# Identifying and exploiting alpha in linear asset pricing models with strong,
semi-strong, and latent factors††thanks: We are grateful to Juan Espinosa
Torres and Jing Kong for carrying out the computations and to Hayun Song for
excellent research assistance, all are PhD students at USC. We are also
grateful for helpful comments from the editor, Fabio Trojani, two anonymous
referees, Natalia Bailey, Alessio Sancetta, Takashi Yamagata and participants
at the 2023 Kansas Econometrics Institute, the 2023 NBER Summer Institute and
the 2023 IAAE Annual Conference in Oslo.

M. Hashem Pesaran
University of Southern California, and Trinity College, Cambridge  Ron P.
Smith
Birkbeck, University of London Corresponding author, email<EMAIL_ADDRESS>
###### Abstract

The risk premia of traded factors are the sum of factor means and a parameter
vector we denote by $\boldsymbol{\phi}$ which is identified from the cross
section regression of $\alpha_{i}$ on the vector of factor loadings,
$\boldsymbol{\beta}_{i}$. If $\boldsymbol{\phi}$ is non-zero one can construct
”phi-portfolios” which exploit the systematic components of non-zero alpha. We
show that for known values of $\boldsymbol{\beta}_{i}$ and when
$\boldsymbol{\phi}\ $is non-zero there exist phi-portfolios that dominate
mean-variance portfolios. The paper then proposes a two-step bias corrected
estimator of $\boldsymbol{\phi}$ and derives its asymptotic distribution
allowing for idiosyncratic pricing errors, weak missing factors, and weak
error cross-sectional dependence. Small sample results from extensive Monte
Carlo experiments show that the proposed estimator has the correct size with
good power properties. The paper also provides an empirical application to a
large number of U.S. securities with risk factors selected from a large number
of potential risk factors according to their strength and constructs phi-
portfolios and compares their Sharpe ratios to mean variance and S&P
portfolios.

JEL Classifications: C38, G10

Key Words: Factor strength, pricing errors, risk premia, missing factors,
pooled Lasso, mean-variance and phi-portfolios.

##  1 Introduction

There are two approaches to the estimation of risk premia and testing of
market efficiency, often referred as the beta and the SDF (stochastic discount
factor) methods, see Jagannathan and Wang (2002). This paper adopts the beta
method, and following the literature uses a linear factor pricing model (LFPM)
to explain the time series of excess returns on individual securities,
$r_{it}=R_{it}-r_{t}^{f}$, where $R_{it}$ is the return and $r_{t}^{f}$ the
risk free rate for $i=1,2,...,n;$ $t=1,2,....,T,$ by a set of observed
tradable risk factors. We use individual securities rather than portfolios
since, as we will show, if risk factors are not strong, large $n$ is required
for accurate estimation. Ang, Liu, and Schwarz (2020) discuss the general
issues in the choice between portfolios and individual stocks. Pesaran and
Smith (2023) discuss both the use of portfolios and the relationship between
the SDF and LFPM approaches.

The LFPM explains the excess return on each security, $r_{it},$ by an
intercept, $\mathit{a}_{i},$ labelled alpha, and a $K\times 1$ vector of
traded risk factors, $\mathbf{f}_{t}=(f_{1t},f_{2t},...,f_{Kt})^{\prime}$ with
loadings, $\boldsymbol{\beta}_{i}$:

$r_{it}=\mathit{a}_{i}+\boldsymbol{\beta}_{i}^{\prime}\mathbf{f}_{t}+u_{it}.$
(1)

Under the Arbitrage Pricing Theory, APT, due to Ross (1976) we have

$E\left(r_{it}\right)=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\lambda}+\eta_{i},$
(2)

Taking unconditional expectations of (1) and using (2) we have

$E(r_{it})=\mathit{a}_{i}+\boldsymbol{\beta}_{i}^{\prime}\mathbf{\mu}=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\lambda}+\eta_{i}.$

where $E(\mathbf{f}_{t})=\boldsymbol{\mu}$ which in turn yields

$\mathit{a}_{i}=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\phi}+\eta_{i}\text{,
}$ (3)

where $\boldsymbol{\phi}=\boldsymbol{\lambda}-\boldsymbol{\mu}$. To identify
the risk premia $\boldsymbol{\lambda}$, most of the literature assumes
$\eta_{i}=0$ for all $i$, but to allow for non-zero $\eta_{i}$ strong
restrictions must be imposed on the degree to which such errors are pervasive.
Ross (1976) requires that $\eta_{i}$ to be square-summable. In this paper we
relax this condition and impose restrictions on the degree to which
idiosyncratic pricing errors can be pervasive.

If $\boldsymbol{\phi}\neq 0$ one can construct what we call ”phi-portfolios”
that exploit the systematic components of $\mathit{a}_{i}$, as captured by the
non-zero elements of $\boldsymbol{\phi}$.111We are grateful to one of the
anonymous reviewers for suggesting the idea of phi-portfolios, as a way of
motivating and interpreting the role of $\boldsymbol{\phi}$ in asset pricing
models. The idiosyncratic pricing errors, $\eta_{i}$, cannot be exploited, and
our proposed phi-portfolios exist only if $\boldsymbol{\phi\neq 0}$. The focus
of much of the literature has been on testing $\mathit{a}_{i}=0$, or on
estimating the risk premium $\boldsymbol{\lambda}$ and mean-variance (MV)
portfolios. But given that estimating a non-zero $\boldsymbol{\phi}$ provides
a way to identify and exploit the alpha in a linear factor pricing model for
large $n,$ $\boldsymbol{\phi}$ is an interesting object in itself, which will
be the focus of this paper.

First, we introduce our proposed phi-portfolio in terms of the factor loadings
$\mathbf{B}_{n}=\left(\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2},...,\boldsymbol{\beta}_{n}\right)^{\prime}$
and $\boldsymbol{\phi}$, and compare its limiting properties (as
$n\rightarrow\infty$) with the standard MV portfolio. We show that for known
factor loadings and non-zero $\boldsymbol{\phi}$ there exist phi-portfolios
with strictly positive returns, given by
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}>0$, that are fully diversified
(their variance tends to zero with $n$). The rate at which the variance of
phi-portfolio returns tends to zero will depend on the strength of the traded
risk factors compared to the strength of the missing (latent) factors,
highlighting the importance of factor strengths in portfolio analysis. Since
the phi-portfolios have Sharpe ratios that tend to infinity with $n$, they
dominate MV portfolios whose Sharpe ratio is bounded in $n$. Note that unlike
MV portfolios, the phi-portfolios do not require knowledge of the inverse of
the covariance matrix of returns, which is particularly difficult to estimate
accurately when $n$ is large.

Next we consider estimation of and inference about $\boldsymbol{\phi}$ using a
large number of individual securities, taking account of firm-specific pricing
errors, traded factors that are not strong, missing latent factors, and panel
data sets where the time dimension, $T$, is small relative to $n$. Factor
strength plays a central role in our analysis of $\boldsymbol{\phi}$. We use a
measure of factor strength, $\alpha_{k},$ developed in Bailey, Kapetanios, and
Pesaran (2016, 2021), which is defined in terms of the proportion of non-zero
factor loadings, $\beta_{ik}$.222Alpha is used both for the LFPM intercepts
and the measure of factor strength, because these are established usages, but
it will be clear from context and subscripts which is being referred to. A
factor is strong if this proportion is very close to unity, it is semi-strong
if $1>\alpha_{k}>1/2$, and it is weak if $\alpha_{k}<1/2$. Use of this measure
allows us to be precise about the degree of pervasiveness and show how the
strengths of the observed factors, the missing factors and the pricing errors,
each influence estimation and inference about $\boldsymbol{\phi}$.

In practice, exploitation of a non-zero $\boldsymbol{\phi}$ requires $n$ to be
large and rebalancing such long-short portfolios for so many securities may
incur high transactions costs or not be feasible. In addition, model
uncertainty, estimation uncertainty, time variation in both
$\boldsymbol{\beta}_{i}$ and in conditional volatility pose additional
difficulties in implementing a strategy to exploit the potential returns
revealed by $\boldsymbol{\phi}$. In developing the theory we will abstract
from such practical difficulties, but in the empirical section we illustrate
some of these issues with a comparison of the performance of
$\boldsymbol{\phi}$ based portfolios relative to MV portfolios which would
face similar difficulties.

We estimate $\boldsymbol{\phi}=(\phi_{1},\phi_{2},...,\phi_{K})^{\prime}$,
using a two-step estimator. In the first step we estimate the intercepts,
$\hat{\alpha}_{i}$, and the factor loadings, $\boldsymbol{\hat{\beta}}_{i}$,
from least squares regressions of excess returns on an intercept and risk
factors. In the second step, $\boldsymbol{\phi}$ is estimated from the cross
section regression of $\hat{\alpha}_{i}$ on $\boldsymbol{\hat{\beta}}_{i}$. As
with the two-step estimator of $\boldsymbol{\lambda}$, such a two-step
estimator of $\phi_{k}$ will also be biased, and requires bias-correction.
Following Shanken (1992), we consider a bias-corrected version of the two-step
estimator of $\boldsymbol{\phi}$, which we denote by
$\boldsymbol{\tilde{\phi}}_{nT}$. We develop the asymptotic distribution of
$\boldsymbol{\tilde{\phi}}_{nT}$ under quite general set of assumptions
regarding the idiosyncratic pricing errors, error cross-sectional dependence,
and the presence of missing (latent) factors. The paper also investigates the
implications of factor strengths for the precision with which
$\boldsymbol{\phi}$ can be estimated. The LFPM, following Chamberlain and
Rothschild (1983), assumes that all the observed factors are strong and the
eigenvalues of the covariance matrix of the errors are bounded.

In developing the arbitrage pricing theory, APT, Ross (1976), whose concerns
were primarily theoretical, assumed the factors had mean zero:
$\boldsymbol{\mu}=0,$ so $\boldsymbol{\phi}=\boldsymbol{\lambda}$, is the risk
premium. For traded factors under market efficiency, where
$\boldsymbol{\phi}=\mathbf{0}$, the risk premium is the factor mean
$\boldsymbol{\mu}=\boldsymbol{\lambda}$. Were one interested in estimating
$\boldsymbol{\lambda}$ there may be statistical reasons to estimate $\phi_{k}$
and $\mu_{k}$ separately, then summing them to obtain an estimate of
$\lambda_{k}$. The factor mean, $\mu_{k}=E(f_{kt})$, can be estimated
consistently at the regular $\sqrt{T}$ rate directly using time series data on
the risk factors, $f_{kt}$, for $t=1,2,...,T,$ and does not require knowledge
of the factor loadings or $n$. In contrast, estimation of $\phi_{k}$ requires
panel data to estimate the factor loadings and hence both $n$ and $T$
dimensions are important. In some cases it may be beneficial to use different
time series dimensions, $T_{\mu}$ and $T_{\phi},$ to estimate $\mu_{k}$ and
$\phi_{k}$, respectively. For example, when factor loadings are subject to
breaks it is advisable to use a relatively short sample, and when some factors
are not sufficiently strong a large $n$ is required. Thus one could use large
$T$ to estimate $\mu_{k}$ and small $T$ to estimate $\phi_{k}$ and
$\lambda_{k}$ can be simply estimated by adding the estimates of $\mu_{k}$ and
$\phi_{k}.$ We do not pursue the use of different $T$, but if the same $T$ is
used to estimate the mean of factor $k,$ say $\hat{\mu}_{k,T},$ and the bias-
corrected $\tilde{\phi}_{k,nT}$ then their sum is the same as the Shanken
(1992) bias-corrected risk premium for factor $k,$ $\tilde{\lambda}_{k,nT}.$
This decomposition can be used to obtain the rate at which
$\tilde{\lambda}_{k,nT}$ converges to its true value,
$\lambda_{0,k}=\phi_{0,k}+\mu_{0,k}$.

The main theoretical results of the paper are set out around five theorems
under a number of key assumptions, with proofs provided in the mathematical
appendix. The small sample properties of $\boldsymbol{\tilde{\phi}}_{nT}$ are
investigated using extensive Monte Carlo experiments, allowing for a mixture
of strong and semi-strong observed factors, latent factors, pricing errors,
GARCH effects and non-Gaussian errors. Small sample results are in line with
our theoretical findings, and confirm that the bias-corrected estimator,
$\boldsymbol{\tilde{\phi}}_{nT}$, has the correct size and good power
properties for samples with time series dimensions of $T=120$ and $T=240$.
They also show, in accord with the theory, that the precision with which
$\phi_{k}$ is estimated falls with $\alpha_{k}$, the strength of the $k^{th}$
factor.

Our theoretical derivations and Monte Carlo simulations assume that the list
of relevant observed factors is known. But in practice the relevant factors
must be selected. Extending the theory to the high dimensional case where
factors are selected, rather than given, is beyond the scope of the present
paper. Since the rate of convergence of $\boldsymbol{\tilde{\phi}}_{nT}$ to
its true value is given by $\sqrt{T}n^{(\alpha_{k}+\alpha_{\min}-1)/2}$, where
$\alpha_{\min}$ is the strength of the least strong factor included, it seems
sensible to select factors on the basis of their factor strength. Weak factors
whose strength is around $1/2$ can be ignored and absorbed into the error
term.

The above selection procedure is applied in a high dimensional setting with
both a large number of securities ($n$ from $1,090$ to $1,175)$ and a large
number of potential risk factors ($m$ from $177$ to $189)$, taken from the
Chen and Zimmermann (2022), which can all be traded. We used monthly data over
the period $1996m1-2022m12$ and considered balanced panels obtained by
including all existing stocks in a given month for which there are $T$
observations. We considered $T=120$ and $240$ months, and focus on the latter
which we found to be more reliable, given the large number of securities being
considered. Various procedures could be used to select risk factors for a
given security, $i$. We used Lasso for this purpose and then selected a subset
of these factors that were chosen by a sufficiently large number of securities
in the sample, and whose estimated strength were above the given threshold
value of $\alpha_{k}>0.75$. We refer to this selection procedure as pooled
Lasso. Using this procedure with $T=240$ we ended up with $7$ risk factors for
the sample ending in $2015$, declining to $4$ in $2021$. Interestingly, the
three Fama-French factors were always included in the set of factors selected
by pooled Lasso.

Accordingly, we considered three linear asset pricing models for our portfolio
analysis: the pooled Lasso selected at the start of our evaluation sample,
denoted by PL7, the Fama-French three factors model, FF3, and the Fama-French
five factors model, FF5, which includes two factors not in PL7. The three
models are estimated using rolling samples of size $T=240$, starting with a
sample ending in $2015m12$ and finishing with a sample ending in $2022m11$.
The hypothesis that $\boldsymbol{\phi}=\mathbf{0}$ was rejected for all $84$
rolling samples and all three models, albeit less strongly over the post
Covid-19 period. The test results suggested possible unexploited return
opportunities, and to investigate this possibility further, we constructed
phi-portfolios and compared their Sharpe ratios (SR) with the ones based on
standard MV portfolios over the full sample evaluation sample,
$2016m1-2022m12$, and sample ending $2019m12$, that excludes the Covid-19
period. We find that in five out of the six cases (3 models 2 samples) the
phi-portfolio has a higher SR than the corresponding MV portfolio. The
exception is the FF5 pre Covid-19. This illustrates that if
$\boldsymbol{\phi}\neq 0,$ it is possible to construct a portfolio that
outperforms the mean variance portfolio. In both the pre Covid-19 sample and
the full sample the highest SR was obtained by the PL7 phi-portfolios, which
also outperformed the S&P500. The SRs for the sample ending in $2022$ were
substantially lower than the sample ending in $2019$, consistent with a
falling value of the probability that $\boldsymbol{\phi}$ was non-zero.

Related literature: On estimation of risk premia, following Fama and MacBeth
and Shanken (1992), estimation of risk premia is further examined by Shanken
and Zhou (2007), Kan, Robotti, and Shanken (2013), and Bai and Zhou (2015).
The survey paper by Jagannathan, Skoulakis, and Wang (2010) provide further
references.

Testing for market efficiency dates back to Jensen (1968) who proposes testing
a${}_{i}=0$ for each $i$ separately. Gibbons, Ross, and Shanken (1989) provide
a joint test for the case where the errors are Gaussian and $n<T$.
Gagliardini, Ossola, and Scaillet (2016) develop two-pass regressions of
individual stock returns, allowing time-varying risk premia, and propose a
standardised Wald test. Raponi, Robotti, and Zaffaroni (2019) propose a test
of pricing error in cross section regression for fixed number of time series
observations. They use a bias-corrected estimator of Shanken (1992) to
standardise their test statistic. Ma, Lan, Su, and Tsai (2020) employ
polynomial spline techniques to allow for time variations in factor loadings
when testing for alphas. Feng, Lan, Liu, and Ma (2022) propose a max-of-square
type test of the intercepts instead of the average used in the literature, and
recommend using a combination of the two testing procedures. He, He, Huang,
and Zhou (2022) propose two statistics, a Wald type statistic which require
$n$ and $T$ to be of the same order of magnitude and a standardised t-ratio.
Kleibergen (2009) considers testing in the case where the loadings are small.
Pesaran and Yamagata (2012, 2024) consider testing that the intercepts in the
LFPM are zero when $n$ is large relative to $T$ and there may be non Gaussian
errors and weakly cross-correlated errors.

A large number of risk factors have been considered in the empirical
literature. We use the Fama and French (1993) three factors in our Monte Carlo
design. In our empirical application we use the five factors proposed by Fama
and French (2015) and the large set of factors provided by Chen and Zimmermann
(2022). Harvey and Liu (2019) document over 400 factors published in top
finance journals. Dello Preite, Uppal, Zaffaroni, and Zviadadze (2022) argue
that despite the hundreds of systematic risk factors considered in the
literature, there is still a sizable pricing error and that this can be
explained by asset specific risk that reflects market frictions and behavioral
biases. There is a large Bayesian literature, including Chib, Zeng, and Zhao
(2020), and Hwang and Rubesam (2022) on selecting factor models. The issue of
factor selection is also addressed by Fama and French (2018).

