a test, we add randomness in the form of (i.i.d. and Gaussian) noise to each ${\Greekmath 0120} _{i,NT}$, thus constructing the sequence $ \left\{ z_{i,NT},1\leq i\leq N\right\} $\ defined in ((ref)). By ( (ref)), we can derive the limiting law of $Z_{N,T} = \max_{1\leq i\leq N}z_{i,NT}$ , and - crucially - show that this depends solely on the probability measure of the added randomness. Given that this is the only driver of the asymptotic behavior of $Z_{N,T}$, and that it is fully under the researcher's control, it can be studied straightforwardly, with no need for further assumptions on the data apart from the ones required to derive the rates of ${\Greekmath 0120} _{i,NT}$. This is the key argument whereby our test can be employed, among other things, under arbitrary levels of cross-sectional dependence: the only cross-sectional dependence that matters for the asymptotics of $Z_{N,T}$ is the one in the added random noise - which can be forced to equal zero by the researcher. Considering as a leading term of comparison the max-type test by feng2022high, in that case the asymptotic behavior of the test statistic is determined by the probability measure of the data, and therefore it is affected by the presence and extent of cross-sectional dependence across $i$. By the same token, in ((ref)) we do not require a consistent estimator of the variance of the estimated ${\Greekmath 010B}_i$, but merely a rescaling factor such as $\widehat{s}_{NT}$, designed to make ${\Greekmath 0120}_{i,NT}$ adimensional. An immediate consequence is that - as mentioned in Section (ref) - in our set-up we can readily allow for unconditional heteroskedasticity in both factors and errors. Seeing as we rely only on rates of convergence for $\widehat{{\Greekmath 010B}}_i$, and no estimation of the asymptotic variance of the estimators is required, our results would hold even in this case, without requiring any modifications or even prior knowledge as to the presence of heteroskedasticity. Conversely, feng2022high use a sequence of t-statistics across $i$, and thus a consistent estimator of the variance of $\widehat{{\Greekmath 010B}}_i$ is required - which, especially in the presence of heteroskedasticity, is well known to be fraught with difficulties (see e.g. the review, and the proposed solution, in casini2024fixed).

Further light on our approach can be shed by comparing it with approaches where the randomness is added to the data, a prime example being the bootstrap. The mode of convergence in Theorem (ref) - where, under the null, $ Z_{N,T}$ converges \textquotedblleft in distribution, almost surely conditional on the sample\textquotedblright\ - is the same as one would find in the case of the bootstrap (bickelfreedman). Notwithstanding this analogy, the way in which randomness is added, the way in which the theory works, and the assumptions required on the data are profoundly different to the randomization method proposed herein. In the bootstrap, randomness is added by resampling the data multiple times, and constructing a (pseudo) version of the test statistic at each resampling: the randomness of the data is not washed away. Hence, the asymptotic behavior of the resampled statistic is still affected by the features of the data - e.g., by serial and/or cross-sectional dependence.\footnote{E.g. in a \textquotedblleft traditional\textquotedblright\ resampling scheme, given data $\left\{ y_{i},1\leq i\leq n\right\} $, at each iteration $ 1\leq b\leq B$ the pseudosample $\left\{ y_{i,b}^{\ast },1\leq i\leq n\right\} $ is constructed such that $\mathbb{P}\left( y_{i,b}^{\ast }=y_{j}|\left\{ y_{i},1\leq i\leq n\right\} \right) =1/n$. Hence, the (conditional) law of $y_{i,b}^{\ast }$ is $\mathbb{P}_{n}\left( y\right) =n^{-1}\sum_{i=1}^{n}\mathbb{I}\left( y_{i}\leq y\right)$, where $\mathbb{I}\left( \cdot \right) $ denotes the indicator function, which clearly depends on the features of the data $\left\{ y_{i},1\leq i\leq n\right\} $.} Furthermore, a typical way of proving the validity of the bootstrap is to show that the distribution of the resampled test statistic, conditional on the sample, converges in some sense (e.g., in distribution a.s. conditional on the sample) to the asymptotic distribution of the original test statistic.

This, however, requires deriving such asymptotic distribution, which is likely to be more complicated and to require stronger assumptions than simply deriving its convergence rate.\footnote{Similar considerations also hold for other approaches based on adding randomness to the data. For example, the randomized tests studied in canay2017randomization require that the distribution of the data be \textquotedblleft approximately symmetric\textquotedblright\ - that is, invariant under certain transformations. Such shape restrictions are not required by our approach. } Moreover, applying the bootstrap in our context is fraught with difficulties: Huang2023 show that the approaches proposed by kosowski2006can and fama2010luck may suffer from (even severe) undersizing and low power, especially when the data exhibit features that are typical of financial returns (e.g., skewed unconditional distributions and large cross-sectional sizes). The bootstrap corrections suggested in Huang2023 ameliorate these issues, but still require weak cross-sectional dependence. \\ Finally, a crucial difference between our approach and approaches based on adding randomness to the data is that, in the latter case, the added randomness vanishes in the limit, and thus it does not affect the limiting behavior of the resulting test statistic. Conversely, the randomness added in our method does not vanish asymptotically, which is a well-known feature of this type of randomized tests (see e.g. corradi2006). We propose a solution in the next section.

Derandomized inference

The discussion above indicates that the results in Theorem (ref) are different to \textquotedblleft standard\textquotedblright\ inferential theory. In particular, ((ref) ) entails $\lim_{\min \left\{ N,T\right\} \rightarrow \infty }$ $\mathbb{P}^{\ast }\left( Z_{N,T}\geq c_{{\Greekmath 011C} }|\mathbb{H}_{A}\right) =1$, which corresponds to the notion of power. The result under the null is more delicate: whilst it holds that $\lim_{\min \left\{ N,T\right\} \rightarrow \infty }\mathbb{P}^{\ast }\left( Z_{N,T}\geq c_{{\Greekmath 011C} }|\mathbb{H}_{0}\right) ={\Greekmath 011C}$, this result is not the standard notion of size. The fact that the added randomness in the construction of $Z_{N,T}$ does not vanish asymptotically entails that, under the null, different researchers using the same data will obtain different values of $Z_{N,T}$, and thus different p-values. \newline We propose a decision rule to discern between $\mathbb{H}_{0}$ and $ \mathbb{H}_{A}$ which is not driven by the added randomness, and is therefore the same across all researchers using the same dataset. Following HT2019, each researcher will compute $Z_{N,T}$ over $B$ replications, at each replication $1\leq b\leq B$ constructing a statistic $ Z_{N,T}^{\left( b\right) }$ using a random sequence ${\Greekmath 0121} _{i}^{\left( b\right) }\overset{i.i.d.}{\sim }\mathcal{N}\left( 0,1\right) $ for $1\leq i\leq N$, independent across $1\leq b\leq B$ and of the sample. Let

be the percentage of times that the researcher does not reject the null at nominal significance level ${\Greekmath 011C} $. An immediate consequence of Theorem (ref) is that, as $\min \left\{ N,T,B\right\} \rightarrow \infty$

