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# Statistical Arbitrage Risk Premium by Machine Learning

Raymond C. W. Leung and Yu-Man Tam 111Leung is with the Cheung Kong Graduate
School of Business, and Tam is with the Office of the Comptroller of the
Currency, U.S. Department of Treasury. This paper is a substantially revised
version of a paper previously circulated under the title “Asset Insurance
Premium in the Cross-Section of Asset Synchronicity”. Please send all
correspondences to Leung at<EMAIL_ADDRESS>Part of this
research is done while Leung was visiting UC Berkeley. We thank Robert M.
Anderson, Lisa Goldberg, Alex Michaelides, Neal Stoughton, Baolian Wang,
Hongjun Yan, and Wenhao Yang for helpful discussions. We also thank the
seminar participants of the Five-Star Forum, Internal Machine Learning
Workshop at the OCC, SWUFE-CDAR 2018 Symposium and the 2019 FMA Annual Meeting
in New Orleans for helpful comments and suggestions. We thank an anonymous
referee who made several suggestions that significantly improved the paper.
Views and opinions expressed are those of the authors and do not necessarily
represent official positions or policy of the OCC. All errors are ours.

(March 2021)

###### Abstract

How to hedge factor risks without knowing the identities of the factors? We
first prove a general theoretical result: even if the exact set of factors
cannot be identified, any risky asset can use some portfolio of similar peer
assets to hedge against its own factor exposures. A long position of a risky
asset and a short position of a “replicate portfolio” of its peers represent
that asset’s factor residual risk. We coin the expected return of an asset’s
factor residual risk as its _Statistical Arbitrage Risk Premium_ (SARP). The
challenge in empirically estimating SARP is finding the peers for each asset
and constructing the replicate portfolios. We use the elastic-net, a machine
learning method, to project each stock’s past returns onto that of every other
stock. The resulting high-dimensional but sparse projection vector serves as
investment weights in constructing the stocks’ replicate portfolios. We say a
stock has high (low) _Statistical Arbitrage Risk_ (SAR) if it has low (high)
R-squared with its peers. The key finding is that “unique” stocks have both a
higher SARP and higher excess returns than “ubiquitous” stocks: in the cross-
section, high SAR stocks have a monthly SARP (monthly excess returns) that is
1.101% (0.710%) greater than low SAR stocks. The average SAR across all stocks
is countercyclical. Our results are robust to controlling for various known
priced factors and characteristics.

Keywords: cross-sectional, elastic-net, empirical asset pricing, machine
learning, portfolio construction, statistical arbitrage risk premium,
statistical arbitrage risk

JEL classification: G11, G12

_Given any stock, how can one hedge against its factor risks?_ This question
is simple to answer with a linear factor model structure. For instance, under
the Markowitz mean-variance portfolio theory and its resulting equilibrium
capital asset pricing model (CAPM) (Sharpe (1964), Lintner (1965)), any given
stock’s returns can be explained by some linear combination of a risk free
asset return and its beta loading on the market portfolio return. Hence, the
factor risk of any stock can be hedged out by shorting its beta loading
multiplied by the market factor. However, with the large number of tradable
and non-tradable factors that have been documented in the literature (Harvey,
Liu, and Zhu (2016)), our question becomes difficult to approach because its
answer then significantly depends on which factors the researcher decides to
include in his empirical study, and is also affected by the empirical
uncertainty with estimating the factor loadings.

The first main contribution of this paper is to _theoretically_ argue an
effective way to hedge against potentially unidentifiable factor risks of a
stock is to answer this dual question: _Given any stock, what portfolio of all
other stocks is most similar to it?_ Suppose all stocks are exposed to the
same set of linear factors but with heterogeneous factor loadings. If one can
identify a group of peers that is the most “similar” to a given stock $i$,
then this portfolio is also exposed to similar factor loadings of this stock
$i$. We view this portfolio of peer stocks as the _replicate_ of stock $i$. A
long position on stock $i$ and a short position on its replicate will expose
the holder to any remaining factor risks of stock $i$ that cannot be
completely hedged out by its peer stocks. We show this long-short position
exactly equates to the residual factor risks of stock $i$. Provided an
econometrician has a method to find these peer stocks, this long-short
position does not require the econometrician to know the true underlying
factor structure of the economy. We call the expected returns of this long-
short position the _Statistical Arbitrage Risk Premium_ (SARP) of stock $i$,
and SARP is the key object of study in this paper.

How do we _empirically_ study SARP? Is there a cross-sectional difference in
SARP? The second main contribution of the paper is answering these questions.
For each month end, we use the _elastic-net_ estimator, a machine learning
method, to project each stock $i$’s past twelve months’ daily returns onto the
returns of _every other_ stock in the market. The resulting elastic-net
projection vector is high-dimensional but very sparse. After a suitable
normalization, the projection vector is then used as investment weights into
all stocks other than $i$. The resulting portfolio is hence a machine learning
constructed _replicate_ of stock $i$. As theoretically motivated above, the
time-series average return from a long position of stock $i$ and a short
position of its replicate is the SARP of stock $i$. Moreover, we call the
elastic-net projection $\mathsf{R^{2}}$ of each stock $i$ as the _Statistical
Arbitrage Risk_ (SAR) of stock $i$; we say a stock has high SAR if it has a
low elastic-net $\mathsf{R^{2}}$, while a stock has low SAR if it has high
$\mathsf{R^{2}}$. The core empirical message of this paper can be succinctly
summarized as:

$\text{\emph{SARP is increasing in SAR}}.$

That is in the cross-section, “unique” stocks (i.e. so having low
$\mathsf{R^{2}}$, and hence high SAR) have a higher SARP than “ubiquitous”
stocks (i.e. high $\mathsf{R^{2}}$, so low SAR). Over the sample period of
January 31, 1976 to December 31, 2020, high SAR stocks have a monthly SARP of
1.368% and low SAR stocks have a monthly SARP of 0.267%, and the difference
1.101% is highly statistically significant. And even without studying SARP, we
have the important corollary that high SAR stocks have a monthly return of
1.481% and low SAR stocks have a monthly return of 0.771%, and the difference
0.710% is also highly statistically significant.

Our paper belongs to a growing literature of applying machine learning methods
to study empirical asset pricing questions. Broadly speaking, many recent
papers in this literature use machine learning methods for factor selection
and/or forecasting. We do neither in this paper. There are only two purposes
of using a machine learning method in this paper: to identify the SAR of each
stock, and to construct the replicate portfolio of each stock. The estimation
and inference of SARP for each stock use conventional empirical asset pricing
procedures.

Recent papers have applied variants of the least absolute shrinkage and
selection operator (LASSO) estimator of Tibshirani (1996). Feng, Giglio, and
Xiu (2020) take advantage of the sparsity property of LASSO and develop a
multi-step approach to evaluate the price of risk of a given new factor above
and beyond an existing set of factors. Freyberger, Neuhierl, and Weber (2017)
uses adaptive group LASSO to select characteristics that provides marginal
information for the cross section of expected stock returns. The literature
has documented a large set of factors or characteristics (Harvey, Liu, and Zhu
(2016)), and it is hoped that machine learning methods can substantially
shrink down the number of factors that can explain the cross-section of
returns. Chinco, Clark-Joseph, and Ye (2019) use the LASSO to predict one-
minute-ahead return using lagged high frequency returns of other stocks as
regressors. Gu, Kelly, and Xiu (2020) apply an extensive battery of machine
learning methods and discover that such methods improve predictability
accuracy over traditional methods. Shu et al. (2020) is a recent paper that
uses an adaptive elastic-net estimator to construct a sparse portfolio that
can track a large index portfolio. While both of our papers use the elastic-
net estimator, our paper emphasizes the use of this estimator to discover a
new asset pricing anomaly, while Shu et al. (2020) emphasizes the use of this
estimator to mimic and dimension-reduce a large portfolio.

Our paper is also related to the literature of pairs trading, substitutability
of risky assets and statistical arbitrage. Gatev, Goetzmann, and Rouwenhorst
(2006) finds pairs of similar stocks using the minimal distance between
normalized historical prices, and argue that the resulting pairs trading
strategy generates abnormal returns and that the source of this profit is the
mispricing of close substitutes. Krauss (2017) is a recent survey of the pairs
trading literature. Wurgler and Zhuravskaya (2002) similarly also argue that
stocks without close substitutes are likely to have large mispricings; the
authors identify a few predefined number of similar stocks using a predefined
sorting method. In contrast by using a machine learning method in this paper,
the selection of a stock’s risky peers is completely data driven. Indeed, the
closest substitute of a given stock could potentially be hundreds of all other
stocks. Huck (2019) and Avellaneda and Lee (2010) are two recent studies on
statistical arbitrage.

Section 1 lays out the theoretical framework of the paper. Section 2 explains
our estimation methodology. The main empirical results of the paper are in
Section 3. Section 3.6 show additional empirical robustness checks. We
conclude in Section 4. We defer the details of the elastic-net procedure to
Section A. All proofs to Section 1 are in the Online Supplementary Materials
Leung and Tam (2021).

##  1 Theoretical motivation

We first prove a general theoretical asset pricing result that will guide our
empirical research design.

######  Theorem 1.1.

Suppose there are $N+1$ risky assets and a single risk-free asset. Assume all
of these risky assets are governed by a linear factor structure with $K$
number of factors with risky returns $\boldsymbol{F}$,

$R_{i}=\alpha_{i}+\boldsymbol{\beta}_{i}^{\top}\boldsymbol{F}+\varepsilon_{i}$
(1)

and where the idiosyncratic risk $\varepsilon_{i}$ of the $i$th risky asset is
assumed to have zero mean and is independent of $\boldsymbol{F}$ for all
$i=1,\ldots,N+1$. Suppose there are strictly more risky assets than factors,
so $N>K$.

Then the excess returns $R_{i}$ of any individual risky asset $i$ can be
expressed as a linear combination of other risky asset returns as.

$R_{i}=\mathbf{b}_{i}^{\top}\boldsymbol{R}_{-i}+\mathbf{a}_{i}^{\top}\boldsymbol{\Phi}_{i}-\mathbf{b}_{i}^{\top}\boldsymbol{\varepsilon}_{-i}+\varepsilon_{i}.$
(2)

That is, the excess returns $R_{i}$ of any individual asset $i$ can be
expressed as a combination of: (i) the $N\times 1$ vector of excess returns of
_all other_ of risky assets $\boldsymbol{R}_{-i}$; (ii) the factor loadings on
some $K$ risky asset returns $\boldsymbol{\Phi}_{i}$, and we will call these
$K$ assets the _factor residuals of asset $i$_; (iii) the $N\times 1$ vector
of idiosyncratic risks $\boldsymbol{\varepsilon}_{-i}$ of all other $N$ risky
assets; and (iv) the idiosyncratic risk $\varepsilon_{i}$ of asset $i$ itself.

The $N\times 1$ vector $\mathbf{b}_{i}$ is dependent on the intercepts
$\\{\alpha_{j}\\}_{j=1,j\neq i}^{N+1}$ and the entire factor loadings
$\\{\boldsymbol{\beta}_{j}\\}_{j=1,j\neq i}^{N+1}$ of the economy and whose
analytical expression is in the Internet Appendix, and where
$\mathbf{a}_{i}^{\top}:=[\alpha_{i},\boldsymbol{\beta}_{i}^{\top}]$.

This result tells us given _any_ factor structure in the financial markets
like (1), of which its theoretical existence can always be justified via Ross
(1976), the returns of a single risky asset $R_{i}$ can be expressed as a
linear combination $\mathbf{b}_{i}$ of returns of _all other_ risky assets
$\boldsymbol{R}_{-i}$ like (2). The key intuition of this result is a hedging
and replication argument. Suppose the researcher does _not_ know the exact
identifies of the factors $\boldsymbol{F}=[F_{1},\ldots,F_{K}]^{\top}$ in the
economy. But as long as all risky assets have an exposure to these factors,
then any particular risky asset $i$ can use some combination $\mathbf{b}_{i}$
of other risky stocks to hedge against risky asset $i$’s factor risks. 222
Some readers may think that the result in Theorem 1.1 and the subsequent
Corollary 1.2 can be trivially derived by “inverting the factor loadings” in
(1) to derive (2). However, we should note that as there are (significantly)
more assets $N$ than the number of factors $K$ in this economy. Thus the
economy-wide factor loading matrix is necessarily of dimensions $N\times K$,
meaning the matrix is not square and thus not invertible. We take extra care
in the proofs to make sure that a proper sense of “factor inversion” is
possible in this general setup.  For the empirical component of our paper,
this means projecting one stock’s returns onto the returns of all other stocks
is not a naive statistical exercise but actually has concrete microeconomic
foundations.

The next result will tell us the economic content of Theorem 1.1.

######  Corollary 1.2.

Suppose the conditions of Theorem 1.1 hold. For each fixed risky asset $i$, we
can find $K+1$ risky assets whose returns are given by

$\displaystyle\Phi_{i,0}$ $\displaystyle=\begin{cases}1,&\text{if
$\alpha_{j}=0$ for all $j\neq i$}\\\
1-\frac{\boldsymbol{\alpha}_{-i}^{\top}}{\left|\left|\boldsymbol{\alpha}_{-i}\right|\right|_{2}^{2}}(\boldsymbol{R}_{-i}-\boldsymbol{\varepsilon}_{-i}),&\text{if
otherwise}\end{cases}$ (3a) $\displaystyle\Phi_{i,k}$
$\displaystyle=F_{k}-\mathbf{c}_{i,k}^{\top}(\boldsymbol{R}_{-i}-\boldsymbol{\varepsilon}_{-i}),\quad
k=1,\ldots,K$ (3b)

where $\mathbf{c}_{i,k}$ is some $N\times 1$ deterministic vector such that
only depends on the factor loadings $\\{\boldsymbol{\beta}_{j}\\}_{j=1,j\neq
i}^{N+1}$ and intercept $\\{\alpha_{j}\\}_{j=1,j\neq i}^{N+1}$ structure of
the economy (its analytical expression is in the Internet Appendix). Here we
denote
$\boldsymbol{\alpha}_{-i}^{\top}:=[\alpha_{1},\ldots,\alpha_{i-1},\alpha_{i+1},\ldots,\alpha_{N+1}]$
and $\left|\left|\cdot\right|\right|_{2}$ is the Euclidean norm on
$\mathbb{R}^{N}$. We will denote
$\boldsymbol{\Phi}_{i}^{\top}:=[\Phi_{i,0},\Phi_{i,1},\ldots,\Phi_{i,K}]$.

Corollary 1.2 is what motivates us to refer to those $K$ risky assets with
returns $\boldsymbol{\Phi}_{i}$ as _factor residuals for asset $i$_. From (2),
we see the following decomposition,

$\mathbf{a}_{i}^{\top}\boldsymbol{\Phi}_{i}=\alpha_{i}\Phi_{i,0}+\boldsymbol{\beta}_{i}^{\top}\boldsymbol{\Phi}_{i,1:K},$
(4)

where $\boldsymbol{\Phi}_{i,1:K}^{\top}:=[\Phi_{i,1},\ldots,\Phi_{i,K}]$.

The first part $\Phi_{i,0}$ adjusts for any potential mispricings in the
economy. As an important special case when there is no pricing at all in the
economy 333 It is well known that if all risky assets are mean-variance
efficient then necessarily there is no mispricing in the economy. See an early
discussion of this classical empirical asset pricing test in Black, Jensen,
and Scholes (1972).  meaning $\alpha_{i}=0$ and $\alpha_{j}=0$ for all assets
$j\neq i$, then $\Phi_{i,0}=1$ and $\alpha_{i}\Phi_{i,0}=0$. If on the other
hand, when the intercepts $\alpha_{i},\alpha_{j}$’s are generically non-zero,
then the factor residuals of asset $i$ will explicitly hedge out any level of
mispricing in the economy through $\Phi_{i,0}$, and asset $i$’s net mispricing
exposure is
$\alpha_{i}\Phi_{i,0}=\alpha_{i}-\alpha_{i}\boldsymbol{\alpha}_{-i}^{\top}(\boldsymbol{R}_{-i}-\boldsymbol{\varepsilon}_{-i})/\left|\left|\boldsymbol{\alpha}_{-i}\right|\right|_{2}^{2}$.
Secondly, the next $K$ parts $\Phi_{i,k}$ adjust for factor exposures. From
the perspective of an agent who owns asset $i$, he is exposed to each of the
$k=1,\ldots,K$ factor returns through the factor loadings
$\boldsymbol{\beta}_{i}^{\top}=[\beta_{i,1},\ldots,\beta_{i,K}]$ of (1).
However, all the other $N$ risky assets will also have some exposure to the
$k$-th factor. Suppose the agent constructs an “artificial asset” that loads
into the return of the $k$-th factor, while shorting some combination
$\mathbf{c}_{i,k}$ of the factor contributions to all the other $N$ risky
assets $\boldsymbol{R}_{-i}-\boldsymbol{\varepsilon}_{-i}$. The resulting
artificial asset has the returns $\Phi_{i,k}$. Hence $\Phi_{i,k}$ is precisely
the residual exposure of asset $i$ to the $k$-th factor, after using the
returns of all other assets to “hedge” as much as possible this factor risk.
The “hedging” nature of these artificial assets motivates us to call them as
“factor residuals” for asset $i$. The holder of asset $i$ is exposed to $K$
number of factor risks, and so he will have to construct $K$ number of these
factor residuals. And since asset $i$ has factor exposure to $K$ number of
factors, the agent will weigh $\beta_{i,k}$ loadings into the $k$-th factor
residual return $\Phi_{i,k}$, as in the
$\boldsymbol{\beta}_{i}^{\top}\boldsymbol{\Phi}_{i,1:K}$ term of (4).