Strong and weak factors in asset returns are considered by Anatolyev and
Mikusheva (2022), Connor and Korajczyk (2022), and Giglio, Xiu, and Zhang
(2023). Beaulieu, Dufour, and Khalaf (2020) discuss the lack of identification
of risk premia when many of the loadings are zero. There has also been concern
about the consequences of omitted factors. Giglio and Xiu (2021) discuss the
problem and try to deal with it using a three-pass method which is valid even
when not all factors in the model are specified or observed using principal
components of the test assets. Onatski (2012) and Lettau and Pelger (2020a, b)
provide extensive discussions of weak factor and latent factors, respectively.
More recent contributions include Bai and Ng (2023) and Uematsu and Yamagata
(2023).

There is also a large literature on portfolio construction. Herskovic,
Moreira, and Muir (2019) discuss low cost methods of hedging risk factors.
Dello Preite, Uppal, Zaffaroni, and Zviadadze (2024) derive an SDF in which
there is compensation for unsystematic risk within the framework of the APT.
Korsaye, Quaini, and Trojani (2021) introduce model-free smart SDFs which give
rise to non-parametric SDF bounds for testing asset pricing models. Daniel,
Mota, Rottke, and Santos (2020) discuss the common practice of creating
characteristic portfolios by sorting on characteristics associated with
average returns and show that these portfolios capture not only the priced
risk associated with the characteristic but also unpriced risk. Quaini,
Trojani, and Yuan (2023) propose an estimator of tradable factor risk premia.

Paper’s outline: The rest of the paper is organized as follows: Section 2
provides the framework for estimation of $\boldsymbol{\phi}.$ Section 3 sets
out the assumptions and states the main theorems. Theorem 1 shows that the
standard Fama-MacBeth estimator is valid only when there are no pricing errors
and $n/T\rightarrow 0$. Theorem 2 shows that the Shanken bias-corrected
estimator of $\lambda_{k}$ continues to be consistent for a fixed $T$ as
$n\rightarrow\infty$, even in presence of weak pricing errors and weak missing
common factors. Theorem 3 provides conditions under which the bias-corrected
estimator, $\boldsymbol{\tilde{\phi}}_{nT}$, is consistent for
$\boldsymbol{\phi}_{0}$, and derives the asymptotic distribution of
$\boldsymbol{\tilde{\phi}}_{nT},$ assuming the observed factors are strong
$(\alpha_{k}=1$ for all $k$). Theorem 4 extends the results to cases where one
or more risk factors are semi-strong, and establishes the rate at which
$\boldsymbol{\tilde{\phi}}_{nT}$ converges to its true value, assuming the
idiosyncratic pricing errors are sufficiently weak, as discussed after the
theorem. For example, for a factor with strength $\alpha_{k}$ we show that
$\tilde{\phi}_{k,nT}-\phi_{0,k}=O_{p}\left(T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\right)$,
and as a consequence

$\tilde{\lambda}_{k,nT}-\lambda_{0,k}=O_{p}\left(T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\right)+O_{p}\left(T^{-1/2}\right),$

where $\lambda_{0,k}$ is the true value of the risk premia associated to
factor $f_{kt}$, and $\alpha_{\min}$ is the strength of the least strong
factor included. This consistency condition is weaker than the one derived by
Giglio, Xiu, and Zhang (2023) who also assume $\eta_{i}=0$, for all $i$.
Finally, Theorem 5 gives conditions for consistent estimation of the
asymptotic variance of $\boldsymbol{\tilde{\phi}}_{nT},$ using a suitable
threshold estimator of the covariance matrix. Section 4 presents the Monte
Carlo (MC) design, its calibration and a summary of the main findings. Section
5 discusses the problem of factor selection from a large number of potential
factors. Section 6 gives the empirical application using monthly data on a
large number of individual US stocks and risk factors over the period
1996-2021. It selects factors, estimates $\boldsymbol{\phi}\mathbf{,}$ and
compares the performance of MV and phi-portfolios. Section 7 provides some
concluding remarks.

Detailed mathematical proofs are provided in a mathematical appendix. Further
information on data sources, MC calibration plus some supplementary material
for the empirical application are provided in the online supplement A. To save
space all MC results are given in the online supplement B.

##  2 Identification and estimation of $\boldsymbol{\phi}$

Let $R_{it}$ denote the holding period return on traded security $i$, which
can be bought long or short without transaction costs, $r_{t}^{f}$ is the risk
free rate, and $r_{it}=R_{it}-r_{t}^{f}$ is the excess return. We start with
the linear factor pricing model (LFPM)

$r_{it}=\mathit{a}_{i}+\boldsymbol{\beta}_{i}^{\prime}\mathbf{f}_{t}+u_{it},$
(4)

for $i=1,2,...,n$ and $t=1,2,...,T$, where $r_{it}$ is explained in terms of
the $K\times 1$ vector of factors
$\mathbf{f}_{t}=(f_{1t},f_{2t},...,f_{Kt})^{\prime}$. The intercept
$\mathit{a}_{i}$ and the $K\times 1$ vector of factor loadings,
$\boldsymbol{\beta}_{i}=(\beta_{i1},\beta_{i2},...,\beta_{iK})^{\prime}$, are
unknown. The idiosyncratic errors, $u_{it}$ have zero means and are assumed to
be serially uncorrelated. The factors, $\mathbf{f}_{t}$, are assumed to be
covariance stationary with the constant mean
$\boldsymbol{\mu}=E\left(\mathbf{f}_{t}\right)$, and
$\boldsymbol{\Sigma}_{f}=E\left[\left(\mathbf{f}_{t}-\boldsymbol{\mu}\right)\left(\mathbf{f}_{t}-\boldsymbol{\mu}\right)^{\prime}\right]$.

Under the Arbitrage Pricing Theory (APT) due to Ross (1976) the pricing
errors, $\eta_{i}$ for $i=1,2,...,n$ defined by

$\eta_{i}=E\left(r_{it}\right)-c-\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\lambda}\mathbf{,}$
(5)

are bounded such that

$\sum_{i=1}^{n}\eta_{i}^{2}<C<\infty,$ (6)

where $c$ is zero-beta expected excess return, $\boldsymbol{\lambda}$ is the
$K\times 1$ vector of risk premia. Under APT, we have

$E\left(r_{it}\right)=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\lambda}+\eta_{i},$
(7)

and reduces to the standard beta representation of an unconditional asset
pricing model, if $c+\eta_{i}=0$. In this case,
$\boldsymbol{\beta}_{i}=-Cov(r_{it},m_{t})/var(m_{t})$, where $m_{t}$ is the
stochastic discount factor (SDF), which satisfies the moment condition
$E_{t}(r_{i,t+1}m_{t+1})=0$. See Pesaran and Smith (2023) for a discussion of
the link. For empirical assessment, we consider the more general unconditional
pricing model (7), and relate to the linear factor pricing model. To this end,
taking unconditional expectations of (4) and using the APT condition we have

$E(r_{it})=\mathit{a}_{i}+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\mu}=\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\lambda}+c+\eta_{i}.$
(8)

which in turn yields

$\mathit{a}_{i}=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\phi}+\eta_{i}\text{,
for }i=1,2,...,n.$ (9)

where

$\boldsymbol{\phi}=\boldsymbol{\lambda}-\boldsymbol{\mu},$ (10)

Under the APT condition, (6), $\boldsymbol{\phi}$ can be identified from the
cross section regression of $\mathit{a}_{i}$ on $\boldsymbol{\beta}_{i}$. We
relax the APT condition and derive restrictions on the degree to which
idiosyncratic pricing errors, $\eta_{i}$, can be pervasive to achieve
identification.

The focus of the literature has been on testing for alpha, $\mathit{a}_{i}=0$,
and the estimation of the risk premia, $\boldsymbol{\lambda}$, using panel
data on excess returns, $\left\\{r_{it},1,2,...,n\text{;
}t=1,2,...,T\right\\},$ and $\mathbf{F}$, the $T\times K$ matrix of
observations on the factors. It is clear that $\boldsymbol{\phi}$ plays an
important role in tests for alpha in LFPM, and a non-zero $\boldsymbol{\phi}$
implies non-zero alphas, which in turn implies exploitable excess profitable
opportunities. More specifically, we show that for known values of
$\boldsymbol{\beta}_{i},$ $i=1,2,...,n\,$ and when $\boldsymbol{\phi}$ is non-
zero there exist phi-based portfolios with non-zero means that are fully
diversified (their variance tends to zero with $n$), namely have Sharpe ratios
that tend to infinity with $n$, and hence dominate mean-variance (MV)
portfolios. For the MV portfolios to be efficient it is necessary that
$\boldsymbol{\phi=0}$.

###  2.1 Why $\boldsymbol{\phi}$ matters: introduction of the phi-portfolios

Substitute the expression for $\mathit{a}_{i}$ given by (9) in (4) to obtain

$r_{it}=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\phi}+\eta_{i}+\boldsymbol{\beta}_{i}^{\prime}\mathbf{f}_{t}+u_{it},\text{
for }i=1,2,...,n,$ (11)

and write the $n$ return equations more compactly as

$\mathbf{r}_{\circ
t}=c\boldsymbol{\tau}_{n}+\mathbf{B}_{n}\boldsymbol{\phi}+\mathbf{B}_{n}\mathbf{f}_{t}+\boldsymbol{\eta}_{n}+\mathbf{u}_{\circ
t},$ (12)

where $\mathbf{r}_{\circ t}=(r_{1t},r_{2t},....,r_{nt})^{\prime},$
$\boldsymbol{\tau}_{n}$ is an $n$-dimensional vector of ones,
$\mathbf{B}_{n}=(\boldsymbol{\beta}_{\circ 1},\boldsymbol{\beta}_{\circ
2},...,\boldsymbol{\beta}_{\circ K}),$ $\boldsymbol{\beta}_{\circ
k}=(\beta_{1k},\beta_{2k},...,\beta_{nk})^{\prime},$ $\mathbf{u}_{\circ
t}=(u_{1t},u_{2t},....,u_{nt})^{\prime}$,
$\boldsymbol{\eta}_{n}=\left(\eta_{1},\eta_{2},...,\eta_{n}\right)^{\prime}$,
and $\mathbf{V}_{u}=E\left(\mathbf{u}_{\circ t}\mathbf{u}_{\circ
t}^{\prime}\right)$. Suppose that the factors, $\mathbf{f}_{t},$ are traded,
and $\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}>0$. Consider the $n\times 1$
vector of phi-portfolio weights,$\mathbf{w}_{\phi}$, given by

$\mathbf{w}_{\phi}=\mathbf{M}_{n}\mathbf{B}_{n}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)^{-1}\boldsymbol{\phi},$
(13)

where
$\mathbf{M}_{n}=\mathbf{I}_{n}-n^{-1}\boldsymbol{\tau}_{n}\boldsymbol{\tau}_{n}^{\prime}$.
Finally, consider the long-short hedged portfolio return

$\rho_{t,\phi}=\mathbf{w}_{\phi}^{\prime}\mathbf{r}_{\circ
t}-\boldsymbol{\phi}^{\prime}\mathbf{f}_{t}=\boldsymbol{\phi}^{\prime}\left[\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)^{-1}\mathbf{B}_{n}\mathbf{M}_{n}\mathbf{r}_{\circ
t}-\mathbf{f}_{t}\right],$ (14)

and using (12) note that

$\rho_{t,\phi}=\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}+\mathbf{w}_{\phi}^{\prime}\boldsymbol{\eta}_{n}+\mathbf{w}_{\phi}^{\prime}\mathbf{u}_{\circ
t}\mathbf{.}$ (15)

For a given vector of pricing errors, $\boldsymbol{\eta}_{n}$,
$E\left(\rho_{t,\phi}\right)=\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}+\mathbf{w}_{\phi}^{\prime}\boldsymbol{\eta}_{n}$,
and
$Var\left(\rho_{t,\phi}\right)=\mathbf{w}_{\phi}^{\prime}\mathbf{V}_{u}\mathbf{w}_{\phi}\mathbf{,}$
where $\mathbf{V}_{u}=E(\mathbf{u}_{\circ t}\mathbf{u}_{\circ t}^{\prime})$.
Using (13) it follows that

$\left\|\mathbf{w}_{\phi}^{\prime}\boldsymbol{\eta}_{n}\right\|\leq\left\|\boldsymbol{\eta}_{n}\right\|\left\|\mathbf{w}_{\phi}\right\|\leq\left\|\boldsymbol{\eta}_{n}\right\|\left\|\boldsymbol{\phi}\right\|\lambda_{max}^{1/2}\left[\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)^{-1}\right]=\frac{\left\|\boldsymbol{\eta}_{n}\right\|\left\|\boldsymbol{\phi}\right\|}{\lambda_{min}^{1/2}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}.$
(16)

Similarly,

$Var\left(\rho_{t,\phi}\right)\leq\left\|\mathbf{w}_{\phi}\right\|^{2}\lambda_{max}\left(\mathbf{V}_{u}\right)=\frac{\lambda_{max}\left(\mathbf{V}_{u}\right)\left\|\boldsymbol{\phi}\right\|^{2}}{\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}.$
(17)

Hence,
$E\left(\rho_{t,\phi}\right)\rightarrow\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}$,
and $Var\left(\rho_{t,\phi}\right)\rightarrow 0$, as $n\rightarrow\infty$, if

$\frac{\left\|\boldsymbol{\eta}_{n}\right\|}{\lambda_{min}^{1/2}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}\rightarrow
0\text{, and
}\frac{\lambda_{max}\left(\mathbf{V}_{u}\right)}{\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}\rightarrow
0.$ (18)

The conditions are met if
$\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)\rightarrow\infty$,
$\lambda_{max}\left(\mathbf{V}_{u}\right)<C$, and the APT condition (6) are
met. The first two conditions follow if the LPFM given by (4) is an
approximate factor model as assumed by Chamberlain and Rothschild (1983),
namely when the factors are strong and the errors, $u_{it}$, are weakly cross
correlated.333The diversification conditions in (18) are met more generally,
and accommodate the presence of semi-strong traded factors in
$\mathbf{f}_{t}$, and less restrictive conditions on $\mathbf{V}_{u}$ and the
pricing errors $\eta_{i}.$ See Remark 1. Therefore, knowledge of
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}$ and its statistical significance
can play an important role in portfolio analysis.

In the case where $c=0,$ $\boldsymbol{\phi}=\mathbf{0}$, and
$\boldsymbol{\eta}_{n}=\mathbf{0}$, there are no exploitable excess profit
opportunities, and portfolios formed using factor loadings reduce to the
standard mean-variance (MV) portfolio, given by
$\rho_{MV,t}=\boldsymbol{\mu}_{R}^{\prime}\mathbf{V}_{R}^{-1}\mathbf{r}_{\circ
t}$, where $\boldsymbol{\mu}_{R}=\mathbf{B}_{n}\boldsymbol{\mu}$ and
$\mathbf{V}_{R}=\mathbf{B}_{n}\mathbf{\Sigma}_{f}\mathbf{B}_{n}^{\prime}+\mathbf{V}_{u}$.
As shown in Lemma A.1, the squared Sharpe ratio of MV portfolio,
$SR_{MV}^{2}=\boldsymbol{\mu}_{R}^{\prime}\mathbf{V}_{R}^{-1}\boldsymbol{\mu}_{R}$,
is bounded by
$\boldsymbol{\mu}^{\prime}\boldsymbol{\Sigma}_{f}^{-1}\boldsymbol{\mu}$, and
this upper bound is achieved only if
$\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)\rightarrow\infty$,
as $n\rightarrow\infty$. Specifically

$\displaystyle SR_{MV}^{2}$
$\displaystyle=\boldsymbol{\mu}_{R}^{\prime}\mathbf{V}_{R}^{-1}\boldsymbol{\mu}_{R}=\boldsymbol{\mu}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\mu-\mu}^{\prime}\mathbf{B}_{n}^{\prime}\left(\mathbf{B}_{n}\mathbf{\Sigma}_{f}\mathbf{B}_{n}^{\prime}+\mathbf{V}_{u}\right)^{-1}\mathbf{B}_{n}\boldsymbol{\mu}$
(19)
$\displaystyle\leq\boldsymbol{\mu}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\mu}$

In contrast, if it is similarly assumed that $c=0$ and
$\boldsymbol{\eta}_{n}=\mathbf{0}$, but
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}>0$, then the following hedged
portfolio can be formed444When $c\neq 0$, it is not possible to eliminate $c$
(which is unpriced), and at the same time exploit the error covariance matrix
$\mathbf{V}_{u}$.

$\displaystyle\rho_{t,\phi}$
$\displaystyle=\boldsymbol{\phi}^{\prime}\left[\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\mathbf{B}_{n}\mathbf{V}_{u}^{-1}\mathbf{r}_{\circ
t}-\mathbf{f}_{t}\right]=\boldsymbol{\phi}^{\prime}\left[\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\left(\mathbf{B}_{n}\boldsymbol{\phi}_{0}+\mathbf{B}_{n}\mathbf{f}_{t}+\mathbf{u}_{\circ
t}\right)-\mathbf{f}_{t}\right]$
$\displaystyle=\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}+\boldsymbol{\phi}^{\prime}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{u}_{\circ
t},$

and its squared Sharpe ratio is given by

$SR_{\phi}^{2}=\frac{\left(\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}\right)^{2}}{\boldsymbol{\phi}^{\prime}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\boldsymbol{\phi}}.$
(20)

Since,

$\boldsymbol{\phi}^{\prime}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\boldsymbol{\phi}\leq\left(\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}\right)\lambda_{\max}\left[\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)^{-1}\right]=\frac{\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}}{\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)},$

then

$SR_{\phi}^{2}\geq\left(\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}\right)\text{
}\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right).$
(21)

As a result, if
$\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)\rightarrow\infty$
as $n\rightarrow\infty$, $SR_{\phi}^{2}$ increases in $n$ without bounds if
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}>0$, and $SR_{\phi}^{2}$ will
eventually dominate $SR_{MV}^{2}$, since the latter is bounded in $n$.

