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## Internet Appendix

###  Appendix A Statistical Arbitrage Models

####  A.1 Portfolio Collapse Methods that Generate $b$

I describe the Brandt et al. (2009) (BSV), Kozak et al. (2020) (KNS),
Bryzgalova et al. (2020) (BPZ), and then the DeMiguel et al. (2009) (DGU)
methods of generating $b$ from $\bar{F}_{t-1}$ and $\bar{\Omega}_{t-1}$.

#####  A.1.1 Brandt et al. (2009) \- BSV

The main objective of Brandt et al. (2009) is to optimize portfolios of a
large number of assets, and the authors assume that optimal weights are linear
in a set of predictors. My implementation of the BSV model similarly assumes
that $f(\cdot)$ is linear in predictors—i.e. $f(Z_{t})=\breve{Z}_{t}$.

While Brandt et al. (2009) discuss a variety of objective functions to
estimate $b$, I focus on their model that maximizes the empirical Sharpe
ratio. The maximal empirical Sharpe ratio estimate of $b$ of these authors can
be written as:

$b=\underset{\tilde{b}}{\text{argmin}}\;(\bar{F}_{t-1}-\bar{\Omega}_{t-1}\tilde{b})^{\prime}\bar{\Omega}_{t-1}^{-1}(\bar{F}_{t-1}-\bar{\Omega}_{t-1}\tilde{b})\;\;\text{
which implies }\;\;b=\bar{\Omega}_{t-1}^{-1}\bar{F}_{t-1}.$ (A.1)

#####  A.1.2 Kozak et al. (2020) \- KNS

The objective of Kozak et al. (2020) is to create an SDF. The KNS method first
performs a standard principle components analysis (PCA) dimension reduction on
the factor $F_{t-1}$ generated by the linear function
$f(Z_{t})=\breve{Z}_{t}$. The number of principle components to use is a
hyperparameter of the model. This produces a smaller set of factors, which are
then reduced to a single portfolio with the vector $b$.

The Kozak et al. (2020) estimator of $b$ is given by

$b=\underset{\tilde{b}}{\text{argmin}}(\bar{F}_{t-1}-\bar{\Omega}_{t-1}\tilde{b})^{\prime}\bar{\Omega}_{t-1}^{-1}(\bar{F}_{t-1}-\bar{\Omega}_{t-1}\tilde{b})+\lambda_{1}||\tilde{b}||_{1}+\lambda_{2}||\tilde{b}||_{2},$
(A.2)

where $||\cdot||_{1}$ and $||\cdot||_{2}$ are the $L^{1}$ and $L^{2}$ norms
respectively, and $\lambda_{1}$ and $\lambda_{2}$ are additional penalty
hyperparameters that shrink the $b$ values towards zero. This has similar
intuition as a standard elastic net.

#####  A.1.3 Bryzgalova et al. (2020) \- BPZL

Bryzgalova et al. (2020) have two methodological contributions in their paper
I discuss: first their random forest techniques and second their method of
obtaining $b$. I discuss the latter here, which is given by:

$b=\underset{\tilde{b}}{\text{argmin}}((\bar{F}_{t-1}+\lambda_{0}\iota_{M})-\bar{\Omega}_{t-1}\tilde{b})^{\prime}\bar{\Omega}_{t-1}^{-1}((\bar{F}_{t-1}+\lambda_{0}\iota_{M})-\bar{\Omega}_{t-1}\tilde{b})+\lambda_{1}||\tilde{b}||_{1}+\lambda_{2}||\tilde{b}||_{2},$
(A.3)

where $\iota_{M}$ is an $M$ dimensional column vector of ones. This extra term
is used to shrink the mean of portfolio returns, potentially eliminating
estimation error.

As further discussed below in this Appendix, $b$ in Equation (A.3) can either
be estimated with a classic Lasso regression with an $L^{1}$ penalty parameter
$\lambda_{1}$, or can be calculated with a LARS regression where the number of
nonzero terms in $\hat{b}\tau$ is explicitly chosen. Choosing the number of
nonzero weights effectively chooses the $\lambda_{1}$ parameter. I follow
Bryzgalova et al. (2020) by using LARS regression with the number of nonzero
values of $b$ as a hyperparameter.

#####  A.1.4 DeMiguel et al. (2009) \- DGU

DeMiguel et al. (2009) choose $b$ by setting:

$b=\underset{\begin{subarray}{c}\lambda_{2}/\lambda_{0}=M\\\
\lambda_{0},\lambda_{2}\to\infty\end{subarray}}{\lim}\underset{b}{\text{argmin}}((\bar{F}_{t-1}+\lambda_{0}\text{sign}(\bar{F}_{t-1}))-\bar{\Omega}_{t-1}b)^{\prime}\bar{\Omega}_{t-1}^{-1}((\bar{F}_{t-1}+\lambda_{0}\text{sign}(\bar{F}_{t-1}))-\bar{\Omega}_{t-1}b)+\lambda_{2}||b||_{2}$
(A.4)

which implies that

$b=\frac{1}{M}\text{sign}(\bar{F}_{t-1}),$ (A.5)

In other words, this gives equal weight to each portfolio. It is an extreme
form of shrinking, where the shrinking parameters go to infinity in the limit.
This is shown in their paper to yield a high Sharpe ratio, in many cases
better than other proposed statistical arbitrageurs. Thus, I use this as a
contender statistical arbitrageur.

####  A.2 Factor Weight Function $f(Z_{t})$

In this subsection, I summarize the various factor weight functions $f(Z_{t})$
other than the statistical arbitrageurs that use the linear predictor weight
functions—which were discussed above.

#####  A.2.1 Classic Asset Pricing Factor Models: FF3, FF6, and HXZ

The FF3, FF6, and HXZ are classic asset pricing factor models. Each of these
models use only a subset of the 62 predictors from the data, which I describe
here. The FF3 uses only the market ($k=1$ predictor), size (lme), and the
book-to-market ratio (beme). The FF6 uses these same predictors, in additional
to investment (investment), profitability (prof), and momentum
(cum_return_12_2). Finally, the HXZ model uses the market ($k=1$), investment
(investment), return on equity (roe), and size (lme). Thus, for these models,
$f(Z_{t})$ is the subset of $\breve{Z}_{t}$ of just these predictors.

These models use different weights on these portfolios than the linear
predictor portfolios. These bins create demand functions that are very
discontinuous, while continuous demand functions are needed for this analysis.
More importantly, using different weighting schemes decreases the
comparability of these models and the models that use all predictors. Thus,
the linear predictor-weighted portfolios should be thought of as
approximations to these models’ different portfolio weighting schemes.

#####  A.2.2 Kelly et al. (2019) \- KPS

Kelly et al. (2019) consider the model:

$r_{\tau+1}=\beta_{\tau}F_{\tau+1}+\epsilon_{\tau+1}\;\;\text{ where
}\;\;\beta_{\tau}=Z_{\tau}\Gamma_{\beta},\;\;F_{\tau+1}=\left(\Gamma_{\beta}^{\prime}Z_{\tau}^{\prime}Z_{\tau}\Gamma_{\beta}\right)^{-1}\Gamma_{\beta}^{\prime}Z_{\tau}^{\prime}r_{\tau+1},$

and $\Gamma_{\beta}$ is $K\times M$ dimensional. They choose $\Gamma_{\beta}$
to minimize:

$\sum_{\tau=1}^{t-1}(r_{\tau}-\beta_{\tau-1}F_{\tau})^{\prime}(r_{\tau}-\beta_{\tau-1}F_{\tau})$

Thus, in this model,

$f(Z_{t})=Z_{t}\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}.$
(A.6)

I estimate $b$ for the Kelly et al. (2019) model using the Kozak et al. (2020)
method described above. The number of factors is a hyperparameter.

#####  A.2.3 Gu et al. (2021) \- GKX

Gu et al. (2021) propose a latent factor model in which the portfolio weights
are

$f(Z_{t})=Z_{t}(Z_{t}^{\prime}Z_{t})^{-1}\Gamma_{A}+\iota_{N}\Gamma_{B}^{\prime},$
(A.7)

where $\Gamma_{A}$ is a $K\times M$ matrix and $\Gamma_{B}$ is a $M$
dimensional column vector of parameters. In the language of neural networks,
$\Gamma_{A}$ is a matrix of weights and $\Gamma_{B}$ is a vector of biases. Gu
et al. (2021) also allows the betas of these factors to be have a neural
network structure as a function of $Z_{t}$, $\beta(Z_{t})$, which is $N\times
M$ dimensional. Model parameters are essentially estimated to achieve a low
sum of squared errors:111There are also penalties in the objective function
and other details typical of neural network backpropagation. See Gu et al.
(2021) for more details

$\sum_{t=0}^{T-1}(r_{t+1}-\beta(Z_{t})F_{t+1})^{\prime}(r_{t+1}-\beta(Z_{t})F_{t+1}).$
(A.8)

I estimate $b$ for the Gu et al. (2021) model using the Bryzgalova et al.
(2020) method described above.

Similar to the model of Kelly et al. (2019), this model requires the number of
factors $M$, as well as $\lambda_{0}$ and $\lambda_{2}$ as hyperparameters.
The standard hyperparameters associated with neural networks, including the
number of layers and nodes, are hyperparameters in this model. Table A.3
displays various amounts of hidden layers and nodes that I use during cross
validation. For example, $(25,5)$ signifies two hidden layers with the first
hidden layer containing 25 nodes and the second containing 5 nodes. The value
$()$ signifies zero hidden layers. As described in Gu et al. (2021), I also
use an $L^{1}$ penalty for the weights in the neural network, which adds
another hyperparameter.

#####  A.2.4 Bryzgalova et al. (2020) \- BPZF

In their effort to create a set of test assets that spans the mean-variance
efficient frontier, Bryzgalova et al. (2020) describe how they create a forest
of decision trees to map a set of $K$ predictors to the portfolio weights,
$f(Z_{t})$, of $M$ test assets such that the mean-variance combination of the
portfolios has the highest Sharpe ratio possible. For a given $K$ and tree
depth, $d$, the authors advocate creating trees based on all possible
combinations of conditional sorts, where at each level of a given tree from 1
to $d$, stocks are mechanically sorted into two portfolios based on the median
value of some predictor across stocks. Portfolios at every intermediate and
final node of every tree are equally weighted. The authors then prune the
trees by choosing weights to maximize the Sharpe ratio using their approached
described in Equation (A.3). The total number of portfolios formed is given by
$K^{d}2^{d}$, which for large values of $K$ becomes untenable.

I employ an alternative method based on Bryzgalova et al. (2020). I stack

$X_{\tau}=\begin{bmatrix}Z_{0}\\\ Z_{1}\\\ \vdots\\\
Z_{\tau-1}\end{bmatrix},\;\;\;Y_{\tau}=\begin{bmatrix}r_{1}\\\ r_{2}\\\
\vdots\\\ r_{\tau}\end{bmatrix}$ (A.9)

and fit a random forest of regression trees with feature matrix $X$ to predict
the target vector $Y$. The number of trees in the forest, the maximum depth of
each tree, and the number of features to consider at each tree are all
hyperparameters of this model.