Let’s rearrange (2) and take its expectation,

$\mathbb{E}[R_{i}]-\mathbf{b}_{i}^{\top}\mathbb{E}[\boldsymbol{R}_{-i}]=\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]-\mathbf{b}_{i}^{\top}\mathbb{E}[\boldsymbol{\varepsilon}_{-i}]+\mathbb{E}[\varepsilon_{i}]=\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}],$
(5)

where $\mathbb{E}[\boldsymbol{\varepsilon}_{-i}]=\boldsymbol{0}_{N}$ and
$\mathbb{E}[\varepsilon_{i}]=0$ because idiosyncratic risks are not priced.
The overall term $\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]$ in
(5) is exactly the expected portfolio return into these $K$ factor residuals.
This is why we will call
$\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]$ as the _Statistical
Arbitrage Risk Premium (SARP) for asset $i$_. The key objective of this paper
is to empirically study SARP.

The next result shows that, under weak economic and technical conditions, SARP
is non-zero.

######  Corollary 1.3.

The SARP of any non-redundant asset $i$ is almost surely non-zero when some of
the $K$ factors are correlated.

The following result relates the regression $\mathsf{R}$-squared to Theorem
1.1.

######  Corollary 1.4.

Suppose we view (2) as a linear regression of $R_{i}$ onto the set of
regressors $\boldsymbol{R}_{-i}$, and $\mathbf{b}_{i}$ as the vector of
regression coefficients. Assume further: (i)
$\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]\neq 0$; (ii) the
idiosyncratic risks are homoskedastic (i.e.
$\mathbb{E}[\varepsilon_{i}\varepsilon_{j}]=0$ if $i\neq j$ and
$=\sigma_{\varepsilon}^{2}$ if $i=j$); and (iii) the variance-covariances
$\mathrm{Var}(\boldsymbol{F})$ and $\mathrm{Var}(\boldsymbol{\Phi}_{i})$ for
all $i$ are positive definite. Then,

  1. (a)

The regression $\mathsf{R}$-squared decreases (increases) as
$\mathbf{a}_{i}^{\top}(\mathrm{Var}(\boldsymbol{\Phi}_{i})-\mathrm{Var}(\boldsymbol{F}))\mathbf{a}_{i}-\sigma_{\varepsilon}^{2}$
becomes more positive (more negative).

  2. (b)

If moreover $\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]>0$, then
the regression $\mathsf{R}$-squared is decreasing in
$\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]$.

In the remainder of this paper, we will identify the regression
$\mathsf{R^{2}}$ with the _Statistical Arbitrage Risk_ (SAR) of an asset $i$.
That is, we will say an asset $i$ has a high (low) SAR if it has low (high)
$\mathsf{R^{2}}$.

Corollary 1.4(a) provides another way to view the $K$ factor residuals of
stock $i$ from (3b). Condition (ii) is used to simplify the equations and
condition (iii) is a mild technical assumption. For the sake of exposition,
consider the case when $\sigma_{\varepsilon}^{2}$ is negligible, and so the
magnitude of the term in Corollary 1.4(a) is driven by the positive- or
negative-definiteness of the matrix
$\mathrm{Var}(\boldsymbol{\Phi}_{i})-\mathrm{Var}(\boldsymbol{F})$. The case
where this $K\times K$ matrix is _negative-definite_ is when the volatility of
the factor residuals of asset $i$ is lower than the volatility of the factors
themselves. This happens when the factor residuals of asset $i$ do a good job
in hedging asset $i$ against its exposure to the $K$ factor risks. Recall from
(3b) these $K$ factor residuals are dependent on the factor structure of all
the other $N$ risky assets. This implies these $K$ factor residuals can only
do a good job in insuring asset $i$ against factor risks if the $N$ other
risky assets also highly co-move with asset $i$ itself, which implies a _high_
regression $\mathsf{R}$-squared. Given the desirability of these $K$ factor
residuals, the holder of asset $i$ will be willing to pay a high price for
these $K$ factor residuals, which then pushes _down_ their expected returns.
This explains why in Corollary 1.4(b), there is a negative relationship
between the regression $\mathsf{R}$-squared and the SARP. The discussion for
the case when that $K\times K$ matrix is positive-definite is analogous. The
above discussions are still contingent upon the existence and positivity of
such a SARP, which again, is entirely an empirical question we now proceed to
answer.

##  2 Empirical hypothesis and methodology

We summarize the empirical implications of our theoretical discussions.

######  Empirical Hypothesis.

  1. (1)

The expected difference between a given asset’s return and some linear
combination of other assets’ returns can be seen as a _Statistical Arbitrage
Risk Premium (SARP)_. Under weak economic and technical conditions, the SARP
is non-zero. Moreover, one does _not_ need to know a priori what are the
underlying factors that drive the economy to compute SARP.

  2. (2)

We can identify regression $\mathsf{R^{2}}$ with _Statistical Arbitrage Risk
(SAR)_. We anticipate assets with low SAR (i.e. high $\mathsf{R^{2}}$) to have
a low SARP, while assets with a high SAR (i.e. low $\mathsf{R^{2}}$) to have a
high SARP in the cross-section.

The empirical methodology of the paper is separated into two distinct steps.
In the first step, we project a given stock’s return onto the span of all
other stocks’ returns to get an empirical approximation of $\mathbf{b}_{i}$
from (2). However, despite the microfoundations of Theorem 1.1, we shall argue
it is econometrically non-trivial to execute this projection. We will apply a
machine learning method to overcome a critical technical hurdle. In the second
step, we will use standard portfolio sort methods from the empirical asset
pricing literature to estimate the expectation
$\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]$ of (5), which is
again the SARP of asset $i$.

###  2.1 Data and projection procedure via the elastic-net

####  2.1.1 Data sources

Our data source is standard. We use both the CRSP daily and monthly data from
December 31, 1974 to December 31, 2020 with the standard filters. In all of
the subsequent empirical analysis, we identify a stock by its PERMNO number as
recorded in the CRSP database. 444 As is well known with the CRSP and
Compustat databases, there are several unique identifiers of equities: PERMNO,
PERMCO and GVKEY. Each have their different strengths and weaknesses. We
recognize some — but few — firms have dual class shares which implies a single
firm can have having multiple PERMNOs. Rather than making arbitrary
corrections to somehow “merge” the time series of returns of these dual class
shares, we simply leave the PERMNOs “as is” in CRSP. In all, this means our
paper focuses on tracking an individual security rather than the individual
firm — although they are synonymous with each other except for those few
special cases.  Moreover, we only include stocks that, in the past twelve
months for any given month end, there are at least 60 days of valid trading
returns. This 60 days choice ensures that we do include effectively all
stocks, except for the most extremely illiquid or dead stocks, so that our
results are not driven only by the liquid and hence most likely large stocks.
We also obtain the Fama-French data from Kenneth French’s website.

Dec 31, 1974 Month end $t-1=$ Dec 31, 1975 $t=$ Jan 31, 1976Twelve months of
daily observations $\footnotesize{d_{1},\ldots,D_{i,t-1}}$One month ahead
1$\cdots$$i-1$$i+1$$\cdots$$N_{t-1}$$i$$\boldsymbol{\hat{\beta}}_{i,t-1}\;\text{and}\;\mathsf{R^{2}}_{i,t-1}$Return
of (focal) stock $i$ at month $t$$R^{\textrm{Foc}}_{i,t}\equiv
R_{i,t}=\frac{P_{i,t}}{P_{i,t-1}}-1$Return of the replicate of stock $i$ at
month
$t$$R^{\textrm{Rep}}_{i,t}=(1-\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1})r_{f,t}+\boldsymbol{R}^{\top}_{-i,t}\boldsymbol{\tilde{\beta}}_{i,t-1}$,
where
$\boldsymbol{\tilde{\beta}}_{i,t-1}:=\boldsymbol{\hat{\beta}}_{i,t-1}/||\boldsymbol{\hat{\beta}}_{i,t-1}||_{1}$
if $\boldsymbol{\hat{\beta}}_{i,t-1}\neq\boldsymbol{0}$, and
$:=\boldsymbol{0}$ if otherwise. Return of longing stock $i$ and shorting its
replicate at month
$t$$R^{\textrm{LS}}_{i,t}=R^{\textrm{Foc}}_{i,t}-R^{\textrm{Rep}}_{i,t}$
Figure 1: Projection and replicate construction procedure. For each month
ending at December 31, 1975, January 31, 1976, …, November 30, 2020, we use
the past twelve months’ worth of daily observations to project each of stock
returns onto the returns span of every other stock. We only consider stocks
have at least 60 days worth for daily trading data. That is, for each stock
$i=1,\ldots,N_{t-1}$, suppose $\\{d_{1},\ldots,D_{i,t-1}\\}$ with $D_{i}\geq
60$ are past twelve months’ worth of trading days that end at month $t-1$.
Note that the the number of stocks $N_{t-1}$ in the market at each month end
$t-1$ may vary. The returns span for stock $i$ are all those other stocks
$j=1,\ldots,i-1,i+1,\ldots,N_{t-1}$ that have trading days that at least
overlap with that of stock $i$, so
$\\{d_{1},\ldots,D_{i,t-1}\\}\subseteq\\{d_{1},\ldots,D_{j,t-1}\\}$.

######  Remark 2.1 (Filters and sampling period).

We subset only for US common equities (i.e. SHRCD code of 10 or 11), and only
those that are listed in the NYSE, AMEX or NASDAQ (i.e. EXCHCD code of 1, 2 or
3). In addition, we start our data sample from 1974 because the CRSP datasets
only started to include NASDAQ stocks in December 1972. We start in December
1974 to allow an additional year of buffering for good measure. Effectively
this means the number of stocks in CRSP pre-1974 and post-1974 are
structurally different in several magnitudes; see (Bali et al., 2016, §7.1.2)
for a detailed discussion. Our estimation and projection method is clearly
sensitive to total number of stocks. Starting our analysis pre-1974 might bias
our results simply due to CRSP data limitations.

####  2.1.2 Projection

The first step in the empirical test of Theorem 1.1 is to project a given
stock’s returns onto the returns span of all other stocks. Our projection
procedure is summarized in Figure 1. For months ending $t-1=$ December 31,
1975, January 31, 1976, …, November 30, 2020, we use the past twelve months’
worth of daily observations to project each of stock $i$’s returns onto the
returns of all other stocks. We reserve a one month gap for returns
realization in a procedure to be described in Section 2.3. Unless specified
otherwise, we will denote $t-1$ as the end of the projection month, and $t$ as
the one-month ahead returns realization date; that is, we set $t=$ January 31,
1976, February 29, 1976, …, December 31, 2020. Figure 3(a) shows the number of
stocks that have at least 60 past trading days in for each given month $t-1$.
Observe since we start our projection procedure on December 31, 1975, we
require daily data starting from December 31, 1974.

Let $N_{t-1}$ be the total number of stocks traded in the market at month
$t-1$. The return vector of stock $i$ at month $t-1$ and the returns span of
all other assets are, respectively:

$\displaystyle y_{i,t-1}=\begin{pmatrix}R_{i,d_{1}}\\\ \vdots\\\
R_{i,d_{D_{i,t-1}}}\end{pmatrix}_{D_{i,t-1}\times 1}$ (6a)
$\displaystyle\mathbf{X}_{i,t-1}=\begin{pmatrix}R_{1,d_{1}}&\ldots&R_{i-1,d_{1}}&R_{i+1,d_{1}}&\ldots&R_{N_{t-1},d_{1}}\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{1,d_{D_{i,t-1}}}&\ldots&R_{i-1,d_{D_{i,t-1}}}&R_{i+1,d_{D_{i,t-1}}}&\ldots&R_{N_{t-1},d_{D_{i,t-1}}}\end{pmatrix}_{D_{i,t-1}\times(N_{t-1}-1)}$
(6b)

where $R_{i,d}$ is the daily return of stock $i$, $D_{i,t-1}$ is the number of
trading days of stock $i$ in the past 12 months ending at month $t-1$. The
dimensions of (6b) are approximately $250\times 5000$ for each stock $i$.
There are $T=539$ number of months from December 31, 1975 to November 30,
2020. Thus we run a total of approximately $T\times 5000\approx 2.7$ million
projections in this paper.

In this paper we will use the _elastic-net estimator_ developed by Zou and
Hastie (2005) to empirically project a given stock’s return onto the returns
span of all other stocks. We defer a detailed and technical discussion of the
elastic-net in our context to Section A.1. But if we are interested in
estimating _linear_ relationships as in Theorem 1.1, why do we not use the
workhorse _ordinary least squares_ (OLS) estimator? The design matrix (6b) of
returns to evaluate our empirical hypothesis is necessarily a $T\times N$
matrix, where $T\approx 250$ is the number of days, and $N\approx 5000$ is the
total number of traded stocks. This is a case where $T\ll N$. This means the
$N\times N$ matrix $\mathbf{X}_{i,t-1}^{\top}\mathbf{X}_{i,t-1}$ is _not_ full
rank. The OLS estimator is thus necessarily _not_ well defined. In contrast,
the elastic-net is a machine learning method that explicitly allows for “wide”
$T\ll N$ regressors.

We denote the elastic-net projection coefficient vector of stock $i$ at month
$t-1$ as $\boldsymbol{\hat{\beta}}_{i,t-1}\in\mathbb{R}^{N_{i,t-1}}$, and the
resulting coefficient of determination (“R-squared”) as
$\mathsf{R^{2}}_{i,t-1}$. Note and recall the conventional definition of
$\mathsf{R^{2}}$ is,

$\mathsf{R^{2}}_{i,t-1}:=1-\frac{\sum_{s=1}^{D_{i,t-1}}(R_{i,d_{s}}-\hat{y}_{i,t-1})^{2}}{\sum_{s=1}^{D_{i,t-1}}(R_{i,d_{s}}-\hat{\mu}_{i,t-1})^{2}},$

where $\hat{y}_{i,t-1}:=\mathbf{X}_{i,t-1}\boldsymbol{\hat{\beta}}_{i,t-1}$ is
the fitted value, and $\hat{\mu}_{i,t-1}$ is the sample mean; they are
explicitly given by

$\hat{y}_{i,t-1}=\mathbf{X}_{i,t-1}\boldsymbol{\hat{\beta}}_{i,t-1},\quad\hat{\mu}_{i,t-1}=\frac{1}{D_{i,t-1}}\sum_{s=1}^{D_{i,t-1}}R_{i,d_{s}}.$

We emphasize we are only using the elastic-net as a projection method for
constructing the replicates, and we do _not_ use it for statistical inference.
The statistical inference claims are on the expected returns of SARP, which we
will discuss beginning in Section 2.3. As a result, despite the large number
of projections that we run at this stage, we do not suffer from the multiple
hypothesis testing problem that has been discussed in the recent empirical
asset pricing literature by Harvey et al. (2016). Other than the use in
calculating $\mathsf{R^{2}}$, we do _not_ use the fitted value
$\hat{y}_{i,t-1}$ in any other subsequent steps. This is in contrast to the
recent literature on financial applications of machine learning (say Gu et al.
(2020) and others) where they use the fitted value (or predicted value when
one uses $\mathbf{X}_{i,t}$ instead of $\mathbf{X}_{i,t-1}$) in assessing
model forecasting accuracy.

######  Remark 2.2 (Not projecting onto the intercept).

We deliberately do _not_ include an intercept as a regressor in (6b).
Including an intercept together with the sparseness property of the elastic-
net estimator will attribute a stock with stale prices with extremely high
$\mathsf{R^{2}}$. See Section A.1.2 for a more detailed discussion. Other than
the projection method here and in Section 3.6.2, unless noted otherwise, all
subsequent statistical inference tests that employ a linear regression will
include an intercept.

######  Remark 2.3 (Why elastic-net and not other machine learning methods?).

Out of a myriad of machine learning methods, why did we choose the elastic-net
estimator to test our empirical implication? Simply put, we regard the
elastic-net estimator as a parsimonious method that allows us to test our
theoretical prediction. We defer to Section A.1 for detailed technical
discussions of why we particularly use and prefer the elastic-net estimator in
this paper.