It is also worth noting that the Sharpe ratio of the MV portfolio continues to
be bounded in $n$ even if $\boldsymbol{\phi\neq 0}$. In this case we have

$SR_{MV}^{2}=\boldsymbol{\lambda}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda-\lambda}^{\prime}\mathbf{B}_{n}^{\prime}\left(\mathbf{B}_{n}\mathbf{\Sigma}_{f}\mathbf{B}_{n}^{\prime}+\mathbf{V}_{u}\right)^{-1}\mathbf{B}_{n}\boldsymbol{\lambda},$

and
$SR_{MV}^{2}\leq\left(\boldsymbol{\mu+\phi}\right)^{\prime}\mathbf{\Sigma}_{f}^{-1}\left(\boldsymbol{\mu+\phi}\right)<C$.
Whether $\boldsymbol{\phi=0}$ or not affects the magnitude of $SR_{MV}^{2}$
but does not alter the fact that $SR_{MV}^{2}$ will be bounded if
$\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{V}_{u}^{-1}\mathbf{B}_{n}\right)\rightarrow\infty$
as $n\rightarrow\infty$.

To make sure that the mean-variance portfolio is efficient we must have
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}=0$, and it is of special interest
to estimate $\boldsymbol{\phi}$ and develop reliable procedures for testing
its statistical significance, using a large number of securities. In practice,
the number of tradeable securities, $n$, might not be sufficiently large,
there are important specification and estimation uncertainties, and the Sharpe
ratio of the phi-based portfolio, $\rho_{t,\phi}$, is likely to be bounded
even if $\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}>0$. We turn to these
issues in the empirical application provided in Section 6.

###  2.2 Fama-MacBeth and Shanken estimators of risk premia

It will prove convenient to write (4) in matrix notation by stacking the
excess returns by $t=1,2,...,T$, for each security $i$

$\mathbf{r}_{i\circ}=\mathit{a}_{i}\boldsymbol{\tau}_{T}+\mathbf{F}\boldsymbol{\beta}_{i}+\mathbf{u}_{i\circ},\text{
for }i=1,2,...,n,$ (22)

where $\mathbf{r}_{i\circ}=(r_{i1},r_{i2},...,r_{iT})^{\prime}$,
$\mathbf{F=(f}_{1},\mathbf{f}_{2},...,\mathbf{f}_{T})^{\prime}$,
$\mathbf{u}_{i\circ}=\left(u_{i1},u_{i2},...,u_{iT}\right)^{\prime}$, and
$\boldsymbol{\tau}_{T}$ is a $T\times 1$ vector of ones. Similarly, stacking
the excess returns by $i$ for each $t$ we have (12) which we rewrite as

$\mathbf{r}_{\circ
t}=\boldsymbol{\alpha}_{n}+\boldsymbol{B}_{n}\mathbf{f}_{t}+\mathbf{u}_{\circ
t},\text{ for }t=1,2,...,T,$ (23)

where
$\boldsymbol{a}_{n}=(\mathit{a}_{1},\mathit{a}_{2},...,\mathit{a}_{in})^{\prime}=c\boldsymbol{\tau}_{n}+\mathbf{B}_{n}\boldsymbol{\phi}+\boldsymbol{\eta}_{n}$.

The risk premia are usually estimated using a two-pass procedure suggested by
Fama and MacBeth (1973). The first-pass runs time series regressions of excess
returns, $r_{it}$, on the $K$ observed factors to give estimates of the factor
loadings, $\boldsymbol{\beta}_{i}:$

$\boldsymbol{\hat{\beta}}_{iT}=\left(\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}\right)^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{r}_{i\circ}.$
(24)

The second-pass runs a cross section regression of average returns,
$\bar{r}_{i\circ}=T^{-1}\sum_{t=1}^{T}r_{it}$, on the estimated factor
loadings, to obtain the FM estimator of $\boldsymbol{\lambda}$, namely

$\boldsymbol{\hat{\lambda}}_{nT}=\left(\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\right)^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\bar{r}}_{n\circ},$
(25)

where
$\mathbf{\hat{B}}_{nT}=(\boldsymbol{\hat{\beta}}_{1T},\boldsymbol{\hat{\beta}}_{2T},...,\boldsymbol{\hat{\beta}}_{nT})^{\prime},$
$\mathbf{\bar{r}}_{n\circ}=(\bar{r}_{1T},\bar{r}_{2T},...,\bar{r}_{nT})^{\prime},$
$\mathbf{M}_{T}=\mathbf{I}_{T}-T^{-1}\boldsymbol{\tau}_{T}\boldsymbol{\tau}_{T}^{\prime}$,
$\boldsymbol{\tau}_{T}$ is a $T$-dimensional vector of ones,
$\mathbf{M}_{n}=\mathbf{I}_{n}-n^{-1}\boldsymbol{\tau}_{n}\boldsymbol{\tau}_{n}^{\prime}$,
and $\boldsymbol{\tau}_{n}$ is an n-dimensional vector of ones.

As is well known, when $T$ is finite FM’s two-pass estimator is biased due the
errors in estimation of factor loadings that do not vanish. The small $T$ bias
of the two-pass estimator of $\boldsymbol{\lambda}$ has been a source of
concern in the empirical literature. Under standard regularity conditions and
as $n\rightarrow\infty$, we have

$\boldsymbol{\hat{\lambda}}_{nT}-\boldsymbol{\lambda}_{0}\rightarrow_{p}\left[\mathbf{\Sigma}_{\beta\beta}+\frac{\overline{\sigma}^{2}}{T}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\right]^{-1}\left(\mathbf{\Sigma}_{\beta\beta}\mathbf{d}_{fT}-\frac{\overline{\sigma}^{2}}{T}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\lambda}_{0}\right),$
(26)

where $\mathbf{d}_{fT}=\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0},$
$\boldsymbol{\lambda}_{0}$ and $\boldsymbol{\mu}_{0}$ are the true values of
$\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$, respectively,
$\mathbf{\Sigma}_{\beta\beta}=\lim_{n\rightarrow\infty}\left(n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)$,
and
$\overline{\sigma}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{i}^{2}>0$.
Following Shanken (1992), $\overline{\sigma}_{n}^{2}$ can be consistently
estimated (for a fixed $T$) by

$\widehat{\bar{\sigma}}_{nT}^{2}=\frac{\sum_{t=1}^{T}\sum_{i=1}^{n}\hat{u}_{it}^{2}}{n(T-K-1)},$
(27)

where

$\hat{u}_{it}=r_{it}-\hat{\alpha}_{iT}-\boldsymbol{\hat{\beta}}_{iT}^{\prime}\mathbf{f}_{t},$
(28)

and as before $\hat{\alpha}_{iT}$ and $\boldsymbol{\hat{\beta}}_{iT}$ are the
OLS estimators of $\alpha_{i}$ and $\boldsymbol{\beta}_{i}$. Using these
results the bias-corrected version of the two-pass estimator is given by555See
also Shanken and Zhou (2007), Kan, Robotti, and Shanken (2013), and Bai and
Zhou (2015), and the survey paper by Jagannathan, Skoulakis, and Wang (2010)
for further references.

$\boldsymbol{\tilde{\lambda}}_{nT}=\mathbf{H}_{nT\
}^{-1}\left(\frac{\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\bar{r}}_{n\circ}}{n}\right),$
(29)

where

$\mathbf{H}_{nT\
}=\frac{\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}}{n}-T^{-1}\widehat{\bar{\sigma}}_{nT}^{2}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}.$
(30)

When all the risk factors are strong, under certain regularity conditions,
there exists a fixed $T_{0}$ such that for all $T>T_{0}$, then

$p\lim_{n\rightarrow\infty}\left(\boldsymbol{\tilde{\lambda}}_{nT}\right)=\boldsymbol{\lambda}_{T}^{\ast}=\boldsymbol{\lambda}_{0}+\left(\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0}\right),$
(31)

where $\boldsymbol{\mu}_{0}$ indicates the true value of the factor mean.
Shanken refers to $\boldsymbol{\lambda}_{T}^{\ast}$ as ”ex-post” risk premia
to be distinguished from $\boldsymbol{\lambda}_{0}$, referred to as ”ex ante”
risk premia. See also Section 3.7 of Jagannathan, Skoulakis, and Wang (2010).

In this paper we exploit Shanken’s bias correction procedure by applying it to
$\boldsymbol{\phi}=$ $\boldsymbol{\lambda}-\boldsymbol{\mu}$ which we identify
directly using (9) from the regression of $\alpha_{i}$ on
$\boldsymbol{\beta}_{i}$ for $i=1,2,...,n$, assuming the idiosyncratic pricing
errors, $\eta_{i}$, are sufficiently weak relative to the strengths of the
risk factors in a sense which will be made precise below.

###  2.3 Estimation of $\boldsymbol{\phi}$

In view of (9), the estimation of $\boldsymbol{\phi}$ can be carried out
following a two-step procedure whereby in the first step ai and
$\boldsymbol{\beta}_{i}$ are estimated from the least squares regressions of
$r_{it}$ on an intercept and $\mathbf{f}_{t}$, and these are then used in a
second step regression to estimate $\boldsymbol{\phi}$, namely,

$\boldsymbol{\hat{\phi}}_{nT}\text{
}=\left(\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\right)^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\hat{a}_{nT},$
(32)

where
â${}_{nT}=(\hat{a}_{1T},\hat{a}_{2T},...,\hat{a}_{nT})^{\prime}=\mathbf{\bar{r}}_{nT}-\mathbf{\hat{B}}_{nT}\boldsymbol{\hat{\mu}}_{T},$
and as before
$\mathbf{\hat{B}}_{nT}=(\boldsymbol{\hat{\beta}}_{1T},\boldsymbol{\hat{\beta}}_{2T},...,\boldsymbol{\hat{\beta}}_{nT})^{\prime}$.
This estimator is consistent for $\boldsymbol{\phi}_{0}$ so long as $n$ and
$T\rightarrow\infty$, and bias-corrections are necessary to ensure the large
$n$ consistency of the estimator when $T$ is fixed. A Shanken type bias-
corrected estimator of $\boldsymbol{\phi}_{0}$ is given by

$\boldsymbol{\tilde{\phi}}_{nT}=\mathbf{H}_{nT\
}^{-1}\left[\frac{\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\text{\^{a}}_{nT}}{n}+T^{-1}\widehat{\bar{\sigma}}_{nT}^{2}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\hat{\mu}}_{T}\right],$
(33)

where $\mathbf{H}_{nT\ }$ and $\widehat{\bar{\sigma}}_{nT}^{2}$ are given by
(30) and (27), respectively. It is also easily established that

$\boldsymbol{\tilde{\phi}}_{nT}=\boldsymbol{\tilde{\lambda}}_{nT}-\boldsymbol{\hat{\mu}}_{T},$
(34)

and for a fixed $T$ and as $n\rightarrow\infty$, we have

$p\lim_{n\rightarrow\infty}\boldsymbol{\tilde{\phi}}_{nT}=p\lim_{n\rightarrow\infty}\boldsymbol{\tilde{\lambda}}_{nT}-\boldsymbol{\hat{\mu}}_{T}.$

Hence, upon using (31)

$p\lim_{n\rightarrow\infty}\boldsymbol{\tilde{\phi}}_{nT}=\boldsymbol{\lambda}_{0}+\left(\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0}\right)-\boldsymbol{\hat{\mu}}_{T}=\boldsymbol{\lambda}_{0}-\boldsymbol{\mu}_{0}=\boldsymbol{\phi}_{0}\text{,
}$ (35)

and there exists a fixed $T_{0}$ such that for all $T>T_{0},$
$\boldsymbol{\tilde{\phi}}_{nT}$ converges to $\boldsymbol{\phi}_{0}$ as
$n\rightarrow\infty$. Also using (31) and (34), and noting that
$\boldsymbol{\lambda}_{0}-\boldsymbol{\mu}_{0}=\boldsymbol{\phi}_{0}$,
interestingly we have

$\boldsymbol{\tilde{\lambda}}_{nT}-\boldsymbol{\lambda}_{T}^{\ast}=\boldsymbol{\tilde{\phi}}_{nT}+\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\lambda}_{T}^{\ast}=\boldsymbol{\tilde{\phi}}_{nT}-\boldsymbol{\phi}_{0}.$

So inference using the Shanken bias-corrected estimator of
$\boldsymbol{\lambda}$ around $\boldsymbol{\lambda}_{T}^{\ast}$, is the same
as making inference using $\boldsymbol{\tilde{\phi}}_{nT}$ around
$\boldsymbol{\phi}_{0}$.

The asymptotic distribution of $\boldsymbol{\tilde{\phi}}_{nT}$ depends on
both $n$ and $T$. Assuming the observed factors are strong and under certain
regularity conditions, to be introduced below, we have

$\sqrt{nT}\left(\boldsymbol{\tilde{\phi}}_{nT}-\boldsymbol{\phi}_{0}\right)\rightarrow_{d}N\left(\mathbf{0,\Sigma}_{\beta\beta}^{-1}\mathbf{V}_{\xi}\mathbf{\Sigma}_{\beta\beta}^{-1}\right),$
(36)

where

$\mathbf{V}_{\xi}=\left(1+\boldsymbol{\lambda}_{0}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}\right)p\lim_{n\rightarrow\infty}\left[n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{V}_{u}\mathbf{M}_{n}\mathbf{B}_{n\
}\right]\emph{.}$

The variance of $\boldsymbol{\tilde{\phi}}_{nT}$ is consistently estimated by

$\widehat{Var\left(\boldsymbol{\tilde{\phi}}_{nT}\right)}=T^{-1}n^{-1}\mathbf{H}_{nT\
}^{-1}\mathbf{\hat{V}}_{\xi,nT}\mathbf{H}_{nT\ }^{-1},$ (37)

where $\mathbf{H}_{nT\ }$ is given by (30),

$\mathbf{\hat{V}}_{\xi,nT}=\left(1+\hat{s}_{nT}\right)\left(n^{-1}\mathbf{\hat{B}}_{n}^{\prime}\mathbf{M}_{n}\mathbf{\tilde{V}}_{u}\mathbf{M}_{n}\mathbf{\hat{B}}_{n}\right),$
(38)

and

$\hat{s}_{nT}=\boldsymbol{\tilde{\lambda}}_{nT}^{{}^{\prime}}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\tilde{\lambda}}_{nT},$
(39)

$\boldsymbol{\tilde{\lambda}}_{nT}$ is defined by (29), and
$\mathbf{\tilde{V}}_{u}\mathbf{\ }$is a suitable estimator of
$\mathbf{V}_{u}=E\left(\mathbf{u}_{\circ t}\mathbf{u}_{\circ
t}^{\prime}\right)$. How to estimate $\mathbf{V}_{u}$ and the conditions under
which $Var\left(\boldsymbol{\tilde{\phi}}_{nT}\right)$ is consistently
estimated by $\widehat{Var\left(\boldsymbol{\tilde{\phi}}_{nT}\right)}$ is
discussed in sub-section 3.2.

Note that even when $\mathbf{V}_{u}=\sigma^{2}\mathbf{I}_{T}$ the variance of
$\boldsymbol{\tilde{\phi}}_{nT}$ does not reduce to
$\sigma^{2}\boldsymbol{\Sigma}_{\beta\beta}^{-1}$, the standard least squares
formula used for the case of known factor loadings. When the loadings are
estimated the scaling term
$\left(1+\boldsymbol{\lambda}_{0}^{\prime}\boldsymbol{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}\right)$
is required and its neglect can lead to serious over-rejection even if
$n/T\rightarrow 0$ as $n$ and $T\rightarrow\infty$.

###  2.4 Factor strength

In this paper we deviate from the standard literature and allow the observed
and latent factors to have different degrees of strength, depending on how
pervasively they impact the security returns. Bailey, Kapetanios, and Pesaran
(2021) define the strength of factor, $f_{kt}$, in terms of the number of its
non-zero factor loadings. For a factor to be strong almost all of its $n$
loadings must differ from zero. Given our focus on estimation of risk premia,
we adopt the following definition which directly relates to the covariance of
$\boldsymbol{\beta}_{i}.$ See also Chudik, Pesaran, and Tosetti (2011).

###### Definition 1

(Factor strengths) The strength of factor $f_{kt}$ is measured by its degree
of pervasiveness as defined by the exponent $\alpha_{{}_{k}}$ in

$\sum_{i=1}^{n}\left(\beta_{ik}-\bar{\beta}_{k}\right)^{2}=\ominus(n^{\alpha_{k}}),$
(40)

and $0<\alpha_{k}\leq 1$. We refer to $\left\\{\alpha_{k},\text{
}k=1,2,...,K\right\\}$ as factor strengths. Factor $f_{kt}$ is said to be
strong if $\alpha_{k}=1$, semi-strong if $1>\alpha_{k}>1/2$, and weak if
$0\leq\alpha_{k}\leq 1/2$. Condition (40) applies irrespective of whether the
loadings, $\beta_{ik}$, are viewed as deterministic or stochastic.