for almost all realizations of $\left\{ \left( u_{i,t},f_{t}^{\prime }\right) ^{\prime },1\leq i\leq N,1\leq t\leq T\right\} $. This result holds for all different researchers, and therefore averaging across the replications $1\leq b\leq B$ removes the added randomness in $ Q_{N,T,B}\left( {\Greekmath 011C} \right) $: hence, all researchers will obtain the same value of $Q_{N,T,B}\left( {\Greekmath 011C} \right) $. As noted in he2023one, $Q_{N,T,B}\left( {\Greekmath 011C} \right) $ corresponds to (the complement to one of) the \textquotedblleft fuzzy decision\textquotedblright\ in equation (1.1a) in geyer. This notion can be illustrated by considering a random variable, say $\mathcal{D}$, which takes two values: \textquotedblleft do not reject $\mathbb{H}_{0}$\textquotedblright \thinspace\ with probability $Q_{N,T,B}({\Greekmath 011C} )$, and \textquotedblleft reject $\mathbb{H}_{0}$\textquotedblright . According to ((ref)), asymptotically it holds that, a.s. conditionally on the sample $\mathbb{P} ^{\ast }\left( {\Greekmath 0121} :\mathcal{D}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textquotedblleft reject }\mathbb{H} _{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textquotedblright }|\mathbb{H}_{0}\right) ={\Greekmath 011C} $, across all researchers, which reconciles the procedure with the notion of size of a test. Similarly, ((ref)) states that, asymptotically, $\mathbb{P} ^{\ast }\left( {\Greekmath 0121} :\mathcal{D}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textquotedblleft reject }\mathbb{H} _{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textquotedblright }|\mathbb{H}_{A}\right) =1$ a.s. conditionally on the sample, which corresponds to the notion of power of a test.

Building on Theorem (ref), a \textquotedblleft derandomized\textquotedblright\ decision rule can be proposed. By ((ref) ), under $\mathbb{H}_{0}$ there exists a triplet of random variables $\left( N_{0},T_{0},B_{0}\right) $ such that

for all $\left( N,T,B\right) $ with $N\geq N_{0}$, $T\geq T_{0}$, and $B\geq B_{0}$. Similarly, under $\mathbb{H}_{A}$ there exists a triplet of random variables $\left( N_{0},T_{0},B_{0}\right) $ such that $Q_{N,T,B}\left( {\Greekmath 011C} \right) \leq {\Greekmath 010F} $, for all ${\Greekmath 010F} >0$ and $\left( N,T,B\right) $ with $N\geq N_{0}$, $T\geq T_{0}$, and $B\geq B_{0}$. This dichotomous behavior can be exploited to construct a decision rule based on $ Q_{N,T,B}\left( {\Greekmath 011C} \right) $, in a way that is more akin to information criteria than tests: $\mathbb{H}_{0}$ is not rejected when $ Q_{N,T,B}\left( {\Greekmath 011C} \right) $ exceeds a threshold, whereas it is rejected otherwise. In theory, one could use the threshold based on the Law of the Iterated Logarithm (LIL) in ((ref)); under $\mathbb{H}_0$, it follows that $\mathbb{P}^{*}\left[ Q_{N,T,B} < \ell_{{\Greekmath 011C}} \right]=0$ as $\min\{N,T,B\} \rightarrow \infty$. Albeit valid asymptotically, this criterion turns out to be biased against $\mathbb{H}_0$ in simulations, especially in small samples; this is not entirely surprising, since the bound induced by the LIL is not likely to \textquotedblleft bite\textquotedblright\ unless $B$ is large (which, in light of the restriction $B=O((\log N)^2)$, requires $N$ to be \textquotedblleft very large\textquotedblright). A decision rule that is more favorable towards the null could be

with $f\left( B\right) $ a user-specified, non-increasing function of $B$ such that $\lim_{B\rightarrow \infty }f\left( B\right) =0$, and $\limsup_{B\rightarrow \infty }\left( f\left( B\right) \right) ^{-1}\sqrt{ 2 \log \log B /B}=0$.

From theory to practice: guidelines and recommendations

The procedure proposed in Section (ref) depends on the nuisance parameter ${\Greekmath 0117}$ and on the tuning quantities $B$ and $f\left( B\right) $, i.e.\ the number of trials and the threshold in the derandomized approach. We offer a set of guidelines/suggestions which could inform the practical application of these procedures.\\

There are at least two ways in which ${\Greekmath 0117} $ in ((ref)) can be determined:

In addition --- as we do in our empirical illustration --- one can use ${\Greekmath 0117}_0 =4$ when constructing ${\Greekmath 0120}_{i,NT}$, i.e. the smallest finite moment prescribed by our theory, which we would recommend when the sample size $T$ does not afford reliable inference.

Turning to the specifics of the derandomization, we note that:

Simulations

We use a similar DGP to feng2022high:

where $f_{t}=(f_{1,t},f_{2,t},f_{3,t})^{\prime }$, $\overline{f} =(0.53,0.19,0.19)^{\prime }$, $\Phi =\mathrm{diag}\left\{ -0.1,0.2,-0.2\right\} $ and ${\Greekmath 0110} _{t}\overset{i.i.d.}{\sim }\mathcal{N} _{3}(0,I_{3})$. Loadings are generated as ${\Greekmath 010C} _{i,1}\overset{i.i.d.}{\sim }\mathcal{U}(0.3,1.8)$, ${\Greekmath 010C} _{i,2}\overset{i.i.d.}{\sim }\mathcal{U} (-1,1) $, and ${\Greekmath 010C} _{i,3}\overset{i.i.d.}{\sim }\mathcal{U}(-0.6,0.9)$ for all $i$. We allow for strong cross-sectional dependence in the innovations $ \mathbf{u}_{t}=(u_{1,t},\dots ,u_{N,t})^{\prime }$ via a factor model:

where $\boldsymbol{{\Greekmath 010D} }=({\Greekmath 010D} _{1},\dots ,{\Greekmath 010D} _{N})^{\prime }$ for $ {\Greekmath 010D} _{i}\overset{i.i.d.}{\sim }\mathcal{U}(0.7,0.9)$, ${\Greekmath 011E} _{g}=0.4$, and ${\Greekmath 011F} _{t}\overset{i.i.d.}{\sim }\mathcal{N}(0,1)$, with $\left\{ {\Greekmath 011F} _{t},1\leq t\leq T\right\} $ generated independently of $\left\{ {\Greekmath 0110} _{t},1\leq t\leq T\right\} $. In ((ref)), the $N$-dimensional random vectors $\left\{ \boldsymbol{{\Greekmath 0118} } _{t},1\leq t\leq T\right\} $ are generated independently of $\left\{ \left({\Greekmath 0110} _{t}, {\Greekmath 011F}_t\right),1\leq t\leq T\right\} $ under the following three set-ups (all with mean zero and covariance matrix $\Sigma _{{\Greekmath 0118} }$):