######  Remark 2.4 (Overlapping data).

It is evident we are using overlapping time data. Overlapping returns data in
traditional empirical asset pricing raises several technical inference issues
revolving around autocorrelations (see Hansen and Hodrick (1980) and a recent
discussion by Hedegaard and Hodrick (2016)). However, the “wide” regressors in
our setting imply we actually gain $\approx 5000$ new cross-sectional data
points for each month advance. Because of the sheer amount of new entering
cross sectional data, it is not necessarily true that the projected
coefficient ending at months $t-2$ and $t-1$ would be quantitatively similar
even if they only differ by a one month step.

###  2.2 Replicate construction

For each month end $t-1$ and each stock $i=1,\ldots,N_{t-1}$, we collect the
projected coefficient $\boldsymbol{\hat{\beta}}_{i,t-1}$ and the
$\mathsf{R^{2}}_{i,t-1}$. We track stock $i$’s _one-month ahead_ return
$R_{i,t}$ from $t-1$ to $t$. For subsequent exposition clarity, we will
sometimes call stock $i$ as the _focal stock_ and write it’s return at month
$t$ as,

$R_{i,t}^{\textrm{Foc}}\equiv R_{i,t}.$ (7)

Next, we introduce the _replicate_ (portfolio) of a stock $i$. This is a key
idea of this paper. Let
$\boldsymbol{R}_{-i,t}:=[R_{1,t},\ldots,R_{i-1,t},R_{i+1,t},\ldots,R_{N_{t-1},t}]^{\top}$
be the vector of month $t$ returns for all stocks except stock $i$. We wish to
treat the projected coefficients $\boldsymbol{\hat{\beta}}_{i,t-1}$ as
investment weights into each of the $N_{t-1}-1$ number of stocks. However, the
regularization nature of the elastic-net causes the entries of these estimated
coefficients to be small in magnitude. So if we were to directly use
$\boldsymbol{\hat{\beta}}_{i,t-1}$ as investment weights, the result would be
a very small allocation into risky component of the portfolio. To have a more
reasonable risky component, we normalize $\boldsymbol{\hat{\beta}}_{i,t-1}$ by
its $L^{1}$ norm and write, 555 For any vector
$\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$, we define its $L^{1}$ norm
as
$\left|\left|\mathbf{x}\right|\right|_{1}:=\sum_{j=1}^{n}\left|x_{j}\right|$.
Note that the $L^{1}$ norm shows up explicitly in the objective function of
the elastic-net estimator. The $L^{2}$ norm also shows up in the elastic-net
objective function; as is usual,
$\left|\left|\mathbf{x}\right|\right|_{2}:=\sqrt{\sum_{j=1}^{n}x_{j}^{2}}$.
However, we choose to normalize by $L^{1}$ and not $L^{2}$ because of scaling.
We have inequality
$\left|\left|\mathbf{x}\right|\right|_{2}\leq\left|\left|\mathbf{x}\right|\right|_{1}$.
Economically and empirically, this implies scaling by the $L^{2}$ norm will
result in a highly levered equity component for the replicate for stock $i$.
The $L^{1}$ scaling generates more a more moderate result.

$\displaystyle\boldsymbol{\tilde{\beta}}_{i,t-1}:=\begin{dcases}\boldsymbol{0}&\text{if
$\boldsymbol{\hat{\beta}}_{i,t-1}=\boldsymbol{0}$,}\\\
\frac{\boldsymbol{\hat{\beta}}_{i,t-1}}{||\boldsymbol{\hat{\beta}}_{i,t-1}||_{1}}&\text{otherwise.}\end{dcases}$
(8)

Since investment weights must sum to one, we place the remainder of the
weights $\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}$ into the risk
free asset with returns $r_{f,t}$, and here $\mathbf{1}$ is a vector of ones
of conformable dimensions. In all, the month $t$ return of stock $i$’s
replicate is,

$R_{i,t}^{\textrm{Rep}}:=(1-\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1})r_{f,t}+\boldsymbol{R}_{-i,t}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}.$
(9)

Figure 2:  Proportion of equity invested in the replicates.  We consider the
average amount of equity-only components that are invested by each replicate
portfolio. Stocks are sorted into their method $m$ $\mathsf{R^{2}}$ decile
bins $k=\textrm{'Lo'},\ldots,\textrm{'Hi'}$. At the end of the estimation
month $t-1$, we have the normalized vector
$\boldsymbol{\tilde{\beta}}_{i,t-1}$ of asset $i$. The quantity
$\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}$ is the proportion
allocated to the equity-only components of stock $i$’s replicate as from (9).
The average equity-only proportion across all the replicates in bin $k$ at
month $t-1$ is the quantity $\frac{1}{|B_{t-1}^{k}|}\sum_{i\in
B_{t-1}^{k}}\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}$. Here we plot
time series of this average equity-only proportion for the bins
$k=\textrm{'Lo'},5$ and ’Hi’, and the $y$-axis are expressed in decimals (e.g.
0.10 means 10%). The plot labeled “Overall average” plots the overall average
of these equity-only components across all bins, which is the quantity
$\frac{1}{10}\frac{1}{|B_{t-1}^{k}|}\sum_{k=1}^{10}\sum_{i\in
B_{t-1}^{k}}\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}$. The sampling
period is from December 1975 to November 2020.

Finally, we track the long-short return of stock $i$ against its replicate,

$R_{i,t}^{\textrm{LS}}:=R_{i,t}^{\textrm{Foc}}-R_{i,t}^{\textrm{Rep}}.$ (10)

This long-short return (10) will proxy for the difference
$R_{i}-\mathbf{b}_{i}^{\top}\boldsymbol{R}_{-i}$ in Theorem 1.1, and is the
key emphasis of study in this paper. We will use conventional portfolio sort
methods of the empirical asset pricing literature to estimate the expectation
$\mathbb{E}[R_{i}]-\mathbf{b}_{i}^{\top}\mathbb{E}[\boldsymbol{R}_{-i}]$,
which then is equal to the theoretically motivated SARP
$\mathbf{a}_{i}^{\top}\mathbb{E}[\boldsymbol{\Phi}_{i}]$ of stock $i$.

######  Remark 2.5 (Projection method for constructing mimicking portfolios).

Breeden et al. (1989) and Lamont (2001) use analogous methods to (9) to
construct a mimicking factor out of tradable base assets. Ang et al. (2006)
use also an analogous method to construct a factor mimicking aggregate
volatility risk. While our approach in (9) seems identical to these previous
methods, we stress the dimensionality of the regressors is substantially
different. The number of base assets in those aforementioned methods is small,
usually in the range of five to ten. In contrast, our base assets are
effectively every other stock other than the stock $i$ itself, which number in
the thousands.Wurgler and Zhuravskaya (2002) 666 We are grateful for an
anonymous referee for pointing out this useful reference to us.  uses an idea
that is conceptually similar — but operationally quite different — to this
paper in evaluating arbitrage risk. Wurgler and Zhuravskaya (2002) first use a
sorting method to closely match a stock $i$ with three other stocks that is
most closest to it on size and book-to-market, and then use a linear
regression on of the returns of stock $i$ onto these three stocks. Similar to
us, they use the regression coefficients to construct a replicate of stock
$i$. However, the selection procedure of Wurgler and Zhuravskaya (2002) by
portfolio sort is not entirely statistical in nature. Indeed, a priori there’s
no reason why the time series statistical properties of any given stock $i$
would be best matched by the top three stock matched on size and value, even
if size and value are well known priced factors. Moreover, the focus of
Wurgler and Zhuravskaya (2002) is to study “demand curves” for stocks and to
identify this effect, the authors only constrain their study to 259 stocks
addition into the S&P 500 index over a 13 year period. In contrast, the focus
of our study is SARP and we consider effectively all stocks (so not just those
in the S&P 500) at any given point in time and over a 45 year period.

###  2.3 Portfolio formation and sort

Having defined three types of returns (7), (9) and (10) associated with stock
$i$, we now proceed to construct portfolios. We use our theoretical
discussions as guidance: Corollary 1.4 anticipates we should find a negative
relationship between the regression $\mathsf{R}$-squared (the SAR of stock
$i$) and the SARP of stock $i$. At the end of the month $t-1$, we sort each
stock $i$ by its $\mathsf{R^{2}}_{i,t-1}$ into decile bins. Let $B_{t-1}^{k}$
be the set of stocks in the $k$-th bin at month $t-1$. We organize the bins in
ascending order, so bin $k=1$ (labeled “Lo”) consists of stocks with the
lowest $\mathsf{R^{2}}$’s, and bin $k=10$ (labeled “Hi”) consists of stocks
with the highest $\mathsf{R^{2}}$’s.

We focus on equal-weighted portfolios in this paper (see Remark 2.6 for a
discussion of the issues in evaluating value-weighted SARP). The equal-
weighted excess returns of bin $k=\textrm{Lo},2,\ldots,\textrm{Hi}$ of the
stocks is standard:

$\bar{R}^{k}_{\textrm{Foc},t}=\frac{1}{\left|B_{t-1}^{k}\right|}\sum_{i\in
B_{t-1}^{k}}(R_{i,t}-r_{f,t}),\\\ $ (11)

where we denote $\left|B\right|$ as the cardinality of a set $B$. In this
paper, we will view and assume the replicate of stock $i$ to have the same
equal-weight as its focal counterpart,

$\bar{R}^{k}_{\textrm{Rep},t}:=\frac{1}{\left|B_{t-1}^{k}\right|}\sum_{i\in
B_{t-1}^{k}}(R_{i,t}^{\textrm{Rep}}-r_{f,t}).$ (12)

Finally, the construction of the equal-weighted portfolio of the long-short of
the focal stocks versus their replicates is now immediate thanks to the
aforementioned equivalent weighting assumption. We define the equal-weighted
portfolio return of the long-focal, short-replicate position in bin $k$ as,

$\bar{R}^{k}_{\textrm{LS},t}:=\bar{R}^{k}_{\textrm{Foc},t}-\bar{R}^{k}_{\textrm{Rep},t}=\frac{1}{\left|B_{t-1}^{k}\right|}\sum_{i\in
B^{k}_{t-1}}R_{i,t}^{\textrm{LS}}.$ (13)

Note (13) does not subtract a risk-free rate as it is the difference of two
excess returns.

######  Remark 2.6 (Ambiguity in value-weighted SARP).

Value-weighting causes several causes for concern in evaluating SARP. Value-
weights for focal stocks in each bin $k$ is conventional. The cause for
caution is what should one assume for the value-weights of the replicates.
Perhaps the most natural way is just assume the replicate of stock $i$ has the
same value-weight as stock $i$ itself. While this value-weight assumption for
the replicates might seem natural, it is also ambiguous. For stocks with high
SAR there is no contention: for these high SAR stocks, their replicates are
effectively just the risk-free asset, and so using value-weights in this way
would just produce the equity risk premium. But for stocks with low SAR, there
will be many similar peer stocks in their replicates. It is possible that a
focal stock has large size, but the elastic-net statistically matches its
peers with small stocks, and vice-versa. Thus the total value of the replicate
portfolio of stock $i$ could be substantially different than that of the focal
stock $i$ itself. Assigning the replicate of stock $i$ to have the same value-
weight of the focal stock $i$ in bin $k$ may or may not reflect the size of
the latter. Indeed, the same concern also holds in converse. This ambiguity in
what “value-weighting” means for the replicates is what drives our paper to
focus on the most parsimonious weighting scheme — equal-weights.

##  3 Empirical results

We can now translate our theoretical predictions of Section 1 to concrete
quantities for an empirical investigation: (i) By Corollary 1.3, SARP should
be non-zero (and indeed positive). That is, we should expect the time-series
average of $\bar{R}^{k}_{\textrm{LS},t}$ of (10) to be non-zero in the data;
and (ii) More importantly by Corollary 1.4, SARP should be increasing with
SAR. In other words, stocks that have high SAR (low $\mathsf{R^{2}}$) should
have high SARP, and stocks with low SAR (high $\mathsf{R^{2}}$) should have
low SARP.

###  3.1 Summary statistics

####  3.1.1 Characteristics summary statistics

Let’s first investigate the average characteristics of stocks that stocks that
are decile sorted by elastic-net $\mathsf{R^{2}}$’s. Let $Z_{i,t-1}$ be some
scalar characteristic of stock $i$ at month $t-1$. For each bin $k$, we
compute the simple average of stocks’ characteristics
$\bar{Z}_{t-1}^{k}:=\frac{1}{\left|B_{t-1}^{k}\right|}\sum_{i\in
B_{t-1}^{k}}Z_{i,t-1}$. The numbers in Table 1 show the time-series mean,
standard deviation, $5^{\mathrm{th}}$ and $95^{\mathrm{th}}$ percentiles of
these characteristics $\bar{Z}_{t-1}^{k}$ for each bin $k$.

Result (i) of Table 1 shows the elastic-net $\mathsf{R^{2}}$ characteristic
$Z_{i,t-1}=\mathsf{R^{2}}_{i,t-1}$. The elastic-net is capturing a wide range
of projection $\mathsf{R^{2}}$’s from effectively 0% in the lowest bin to 62%
in the highest bin. 777 While there is some resemblance, we caution that the
positive association of $\mathsf{R^{2}}$ and number of non-zero elements in
the estimated elastic-net coefficient vector is different from the OLS case
where including more regressors necessarily and mechanically raises
$\mathsf{R^{2}}$. In our elastic-net application, the number of regressors
$N_{t-1}$ remains fixed, and it is an estimation outcome that only few of them
have non-zero coefficient loadings.  As a matter of comparison and for
subsequent robustness discussions in Section 3.6.2, at the end of month $t-1$
we take each stock $i$’s past twelve months’ of daily returns and project them
onto the daily returns of the Fama and French (2015) five factors using OLS.
We collect the resulting FF5 OLS $\mathsf{R^{2}}$ and this is also a
characteristic for each stock $i$ for month $t-1$. Result (ii) shows the
summary statistics of the FF5 OLS $\mathsf{R^{2}}$ characteristic for stocks
sorted by elastic-net $\mathsf{R^{2}}$. It is quite interesting to observe
that stocks with high SAR (low elastic-net $\mathsf{R^{2}}$) have also low
corresponding FF5 OLS $\mathsf{R^{2}}$, and likewise stocks with low SAR (high
elastic-net $\mathsf{R^{2}}$) have high FF5 OLS $\mathsf{R^{2}}$.

Results (iii) and (iv) examine the quality of our replicate construction
procedure. Recall (9). The characteristic $Z_{i,t-1}=\textrm{{Number of non-
zero entries in $\boldsymbol{\hat{\beta}}_{i,t-1}$}}$ is the number of risky
peers used to construct the replicate of stock $i$ and (iii) shows this
result. Note the sparseness of the elastic-net projection vector: while
$N_{t-1}$ ranges in the thousands, the number of non-zero entries in the
projection vector is remarkably low. For stocks with the highest SAR, it is
essentially so “unique” that the replicate of this stock is just the risk-free
asset. In other words, the next best alternative of not investing into a truly
unique stock is simply the risk-free asset. In contrast for stocks with the
lowest SAR, one can statistically identify about 82.58 risky peers. So even if
one forgoes investing into a “ubiquitous” stock, one can construct its
replicate consisting of 82.58 peer stocks that have similar statistical
properties. Next, the characteristic
$Z_{i,t-1}=\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}$ is the
proportion of equity that is used to construct the replicate of stock $i$.
Result (iv) shows we allocate only 4% into equity components (so 96% into the
risk-free asset) of the replicates for stocks with the highest SAR, but we
allocate on average 60% into equity components of the replicates for stocks
with the lowest SAR. At least with our elastic-net projection procedure and
even for stocks with the highest SAR, we basically never achieve an 100%
allocation into equity for the replicates of any stock; an 100% allocation can
happen if there is a stock $l$ physically different from stock $i$ but are
otherwise statistically equivalent. That is to say, _all_ stocks are
effectively unique but there’s still nonetheless a wide range of
heterogeneity.

Results (v) to (vii) show the market capitalization, book-to-market and dollar
volume liquidity characteristics. From result (v), low SAR stocks tend to be
smaller stocks while high SAR stocks tend to be bigger stocks. Nonetheless, we
should keep in mind the magnitudes and also within decile bin heterogeneity.
For low SAR stocks the average market capitalization is about $1.2 billion
with a standard deviation of $9.3 billion, and for the high SAR stocks the
average is $6.1 billion with a standard deviation of $26.2 billion. In all,
while the high SAR bin holds on average “mega-sized” stocks, the stocks in the
low SAR bin are not exclusively microcap stocks either. Hence our subsequent
cross-sectional results are unlikely to be driven by a size effect. Result
(vi) shows an interesting value tilt for stocks with high SAR and a growth
tilt for stocks with low SAR. Finally, result (vi) shows the dollar volume
liquidity (VOLD) characteristic. In particular, $\textrm{VOLD}_{i,t-1}$ is
defined as the product of trading volume of stock $i$ on the last day of month
$t-1$ and the closing price of stock $i$ on the last day of month $t-1$, then
all divided by 1,000,000. We see that on average, high SAR stocks have a lower
liquidity than low SAR stocks. On the surface, results (v) to (vii), it
appears size, book-to-market and liquidity are related to a stock’s SAR. We
will explicitly control for these three characteristics, among others, in
Section 3.4 to ensure our results are not driven by the well-known pricing
qualities of these characteristics in the cross-section.