In the above definition $\ominus_{p}\left(n^{\alpha_{k}}\right)$ denotes the
rate at which additional securities add to the factor’s strength and
$\alpha_{k}$ can be viewed as a logarithmic expansion rate in terms of $n$ and
relates to the proportion of non-zero factor loadings. In the literature it is
commonly assumed that the covariance matrix of factor loadings defined by

$\boldsymbol{\Sigma}_{\beta\beta}=p\lim_{n\rightarrow\infty}\left[n^{-1}\sum_{i=1}^{n}\left(\boldsymbol{\beta}_{i}-\boldsymbol{\bar{\beta}}_{n}\right)\left(\boldsymbol{\beta}_{i}-\boldsymbol{\bar{\beta}}_{n}\right)^{\prime}\right],$
(41)

is positive definite, where
$\boldsymbol{\bar{\beta}}_{n}=n^{-1}\sum_{i=1}^{n}\boldsymbol{\beta}_{i}=(\bar{\beta}_{1},\bar{\beta}_{2},...,\bar{\beta}_{k})^{\prime}$.
For $\mathbf{\Sigma}_{\beta\beta}$ to be positive definite matrix it is
necessary that all the $K$ risk factors under consideration are strong in the
sense that

$p\lim_{n\rightarrow\infty}\left[n^{-1}\sum_{i=1}^{n}\left(\beta_{ik}-\bar{\beta}_{k}\right)^{2}\right]>0\text{,
for }k=1,2,...,K.$ (42)

In terms of our definition of factor strength, $\mathbf{\Sigma}_{\beta\beta}$
will be positive definite if all the observed factors are strong, namely if
$\alpha_{k}=1$ for $k=1,2,...,K$. However, such an assumption is quite
restrictive and is unlikely to be satisfied for many risk factors being
considered in the literature. Bailey, Kapetanios, and Pesaran (2021) show
that, apart from the market factor, only a handful of 144 factors in the
literature considered by Feng, Giglio, and Xiu (2020) come close to being
strong. Giglio, Xiu, and Zhang (2023) consider the estimation of PCA-based
risk premia in the presence of weak factors. However, their definition of
factor strength involves both $n~{}$and $T$, and is best viewed as a
consistency condition rather than factor strength as such. See the discussion
following Theorem 4. Our notion of factor strength, $\alpha_{k}$, is in line
with the recent literature. See, for example, Bai and Ng (2023) and Uematsu
and Yamagata (2023).

###  2.5 Missing factor

We now turn to the structure of the errors, $u_{it}$, in the returns
equations, and consider two possible sources of error cross-sectional
dependence: a missing or latent factor and production networks. The issue of
missing factors has been investigated in the recent literature by Giglio and
Xiu (2021) and Anatolyev and Mikusheva (2022). The issue of production
networks has been investigated in the recent literature by Herskovic (2018),
who derives two risk factors based on the changes in network concentration and
network sparsity, and Gofman, Segal, and Wu (2020), who focus on the vertical
dimension of production by modeling a supply chain, in terms of supplier-
customer links. They find that the further away a firm is from final consumers
the higher its return. They use this to create a factor TMB (top minus
bottom). Both sources of cross-sectional error dependence could be important,
since network dependence cannot be represented using latent factor models. See
Section 3 of Chudik, Pesaran, and Tosetti (2011).

To allow for both forms of error cross-sectional dependence we consider the
following decomposition of $u_{it}$

$u_{it}=\gamma_{i}g_{t}+v_{it},$ (43)

where $g_{t}$ is the missing (latent) factor and $v_{it}$ is weakly cross-
correlated in the sense of approximate factor models due to Chamberlain (1983)
and Chamberlain and Rothschild (1983). Here we allow for a single missing
factor to simplify the exposition, but note that increasing the number of
missing factors has little impact on our analysis, so long as the number of
missing factors is fixed. Using the normalization $E(g_{t}^{2})=1$, and
assuming that $\gamma_{i}g_{t}$ and $v_{it}$ are independently distributed
then $E(u_{it}u_{jt})=\sigma_{ij}=\gamma_{i}\gamma_{j}+\sigma_{v,ij}$, and as
shown in Lemma A.2,
$n^{-1}\sum_{i=1}^{n}\sum_{j=1}^{n}\left|\sigma_{ij}\right|=O(1)$ so long as
the strength of $g_{t},$ $\alpha_{\gamma}<1/2$ and
$\lambda_{\max}(\mathbf{V}_{v})<\infty.$ This is despite the fact that
$\lambda_{\max}(\mathbf{V}_{u})=O(n^{\alpha_{\gamma}})$, where
$\mathbf{V}_{v}=E\left(\mathbf{v}_{i}\mathbf{v}_{i}^{\prime}\right)$ and
$\mathbf{V}_{u}=E\left(\mathbf{u}_{i}\mathbf{u}_{i}^{\prime}\right)$.666Note
that Chamberlain’s approximate factor model specification requires
$\lambda_{\max}(\mathbf{V}_{u})=O(1)$ and is violated if $\alpha_{\gamma}>0$.
In the Monte Carlo experiments, we consider the possibility of missing
factors, as well as weak spatial and network cross-dependence that satisfy
conditions of approximate factor models.

###  2.6 Pricing errors

The APT condition (6), given by (18) in Theorem II of Ross (1976), ensures
that under APT the (idiosyncratic) pricing errors are sparse. In this paper we
relax Ross’s condition to

$\sum_{i=1}^{n}\eta_{i}^{2}=O(n^{\alpha_{\eta}}),$ (44)

where the exponent $\alpha_{\eta}$ measures the degree of pervasiveness of
pricing errors. Deviations from APT are measured in terms of $\alpha_{\eta}$
($0\leq\alpha_{\eta}<1$). We investigate the robustness of our proposed
estimator of $\boldsymbol{\phi}$ to $\alpha_{\eta}$. This extension is
important for tests of market efficiency where the null of interest is
$H_{0}:$ $\mathit{a}_{i}=c$ for all $i$ in (9). We note that under the
alternative hypothesis $H_{1}:$
$\mathit{a}_{i}=c+\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\phi}+\eta_{i}$,
therefore it is desirable to develop a test of
$\boldsymbol{\phi}=\boldsymbol{0}$ which is robust to a wider class of pricing
errors than those entertained originally by Ross, where $\alpha_{\eta}=0$.

###### Remark 1

Having formalized the concepts of factor strength, missing factors, and the
less restrictive APT condition given by (44), it is now of interest to revisit
the conditions under which the phi-portfolio fully diversifies. Consider (18)
and note that

$\frac{\left\|\boldsymbol{\eta}_{n}\right\|}{\lambda_{min}^{1/2}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}=O\left[n^{-\left(\alpha_{min}/2-\alpha_{\eta}\right)}\right]\text{,
and
}\frac{\lambda_{max}\left(\mathbf{V}_{u}\right)}{\lambda_{min}\left(\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\right)}=O\left[n^{-\left(\alpha_{min}-\alpha_{\gamma}\right)}\right].$

Using these results in (15) and (17) and we have

$E\left(\rho_{t,\phi}\right)=\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}+O\left[n^{-\left(\alpha_{min}/2-\alpha_{\eta}\right)}\right]\text{,
and
}Var\left(\rho_{t,\phi}\right)=O\left[n^{-\left(\alpha_{min}-\alpha_{\gamma}\right)}\right]$

Therefore, for phi-portfolio to dominate the MV portfolio in addition to
$\boldsymbol{\phi}^{\prime}\boldsymbol{\phi}\neq 0$, it is also required that
the strengths of the traded factors $\alpha_{k}$, $k=1,2,...,K$ are strictly
larger than the strength of the missing factor, $\alpha_{\gamma},$ as well as
twice the strength of the idiosyncratic pricing errors, $\alpha_{\eta}$.

##  3 Assumptions and theorems

We make the following standard assumptions about $\mathbf{f}_{t},g_{t}$,
$v_{it}$, $\boldsymbol{\beta}_{i}$, $\eta_{i}$, and $\gamma_{i}$ (the drivers
of asset returns):

###### Assumption 1

(Observed common factors) (a) The $K\times 1$ vector of observed risk factors,
$\mathbf{f}_{t}$, follows the general linear process

$\mathbf{f}_{t}=\boldsymbol{\mu}+\sum_{\ell=0}^{\infty}\mathbf{\Psi}_{\ell}\mathbf{\zeta}_{t-\ell},$
(45)

where $\left\|\boldsymbol{\mu}\right\|<C$, $\mathbf{\zeta}_{t}\thicksim
IID(\mathbf{0},\mathbf{I}_{K})$, and $\mathbf{\Psi}_{\ell}$ are $K\times K$
exponentially decaying matrices such that
$\left\|\mathbf{\Psi}_{\ell}\right\|<C\rho^{\ell}$ for some $C>0$ and
$0<\rho<1$. (b) The $T\times K$ data matrix
$\mathbf{F=(f}_{1},\mathbf{f}_{2},...,\mathbf{f}_{T})^{\prime}$ is full column
rank and there exists $T_{0}$ such that for all $T>T_{0}$,
$\boldsymbol{\hat{\Sigma}}_{f}=T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}$
is a positive definite matrix,
$\lambda_{\max}\left[(T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F)}^{-1}\right]<C$,
$\boldsymbol{\hat{\Sigma}}_{f}\rightarrow_{p}\boldsymbol{\Sigma}_{f}=E\left(\mathbf{f}_{t}-\boldsymbol{\mu}_{0}\right)\left(\mathbf{f}_{t}-\boldsymbol{\mu}_{0}\right)^{\prime}>\mathbf{0},$
where $\boldsymbol{\mu}_{0}$ is the true value of $\boldsymbol{\mu}$.

###### Assumption 2

(Observed factor loadings) (a) The factor loadings $\beta_{ik}$ for
$i=1,2,...,n$ and $k=1,2,...,K$ are stochastically bounded such that
$sup_{ik}E\left(\beta_{ik}^{2}\right)<C$,

$\sum_{i=1}^{n}\left(\beta_{ik}-\bar{\beta}_{k}\right)^{2}=\ominus_{p}(n^{\alpha_{k}})\text{,
for }k=1,2,...,K\text{, }0\leq\alpha_{k}\leq 1,$ (46)

(b)The $n\times K$ matrix of factor loadings,
$\mathbf{B}_{n}=(\boldsymbol{\beta}_{\circ 1},\boldsymbol{\beta}_{\circ
2},...,\boldsymbol{\beta}_{\circ K}),$ where $\boldsymbol{\beta}_{\circ
k}=(\beta_{1k},\beta_{2k},...,\beta_{nk})^{\prime}$ satisfy

$0<c<\lambda_{min}\left(\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}\right)<\lambda_{max}\left(\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}\right)<C<\infty,$
(47)

for some small and large positive constants, $c$ and $C$, where
$\mathbf{M}_{n}=\mathbf{I}_{n}-n^{-1}\boldsymbol{\tau}_{n}\boldsymbol{\tau}_{n}^{\prime}$,
$\boldsymbol{\tau}_{n}=(1,1,...,1)^{\prime},$ and $\mathbf{D}_{\alpha}\,$ is
the $n\times n$ diagonal matrix

$\mathbf{D}_{\alpha}=Diag(n^{\alpha_{1}/2},n^{\alpha_{2}/2},....,n^{\alpha_{K}/2}).$
(48)

###### Assumption 3

(latent factor) (a) The latent factor, $g_{t}$, in (43) is distributed
independently of $\mathbf{f}_{t^{\prime}},$ for all $t$ and $t^{\prime}$,
$g_{t}$ is serially independent with mean zero, $E(g_{t})=0,$
$E(g_{t}^{2})=1$, and a finite fourth order moment, $sup_{t}E(g_{t}^{4})<C$.
(b) The loadings $\gamma_{i}$ are such that $sup_{i}\left|\gamma_{i}\right|<C$
and

$\sum_{i=1}^{n}\left|\gamma_{i}\right|=O(n^{\alpha_{\gamma}}).$ (49)

###### Assumption 4

(idiosyncratic errors) (a) The errors $\left\\{v_{it}\text{,
}i=1,2,...,n;\text{ }t=1,2,...,T\right\\}$ are distributed independently of
the factors $f_{k,t^{\prime}}$, and $g_{t}$, for all $i,t,t^{\prime}$ and
$k=1,2,...,K$, and their associated loadings $\beta_{ik},$ and $\gamma_{i}$.
They are serially independent with $E(v_{it})=0$ and finite fourth order
moments $E(v_{it}^{4})<\infty$, and covariances
$E(v_{it}v_{jt})=\sigma_{v,ij}$, such that

$\sup_{i}\sum_{j=1}^{n}\left|\sigma_{v,ij}\right|<\infty,\text{ and
}\sup_{i}\sum_{j=1}^{n}Cov(v_{it}^{2},v_{jt}^{2})<\infty,\text{ }$ (50)

with $\lambda_{\min}\left(\mathbf{V}_{v}\right)>0$, where
$\mathbf{V}_{v}=(\sigma_{v,ij})$. (b) The degree of cross-sectional dependence
of $v_{it}$ is sufficiently weak so that

$T^{-1/2}n^{-1/2}\sum_{t=1}^{T}\sum_{i=1}^{n}(\beta_{ik}-\bar{\beta}_{k})v_{it}\rightarrow_{d}N(0,\omega_{k}^{2})\text{,
for }k=1,2,...,K,$ (51)

where

$\omega_{k}^{2}=p\lim_{n\rightarrow\infty}n^{-\alpha_{k}}\sum_{i=1}^{n}\sum_{j=1}^{n}(\beta_{ik}-\bar{\beta}_{k})(\beta_{jk}-\bar{\beta}_{k})\sigma_{v,ij}.$
(52)

###### Assumption 5

(Pricing errors) The pricing errors, $\eta_{i}$, for $i=1,2,...,n$ are
individually bounded, $sup_{j}\left|\eta_{j}\right|<C$ , and are distributed
independently of the factor loadings, $\beta_{jk}$, and $\gamma_{j}$ for all
$i,j$ and $k=1,2,...,K$, as well as satisfying the condition

$\sum_{i=1}^{n}\left|\eta_{i}\right|=O\left(n^{\alpha_{\eta}}\right),$ (53)

with $\alpha_{\eta}<1/2$.

###### Remark 2

Under Assumption 1, $E\left(\mathbf{f}_{t}\right)=\boldsymbol{\mu},$ and
$Var(\mathbf{f}_{t})=\boldsymbol{\Sigma}_{f}=\sum_{\ell=0}^{\infty}\mathbf{\Psi}_{\ell}\mathbf{\Psi}_{\ell}^{\prime}$.
Also since
$\left\|\boldsymbol{\Sigma}_{f}\right\|\leq\sum_{\ell=0}^{\infty}\left\|\mathbf{\Psi}_{\ell}\right\|^{2}$
it then follows from part (a) of Assumption 1 that
$\left\|\boldsymbol{\Sigma}_{f}\right\|<C$.

###### Remark 3

Under Assumption 2

$\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}\rightarrow_{p}\mathbf{\Sigma}_{\beta\beta}(\boldsymbol{\alpha}\mathbf{)}>0,$
(54)

where $\mathbf{\Sigma}_{\beta\beta}(\boldsymbol{\alpha}\mathbf{)}$ is a
$k\times k$ symmetric positive definite matrix which is a function of
$\boldsymbol{\alpha}=\mathbf{(}\alpha_{1},\alpha_{2},...,\alpha_{K})^{\prime}$.
This follows from (47) since for any non-zero $n\times 1$ vector
$\boldsymbol{\kappa}$,

$\boldsymbol{\kappa}^{\prime}\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}\boldsymbol{\kappa}\geq\left(\mathbf{\kappa}^{\prime}\mathbf{\kappa}\right)\lambda_{min}\left(\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}\right)>0.$

In the standard case where the factors are all strong ($\alpha_{k}=1$ for all
$k$), we have
$n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\rightarrow_{p}\mathbf{\Sigma}_{\beta\beta}(\boldsymbol{\tau}_{K}\mathbf{)=\Sigma}_{\beta\beta}>0$.

###### Remark 4

The high level condition (51) in Assumption 4 is required for establishing the
asymptotic normality of the estimator of $\boldsymbol{\phi}_{0}$, and is
clearly met when $v_{it}$ and/or $\beta_{ik}$ are independently distributed.
It is also possible to establish (51) under weaker conditions assuming that
$v_{it}$ and/or $\beta_{ik}$ satisfy some time-series type mixing conditions
applied to cross section.

###### Remark 5

The exponent parameter, $\alpha_{\eta}$, of the pricing condition in (53), can
be viewed as the degree to which pricing errors are pervasive in large
economies (as $n\rightarrow\infty$). Letting
$\boldsymbol{\eta}_{n}=(\eta_{1},\eta_{2},...,\eta_{n})^{\prime}\,$ we have

$\sum_{i=1}^{n}\eta_{i}^{2}=\left\|\boldsymbol{\eta}_{n}\right\|^{2}\leq\left\|\boldsymbol{\eta}_{n}\right\|_{\infty}\left\|\boldsymbol{\eta}_{n}\right\|_{1}=sup_{j}\left|\eta_{j}\right|\left(\sum_{i=1}^{n}\left|\eta_{i}\right|\right),$
(55)

and under Assumption 5 it also follows that

$\sum_{i=1}^{n}\eta_{i}^{2}=O\left(n^{\alpha_{\eta}}\right).$ (56)

Similarly

$\sum_{i=1}^{n}\gamma_{i}^{2}=O(n^{\alpha_{\gamma}}).$ (57)

###### Remark 6

Whilst (53) implies (56), the reverse does not follow. By allowing for
$\alpha_{\eta}>0$ we are relaxing the Ross’s boundedness condition that
requires setting $\alpha_{\eta}=0$.

###### Remark 7

The assumption that the observed and missing factors, $\mathbf{f}_{t}$ and
$g_{t^{\prime}},$ are distributed independently is not restrictive and can be
relaxed. For example, suppose that

$g_{t}=\mu_{g}+\boldsymbol{\theta}^{\prime}\mathbf{f}_{t}+v_{gt},$

where $\boldsymbol{f}_{t}$ and $v_{gt}$ are independently distributed. Then
using (43) we have

$u_{it}=\gamma_{i}\mu_{g}+\gamma_{i}\left(\boldsymbol{\theta}^{\prime}\mathbf{f}_{t}\right)+\gamma_{i}v_{gt}+v_{it},$

and the return equation (4) can be written as

$r_{it}=\left(\mathit{a}_{i}+\gamma_{i}\mu_{g}\right)+\left(\boldsymbol{\beta}_{i}+\gamma_{i}\boldsymbol{\theta}\right)^{\prime}\mathbf{f}_{t}+\gamma_{i}v_{gt}+v_{it},$

with $v_{gt}$ now acting as the missing common factor, which, by construction,
is distributed independently of $\mathbf{f}_{t}$.