Each intermediate node and end node of each tree represents a portfolio. In
other words, each column of $f(Z_{t})$ represents an equally weighted
portfolio from a node in a tree of the fitted random forest.222For example,
assume that each tree is fitted to a depth of 4. Then there are $2^{4}=16$ end
nodes, and $1+2^{1}+2^{2}+2^{3}=15$ intermediate nodes. Therefore, there are
$15+16=31$ total portfolios in this tree. If there are 100 trees in the
forest, then there are $31*100=3,100$ total portfolios. Since the base node
portfolios are always the same, there are at least $100-1=99$ duplicate
portfolios. Thus, if there are no other duplicate portfolios, then there are
$3,001$ unique portfolios in this forest. Thus, a model with these
hyperparameters assumes that $M=3,001$. Thus,
$F_{t+1}=f(Z_{t})^{\prime}r_{t+1}$ is a potentially large dimensional vector
of portfolio returns.

In estimating $b$ as described using the BPZ method described above,
Bryzgalova et al. (2020) recommend rescaling these forest portfolios to take
into account estimation noise given the varying number of assets in each
portfolio. I do this as they recommend.

The hyperparameters required for this model are the number of trees in the
forest, the maximum depth of the trees, the maximum number of features
considered at each tree, as well as the three BPZ hyperparameters required to
calculate $b$.

#####  A.2.5 Neural Network - NN

This is a neural network model original to this paper. Let
$g:\mathbb{R}^{N}\times\mathbb{R}^{K}\mapsto\mathbb{R}^{N}\times\mathbb{R}^{J+1}$
denote a standard neural network with $J+1\geq 2$ output nodes. Let $\mu_{t}$
denote the first column of $g(Z_{t})$, and $\Gamma_{t}$ denote the remaining
$J$ columns of $g(Z_{t})$. Thus, $\Gamma_{t}$ is a $N\times J$ matrix, while
$\mu_{t}$ is $N\times 1$. Thus, I can write

$\begin{bmatrix}\underbrace{\mu_{t}}_{N\times
1}&\underbrace{\Gamma_{t}}_{N\times
J}\end{bmatrix}=\underbrace{g(Z_{t})}_{N\times(J+1)}$ (A.10)

The model assumes a KY style covariance matrix:

$\Sigma_{t}=\Gamma_{t}\Gamma_{t}^{\prime}+\zeta I_{N}$ (A.11)

where $\zeta$ is a scalar parameter and $I_{N}$ is an $N\times N$ identity
matrix. Finally, the model assumes

$r_{t+1}\sim N(\mu_{t},\Sigma_{t})$ (A.12)

and the neural network parameters associated with $g$, as well as the scalar
$\zeta$, are estimated with a typical neural network maximum a posterior (MAP)
estimation procedure with a multivariate normal log-likelihood function. More
details about this SDF, including equations to efficiently compute the log-
likelihood, are found below in this Appendix.

For this SDF, $f$ is defined such that

$f(Z_{t})=\Sigma_{t}^{-1}\mu_{t}$ (A.13)

where $\Sigma_{t}$ and $\mu_{t}$ are functions of $Z_{t}$ defined in equations
(A.10) and (A.11). Note that $f(Z_{t})$ is actually already an $N\times 1$
matrix. So the collapse method is trivially set to $b=1$.

The hyperparameters required for this algorithm are the number of layers and
nodes in each layer as well as a $L^{2}$ penalty regularization parameter
commonly used in neural networks.

#####  A.2.6 Random Forest - RF

I also consider a random forest model where the means of stocks are predicted
with a random forest. The covariance matrix is just assumed to be proportional
to the identity matrix, which potentially reduces estimation error in the
covariance matrix following the logic of DeMiguel et al. (2009).

For this statistical arbitrageur, consider a fitted regression random forest
that used features $X$ to fit $Y$ as defined in equation (A.9) above. Let
$f(Z_{t})$ be the random forest excess return predictions for each of the rows
of $Z_{t}$. Thus, $f(Z_{t})$ is a $N\times 1$ dimensional vector of excess
return predictions. I take these to be estimates of the expected excess
returns. Since the covariance matrix is assumed to be proportional to the
diagonal matrix, then $f(Z_{t})$ are the MVE weights insofar as $f(Z_{t})$ are
the expected excess returns. Like the above model, this model delivers a
single factor, meaning that trivially $b=1$.

The hyperparameters required for this model are the number of trees in the
forest, the maximum depth of the trees, and the maximum number of features
considered at each tree.

####  A.3 PCA Reduction - Kozak et al. (2020)

This subsection describes in detail how PCA can be used to reduce the
dimension of $f$ before $b$ is computed, as described in Section 4.6.

Let
$f^{1}:\mathbb{R}^{N}\times\mathbb{R}^{K}\mapsto\mathbb{R}^{N}\times\mathbb{R}^{M^{1}}$
denote any given $f$ from Section 4.6. Note that I substitute the notation of
$f$ and $M$ to $f^{1}$ and $M^{1}$ respectively. Define:

$F_{t+1}^{1}\equiv f^{1}(Z_{t})^{\prime}r_{t+1},$ (A.14)
$\bar{F}^{1}_{T}=\frac{1}{T}\sum_{t=1}^{T}F_{t}^{1},\text{ and }$ (A.15)
$\bar{\Omega}^{1}=\frac{1}{T}\sum_{t=1}^{T}(F_{T}-\bar{\Omega}^{1})(F_{T}-\bar{\Omega}^{1})^{\prime}.$
(A.16)

Let $M\leq M^{1}$. Define $Q$ to be the $M^{1}\times M$ matrix whose $m^{th}$
column contains the eigenvector corresponding to the $m^{th}$ largest
eigenvalue of $\bar{\Omega}^{1}$.

Then define

$f(Z_{t})=f^{1}(Z_{t})Q$ (A.17)

Note that $f$ now maps from $\mathbb{R}^{N}\times\mathbb{R}^{K}$ to
$\mathbb{R}^{N}\times\mathbb{R}^{M}$. The matrix $Q$ acts as a dimension
reduction method. This imposes the prior on the data that the eigenvectors
associated with the smallest $M^{1}-M$ eigenvalues are not important for
delivering risk prices in the SDF.

Now that $f$ is in hand, $b$ can be calculated according to Brandt et al.
(2009), Kozak et al. (2020), or Bryzgalova et al. (2020) as described above.

####  A.4 Efficient Computation of KNS - Kozak et al. (2020)

I follow Bryzgalova et al. (2020) in the efficient calculation of $b$, which
can be applied to the KNS version of $b$ trivially.

For simplicity, I drop the $T$ subscripts on $\bar{F}\equiv\bar{F}_{T}$ and
$\bar{\Omega}=\bar{\Omega}_{T}$.

I can write

$\bar{\Omega}=QDQ^{\prime},$ (A.18)

where $Q$ is an $M\times M$ matrix where the $j^{th}$ column is the
eigenvector associated with the $j^{th}$ largest eigenvalue of $\bar{\Omega}$.
$D$ is the corresponding diagonal matrix of eigenvalues. Define
$m=\min\\{T,M,J\\}$ where $J$ is the number of strictly positive eigenvalues
of $\bar{\Omega}$. Let $\tilde{Q}$ denote the $M\times m$ matrix of the first
$m$ columns of $Q$. Let $\tilde{D}$ denote the $m\times m$ diagonal matrix of
the first $m$ terms along the diagonal of $D$. Then define

$\tilde{\Omega}^{1/2}=\tilde{Q}\tilde{D}^{1/2}\tilde{Q}^{\prime},\;\;\;\tilde{\Omega}^{-1/2}=\tilde{Q}\tilde{D}^{-1/2}\tilde{Q}^{\prime},\;\text{
and }\;\tilde{\omega}=\tilde{\Omega}^{-1/2}\bar{\Omega}.$ (A.19)

Let

$X=\begin{bmatrix}\tilde{\Omega}^{1/2}\\\ \sqrt{\lambda_{2}}I_{M}\\\
\end{bmatrix},\;\text{ and }\;y=\begin{bmatrix}\tilde{\omega}\\\ 0_{M}\\\
\end{bmatrix}.$ (A.20)

Then the problem is equivalent to a lasso/LARS regression that minimizes:

$(y-Xb)^{\prime}(y-Xb)+\lambda_{1}||b||_{1}.$ (A.21)

####  A.5 Efficient Computation of BPZ - Bryzgalova et al. (2020)

Efficient computation of $b$ for the BPZ method is identical to the above
subsection except that $\tilde{\omega}$ should be replaced with

$\tilde{\omega}=\tilde{\Omega}^{-1/2}(\bar{\Omega}+\lambda_{0}\iota).$ (A.22)

####  A.6 Efficient Computation of the Neural Network (NN)

In order to calculate the maximum likelihood function, $\Sigma_{t}^{-1}$ and
the determinant of $\Sigma_{t}$, denoted as $|\Sigma_{t}|$ need to be
calculated efficiently, even with large $N$. This section shows formulas that
can be used to do this.

By the Woodbury Matrix Identity:

$\Sigma_{t}^{-1}=\frac{1}{\zeta}\left(I_{N}-\Gamma_{t}(\zeta
I_{J}+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime}\right).$ (A.23)

Note that if $J=1$, then $\zeta I_{J}+\Gamma_{t}^{\prime}\Gamma_{t}$ is a
scalar. If $J$ is much smaller than $N$, then $(\zeta
I_{J}+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}$ is much easier to numerically
calculate than $\Sigma_{t}^{-1}$.

By Sylvester’s matrix identity:

$|\Sigma_{t}|=\zeta^{N}\left|I_{J}+\frac{1}{\zeta}\Gamma_{t}^{\prime}\Gamma_{t}\right|.$
(A.24)

Note that if $J=1$, $I_{J}+\frac{1}{\zeta}\Gamma_{t}^{\prime}\Gamma_{t}$ is a
scalar. If $J$ is much smaller than $N$, then the determinant of
$I_{J}+\frac{1}{\zeta}\Gamma_{t}^{\prime}\Gamma_{t}$ is much easier to
numerically calculate than the determinant of $\Sigma_{t}$.

Using these formulas, it is trivial to calculate the multivariate normal log-
likelihood function. Denote this as $l(\theta)$, where $\theta$ is a vector of
parameters from the neural network $g$ and the scalar $\zeta$. These
parameters are chosen by setting $\theta^{*}$ to

$\theta^{*}=\text{argmax}_{\theta}\;l(\theta)-\lambda||\theta||_{2},$ (A.25)

where $\lambda$ is a hyperparameter. This is a standard maximum a posteriori
(MAP) estimator commonly used with neural networks and in other areas of
statistics and econometrics.

###  Appendix B Additional Details

####  B.1 Kernel Density Function

This kernel function $\mathcal{K}(\cdot)$ has an $N$-dimensional vector input,
with $N$-dimensional vector output. As written above, $\mathcal{K}(x)_{i}$ is
the $i^{th}$ output of $\mathcal{K}(x)$. It is defined as

$\mathcal{K}(x)_{i}=\frac{1}{N}\sum_{j=1}^{N}\Phi\left(\frac{x_{i}-x_{j}}{h}\right)$
(B.1)

where $\Phi$ is the standard normal cumulative distribution function (cdf) and
$h$ is scalar called the bandwidth. I use the standard Silverman’s rule of
thumb bandwidth (Silverman, 1988):

$h=0.9\min\left(\sigma,\frac{\text{IQR}}{1.34}\right)N^{-1/5}$ (B.2)

where $\sigma$ is the typical estimate of the standard deviation of the
elements in $x$, and IQR is the interquartile range of $x$. If the
interquartile range is zero333The dividend-price ratio variable, ldp, is so
full of zeros that sometimes the interquartile range can be zero. then that
term is ignored and $\sigma$ is just used.