In all scenarios, we report rejection frequencies under the null and under the alternative, for nominal level ${\Greekmath 011C} =5\%$, using $N\in \left\{ 100,200,500\right\} $ and $T\in \left\{ 100,200,300,500, 1000, 2000\right\} $. As far as the alternative hypothesis is concerned, we consider a rather sparse alternative where ${\Greekmath 010B} _{i}\overset{i.i.d.}{\sim }\mathcal{N}(0,1)$ for $5\%$ percent of the cross-sectional units $1\leq i\leq N$. We construct the test statistic using a notional value of ${\Greekmath 0117} =5$ in ((ref)), and consider both the \textquotedblleft one-shot\textquotedblright\ test in Section (ref), and the derandomized version in Section (ref) . For the latter, we examine results using both (ref) and (ref) with $f(B)=B^{-1/4}$. We compare our test with the tests by feng2022high, pesaran2023testing, and ardia2024robust.\footnote{We set tuning parameters of the AS test as suggested in their Monte Carlo exercise (in particular, using their notation, we set $L=0$ and ${\Greekmath 0120} =1/3$). When applicable, i.e.\ when $N<T$, we also check performances of the test by gibbons1989test Moreover, Monte Carlo results in feng2022high and pesaran2023testing show that their tests consistently outperform that of gungor2016multivariate. Hence, we omit comparisons with this last approach. The comparison with the approaches of fan2015power and gagliardini2016time is reported in Section (ref) of the Supplement.}

We start from the Gaussian case; results using the one-shot test are in Table (ref), whereas in Table (ref) we report rejection frequencies from the derandomization approach. Our test is the best one at controlling the size for all combinations of $N$ and $T$, whereas the other tests are consistently oversized. Our test becomes slightly undersized, and subsequently plateaus, as $T$ increases. This is not accompanied by any loss of power; further, the alternative critical values presented in Section (ref) of the Supplement yield empirical rejection frequencies that are extremely close to the nominal size when $T\geq 500$. Table (ref) suggest that the derandomization procedure based on $f(B)=B^{-1/4}$ also works very well. Indeed, as predicted by the theory, the empirical rejection frequencies are extremely close to zero under the null and quickly converge to one under the alternative; the latter is particularly true when $N$ gets large, thus showing that our approach is particularly suitable when $N>T$. As expected, the threshold based on the LIL leads to higher empirical rejection frequencies under both the null and the alternative. As far as power is concerned, the right panels of the table show that our tests performs satisfactorily, whilst at the same time guaranteeing size control. We note that the test by FLLM outperforms ours in terms of power in most cases, but it is also oversized. Turning to the case of data with heavier tails, Tables (ref) and (ref) report empirical rejection frequencies for the Student's $t$ case. Results for the randomized test are in line with those of Table (ref). As in the Gaussian case, the other tests are oversized, while ours becomes slighlty undersized when $T$ grows large. Again, this can be solved by using the alternative critical values from Section (ref). The derandomized procedure works as expected also in the case of heavier tails. Similar considerations hold for power as for the Gaussian case. Finally, results under the GARCH case are in Tables (ref) and (ref); size and power of all tests behave as in the other cases, and so does the decision rule based on Theorem (ref). Our test is slighlty oversized when $N=500$ and $T=100$, but still outperforms all the others when sizes and powers are considered. The use of the alternative critical values still improves the (slight) under-rejection for large values of $T$. Notably, GRS performs very well whenever applicable, exhibiting good finite sample properties even when the DGP violates its assumption of i.i.d. Gaussian errors. However, the fact that the test requires $T>N$ makes it inapplicable whenever one deals with a rather large number of test assets.\footnote{For instance, using the GRS with monthly data on $N=200$ test assets implicitly assumes that assets' ${\Greekmath 010C}$s are constant over more than 16 years. This is at odds with all the available empirical findings.}

We conclude this section with a further experiment, where we allow for cross-sectional heteroskedasticity. We simulate assets with higher variance, the higher ${\Greekmath 010B}_i$. The tendency of severely misspriced assets to exhibit large variances is a clear feature of the data employed in our empirical analysis (see Section (ref)), especially for the most severely misspriced assets. Hence, we consider the case where ${\Greekmath 011B}^{2}_{{\Greekmath 0118}_i} = Var\left({\Greekmath 0118}_{i,t}\right)$. In the Gaussian and Student's $t$ DGP, we simulate $\bf{{\Greekmath 0118}}_t$ so that ${\Greekmath 011B}_{{\Greekmath 0118}_i} = 1+{\Greekmath 0123}|{\Greekmath 010B}_{i}|$, while for the GARCH case set ${\Greekmath 0121}_{i} = (1+{\Greekmath 0123}|{\Greekmath 010B}_{i}|)(1-{\Greekmath 010C}_i-{\Greekmath 0119}_i)$ so that $Var\left({\Greekmath 0118}_{i,t}\right) = \left(1+{\Greekmath 0123}|{\Greekmath 010B}_{i}|\right)^2$ under all DGPs, and we consider ten equally spaced values of ${\Greekmath 0123}$ between one and five percent. Results are in Figure (ref), showing that our test has unit power for all levels of asset specific variance. Conversely, the power of all the other tests decreases, with the test by feng2022high exhibiting the largest loss of power under all DGPs. This makes sense since the test by feng2022high is a max-$t^{2}$ test, hence being driven by both the alpha and the asset-specific variance.\footnote{Further results are in Section (ref) of the Supplement, where we report results for: the case ${\Greekmath 011E} _{g}=0$ (Section (ref)); the cases where $g_{t}$ is a weak or semi-strong omitted factor (Section (ref)); additional power analyses for different levels of sparsity under the alternative (Section (ref)); the case when only one factor is strong, while the others are at most semi-strong (Section (ref)); the cases of non-tradable factors (studied in Section (ref)) and latent factors (studied in Section (ref)) (Sections (ref) and (ref) respectively); using alternative critical values in the rejection rule based on Theorem (ref) (Section (ref)).}

Empirical illustration

We illustrate our procedure by testing whether several linear factor pricing models correctly price the constituents of the S{&}P 500 index. We collect monthly data on all stocks that were part of the S&P 500 for at least five years between January 1985 and December 2024, using simple percentage returns on the $i$-th stock gross of dividend yield

where $P_{i,t}$ is the end of the month stock price and $DY_{i,t}$ is the percent per annum dividend yield.\footnote{ To ensure that the index accurately represents the US stock market, S{&}P regularly updates its composition. We account for these changes by revising the set of included assets every month. Security data are sourced from Datastream, while we obtain those on the pricing factors (and on the risk-free rate) from the website of Professor French.} We define the excess returns as $y_{i,t}$, and test six linear factor pricing models that are all encompassed by the following regression

where: $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{MKT}_{t}$ is the excess return on the market, $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{SMB}_{t}$ the size factor, $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{HML}_{t}$ the book-to-market factor, $ \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{RMW}_{t}$ the profitability factor, $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{CMA}_{t}$ the investment strategy factor, and $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{MOM}_{t}$ the momentum factor. \footnote{ All factors are observable and tradable; see the website of Professor French for their description.} Both the excess return $y_{i,t}$ and the market factor use the one-month Treasury bill rate as risk-free rate. The first model that we test is the CAPM, which is obtained when only the market factor is considered. We then consider a two-factor (FF2 henceforth) model based on the market and momentum factors, as well as the usual three- and five-factors models of fama1993common and fama2015five, which we also augment with momentum to obtain a four- (FF4) and a six-factors (FF6) model. In line with fan2015power and pesaran2023testing, we mitigate the impact of possible time variation in the factor loadings by running our inferential procedure on 5-years rolling windows.\footnote{We use $T=60$ (i.e. $ 5 $ years) in estimation, while $N$ ranges between $437$ and $568$. Within each window, we discard securities with at least one missing observation. In Section (ref) in the Supplement, we show that using the derandomized confidence function as suggested in Section (ref) allows to control for the family-wise rejection frequency even when the number of windows passes to infinity, with no changes or corrections required to the nominal level of the individual tests.}