Pursuant to Table 1(i), we directly plot the time series of the cross-
sectionally averaged $\mathsf{R^{2}}$ characteristic in Figure 3(b). Again, as
motivated by our theoretical discussions, we regard assets with low
$\mathsf{R^{2}}$ to have high SAR, and assets with high $\mathsf{R^{2}}$ to
have low SAR. As a consequence, the plots in Figure 3(b) can thus be regarded
as an “average” SAR of the economy at any given point in time. By inspection,
it appears the average SAR of the economy is countercyclical and spikes
considerably during times of financial distress (e.g. 1980-1991 US savings and
loan crisis, 1998 Russian financial, 2008 - 2009 Great Recession, Greek
government debt crisis 2011-2012, March 2020 COVID-19 crash, and others).
These plots strongly suggest SAR (and the resulting SARP of assets) is not
simply a statistical construct but is also capturing systematic shocks. We
formally test the association of SAR with macroeconomic variables in Figure
3(c). The regression results corroborate the visual inspection of Figure 3(b):
shocks to the average SAR is countercyclical, in that it is negatively
associated with shocks to both personal consumption expenditure and consumer
sentiment.

| EN $\mathsf{R^{2}}$ Lo | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | EN $\mathsf{R^{2}}$ Hi
---|---|---|---|---|---|---|---|---|---|---
(i) EN $\mathsf{R^{2}}$ | -0.01 | -0.00 | 0.02 | 0.07 | 0.12 | 0.18 | 0.25 | 0.33 | 0.42 | 0.62
| [0.01] | [0.02] | [0.07] | [0.13] | [0.18] | [0.22] | [0.25] | [0.28] | [0.29] | [0.28]
| (-0.03, -0.00) | (-0.01, 0.01) | (-0.00, 0.14) | (-0.00, 0.37) | (-0.00, 0.52) | (-0.00, 0.62) | (0.00, 0.71) | (0.01, 0.79) | (0.03, 0.86) | (0.11, 0.98)
(ii) FF5 OLS $\mathsf{R^{2}}$ | 0.08 | 0.07 | 0.07 | 0.09 | 0.11 | 0.14 | 0.17 | 0.21 | 0.25 | 0.31
| [0.07] | [0.07] | [0.07] | [0.09] | [0.11] | [0.13] | [0.15] | [0.17] | [0.19] | [0.21]
| (0.01, 0.22) | (0.01, 0.20) | (0.01, 0.21) | (0.01, 0.27) | (0.01, 0.35) | (0.01, 0.42) | (0.01, 0.48) | (0.01, 0.54) | (0.02, 0.60) | (0.02, 0.68)
(iii) Non-zero entries of EN proj vector | 0.50 | 0.72 | 2.34 | 6.17 | 11.37 | 17.76 | 25.22 | 34.07 | 46.85 | 82.58
| [0.60] | [1.63] | [5.73] | [11.66] | [17.00] | [21.89] | [26.84] | [32.28] | [38.45] | [54.92]
| (0.00, 1.00) | (0.00, 2.00) | (0.00, 12.00) | (0.00, 32.00) | (0.00, 46.00) | (0.00, 59.00) | (1.00, 74.00) | (1.00, 92.00) | (3.00, 115.00) | (10.00, 186.00)
(iv) Sum of EN proj vector entries | 0.04 | 0.07 | 0.20 | 0.33 | 0.44 | 0.50 | 0.58 | 0.61 | 0.62 | 0.60
| [0.26] | [0.33] | [0.45] | [0.51] | [0.51] | [0.51] | [0.58] | [0.50] | [0.38] | [0.26]
| (0.00, 0.93) | (0.00, 1.00) | (0.00, 1.00) | (-0.08, 1.00) | (-0.04, 1.00) | (-0.20, 1.00) | (-1.00, 1.00) | (-0.81, 1.00) | (-0.22, 1.00) | (0.11, 0.91)
(v) MktCap (in mil) | 1,214.89 | 863.83 | 888.31 | 1,262.53 | 1,548.21 | 1,817.25 | 2,223.06 | 2,681.28 | 3,450.98 | 6,109.52
| [9,253.69] | [7,449.02] | [8,222.65] | [11,497.41] | [12,337.95] | [13,904.73] | [16,214.65] | [16,726.44] | [19,528.92] | [26,175.47]
| (5.55, 3,644.49) | (4.33, 2,283.37) | (3.99, 2,097.74) | (4.13, 2,828.01) | (4.59, 4,027.00) | (5.27, 5,272.39) | (6.47, 6,889.35) | (7.71, 9,005.63) | (9.48, 12,596.65) | (14.86, 23,868.21)
(vi) Book-to-market | 0.95 | 0.98 | 0.96 | 0.90 | 0.85 | 0.81 | 0.78 | 0.75 | 0.72 | 0.68
| [1.17] | [1.16] | [1.17] | [1.14] | [1.07] | [0.98] | [0.94] | [0.87] | [0.84] | [0.68]
| (0.11, 2.45) | (0.11, 2.57) | (0.10, 2.54) | (0.10, 2.38) | (0.09, 2.24) | (0.09, 2.12) | (0.10, 2.01) | (0.10, 1.89) | (0.10, 1.79) | (0.10, 1.64)
(vii) Dollar volume liquidity (in mil) | 6.96 | 4.67 | 5.46 | 8.71 | 11.39 | 13.70 | 16.80 | 20.39 | 26.66 | 44.11
| [56.09] | [37.73] | [50.05] | [107.06] | [107.76] | [109.11] | [114.71] | [117.61] | [127.33] | [166.52]
| (0.01, 23.11) | (0.01, 13.79) | (0.01, 14.42) | (0.01, 24.60) | (0.01, 37.15) | (0.01, 50.20) | (0.01, 64.98) | (0.02, 86.54) | (0.02, 123.23) | (0.03, 198.25)

Table 1: Summary characteristics of elastic-net $\mathsf{R^{2}}$ sorted
stocks. We project each stock’s past twelve month’s daily returns onto every
other stock using the elastic-net estimator. Stocks are sorted into deciles
based on their elastic-net $\mathsf{R^{2}}$ from the lowest (quantile 1,
labelled “Lo”) to highest (quantile 10, labelled “Hi”). At the end of each
month, we first compute the simple average of the stocks’ characteristics
within each bin. We then take the time-average of these averaged
characteristics for each bin, and the displayed figure shows this cross-
sectional and time-series average for each bin. Brackets show the standard
deviation and the parentheses show the $(5^{\mathrm{th}},95^{\mathrm{th}})$
percentiles of the characteristics. The sampling period is from December 31,
1975 to November 30, 2020.

(a) Dimensionality value $N_{t-1}$

(b) Average EN $\mathsf{R^{2}}$ and lagged its difference

| (1) | (2) | (3)
---|---|---|---
| EN $\Delta\bar{\mathsf{R}}^{2}_{t}$
$\Delta\log\textrm{INDPRO}_{t}$ | 0.092 | 0.305 | 0.229
| (0.278) | (0.951) | (0.871)
$\Delta\log\textrm{PCE}_{t}$ |  $-$0.820 |  $-$1.082 |  $-$0.893
| ($-$2.138) | ($-$2.074) | ($-$2.044)
$\Delta\textrm{UNRATE}_{t}$ |  $-$0.750 |  $-$0.980 |  $-$0.505
| ($-$0.879) | ($-$1.118) | ($-$0.694)
$\Delta\log\textrm{UMCSENT}_{t}$ |  $-$0.081 |  $-$0.075 |  $-$0.047
| ($-$3.079) | ($-$3.294) | ($-$2.262)
$\Delta\log\textrm{INDPRO}_{t-1}$ |  |  $-$0.289 |  $-$0.111
|  | ($-$1.195) | ($-$0.587)
$\Delta\log\textrm{PCE}_{t-1}$ |  |  $-$0.046 | 0.075
|  | ($-$0.148) | (0.270)
$\Delta\textrm{UNRATE}_{t-1}$ |  | 0.110 | 0.472
|  | (0.224) | (1.008)
$\Delta\log\textrm{UMCSENT}_{t-1}$ |  |  $-$0.031 |  $-$0.029
|  | ($-$1.617) | ($-$1.592)
$\textrm{MktRF}_{t}$ |  |  |  $-$0.183
|  |  | ($-$2.746)
Constant | 0.002 | 0.003 | 0.004
| (1.431) | (1.519) | (1.918)
Observations | 514 | 513 | 513
R2 | 0.073 | 0.089 | 0.176
Adjusted R2 | 0.065 | 0.075 | 0.161

(c) Regressing the difference of EN $\mathsf{R^{2}}$ on macro variables

Figure 3: Number of regressors, average elastic-net $\mathsf{R^{2}}$, and
macro regressions. Figure (a) shows the time series of the total number of
traded stocks $N_{t-1}$ in the US at any given point in time. The elastic-net
$\mathsf{R^{2}}$ for each stock $i$ at month $t-1$ is denoted as
$\mathsf{R^{2}}_{i,t-1}$. The top panel of Figure (b) plots the time series of
the averaged $\mathsf{R^{2}}$ across all stocks $\bar{\mathsf{R^{2}}}_{t}$,
and the bottom panel plots the difference
$\Delta\bar{\mathsf{R^{2}}}_{t}:=\mathsf{R^{2}}_{t}-\mathsf{R^{2}}_{t-1}$.
Table (c) regresses the time series $\Delta\bar{\mathsf{R^{2}}}_{t}$ onto the
following regressors’ difference and its lag: industrial production, personal
consumption expenditure, unemployment rate, University of Michigan consumer
sentiment, and the market factor. All macro variables data are from FRED
Economic Data of the St. Louis Federal Reserve. Since UMCSENT is only readily
available from January 1978, the regressions’ sampling period is monthly from
January 31, 1978 to December 31, 2020.

####  3.1.2 Time series correlations and replicates as hedging instruments

How good are the replicates in mimicking the return behavior of the focals?
While we will discuss at length the first moments of the focals and the
replicates starting from Section 3.2, let’s first investigate into their
second cross-moments. For bins
$k_{1},k_{2}\in\\{\mathrm{Lo},2,\ldots,\mathrm{Hi}\\}$ and return type
$m_{1},m_{2}\in\\{\textrm{Foc},\textrm{Rep}\\}$ (recall (11) and (12)), we
track the time series $\bar{R}^{k_{1}}_{m_{1},t}$ and the time series
$\bar{R}^{k_{2}}_{m_{2},t}$. The numbers in Table 2 reports the sample
correlation between the time series $\bar{R}^{k_{1}}_{m_{1},t}$ and
$\bar{R}^{k_{2}}_{m_{2},t}$. The diagonal entries are most important: stocks
with low (high) $\mathsf{R^{2}}$’s have low (high) correlations with their
replicates. Specifically, the correlation between the returns of stocks with
highest SAR (so they have the lowest $\mathsf{R^{2}}$) with their replicates
is only $17.8\%$. In contrast, this correlation rises substantially along the
diagonal of Table 2 to $91.9\%$ for stocks with the lowest SAR (so highest
$\mathsf{R^{2}}$). In all, for stocks with low SAR, the elastic-net replicate
portfolio procedure (that is entirely based on past returns) do a fairly good
job in matching the stock’s one-month ahead return second moments. However,
but expectedly, the replicates do a poor job in mimicking stocks with high
SAR.

| Lo Rep | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Hi Rep
---|---|---|---|---|---|---|---|---|---|---
EN $\mathsf{R^{2}}$ |  |  |  |  |  |  |  |  |  |
Lo Foc | 0.178 | 0.358 | 0.500 | 0.613 | 0.661 | 0.705 | 0.693 | 0.678 | 0.729 | 0.801
2 | 0.208 | 0.363 | 0.499 | 0.642 | 0.709 | 0.744 | 0.738 | 0.724 | 0.770 | 0.826
3 | 0.210 | 0.416 | 0.542 | 0.664 | 0.735 | 0.762 | 0.757 | 0.742 | 0.782 | 0.839
4 | 0.220 | 0.395 | 0.543 | 0.695 | 0.762 | 0.780 | 0.778 | 0.764 | 0.803 | 0.857
5 | 0.187 | 0.370 | 0.525 | 0.688 | 0.769 | 0.790 | 0.782 | 0.769 | 0.811 | 0.868
6 | 0.188 | 0.337 | 0.491 | 0.664 | 0.758 | 0.793 | 0.798 | 0.789 | 0.836 | 0.893
7 | 0.189 | 0.328 | 0.473 | 0.649 | 0.763 | 0.813 | 0.813 | 0.807 | 0.855 | 0.904
8 | 0.185 | 0.306 | 0.450 | 0.629 | 0.760 | 0.819 | 0.828 | 0.830 | 0.879 | 0.918
9 | 0.182 | 0.289 | 0.426 | 0.601 | 0.737 | 0.806 | 0.818 | 0.824 | 0.884 | 0.927
Hi Foc | 0.166 | 0.275 | 0.416 | 0.573 | 0.684 | 0.752 | 0.759 | 0.765 | 0.833 | 0.919
Table 2:  Pairwise correlations of one-month ahead returns between stocks and
their replicates. The $(k_{1},k_{2})$-th entry above represents the
correlation of the time series one month ahead returns between the $k_{1}$-th
elastic-net $\mathsf{R^{2}}$ sorted bin of the stocks and the $k_{2}$-th
portfolio bin its replicate. The entries are expressed in decimals (e.g. 0.10
means 10%). We form equal-weighted portfolios decile portfolios every month by
projecting each stock’s daily return over the past year onto every other stock
using the elastic-net estimator. Stocks are sorted into deciles based on their
elastic-net $\mathsf{R^{2}}$ from the lowest (quantile 1, labelled “Lo”) to
highest (quantile 10, labelled “Hi”). The rows labelled “Foc” report the
portfolio of the focal stocks (7). The columns labelled “Rep” report the
replicate returns of the focal stock that are constructed out of the estimated
elastic-net beta coefficients according to (9). The sampling period is from
January 31, 1976 to December 31, 2020.

###  3.2 Main empirical result: Statistical arbitrage risk premium (SARP)

The univariate portfolio sorts of Table 3 show the main empirical result of
this paper. Let’s first discuss the returns of focal stocks in the first
column of Table 3. Stocks with the _lowest_ elastic-net $\mathsf{R^{2}}$ earn
an excess return of 1.481% ($t$-stat 5.298) per month, while stocks with the
_highest_ elastic-net $\mathsf{R^{2}}$ earn an excess return of 0.771%
($t$-stat 2.882). The _difference_ in returns between stocks with the lowest
and highest elastic-net $\mathsf{R^{2}}$ is 0.710% ($t$-stat 4.386). In other
words, high SAR stocks have a substantially greater return than low SAR
stocks.

| Focal | Replicate | Foc - Rep
---|---|---|---
| mean | $t$ | mean | $t$ | mean | $t$
EN $\mathsf{R^{2}}$ |  |  |  |  |  |
Lo | 1.481 | (5.298) | 0.115 | (2.109) | 1.368 | (4.929)
2 | 1.167 | (4.047) | 0.346 | (2.767) | 0.818 | (3.056)
3 | 1.008 | (3.408) | 0.383 | (1.516) | 0.633 | (2.459)
4 | 0.941 | (3.186) | 0.432 | (1.451) | 0.511 | (2.280)
5 | 0.915 | (3.196) | 0.360 | (1.102) | 0.561 | (2.634)
6 | 0.889 | (3.196) | 0.465 | (1.400) | 0.431 | (2.089)
7 | 0.869 | (3.177) | 0.724 | (2.080) | 0.149 | (0.720)
8 | 0.891 | (3.198) | 0.792 | (2.303) | 0.093 | (0.470)
9 | 0.813 | (2.943) | 0.655 | (2.130) | 0.161 | (1.112)
Hi | 0.771 | (2.882) | 0.503 | (2.070) | 0.267 | (2.581)
Lo - Hi | 0.710 | (4.386) | -0.387 | (-1.662) | 1.101 | (4.132)
Avg | 0.974 | (3.584) | 0.478 | (2.069) | 0.499 | (3.391)
Table 3: (MAIN RESULTS) Univariate portfolio sort by elastic-net
$\mathsf{R^{2}}$. We form equal-weighted decile portfolios every month by
projecting each stock’s past twelve month’s daily returns onto every other
stock using the elastic-net estimator. Stocks are sorted into deciles based on
their elastic-net $\mathsf{R^{2}}$ from the lowest (quantile 1, labelled “Lo”)
to highest (quantile 10, labelled “Hi”). The row labelled “Lo - Hi” is the
monthly return difference between the “Lo” bin and the “Hi” bin. The row
labelled “Avg” is the simple average of the monthly excess returns across the
ten $k=\textrm{`Lo'},2,\ldots,\textrm{`Hi'}$ bins. The column labelled “Focal”
reports the one-month ahead portfolio excess returns (7). The column labelled
“Replicate” reports the one-month ahead excess returns of a portfolio that are
constructed out of the estimated normalized elastic-net coefficients according
to (9). The column labelled “Foc - Rep” reports the one-month ahead returns of
the portfolio of a long position in the focal stocks, and a short position in
the corresponding replicates. We define the _Statistical Arbitrage Risk
Premium_ (SARP) as the returns from “Foc - Rep”. In addition, a stock with low
EN $\mathsf{R^{2}}$ is said to have high _Statistical Arbitrage Risk_ (SAR),
while a stock with high EN $\mathsf{R^{2}}$ is said to have low SAR. The
“mean” column is reported in monthly percentage terms (e.g. 1.0 means 1%).
Robust Newey and West (1987) $t$-statistics with six lags are reported in
column “$t$” in parentheses. The sample period is monthly from January 31,
1976 to December 31, 2020.