###### Remark 8

Assumptions 3 and 4 allow $u_{it}$ to be cross-sectionally weakly correlated,
but require the errors to be serially uncorrelated. This requirement is not
strong for asset pricing models, since realized returns are only mildly
serially correlated and most likely such serial dependence will be captured by
the serial correlation in the observed factors.

As we shall see, to estimate and conduct inference on the risk premia
associated with the observed factors, $f_{kt}$, we require
$\alpha_{k}>\alpha_{\gamma}<1/2$, where $\alpha_{\gamma}$ denotes the strength
of the latent factor, $g_{t},$ and similarly defined by
$\sum_{i=1}^{n}\gamma_{i}^{2}=\ominus(n^{\alpha_{\gamma}})$. Namely, the
latent factor must be sufficiently weak so that ignoring it will be
inconsequential, and observed factors sufficiently strong so that they can be
distinguished from the weak latent factor.

The main theoretical results of the paper are set out around five theorems.
Theorem 1 considers the Fama-MacBeth two-step estimator and derives its
limiting property as $n$ and $T\rightarrow\infty$. To eliminate the bias of
Fama-MacBeth estimator we require $n/T\rightarrow 0$, and to eliminate the
effects of pricing errors we need $Tn^{\alpha_{\eta}}/n\rightarrow 0$, which
results in a contradiction. Thus the Fama-MacBeth estimator is valid only when
there are no pricing errors ($\eta_{i}=0$ for all $i$) and when
$n/T\rightarrow 0$. Theorem 2 provides a proof that the estimator of
$\bar{\sigma}_{n}^{2}$ (denoted by $\widehat{\bar{\sigma}}_{nT}^{2}$) proposed
by Shanken (1992) continues to be unbiased for a fixed $T$ as
$n\rightarrow\infty$, even under the general setting of the current paper that
allows for missing factors as well as pricing errors. Theorem 2 also
establishes that
$\widehat{\bar{\sigma}}_{nT}^{2}-\bar{\sigma}_{n}^{2}\rightarrow
O_{p}(n^{-1/2}T^{-1/2})$, which is essential for establishing the results for
the bias-corrected estimator of $\boldsymbol{\phi}_{0}$, namely
$\boldsymbol{\tilde{\phi}}_{nT}$ given by (33), summarized in Theorem 3. This
theorem provides conditions under which $\boldsymbol{\tilde{\phi}}_{nT}$ is a
consistent estimator of $\boldsymbol{\phi}_{0}$, and derives its asymptotic
distribution assuming the observed factors are strong, again allowing for
pricing errors, a missing factor, and other forms of weak error cross-
sectional dependence. Theorem 4 extends the results of Theorem 3 to the case
where one or more of the observed risk factors are semi-strong and shows how
factor strength impacts the precision with which the elements of
$\boldsymbol{\phi}_{0}$ are estimated. Finally, Theorem 5 presents the
conditions under which the asymptotic variance of
$\boldsymbol{\tilde{\phi}}_{nT}$ can be consistently estimated.

###### Theorem 1

(Small $T$ bias of Fama-MacBeth estimator of $\boldsymbol{\lambda}$) Consider
the multi-factor linear return model (23) with the missing factor $g_{t}$ in
$u_{it}$ as defined by (43) and the associated risk premia,
$\boldsymbol{\lambda}$, defined by (5). Suppose that Assumptions 1, 2, 4, 3
and 5 hold and all observed factors are strong. Suppose further that the true
value of the risk premia, $\boldsymbol{\lambda}_{0}$, is estimated by Fama-
MacBeth two-pass estimator, $\boldsymbol{\hat{\lambda}}_{nT}$, defined by
(25). Then for any fixed $T>T_{0}$ such that
$\lambda_{\min}\left(T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}\right)>0,$
we have (as $n\rightarrow\infty$)

$\boldsymbol{\hat{\lambda}}_{nT}-\boldsymbol{\lambda}_{0}=\left(\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0}\right)-\frac{\bar{\sigma}^{2}}{T}\left[\boldsymbol{\Sigma}_{\beta\beta}+\bar{\sigma}^{2}\frac{1}{T}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\right]^{-1}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\lambda}_{T}^{\ast}+o_{p}(1),$
(58)

where $\boldsymbol{\hat{\mu}}_{T}=T^{-1}\sum_{t=1}^{T}\mathbf{f}_{t}$,

$\mathbf{\Sigma}_{\beta\beta}=\lim_{n\rightarrow\infty}\left(\frac{\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}}{n}\right)\text{,
and
}\overline{\sigma}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{i}^{2}>0.$

The proof is provided in Section B.1 of the mathematical appendix.

To derive the asymptotic distribution of
$\boldsymbol{\hat{\lambda}}_{nT}-\boldsymbol{\lambda}_{0}$ it is required that
both $n$ and $T\rightarrow\infty$, jointly. Also, noting that

$\boldsymbol{\hat{\lambda}}_{nT}-\boldsymbol{\lambda}_{0}=\left(\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0}\right)+\left(\boldsymbol{\hat{\phi}}_{nT}-\boldsymbol{\phi}_{0}\right),$

it is clear that increasing $n$ is not relevant for the distribution of
$\boldsymbol{\hat{\mu}}_{T}-\boldsymbol{\mu}_{0}$, but joint $n$ and $T$
asymptotics are required when investigating the distribution of
$\boldsymbol{\hat{\phi}}_{nT}-\boldsymbol{\phi}_{0}$. Focussing on the latter,
and using result (B.12) in the Appendix, we have

$\displaystyle\left(n^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\right)\sqrt{nT}\left(\boldsymbol{\hat{\phi}}_{nT}-\boldsymbol{\phi}_{0}\right)=n^{-1/2}T^{1/2}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\boldsymbol{\eta}_{n}+n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{M}_{n}\boldsymbol{\eta}_{n}$
$\displaystyle+n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\bar{u}}_{n\circ}-n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{U}_{nT}\mathbf{G}_{T}\boldsymbol{\lambda}_{T}^{\ast}.$

Where
$\boldsymbol{U}_{nT}=\left(\mathbf{u}_{1\circ},\mathbf{u}_{2\circ},...,\mathbf{u}_{n\circ}\right)^{\prime},$
$\mathbf{u}_{i\circ}=\left(u_{i1},u_{i2},...,u_{iT}\right)^{\prime},$
$\mathbf{G}_{T}=$
$\mathbf{M}_{T}\mathbf{F}\left(\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}\right)^{-1}$,
$\overline{\mathbf{u}}_{n\circ}=(\overline{u}_{1\circ},\overline{u}_{2\circ},...,\overline{u}_{n\circ})^{\prime},$
and $\overline{u}_{i\circ}=T^{-1}\sum_{t=1}^{T}u_{it}.$ Consider first the
terms that include the pricing errors, $\boldsymbol{\eta}_{n}$, and using the
results in Lemma A.3 note that

$n^{-1/2}T^{1/2}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\boldsymbol{\eta}_{n}=O_{p}\left(T^{1/2}n^{-1/2+\alpha_{\eta}}\right),\text{
}n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{M}_{n}\boldsymbol{\eta}_{n}=O_{p}\left(n^{-1/2+\frac{\alpha_{\eta}+\alpha_{\gamma}}{2}}\right).$

It is clear that the effects of pricing errors on the distribution of
$\boldsymbol{\hat{\phi}}_{nT}$ vanish only if
$T^{1/2}n^{-1/2+\alpha_{\eta}}\rightarrow 0$, and
$\alpha_{\eta}+\alpha_{\gamma}<1$. Also

$\displaystyle
n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\bar{u}}_{n\circ}$
$\displaystyle=O_{p}\left(T^{-1/2}\right)\text{, }$ $\displaystyle
n^{-1/2}T^{1/2}\mathbf{G}_{T}^{\prime}\mathbf{U}_{nT}^{\prime}\mathbf{U}_{nT}\mathbf{G}_{T}\boldsymbol{\lambda}_{T}^{\ast}$
$\displaystyle=\sqrt{\frac{n}{T}}\bar{\sigma}_{n}^{2}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\lambda}_{T}^{\ast}+O_{p}\left(T^{-1/2}\right).$

Finally, for the first two terms involving $\mathbf{B}_{n}$ and
$\mathbf{U}_{nT}$ we have

$n^{-1/2}T^{1/2}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\left(\mathbf{\bar{u}}_{n\circ}-\mathbf{U}_{nT}\mathbf{G}_{T}\boldsymbol{\lambda}_{T}^{\ast}\right)=O_{p}\left(1\right).$
(59)

It is clear that the small $T$ bias of the asymptotic distribution of the two-
step estimator, given by
$\sqrt{\frac{n}{T}}\bar{\sigma}_{n}^{2}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\lambda}_{T}^{\ast}$,
does not vanish unless, $n/T\rightarrow 0$. At the same time for the pricing
errors to have no impact on the distribution of the two-step estimator we must
have $Tn^{a_{\eta}}/n\rightarrow 0$. Both conditions cannot be met
simultaneously. It is possible to derive the asymptotic distribution of
$\boldsymbol{\hat{\phi}}_{nT}$, and hence that of
$\boldsymbol{\hat{\lambda}}_{nT}$, when $n/T\rightarrow 0$ and
$\mathbf{\eta=0}$, but these are quite restrictive conditions, and to avoid
them we follow Shanken (1992) and instead consider a bias-corrected version of
$\boldsymbol{\hat{\phi}}_{nT}$, namely $\boldsymbol{\tilde{\phi}}_{nT}$ given
by (33). As noted earlier
$\boldsymbol{\tilde{\phi}}_{nT}=\boldsymbol{\tilde{\lambda}}_{nT}-\boldsymbol{\hat{\mu}}_{T}$,
where $\boldsymbol{\tilde{\lambda}}_{nT}$ is the bias-corrected version of
$\boldsymbol{\hat{\lambda}}_{nT}$ originally proposed by Shanken.

To investigate the asymptotic properties of $\boldsymbol{\tilde{\phi}}_{nT}$
we first need to establish conditions under which
$\widehat{\bar{\sigma}}_{nT}^{2}$, defined by (27), is a consistent estimator
of $\bar{\sigma}_{n}^{2}=n^{-1}\sum_{i=1}^{n}\sigma_{i}^{2}$, which enters the
bias-corrected estimator. The proof of consistency in the literature does not
allow for missing factors or pricing errors and only considers the case where
$T$ is fixed as $n\rightarrow\infty$. For derivation of asymptotic
distribution of $\boldsymbol{\tilde{\phi}}_{nT}$ we also need to consider the
limiting properties of $\widehat{\bar{\sigma}}_{nT}^{2}$ under joint $n$ and
$T$ asymptotics. The following theorem provides the required results for
$\widehat{\bar{\sigma}}_{nT}^{2}$ as an estimator of $\bar{\sigma}_{n}^{2}$.

###### Theorem 2

Consider $\widehat{\bar{\sigma}}_{nT}^{2}$, the estimator of
$\bar{\sigma}_{n}^{2}$ given by (see (27)),

$\widehat{\bar{\sigma}}_{nT}^{2}=\frac{\sum_{t=1}^{T}\sum_{i=1}^{n}\hat{u}_{it}^{2}}{n(T-K-1)},$
(60)

and suppose that Assumptions 1, 3, and 4, are satisfied. Then for a fixed $T$

$\lim_{n\rightarrow\infty}E\left(\widehat{\bar{\sigma}}_{nT}^{2}\right)=\bar{\sigma}^{2},$
(61)

where $\bar{\sigma}^{2}=\lim_{n\rightarrow\infty}\bar{\sigma}_{n}^{2}$, and
$\bar{\sigma}_{n}^{2}=n^{-1}\sum_{i=1}^{n}\sigma_{i}^{2}$. Furthermore

$\widehat{\bar{\sigma}}_{nT}^{2}-\bar{\sigma}_{n}^{2}=O_{p}\left(T^{-1/2}n^{-1/2}\right).$
(62)

For a proof see sub-section B.2 in the Appendix.

Result (62) shows that $\widehat{\bar{\sigma}}_{nT}^{2}$ continues to be a
consistent estimator of
$\bar{\sigma}^{2}=\lim_{n\rightarrow\infty}\bar{\sigma}_{n}^{2}$ for a fixed
$T$ as $n\rightarrow\infty$, even in the presence of pricing errors and a
missing common factor. This result also holds when one or more of the factors
are semi-strong.

Equipped with the above result we are now in a position to present the theorem
that sets out the asymptotic distribution of $\boldsymbol{\tilde{\phi}}_{nT}$.

###### Theorem 3

Consider $\boldsymbol{\tilde{\phi}}_{nT}$, the bias-corrected estimators of
$\boldsymbol{\phi}_{0}$ given by (33). Suppose Assumptions 1, 2, 4, 3 and 5
hold, all the observed factors are strong, ($\alpha_{k}=1$, for
$k=1,2,...,K$), and the strength of the missing factor, $\alpha_{\gamma}$
defined by (49), satisfies $\alpha_{\gamma}$ $<1/2$. Then

$\displaystyle\boldsymbol{\tilde{\phi}}_{nT}-\boldsymbol{\phi}_{0}$
$\displaystyle=O_{p}\left(T^{-1/2}n^{{}^{-1/2}}\right)+O_{p}\left(T^{-1/2}n^{-1+\frac{\alpha_{\eta}+\alpha_{\gamma}}{2}}\right)$
(63)
$\displaystyle+O_{p}\left(n^{-1+\alpha_{\eta}}\right)+O_{p}\left(T^{-1}n^{-1/2}\right),$

where $\alpha_{\eta}\,$denotes the degree of pervasiveness of the pricing
errors defined by (44). (a) When $T$ is fixed, $\alpha_{\gamma}<1/2$ and
$\alpha_{\eta}<1$, then there exists $T_{0}$ such that for all $T>T_{0}$

$p\lim_{n\rightarrow\infty}\left(\boldsymbol{\tilde{\phi}}_{nT}\right)=\boldsymbol{\phi}_{0}\text{.}$
(64)

Also

$\sqrt{nT}\left(\boldsymbol{\tilde{\phi}}_{nT}-\boldsymbol{\phi}_{0}\right)=\mathbf{\Sigma}_{\beta\beta}^{-1}\boldsymbol{\xi}_{nT}+O_{p}\left(n^{-\frac{1}{2}+\frac{\alpha_{\eta+\alpha_{\gamma}}}{2}}\right)+O_{p}\left(T^{1/2}n^{-1/2+\alpha_{\eta}}\right)+O_{p}\left(T^{-1/2}\right),$
(65)

where
$\mathbf{\Sigma}_{\beta\beta}=p\lim_{n\rightarrow\infty}\left(n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n\
}\right),$

$\boldsymbol{\xi}_{nT}=n^{-1/2}T^{-1/2}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{U}_{nT}\mathbf{a}_{T},$
(66)

and
$\mathbf{a}_{T}=\boldsymbol{\tau}_{T}-\mathbf{M}_{T}\mathbf{F}(T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F)}^{-1}\boldsymbol{\lambda}_{T}^{\ast}$.
(b) If $\alpha_{\gamma}<1/2$, $\alpha_{\eta}<1/2,$ and
$\sqrt{\frac{T}{n}}n^{\alpha_{\eta}}\rightarrow 0$, as $n$ and
$T\rightarrow\infty$ jointly, then

$\sqrt{nT}\left(\boldsymbol{\tilde{\phi}}_{nT}-\boldsymbol{\phi}_{0}\right)\rightarrow_{d}N\left(\mathbf{0,\Sigma}_{\beta\beta}^{-1}\mathbf{V}_{\xi}\mathbf{\Sigma}_{\beta\beta}^{-1}\right),$
(67)

where

$\mathbf{V}_{\xi}=\left(1+\boldsymbol{\lambda}_{0}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}\right)\text{
}p\lim_{n\rightarrow\infty}\left(n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{V}_{u}\mathbf{M}_{n}\mathbf{B}_{n\
}\right).$ (68)

For a proof see sub-section B.3 in the Appendix.

Result (63) establishes the finite $T$ consistency of
$\boldsymbol{\tilde{\phi}}_{nT}$ for $\boldsymbol{\phi}_{0}$ so long as
$\alpha_{\gamma}<1/2$ and $\alpha_{\eta}<1$, thus extending the Shanken result
to a much more general setting. To the best of our knowledge the asymptotic
distribution in (67) is new and shows that the asymptotic covariance matrix of
$\boldsymbol{\tilde{\phi}}_{nT}$ includes the term
$\boldsymbol{\lambda}_{T}^{\ast\prime}(T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F)}^{-1}\boldsymbol{\lambda}_{T}^{\ast}$,
that arises from the first stage estimation of the factor loadings, and must
be included in the analysis for valid inference. It is also clear that this
additional term does not vanish with $T\rightarrow\infty$, and tends to
$\boldsymbol{\lambda}_{0}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}\geq\left(\boldsymbol{\lambda}_{0}^{\prime}\boldsymbol{\lambda}_{0}\right)\lambda_{\max}\left(\mathbf{\Sigma}_{f}^{-1}\right)=\left(\boldsymbol{\lambda}_{0}^{\prime}\boldsymbol{\lambda}_{0}\right)\lambda_{\min}\left(\mathbf{\Sigma}_{f}\right)>0$,
which is strictly non-zero unless $\boldsymbol{\lambda}_{0}=\mathbf{0}$.
Shanken type bias correction addresses the mean of the asymptotic distribution
of $\boldsymbol{\tilde{\phi}}_{nT}$, but not its covariance.