Importantly, $\mathcal{K}(x)_{i}$ is guaranteed to be in the interval $[0,1]$.
Thus, $\mathcal{K}(x)_{i}-0.5$ is obviously in $[-0.5,0.5]$.

The derivative of this is needed to calculate elasticity values. I can write:

$\frac{\partial\mathcal{K}(x)_{i}}{\partial
x_{i}}=\underbrace{\frac{1}{Nh}\sum_{j=1}^{N}\phi\left(\frac{x_{i}-x_{j}}{h}\right)}_{\text{direct
effect}}\underbrace{-\underbrace{\frac{1}{Nh}\phi\left(0\right)}_{\approx
0}+\frac{\partial\mathcal{K}(x)_{i}}{\partial h}\underbrace{\frac{\partial
h}{\partial x_{i}}}_{\approx 0}}_{\begin{subarray}{c}\text{indirect effect of
a change in }x_{i}\\\ \text{ on the distribution}\end{subarray}}.$ (B.3)

With many assets (large $N$), changing a single observation changes the kernel
distribution—the indirect effect labeled above—very little. For example,
$\partial h/\partial x_{i}$ is how much the estimate of a standard deviation
or interquartile range changes when a single observation changes (adjusted by
a scalar). This effect is very small, in fact this can easily be in the range
of computer rounding error or absolutely zero with many assets. Thus, I can
safely and accurately use the approximation:

$\frac{\partial\mathcal{K}(x)_{i}}{\partial
x_{i}}=\frac{1}{Nh}\sum_{j=1}^{N}\phi\left(\frac{x_{i}-x_{j}}{h}\right)\equiv\mathcal{k}(x)_{i},$
(B.4)

where $\phi(\cdot)$ is the standard normal probability density function (pdf).
Note that $\mathcal{k}(x)$ is the standard kernel density function defined
above, with a similar $N$-dimensional input and $N$-dimensional output. Thus,
this $\mathcal{K}(\cdot)$ is just a standard kernel cdf and the derivative is
a standard kernel pdf.

There are two kinds of endogenous predictors (other than the level term): (1)
predictors with a denominator price term and (2) predictors with a numerator
price term. The denominator price predictors are of the form:

$x_{i,k,t}=\frac{x_{i,k,t}^{n}}{p_{i,t}}+x_{i,k,t}^{a},$ (B.5)

where $x_{i,k,t}^{n}$ and $x_{i,k,t}^{a}$ are exogenous components.

The numerator predictors have the form:

$x_{i,k,t}=\frac{p_{i,t}}{x_{i,k,t}^{d}}+x_{i,k,t}^{a},$ (B.6)

where $x_{i,k,t}^{d}$ and $x_{i,k,t}^{a}$ are exogenous components.

Recall that the normalized predictors have the form:

$\underaccent{\dot}{z}_{i,k,t}=\mathcal{K}(\text{ArcSinh}(x_{k,t}))_{i}-0.5$
(B.7)

Thus, I can calculate

$\frac{\partial\underaccent{\dot}{z}_{i,k,t}}{\partial
x_{i,k,t}}=\frac{\mathcal{k}(\text{ArcSinh}(x_{k,t}))_{i}}{\sqrt{x_{i,k,t}^{2}+1}}.$
(B.8)

Furthermore, predictors with a numerator price term have the following
derivative with respect to the log price term:

$\frac{\partial\underaccent{\dot}{z}_{i,k,t}}{\partial\log(p_{i,t})}=\frac{\partial\underaccent{\dot}{z}_{i,k,t}}{\partial
x_{i,k,t}}\frac{\partial
x_{i,k,t}}{\partial\log(p_{i,t})}=\frac{\mathcal{k}(\text{ArcSinh}(x_{k,t}))_{i}}{\sqrt{x_{i,k,t}^{2}+1}}\left(\frac{p_{i,t}}{x_{i,k,t}^{d}}\right).$
(B.9)

Similarly, predictors with a denominator price have the following derivative:

$\frac{\partial\underaccent{\dot}{z}_{i,k,t}}{\partial\log(p_{i,t})}=-\frac{\mathcal{k}(\text{ArcSinh}(x_{k,t}))_{i}}{\sqrt{x_{i,k,t}^{2}+1}}\left(\frac{x_{i,k,t}^{n}}{p_{i,t}}\right).$
(B.10)

It is the values in (B.9) and (B.10) that are reported in Table 1.

####  B.2 Endogenous Anomaly Predictors

There are two kinds of endogenous predictors (other than the level term): (1)
predictors with a denominator price term and (2) predictors with a numerator
price term. The denominator price predictors are of the form:

$x_{i,k,t}=\frac{x_{i,k,t}^{n}}{p_{i,t}}+x_{i,k,t}^{a}$ (B.11)

where again $x_{i,k,t}^{n}$ and $x_{i,k,t}^{a}$ are exogenous components. For
example, if $x_{i,k,t}$ is the book to price ratio, then $x_{i,k,t}^{n}$ is
the book value divided by shares outstanding and $x_{i,k,t}^{a}=0$. Obviously
the per share stock price divided by the book value divided by the shares
outstanding yields the market to book ratio.

The numerator predictors have the form:

$x_{i,k,t}=\frac{p_{i,t}}{x_{i,k,t}^{d}}+x_{i,k,t}^{a}$ (B.12)

where $x_{i,k,t}^{d}$ and $x_{i,k,t}^{a}$ are exogenous components.

The 15 endogenous variables—along with the 47 exogenous variable
descriptions—are described in detail in Freyberger et al. (2020). I describe
the construction and components of these variables below using this notation,
although I do not describe in detail the construction of every variable
because it is described so well in Freyberger et al. (2020). Some of the
descriptions come directly from Freyberger et al. (2020), and I follow them by
putting CRSP and Compustat variables in parentheses. I follow their lagged
timing convention for when balance sheet variables are used.

Denominator Price Predictors:

a2me

  * •

Description: assets to market equity ratio.

  * •

$x_{i,k,t}^{n}$: total assets (AT) divided by shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

beme

  * •

Description: book equity to market equity ratio.

  * •

$x_{i,k,t}^{n}$: ratio of book value of equity to shares outstanding (SHROUT).
See Freyberger et al. (2020) for the construction of book value of equity.

  * •

$x_{i,k,t}^{a}=0$.

beme_adj

  * •

Description: book equity to market equity ratio minus the industry mean of
this ratio.

  * •

$x_{i,k,t}^{n}$: same as beme above.

  * •

$x_{i,k,t}^{a}$: negative of the average beme ratio within an asset’s (Fama-
French 48) industry.

debt2p

  * •

Description: debt to market equity ratio.

  * •

$x_{i,k,t}^{n}$: debt to shares outstanding ratio (SHROUT). See Freyberger et
al. (2020) for the construction of debt in the debt2p variable.

  * •

$x_{i,k,t}^{a}=0$.

e2p

  * •

Description: earnings to price ratio.

  * •

$x_{i,k,t}^{n}$: earnings (IB) divided by shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

ldp

  * •

Description: dividends to price ratio.

  * •

$x_{i,k,t}^{n}$: Sum (across time and all shares) of dividends paid out over
the last 12 months divided by shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

nop

  * •

Description: net payout ratio.

  * •

$x_{i,k,t}^{n}$: ratio of common dividends (DVC) plus purchase of common and
preferred stock (PRSTKC) minus the sale of common and preferred stock (SSTK)
over shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

o2p

  * •

Description: payout ratio.

  * •

$x_{i,k,t}^{n}$: ratio of common dividends (DVC) plus purchase of common and
preferred stock (PRSTKC) minus the change in value of the net number of
preferred stocks outstanding (PSTKRV) over shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

s2p

  * •

Description: sales to price ratio.

  * •

$x_{i,k,t}^{n}$: common dividends (DVC) plus purchase of common and preferred
stock (PRSTKC) minus the change in value of the net number of preferred stocks
outstanding (PSTKRV) over shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

Numerator Price Predictors:

cum_return_1_0

  * •

Description: return over the prior month.

  * •

$x_{i,k,t}^{d}$: one month lagged price (PRC), adjusted for stock splits.

  * •

$x_{i,k,t}^{a}$: ratio of dividends paid out over the previous month to the
one month lagged price, adjusted for stock splits.

lme

  * •

Description: size—or market equity—defined as price (PRC) times shares
outstanding (SHROUT).

  * •

$x_{i,k,t}^{d}$: one over shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}=0$.

lme_adj

  * •

Description: size—same as lme above—minus the average (Fama-French 48)
industry lme.

  * •

$x_{i,k,t}^{d}$: one over shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}$: Negative of average lme within an asset’s (Fama-French 48)
industry.

q

  * •

Description: Tobin’s Q—see Freyberger et al. (2020) for more details.

  * •

$x_{i,k,t}^{d}$: total assets (AT) divided by shares outstanding (SHROUT).

  * •

$x_{i,k,t}^{a}$: total assets (AT) minus cash and short-term investments
(CEQ), minus deferred taxes (TXDB) scaled by total assets (AT).

rel_to_high_price

  * •

Description: the price to 52 week high stock price ratio.

  * •

$x_{i,k,t}^{d}$: 52 week high stock price.

  * •

$x_{i,k,t}^{a}=0$.

roc

  * •

Description: economic rents over cash ratio.

  * •

$x_{i,k,t}^{d}$: ratio of Short-Term Investments (CHE) to shares outstanding
(SHROUT).

  * •

$x_{i,k,t}^{a}=0$: ratio of long-term debt (DLTT) minus total assets to Cash
and Short-Term Investments (CHE).

####  B.3 Derivation of Demand Function

The derivation of the demand function follows trivially from Koijen and Yogo
(2019) (KY), but I describe it here in detail for the sake of completeness.
Let $\mu_{t}=\mathbb{E}_{t}[r_{t+1}]$ be the $N$-dimensional vector of asset
expected excess returns, and let $\Sigma_{t}=\mathbb{V}\text{ar}_{t}[r_{t+1}]$
be the $N\times N$ conditional covariance matrix. Let $\mu_{t}^{j}$ and
$\Sigma_{t}^{j}$ be similar, but instead let these terms represent the
subjective beliefs of institution $j$ for the mean and covariance
respectively. Like KY, I assume investors believe:

$\mu_{t}^{j}=Z_{t}\Phi_{t}^{j},\;\;\;\Sigma_{t}^{j}=\Gamma_{t}^{j}(\Gamma_{t}^{j})^{\prime}+\zeta_{t}^{j}I,\;\;\;\text{
and }\Gamma_{t}^{j}=Z_{t}\Psi_{t}^{j},$ (B.13)

where $\Phi_{t}^{j}$ and $\Psi_{t}^{j}$ are $K\times 1$ vectors of parameters,
$\zeta_{t}^{j}$ is a positive scalar, $I$ is an $N\times N$ identity matrix,
and here $Z_{t}$ is an $N\times(K+1)$ matrix of $K$ predictors
$\dot{z}_{i,k,t}$ (not $\underaccent{\dot}{z}_{i,k,t}$) and the first column
of $Z_{t}$ is a constant (intercept term). See KY for a discussion about this
standard single factor structure. KY mention "[w]e could relax the one-factor
assumption and generalize to a multifactor case, but the resulting expressions
are less intuitive and less preferable for expositional purposes."