By conventional wisdom in the empirical asset pricing literature, these
results already strongly hint that $\mathsf{R^{2}}$ is a potential priced
factor in the cross-section. However, guided by our theoretical discussions of
Section 1, we are not just interested in testing the cross-sectional
difference in the focal stocks (but see later in Section 3.5 for a factor
discussion). Rather, we are interested in empirically testing for the cross-
sectional presence of SARP. Let’s consider the empirical results of the
replicates in the second column of Table 3. The excess returns of the
replicates are almost monotonically increasing, from $0.115\%$ ($t$-stat
2.109) per month in the _lowest_ $\mathsf{R^{2}}$ bin, to 0.503% ($t$-stat
2.070) in the _highest_ $\mathsf{R^{2}}$ bin, even though their _difference_
-0.387% ($t$-stat -1.662) is only weakly statistically significant. The excess
returns of the replicates in some middle bins (e.g. bins 3 - 6) are
statistically insignificant from zero. Other than these four middle bins, it
appears the projection construction procedure does construct a replicate asset
that has reasonable mean returns.

Finally, we come to the main highlight of our paper: the third column of Table
3 proxies for the SARP. The focals minus replicates have an average return of
1.368% ($t$-stat 4.929) per month for focal stocks with the _lowest_
$\mathsf{R^{2}}$’s, it is 0.267% ($t$-stat 2.581) with the _highest_
$\mathsf{R^{2}}$’s, and the _difference_ of 1.101% ($t$-stat 4.132) is highly
statistically significant. This result is strong evidence of showing the key
hypothesis of our theoretical discussions: assets with high SAR have high
SARP, and assets with low SAR have low SARP. Beyond the aforementioned cross-
sectional results, we also document the presence of an “unconditional SARP”.
The “Avg” portfolio in Table 3 takes the simple average of returns of all of
the ten $k=\textrm{`Lo'},2,\ldots,\textrm{`Hi'}$ bins. We find that “Avg”
enjoys a monthly SARP of 0.499% ($t$-stat 3.391).

In all, these empirical results show strong evidence in support of our core
theoretical predictions: _(i) an unconditional SARP exists and is positive for
the average representative asset; and (ii) SARP is increasing with SAR: stocks
with low (high) SAR earn a low (high) SARP in the cross-section._

###  3.3 Controlling for other risk factors and characteristics

We present two sets of evidence to show our main result — SARP is increasing
in SAR — from Table 3 is robust after controlling for risk factors and other
characteristics.

####  3.3.1 Risk adjusted returns on post-formation portfolios

We show our results still persist even after adjusting for the Fama-French
three factors (Fama and French (1992, 1993)) and Fama-French five factors
(Fama and French (2015)). For each bin $k$ and each return
$r_{t}^{k}\in\\{\bar{R}^{k}_{\mathrm{Foc},t},\bar{R}^{k}_{\mathrm{Rep},t},\bar{R}^{k}_{\mathrm{LS},t}\\}$,
we consider the following two time series factor model regressions:

$\displaystyle
r_{t}^{k}=\alpha^{k}+\beta^{k}_{\textrm{MktRF}}\textrm{MktRF}_{t}+\beta^{k}_{\textrm{SMB}}\textrm{SMB}_{t}+\beta^{k}_{\textrm{HML}}\textrm{HML}_{t}+\varepsilon_{t}^{k},$
(14a) $\displaystyle
r_{t}^{k}=\alpha^{k}+\beta^{k}_{\textrm{MktRF}}\textrm{MktRF}_{t}+\beta^{k}_{\textrm{SMB}}\textrm{SMB}_{t}+\beta^{k}_{\textrm{HML}}\textrm{HML}_{t}+\beta^{k}_{\textrm{CMA}}\textrm{CMA}_{t}+\beta^{k}_{\textrm{RMW}}\textrm{RMW}_{t}+\varepsilon_{t}^{k}.$
(14b)

The regressors are well-known: “MktRF” is the market factor, “SMB” is the size
factor, “HML” is the value factor, “CMA” is the investment factor, and “RMW”
is the profitability factor.

Table 4 shows the FF3 and FF5 $\alpha$ estimates. Even controlling for known
factors, the overall result that SARP is increasing in SAR remains robust. In
particular, the FF3 $\alpha$ of SARP of stocks with the highest SAR is 0.573%
($t$-stat 4.216) per month, while it is 0.020% ($t$-stat 0.207) for stocks
with the lowest SAR, and the cross-sectional difference is statistically
significant at 0.553% ($t$-stat 3.039). The cross-sectional $\alpha$ estimate
of SARP using the FF5 model is analogous.

| Lo EN $\mathsf{R^{2}}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Hi EN $\mathsf{R^{2}}$ | Lo - Hi
---|---|---|---|---|---|---|---|---|---|---|---
Focal |  |  |  |  |  |  |  |  |  |  |
FF3 $\alpha$ | 0.658 | 0.360 | 0.155 | 0.061 | 0.000 | -0.063 | -0.097 | -0.100 | -0.180 | -0.219 | 0.877
| (5.338) | (2.909) | (1.166) | (0.485) | (0.001) | (-0.722) | (-1.500) | (-1.581) | (-2.437) | (-2.376) | (4.908)
FF5 $\alpha$ | 0.700 | 0.383 | 0.140 | 0.044 | -0.014 | -0.051 | -0.028 | 0.060 | 0.061 | -0.033 | 0.733
| (5.684) | (2.911) | (1.012) | (0.318) | (-0.111) | (-0.453) | (-0.265) | (0.529) | (0.485) | (-0.336) | (4.217)
Replicate |  |  |  |  |  |  |  |  |  |  |
FF3 $\alpha$ | 0.087 | 0.198 | -0.031 | -0.200 | -0.396 | -0.353 | -0.150 | -0.061 | -0.169 | -0.239 | 0.326
| (1.526) | (1.859) | (-0.143) | (-0.829) | (-1.676) | (-1.629) | (-0.643) | (-0.253) | (-0.907) | (-2.021) | (2.962)
FF5 $\alpha$ | 0.100 | 0.195 | -0.052 | -0.144 | -0.209 | -0.069 | 0.172 | 0.297 | 0.160 | -0.062 | 0.162
| (1.631) | (1.829) | (-0.253) | (-0.557) | (-0.750) | (-0.275) | (0.643) | (1.081) | (0.758) | (-0.507) | (1.399)
Foc - Rep |  |  |  |  |  |  |  |  |  |  |
FF3 $\alpha$ | 0.573 | 0.163 | 0.197 | 0.264 | 0.403 | 0.299 | 0.060 | -0.045 | -0.008 | 0.020 | 0.553
| (4.216) | (1.142) | (1.040) | (1.271) | (1.913) | (1.437) | (0.275) | (-0.209) | (-0.055) | (0.207) | (3.039)
FF5 $\alpha$ | 0.602 | 0.187 | 0.199 | 0.193 | 0.203 | 0.031 | -0.193 | -0.242 | -0.098 | 0.028 | 0.574
| (4.330) | (1.231) | (1.041) | (0.900) | (0.909) | (0.148) | (-0.854) | (-1.081) | (-0.618) | (0.281) | (3.149)

Table 4: Fama-French 3 and 5 factor regressions on portfolios sorted by
elastic-net $\mathsf{R^{2}}$. We run the Fama-French three (14a) and five
(14b) factor time series regressions onto the elastic-net $\mathsf{R^{2}}$
sorted decile portfolios. Stocks are sorted into deciles based on their
elastic-net $\mathsf{R^{2}}$ from the lowest (quantile 1, labelled “Lo”) to
highest (quantile 10, labelled “Hi”). The column labelled “Lo - Hi” is the
monthly return difference between the “Lo” bin and the “Hi” bin. The row
labelled “Avg” is the simple average of the monthly excess returns across the
ten $k=\textrm{`Lo'},2,\ldots,\textrm{`Hi'}$ bins. The group labelled “Focal”
reports the one-month ahead portfolio excess returns (7). The group labelled
“Replicate” reports the one-month ahead excess returns of a portfolio that are
constructed out of the estimated normalized elastic-net beta coefficients
according to (9). The group labelled “Foc - Rep” reports the one-month ahead
returns of the portfolio of a long position in the focal stocks, and a short
position in the corresponding replicates. We define the _Statistical Arbitrage
Risk Premium_ (SARP) as the returns from “Foc - Rep”. In addition, a stock
with low EN $\mathsf{R^{2}}$ is said to have high _Statistical Arbitrage Risk_
(SAR), while a stock with high EN $\mathsf{R^{2}}$ is said to have low SAR.
The estimated values are reported in monthly percentage points (e.g. 1.0 means
1%) and we report the Newey and West (1987) $t$-statistics with six lags in
parentheses. The sample period is from January 31, 1976 to December 31, 2020.

###  3.4 Bivariate dependent sorts

We show our main result remains robust after we control for various stock
characteristics. At the end of each month, stocks are first sorted by a
characteristic into quintile portfolios. Then for a given characteristic and
within each of its five portfolios, we further sort stocks based on the
elastic-net $\mathsf{R^{2}}$ into deciles bins. Each of these ten elastic-net
$\mathsf{R^{2}}$ bins are then averaged over their respective five
characteristic portfolios. In all, the ten resulting elastic-net
$\mathsf{R^{2}}$ bins represent returns that control for a particular
characteristic. All portfolios are equal-weighted.

Let’s describe the characteristics. _MktCap_ is market capitalization and
_B/M_ is book-to-market, and their quintiles are calculated using the NYSE
breakpoints. _IdioVol_ is Ang et al. (2006)’s idiosyncratic volatility; the
idiosyncratic volatility of stock $i$ at month $t-1$ is the standard deviation
of the residuals, arising from the OLS regression of past one year’s daily
returns leading up until month $t-1$ onto the Fama-French three factors.
_TotalVol_ is total volatility; it is the sample standard deviation of a stock
$i$’s past twelve months of daily returns up until month $t-1$. _IdioSkew_ is
idiosyncratic skewness; it is calculated as the sample Pearson’s moment of
coefficient skewness of the residuals of the OLS regression of a stock’s past
twelve months’ daily returns up until monh $t-1$ onto the Fama-French three
factors (with intercept). _TotalSkew_ is total skewness; it is calculated as
the sample Pearson’s moment of coefficient skewness of a stock’s past twelve
months’ daily returns up until month $t-1$. _AmihudIlliq_ is Amihud (2002)’s
illiquidity measure; it is defined as
$\textrm{AmihudIlliq}_{i,t-1}:=\frac{1}{D_{i,t-1}}\sum_{d=1}^{D_{i,t-1}}\frac{\left|R_{i,d}\right|}{\textrm{VOLD}_{i,d}}$,
where $D_{i,t-1}$ is the total number of trading days of stock $i$ in the past
twelve months leading up to month $t-1$ and VOLD was as defined in Section
3.1.1. _Mom_ is momentum; we define momentum of stock $i$ at the end of month
$t-1$ (i.e. the end of the estimation period) as the return of the stock
during the 11-month period covering months $t-12$ through $t-2$. _STR_ is
Jegadeesh (1990) and Lehmann (1990)’s short-term reversal; it is defined as
$\textrm{STR}_{i,t-1}=100\times R_{i,t-1}$.

Table 5 shows the results. We omit showing the results on the replicates for
brevity. Overall, these results that control for the aforementioned
characteristics are consistent with the unconditional results of Table 3. We
pay special attention our results after controlling for size, value, VOLD,
idiosyncratic volatility and total volatility. In particular, even though
Table 1 shows a size, value and VOLD tilt for the elastic-net $\mathsf{R^{2}}$
sorted stocks, the results here show that SARP is still persistent after we
control for these characteristics.

|  | Lo EN $\mathsf{R^{2}}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Hi EN $\mathsf{R^{2}}$ | Lo - Hi
---|---|---|---|---|---|---|---|---|---|---|---|---
MktCap | Focal | 1.389 | 1.101 | 0.995 | 0.937 | 0.811 | 0.826 | 0.880 | 0.954 | 0.911 | 0.921 | 0.468
|  | (5.527) | (4.335) | (3.829) | (3.521) | (3.018) | (2.974) | (3.078) | (3.162) | (2.997) | (2.917) | (3.070)
| Foc - Rep | 1.085 | 0.680 | 0.616 | 0.642 | 0.420 | 0.481 | 0.236 | 0.180 | 0.205 | 0.404 | 0.681
|  | (4.863) | (3.207) | (3.380) | (3.691) | (2.255) | (2.726) | (1.168) | (0.989) | (1.438) | (3.600) | (3.737)
B/M | Focal | 1.434 | 1.150 | 0.954 | 1.002 | 0.931 | 0.920 | 0.911 | 0.936 | 0.926 | 0.882 | 0.552
|  | (5.074) | (4.074) | (3.250) | (3.586) | (3.346) | (3.324) | (3.347) | (3.422) | (3.461) | (3.404) | (3.835)
| Foc - Rep | 1.293 | 0.820 | 0.577 | 0.608 | 0.524 | 0.461 | 0.198 | 0.135 | 0.242 | 0.353 | 0.941
|  | (4.716) | (3.268) | (2.446) | (3.009) | (2.556) | (2.405) | (0.972) | (0.660) | (1.524) | (3.395) | (3.516)
IdioVol | Focal | 1.416 | 1.090 | 1.015 | 0.952 | 0.985 | 0.855 | 0.945 | 0.923 | 0.827 | 0.781 | 0.635
|  | (5.453) | (4.285) | (3.851) | (3.549) | (3.651) | (3.040) | (3.234) | (3.116) | (2.749) | (2.451) | (4.031)
| Foc - Rep | 1.196 | 0.641 | 0.665 | 0.610 | 0.663 | 0.402 | 0.241 | 0.159 | 0.159 | 0.300 | 0.896
|  | (4.830) | (2.659) | (3.214) | (2.994) | (3.401) | (2.163) | (1.205) | (0.836) | (1.099) | (2.618) | (4.371)
TotalVol | Focal | 1.422 | 1.124 | 1.004 | 0.977 | 0.960 | 0.859 | 0.921 | 0.893 | 0.849 | 0.777 | 0.645
|  | (5.244) | (4.252) | (3.642) | (3.584) | (3.463) | (3.043) | (3.216) | (3.091) | (2.921) | (2.657) | (4.557)
| Foc - Rep | 1.276 | 0.676 | 0.639 | 0.633 | 0.604 | 0.436 | 0.160 | 0.141 | 0.174 | 0.295 | 0.981
|  | (4.888) | (2.772) | (2.870) | (3.148) | (3.103) | (2.379) | (0.788) | (0.743) | (1.194) | (2.851) | (4.301)
AmihudIlliq | Focal | 1.367 | 1.103 | 0.973 | 0.992 | 0.866 | 0.850 | 0.950 | 0.871 | 0.979 | 0.932 | 0.434
|  | (5.436) | (4.261) | (3.747) | (3.653) | (3.175) | (3.116) | (3.362) | (2.970) | (3.390) | (3.258) | (3.071)
| Foc - Rep | 0.996 | 0.730 | 0.583 | 0.724 | 0.510 | 0.473 | 0.300 | 0.099 | 0.295 | 0.419 | 0.577
|  | (4.546) | (3.655) | (3.142) | (4.076) | (2.826) | (2.697) | (1.478) | (0.508) | (2.042) | (3.906) | (3.197)
Mom | Focal | 1.428 | 1.114 | 1.103 | 0.976 | 0.975 | 0.882 | 0.894 | 0.895 | 0.775 | 0.703 | 0.725
|  | (5.151) | (3.901) | (3.856) | (3.429) | (3.470) | (3.167) | (3.267) | (3.288) | (2.779) | (2.626) | (5.282)
| Foc - Rep | 1.311 | 0.739 | 0.725 | 0.561 | 0.611 | 0.432 | 0.155 | 0.110 | 0.092 | 0.216 | 1.095
|  | (4.803) | (2.841) | (3.048) | (2.611) | (2.914) | (2.301) | (0.763) | (0.537) | (0.604) | (2.168) | (4.309)
STR | Focal | 1.461 | 1.171 | 0.983 | 0.945 | 0.881 | 0.833 | 0.888 | 0.901 | 0.834 | 0.822 | 0.638
|  | (5.252) | (4.069) | (3.464) | (3.199) | (3.146) | (3.038) | (3.187) | (3.314) | (3.010) | (3.027) | (4.213)
| Foc - Rep | 1.332 | 0.825 | 0.616 | 0.528 | 0.526 | 0.334 | 0.162 | 0.095 | 0.195 | 0.320 | 1.012
|  | (4.893) | (3.235) | (2.525) | (2.408) | (2.631) | (1.724) | (0.792) | (0.469) | (1.377) | (3.160) | (3.972)
VOLD | Focal | 1.341 | 1.103 | 0.985 | 0.936 | 0.871 | 0.831 | 0.951 | 0.929 | 0.973 | 0.949 | 0.392
|  | (5.153) | (4.142) | (3.700) | (3.427) | (3.153) | (3.020) | (3.337) | (3.246) | (3.491) | (3.511) | (2.782)
| Foc - Rep | 1.033 | 0.745 | 0.610 | 0.603 | 0.528 | 0.459 | 0.247 | 0.170 | 0.303 | 0.423 | 0.611
|  | (4.711) | (3.686) | (3.261) | (3.487) | (3.006) | (2.673) | (1.234) | (0.842) | (2.017) | (4.029) | (3.211)
IdioSkew | Focal | 1.451 | 1.148 | 1.041 | 0.943 | 0.903 | 0.856 | 0.843 | 0.898 | 0.864 | 0.787 | 0.664
|  | (5.408) | (4.159) | (3.494) | (3.180) | (3.218) | (3.098) | (2.983) | (3.195) | (3.035) | (2.885) | (4.310)
| Foc - Rep | 1.300 | 0.773 | 0.711 | 0.538 | 0.563 | 0.375 | 0.202 | 0.061 | 0.169 | 0.287 | 1.013
|  | (4.938) | (2.915) | (3.234) | (2.558) | (2.986) | (1.928) | (1.057) | (0.293) | (1.172) | (2.773) | (4.095)
TotalSkew | Focal | 1.437 | 1.156 | 1.032 | 0.982 | 0.950 | 0.804 | 0.842 | 0.908 | 0.831 | 0.794 | 0.643
|  | (5.438) | (4.175) | (3.478) | (3.397) | (3.348) | (2.926) | (2.973) | (3.207) | (2.898) | (2.897) | (4.267)
| Foc - Rep | 1.271 | 0.758 | 0.685 | 0.613 | 0.596 | 0.342 | 0.175 | 0.064 | 0.185 | 0.290 | 0.980
|  | (4.982) | (2.894) | (3.107) | (3.029) | (3.214) | (1.774) | (0.851) | (0.325) | (1.296) | (2.908) | (4.118)