The $O_{p}\left(T^{-1/2}\right)$ term in (65) arises from the sampling errors
involved in the estimation of the factor loadings and $\bar{\sigma}_{n}^{2}$,
and tends to zero at the regular $\sqrt{T}$ rate. But $n$ has to be
sufficiently large to eliminate the effects of pricing errors on
identification of $\boldsymbol{\phi}_{0}$, as dictated by condition
$\sqrt{\frac{T}{n}}n^{\alpha_{\eta}}\rightarrow 0,$ as $n$ and
$T\rightarrow\infty$.777The condition
$\sqrt{\frac{T}{n}}n^{\alpha_{\eta}}\rightarrow 0$ can be weakened somewhat to
$\sqrt{\frac{T}{n}}n^{\alpha_{\eta}/2}\rightarrow 0$ if we also assume that
$\beta_{ik}-\bar{\beta}_{k}$ are independently distributed over $i$, but will
still require $n$ to be larger than $T$. The requirement that $T$ need not be
too large relative to $n$ for estimation of $\boldsymbol{\phi}_{0}$ is
consistent with separating the estimation of $\boldsymbol{\phi}_{0}$ from that
of $\boldsymbol{\mu}_{0}$, allowing the use a relatively small $T$ and a large
$n$ to estimate $\boldsymbol{\phi}_{0}$ and a relatively large $T$ when
estimating $\boldsymbol{\mu}_{0}.$

###  3.1 What if one or more of the risk factors are semi-strong?

We now turn to an intermediate case where one or more of the observed factors
are semi-strong, in the sense that their factor strength, $\alpha_{k}$ lies
between $1/2$ and $1$. The case of weak risk factors is already covered in the
proceeding analysis, and such factors can be included in the error term,
$u_{it}$, with little consequence for the estimation of risk premia of the
remaining factors that are strong or semi-strong. Weak factors do not have any
explanatory power and can be dropped from the analysis.

When one or more of the observed factors is semi-strong
$\mathbf{\Sigma}_{\beta\beta}$ is no longer positive definite and Theorem 3
does not apply, but it is possible to adapt the proofs to establish the
limiting properties of $\tilde{\phi}_{k,nT}\boldsymbol{\ }$(the $k^{th}$
element of $\boldsymbol{\tilde{\phi}}_{nT}$) for different values of
$\alpha_{k}\,$.

To this end, analogously to $\boldsymbol{\tilde{\phi}}_{nT}$, we introduce the
following estimator of $\boldsymbol{\phi}_{0}$

$\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)=\mathbf{H}_{nT}^{-1}\left(\boldsymbol{\alpha}\right)\left[\mathbf{D}_{\alpha}^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\text{\^{a}}_{nT}+\frac{n}{T}\widehat{\bar{\sigma}}_{nT}^{2}\mathbf{D}_{\alpha}^{-1}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\hat{\mu}}\right],$
(69)

where

$\mathbf{H}_{nT}\left(\boldsymbol{\alpha}\right)=\mathbf{D}_{\alpha}^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\mathbf{D}_{\alpha}^{-1}-\frac{n}{T}\widehat{\bar{\sigma}}_{nT}^{2}\mathbf{D}_{\alpha}^{-1}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\mathbf{D}_{\alpha}^{-1}.$
(70)

It is now easily seen that

$\mathbf{D}_{\alpha}\left(\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)-\boldsymbol{\phi}_{0}\right)=\mathbf{H}_{nT}^{-1}\left(\boldsymbol{\alpha}\right)\mathbf{q}_{nT}\left(\boldsymbol{\alpha}\right),$
(71)

where

$\mathbf{q}_{nT}\left(\boldsymbol{\alpha}\right)=\mathbf{D}_{\alpha}^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\text{\^{a}}_{nT}+\frac{n}{T}\widehat{\bar{\sigma}}_{nT}^{2}\mathbf{D}_{\alpha}^{-1}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}\boldsymbol{\hat{\mu}}_{T}-\mathbf{H}_{nT}(\boldsymbol{\alpha})\mathbf{D}_{\alpha}\boldsymbol{\phi}_{0}.$
(72)

$\mathbf{D}_{\alpha}$ is defined by (48), and
$\boldsymbol{\alpha}=\mathbf{(}\alpha_{1},\alpha_{2},...,\alpha_{K})^{\prime}$.
It is easily established that numerically
$\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)$ is identical
to $\boldsymbol{\tilde{\phi}}_{nT}$, and its introduction is primarily for the
purpose of establishing the limiting properties of
$\tilde{\phi}_{k,nT}-\phi_{0,k}$ that do depend on $\alpha_{k}$. Note that

$\mathbf{H}_{nT}\left(\boldsymbol{\alpha}\right)=n\mathbf{D}_{\alpha}^{-1}\mathbf{H}_{nT}\mathbf{D}_{\alpha}^{-1},\text{
and
}\mathbf{q}_{nT}\left(\boldsymbol{\alpha}\right)=n\mathbf{D}_{\alpha}^{-1}\mathbf{s}_{nT}$

where $\mathbf{s}_{nT}$ and $\mathbf{H}_{nT}$ are already defined by (B.23)
and (B.24). Using these in (71) we have

$\mathbf{D}_{\alpha}\left(\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)-\boldsymbol{\phi}_{0}\right)=\left(n\mathbf{D}_{\alpha}^{-1}\mathbf{H}_{nT}\mathbf{D}_{\alpha}^{-1}\right)^{-1}n\mathbf{D}_{\alpha}^{-1}\mathbf{s}_{nT}=\mathbf{D}_{\alpha}\mathbf{H}_{nT}^{-1}\mathbf{s}_{nT},$

and it follows that
$\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)-\boldsymbol{\phi}_{0}=\mathbf{H}_{nT}^{-1}\mathbf{s}_{nT}=\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\tau}_{K}\right)=\boldsymbol{\tilde{\phi}}_{nT}$.
See (B.22).

The convergence results for
$\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)$ are set out
in the following theorem.

###### Theorem 4

Consider $\boldsymbol{\tilde{\phi}}_{nT}\left(\boldsymbol{\alpha}\right)$, the
bias-corrected estimators of $\boldsymbol{\phi}_{0}$ given by (69), and
suppose Assumptions 1, 2, 4, 3 and 5 hold, the strength of observed factors,
$\boldsymbol{f}_{t}=(f_{1t},f_{2t},...,f_{Kt})^{\prime},$ is given by
$\boldsymbol{\alpha}\mathbf{=(}\alpha_{1},\alpha_{2},...,\alpha_{K})^{\prime}$,
and the strength of the missing factor, $g_{t}$, defined by (49) is
$\alpha_{\gamma}$. Let $\alpha_{\min}=\min_{k}(\alpha_{k})$ and suppose that
$\alpha_{\gamma}<1/2$. Then

$\mathbf{H}_{nT}\left(\boldsymbol{\alpha}\right)=\mathbf{D}_{\alpha}^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{B}_{n}\mathbf{D}_{\alpha}^{-1}+O_{p}\left(T^{-1}n^{-\alpha_{\min}+1/2}\right),$

where $\mathbf{H}_{nT}\left(\boldsymbol{\alpha}\right)$ is given by (70), and
by part (b) of Assumption 2,
$\mathbf{H}_{nT}\left(\boldsymbol{\alpha}\right)\rightarrow_{p}\mathbf{\Sigma}_{\beta\beta}(\boldsymbol{\alpha}\mathbf{)}>0$,
for any fixed $T>T_{0}$ such that
$\lambda_{\max}\left(\frac{\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}}{T}\right)^{-1}<C$
and $\alpha_{\min}>1/2>\alpha_{\gamma}$. Also

$\displaystyle\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,k}$
$\displaystyle=O_{p}\left(n^{-(\alpha_{k}+\alpha_{\min})/2+1/2}T^{-1/2}\right)+O_{p}\left(n^{\frac{-\left(\alpha_{k}+\alpha_{\min}\right)+\left(\alpha_{\eta+\alpha_{\gamma}}\right)}{2}}T^{-1/2}\right)$
(73)
$\displaystyle+O_{p}\left(n^{-\left(\alpha_{k}+\alpha_{\min}\right)/2+\alpha_{\eta}}\right)+O_{p}\left(n^{-(\alpha_{k}+\alpha_{\min})/2+1/2}T^{-1}\right).$

See sub-section B.4 of the Appendix for a proof.

The result in (73) establishes the consistency of
$\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)=\tilde{\phi}_{k,nT}$ even
if $f_{kt}$ is semi-strong so long as $n\rightarrow\infty$, and
$\alpha_{\min}>1/2,$ $\alpha_{k}+\alpha_{\min}>\alpha_{\eta}+\alpha_{\gamma}$,
and $\alpha_{k}+\alpha_{\min}>2\alpha_{\eta}$. Clearly, these results reduce
to the case of strong factors where $\alpha_{\min}=\alpha_{k}=1$. Turning to
the asymptotic distribution of
$\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)$, again only convergence
rates are affected, and instead of the regular rate of $\sqrt{nT}$, we have $\
\sqrt{T}n^{(\alpha_{k}+\alpha_{\min}-1)/2}$, and using (73) we have

$\displaystyle\sqrt{T}n^{(\alpha_{k}+\alpha_{\min}-1)/2}\left(\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,k}\right)$
(74)
$\displaystyle=O_{p}(1)+O_{p}\left(n^{-1/2+\frac{\left(\alpha_{\eta}+\alpha_{\gamma}\right)}{2}}\right)+O_{p}\left(\sqrt{T}n^{-1/2+\alpha_{\eta}}\right)+O_{p}\left(T^{-1/2}\right).$

The conditions needed for eliminating the effects of the pricing errors are
the same as before and are given by $\alpha_{\eta}+\alpha_{\gamma}<1$ and
$\sqrt{T}n^{-1/2+\alpha_{\eta}}\rightarrow 0$. The asymptotic distribution is
unaffected except for the slower rate of convergence alluded to above. It is
also of interest to note that adding semi-strong factors can adversely affect
the convergence rate of the strong factor with $\alpha_{k}=1$. As an example
suppose the asset pricing model contains two factors, one strong,
$\alpha_{1}=1$ and one semi strong with $\alpha_{2}<1$. Then the convergence
rate of $\tilde{\phi}_{1,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,1}$ is
given by$\sqrt{T}$ $n^{(1+\alpha_{2}-1)/2}$ which is slower than the rate we
would have obtained for
$\tilde{\phi}_{1,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,1}$ if both
factors were strong ($\alpha_{min}=\alpha_{k}=1)$, namely the regular rate of
$\sqrt{nT}$.

Furthermore, when conditions $\alpha_{\eta}+\alpha_{\gamma}<1$ and
$\sqrt{T}n^{-1/2+\alpha_{\eta}}\rightarrow 0$ are met we have

$\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,k}=O_{p}\left(T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\right),$
(75)

and $\phi_{0,k}$ is consistently estimated if
$T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\rightarrow 0$. Also, using (34),
an estimator of the risk premia, $\lambda_{k}$, is given by
$\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)=\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)+\hat{\mu}_{k,T}$,
where $\hat{\mu}_{k,T}=T^{-1}\sum_{t=1}^{T}f_{kt}$. Hence

$\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)-\lambda_{0,k}=\left[\tilde{\phi}_{k,nT}\left(\boldsymbol{\alpha}\right)-\phi_{0,k}\right]+\left(\hat{\mu}_{k,T}-\mu_{0,k}\right).$

Under Assumption 1 $\hat{\mu}_{k,T}-\mu_{0,k}=O_{p}\left(T^{-1/2}\right)$, and
using (75) it then follows that

$\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)-\lambda_{0,k}=O_{p}\left(T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\right)+O_{p}\left(T^{-1/2}\right),$
(76)

and $\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)$ is a consistent
estimator of $\lambda_{0,k}$ if $T\rightarrow\infty$ as well as
$T^{-1/2}n^{-(\alpha_{k}+\alpha_{\min}-1)/2}\rightarrow 0$. More specifically,
suppose $T=\ominus\left(n^{d}\right)$ for some $d>0$, where
$\ominus\left(\cdot\right)$ denotes $T$ and $n^{d}$ are of the same order of
magnitude. Then for any $d>0$, the condition for consistency of
$\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)$ is given by
$\alpha_{k}+\alpha_{min}+d>1$ and $d>0$. In the case where all risk factors
have the same strength, $\alpha$, the consistency condition reduces to
$\alpha>\left(1-d\right)/2$, which is weaker than the one derived by Giglio,
Xiu, and Zhang (2023), namely $n/(\left\|\beta\right\|^{2}T)\rightarrow 0$,
where, in terms of our notation,
$\left\|\beta\right\|^{2}=\ominus\left(n^{\alpha}\right)$. This latter
condition will be met if $\alpha>1-d$. In practice where $T$ is small relative
to $n$, the accuracy of
$\tilde{\lambda}_{k,nT}\left(\boldsymbol{\alpha}\right)$ as an estimator
$\lambda_{0,k}$ does depend on $\alpha$, and our weaker condition on
$\alpha>\left(1-d\right)/2$ is advantageous.

###  3.2 Consistent estimation of the variance of
$\boldsymbol{\tilde{\phi}}_{nT}$

To carry out inference on $\boldsymbol{\phi}_{0}$, or any of its elements
individually, we require a consistent estimator of
$Var\left(\boldsymbol{\tilde{\phi}}_{nT}\right)$. Using (67) and (68) we first
note that $\mathbf{\Sigma}_{\beta\beta}$ is consistently estimated by
$\mathbf{H}_{nT\ }$ given by (30). Therefore, it is sufficient to find a
suitable estimator of $\mathbf{V}_{u}=(\sigma_{ij})$ such that
$\mathbf{V}_{\xi}$ given by (68) is consistently estimated. Under suitable
sparsity restrictions $\mathbf{V}_{u}$ can be consistently estimated using the
various thresholding procedures advanced in the statistical literature by
Bickel and Levina (2008a, b), Cai and Liu (2011), and Bailey, Pesaran, and
Smith (2019, BPS). Fan, Liao, and Mincheva (2011, 2013) also show that the
adaptive threshold technique of Cai and Liu applies equally to the residuals
from an approximate factor model. Here we consider the threshold estimator
proposed by BPS which does not require cross-validation and is shown to have
desirable small sample properties. It is given by
$\mathbf{\tilde{V}}_{u}=\left(\tilde{\sigma}_{ij}\right)$

$\displaystyle\tilde{\sigma}_{ii}$ $\displaystyle=\hat{\sigma}_{ii}$
$\displaystyle\tilde{\sigma}_{ij}$
$\displaystyle=\hat{\sigma}_{ij}\mathbf{1}\left[\left|\hat{\rho}_{ij}\right|>T^{-1/2}c_{\alpha}(n,\delta)\right],\text{
}i=1,2,\ldots,n-1,\text{ }j=i+1,\ldots,n,$ (77)

where

$\hat{\sigma}_{ij}=\frac{1}{T}\sum_{t=1}^{T}\hat{u}_{it}\hat{u}_{jt},\text{
}\hat{\rho}_{ij}=\frac{\hat{\sigma}_{ij}}{\sqrt{\hat{\sigma}_{ii}\hat{\sigma}_{jj}}},\text{
\
}\hat{u}_{it}=r_{it}-\hat{\alpha}_{i,T}-\boldsymbol{\hat{\beta}}_{i,T}^{\prime}\mathbf{f}_{t},$
(78)

and $c_{p}(n,\delta)=\Phi^{-1}\left(1-\frac{p}{2n^{\delta}}\right),$ is a
normal critical value function, $p$ is the the nominal size of testing of
$\sigma_{ij}=0$, ($i\neq j$) and $\delta$ is chosen to take account of the
$n(n-1)/2$ multiple tests being carried out. Monte Carlo experiments carried
out by BPS suggest setting $\delta=2$. The variance estimator given by (77)
does not require a knowledge of the factor strength and applies to risk
factors of differing degrees.

Under Assumptions 1, 3, and 4,
$\left\|\mathbf{V}_{u}\right\|=O\left(n^{\alpha_{\gamma}}\right)$, and using
results in Fan, Liao, and Mincheva (2011, 2013) we have

$\left\|\mathbf{\tilde{V}}_{u}-\mathbf{V}_{u}\right\|=O_{p}\left(n^{\alpha_{\gamma}}\sqrt{\frac{\ln(n)}{T}}\right).$
(79)

Consider the following estimator of $\mathbf{V}_{\xi}$

$\mathbf{\hat{V}}_{\xi,nT}=\left(1+\hat{s}_{nT}\right)\left(n^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\tilde{V}}_{u}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\right)\text{.}$

where
$\hat{s}_{nT}=\boldsymbol{\tilde{\lambda}}_{nT}^{{}^{\prime}}\left(T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F}\right)^{-1}\boldsymbol{\tilde{\lambda}}_{nT}$.
Under Assumption 1,
$T^{-1}\mathbf{F}^{\prime}\mathbf{M}_{T}\mathbf{F\rightarrow}_{p}\mathbf{\Sigma}_{f}$
and using the results above we have
$\boldsymbol{\tilde{\lambda}}_{nT}=\boldsymbol{\tilde{\phi}}_{nT}+\boldsymbol{\hat{\mu}}_{T}\rightarrow_{p}\boldsymbol{\phi}_{0}+\boldsymbol{\mu}_{0}=\boldsymbol{\lambda}_{0}$.
Hence,
$\hat{s}_{nT}\rightarrow_{p}\boldsymbol{\lambda}_{0}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}$
as $n,T\rightarrow\infty$, jointly, and it is sufficient to show that

$n^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\tilde{V}}_{u}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}-n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{V}_{u}\mathbf{M}_{n}\mathbf{B}_{n\
}\rightarrow_{p}\mathbf{0}\text{.}$ (80)

The following theorem provides a formal statement of the conditions under
which $\mathbf{\hat{V}}_{\xi,nT}$ is a consistent estimator of
$\mathbf{V}_{\xi}$.