Classic CARA demand with multivariate normal expected returns has the
following functional form:

$w_{t}^{j}=\frac{1}{\gamma_{t}^{j}}\left(\Sigma_{t}^{j}\right)^{-1}\mu_{t}^{j},$
(B.14)

where $\gamma_{t}^{j}$ is the risk aversion of the institution at time $t$.

With these assumptions above—using similar logic to KY—I can write:

$w_{t}^{j}=Z_{t}\hat{\beta}_{t}^{j}$ (B.15)

where the $K\times 1$ vector of parameters $\beta_{t}^{j}$ has the form:

$\beta_{t}^{j}=\frac{1}{\gamma_{t}^{j}\zeta_{t}^{j}}\left(\Phi_{t}^{j}-\kappa_{t}^{j}\Psi_{t}^{j}\right)$
(B.16)

where $\kappa_{t}^{j}$ is a scalar (i.e. common across assets):

$\kappa_{t}^{j}=\frac{(\Gamma_{t}^{j})^{\prime}\mu_{t}^{j}}{(\Gamma_{t}^{j})^{\prime}\Gamma_{t}^{j}+\zeta_{t}^{j}}$
(B.17)

The proof of this—which is just a trivial application of the Woodbury matrix
identity—is so similar to the proof of KY’s proposition 1 that it is excluded
here.

Importantly, KY shows that this linear demand functional is still delivered
when an institution has short selling constraints on some assets. See KY for
more details.

Equation (B.15) can be written, for an individual asset $i$, as:

$\displaystyle w_{i,t}^{j}$
$\displaystyle=\hat{\beta}_{0,t}^{j}+\hat{\beta}_{1,t}^{j}\breve{\underaccent{\dot}{z}}_{i,1,t}+\sum_{k=2}^{K}\hat{\beta}_{k,t}^{j}\dot{z}_{i,k,t}+\epsilon_{i,t}^{j}$
$\displaystyle=\hat{\beta}_{0,t}^{j}+\hat{\beta}_{1,t}^{j}\left(\frac{P_{i,t}}{A_{t}^{j}}\right)+\sum_{k=2}^{K}\hat{\beta}_{k,t}^{j}\left(a_{i,1,k,t}+a_{2,k,t}\log(P_{i,t})\right)+\epsilon_{i,t}^{j},$
(B.18)

where these $\hat{\beta}_{k,t}^{j}$ are the elements of $\hat{\beta}_{t}^{j}$.
Notice that this is equation (5.3). Equation (21) can then be used to deliver
equation (15).

What are the differences between this and KY? All differences are trivial, but
I list them here. First, I derive standard mean-variance demand instead of log
utility demand. Since an SDF/factor model is equivalent to mean-variance
demand, this result is useful (and easily obtainable). Second, KY use log
returns, and I use simple excess returns.444This means their $\mu$ and
$\Sigma$ are slightly different to deal with log returns. I just use standard
excess returns here. Third, I ignore short-selling constraints in the math
here, but this can be trivially added back in.

####  B.4 Equilibrium Solution

Incumbent demand for asset $i$, written as the number of shares is:

$\frac{(1-\underaccent{\dot}{\theta}_{t})A_{t}w_{i,t}}{p_{i,t}}.$ (B.19)

Recall that the portfolio weight of asset $i$ is $w_{i,t}$, and the assets
under management for all incumbents is
$(1-\underaccent{\dot}{\theta}_{t})A_{t}$. Thus, the dollar amount that
incumbents demand to invest in the asset is
$(1-\underaccent{\dot}{\theta}_{t})A_{t}w_{i,t}$. Thus, dividing by the share
price delivers demand in terms of the number of shares as in (B.19).

Similarly, demand for statistical arbitrageurs in aggregate for asset $i$,
written in terms of shares, is:

$\frac{\underaccent{\dot}{\theta}_{t}A_{t}\underaccent{\dot}{w}_{i,t}}{p_{i,t}}.$
(B.20)

Thus, total market demand is the sum of (B.19) and (B.20). Supply is
$S_{i,t}$, which is the number of shares outstanding. Thus, equilibrium is
defined by the equation:

$\frac{(1-\underaccent{\dot}{\theta}_{t})A_{t}w_{i,t}}{p_{i,t}}+\frac{\underaccent{\dot}{\theta}_{t}A_{t}\underaccent{\dot}{w}_{i,t}}{p_{i,t}}=S_{i,t}.$
(B.21)

Multiplying both sides by $p_{i,t}$ delivers the main equilibrium equation in
the text—equation (26).

Using (28), I can rewrite (26) as

$\zeta_{i,0,t}+\zeta_{i,1,t}P_{i,t}+\zeta_{i,2,t}\log(P_{i,t})=P_{i,t}.$
(B.22)

This can be rewritten as

$P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}+\log(P_{i,t})=-\frac{\zeta_{i,0,t}}{\zeta_{i,2,t}}.$
(B.23)

Taking the exponential of both sides yields:

$P_{i,t}\exp\left(P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\right)=\exp\left(-\frac{\zeta_{i,0,t}}{\zeta_{i,2,t}}\right).$
(B.24)

Multiplying both sides by $(\zeta_{i,1,t}-1)/\zeta_{i,2,t}$:

$P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\exp\left(P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\right)=\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\exp\left(-\frac{\zeta_{i,0,t}}{\zeta_{i,2,t}}\right).$
(B.25)

Notice that the left-hand side has the form of $we^{w}$, where
$w=P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}$. Since $\zeta_{i,1,t}<1$ and
$\zeta_{i,2,t}<0$, $\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}$ is positive. Thus,
I can equivalently write:

$P_{i,t}\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}=W\left(\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\exp\left(-\frac{\zeta_{i,0,t}}{\zeta_{i,2,t}}\right)\right).$
(B.26)

where $W(\cdot)$ is the Lambert W function. Thus, I can write:

$P_{i,t}=\frac{\zeta_{i,2,t}}{\zeta_{i,1,t}-1}W\left(\frac{\zeta_{i,1,t}-1}{\zeta_{i,2,t}}\exp\left(-\frac{\zeta_{i,0,t}}{\zeta_{i,2,t}}\right)\right),$
(B.27)

which is equation (27). Recall that positive values of the Lambert W function
are unique; thus this the unique equilibrium. Also, it is easy to see that the
price is guaranteed to be positive. This is close to the KY equilibrium,
except that because their demand function is a bit more complicated, they have
a unique positive price for each asset without a closed-form solution like I
have here.

####  B.5 General Details about the Calculation of statistical arbitrageur
Derivatives

In order to calculate (14), I have to calculate

$\nabla_{{z}_{i,k,t}}(f(Z_{t})_{i}b)\equiv\frac{\partial
f(Z_{t})_{i}b}{\partial{z}_{i,k,t}}.$ (B.28)

Here I drop the dot under $z_{i,k,t}$ for simplicity, and because in the paper
I fill $Z_{t}$ with $\underaccent{\dot}{z}_{i,k,t}$ values (main
specification) and $\dot{z}_{i,k,t}$ values (robustness check).

In words, it is important to know how statistical arbitrageur-implied demand
changes when predictors change. This is because the endogenous predictors
change when prices change, and it is critical to know these values in order to
calculate statistical arbitrageur elasticity.

Let $Y$ be an $N\times J$ matrix with elements $Y_{i,j}$ and let $X$ be an
$N\times 1$ vector with elements $X_{i}$. Assume that each element of
$Y_{i,j}$ is a differentiable function of $X_{i}$. Define the following:

$\nabla_{X}Y=\begin{bmatrix}\frac{\partial Y_{1,1}}{\partial
X_{1}}&\frac{\partial Y_{1,2}}{\partial X_{1}}&\ldots&\frac{\partial
Y_{1,J}}{\partial X_{1}}\\\ \frac{\partial Y_{2,1}}{\partial
X_{2}}&\frac{\partial Y_{2,2}}{\partial X_{2}}&\ldots&\frac{\partial
Y_{2,J}}{\partial X_{2}}\\\ \vdots&\vdots&\ddots&\vdots\\\ \frac{\partial
Y_{N,1}}{\partial X_{N}}&\frac{\partial Y_{N,2}}{\partial
X_{N}}&\ldots&\frac{\partial Y_{N,J}}{\partial X_{N}}\\\ \end{bmatrix}.$
(B.29)

I will assume throughout that

$\frac{\partial Y_{i,j}}{\partial X_{l}}=0\text{ for all }l\neq i.$ (B.30)

Importantly, this is not the standard Jacobian matrix. I give some identities
below that are useful for calculating these derivatives with the models. These
identities differ from standard Jacobian identities because this is a separate
concept (although obviously it is related and borrows similar notation).

Using this matrix notation for example, I can rewrite the elasticity of
statistical arbitrageurs (equation (14)) as the $N$-dimensional vector,
defined over only assets with positive weights:

$\eta_{t}=\iota-\text{diag}(\underaccent{\dot}{w}_{t})^{-1}\nabla_{\log(p_{t})}\underaccent{\dot}{w}_{t},$
(B.31)

where $\iota$ is a vector of ones, $\underaccent{\dot}{w}_{t}$ and $p_{t}$ the
vectorized versions of $\underaccent{\dot}{w}_{i,t}$ and $p_{i,t}$, and
$\text{diag}(\cdot)$ is the diagonal matrix with the input along its diagonal.
Recall that $\underaccent{\dot}{w}_{t}=f(Z_{t})b$.

Let $z_{k,t}$ denote the $k^{th}$ column of $Z_{t}$. Then equation (14) can be
written in vector form for assets with positive weights as

$\displaystyle\eta_{t}$
$\displaystyle=\iota-\text{diag}(\underaccent{\dot}{w}_{t})^{-1}\sum_{k=1}^{K}\text{diag}(\nabla_{z_{k,t}}(f(Z_{t})b))\nabla_{\log(p_{t})}z_{k,t}$
(B.32)
$\displaystyle=\iota-\text{diag}(\underaccent{\dot}{w}_{t})^{-1}\sum_{k=1}^{K}(\nabla_{z_{k,t}}(f(Z_{t})b))\circ(\nabla_{\log(p_{t})}z_{k,t}),$
(B.33)

where $\circ$ denotes the standard Hadamard (element-wise) product. This is a
convenient way of writing this because Appendix B.2 above already gives the
formulas used to calculate the elements of $\nabla_{\log(p_{t})}z_{k,t}$, and
the text gives the formulas for the calculation of
$\underaccent{\dot}{w}_{t}=f(Z_{t})b$. Thus, all that is left is the formulas
for $\nabla_{z_{k,t}}(f(Z_{t})b)$ for the different statistical arbitrageurs.

Here are some simple identities with this matrix derivative that is useful for
the calculations. For all of these identities below, I define $Y$ and $X$ as
above. Let $A$, $B$, $C$, $D$, and $E$ be constant matrices (not a function of
the elements of $X$).

######  Proposition 1.

$\nabla_{X}(YB)=(\nabla_{X}Y)B.$ (B.34)

This result is trivial, but used in these calculations below.