Results

Table (ref) reports the percentage of windows for which we can reject the null that a pricing model is correctly specified. These empirical rejection frequencies are computed using our derandomized procedure based on $B=(\log N)^2$, at $5\%$ nominal significance level. We calculate ${\Greekmath 0120} _{i,NT}$ using both ${\Greekmath 0117}=5$ (in the light of the very good finite sample results found in Section (ref)), and ${\Greekmath 0117}=4$ (by way of robustness check, as suggested in our guidelines in Section (ref)). We also report results obtained with the test by feng2022high, as our closest benchmark.

The first sub-panel of Table (ref) shows rejection frequencies for the whole sample. Results suggest that the most parsimonious models - the CAPM and FF2 - are the least rejected models (i.e.\ the ones that produce the lowest percentage of rejections of the null of zero alphas).\footnote{This result is somewhat surprising in that, if the true model was a larger model that nests the CAPM or FF2, then underconditioning would entail a bias in the alpha estimates of the more parsimonious models. However, moskowitz2025 show that, in the absence of mispricing, the CAPM performs better in terms of alpha tests than several prominent multifactor models. This is because, while multifactor betas can help capture mispricing, persistence in those betas leads the multifactor models to distort expected returns well after that information gets priced correctly. Also, it is well known that individual stocks have poorly estimated betas, with the estimation error acting both as a source of downward bias due to an errors-in-variables problem blume1975betas and as a driver of the aforementioned persistence. However, ang2020using argue that, although the measurement errors in the betas are larger in individual stocks relative to e.g. portfolio returns, the larger cross-sectional spread in the betas of individual stocks more than offsets this error, leading to a more accurate estimate of the market risk premium.} Results using the approach of feng2022high also generally favor the use of more parsimonious models, i.e.\ CAPM and FF2. However, the rejection frequencies appear unrealistically small, as they suggest that a linear model based on market and momentum correctly prices large cap US stocks for almost 90% of the sample, which comprises several periods of market turmoil.

The next four sub-panels of Table (ref) show rejection frequencies over four periods of market turmoil: the Asian financial crisis, the Burst of the Dot-com Bubble, the Global Financial Crisis, and the COVID-19 pandemic.\footnote{ For the COVID-19 pandemic, we consider rolling windows between January 2020 and May 2021, where the end date corresponds to the termination of lockdown policies in most of the world. The dates for the other periods of market turmoil are set as in pesaran2023testing.} Our procedure suggests that these models almost never price large cap US stocks during the Asian and the Global Financial Crisis. The picture is only slightly more positive during the early 2000s crisis. Finally, we see that the size and book-to-market factors play an important role during the COVID-19 pandemic, as the three-factor model is the best over this period.\footnote{ To interpret this result note that, while the COVID-19 period shares with other crises the fact that the stock market yielded low returns and was characterized by high volatility and low liquidity, it also has features that make it very distinct. Indeed, during the COVID-19 pandemic households were required to stay home to slow the spread of the virus and a variety of firms were severely restricted in producing their goods and services, essentially constraining output production and limiting consumption decisions. This distinctive feature of the COVID-19 period implies that, while uncertainty about the end of the pandemic was very high, in the short term recession fears and low growth expectations strongly characterize that period gormsen2023financial. It is not surprising, therefore, that value stocks --- generally considered long-horizon investments --- came under huge pressure as economic uncertainty prompted investors to shorten their time horizons, and indeed we find that the value factor contributes to the slightly better performance of larger models during this period (further results are available upon request). This is consistent with the evidence in campbell2025drives, who document that value experienced a historically striking drop in performance during the pandemic. }

This sub-periods analysis also indicates that some of the results using the approach of feng2022high may be overly optimistic. Indeed, their test suggests that the FF2 model perfectly explains the cross-section of expected excess returns on large cap US stocks during the Dot-com Bubble and the COVID pandemic. More generally, and also looking at the last two sub-panels, their procedure tells us that FF2 almost always prices all large cap US stocks, no matter whether the market is experiencing a period of distress or stability. While possible, this result does not seem plausible, and we argue that it is likely due to the loss of power suffered by the test of feng2022high when the asset-specific variance of the misspriced assets increases, as highlighted in the earlier simulation section. We study this presumption starting from Figure (ref). Its upper panels present the time-series of the three largest estimated alphas for model FF2 (each point of the series represents the absolute value of the largest estimate of ${\Greekmath 010B}_i$ for a given window). We see that these are maximal between January 1999 and June 2009, which is the period corresponding to the red shaded areas. The lower panels report the time series of asset-specific standard deviation for the assets whose estimated alpha is reported in panels (a) - (c). These standard deviations are strongly associated with the estimated alphas during the highlighted period, suggesting that, over this period, we are in a setting which is very similar to that of our Monte Carlo results in Figure (ref), where we would expect the test by feng2022high to display low power. Indeed, their test rejects the FF2 model for 14% of the windows between January 1999 and June 2006, while our procedure based on ${\Greekmath 0117}=4$ (${\Greekmath 0117}=5$, respectively) rejects 92% (74%, respectively) of the times. The results in Table (ref) and Figure (ref) suggest a similar conclusion when looking at the Asian financial crisis and the outbreak of the COVID pandemic. These empirical findings highlight the usefulness of our testing procedure particularly when asset specific variances are highly associated with the size of absolute pricing errors, as seems to be the case for US stocks during periods of market turmoil.

Conclusions

We propose a novel, general methodology to test for \textquotedblleft zero alpha\textquotedblright\ in linear factor pricing models, with observable or latent, tradable or non-tradable pricing factors. Our proposed test relies on a randomization scheme and can be used even when residuals exhibit conditional heteroskedasticity, strong cross-sectional dependence and non-Gaussianity, and allows for the number of assets, $N$, to pass to infinity, also at a faster rate than the sample size $T$. Extensive Monte Carlo analysis shows that the test has very good power properties, and that it is - compared to other extant approaches - the one that better controls the Type I Error probability in all scenarios.