###### Theorem 5

Suppose Assumptions 1, 2, 4, 3 and 5 hold, and all the observed factors are
strong, ($\alpha_{k}=1$, for $k=1,2,...,K$), and the strength of the missing
factor, $g_{t}$, defined by (49), $\alpha_{\gamma}$ $<1/2$. Then

$\left\|\mathbf{\hat{V}}_{\xi,nT}-\mathbf{V}_{\xi}\right\|=O_{p}\left(n^{\alpha_{\gamma}}\sqrt{\frac{\ln(n)}{T}}\right),$
(81)

where

$\mathbf{\hat{V}}_{\xi,nT}=\left(1+\hat{s}_{nT}\right)\left(n^{-1}\mathbf{\hat{B}}_{nT}^{\prime}\mathbf{M}_{n}\mathbf{\tilde{V}}_{u}\mathbf{M}_{n}\mathbf{\hat{B}}_{nT}\right),$
(82)

$\mathbf{\tilde{V}}_{u}=\left(\tilde{\sigma}_{ij}\right)$,
$\tilde{\sigma}_{ij}$ is the threshold estimator of $\sigma_{ij}$ given by
(77), and

$\mathbf{V}_{\xi}=\left(1+\boldsymbol{\lambda}_{0}^{\prime}\mathbf{\Sigma}_{f}^{-1}\boldsymbol{\lambda}_{0}\right)\text{
}p\lim_{n\rightarrow\infty}\left(n^{-1}\mathbf{B}_{n}^{\prime}\mathbf{M}_{n}\mathbf{V}_{u}\mathbf{M}_{n}\mathbf{B}_{n\
}\right),$ (83)

For a proof see sub-section B.5 in the Appendix.

This theorem shows that consistent estimation of
$Var\left(\boldsymbol{\tilde{\phi}}_{nT}\right)$ can be achieved by using a
suitable threshold estimator of $\mathbf{V}_{u}$, so long as the strength of
the missing factor, $\alpha_{\gamma}$, is sufficiently weak in the sense that
$n^{\alpha_{\gamma}}\sqrt{\ln(n)/T}\rightarrow 0$ as $n,T\rightarrow\infty$.

##  4 Small sample properties of the estimators and tests for
$\boldsymbol{\phi}$

###  4.1 Monte Carlo Design

This section presents Monte Carlo simulations to investigate the small sample
properties of estimators and tests for $\boldsymbol{\phi}_{0}$. In the
empirical application of the next section the factors are selected from a
large list. But here we assume $K=3$ and mimic the 3 Fama-French factors,
namely the market return minus the risk free rate, MKT, the value factor (high
minus low book to market portfolios, HML) and the size factor (small minus big
portfolios, SMB). These are denoted by $f_{kt},$ $k=M,H,S$.888Data on factors
and the risk free rate are downloaded from Kenneth French’s data library:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html For
further details see Section S3 of the online supplement A.

####  4.1.1 Loadings and factor strengths

To calibrate the loadings, $\beta_{ik}$, we used excess returns on a large
number securities observed over the shorter sample covering the 20 years
$2002m1$ $-2021m12$ ($T=240$). Monthly returns for NYSE and NASDAQ stocks code
10 and 11 from CRSP were downloaded from Wharton Research Data Services and
converted to excess returns over the risk free rate, taken from Kenneth
French’s webpages, in percent per month. Only stocks with available data for
the full sample were included, yielding a balanced panel, and to avoid
outliers influencing the results, stocks with a kurtosis greater than 16 were
excluded. There were $1289$ stocks before exclusion on the basis of kurtosis
and $1175$ after. The summary statistics giving mean, median, standard
deviation of the estimates of $\beta_{ik}$ and their histograms are provided
in the online supplement A.

For factor strength, we considered a range of DGPs. Given the evidence that
most factors, other than the market factor, are not strong, we focus on the
case where there is one strong factor, namely the market factor with
$\alpha_{M}=1,$ plus two semi-strong factors, with the value factor, $HML$,
being quite strong with $\alpha_{H}=0.85,$ and the size factor, $SMB$, being
only moderately strong with $\alpha_{S}=0.65$. These estimates are also
informed by the results provided in Bailey, Kapetanios, and Pesaran (2021) who
propose methods for estimation of factor strength. For a given factor
strength, $\alpha_{k}$, the associated loadings, $\beta_{ik}$, are generated
as
$\boldsymbol{\beta}_{k}=(\beta_{1k},\beta_{2k},,...,\beta_{\alpha_{k}},0,0,...0)$
where $n_{\alpha_{k}}=\lfloor n^{\alpha_{k}}\rfloor$ the integer part of
$n^{\alpha_{k}}$, with non-zero and zero values of $\boldsymbol{\beta}_{k}$
given by

$\displaystyle\beta_{ik}$ $\displaystyle\sim
IIDN(\mu_{\beta_{k}},\sigma_{\beta_{k}}^{2})\text{, for }i=1,2,....,\lfloor
n^{\alpha_{k}}\rfloor,$ $\displaystyle\beta_{ik}$ $\displaystyle=0\text{ for
}i=\lfloor n^{\alpha_{k}}\rfloor+1,\lfloor n^{\alpha_{k}}\rfloor+2,...,n,$

where $\lfloor n^{\alpha_{k}}\rfloor$ denotes the integer part of
$n^{\alpha_{k}}$. Since the security returns are randomly generated, it does
not matter how zero and non-zero values of $\beta_{ik}$ are distributed across
$i$. Also, the zero loadings can also be replaced by an exponentially decaying
sequence without any implications for the simulation results.999See also
footnote 5 of Bailey, Kapetanios, and Pesaran (2016, p.942). We also set

$\displaystyle\mu_{\beta_{M}}$ $\displaystyle=1\text{,
}\sigma_{\beta_{M}}=0.4;\text{ }\mu_{\beta_{H}}=0.2\text{,
}\sigma_{\beta_{H}}=0.5$ $\displaystyle\mu_{\beta_{S}}$
$\displaystyle=0.6\text{, }\sigma_{\beta_{S}}=0.5,$

which match the mean and standard deviation of the estimates of $\beta_{ik}$.
See above.

####  4.1.2 Generation of pricing errors

The pricing errors in (6) can be considered as firm-specific characteristics
and are set as
$\boldsymbol{\eta}_{n}=\left(\eta_{1},\eta_{2},...,\eta_{n_{\eta}},0,0,...,0\right)^{\prime}.$
The non-zero loadings of $\boldsymbol{\eta}_{n}$ for $i\leq n_{\eta}=\lfloor
n^{\alpha_{\eta}}\rfloor$ are drawn from $IIDU(0.7,0.9)$, and $\eta_{i}=0$ for
$i=n_{\eta}+1,n_{\eta}+2,....,n.$ We consider $\alpha_{\eta}=(0,0.3).$ When
$\alpha_{\eta}=0$ we have $\eta_{i}=0$ , for all $i.$ As in the case of factor
loadings the non-zero values of $\boldsymbol{\eta}_{n}$ must be randomly
allocated to different groups.

####  4.1.3 Generation of return equation errors

The return equation errors, $u_{it}$, are generated following (43) as a
combination of a missing factor, $g_{t}\sim IIDN\left(0,1\right)$ plus an
idiosyncratic error, $v_{jt}.$ The loadings
$\boldsymbol{\gamma}=\left(\gamma_{1},\gamma_{2},...,\gamma_{n_{\gamma}},0,0,...,0\right)^{\prime}$
of the missing factor are set as

$\displaystyle\gamma_{i}$ $\displaystyle\sim IIDU(0.7,0.9)\text{, for
}i=1,2,....,\lfloor n^{\alpha_{\gamma}}\rfloor,$ $\displaystyle\gamma_{i}$
$\displaystyle=0\text{, for }i=\lfloor n^{\alpha_{\gamma}}\rfloor+1,\lfloor
n^{\alpha_{\gamma}}\rfloor+2,...,n,$

where $\alpha_{\gamma}$ is the strength of the missing factor $g_{t}$. We
consider $\alpha_{\gamma}=1/4$ and $1/2$.

For the idiosyncratic errors, $v_{it}$, we consider spatial as well as a block
diagonal specification, with the spatial specification including a diagonal
specification as the special case. Under the spatial specification the
idiosyncratic errors are generated as the first order spatial autoregressive
model
$v_{it}=\rho_{\varepsilon}\sum_{j=1}^{n}w_{ij}v_{jt}+\kappa\varepsilon_{it},$
which can be written in matrix notation as
$\mathbf{v}_{t}=\rho_{\varepsilon}\mathbf{Wv}_{t}+\kappa\boldsymbol{\varepsilon}_{t},$
and solved for as
$\mathbf{v}_{t}=\kappa\left(\mathbf{I}_{n}-\rho_{\varepsilon}\mathbf{W}\right)^{-1}\boldsymbol{\varepsilon}_{t}$.
Adding the missing factor now yields

$\mathbf{u}_{t}=\boldsymbol{\gamma}\text{
}g_{t}+\kappa\left(\mathbf{I}_{n}-\rho_{\varepsilon}\mathbf{W}\right)^{-1}\boldsymbol{\varepsilon}_{t}.$
(84)

The spatial coefficient $\rho_{\varepsilon}$ is such that
$|\rho_{\varepsilon}|<1$, $\mathbf{W=(}w_{ij})$ with $w_{ii}=0$, and
$\sum_{j=1}^{n}w_{ij}=1$. The diagonal case is obtained by setting
$\rho_{\varepsilon}=0$, with $\rho_{\varepsilon}=0.5$ characterizing the SAR
specification. The weight matrix $\mathbf{W}=(w_{ij})$ is set to follow the
familiar rook pattern where all its elements are set to zero except for
$w_{i+1,i}=w_{j-1,j}=0.5$ for $i=1,2,...,n-2$ and $j=3,4...,n$, with
$w_{1,2}=w_{n,n-1}=1$.

Under the block error covariance specification, $\mathbf{v}_{t}$ is generated
as $\mathbf{v}_{t}=\kappa\mathbf{\hat{S}}\boldsymbol{\varepsilon}_{t},$ where
$\mathbf{\hat{S}}$ is a block diagonal matrix with its $b^{th}$ block given by
$\mathbf{\hat{S}}_{b}$ for $b=1,2,...,B$, and
$\boldsymbol{\varepsilon}_{t}=(\mathbf{\varepsilon}_{1t}^{\prime},\mathbf{\varepsilon}_{2t}^{\prime},...,\mathbf{\varepsilon}_{Bt}^{\prime})^{\prime}$,
and
$\boldsymbol{\varepsilon}_{bt}=(\varepsilon_{b,1t},\varepsilon_{b,2t},...,\varepsilon_{b,n_{b},t})^{\prime}$.
$\mathbf{\hat{S}}$ is set as a Cholesky factor of the correlation matrix of
$\mathbf{u}_{t}$. Denoting this correlation matrix by $\mathbf{\hat{R}}_{u},$

$\mathbf{\hat{R}}_{u}=\left[Diag(\mathbf{\hat{V}}_{Bu})\right]^{-1/2}\mathbf{\hat{V}}_{Bu}\left[Diag(\mathbf{\hat{V}}_{Bu})\right]^{-1/2}=Diag(\mathbf{\hat{R}}_{bu},\text{
}b=1,2,...,B),$

where $\mathbf{\hat{V}}_{Bu}$ is the threshold estimator of $\mathbf{V}_{u}$
subject to the additional restriction that $\mathbf{V}_{u}$ is block diagonal.
For each block $\mathbf{\hat{R}}_{bu}$ we set the number of distinct non-zero
elements of this block equal to the integer part of $[n_{b}(n_{b}-1)/2]\times
q_{b}$ where $q_{b}$ is the proportion of non-zero distinct elements in block
$b$ of our calibrated sample and computed by the calibration over the sample
$2001m10-2021m9$. The non-zero elements are drawn randomly from $IIDU(0,0.5)$.
Similarly, adding the missing factor, we have

$\mathbf{u}_{t}=\boldsymbol{\gamma}g_{t}+\kappa\mathbf{\hat{S}}\boldsymbol{\varepsilon}_{t}.$
(85)

The block diagonal structure is intended to capture possible within industry
correlations not picked up by observed or weak missing factors, with each
block representing an industry or sector. To calibrate the block structure
estimates of the pair-wise correlations between the residuals of the return
regressions using the Fama-French three factors of the $T=240$ sample ending
in $2021$ were obtained. Then all the statistically insignificant correlations
were set to zero, allowing for the multiple testing nature of the tests. For
the majority of securities (668 out of the 1168), the pair-wise return
correlations were not statistically significant. The securities with a
relatively large number of non-zero correlations were either in the banking or
energy related industries. Considering stocks by 2-digit SIC classifications,
a division into $B=14$ contiguous groups ranging in size from $33$ to $145$
stocks, seemed sensible. More detail on the process is given in Section S4 of
the online supplement A.

The primitive errors, $\varepsilon_{it}$ for $i=1,2,...,n$ in (84) and (85)
are generated as $\varepsilon_{it}=\sqrt{\sigma_{ii}}\varpi_{it}$, where
$\varpi_{it}\sim IIDN\left(0,1\right)$, and
$\varepsilon_{it}=\sqrt{\sigma_{ii}}\left[\sqrt{\frac{v-2}{v}}\varpi_{it}\right],$
where $\varpi_{it}\sim IID$ $t(v)$, with $t(v)$ denotes a standard $t$
distributed variate with $v=5$ degrees of freedom. Also $\sigma_{ii}\thicksim
IID$ $0.5(1+\chi_{1}^{2})$ $,\ for$ $i=1,2,...,n_{b}$ and $b=1,2,...,B$. In
this way, it is ensured that $Var(\varepsilon_{b,it})=\sigma_{b,ii}$, and on
average $E\left[Var(\varepsilon_{b,it})\right]=E(\sigma_{ii})=1$, under both
Gaussian and t-distributed errors. Note that
$Var\left(\nu_{b,it}\right)=v/(v-2)$. All the experiments are designed to give
an $R^{2}$ of about $0.3,$ similar to that obtained in the empirical
applications. For further details see sub-section S-3.5 of the online
supplement A.

####  4.1.4 Experiments

In total, we consider 12 experimental designs: six designs with Gaussian
errors and six with $t(5)$ distributed errors. We considered designs with
GARCH effects, with and without pricing errors, $\eta_{i}$, and with and
without the missing factor, $g_{t}$. We also considered designs with spatial
patterns in the idiosyncratic errors, $v_{it}$. All experiments are
implemented using $R=2,000$ replications. Details of of the 12 experiments are
summarized in Table S-1 of the online supplement B (MC results).

####  4.1.5 Alternative estimators of $\mathbf{V}_{u}$

Subsection 3.2 considered consistent estimation of the variance of
$\tilde{\phi}_{nT}$ using $\mathbf{\tilde{V}}_{u}$ a threshold estimator for
$\mathbf{V}_{u},$ given by equation (77). For comparison purposes we also
considered two other estimators of $\mathbf{V}_{u}.$ These were the sample
covariance matrix
$\mathbf{\hat{V}}_{u}=\sum_{t=1}^{T}\mathbf{\hat{u}}_{t}\mathbf{\hat{u}}_{t}^{{}^{\prime}}/T$
and a diagonal covariance matrix, where the off-diagonal elements of
$\mathbf{\hat{V}}_{u},$ $\hat{\sigma}_{ij},$ are set to zero. Thus we have
three designs for the return error covariance matrix, $\mathbf{V}_{u},$ and
three different estimators of it. A comparison of the results for the
different covariance matrices is available on request. The diagonal estimator,
as to be expected, performed poorly when the true covariance matrix was not
diagonal, particularly for the spatial error covariance matrix and when the
strength of the missing factor was close to $1/2$. For these designs the
sample and threshold estimators of the covariance matrix generally performed
similarly and given that there is a theoretical justification for the
threshold estimator and there are structures of the error covariance matrix
for which the sample estimator is unlikely to perform well we report the
results using the threshold estimator in the simulations below.

###  4.2 Monte Carlo results

We focus on a comparison of two-step (defined by (32) and the bias-corrected
(BC) estimator (defined by (33)), and report bias, root mean square error
(RMSE) and size for testing $H_{0j}\,:\phi_{0k}=0,$ $k=M,H,S$ at the five per
cent nominal level, for all $n=100,500,1,000,3,000$ and $T=60,$ $120,$ $240$
combinations. The results for all 12 experiments are summarized in Tables
S-A-E1 to S-A-E12 in the online supplement B. In terms of bias and RMSE the
two-step estimator does much better than the bias-corrected (BC) estimator
when $T=60$ and $n=100$, but this gap closes quickly as $n$ is increased. In
fact for $T=60$ and $n=3,000,$ the bias and RMSE of the BC estimator (at
$0.0010$ and $0.0607)$ are much less than those of the two-step estimator (at
-$0.0080$ and $0.1489)$. This pattern continues to hold when $T=120$ and
$240$. Bias correction can cause the RMSE to ”blow up” for small samples, such
as $n=100$ and $T=60$, but for $n=500$ and above the bias-corrected estimator
always has a smaller RMSE than the two-step estimator. As discussed in the
theoretical section, having a large $n$ is important for the properties of the
estimators.

But most importantly, the two-step estimator is subject to substantial size
distortions, particularly when $T$ is small relative to $n$. As predicted by
the theory, the degree of over-rejection of the tests based on FM estimator
falls with $T,$ but increases with $n$. For example, the two-step test sizes
rise from $11.1\%$ when $T=60$ and $n=100$ to $60.9\%$ when $T=60$ and
$n=3,000$. Increasing $T$ reduces the size distortion of the two-step
estimator but test sizes are still substantially above the $5\%$ nominal value
when $n$ is large. The strong tendency of the tests based on the two-step
estimator to over-reject could be an important contributory factor leading to
false discovery of a large number of apparently significant factors in the
literature. In contrast, sizes of the tests based on the BC estimator, using
the variance estimator given by (37), are all close to its nominal value,
irrespective of the factor strength or sample size combinations. We only note
some elevated test sizes in the case of the experimental design 12, and when
we consider the semi-strong factors. The highest test size of $7.85$ per cent
is obtained for the least strong factor, $f_{st}$, when $n=3000$ and $T=60$.
See Table S-A-E12 of the online supplement B.