######  Proposition 2.

Let $A$ be $N\times J$ dimensional, and $B$ be $N\times 1$ dimensional. Then:

$\nabla_{X}(AY^{\prime}B)=\text{Diag}(A(\nabla_{X}Y)^{\prime})\circ B,$ (B.35)

where $\text{Diag}(\cdot)$ denotes the function that takes the diagonal of its
matrix input and makes it a column vector (the off diagonal terms are
discarded).

###### Proof.

I can write:

$(AY^{\prime}B)_{i}=\sum_{l}\sum_{m}A_{i,l}Y^{\prime}_{l,m}B_{m}=\sum_{l}\sum_{m}A_{i,l}Y_{m,l}B_{m}.$
(B.36)

Then I can write:

$\frac{\partial(AY^{\prime}B)_{i}}{\partial X_{i}}=\frac{\partial}{\partial
X_{i}}\sum_{l}\sum_{m}A_{i,l}Y_{m,l}B_{m}$ $=\sum_{l}A_{i,l}\frac{\partial
Y_{i,l}}{\partial X_{i}}B_{i}=B_{i}\sum_{l}A_{i,l}\frac{\partial
Y_{i,l}}{\partial X_{i}}.$ (B.37)

and writing this back as matrices implies that

$\nabla_{X}(AY^{\prime}B)=\text{Diag}(A(\nabla_{X}Y)^{\prime})\circ B.$ (B.38)

∎

######  Proposition 3.

Let $A$ be $N\times N$ and $B$ be $J\times 1$. Then

$\nabla_{X}(AYB)=\text{Diag}(A)\circ((\nabla_{X}Y)B).$ (B.39)

###### Proof.

Again write:

$(AYB)_{i}=\sum_{l}\sum_{m}A_{i,l}Y_{l,m}B_{m}$ (B.40)

Then I can write:

$\frac{\partial(AYB)_{i}}{\partial X_{i}}=\frac{\partial}{\partial
X_{i}}\sum_{l}\sum_{m}A_{i,l}Y_{l,m}B_{m}$ $=\sum_{m}A_{i,i}\frac{\partial
Y_{i,m}}{\partial X_{i}}B_{m}=A_{i,i}\sum_{m}\frac{\partial Y_{i,m}}{\partial
X_{i}}B_{m}.$ (B.41)

Writing this back in matrix form gives:

$\nabla_{X}(AYB)=\text{Diag}(A)\circ((\nabla_{X}Y)B).$ (B.42)

∎

It is well-known result that if $g(x)$ is a differentiable matrix-valued
function of the scalar $x$, then

$\frac{\partial g(x)^{-1}}{\partial x}=-g(x)^{-1}\frac{\partial g(x)}{\partial
x}g(x)^{-1}.$ (B.43)

I use this result in the calculations below.

######  Proposition 4.

Assume the first dimension of $A$ is $N$, $E$ is a column vector,
$B+CY^{\prime}D$ is a square invertible matrix, and the rest of the dimensions
of the matrices are such that the multiplication is well-defined. Then:

$\nabla_{X}(A(B+CY^{\prime}D)^{-1}E)=-\text{Diag}(A(B+CY^{\prime}D)^{-1}C(\nabla_{X}Y)^{\prime})\circ(D(B+CY^{\prime}D)^{-1}E).$
(B.44)

###### Proof.

$\frac{\partial}{\partial
X_{i}}(A(B+CY^{\prime}D)^{-1}E)=-A(B+CY^{\prime}D)^{-1}\left(\frac{\partial}{\partial
X_{i}}CY^{\prime}D\right)(B+CY^{\prime}D)^{-1}E.$ (B.45)

Using this combined with Proposition 2 gives the stated result. ∎

######  Proposition 5.

Assume the first dimension of $A$ is $N$, $E$ is a column vector, $B+CYD$ is a
square invertible matrix, and the rest of the dimensions of the matrices are
such that the multiplication is well-defined. Then

$\nabla_{X}(A(B+CYD)^{-1}E)=-\text{Diag}(A(B+CYD)^{-1}C)\circ((\nabla_{X}Y)D(B+CYD)^{-1}E).$
(B.46)

###### Proof.

$\frac{\partial}{\partial
X_{i}}(A(B+CYD)^{-1}E)=-A(B+CYD)^{-1}\left(\frac{\partial}{\partial
X_{i}}CYD\right)(B+CYD)^{-1}E.$ (B.47)

Using this combined with Proposition 3 gives the stated result. ∎

All of the statistical arbitrageur derivative calculations below simply use
these five propositions combined with the product rule.

####  B.6 Linear statistical arbitrageur Derivatives

I use the notation and propositions from Appendix B.5 in this section of the
Appendix.

The linear statistical arbitrageurs—BPZL, BSV, DGU, FF3, FF6, HXZ, and KNS
models—have trivial derivative calculations. For all models (not just these
linear models), I can calculate:

$\nabla_{z_{k,t}}w_{t}=(\nabla_{z_{k,t}}f(Z_{t}))b.$ (B.48)

However, for these models $\nabla_{z_{k,t}}f(Z_{t})$ is also trivial to
calculate.

For example, for the BPZL, BSV, DGU, and the KNS models,
$f(Z_{t})=\breve{Z}_{t}$. Thus, for these models

$\nabla_{z_{k,t}}f(Z_{t})=\nabla_{z_{k,t}}\breve{Z}_{t}=\left(\frac{4}{N}\right)\nabla_{z_{k,t}}Z_{t},$
(B.49)

where $\nabla_{z_{k,t}}Z_{t}$ is simply an $N\times K$ matrix of zeros, except
the $k^{th}$ column is filled with ones.

The other linear models—FF3, FF6, and HXZ—are identical except that the
derivative $\nabla_{z_{k,t}}f(Z_{t})$ equals a matrix of zeros if predictor
$k$ is not included in the model at all.

####  B.7 KPS Model Derivatives

I use the notation and propositions from Appendix B.5 in this section of the
Appendix.

In Kelly et al. (2019), the MVE portfolio is:

$w_{t}=f(Z_{t})b.$ (B.50)
$f(Z_{t})=Z_{t}\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}.$
(B.51)

I can calculate:

$\nabla_{z_{k,t}}w_{t}=(\nabla_{z_{k,t}}Z_{t})\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}b$
$-\text{Diag}(Z_{t}\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}\Gamma_{\beta}^{\prime}(\nabla_{z_{k,t}}Z_{t})^{\prime})\circ(Z_{t}\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}b)$
$-\text{Diag}(Z_{t}\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}\Gamma_{\beta}^{\prime}Z_{t}^{\prime})\circ((\nabla_{z_{k,t}}Z_{t})\Gamma_{\beta}\left(\Gamma_{\beta}^{\prime}Z_{t}^{\prime}Z_{t}\Gamma_{\beta}\right)^{-1}b).$
(B.52)

####  B.8 GKX Model Derivatives

I use the notation and propositions from Appendix B.5 in this section of the
Appendix.

In Gu et al. (2021), the portfolio weights are

$f(Z_{t})=Z_{t}(Z_{t}^{\prime}Z_{t})^{-1}\Gamma_{A}+\iota_{N}\Gamma_{B}^{\prime},$
(B.53)

where $\Gamma_{A}$ is a $K\times M$ matrix and $\Gamma_{B}$ is a $M$
dimensional column vector of parameters. In the language of neural networks,
$\Gamma_{A}$ is a matrix of weights and $\Gamma_{B}$ is a vector of biases.

Then I have:

$\nabla_{z_{k,t}}w_{t}=(\nabla_{z_{k,t}}Z_{t})(Z_{t}^{\prime}Z_{t})^{-1}\Gamma_{A}b$
$-\text{Diag}(Z_{t}(Z_{t}^{\prime}Z_{t})^{-1}(\nabla_{z_{k,t}}Z_{t})^{\prime})\circ(Z_{t}(Z_{t}^{\prime}Z_{t})^{-1}\Gamma_{A}b)$
$-\text{Diag}(Z_{t}(Z_{t}^{\prime}Z_{t})^{-1}Z_{t}^{\prime})\circ((\nabla_{z_{k,t}}Z_{t})(Z_{t}^{\prime}Z_{t})^{-1}\Gamma_{A}b).$
(B.54)

####  B.9 NN Model Derivatives

I use the notation and propositions from Appendix B.5 in this section of the
Appendix.

I have:

$w_{t}=\frac{1}{\zeta}\mu_{t}-\frac{1}{\zeta}\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime}\mu_{t}.$

Then I can calculate:

$\nabla_{z_{k,t}}w_{t}=\frac{1}{\zeta}(\nabla_{z_{k,t}}\mu_{t})-\frac{1}{\zeta}(\nabla_{z_{k,t}}\Gamma_{t})(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime}\mu_{t}$
$+\frac{1}{\zeta}\text{Diag}(\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}(\nabla_{z_{k,t}}\Gamma_{t})^{\prime})\circ(\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime}\mu_{t})$
$+\frac{1}{\zeta}\text{Diag}(\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime})\circ((\nabla_{z_{k,t}}\Gamma_{t})(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime}\mu_{t})$
$-\frac{1}{\zeta}\text{Diag}(\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}(\nabla_{z_{k,t}}\Gamma_{t})^{\prime})\circ\mu_{t}$
$-\frac{1}{\zeta}\text{Diag}(\Gamma_{t}(\zeta
I+\Gamma_{t}^{\prime}\Gamma_{t})^{-1}\Gamma_{t}^{\prime})\circ(\nabla_{z_{k,t}}\mu_{t}).$
(B.55)

In the case of zero hidden layers, then

$\mu_{t}=Z_{t}\beta_{\mu},\;\text{ and }\;\Gamma_{t}=Z_{t}\beta_{\Sigma},$
(B.56)

for some parameter vectors $\beta_{\mu}$ and $\beta_{\Sigma}$.

Then

$\nabla_{z_{k,t}}\mu_{t}=(\nabla_{z_{k,t}}Z_{t})\beta_{\mu},\;\;\;\nabla_{z_{k,t}}\Gamma_{t}=(\nabla_{z_{k,t}}Z_{t})\beta_{\Gamma}.$
(B.57)

In the more general case,

$\mu_{i,t}=g_{\mu}(Z_{i,t}),\;\text{ and }\;\Gamma_{i,t}=g_{\Sigma}(Z_{i,t}).$
(B.58)

where $g_{\mu}:\mathbb{R}^{K}\to\mathbb{R}$ and $g_{\Sigma}:\mathbb{R}^{K}\to
V$ are the standard neural network functions. Then let $Dg_{\mu}$ be the
$1\times K$ Jacobian matrix of derivatives of $g_{\mu}$ and $Dg_{\Sigma}$ be
the $V\times K$ Jacobian matrix of $g_{\Sigma}$. Then I can write:

$\nabla_{z_{k,t}}\mu_{i,t}=Dg_{\mu}(\nabla_{z_{k,t}}Z_{i,t}),\;\;\;\nabla_{z_{k,t}}\Gamma_{i,t}=Dg_{\Sigma}(\nabla_{z_{k,t}}Z_{i,t}).$
(B.59)

####  B.10 BPZF and RF Model Derivatives

These models are not continuous functions of $Z_{t}$. Ignoring the break
points severely attenuates the any measure of how much $f(Z_{t})$ varies when
$Z_{t}$ varies. In order to have a continuous demand function (necessary for
equilibrium to exist) and to have reasonable estimates of the elasticity of
these statistical arbitrageurs, I am forced to approximate these derivatives
with numerical derivatives. This is done as describe below.