Table 5: Dependent bivariate sorts by characteristics and elastic-net
$\mathsf{R^{2}}$. This table reports mean returns in monthly percentage points
(e.g. 1.0 means 1%) and Newey and West (1987) robust $t$-statistics with 6
lags in parentheses. We perform a bivariate dependent sort. Each month, we
first sort stocks based on a characteristic (i.e. size, book-to-market,
idiosyncratic volatility, total volatility, Amihud’s illiquidity, momentum,
short-term reversal, dollar volume liquidity, idiosyncratic skewness, and
total skewness; see Section 3.4 for details and references on these
characteristics) into quintiles portfolios. For a given characteristic and
within each of its five portfolios, we sort stocks based on elastic-net
$\mathsf{R^{2}}$ into decile bins, from the lowest (quantile 1, labelled “Lo”)
to highest (quantile 10, labelled “Hi”). The column labelled “Lo - Hi” is the
monthly return difference between the “Lo” bin and the “Hi” bin. These ten
elastic-net $\mathsf{R^{2}}$ bins are then averaged over each of the five
characteristic portfolios. Thus these ten elastic-net $\mathsf{R^{2}}$ bins
represent returns that control for a particular characteristic. All portfolios
are equal-weighted. The sample period is monthly from January 31, 1976 to
December 31, 2020.

###  3.5 Statistical Arbitrage Risk factor (SAR factor)

The main empirical objective of this paper is to show assets’ SARP increases
with SAR. However as hinted in the main results Section 3.2, conventional
empirical asset pricing results would suggest that SAR is a priced factor. We
can show the following corollary factor result. Let us define the _Statistical
Arbitrage Risk factor (SAR factor)_ with returns at month $t$ as

$R_{\mathrm{SAR},t}:=\bar{R}^{\mathrm{Lo}}_{\mathrm{LS},t}-\bar{R}^{\mathrm{Hi}}_{\mathrm{LS},t}.$
(15)

where recall the definition of $\bar{R}^{k}_{\mathrm{LS},t}$ in (13). In other
words, the SAR factor is simply the difference in the SARP of the high SAR and
low SAR stocks.

Let’s investigate the price of risk $\lambda_{\mathrm{SAR}}$ of our SAR factor
by the classical Fama and MacBeth (1973) regressions. We run the Fama and
MacBeth (1973) with four different models: (i) SAR factor + CAPM; (ii) SAR
factor + FF3 + momentum; (iii) SAR factor + FF5 + momentum; and (iv) FF5 +
momentum as a benchmark. The data of the momentum factor is available from
Kenneth French’s website. Table 6(a) shows the correlations of our SAR factor
along with the other asset pricing factors. Next is the choice of test assets.
We first evaluate the price of risk of our SAR factor on the classical 25
portfolios formed on size and book-to-market ($5\times 5)$. In addition, given
our projection and replicate construction procedure of Section 2 is explicitly
dependent on past returns, it is prudent to evaluate our SAR factor on test
assets that are also sorted along a past return dependent characteristic. To
this end, we will also consider the 25 portfolios formed on size and momentum
($5\times 5$) and the 25 portfolios formed on size and residual variance
($5\times 5)$. Finally, we also evaluate our SAR factor on portfolios that are
further sorted along corporate fundamentals, and consider the 25 portfolios
formed on book-to-market and investments ($5\times 5$), and the 36 portfolios
formed on size, operating profitability and investments ($2\times 4\times 4$).
All test portfolios are equal-weighted and are obtained from French’s website.
Table 6 shows the main result of this section and the focus is the value of
$\lambda_{\mathrm{SAR}}$. The SAR factor has a positive price of risk, is
statistically significant, and is robust across the various model
specifications and test assets. Focusing on the SAR factor + FF5 + momentum
model, we see the price of risk estimate for the SAR factor range from 0.990%
to 1.631% per month, depending on the test asset.

Test asset | $\lambda_{\mathrm{SAR}}$ | $\lambda_{\mathrm{MktRF}}$ | $\lambda_{\mathrm{SMB}}$ | $\lambda_{\mathrm{HML}}$ | $\lambda_{\mathrm{RMW}}$ | $\lambda_{\mathrm{CMA}}$ | $\lambda_{\mathrm{MOM}}$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---
Size and B/M ($5\times 5$) |  |  |  |  |  |  |  |
|  | 0.993 | 0.445 | 0.215 | -0.171 | 0.326 | 2.286 | 96.91
|  | (4.695) | (3.308) | (1.575) | (-0.724) | (1.861) | (4.122) | [p = 0.000]
| 0.652 | 0.777 |  |  |  |  |  | 64.66
| (2.354) | (3.817) |  |  |  |  |  | [p = 0.000]
| 1.099 | 0.997 | 0.438 | 0.215 |  |  | 2.489 | 56.24
| (3.989) | (4.774) | (3.267) | (1.589) |  |  | (4.449) | [p = 0.000]
| 1.224 | 0.861 | 0.434 | 0.184 | 0.366 | -0.037 | 1.876 | 161.15
| (4.288) | (4.120) | (3.234) | (1.348) | (1.821) | (-0.196) | (3.577) | [p = 0.000]
Size and Momentum ($5\times 5$) |  |  |  |  |  |  |  |
|  | 0.717 | 0.308 | 1.053 | -0.467 | 0.894 | 0.615 | 64.50
|  | (3.618) | (2.203) | (3.397) | (-1.763) | (2.907) | (3.209) | [p = 0.000]
| 0.948 | 0.666 |  |  |  |  |  | 88.03
| (3.492) | (3.291) |  |  |  |  |  | [p = 0.000]
| 1.097 | 0.674 | 0.334 | 0.782 |  |  | 0.616 | 59.11
| (4.010) | (3.403) | (2.486) | (3.707) |  |  | (3.215) | [p = 0.000]
| 1.031 | 0.688 | 0.306 | 0.972 | -0.206 | 0.461 | 0.618 | 229.47
| (3.633) | (3.462) | (2.192) | (3.121) | (-0.738) | (1.370) | (3.223) | [p = 0.000]
Size and Residual Variance ($5\times 5$) |  |  |  |  |  |  |  |
|  | 0.555 | 0.257 | 1.216 | 0.052 | 0.295 | -0.541 | 39.07
|  | (2.809) | (1.705) | (5.292) | (0.340) | (0.968) | (-0.822) | [p = 0.027]
| 0.406 | 0.832 |  |  |  |  |  | 581.88
| (1.462) | (4.134) |  |  |  |  |  | [p = 0.000]
| 1.898 | 0.710 | 0.618 | 0.775 |  |  | 3.380 | 84.35
| (4.998) | (3.595) | (3.967) | (3.601) |  |  | (4.358) | [p = 0.000]
| 1.631 | 0.678 | 0.434 | 1.037 | 0.225 | -0.453 | 2.379 | 74.61
| (4.507) | (3.450) | (2.835) | (4.500) | (1.443) | (-1.348) | (3.462) | [p = 0.000]
B/M and Investment ($5\times 5$) |  |  |  |  |  |  |  |
|  | 0.921 | 0.177 | 0.638 | -0.202 | 0.826 | 0.751 | 11.53
|  | (3.121) | (0.533) | (3.372) | (-0.875) | (7.591) | (1.496) | [p = 0.985]
| 1.021 | 0.681 |  |  |  |  |  | 72.78
| (3.666) | (3.240) |  |  |  |  |  | [p = 0.000]
| 1.868 | 1.190 | -0.083 | 0.490 |  |  | 2.615 | 84.89
| (5.428) | (4.846) | (-0.328) | (3.084) |  |  | (4.581) | [p = 0.000]
| 1.000 | 0.960 | 0.106 | 0.607 | -0.116 | 0.807 | 0.892 | 18.96
| (2.573) | (3.262) | (0.326) | (3.088) | (-0.458) | (7.129) | (1.567) | [p = 0.754]
Size, Operating profitability and Investment ($2\times 4\times 4$) |  |  |  |  |  |  |  |
|  | 0.820 | 0.425 | 0.534 | -0.007 | 0.525 | 1.317 | 159.07
|  | (3.998) | (2.726) | (2.677) | (-0.061) | (5.034) | (3.020) | [p = 0.000]
| 0.766 | 0.836 |  |  |  |  |  | 223.96
| (2.772) | (4.143) |  |  |  |  |  | [p = 0.000]
| 1.169 | 0.786 | 0.236 | 0.879 |  |  | 1.106 | 491.33
| (4.145) | (3.801) | (1.556) | (5.312) |  |  | (2.287) | [p = 0.000]
| 0.990 | 0.808 | 0.339 | 0.668 | 0.019 | 0.477 | 1.232 | 136.89
| (3.376) | (3.943) | (2.077) | (3.093) | (0.166) | (4.445) | (2.814) | [p = 0.000]

Table 6: SAR factor price of risk. This table shows the market price of risk
$\lambda_{\mathrm{SAR}}$ in monthly percentage points (i.e. 1.0 means 1%) of
the SAR factor and various other factors, and where the parentheses show the
Fama and MacBeth (1973) $t$-statistics. The last column shows the $\chi^{2}$
statistic and the brackets show its $p$-value. We define our _Statistical
Arbitrage Risk factor_ (SAR factor) in (15); the factor return
$R_{\mathrm{SAR},t}$ is a result of a long position on the Lo elastic-net
$\mathsf{R^{2}}$ bin with returns $\bar{R}^{\mathrm{Lo}}_{\mathrm{LS},t}$ and
a short position on the Hi elastic $\mathsf{R^{2}}$ bin with returns
$\bar{R}^{\mathrm{Hi}}_{\mathrm{LS},t}$. We focus on five equal-weighted tests
assets: the 25 portfolios formed by the 5 size and 5 book-to-market bins; the
25 portfolios formed by the 5 size and 5 momentum bins; the 25 portfolios
formed by size and Ang et al. (2006) residual variance bins; 25 portfolios
formed by 5 book-to-market and 5 investment bins; and 36 portfolios formed by
2 size, 4 operating profitability and 4 investment bins. We use the Fama and
MacBeth (1973) two-pass procedure by first running a monthly time series
regression of the test assets onto the proposed factor models to obtain the
betas, and then run a cross-sectional regression to obtain the prices of risk.
The sampling period is monthly from January 31, 1976 to December 31, 2020.

####  3.5.1 Investment strategy

The SAR factor is clearly a tradable portfolio. Figure 7(a) shows the
cumulative returns (not inflation adjusted) from December 31, 1975 to December
31, 2020 of an initial $\$100$ investment on our SAR factor and other factors,
and Figure 7(b) plots the log cumulative returns. The strategy “(Lo - Hi) Foc”
is the cumulative returns from
$\bar{R}^{\mathrm{Lo}}_{\mathrm{Foc},t}-\bar{R}^{\mathrm{Hi}}_{\mathrm{Foc},t}$;
the strategy “(Lo - Hi) Rep” is that of
$\bar{R}^{\mathrm{Lo}}_{\mathrm{Rep},t}-\bar{R}^{\mathrm{Hi}}_{\mathrm{Rep},t}$;
and “SAR factor” is that of (15). Following the “(Lo - Hi) Foc” strategy will
result in a $\$3,177.63$ cumulative return on December 31, 2020, “(Lo - Hi)
Rep” will result in $\$5.70$ and SAR factor will result in $\$18,219.66$. The
amazing performance of SAR factor is firstly because the “(Lo - Hi) Foc”
strategy itself already has a high return. More importantly, however, the SAR
factor strategy simultaneously takes advantage of both the low returns of “(Lo
- Hi) Rep” as leverage and also its high correlations with the focal stocks.
As a matter of comparison, investing into the risk-free asset would result in
the cumulative return of $\$687.29$; the market factor returns $\$2,505.58$;
the SMB factor $\$264.59$; the HML factor $\$220.58$; the CMA factor
$\$318.83$; the RMW factor $\$467.17$; and the MOM factor $\$1,547.36$. See
Figure 4 for a plot of the monthly returns of the SAR factor. See Figure 5 for
a plot of the rolling cumulative returns of the SAR factor.

| Cum. return | Ann. Sharpe ratio | Mean | Std | Skewness | Kurtosis | (5, 95)-th pct |  $\textrm{ARMA}(1,\cdot)$ coef |  $\textrm{ARMA}(\cdot,1)$ coef
---|---|---|---|---|---|---|---|---|---
(Lo - Hi) Foc | 3,177.63 | 0.329 | 0.71 | 3.699 | 0.315 | 3.523 | (-5.325, 6.239) | -0.95 (-11.87) | 0.93 (10.53)
(Lo - Hi) Rep | 5.70 | -0.489 | -0.387 | 5.308 | 0.053 | 2.258 | (-8.898, 7.904) | -0.09 (-0.29) | 0.20 (0.68)
SAR factor | 18,219.66 | 0.495 | 1.101 | 5.169 | 0.07 | 3.456 | (-6.293, 9.734) | 0.36 (3.05) | -0.12 (-0.99)
Mkt-RF | 2,505.58 | 0.263 | 0.7 | 4.477 | -0.654 | 2.202 | (-7.212, 7.191) | -0.68 (-3.24) | 0.75 (3.92)
SMB | 264.59 | -0.160 | 0.223 | 2.914 | 0.207 | 3.83 | (-4.111, 4.740) | 0.34 (0.62) | -0.31 (-0.56)
HML | 220.58 | -0.199 | 0.19 | 2.948 | 0.079 | 2.463 | (-4.082, 5.213) | 0.58 (3.29) | -0.40 (-2.05)
RMW | 467.17 | -0.070 | 0.312 | 2.279 | -0.4 | 12.862 | (-2.583, 3.482) | 0.07 (0.32) | 0.07 (0.30)
CMA | 318.83 | -0.222 | 0.234 | 1.936 | 0.418 | 2.069 | (-2.521, 3.284) | -0.58 (-2.87) | 0.68 (3.80)
MOM | 1,547.36 | 0.199 | 0.608 | 4.366 | -1.335 | 10.738 | (-6.555, 6.701) | -0.41 (-1.51) | 0.49 (1.87)
RF | 687.29 |  | 0.358 | 0.291 | 0.737 | 0.351 | (0.000, 0.870) | 0.98 (128.39) | -0.13 (-2.54)