We also experimented with raising $\alpha_{\eta}$ from $0.3$ to $0.5,$ making
the pricing errors much more pervasive and $\rho_{\varepsilon}$ from $0.5$ to
$0.85,$ introducing more spatial correlation. This increased the rejection
rate in experiments 9 and 10.

####  4.2.1 Empirical power functions

Plots of the empirical power functions for testing the null hypothesis
$H_{0j}\,:\phi_{0k}=0,$ $k=M,H,S$, are also provided in the online supplement
B (Figures S-A-E1 to S-A-E12). All power curves have the familiar bell curve
shape and tend to unity as $n$ and/or $T$ are increased, showing the test has
satisfactory power, particularly for $n$ sufficiently large even when $T=60$.
Again the power functions are quite similar across the 12 different
experiments and show similar patterns for strong and semi-strong factors.
However, this similarity hides the fact that the test of $\phi_{k}=0$ for the
strong factor is much more powerful than corresponding tests for the semi-
strong factors, with the test power declining as factor strength is reduced.

####  4.2.2 Differences in performance of strong and semi-strong factors

These differences in the effects of factor strength on the power of the test
of $\phi_{k}=0$ are in line with our theoretical results, and are also
reflected in the rate at which the RMSE of the estimators of $\phi_{M}$,
$\phi_{H}$ and $\phi_{S}$ fall with $n$. For example, using results in Table
S-A-E12, for the bias-corrected estimator and design 12 with $T=240,$ the
ratio of RMSE of $n=3,000$ to $n=100$ is $17\%$ for the strong factor
$\left(\alpha_{M}=1\right)$, $23\%$ for the first semi-strong factor with
$\alpha_{H}=0.85$, and $36\%$ for the second semi-strong factor with
$\alpha_{S}=0.65.$ As strength falls one needs larger cross section samples of
securities to attain the same level of precision. In the case of the two-step
estimator there was the same pattern, but the fall in the RMSE with $n$ was
much slower. The ratio for $T=240$ of RMSE of $n=3,000$ to $n=100$ is $25\%$
for $\phi_{M}$, rather than $17\%,$ which it was for the bias-corrected
estimator; $48\%$ rather than $23\%$ for $\phi_{H}$, and the coefficient of
the third least strong factor the RMSE fell then increased with $n$.

####  4.2.3 Semi-strong versus weak factors

So far, we have assumed that the DGP is correctly specified with two semi-
strong and no observed weak factors. Here we consider the implications of
incorrectly excluding semi-strong factors or correctly including weak factors
on the small sample properties of the BC estimator of $\phi_{M},$ the
coefficient of the strong factor. Using the same DGP (which includes one
strong factor and two semi-strong factors), we carried out additional MC
experiments (designs 1-12) where we also estimated $\phi_{M}$ without the
semi-strong factors being included in the regression. Comparative results,
with and without the semi-strong factors, are summarized in Tables S-B-E1 to
S-C-E12 of the online supplement B. We find that incorrectly excluding semi-
strong factors can be quite costly, both in terms of bias and RMSE as well as
size distortions. In terms of RMSE it was almost always better to estimate the
model with the semi strong factors included. The exception was for the case of
$T=60,$ $n=100,$ where including the semi-strong factors with a very small
sample size caused the RMSE to blow up. Size distortions resulting from the
exclusion of the semi-strong factors tended to be more pronounced for large
$n$ and $T$ samples. These conclusions were not sensitive to the choice of the
experimental design. For these experiments the lesson seems to be that it is
important to have $n$ large and include relevant semi-strong factors providing
that they are sufficiently strong.

When the DGP includes one strong factor ($\alpha_{M}=1$) and two weak factors
($\alpha_{H}=$ $\alpha_{S}=0.5)$, in terms of the bias and RMSE for $\phi_{M}$
it is unambiguously better to exclude the weak factors from the regression,
even though they are in the DGP. Weak factors are best treated as missing and
absorbed in the error term.

###  4.3 Main conclusions from MC experiments

The conclusions from the Monte Carlo simulations are that the bias corrected
estimator of $\boldsymbol{\phi}_{0}\boldsymbol{,}$ generally works well.
Although, it can generate a large RMSE for small $n$ and $T$, this can be
solved by increasing $n.$ This performance is robust to non-Gaussian errors,
GARCH effects, missing weak factors, pricing errors and cross-sectional
dependence of the types considered in these experiments. The size is generally
correct and the power good. The rate at which RMSEs decline with $n$ depends
on the strength of the underlying factors. Semi-strong factors need much
larger values of $n$ for precise estimation. Tests of the joint significance
of $\phi_{M}=\phi_{H}=\phi_{S}=0,$ not reported here, also performed well, as
might be expected given the good power performance of the separate induced
tests which are reported. Including weak factors could be harmful, but there
are potential advantages of adding semi-strong factors, although issue of how
best to select such factors is an open question to which we now turn.

##  5 Factor selection when the number of securities and the number of factors
are both large

Our theoretical derivations and Monte Carlo simulations both assume that the
number of risk factors included in the return regressions is fixed and the
factors are known. For the empirical application we face the additional
challenge of selecting a small number of relevant risk factors from a possibly
large number of potential factors, $m$. This problem has been the subject of a
number of recent studies. Harvey, Liu, and Zhu (2016) propose a multiple
testing approach aimed at controlling the false discovery rate in the process
of factor selection, and in a more recent paper Harvey and Liu (2020) suggest
using a double-bootstrap method to calibrate the t-statistic used for
controlling the desired level of the false discovery rate. Giglio and Xiu
(2021) suggest applying the double-selection Lasso procedure by Belloni,
Chernozhukov, and Hansen (2014) to second pass regressions. None of these
methods distinguish between strong, semi-strong or weak factors in their
selection process, whereas the theory and simulations presented above indicate
the importance of factor strength for estimation and inference.

Here we propose an alternative selection procedure where we first estimate the
strength of all the $m$ factors under consideration, and then select factors
with strength above a given threshold, the value of which is informed by the
convergence results of Theorem 4. This theorem showed that if factor $f_{kt}$
has strength $\alpha_{k},$ then for a given $T$ the BC estimator of $\phi_{k}$
converges to its true value, $\phi_{0k}$, at the rate of
$n^{(\alpha_{k}+\alpha_{\min}-1)/2}$, where
$\alpha_{\min}=\min_{i}(\alpha_{i})$. As is recognized in non-parametric
estimation literature, if the rate of convergence is less than $1/3,$ the gain
in precision with $n$ is so slow that the estimator may not be that
useful.101010The Manski (1985) maximum score estimator for a binary response
model has $n^{1/3}$ convergence and this is regarded as very slow and there
are suggested modifications such as Horowitz (1992) to increase the rate of
convergence to $n^{2/5}$. To achieve rate of $n^{1/3}$ we need to set the
threshold value of $\alpha_{k}$, denoted by $\underline{\alpha}$, such that
$\underline{\alpha}+\alpha_{\min}>1+2/3$. The smallest value of such a
threshold is obtained when $\alpha_{\min}=\underline{\alpha}$ or if
$\underline{\alpha}>1/2+1/3.$ Given the threshold the main issue is how to
estimate factor strength. This problem is already addressed in Bailey,
Kapetanios, and Pesaran (2021, BKP) when $m$ is fixed. In this setting they
base their estimation on the statistical significance of $f_{kt}$ in the first
stage time-series regressions of excess returns on all the factors under
consideration, whilst allowing for the $n$ multiple testing problem which
their approach entails. When $m\,$(the number of factors) is also large the
first stage regressions will also be subject to the multiple testing problem
and penalized regression techniques such as Lasso or the one covariate at a
time (OCMT) selection procedure technique proposed by Chudik, Kapetanios, and
Pesaran (2018) could be used. Irrespective of selection technique used at the
level of individual security returns, we end up with $n$ different subsets of
the $m$ factors under consideration. Factor strengths can then be estimated
similarly to BKP from their selection frequencies across the $n$ securities.

To be more specific, denote the set of $m$ factors under consideration by
$\mathcal{S}$ and denote the set of selected factors for security $i$ by
$\hat{S}_{i}$ and their numbers by $\hat{m}_{i}=|\hat{S}_{i}|$. Clearly
$\hat{S}_{i}\subseteq\mathcal{S}$, and $\hat{m}_{i}\leq m$, for $i=1,2,...,n$.
Then compute the proportion of stocks in which the $k^{th}$ factor is
selected, $\hat{\pi}_{k},$ for $k\in\\{1,2,...,m\\}$ based on $\hat{S}_{i},$
$i=1,2,...,n$ by
$\hat{\pi}_{k}=\frac{1}{n}\sum_{i=1}^{n}\mathcal{I}\\{k\in\hat{S}_{i}\\}$.
Then the strength of the $k^{th}$ factor is measured by

$\hat{\alpha}_{k}=\begin{cases}1+\frac{\ln\hat{\pi}_{k}}{\ln n},\text{ if
}\hat{\pi}_{k}>0,\\\ 0,\quad\quad\quad\text{ if }\hat{\pi}_{k}=0.\end{cases}$
(86)

The transformation from $\hat{\pi}_{k}$ to $\hat{\alpha}_{k}$ is explained and
justified in BKP, where it is shown that considering strength aids
interpretation because it is not dependent on $n$. It is beyond the scope of
the present paper to provide theoretical justification for the proposed factor
selection procedure, but using extensive Monte Carlo experiments Yoo (2022)
has shown that the proposed method has desirable small sample properties
whether Lasso or OCMT is used for factor selection at the level of individual
security returns.

##  6 An empirical application using a large number of U.S. securities and a
large number of risk factors

This section uses the results above in the explanation of monthly returns for
a large number, $n,$ of U.S. securities, by a large active set of $m$
potential risk factors. We first briefly describe the sources and
characteristics of the data for the stock returns and factors, which cover
different sub-samples over the period $1996m1-2022m12.$ We then consider the
selection of a subset of $K$ factors from the active set. Finally we test
$\boldsymbol{\phi}=\boldsymbol{0}$, and construct and evaluate phi-portfolios
and corresponding mean-variance portfolios for alternative models.

Monthly returns (inclusive of dividends) for NYSE and NASDAQ stocks from CRSP
with codes $10$ and $11$ were downloaded from Wharton Research Data Services.
They were converted to excess returns by subtracting the risk free rate, which
was taken from Kenneth French’s data base. To obtain balanced panels of stock
returns and factors, only variables for which there was data for the full
sample under consideration were used. Excess returns are measured in percent
per month. To avoid outliers influencing the results, stocks with a kurtosis
greater than 16 were excluded. To examine factor selection, four samples were
considered, each had $20$ years of data, $T=240$, ending in $2015m12$,
$2017m12$, $2019m12$, $2021m12$. Filtering out the stocks with kurtosis larger
than $16$ removed about $100$ of the roughly $1200$ stocks. The number of
stocks ($n$) considered for each of the four $T=240$ samples are given in
panel A of Table 1. Further detail is given in Section S5 of the online
supplement A.111111Summary statistics for the excess returns across the
different samples are given in Table S-5.1 of the online supplement A. For
analysis of the phi-portfolios, the factors selected in the sample ending in
$2015m12$ were used to construct portfolios up to $2022m12.$

###  6.1 Factor selection

For factor selection we used a sample of $T=240$ observations121212Some
results for $T=120$ are included in the online supplement.. The set of factors
considered combine the $5$ Fama-French factors with the $207$ factors from the
Chen and Zimmermann (2022), Open Source Asset Pricing webpages, both
downloaded July 6 2022. Only factors with data for the full sample were
considered so the return regressions constitute a balanced panel. The number
of factors in each of the four $20$ year samples ending in the years $2015$,
$2017$, $2019$, $2021$ is also given in panel A of Table 1, and range between
$187$ to $199$. Summary statistics for the factors in the active set are given
in S-5.2 of the online supplement A.

To implement the factor selection procedure set out in Section 5, Lasso is
used to carry out selection in the return regressions for individual
securities and we refer to the factor selection procedure as pooled Lasso
(PL). As is well known, Lasso does not work well with too many highly
correlated regressors, therefore, factors with an absolute correlation with
the market factor greater than $0.70$ were dropped. This still left between
$177$ and $190$ risk factors in the active set $\mathcal{S}$, depending on the
sample period (see panel A of Table 1).

Specifically, Lasso was applied $n$ times to the regressions of excess
returns, $r_{it}=R_{it}-r_{t}^{f}$, for $i=1,2,...,n$, on the $177$ to $190$
factors in the active set, $\mathcal{S}$, to select the sub-set $\hat{S}_{i}$
for each $i$ over the four 20-year samples, separately. Following the
literature, the tuning parameters in the Lasso algorithm were set by ten-fold
cross-validation.131313The post-Lasso and one covariate multiple testing
(OCMT) approach of Chudik, Kapetanios, and Pesaran (2018, CKP). were also
investigated, but Lasso seemed to work reasonably well. The details of the
Lasso procedure used are given in Section 2.2 of the online supplement of CKP
paper. Interestingly, the market factor was selected by Lasso for almost all
the securities, thus confirming the pervasive nature of the market factor. No
other factor came close to being selected for all the securities. Lasso tended
to choose a lot of non-market factors and every factor got chosen in at least
one return regression. The mean number of non-market factors chosen by Lasso
fell from $11.6$ in the $2015$ sample to $9.9$ in the $2021$ sample. The
median was lower, falling from $10$ to $8$. There was a long right tail
because Lasso tended to choose a very large number of non-market factors for
some securities, ranging from a maximum of $48$ in the 2021 sample to $54$ in
the $2017$ sample.

Apart from the market factor, there are no systematic patterns for the rest of
selected factors in the return equations for the individual securities.
Following the theory set out in Section 5, the $K$ factors used to estimate
$\boldsymbol{\phi}$ are chosen on the basis of their factor strength.141414The
idea of using factor strength could also be viewed as a kind of averaging of
the factors selected in individual return regressions. A minimum threshold of
$0.7$ was used. This is below the threshold value of
$\underline{\alpha}=1/2+1/3$ required to achieve the convergence rate of
$n^{1/3}$, and is intended to capture borderline semi-strong factors. We also
consider the values of $0.75,$and $0.80$ that are quite close to
$\underline{\alpha}$. The number of selected factors for different choices of
factor strength threshold is given in panel B of Table 1, for the four
different samples.

Using the threshold of $0.70$, $17$ factors (inclusive of the market factor)
were selected for the sample ending in $2015$, with the number of selected
factors declining to $15,13$ and $11,$ for the samples ending $2017,$ $2019$
and $2021$, respectively. At the other extreme, setting the threshold at
$0.80$, the number of selected factors dropped to $4$ for the samples ending
in $2015$, $2017$ and $2021$, and $2$ for the sample ending in $2021$. Since
$17$ factors seemed too large and $2$ factors too small, the threshold value
was set at the intermediate value of $0.75$. We considered always conditioning
on the market factor, but since Lasso almost always selected it, this was
unnecessary.

$T=240$ with end dates | 2021 | 2019 | 2017 | 2015
---|---|---|---|---
Panel A: Number of stocks and factors under consideration
Number of stocks | 1289 | 1276 | 1243 | 1181
Number of stocks with kurtosis $<16$ | 1175 | 1143 | 1132 | 1090
Number of non-market factors | 187 | 198 | 199 | 197
Number of non-market factors with $r<0.70$ | 177 | 189 | 190 | 189
Panel B: Number of selected factors by strength threshold
Number with strength $>0.80$ | 2 | 4 | 4 | 4
Number with strength $>0.75$ | 4 | 6 | 7 | 7
Number with strength $>0.70$ | 11 | 13 | 15 | 17

Note: Panel A shows the number of stocks and risk factors used before and
after filtering by the specified criterion. Panel B shows the number of risk
factors selected with strength greater than the specified threshold level
using Lasso to select factors at the level of individual securities.

Table 1: Summary statistics for the number of stocks and number of selected
factors using the factor strength threshold of 0.75, for four twenty years
($T=240$) samples ending in 2021, 2019, 2017 and 2015

The list of factors selected by pooled Lasso for the four samples are given in
Table 2. The three Fama-French factors, Market, HML and SMB, are all selected
in all four periods.151515We also considered selecting the risk factors using
the generalized one covariate at a time (OCMT) method proposed by
Sharifvaghefi (2023). Using GOCMT the Fama-French three factors were again
amongst the five strongest factors selected. The use of GOCMT for factor
selection is also investigated by Yoo (2022), using Monte Carlo and empirical
applications. Of the Fama-French three, only the market factor is strong, with
estimated strength in excess of $0.98$ across the four periods.161616These
results also support the choice of 3 Fama-French factors and their strength
used in our Monte Carlo simulations. The other factor which is selected across
all the four periods is ”short selling”. This is proposed by Dechow, Hutton,
Meulbroek, and Sloan (2001) who argue that short-sellers target firms that are
priced high relative to fundamentals. It measures the extent to which
investors are shorting the market as reflected in Compustat data. Two
additional factors are selected in periods ending in $2019$ and earlier. One
is ”Beta Tail Risk” proposed by Kelly and Jiang (2014) which estimates a time-
varying tail exponent from the cross section of returns. The other is ”Cash
Based Operating Profitability” (CBOP) suggested by Ball, Gerakos, Linnainmaa,
and Nikolaev (2016). This is operating profit less accruals, with working
capital and R&D adjustments. For periods ending in 2017 and 2015 the ”Sin
Stock” indicator proposed by Hong and Kacperczyk (2009) is also selected. It
takes the value of unity if the stock in question is involved in producing
alcohol, tobacco, and gaming. They find that such stocks are held less by
norm-constrained institutions such as pension plans.

The factor strengths are relatively stable across the periods, with many of