I calculate $\nabla_{\log(p_{t})}f(Z_{t})b$ directly. Here I use
$\nabla_{\log(p_{t})}Z_{t}$ (see Appendix B.5 to see how this is calculated).
In particular, for these two models, I approximate:

$\nabla_{\log(p_{t})}f(Z_{t})b\approx\frac{f(Z_{t}+h\nabla_{\log(p_{t})}Z_{t})b-f(Z_{t}-h\nabla_{\log(p_{t})}Z_{t})b}{2h}.$
(B.60)

As $h$ goes to zero, this becomes a better and better approximation of the
derivative (where it is defined). However, the actual derivative understates
how much $f(Z_{t})b$ changes when prices change because it ignores the large
breakpoint jumps. Thus, I calculate this with values of $h$ that are not too
small and not too large.

I calculate this with $h\in\\{0.25,0.5,0.75,1\\}$, and then average over these
four numerical derivatives. This obviously corresponds to a 0.25%, 0.5%,
0.75%, and 1% price change, which seems like a reasonable degree of price
change to approximate how much $f(Z_{t})b$ reacts to changes in prices.

####  B.11 Recursive Experiments

In subsection B.11.1 below I describe how the arbitrageur fraction is
endogenized to have both wealth effects and fund flow effects. In subsection
B.11.2 I describe the results of these recursive experiments.

#####  B.11.1 Recursive Arbitrageur Fraction

In this subsection, I consider recursive experiments where the arbitrageur
fraction $\theta_{t}$ is updated through time and the arbitrageurs re-estimate
their models. I consider arbitrageurs who re-estimate their models annually,
since this reduces computational costs and only a single month tends to change
model parameters relatively little.

I follow Berk and Green (2004), who have a simple fund flow relationship
similar to:

$\theta_{t}=\Bigg{(}\underbrace{\underaccent{\dot}{\phi}\frac{1}{11}\sum_{\tau=t-12}^{t-1}\underaccent{\dot}{r}_{\tau}}_{\begin{subarray}{c}\text{fund}\\\
\text{flow
term}\end{subarray}}+\underbrace{\underaccent{\dot}{r}_{t-1}}_{\begin{subarray}{c}\text{wealth}\\\
\text{effect}\end{subarray}}\Bigg{)}\theta_{t-1}$ (B.61)

where $\underaccent{\dot}{\phi}$ is an exogenous fund flow parameter based on
the previous year’s return excluding the previous month, and
$\underaccent{\dot}{r}_{\tau}$ are the statistical arbitrageur excess returns.
The wealth effect captures the simple dynamics that if the statistical
arbitrageur funds earn more, they have more money to manage. In other words,
$\theta_{t+1}A_{t+1}-\theta_{t}A_{t}$ is not a typical fund flow measure, but
rather a change in AUM. Thus, this wealth effect should be added as shown, and
without it the assumption would be that extra returns are withdrawn by
investors. The fund flow term comes from Berk and Green (2004), where flows
are a function of the previous year’s returns. Like Berk and Green (2004),
there is no intercept in this equation.

Note that $\theta_{t+1}$ is not a function of $\underaccent{\dot}{r}_{t}$, but
only earlier period returns. Thus, both the fund flow term and wealth effects
are in some sense lagged or delayed. This is because
$\underaccent{\dot}{r}_{t}$ is a function of prices at time $t$, meaning that
$\theta_{t}$ would be a function of prices at time $t$ as well. This would
invalidate the KY assumption that AUM are exogenous to time $t$ prices, as
well as make the exercise less tractable. Frictions that slow the deployment
of capital immediately, as in Duffie (2010) for example, are enough to justify
this.

I use the CRSP fund flow data to estimate the parameter
$\underaccent{\dot}{\phi}$, by simply running a regression of monthly flows in
the previous average excess return of the fund excluding the previous month. I
estimate the regression without an intercept since there is no intercept in
equation (B.61). Table A.8 in the Appendix shows the results, giving a value
of $\underaccent{\dot}{\phi}=0.38$ to plug into the counterfactual
experiments.

In order to reduce extreme flows in the experiments, I make the maximum
possible rise in $\theta_{t}-\theta_{t}$ to be 0.01 per month. A rise of 1% of
the entire stock market in a single month for a group of asset managers would
constitute extremely large flows, and thus this is a relatively conservative
restriction.

#####  B.11.2 Recursive Experiment Results

In the recursive experiments, I set the initial arbitrageur fraction (ILF),
denoted as $\theta_{0}$, to a range of values. Table A.1 shows the recursive
counterfactual results. While the extreme alpha increases from the simple
experiments are significantly tamed, most statistical arbitrageurs still fail
to reduce alpha. The statistical arbitrageurs that do reduce alpha fail to
eliminate even half of it.

In many cases, the alpha is worse than in the simple experiments. This is
because the statistical arbitrageurs in aggregate might cause a severe
overcorrection, which causes them to reduce or reverse their exposure to the
anomaly, causing oscillation. A naive econometrician might call these
oscillations noise, but in reality the actual sign of the alpha is
oscillating. In a world with inelastic demand, it is hard to eliminate alpha.
This especially holds true with poorly aligned demand. Learning is no panacea
for alpha, and in fact can make it worse.

| Panel A: Log GRS Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.112^{***}$ | $0.373^{***}$ | $0.121^{***}$ | $0.189^{***}$ | $-0.267^{***}$ | $0.91^{***}$ | $-0.037^{***}$ | $-0.251^{***}$ | $0.482^{***}$ | $0^{***}$ | $0.04^{***}$ | $-0.014^{***}$
0.0005 | $-0.122^{***}$ | $0.523^{***}$ | $0.174^{***}$ | $0.393^{***}$ | $-0.248^{***}$ | $1.105^{***}$ | $-0.037^{***}$ | $-0.216^{***}$ | $0.544^{***}$ | $-0.001^{***}$ | $-0.139^{***}$ | $-0.025^{***}$
0.001 | $-0.024^{***}$ | $0.616^{***}$ | $0.188^{***}$ | $0.443^{***}$ | $-0.229^{***}$ | $1.164^{***}$ | $-0.041^{***}$ | $-0.197^{***}$ | $0.561^{***}$ | $-0.003^{***}$ | $0.083^{***}$ | $-0.026^{***}$
0.005 | $0.018^{***}$ | $0.685^{***}$ | $0.203^{***}$ | $0.648^{***}$ | $-0.18^{***}$ | $1.264^{***}$ | $-0.056^{***}$ | $-0.148^{***}$ | $0.558^{***}$ | $-0.009^{***}$ | $-0.034^{***}$ | $-0.035^{***}$
0.01 | $-0.081^{***}$ | $0.879^{***}$ | $0.252^{***}$ | $0.695^{***}$ | $-0.16^{***}$ | $1.297^{***}$ | $-0.061^{***}$ | $-0.125^{***}$ | $0.53^{***}$ | $-0.015^{***}$ | $-0.05^{***}$ | $-0.054^{***}$
0.05 | $-0.04^{***}$ | $1.701^{***}$ | $0.407^{***}$ | $0.833^{***}$ | $-0.097^{***}$ | $1.363^{***}$ | $-0.038^{***}$ | $-0.056^{***}$ | $0.529^{***}$ | $-0.117^{***}$ | $0.059^{***}$ | $-0.085^{***}$
0.1 | $-0.012^{***}$ | $0.852^{***}$ | $0.411^{***}$ | $0.966^{***}$ | $-0.058^{***}$ | $1.379^{***}$ | $-0.01^{***}$ | $-0.018^{***}$ | $0.566^{***}$ | $-0.073^{***}$ | $0.018^{***}$ | $-0.064^{***}$
0.25 | $0.091^{***}$ | $0.54^{***}$ | $0.555^{***}$ | $1.102^{***}$ | $-0.032^{***}$ | $1.386^{***}$ | $0.006^{***}$ | $0.008^{***}$ | $0.612^{***}$ | $-0.065^{***}$ | $0.004^{***}$ | $-0.034^{***}$
0.5 | $0.18^{***}$ | $0.381^{***}$ | $0.677^{***}$ | $1.094^{***}$ | $-0.023^{***}$ | $1.39^{***}$ | $0.014^{***}$ | $0.018^{***}$ | $0.582^{***}$ | $-0.403^{***}$ | $-0.039^{***}$ | $-0.022^{***}$
0.75 | $0.056^{***}$ | $0.901^{***}$ | $0.59^{***}$ | $1.109^{***}$ | $-0.019^{***}$ | $1.393^{***}$ | $0.017^{***}$ | $0.022^{***}$ | $0.604^{***}$ | $-0.153^{***}$ | $-0.007^{***}$ | $-0.018^{***}$
0.9 | $0.298^{***}$ | $0.545^{***}$ | $0.585^{***}$ | $1.135^{***}$ | $-0.016^{***}$ | $1.394^{***}$ | $0.019^{***}$ | $0.024^{***}$ | $0.612^{***}$ | $-0.128^{***}$ | $-0.057^{***}$ | $-0.016^{***}$
0.99 | $0.328^{***}$ | $0.494^{***}$ | $0.584^{***}$ | $1.138^{***}$ | $-0.015^{***}$ | $1.395^{***}$ | $0.02^{***}$ | $0.025^{***}$ | $0.615^{***}$ | $-0.196^{***}$ | $-0.075^{***}$ | $-0.019^{***}$
| Panel B: Log GRS Ratio, February 2000 - January 2010
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $0.107^{***}$ | $0.525^{***}$ | $0.282^{***}$ | $0.007^{***}$ | $-0.025^{***}$ | $0.059^{***}$ | $0.009^{***}$ | $-0.004^{***}$ | $0.581^{***}$ | $0^{***}$ | $-0.015^{***}$ | $0.001^{***}$
0.0005 | $0.076^{***}$ | $0.54^{***}$ | $0.287^{***}$ | $0.038^{***}$ | $-0.065^{***}$ | $0.125^{***}$ | $0.011^{***}$ | $-0.008^{***}$ | $0.568^{***}$ | $0^{***}$ | $-0.022^{***}$ | $0.001^{***}$
0.001 | $0.05^{***}$ | $0.545^{***}$ | $0.262^{***}$ | $0.09^{***}$ | $-0.079^{***}$ | $0.145^{***}$ | $0.012^{***}$ | $-0.006^{***}$ | $0.556^{***}$ | $0^{***}$ | $0.021^{***}$ | $0^{***}$
0.005 | $0.101^{***}$ | $0.688^{***}$ | $0.104^{***}$ | $0.199^{***}$ | $-0.099^{***}$ | $0.181^{***}$ | $0.015^{***}$ | $0.006^{***}$ | $0.404^{***}$ | $0.001^{***}$ | $0.039^{***}$ | $-0.002^{***}$
0.01 | $-0.031^{***}$ | $0.727^{***}$ | $0.132^{***}$ | $0.262^{***}$ | $-0.103^{***}$ | $0.198^{***}$ | $0.014^{***}$ | $0.012^{***}$ | $0.341^{***}$ | $0.003^{***}$ | $0.109^{***}$ | $-0.004^{***}$
0.05 | $-0.022^{***}$ | $0.765^{***}$ | $0.185^{***}$ | $0.386^{***}$ | $-0.112^{***}$ | $0.262^{***}$ | $0.01^{***}$ | $0.028^{***}$ | $0.295^{***}$ | $0.01^{***}$ | $0.076^{***}$ | $-0.009^{***}$
0.1 | $0.077^{***}$ | $0.721^{***}$ | $0.144^{***}$ | $0.523^{***}$ | $-0.116^{***}$ | $0.311^{***}$ | $0.006^{***}$ | $0.031^{***}$ | $0.274^{***}$ | $-0.021^{***}$ | $0.04^{***}$ | $-0.018^{***}$
0.25 | $0.07^{***}$ | $0.694^{***}$ | $0.109^{***}$ | $0.594^{***}$ | $-0.122^{***}$ | $0.353^{***}$ | $0.002^{***}$ | $0.028^{***}$ | $0.098^{***}$ | $-0.194^{***}$ | $-0.012^{***}$ | $-0.029^{***}$
0.5 | $0.083^{***}$ | $0.698^{***}$ | $-0.016^{***}$ | $0.608^{***}$ | $-0.125^{***}$ | $0.374^{***}$ | $0^{***}$ | $0.027^{***}$ | $0.087^{***}$ | $-0.19^{***}$ | $0.025^{***}$ | $-0.026^{***}$
0.75 | $-0.021^{***}$ | $0.775^{***}$ | $0.039^{***}$ | $0.646^{***}$ | $-0.127^{***}$ | $0.385^{***}$ | $0^{***}$ | $0.026^{***}$ | $0.165^{***}$ | $-0.045^{***}$ | $-0.026^{***}$ | $-0.027^{***}$
0.9 | $0.061^{***}$ | $0.744^{***}$ | $0.046^{***}$ | $0.651^{***}$ | $-0.128^{***}$ | $0.39^{***}$ | $0^{***}$ | $0.025^{***}$ | $0.187^{***}$ | $-0.1^{***}$ | $-0.008^{***}$ | $-0.03^{***}$
0.99 | $0.093^{***}$ | $0.328^{***}$ | $0.049^{***}$ | $0.654^{***}$ | $-0.128^{***}$ | $0.393^{***}$ | $0^{***}$ | $0.025^{***}$ | $0.197^{***}$ | $-0.037^{***}$ | $0.084^{***}$ | $-0.032^{***}$
|  Note: ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