(a) Cumulative returns, Sharpe ratios and moments

(a) Plot of log cumulative returns

| SAR | MktRF | SMB | HML | RMW | CMA | MOM | (Lo - Hi) Foc | (Lo - Hi) Rep
---|---|---|---|---|---|---|---|---|---
SAR | 1.000 |  |  |  |  |  |  |  |
MktRF | 0.555 | 1.000 |  |  |  |  |  |  |
SMB | 0.593 | 0.257 | 1.000 |  |  |  |  |  |
HML | -0.099 | -0.209 | -0.053 | 1.000 |  |  |  |  |
RMW | -0.385 | -0.259 | -0.399 | 0.198 | 1.000 |  |  |  |
CMA | -0.107 | -0.359 | -0.051 | 0.674 | 0.113 | 1.000 |  |  |
MOM | -0.090 | -0.133 | 0.002 | -0.225 | 0.098 | -0.018 | 1.000 |  |
(Lo - Hi) Foc | 0.315 | -0.361 | 0.025 | 0.142 | 0.107 | 0.285 | 0.181 | 1.000 |
(Lo - Hi) Rep | -0.752 | -0.790 | -0.559 | 0.193 | 0.448 | 0.301 | 0.213 | 0.389 | 1.000

(b) Correlations

Table 7: Cumulative returns and various descriptive statistics of the SAR
factor and other assets. The column “Cum. return” of Table (a) shows the
cumulative returns from an initial $\$100$ investment on December 31, 1975 to
December 31, 2020 (the final amount is not inflation adjusted). The column
“Ann. Sharpe ratio” shows the average monthly excess returns of various
strategies divided by its standard deviation and then multiplied by
$\sqrt{12}$. The third to sixth columns of (a) show the first four moments of
the monthly excess returns time series; the mean and standard deviation
columns are expressed in percentage points (e.g. 1.0 means 1%). The seventh
column shows the 5-th and 95-th percentile of the monthly returns in
percentage points. The column “ARMA(1, $\cdot$) coef” shows the estimated
coefficient $a$ while the column “ARMA($\cdot$, 1) coef” shows the estimated
coefficient $b$ of the ARMA(1,1) model
$r_{t}=c+ar_{t-1}+b\epsilon_{t-1}+\epsilon_{t}$, and the parentheses show the
associated $t$-statistic. Figure (b) shows the log cumulative returns from
said initial investment across various investment strategies. The red shaded
regions are NBER recession dates. Table (c) shows the time series correlations
of our SAR factor against various other factors. The sampling period is
monthly from January 31, 1976 to December 31, 2020.  Figure 4: Time series
plot of monthly returns of the SAR factor. The top panel plots the monthly
returns of the SAR factor of (15) in blue, while the orange line is that of
the market factor for comparison. Returns are expressed in monthly percentage
points (e.g. 1.0 means 1%). The bottom left panel plots the strategy “(Lo -
Hi) Foc”, while the bottom right panel plots the strategy “(Lo - Hi) Rep”. The
red shaded regions are NBER recession dates. The sampling period is monthly
from January 31, 1976 to December 31, 2020.  Figure 5: Rolling window
cumulative returns of the SAR factor. We plot the cumulative returns of
rolling $n=$ 12, 36 and 60 months of an initial $\$1$ investment of the SAR
factor, the “(Lo - Hi) Foc” strategy, and the market factor for comparison
(e.g. 1.5 means a total of return of 150% over the past $n$ many months). The
red shaded regions are NBER recession dates. The sampling period is monthly
from January 31, 1976 to December 31, 2020.

###  3.6 Additional robustness checks

We run several robustness checks on our main result and they are summarized in
Table 8.

|  | Lo | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Hi | Lo - Hi
---|---|---|---|---|---|---|---|---|---|---|---|---
(i) EN $\mathsf{R^{2}}$ with min one risky asset | Focal | 1.440 | 1.001 | 1.026 | 0.888 | 0.987 | 0.946 | 0.864 | 0.865 | 0.793 | 0.782 | 0.658
|  | (5.051) | (3.444) | (3.407) | (3.128) | (3.514) | (3.425) | (3.104) | (3.143) | (2.817) | (2.958) | (3.901)
| Foc - Rep | 1.118 | 0.617 | 0.560 | 0.410 | 0.253 | 0.155 | 0.043 | 0.134 | 0.252 | 0.303 | 0.815
|  | (4.077) | (2.382) | (2.468) | (1.933) | (1.109) | (0.674) | (0.201) | (0.817) | (2.058) | (2.994) | (3.147)
(ii) ———— controlling for MktCap | Focal | 1.332 | 1.019 | 0.893 | 0.837 | 0.930 | 0.886 | 0.942 | 0.950 | 0.898 | 0.894 | 0.438
|  | (5.330) | (4.027) | (3.486) | (3.227) | (3.423) | (3.123) | (3.228) | (3.148) | (2.959) | (2.858) | (2.826)
| Foc - Rep | 0.835 | 0.580 | 0.477 | 0.443 | 0.312 | 0.080 | 0.205 | 0.177 | 0.297 | 0.419 | 0.416
|  | (3.589) | (2.960) | (2.488) | (2.194) | (1.420) | (0.366) | (1.064) | (1.080) | (2.657) | (3.744) | (2.163)
(iii) ———— controlling for B/M | Focal | 1.380 | 0.988 | 0.998 | 0.952 | 0.981 | 0.912 | 0.916 | 0.931 | 0.912 | 0.888 | 0.492
|  | (4.868) | (3.388) | (3.471) | (3.426) | (3.592) | (3.288) | (3.329) | (3.411) | (3.404) | (3.418) | (3.350)
| Foc - Rep | 1.068 | 0.596 | 0.543 | 0.442 | 0.259 | 0.122 | 0.101 | 0.201 | 0.289 | 0.396 | 0.672
|  | (4.110) | (2.484) | (2.467) | (2.155) | (1.188) | (0.518) | (0.494) | (1.139) | (2.218) | (3.831) | (2.729)
(iv) ———— controlling for IdioVol | Focal | 1.354 | 1.003 | 1.013 | 0.929 | 0.950 | 0.933 | 0.958 | 0.869 | 0.859 | 0.759 | 0.595
|  | (5.278) | (3.887) | (3.895) | (3.444) | (3.513) | (3.268) | (3.291) | (2.966) | (2.825) | (2.418) | (3.669)
| Foc - Rep | 0.916 | 0.543 | 0.606 | 0.529 | 0.215 | 0.159 | 0.168 | 0.173 | 0.252 | 0.315 | 0.601
|  | (3.593) | (2.446) | (2.745) | (2.603) | (0.965) | (0.736) | (0.832) | (1.071) | (2.133) | (2.797) | (2.672)
(v) ———— controlling for Mom | Focal | 1.349 | 1.130 | 1.003 | 0.991 | 0.964 | 0.919 | 0.924 | 0.830 | 0.790 | 0.703 | 0.646
|  | (4.794) | (3.955) | (3.464) | (3.540) | (3.463) | (3.333) | (3.384) | (3.042) | (2.809) | (2.628) | (4.617)
| Foc - Rep | 1.008 | 0.736 | 0.593 | 0.488 | 0.201 | 0.132 | 0.139 | 0.087 | 0.228 | 0.228 | 0.780
|  | (3.771) | (3.068) | (2.782) | (2.238) | (0.954) | (0.570) | (0.638) | (0.488) | (1.891) | (2.278) | (3.216)
(vi) ———— controlling for STR | Focal | 1.435 | 1.008 | 0.925 | 0.934 | 0.954 | 0.849 | 0.949 | 0.879 | 0.830 | 0.809 | 0.626
|  | (4.999) | (3.611) | (3.116) | (3.319) | (3.429) | (3.072) | (3.471) | (3.183) | (2.947) | (3.004) | (3.820)
| Foc - Rep | 1.116 | 0.606 | 0.486 | 0.445 | 0.247 | 0.005 | 0.133 | 0.148 | 0.302 | 0.322 | 0.795
|  | (4.087) | (2.490) | (2.168) | (2.146) | (1.121) | (0.023) | (0.628) | (0.920) | (2.555) | (3.209) | (3.156)
(vii) ———— controlling for VOLD | Focal | 1.259 | 1.010 | 0.921 | 0.873 | 0.959 | 0.871 | 0.971 | 0.929 | 0.958 | 0.928 | 0.331
|  | (4.905) | (3.807) | (3.362) | (3.296) | (3.520) | (3.097) | (3.367) | (3.228) | (3.491) | (3.443) | (2.329)
| Foc - Rep | 0.800 | 0.544 | 0.534 | 0.507 | 0.223 | 0.132 | 0.197 | 0.200 | 0.383 | 0.410 | 0.390
|  | (3.623) | (2.671) | (2.745) | (2.675) | (0.968) | (0.643) | (0.939) | (1.182) | (3.028) | (3.899) | (2.197)
(viii) FF3 OLS $\mathsf{R^{2}}$ | Focal | 1.260 | 1.140 | 1.006 | 0.971 | 0.974 | 0.907 | 0.907 | 0.880 | 0.880 | 0.806 | 0.455
|  | (3.961) | (3.667) | (3.245) | (3.131) | (3.363) | (3.197) | (3.421) | (3.497) | (3.482) | (3.165) | (1.944)
| Foc - Rep | 1.121 | 0.821 | 0.630 | 0.487 | 0.386 | 0.416 | 0.364 | 0.319 | 0.245 | 0.193 | 0.928
|  | (4.353) | (3.787) | (2.998) | (2.546) | (2.087) | (2.490) | (2.092) | (1.749) | (1.399) | (1.216) | (2.882)
(ix) ———— controlling for MktCap | Focal | 0.872 | 0.961 | 0.977 | 1.002 | 1.026 | 0.994 | 0.997 | 1.015 | 0.953 | 0.923 | -0.052
|  | (3.678) | (3.801) | (3.754) | (3.635) | (3.755) | (3.654) | (3.530) | (3.413) | (3.074) | (2.879) | (-0.274)
| Foc - Rep | 0.606 | 0.511 | 0.542 | 0.457 | 0.550 | 0.519 | 0.481 | 0.494 | 0.414 | 0.371 | 0.235
|  | (3.645) | (3.046) | (3.115) | (2.713) | (3.291) | (3.606) | (3.004) | (3.336) | (2.778) | (2.620) | (1.672)
(x) FF5 OLS $\mathsf{R^{2}}$ | Focal | 1.245 | 1.147 | 1.009 | 0.989 | 0.991 | 0.916 | 0.886 | 0.874 | 0.880 | 0.792 | 0.453
|  | (4.029) | (3.612) | (3.237) | (3.213) | (3.396) | (3.246) | (3.350) | (3.482) | (3.509) | (3.097) | (1.998)
| Foc - Rep | 1.093 | 0.868 | 0.626 | 0.460 | 0.433 | 0.404 | 0.335 | 0.313 | 0.255 | 0.191 | 0.902
|  | (4.484) | (3.838) | (3.058) | (2.280) | (2.502) | (2.340) | (1.927) | (1.729) | (1.461) | (1.247) | (2.960)
(xi) ———— controlling for MktCap | Focal | 0.881 | 0.925 | 1.048 | 1.010 | 1.023 | 0.981 | 0.953 | 1.019 | 0.962 | 0.916 | -0.035
|  | (3.707) | (3.732) | (4.086) | (3.702) | (3.699) | (3.529) | (3.363) | (3.408) | (3.171) | (2.819) | (-0.182)
| Foc - Rep | 0.584 | 0.554 | 0.548 | 0.502 | 0.514 | 0.510 | 0.432 | 0.448 | 0.451 | 0.400 | 0.185
|  | (3.516) | (3.294) | (3.190) | (2.884) | (3.137) | (3.311) | (2.678) | (3.064) | (3.160) | (2.877) | (1.360)

Table 8: Robustness checks. The reported numbers are monthly returns in
percentage points (i.e. 1.0 means 1.0%) and the parentheses show Newey and
West (1987) robust $t$-statistics with six lags. Result (i) repeats the
procedure that shows the main result Table 3 except we restrict each stock to
have a replicate that must contain at least one single risky asset, so the
subsetted stocks’ replicates cannot simply just be the risk-free asset.
Results (ii) to (iv) repeat the bivariate dependent sort procedure of Table 5
but enforces the aforementioned minimum one risky asset requirement on the
stocks. Results (viii) and (x) are analogous to the main result of Table 3
except we sort stocks by deciles by their FF3 and FF5 $\mathsf{R^{2}}$’s,
respectively. FF3 and FF5 replicates for each stock are constructed out of
projected normalized OLS coefficients according to Section 3.6.2. Results (ix)
and (xi) show the corresponding FF3 and FF5 $\mathsf{R^{2}}$ decile sort
results that first sort stocks into their market capitalization into
quintiles; compare analogously to Table 5 for bivariate elastic-net
$\mathsf{R^{2}}$ sort results. The sample period is monthly from January 31,
1976 to December 31, 2020.

####  3.6.1 SARP is not driven by the equity risk premium

Is SARP just the equity risk premium in disguise? From Table 1(iv) and the
replicate construction procedure of Section 2.2, it is evident that stocks
with high SAR have replicates that are essentially just the risk-free asset.
The SARP for high SAR stocks is simply just the equity premium. However, the
SARP for low SAR stocks are distinctively different from their equity premia;
there are a plethora of risky peers in the replicates of low SAR stocks.
Nonetheless, as constructed, there is potential concern the main empirical
message of this paper — stocks with high (low) SAR have high (low) SARP — is
driven by the equity premium of low SAR stocks. To rule out this concern, we
follow the projection and replicate construction procedures of Section 2, but
then drop all stocks whose replicates only consist of the risk-free assets
(that is, the replicates now must contain at least one risky peer). Table 8(i)
shows the result; compare this against the main result Table 3. We see high
SAR stocks still have a monthly return of 0.658% ($t$-stat 3.901) higher than
that of low SAR stocks. More importantly, we see the SARP of high SAR stocks
(which is now distinctly different from the equity risk premium) is still
substantially higher at 0.815% ($t$-stat 3.147) per month than the SARP of low
SAR stocks. Hence the cross-sectional difference in SARP is not driven by the
equity risk premium of high SAR stocks. In results (ii) to (vii), we repeat
the bivariate sort procedure of Table 5 and sort various characteristics into
quintiles. Our main result remains robust after subsetting for stocks to have
at least one risky peer in its replicate, and after controlling for these
characteristics.

####  3.6.2 Weak SARP when using FF3 and FF5 for replicate construction

Is there any value at all in running a computationally expensive elastic-net
projection procedure of Section 2.1.2? Can we get analogous results using
simple OLS projections with much fewer regressors? Instead of projecting each
stock $i$ onto the past twelve months’ daily returns of every other stock
using elastic-net, let’s simply project each stock $i$’s past twelve months’
daily returns onto the FF3 and FF5 factors using OLS. 888 Since we did not
project onto an intercept using the elastic-net estimator, we will also not
project an intercept when using the FF3 and FF5 OLS regressions (14a) and
(14b). Recall Remark 2.2.  For each stock $i$ and at month $t-1$, we collect
the $3\times 1$ FF3 OLS projection vector
$\boldsymbol{\hat{\beta}}_{i,t-1}^{\mathrm{FF3}}$ and the $5\times 1$ FF5 OLS
projection vector $\boldsymbol{\hat{\beta}}_{i,t-1}^{\mathrm{FF5}}$. Symmetric
to the normalizing procedure of Section 2.2, we will normalize the projection
vectors by their $L^{1}$ norm as
$\boldsymbol{\tilde{\beta}}_{i,t-1}^{\mathrm{FF}z}:=\boldsymbol{\hat{\beta}}_{i,t-1}^{\mathrm{FF}z}/||\boldsymbol{\hat{\beta}}_{i,t-1}^{\textrm{FF}z}||_{1}$
for $z=3,5$. Analogous to (9), the returns of the FF$z$ replicate of stock $i$
is

$R_{i,t}^{\textrm{Rep,
FF}z}:=(1-\mathbf{1}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}^{\textrm{FF}z})r_{f,t}+\boldsymbol{R}_{\mathrm{FF}z,t}^{\top}\boldsymbol{\tilde{\beta}}_{i,t-1}^{\textrm{FF}z},$
(16)

where
$\boldsymbol{R}_{\mathrm{FF}3,t}=[R_{\mathrm{MktRF},t},R_{\mathrm{SMB},t},R_{\mathrm{HML},t}]^{\top}$
and
$\boldsymbol{R}_{\mathrm{FF}5,t}=[R_{\mathrm{MktRF},t},R_{\mathrm{SMB},t},R_{\mathrm{HML},t},R_{\mathrm{CMA},t},R_{\mathrm{RMW},t}]^{\top}$
are the month $t$ returns of the FF3 and FF5 factors. Finally, we now sort all
stocks on their FF$z$ OLS $\mathsf{R^{2}}$’s into deciles.