We discuss two possible extensions which are under investigation by the authors. First, building on the theory developed herein, a randomized test could be developed to test for the null hypotheses $\mathbb{H} _{0,i}\,:\,{\Greekmath 010B} _{i}=0$ for $i=1,\dots ,N$, while controlling for multiple testing.\footnote{By the same token, the test statistic could be constructed also when the equality is replaced by an inequality, such as $\mathbb{H}_{0,i}\,:\,{\Greekmath 010B} _{i}\leq 0$, as in giglio2021thousands.} Indeed, the individual test statistics would be perturbed by adding randomness independently across $i$, thus making the randomized statistics (conditionally) independent across $i$, which would facilitate the application of customarily employed procedures for size control under multiple testing. This would find a natural application in testing mutual/hedge-funds performances, as well as assessing trading strategies. \\ A second extension/application of randomized tests based on rates of convergence is model comparison. Consider two models for $y_{i,t}$, which will be denoted henceforth using the superscripts $^{\left( 1\right) }$ and $^{\left( 2\right) }$, with ${\Greekmath 010B} _{i}^{\left( 1\right) } $ and ${\Greekmath 010B} _{i}^{\left( 2\right) } $ respectively. The two models can be compared to check whether ${\Greekmath 010B} _{i}^{\left( 1\right) } $ and ${\Greekmath 010B} _{i}^{\left( 2\right) } $ differ in some sense; in the spirit of the max-type test proposed in this paper, one can test whether the maximally selected difference of absolute alphas is zero, viz.

Under ((ref)), the worst-case scenario discrepancy between the alphas of the two models is zero, indicating a comparable performance. A test can be constructed - adapting our methodology - by defining $ {\Greekmath 0120} _{i,NT}^{\left( 1,2\right) }=\left\vert T^{1/{\Greekmath 0117} }\left[ \left\vert \widehat{{\Greekmath 010B} }_{i}^{\left( 1\right) }\right\vert -\left\vert \widehat{ {\Greekmath 010B} }_{i}^{\left( 2\right) }\right\vert \right] /\widehat{s}_{NT}^{\left( 1,2\right) }\right\vert ^{{\Greekmath 0117} /2}, $ where $\widehat{s}_{NT}^{\left( 1,2\right) }$ denotes a scaling factor. \footnote{ Even in this case, any scaling factor which is bounded away from zero and infinity, and which renders the estimators $\widehat{{\Greekmath 010B} }_{i}^{\left( 1\right) }$ and $\widehat{{\Greekmath 010B} }_{i}^{\left( 2\right) }$ scale free, can be used e.g., $ \widehat{s}_{NT}^{\left( 1,2\right) }=\sqrt{\sum_{i=1}^{N}\left( \left( \widehat{u}_{i,t}^{\left( 1\right) }\right) ^{2}+\left( \widehat{u} _{i,t}^{\left( 2\right) }\right) ^{2}\right) /NT,}$ where $\widehat{u}_{i,t}^{\left( 1\right) }$ and $\widehat{u}_{i,t}^{\left( 2\right) }$\ are the residuals from models $^{\left( 1\right) }$ and $ ^{\left( 2\right) }$ respectively, by adapting the definition of $\widehat{s} _{NT}$ in ((ref)).} It is immediate to see that, under the null, ${\Greekmath 0120} _{i,NT}^{\left( 1,2\right) }=o_{a.s.}\left( 1\right) $, whereas it would diverge at a rate $T^{1/2}$ as long as $\left\vert {\Greekmath 010B} _{i}^{\left( 1\right) }\right\vert \neq \left\vert {\Greekmath 010B} _{i}^{\left( 2\right) }\right\vert $\ for at least one $i$. Hence, our approach can be applied verbatim to ${\Greekmath 0120} _{i,NT}^{\left( 1,2\right) }$, perturbing each $ {\Greekmath 0120} _{i,NT}^{\left( 1,2\right) }$ by an i.i.d. standard Gaussian shock ${\Greekmath 0121} _{i}$ and using $ Z_{NT}^{\left( 1,2\right) }=\max_{1\leq i\leq N}\left\vert {\Greekmath 0120} _{i,NT}^{\left( 1,2\right) }+{\Greekmath 0121} _{i}\right\vert$. This test could be applied to compare two non-nested models, but also to compare a nested and a nesting model - in the latter case, upon not rejecting $\mathbb{H}_{0}$ of ((ref)), the applied user can conclude that the nested, \textquotedblleft smaller\textquotedblright\ model does not result in any worsening. Building on the previous paragraph, model comparison may also be approached as a multiple testing problem in the cross-section. In particular, we may consider the $N$-dimensional sequence of pair-wise null hypotheses $\left\{\mathbb{H}^{(m,j)}_{0,1},\dots,\mathbb{H}_{0,N}^{(m,j)}\right\}$ with generic entry $ \mathbb{H}^{(m,j)}_{0,i}\,:\, |{\Greekmath 010B}_{i,m}| = |{\Greekmath 010B}_{i,j}| $, and test it against the one-sided alternative $|{\Greekmath 010B}_{i,j}|>|{\Greekmath 010B}_{i,m}|$. \\ In conclusion, we would like to note that our approach - whilst focused here on the issue of testing for alpha in asset pricing models - can be exported to other set-ups. Using essentially the same arguments as in this paper, for example, one could test for the two sample problem in high dimension, or for the equality of intercept and slope in a high-dimensional regression model, or for \textquotedblleft pooling versus not pooling\textquotedblright\ in a large panel data model. In all cases, our approach would be applicable even in the presence of a whole world of misspecification for the errors - thus, being a viable alternative to other, more standard, methodologies when these fail due to their underlying assumptions not being satisfied or due to the need for a robust estimator of the long-run covariance, or for a large covariance matrix estimation.

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Further simulations

This section extends the Monte Carlo analysis of the main body by studying the finite sample performances of the test based on Theorem (ref) under different data generating processes. Specifically, Section (ref) displays empirical rejection frequencies for other competing approaches that were not included in the main body; Sections (ref) and (ref) modify the persistence and factor structure for the error terms of the pricing models $u_{i,t}$, while Section (ref) discusses the implications of changes in the strength of the pricing factors $ f_{t}=(f_{1,t},f_{2,t},f_{3,t})^{\prime }$. Section (ref) studies how the power of the tests varies as a function of the percentage of assets that are misspriced under the alternative. Sections (ref) and (ref) present finite results when pricing factors are non-tradable and latent, respectively. Section (ref) contains size and power results for the test in Theorem (ref) using alternative critical values.

Further competing tests

Tables (ref) to (ref) report empirical rejection frequencies for the tests of fan2015power, gagliardini2016time under the DGPs of the main body.\footnote{ The test of FLY is based on estimating the covariance matrix of $u_{t}$ with the POET method of fan2013large. As in fan2015power, we consider the soft thresholding function; results based on hard and SCAD thresholding fan2001variable are numerically identical and available upon request.} Here and in what follows, we do not report results for sample sizes $T=1000$ and $T=2000$ as the computational cost of both approaches was excessively demanding. Finally, for completeness, we also repeat the results of our testing procedure based on Theorem (ref). As for the average-type test of pesaran2023testing, these approaches substantially over-reject the null under $\mathbb{H}_0$. Estimation of a large dimensional covariance matrix in the presence of strongly cross-sectionally correlated and mildly persistent residuals drives these over-rejections.