Table A.1: Recursive Experiments GRS. This table shows the log ratio of the
GRS test $F$ statistics of the counterfactual experiment anomaly portfolio
outcomes to the baseline GRS test statistic. Thus, positive values implies
that alpha increased relative to baseline, while negative values implies that
alpha decreased. Small values (close to zero) are approximately percentage
changes. ILF ($\theta_{0}$) represents the initial arbitrageur fraction used
in the counterfactual experiments (the arbitrageur fraction as of January 1990
when the statistical arbitrageur starts managing capital counterfactually).
Each column represents a different statistical arbitrageur model, where Table
2 describes the initialisms of these models. The GRS statistics are calculated
using one decade: either February 2010 - January 2020 (Panel A) or February
2000 - January 2010 (Panel B). Stars are given to represent whether the alpha
remaining counterfactually is different from zero (alpha remained
statistically significant), not whether baseline and counterfactual alpha are
statistically different from one another. The critical values for the log GRS
ratio are $-2.008$, $-1.937$, and $-1.805$ for 10%, 5%, and 1% significance
respectively for the 2010 - 2020 decade. The critical values for the log GRS
ratio are $-1.976$, $-1.905$, and $-1.772$ for 10%, 5%, and 1% significance
respectively for the 2000 - 2010 decade.

In these counterfactual experiments, the outcomes are sensitive to initial
conditions. For example, a statistical arbitrageur who with small amounts of
capital increases alpha can actually decrease alpha with more capital. Also,
with some initial capital, a statistical arbitrageur might decrease alpha,
with more initial capital increase alpha, and then worsen alpha with even more
capital. It should be remembered that these are not simulations, but rather
counterfactual experiments. There is a small random component in the neural
network statistical arbitrageurs (GKX and NN) due to the initialization of the
parameters and in the random forest statistical arbitrageurs (BPZF and RF),
but these are relatively small effects. This sensitivity to initial conditions
is largely due to positive feedback loops between the price impact of
statistical arbitrageurs, arbitrageur AUM, arbitrageur beliefs, and
arbitrageur demand. In other words, there are strong hysteresis effects in the
experiments, which makes asset prices relatively chaotic. This happens even
though there is a unique equilibrium every period.

The takeaway is simple: most statistical arbitrageurs fail to eliminate alpha
and alpha often increases.

Which experiments have more alpha? The experiments where the statistical
arbitrageur’s portfolio weights misalign with the alpha and the price impact
is the largest. I estimate a regression of the log alpha ratio in every month
and every experiment on the measure of alignment estimated with ten year
rolling windows, the arbitrageur fraction $\theta_{t}$, and the median price
impact across assets—inverse of the elasticity of the assets. I include fixed
effects for every month, and cluster the standard errors by both experiment
and month. Thus, the regression compares the same month across experiments.
The results are shown in Table A.2.

| Dependent variable: Log GRS Ratio
---|---
| (1)
Alignment | -0.320∗∗∗
| (0.079)
Price Impact | 0.121∗
| (0.069)
$\theta_{t}$ | 0.133∗∗∗
| (0.036)
Observations | 34560
$R^{2}$ | 0.118
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

Table A.2: What Experiments Have More Mispricing? This plot regresses rolling
ten year anomaly GRS ratios on alignment (discussed in Table 6), median price
impact (inverse of elasticity), and the arbitrageur fraction, with monthly by
model fixed effects. In other words, this regression compares alpha across
counterfactual experiments within model. Standard errors, clustered at both
the month and experiment level (which takes into account the overlapping
periods), are shown in parentheses below the estimates.

Table A.2 shows that in experiments with larger price impacts, alpha relative
to baseline is higher. Also, the experiments with higher rates of
amplification and severe alpha—more misalignment between statistical
arbitrageur weights and alphas—have higher alpha relative to baseline. The
purpose of this regression is simply to illustrate the main points that are
theoretically shown above: the experiments with higher price impact and more
misalignment between statistical arbitrageur weights and alphas have higher
alpha outcomes.555The price impact is only statistically significant at the
10% level because there is relatively little variation in the elasticity of
statistical arbitrageurs within arbitrageur types across experiments. Also,
this is median price impact, which is only a crude measure of impact on assets
that affect alpha the most. I use this simple measure for transparency.

###  Appendix C Additional Tables and Figures

Figure A.1: Value $\alpha$. This plot shows the $\alpha$ of value (HML), when
value is regressed on the market with various rolling windows, labeled in the
legend. The monthly factor returns data is downloaded from Kenneth French’s
website.

Model | Hyperparameter | Search Values | Selected
---|---|---|---
BPZF | Trees |  | 100
| Max depth |  | 4
| Max features |  | $\approx\log_{2}(K)$
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-3}$, $10^{-1}$, …, $10^{5}$ | $10^{-1}$
| LARS nonzero coefficients | 5, 10, 15, …, 50 | 10
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-3}$, $10^{-1}$, …, $10^{5}$ | $10^{3}$
BPZL | No. of PCA factors | 10, 25, 40, 55, $K$ (no PCA) | 40
| Mean penalty - $\lambda_{0}$ | 0, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-1}$
| LARS nonzero coefficients | 2, 4, 6, …, 20 | 12
|  $L^{2}$ penalty - $\lambda_{2}$ | 0, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | 0
FF3 |  $L^{1}$ penalty - $\lambda_{1}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-1}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{0}$
FF6 |  $L^{1}$ penalty - $\lambda_{1}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-1}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{0}$
GKX | NN nodes and layers |  $()$, $(32)$, $(32,16)$, $(32,16,8)$ | $(32)$
|  $L^{1}$ penalty of NN weights |  $10^{-5}$, $10^{-4}$, $10^{-3}$ | $10^{-4}$
| Learning Rate |  $0.01$, $0.001$ | $0.001$
| No. of Factors | 1, 3, 5, 10 | 1
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
HXZ |  $L^{1}$ penalty - $\lambda_{1}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-1}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{0}$
KNS | No. of PCA factors | 10, 25, 40, 55, $K$ (no PCA) | $K$
|  $L^{1}$ penalty - $\lambda_{1}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-2}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-1}$
KPS | No. of PCA factors | 1, 2, 3, …, 10 | 5
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $10^{0}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $10^{-1}$
NN | NN nodes and layers |  $()$, $(5)$, $(25)$, $(25,5)$ | $()$
|  $L^{2}$ penalty of NN weights | 0, 50, 100, …, 1000 | 600
RF | Trees |  | 100
| Max depth |  | 4
| Max features |  | $\approx\log_{2}(K)$

Table A.3: Hyperparameters. This table shows the hyperparamter search values
and selected values from the 4-fold cross validation design.

Model | Hyperparameter | Search Values | Selected
---|---|---|---
BPZF | Trees |  | 100
| Max depth |  | 4
| Max features |  | $\approx\log_{2}(K)$
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-3}$, $10^{-1}$, …, $10^{5}$ | $10^{-3}$
| LARS nonzero coefficients | 5, 10, 15, …, 50 | 50
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-3}$, $10^{-1}$, …, $10^{5}$ | $10^{3}$
BPZL | No. of PCA factors | 10, 25, 40, 55, $K$ (no PCA) | $K$
| Mean penalty - $\lambda_{0}$ | 0, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-2}$
| LARS nonzero coefficients | 2, 4, 6, …, 20 | 18
|  $L^{2}$ penalty - $\lambda_{2}$ | 0, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{0}$
GKX | NN nodes and layers |  $()$, $(32)$, $(32,16)$, $(32,16,8)$ | $(32,16,8)$
|  $L^{1}$ penalty of NN weights |  $10^{-5}$, $10^{-4}$, $10^{-3}$ | $10^{-4}$
| Learning Rate |  $0.01$, $0.001$ | $0.001$
| No. of Factors | 1, 3, 5, 10 | 6
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
KNS | No. of PCA factors | 10, 25, 40, 55, $K$ (no PCA) | $K$
|  $L^{1}$ penalty - $\lambda_{1}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-2}$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-10}$, $10^{-9}$, $10^{-8}$, …, $10^{0}$ | $10^{-2}$
KPS | No. of PCA factors | 1, 2, 3, …, 10 | 1
| Mean penalty - $\lambda_{0}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
|  $L^{2}$ penalty - $\lambda_{2}$ |  $0$, $10^{-5}$, $10^{-4}$, $10^{-3}$, …, $10^{5}$ | $0$
NN | NN nodes and layers |  $()$, $(5)$, $(25)$, $(25,5)$ | $()$
|  $L^{2}$ penalty of NN weights | 0, 50, 100, …, 1000 | 250
RF | Trees |  | 100
| Max depth |  | 4
| Max features |  | $\approx\log_{2}(K)$

Table A.4: Hyperparameters. This is similar to Table A.3, except that these
are the linear factor models where there is no level/market term (just cross-
sectional models).