Table 8(viii) shows the FF3 results while (x) shows the FF5 results. At first
glance, it seems we can still identify SARP in the cross-section of FF$z$
$\mathsf{R^{2}}$ decile sorted stocks. The FF3 procedure shows that high FF3
SAR stocks have a SARP of 1.121% ($t$-stat 4.353) per month while low FF3 SAR
stocks have a 0.193% ($t$-stat 1.216), with a cross-sectional difference of
0.928% ($t$-stat 2.882). We can see a similar result in the FF5 projections in
(iv). However upon closer inspection, the FF$z$ procedures are not robust to
size effects, while it is robust for the elastic-net counterpart of Section
2.1.2. In Table 8(ix), we repeat the bivariate sort procedure to that of Table
5 but with FF3 $\mathsf{R^{2}}$ decile bins. After controlling for stocks’
market capitalization quintiles, there is no cross-sectional SARP: stocks with
high FF3 SAR have a SARP of 0.606% ($t$-stat 3.645) while stocks with low FF3
SAR have a SARP of 0.371% ($t$-stat 2.620), but the cross-sectional difference
of 0.235% ($t$-stat 1.672) is statistically insignificant. Likewise in result
(xi), we see no cross-sectional SARP in stocks that are constructed out of the
aforementioned FF5 projection and sorting procedure, after controlling for
size. Overall, these results show SARP can still be found using conventional
asset pricing factor models and the standard OLS procedure. However, the
“quality” of this SARP is rather low compared to the SARP that is identified
and constructed via our elastic-net procedure.

##  4 Conclusion

This paper theoretically and empirically show the relationship between the
_Statistical Arbitrage Risk_ (SAR) and _Statistical Arbitrage Risk Premium_
(SARP) of a stock. Theoretically, SARP is the expected return of the residual
factor risks of a given stock. Empirically we use the elastic-net, a machine
learning method, to project a given stock’s returns onto the span of every
other stock in the market. The projection $\mathsf{R^{2}}$ is SAR, and the
normalized projection coefficient entries serve as investment weights in
constructing the replicate portfolio of a given stock. The core message of
this paper is: _SARP is increasing in SAR_.

We see several interesting directions to further study SAR and SARP by machine
learning (ML). In this paper, the elastic-net serves two simultaneous roles
and steps: (1) _fitting and variable selection_ and (2) _portfolio
construction_. In Step (1), the elastic-net $\mathsf{R^{2}}$ is used to
measure the SAR of a stock $i$, and the non-zero entries of the projection
vector are used to identify the risky peers of stock $i$. In Step (2), the
normalized values of the non-entries are used as investment weights to
construct the replicate of stock $i$. A further study of SAR and SARP would be
to disentangle these two roles by using two different ML methods. Step (1) can
benefit from using a more sophisticated method $\mathrm{ML}_{1}$ that take in
“wide regressors” as inputs but has better sparse variable selection and
goodness-of-fit $\mathsf{R^{2}}$ (and hence SAR) properties than the elastic-
net. For step (2), we see benefits in designing a $\mathrm{ML}_{2}$ replicate
portfolio construction method that also takes into account the relative
importance of these peers from step (1). Once these two improved steps are
complete, the estimation of SARP and other analyses can follow conventional
empirical asset pricing procedures as outlined in this paper. Tangential to
using ML methods for forecasting and factors selection in finance, we feel a
novel avenue for ML applications in finance is this sense of portfolio
construction. In closing, we strongly believe this paper has merely scratched
the surface in the study of SAR and SARP by machine learning.

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##  Appendix A Appendix

###  A.1 Discussions on the projection procedure

Here we outline the details of our projection procedure.

####  A.1.1 Elastic-net estimator

Let’s introduce the optimization problem of the elastic-net estimator
developed by Zou and Hastie (2005). Let $y$ be a $T\times 1$ vector,
$\mathbf{X}$ be a $T\times N$ matrix and let $\boldsymbol{\beta}$ be a
$N\times 1$ vector. Consider the optimization problem,

$\boldsymbol{\hat{\beta}}=\operatorname*{arg\,min}_{\boldsymbol{\beta}\in\mathbb{R}^{N}}\;\left\\{\frac{1}{2T}\left|\left|y-\mathbf{X}\boldsymbol{\beta}\right|\right|_{2}^{2}+\lambda_{1}\left|\left|\boldsymbol{\beta}\right|\right|_{1}+\lambda_{2}\left|\left|\boldsymbol{\beta}\right|\right|_{2}^{2}\right\\},\quad\lambda_{1},\lambda_{2}\geq
0,$ (17)

where
$\left|\left|\mathbf{x}\right|\right|_{1}:=\sum_{j=1}^{N}\left|x_{j}\right|$
is the $L^{1}$-norm on $\mathbb{R}^{N}$, and
$\left|\left|\mathbf{x}\right|\right|_{2}:=\sqrt{\sum_{j=1}^{N}x_{j}^{2}}$ is
the $L^{2}$-norm on $\mathbb{R}^{N}$. In our application, $y$ will be a time
series vector of $T$ days of returns of a particular stock, and $\mathbf{X}$
will be the concatenation of the time series of $N$ number of other stocks.
Note this means there are $N+1$ total number of stocks.

The solution $\boldsymbol{\hat{\beta}}$ is called the elastic-net estimator.
This estimator encompasses the special cases of the ordinary least squares
(OLS) estimator (when $\lambda_{1}=0,\lambda_{2}=0$), least absolute shrinkage
and selection operator (LASSO) estimator of Tibshirani (1996) (when
$\lambda_{1}>0,\lambda_{2}=0$), and the ridge estimator (when
$\lambda_{1}=0,\lambda_{2}>0$). The hyperparameters $\lambda_{1},\lambda_{2}$
control the strength of the $L^{1}$\- and $L^{2}$-norm penalties,
respectively. In this paper when we refer to the elastic-net estimator, we
always refer to the case when $\lambda_{1},\lambda_{2}$ are both strictly
positive. In our actual implementation, we use a 3-fold cross-validation
procedure to empirically select the hyperparameters
$\lambda_{1},\lambda_{2}>0$.

######  Remark A.1.

To reduce the already lengthy computational time in estimating (17), we cross-
validate for only one hyperparameter rather than two in our empirical studies
by making the following simplifying assumption on $\lambda_{1},\lambda_{2}$.
We set $\lambda_{1}=\lambda\ell$ and $\lambda_{2}=\frac{1}{2}\lambda(1-\ell)$,
and set $\ell=1/2$, and only cross-validate for the single $\lambda>0$
parameter.

####  A.1.2 Intercept estimation, stale prices and sparsity

We deliberately do not estimate an intercept in (17). 999 The elastic-net
estimator that contains an intercept is given by,
$\hat{c},\boldsymbol{\hat{\beta}}\in\operatorname*{arg\,min}_{c\in\mathbb{R},\boldsymbol{\beta}\in\mathbb{R}^{p}}\left\\{\left|\left|y-c-\mathbf{X}\boldsymbol{\beta}\right|\right|_{2}^{2}+\lambda_{1}\left|\left|\boldsymbol{\beta}\right|\right|_{1}+\lambda_{2}\left|\left|\boldsymbol{\beta}\right|\right|_{2}^{2}\right\\},\quad\lambda_{1},\lambda_{2}\geq
0,$ where by convention, $L^{1}$\- and $L^{2}$-penalties are only applied on
the $\boldsymbol{\beta}$ coefficients and not on the intercept $c$.  This is
to avoid attributing a price with very stale prices with high
$\mathsf{R^{2}}$.

Imagine a given stock $i$ has a vector of 12 months’ worth daily returns
$y_{i,t-1}$, where all the daily returns are almost all $0$’s except on a
handful of days. As an extreme, suppose all other stocks has a return matrix
$\mathbf{X}_{i,t-1}$ of (6) with rank $D_{i,t-1}$. 101010 In the actual
empirical implementations, we do not impose nor check for any such condition.
Then using the elastic-net estimator without intercept (17), the squared error
term would be high, and thus leading to an overall low
$\mathsf{R^{2}}_{i,t-1}$. Observe that in this case
$\boldsymbol{\hat{\beta}}_{i,t-1}=\boldsymbol{0}$ is not necessarily an
optimal solution because $\mathbf{X}_{i,t-1}$ is of rank $D_{i,t-1}$ while
$\boldsymbol{\hat{\beta}}_{i,t-1}$ is $N_{t-1}\times 1$, and we have that
$D_{i,t-1}\ll N_{t-1}$. This means there could exist some sparse
$\boldsymbol{\hat{\beta}}_{i,t-1}\neq\boldsymbol{0}$ that achieves a smaller
value in (17) than that of the zero vector. This is so when
$||y_{i,t-1}-\mathbf{X}_{i,t-1}\boldsymbol{\hat{\beta}}_{i,t-1}||_{2}^{2}+\lambda_{1}||\boldsymbol{\hat{\beta}_{i,t-1}}||_{1}+\lambda_{2}||\boldsymbol{\hat{\beta}}_{i,t-1}||_{2}^{2}\ll||y_{i,t-1}||_{2}^{2}$,
especially when the penalty weights $\lambda_{1},\lambda_{2}$ are small. In
contrast, if one were to use the elastic-net estimator with an intercept, then
setting $\boldsymbol{\hat{\beta}}_{i,t-1}=\boldsymbol{0}$ with intercept
$\hat{c}\approx 0$ is an optimal solution. As a result, this would lead to the
numerator term in the calculation of $\mathsf{R^{2}}$ to approximately equal
to zero, and thus resulting in a high $\mathsf{R^{2}}$. We want to avoid the
latter case in our results.

In this paper, we want to exclusively reserve “high $\mathsf{R^{2}}$” to mean
stock $i$ can be well explained by other risky assets, and “low
$\mathsf{R^{2}}$” to mean stock $i$ cannot be explained by other risky assets.

####  A.1.3 Parsimony: Why elastic-net and not other machine learning methods?

Out of a myriad of machine learning methods, why did we choose the elastic-net
estimator to test our empirical implication? In Theorem 1.1, we motivated the
need to _linearly_ regress a stock return $R_{i}$ onto the returns
$\boldsymbol{R}_{-i}$ of all other stocks. The elastic-net estimator can
actually be seen as a linear regression problem with constraints. By Lagrange-
duality, (17) is identical to the following constrained least squares problem,

$\displaystyle\min_{\boldsymbol{\beta}\in\mathbb{R}^{N}}$
$\displaystyle\frac{1}{2T}\left|\left|y-\mathbf{X}\boldsymbol{\beta}\right|\right|^{2}_{2}$
(18) subject to
$\displaystyle\left|\left|\boldsymbol{\beta}\right|\right|_{1}\leq\eta_{1},$
$\displaystyle\left|\left|\boldsymbol{\beta}\right|\right|_{2}^{2}\leq\eta_{2},$

for some $\eta_{1},\eta_{2}\geq 0$ that is dependent on the values of
$\lambda_{1},\lambda_{2}\geq 0$. Most references in the statistical literature
(e.g. Zou and Hastie (2005)) prefers the unconstrained Lagrangian-form (17)
over the constrained optimization form (18) for theoretical and computational
reasons. Here, we explicitly draw out the “linear nature” of the elastic-net
estimator via (18) because Theorem 1.1 indeed predicts a linear relationship.

We can now justify why we choose to implement the elastic-net estimator in
this paper out of the myriad of machine learning methods. Without a doubt,
there are more advanced machine learning and econometric methods that can
increase the in-sample fit and thereby boost the in-sample $\mathsf{R^{2}}$. A
recent extensive study by Gu, Kelly, and Xiu (2020) shows that many possible
machine learning methods can achieve high in-sample and out-of-sample
$\mathsf{R^{2}}$’s. However, the issue with these “black box” methods is that
regardless of their in-sample or even out-of-sample performance, their
estimated model parameters often do not have a transparent link to the
regressors themselves. In contrast, although the elastic-net estimator is
inherently non-linear, its estimated coefficients can be applied in linearly
back to the regressors. This parsimonious nature of the elastic-net allows us
to explicitly and linearly construct the replicates as in (9).

We emphasize that the elastic-net machine learning method is simply a tool to
test the prediction of Theorem 1.1 via the construction of the replicates (9).
We have no intentions to conduct statistical inference on the estimated
elastic-net coefficients. Our statistical inference claims are still based
upon well understood portfolio sort methods in the empirical asset pricing
literature as outlined in Section 2.3.

The consideration of the replicates is what drives us to prefer the elastic-
net over its close cousin, the LASSO. The LASSO enjoys a “sparsity property”,
whereby the number of estimated coefficients tend to be small, even though the
set of regressors could be large. Sparsity means there are only a handful of
stocks that have to be considered in order to construct the replicate of stock
$i$. This is important because if the number of stocks needed to construct the
replicate of stock $i$ is large, then whatever statistical results we claim to
find may be economically infeasible due to trading costs and other market
frictions. Unfortunately, if a group of regressors are highly correlated with
each other then LASSO has a tendency to only select one regressor effectively
at random. This makes for a poor portfolio construction of the replicates for
numerous reasons, and loss of potential diversification is an obvious one. The
elastic-net remedies this problem by inheriting the grouping property of the
ridge estimator. In all, this means the elastic-net is a good candidate for
our consideration of the replicates because: (i) it can fit the data (i.e.,
from the least squares property of the OLS); (ii) it can linearly apply its
estimated coefficients over the regressors; (iii) it has a sparsity property
(i.e., from LASSO); and (iv) it has a grouping property (i.e., from ridge).

Would a more general machine learning method be useful for our purposes? The
answer is mixed. A general form of a machine learning estimator has the form
$y=g(\mathbf{X};\theta,\lambda)+\varepsilon$, where $y$ is the response
variable, $g$ could be a parameterized or non-parametric function,
$\mathbf{X}$ is the set of regressors, $\theta$ parameterizes $g$, $\lambda$
is a hyperparameter, and $\varepsilon$ is a nuisance parameter. Generally
speaking, the relationship between $\mathbf{X}$, $\theta$ and $\lambda$ could
be highly non-linear. The data is typically split into three sets
$[y_{\textrm{train}},y_{\textrm{validate}},y_{\textrm{test}}]$ and
$[\mathbf{X}_{\textrm{train}},\mathbf{X}_{\textrm{validate}},\mathbf{X}_{\textrm{test}}]$.
For a given hyperparameter $\lambda$, the machine learning method uses the
_training set_ $(y_{\textrm{train}},\mathbf{X}_{\textrm{train}})$ to fit the
data to find an optimal parameter $\hat{\theta}(\lambda)$. Using the
_validation set_ $(y_{\textrm{valid}},\mathbf{X}_{\textrm{valid}})$, the
method then finds an optimal $\hat{\lambda}$ that fits the validation data,
and a trained model is then given by the parameters
$\hat{\theta}(\hat{\lambda})$ and $\hat{\lambda}$. Finally, the forecast
accuracy of the model is tested against the _testing set_
$(y_{\textrm{test}},\mathbf{X}_{\textrm{test}})$ by comparing against the
predicted value
$\hat{y}=g(\mathbf{X}_{\textrm{train}};\hat{\theta}(\hat{\lambda}),\hat{\lambda})$
against its realization $y_{\textrm{train}}$.

In our context, if we were to apply such general machine learning method to
asset $i$, then we would still use the same response variable and regressors
as in (6). However, the fitted value would be of very little use to us when
constructing the replicates (9) from a finance perspective. There are two
issues. Firstly, we are _not_ using a forecast value $\hat{y}$ to construct
portfolios. We are using the fitted coefficients
$\hat{\theta}(\lambda)=\boldsymbol{\hat{\beta}}_{i,t-1}$ (which is then
subsequently normalized by its $L^{1}$ norm) as investment weights for the
replicate of stock $i$. In order to use fitted coefficients as investment
weights, then it is almost _necessary_ that these fitted coefficients have a
clear and linear relationship to the one-month ahead returns
$\mathbf{X}_{\textrm{train}}=\boldsymbol{R}_{-i,t}$. The OLS, LASSO and
elastic-net certainly have this property, as seen in (9). However, a general
machine learning method does _not_ have this linear relationship between the
fitted coefficients and its out-of-sample regressors. This non-linearity
between fitted coefficients and its out-of-sample regressors explicitly
prevent us from directly using these general machine learning methods in
constructing a portfolio.

Nonetheless, as discussed at the conclusion of the main text, there are many
research avenues to disentangle the fitting role and replicate portfolio
construction role of the elastic-net. By designing a method $\mathrm{ML}_{1}$
for fitting and another method $\mathrm{ML}_{2}$ for replicate portfolio
construction, there are many other properties of the SAR and SARP to explore.
Regardless, and at least to us, seems to be the most parsimonious and is a
good baseline.