No persistence in the omitted factor

We complement the results of Section (ref) under ${\Greekmath 011E} _{g}=0$ in ((ref)) - that is, the omitted (strong) common factor has no persistence.

Tables (ref), (ref) and (ref) contain the empirical rejection frequencies under the null (left panels) and the alternative (right panels) for our one-shot test. For average type tests (FLY, GOS and PY) results are in line with those of the main body, though the degree of over-rejection under $\mathbb{H}_{0}$ is smaller. The $p$-value combination of AS is still mildly oversized, while the max-type approach of FLLM performs on par with ours.

Results for the \textquotedblleft strong\textquotedblright\ decision rule of Theorem (ref) are in Tables (ref), (ref) and (ref). As desired, empirical rejection frequencies quickly converge to zero under the null and for all cases considered. As in the main body, convergence is faster when using $ f(B)=B^{-1/4}$. Similarly good results hold under the alternative, where the rejection frequencies always converge to one.

Different strengths of the omitted factor

We consider the same three-factor pricing model of Section (ref) . In the main body, ${\Greekmath 010D} _{i}\overset{i.i.d.}{\sim }\mathcal{U}(0.7,0.9)$ which implies that ${\Greekmath 010D} _{i}>0$ for any cross-sectional unit. We now depart from that assumption in two ways: first, we consider the case where only $\lfloor N^{0.4}\rfloor $ assets have a non-zero loading on $g_{t}$; secondly, we look at a situation where $\lfloor N^{0.8}\rfloor $ cross-sectional units have a non-zero loading on $g_{t}$. These two experiments correspond to the case of a weak and of a semi-strong omitted pricing factors, respectively. Notably, the largest eigenvalue of the covariance matrix of $\mathbf{u}_{t}$ is bounded in the weak factor case, while it diverges to infinity in the semi-strong one. We consider the same sample sizes and alternative hypothesis as in Section (ref).

Empirical rejection frequencies for the weak factor case are in Table (ref). Results for our test and for those of feng2022high and fan2015power are very similar to those in the main body (Table (ref)) both under both the null and the alternative hypothesis. Actual sizes of the GOS and PY tests are much closer to the nominal one, suggesting that strong-cross sectional dependence in the residuals was the driver of their overrejections. Their powers are substantially unaltered. The test of ardia2024robust performs well in terms of size, but still exhibits a lack of power when $T$ is small. All previous considerations hold unchanged with respect to the value of ${\Greekmath 011E}_g$.

Table (ref) shows results for the semi-strong case. We would like to point out the importance of this data generating process, as bailey2021measurement described tens of semi-strong pricing factor for the cross-section of US excess returns. Empirical rejection frequencies are very close to those of the main body, as the FLY, GOS, PY, and FLLM tests all become oversized when ${\Greekmath 011E}_g = 0.4$. This is most likely an effect of strong cross-sectional dependence in the residuals, as implied by a diverging eigenvalue in their covariance matrix. Results on the test by of ardia2024robust are equivalent to those of Tables (ref) and (ref).

Different levels of sparsity under the alternative

In this section, we consider power under twelve different alternatives, each based on a different percentage of mis-priced assets: $1\%,$ $2\%,$ $3\%,$ $\dots ,$ $9\%,$ $10\%,$ $15\%$ and $20\%$. For brevity, we only focus on sample sizes $T=100$ and $N=500$, which are the most relevant for the empirical analysis of Section (ref). Figure (ref) reports the empirical rejection frequencies for the twelve levels of sparsity (the horizontal axis reports the percentage of mis-priced assets) across all DGPs. The upper panels consider ${\Greekmath 011E}_g = 0$, while the case ${\Greekmath 011E}_g = 0.40$ is in the lower ones. Tests by FLY and FLLM always achieve unit power, while our approach performs almost equally well (and better than the other tests), as its power converges to one almost immediately.

Strong and semi-strong pricing factors

We now consider the same data generating process of Section (ref) but assuming that the omitted factor $g_{t}$ is strong. This time, however, we assume that ${\Greekmath 010C} _{i,2}={\Greekmath 010C} _{i,3}=0$ for $N-\lfloor N^{0.8}\rfloor $ randomly chosen assets (cf. bailey2021measurement, who found that only the market factor is strong, while other $140$ factors are at most semi-strong). Empirical rejection frequencies for this data generating process are reported in Table (ref). Results on all tests are substantially equivalent to those of Table (ref), thus validating results of the main body also in the case where only one pricing factor is strong. This holds true irrespectively of the value of ${\Greekmath 011E} _{g }$.

Non-tradable factors

We now study the finite sample properties of the testing procedure described in Theorem (ref). To do it, we consider a three-factor pricing model similar to that of Section (ref) but with no factor structure in $u_{i,t}$, viz.

where we have constructed the DGP in terms of $v_{t}$ for coherence with Section (ref) (see equation (ref), in particular). Similarly to the main body, $\mathbf{u}_{t}=\left( u_{1,t},\dots ,u_{N,t}\right) ^{\prime }$ follows one of the following three specifications:

Values of $\Phi$, ${\Greekmath 010C} = \left({\Greekmath 010C}_1,{\Greekmath 010C}_2,{\Greekmath 010C}_3\right)^{\prime}$, and of GARCH parameters are set as in Section (ref), while $ {\Greekmath 0115}_p \overset{i.i.d.}{\sim}\mathcal{U}(0,1/2)$ for $p=1,2,3$.

Table (ref) reports empirical rejection frequencies under the null (left panels) and under the alternative (right panel) for the test in Theorem (ref). As in Section (ref), we set the nominal size to ${\Greekmath 011C} = 5\%$ and study powers for the same significance level. Because our test is the only one with a fully-fledged asymptotic theory for non-tradabale factors, we do not report rejection frequencies for other approaches. Our procedure satisfactorily controls the size for any sample size and specification of the residuals $u_{i,t}$. Empirical powers are consistently one. Table (ref) presents rejection frequencies for a derandomized procedure similar to that of Theorem (ref) but based on Theorem (ref) rather than (ref). No matter the residuals' properties, empirical rejection frequencies always converge to zero as the sample size increases. As for the analysis in Section (ref), this convergence is much faster using critical values based on $f(B) = B^{-1/4}$.

Latent factors

We now investigate the finite sample properties of the test in Theorem (ref) when considering the DGP of Section (ref) with unobserved factors. Again, size and power are studied for nominal level ${\Greekmath 011C} = 5\%$. Results for empirical rejections frequencies under the null (left panels) and under the alternative (right panels) are reported in Table (ref). Again, the testing procedure exhibits satisfactory size control and excellent power properties. The test is a bit oversized in the GARCH case, particularly when $T=100$. Given the results on the other estimators (standard OLS and Fama-MacBeth), this over-rejection seem to be due to problems in PCA-based estimation of factors and loadings under this particular GARCH DGP. Similar conclusions hold when we look at results on the derandomized procedure in Table (ref), where we see that the approach works fine up to some over-rejection of the null in the GARCH case.

Alternative critical values

The results in Section (ref) suggest that a test based on $c_{\Greekmath 011C}$ becomes slightly undersized as $T$ grows large (for any given $N$). In this section, we consider an alternative, fixed $N$ family of critical values

These critical values are theoretically supported by the fact that the Gumbel cumulative distribution function appears as the limit of the term $\Phi^{N}\left(a_{n}x +b_{n}\right)$ in the proof of Theorem (ref). Hence, these Gaussian quantiles area a natural finite sample counterpart to the Gumbel ones.

Results for the DGPs of the main body, even when ${\Greekmath 011E}_g = 0$, are in Table (ref). Empirical rejection frequencies are always larger than those based on asymptotic critical values. This is beneficial when $T$ gets larger, where asymptotic critical values returned empirical sizes lower than the nominal one (i.e.\ 5%). Notably, large $T$ results (say, $T$ at most 300), are now on par with, if not better than, those for the GRS approach. Hence, we suggest using our test with asymptotic critical values when $T$ is small. When $T$ is larger, the user should employ either our test with adjusted critical values or, if the sample size permits it, the GRS one.

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Extensions

As mentioned in the introduction to the main paper, our main contribution is a methodology to test for no pricing errors, and we have primarily focused on ((ref)) and assumed factors are observable and tradable only for simplicity.

In this extension, we show that our methods can be readily extended to more complex settings. In essence, we show that - as long as a consistent estimator of the ${\Greekmath 010B} _{i}$s is available - our methodology can be applied even to the case of latent or non-tradable, obtaining the same results as in the case of observable and tradable ones. As illustrative examples, we consider Fama-MacBeth estimation for the case of observable but non-tradable factors (Section (ref)), and principal component analysis (PCA) estimation for latent factors (Section (ref)); in both cases, we directly use the estimation techniques proposed by giglio2021thousands for the ${\Greekmath 010B} _{i}$s. We only report the main results on the \textquotedblleft one shot\textquotedblright\ tests; the extension to derandomization can be done by following verbatim Section (ref). We note, however, that strong factors are required in these cases. \\ We also consider the extension of the derandomized decision rule in the case of multiple testing (Section (ref)), and of Assumption (ref) (Section (ref)).

Henceforth, we define $\mathbb{M}_{1_{N}}=\mathbb{I}_{N}-N^{-1}\mathbf{{\Greekmath 0113} }_{N}\mathbf{{\Greekmath 0113} }_{N}^{\prime }$, where $\mathbf{{\Greekmath 0113} }_{N}$ is an $ N\times 1$ vector of ones.

Non-tradable factors and Fama-MacBeth estimation

Consider first the case of a linear factor pricing model based on $K$ observable, non-tradable factors

where $v_{t}=f_{t}-\mathbb{E}\left( f_{t}\right) $ and ${\Greekmath 0115} \in \mathbb{R }^{K}$ is the vector of risk premia for the $K$ factors $f_{t}$.

Estimation of ${\Greekmath 010B} _{i}$ is based on Algorithm 3 in giglio2021thousands.

Based on $\widehat{{\Greekmath 010B} }_{i}^{FM}$ defined in ((ref)), we can construct the same test statistic as before, based on

where the rescaling sequence $\widehat{s}_{NT}^{FM}$ is constructed as in ( (ref)), using the residuals $\widehat{u}_{i,t}^{FM}=y_{i,t}-\left( \widehat{{\Greekmath 010B} }_{i}^{FM}+\widehat{{\Greekmath 010C} }_{i}^{\prime }f_{t}\right) $. Defining $z_{i,NT}^{FM}={\Greekmath 0120} _{i,NT}^{FM}+{\Greekmath 0121} _{i}$,\footnote{ As before, ${\Greekmath 0121} _{i}\overset{i.i.d.}{\sim }\mathcal{N}\left( 0,1\right) $ generated independently of the sample $\left\{ \left( u_{i,t},f_{t}^{\prime }\right) ^{\prime },1\leq i\leq N,1\leq t\leq T\right\} $.} our test can be based on

In order to derive our asymptotic theory, we consider the following assumptions. Let

where we define

It holds that

Latent factors

Consider now a linear factor pricing model based on $K$ latent factors

where $v_{t}=f_{t}-\mathbb{E}\left( f_{t}\right) $ is not observable , ${\Greekmath 010C} _{i}$ is a $K\times 1$ vector of loadings and, as above, ${\Greekmath 0115} $ is the vector of risk premia for the $K$ latent factors $f_{t}$.

Write $\widetilde{\mathbf{y}}_{t}=\mathbf{{\Greekmath 010C} }\widetilde{v}_{t}+ \widetilde{\mathbf{u}}_{t}$, where $\widetilde{\mathbf{y}}_{t}=\mathbf{y} _{t}-\overline{\mathbf{y}}$ with $\mathbf{y}_{t}=\left( y_{1,t},...,y_{N,t}\right) ^{\prime }$, $\widetilde{v}_{t}=v_{t}-\left( T^{-1}\sum_{t=1}^{T}v_{t}\right) $, $\widetilde{\mathbf{u}}_{t}$ is defined analogously, and $\mathbf{{\Greekmath 010C} }=\left( {\Greekmath 010C} _{1},...,{\Greekmath 010C} _{N}\right) ^{\prime }$. Define also the $N\times N$ sample second moment matrix

The estimation of ${\Greekmath 010B} _{i}$ follows Algorithm 4 in giglio2021thousands.

Let $C_{N,T}=\min \left\{ N,T\right\} $. Based on $\widehat{{\Greekmath 010B} }_{i}^{PC} $ defined in ((ref)), we define

where $\widehat{s}_{NT}^{PC}$ is constructed as in ((ref)), using $ \widehat{u}_{i,t}^{PC}=y_{i,t}-\left( \widehat{{\Greekmath 010B} }_{i}^{PC}+\widehat{ {\Greekmath 010C} }_{i}^{PC\prime }\widehat{f}_{t}^{PC}\right) $ and $\widehat{f} _{t}^{PC}=N^{-1}\widehat{\mathbf{{\Greekmath 010C} }}^{PC\prime }\widetilde{\mathbf{y}} _{t}$. Letting $z_{i,NT}^{PC}={\Greekmath 0120} _{i,NT}^{PC}+{\Greekmath 0121} _{i},$ with ${\Greekmath 0121} _{i}$ defined as above, the test is based on

We consider the following assumptions - which complement and extend the assumptions in the main paper - for the case of latent factors.

It holds that

We conjecture that Theorem (ref) holds with minor modifications to Assumptions (ref)-(ref) when one estimates factors and loadings with the Risk-Premium PCA (RP-PCA) approach of lettau2020estimating, lettau2020factors. In fact, just like our Theorem (ref), their theory for the strong factors case relies on conditions that are extremely similar to those of bai03. Similar considerations also hold for the Projected PCA approach of fan2016projected, which is increasingly being used in asset pricing studies (see kim2021arbitrage and hong2025dynamic, among others).

On multiple testing and the derandomized confidence function

We consider the use of the derandomized rule discussed in Section (ref) in the main paper in the presence of multiple testing. Whilst this extension is specifically designed to address the presence of multiple tests carried out across multiple windows as in Section (ref), it can be extended beyond this specific case.

Consider the case where the test is repeated across $1\leq v\leq W$ windows, whose data may overlap fully, partly, or not at all, for the null hypotheses that \[ \mathbb{H}_{0}^{\left( v\right) }:\max_{1\leq i\leq N}\left\vert {\Greekmath 010B} _{i}^{\left( v\right) }\right\vert =0, \] where ${\Greekmath 010B} _{i}^{\left( v\right) }$ is the intercept estimated for unit $i $ with the dataset pertaining to window $v$. A natural question is how frequently - in the case whereby $\mathbb{H}_{0}^{\left( v\right) }$ is satisfied across all $1\leq v\leq W$ - can one null $\mathbb{H}_{0}^{\left( v\right) }$ be rejected. This question corresponds to the idea of family-wise size control.

Suppose that, at each window $v$, the test is carried out using the randomized confidence function defined in equation ((ref)), which we denote with the short-hand notation $Q_{v}\left( {\Greekmath 011C} \right) $. At each $v$, the randomness added to the statistic ${\Greekmath 0120} _{i,NT}^{\left( v\right) }$, say $ {\Greekmath 0121} _{i}^{\left( b\right) ,\left( v\right) }$, is independent across $ 1\leq b\leq B$ and across $v$. Then, conditional on the whole sample, $ Q_{v}\left( {\Greekmath 011C} \right) $ is independent across $v$. Our question can now be formalised by calculating \[ \mathbb{P}^{\ast }\left[ \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{reject any null hypothesis}|\left\{ \mathbb{H} _{0}^{\left( v\right) }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ is true, }1\leq v\leq W\right\} \right] . \] The following theorem stipulates that, in essence, family-wise rejection probability is controlled, as long as there are not too many windows.

The theorem states that family-wise size control is guaranteed, with the probability of rejecting any null tending to zero. This requires the condition in ((ref)), which is stated generally, and in which, for example, one could choose a sample size-adjusted nominal level ${\Greekmath 011C} $, akin to a Bonferroni condition. On the other hand, keeping ${\Greekmath 011C} $ fixed, as we do in Section (ref), and using, as suggested in Section (ref), $f\left( B\right) =B^{-1/4}$, ((ref)) would be satisfied as long as \[ \log W-B^{1/2}\rightarrow -\infty ; \] seeing as we recommend using $B=\left\lfloor \left( \log N\right) ^{2}\right\rfloor $, the above would be equivalent to \[ W=o\left( N\right) . \]

Extension of the asymptotic regime in Assumption (ref)

Here and henceforth, the Euclidean norm of a vector is denoted as $ \left\Vert \mathbf{\cdot }\right\Vert $. Given an $m\times n$ matrix $ \mathbf{A}$ with element $a_{ij}$\ we use the following notation for its norms: $\left\Vert \mathbf{A}\right\Vert $ is the Euclidean/spectral norm, defined as $\left\Vert \mathbf{A}\right\Vert \leq \sqrt{{\Greekmath 0115} _{\max }\left( \mathbf{A}^{\prime }\mathbf{A}\right) }$; $\left\Vert \mathbf{A} \right\Vert_{F}$ is the Frobenious norm; $\left\Vert \mathbf{A} \right\Vert_{1}$ is the $\mathcal{L}_{1}$-norm defined as $\left\Vert \mathbf{A}\right\Vert _{1}=\max_{1\leq j\leq n}\sum_{i=1}^{m}\left\vert a_{ij}\right\vert $; the $\mathcal{L}_{\infty }$-norm$\ \left\Vert \mathbf{A} \right\Vert _{\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }}$is defined as $\left\Vert \mathbf{A} \right\Vert _{\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }}=\max_{1\leq i\leq m}\sum_{j=1}^{n}\left\vert a_{ij}\right\vert $.

We discuss how to extend the asymptotic regime required in Assumption (ref). In particular, we show how we can further relax the assumption to $N=O\left( T^{{\Greekmath 0117} /4-{\Greekmath 0122} }\right) $, which allows for larger values of $N$ compared to Assumption (ref), and to the corresponding Assumption A1(iii) in feng2022high. In such a case, we would need to redefine ${\Greekmath 0120} _{i,NT}$ as

where ${\Greekmath 010E} $ is a user-chosen quantity such that

Equation ((ref)) does not suggest a decision rule per se, but only an upper bound for ${\Greekmath 010E} $, which is a tuning parameter. The rationale underpinning ((ref)) is based on the fact that - upon inspecting the proofs of Theorem (ref) and Lemma (ref) - we require that, under the null, $\sum_{i=1}^{N}{\Greekmath 0120} _{i,NT}=o_{a.s.}\left( 1\right) $. In turn, this follows as long as $N\left\vert T^{{\Greekmath 010E} } \widehat{{\Greekmath 010B} }_{i,T}\right\vert ^{{\Greekmath 0117} /2}$ drifts to zero; intuitively, under the null $\widehat{{\Greekmath 010B} }_{i,T}$ drifts to zero at a rate $ O_{a.s.}\left( T^{-1/2}\right) $, and therefore $N\left\vert T^{{\Greekmath 010E} } \widehat{{\Greekmath 010B} }_{i,T}\right\vert ^{{\Greekmath 0117} /2}=O_{a.s.}\left( NT^{\left( {\Greekmath 010E} -1/2\right) {\Greekmath 0117} /2}\right) =o_{a.s.}\left( 1\right) $ by the definition of ${\Greekmath 010E} $\ in ((ref)).

The same arguments hold in the case of nontradable and latent factors, upon replacing $T^{1/{\Greekmath 0117} }$ with $T^{{\Greekmath 010E} }$ in the definition of ${\Greekmath 0120} _{i,NT}^{FM}$, and $C_{NT}^{1/{\Greekmath 0117} }$ with $C_{NT}^{{\Greekmath 010E} }$\ in the definition of ${\Greekmath 0120} _{i,NT}^{PC}$, respectively.

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Technical lemmas

Henceforth, we denote the distribution function of the standard normal calculated at $-\infty <x<\infty $ as $\Phi \left( x\right) $.

Preliminary lemmas

We begin with a Baum-Katz-type theorem which is also reported in massacci2022.

The following estimate on the growth rate of moments of partial sums will be used throughout the paper, and it can be contrasted with Proposition 4.1 in berkeshormann.

Lemmas for Section (ref)

Lemmas for Section (ref)

We now report a series of lemmas for the case, discussed in Section (ref), of nontradable factors. In order for the notation not to be overly burdensome, we will assume - unless otherwise stated - $K=1$ whenever possible and with no loss of generality.

Recall the short-hand notation $\mathbf{S}_{{\Greekmath 010C} }$ defined in ((ref) ), let $\mathbf{{\Greekmath 010B} }=\left( {\Greekmath 010B} _{1},...,{\Greekmath 010B} _{N}\right) ^{\prime } $ and

and

and note that, after standard passages

Finally, under both $\mathbb{H}_{0}$ and $\mathbb{H}_{A}$ we have

We now report two lemmas on the rates of $\widehat{{\Greekmath 0115} }-{\Greekmath 0115} $ under the null and under the alternative.