| Panel A: Log Mispricing Ratio, February 2010 - January 2020
---|---
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.16^{***}$ | $-0.162^{***}$ | $-0.086^{***}$ | $-0.071^{***}$ | $-0.006^{***}$ | $-0.076^{***}$ | $-0.012^{***}$ | $0.511^{***}$ | $-0.013^{***}$
0.0005 | $0.207^{***}$ | $0.26^{***}$ | $0.345^{***}$ | $-0.14^{***}$ | $-0.02^{***}$ | $0.418^{***}$ | $-0.018^{***}$ | $0.876^{***}$ | $-0.018^{***}$
0.001 | $0.612^{***}$ | $0.632^{***}$ | $0.668^{***}$ | $-0.03^{***}$ | $-0.027^{***}$ | $0.681^{***}$ | $0.006^{***}$ | $1.065^{***}$ | $0.014^{***}$
0.005 | $1.27^{***}$ | $1.612^{***}$ | $1.176^{***}$ | $0.641^{***}$ | $-0.031^{***}$ | $1.186^{***}$ | $0.227^{***}$ | $1.946^{***}$ | $0.287^{***}$
0.01 | $1.64^{***}$ | $2.29^{***}$ | $1.545^{***}$ | $0.988^{***}$ | $-0.031^{***}$ | $1.736^{***}$ | $0.388^{***}$ | $2.102^{***}$ | $0.467^{***}$
0.05 | $2.646^{***}$ | $3.304^{***}$ | $2.982^{***}$ | $1.856^{***}$ | $-0.04^{***}$ | $3.093^{***}$ | $0.762^{***}$ | $2.747^{***}$ | $0.829^{***}$
0.1 | $3.069^{***}$ | $3.58^{***}$ | $3.372^{***}$ | $2.121^{***}$ | $-0.039^{***}$ | $3.425^{***}$ | $0.874^{***}$ | $3.019^{***}$ | $0.921^{***}$
0.25 | $3.371^{***}$ | $3.953^{***}$ | $3.654^{***}$ | $2.294^{***}$ | $-0.001^{***}$ | $3.715^{***}$ | $0.997^{***}$ | $3.289^{***}$ | $1.004^{***}$
0.5 | $3.464^{***}$ | $4.148^{***}$ | $3.759^{***}$ | $2.343^{***}$ | $0.041^{***}$ | $3.815^{***}$ | $1.083^{***}$ | $3.355^{***}$ | $1.043^{***}$
0.75 | $3.492^{***}$ | $4.217^{***}$ | $3.793^{***}$ | $2.36^{***}$ | $0.058^{***}$ | $3.842^{***}$ | $1.121^{***}$ | $3.371^{***}$ | $1.058^{***}$
0.9 | $3.501^{***}$ | $4.24^{***}$ | $3.804^{***}$ | $2.365^{***}$ | $0.063^{***}$ | $3.849^{***}$ | $1.134^{***}$ | $3.374^{***}$ | $1.063^{***}$
0.99 | $3.505^{***}$ | $4.25^{***}$ | $3.808^{***}$ | $2.367^{***}$ | $0.065^{***}$ | $3.852^{***}$ | $1.14^{***}$ | $3.375^{***}$ | $1.065^{***}$
| Panel B: Log Mispricing Ratio, February 2000 - January 2010
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.002^{***}$ | $-0.026^{***}$ | $-0.034^{***}$ | $-0.001^{***}$ | $0.002^{***}$ | $-0.034^{***}$ | $0.001^{***}$ | $0.239^{***}$ | $0.004^{***}$
0.0005 | $0.048^{***}$ | $0.014^{***}$ | $0.004^{***}$ | $-0.016^{***}$ | $0.005^{***}$ | $0.074^{***}$ | $0.002^{***}$ | $0.858^{***}$ | $0.013^{***}$
0.001 | $0.177^{***}$ | $0.256^{***}$ | $0.258^{***}$ | $-0.031^{***}$ | $0.007^{***}$ | $0.425^{***}$ | $0^{***}$ | $1.179^{***}$ | $0.022^{***}$
0.005 | $0.936^{***}$ | $1.463^{***}$ | $1.505^{***}$ | $0.081^{***}$ | $0.009^{***}$ | $1.675^{***}$ | $0.002^{***}$ | $1.595^{***}$ | $0.086^{***}$
0.01 | $1.236^{***}$ | $1.916^{***}$ | $2.016^{***}$ | $0.304^{***}$ | $0.011^{***}$ | $2.124^{***}$ | $0.025^{***}$ | $1.726^{***}$ | $0.148^{***}$
0.05 | $1.839^{***}$ | $2.741^{***}$ | $2.872^{***}$ | $0.903^{***}$ | $0.012^{***}$ | $2.923^{***}$ | $0.145^{***}$ | $2.105^{***}$ | $0.337^{***}$
0.1 | $2.056^{***}$ | $2.992^{***}$ | $3.076^{***}$ | $1.056^{***}$ | $0.01^{***}$ | $3.125^{***}$ | $0.203^{***}$ | $2.477^{***}$ | $0.406^{***}$
0.25 | $2.215^{***}$ | $3.207^{***}$ | $3.223^{***}$ | $1.152^{***}$ | $0.005^{***}$ | $3.265^{***}$ | $0.265^{***}$ | $3.02^{***}$ | $0.47^{***}$
0.5 | $2.26^{***}$ | $3.274^{***}$ | $3.272^{***}$ | $1.176^{***}$ | $0.003^{***}$ | $3.31^{***}$ | $0.294^{***}$ | $3.283^{***}$ | $0.501^{***}$
0.75 | $2.273^{***}$ | $3.286^{***}$ | $3.287^{***}$ | $1.184^{***}$ | $0.003^{***}$ | $3.321^{***}$ | $0.305^{***}$ | $3.345^{***}$ | $0.514^{***}$
0.9 | $2.278^{***}$ | $3.288^{***}$ | $3.292^{***}$ | $1.187^{***}$ | $0.003^{***}$ | $3.325^{***}$ | $0.308^{***}$ | $3.356^{***}$ | $0.519^{***}$
0.99 | $2.279^{***}$ | $3.288^{***}$ | $3.294^{***}$ | $1.188^{***}$ | $0.003^{***}$ | $3.326^{***}$ | $0.309^{***}$ | $3.358^{***}$ | $0.521^{***}$
|  Note: ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

Table A.5: Simple Experiments GRS. This is similar to Table 7, except this
shows the counterfactual experiment results for statistical arbitrageurs
without a level/market term.

| Panel A: Log Fixed Cov. Mispricing Ratio, February 2010 - January 2020
---|---
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.057$ | $-0.034$ | $0.008$ | $-0.014$ | $-0.002$ | $-0.054$ | $0.$ | $-0.008$ | $-0.018$ | $-0.013$ | $0.26$ | $-0.001$
0.0005 | $-0.09$ | $0.004$ | $0.164$ | $-0.04$ | $-0.014$ | $-0.155$ | $0.001$ | $-0.046$ | $0.188$ | $-0.048$ | $2.789$ | $-0.001$
0.001 | $0.155$ | $0.231$ | $0.486$ | $-0.031$ | $-0.023$ | $0.018$ | $0.002$ | $-0.072$ | $0.623$ | $-0.073$ | $4.139$ | $0.006$
0.005 | $1.909$ | $1.631$ | $1.992$ | $0.312$ | $-0.002$ | $1.715$ | $0.004$ | $0.094$ | $2.328$ | $-0.079$ | $6.52$ | $0.086$
0.01 | $2.679$ | $2.296$ | $2.638$ | $0.641$ | $0.038$ | $2.463$ | $0.005$ | $0.344$ | $3.003$ | $-0.012$ | $7.241$ | $0.165$
0.05 | $3.85$ | $3.395$ | $3.649$ | $1.394$ | $0.113$ | $3.606$ | $-0.018$ | $1.007$ | $4.06$ | $0.283$ | $8.376$ | $0.373$
0.1 | $4.107$ | $3.661$ | $3.881$ | $1.604$ | $0.125$ | $3.864$ | $-0.048$ | $1.207$ | $4.304$ | $0.393$ | $8.674$ | $0.437$
0.25 | $4.298$ | $3.867$ | $4.057$ | $1.771$ | $0.131$ | $4.058$ | $-0.09$ | $1.366$ | $4.491$ | $0.489$ | $8.93$ | $0.495$
0.5 | $4.375$ | $3.949$ | $4.125$ | $1.833$ | $0.139$ | $4.132$ | $-0.124$ | $1.433$ | $4.565$ | $0.517$ | $9.048$ | $0.517$
0.75 | $4.403$ | $3.979$ | $4.149$ | $1.853$ | $0.161$ | $4.155$ | $-0.147$ | $1.462$ | $4.592$ | $0.519$ | $9.097$ | $0.523$
0.9 | $4.413$ | $3.99$ | $4.158$ | $1.859$ | $0.184$ | $4.159$ | $-0.156$ | $1.469$ | $4.601$ | $0.517$ | $9.117$ | $0.524$
0.99 | $4.417$ | $3.994$ | $4.161$ | $1.862$ | $0.205$ | $4.159$ | $-0.161$ | $1.456$ | $4.605$ | $0.517$ | $9.128$ | $0.525$
| Panel B: Log Fixed Cov. Mispricing Ratio, February 2000 - January 2010
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.038$ | $-0.063$ | $-0.071$ | $-0.036$ | $-0.051$ | $-0.062$ | $-0.001$ | $-0.041$ | $-0.1$ | $-0.005$ | $-0.182$ | $-0.004$
0.0005 | $-0.111$ | $-0.224$ | $-0.224$ | $-0.142$ | $-0.16$ | $-0.135$ | $-0.004$ | $-0.087$ | $-0.32$ | $-0.017$ | $1.912$ | $-0.011$
0.001 | $-0.095$ | $-0.295$ | $-0.257$ | $-0.222$ | $-0.217$ | $-0.05$ | $-0.005$ | $-0.067$ | $-0.314$ | $-0.026$ | $3.244$ | $-0.011$
0.005 | $0.654$ | $0.312$ | $0.587$ | $-0.357$ | $-0.37$ | $0.693$ | $-0.01$ | $0.084$ | $0.944$ | $-0.036$ | $5.667$ | $0.034$
0.01 | $1.218$ | $0.891$ | $1.214$ | $-0.294$ | $-0.438$ | $1.184$ | $-0.014$ | $0.187$ | $1.65$ | $-0.027$ | $6.413$ | $0.085$
0.05 | $2.235$ | $1.959$ | $2.278$ | $0.109$ | $-0.545$ | $2.111$ | $-0.032$ | $0.434$ | $2.769$ | $0.026$ | $7.567$ | $0.237$
0.1 | $2.476$ | $2.229$ | $2.531$ | $0.267$ | $-0.566$ | $2.342$ | $-0.041$ | $0.509$ | $3.032$ | $0.055$ | $7.86$ | $0.285$
0.25 | $2.666$ | $2.45$ | $2.733$ | $0.412$ | $-0.583$ | $2.532$ | $-0.053$ | $0.573$ | $3.245$ | $0.091$ | $8.11$ | $0.325$
0.5 | $2.748$ | $2.546$ | $2.819$ | $0.48$ | $-0.589$ | $2.616$ | $-0.059$ | $0.608$ | $3.336$ | $0.111$ | $8.227$ | $0.343$
0.75 | $2.779$ | $2.582$ | $2.851$ | $0.507$ | $-0.588$ | $2.648$ | $-0.061$ | $0.631$ | $3.371$ | $0.119$ | $8.275$ | $0.351$
0.9 | $2.79$ | $2.595$ | $2.862$ | $0.516$ | $-0.586$ | $2.66$ | $-0.061$ | $0.648$ | $3.383$ | $0.122$ | $8.293$ | $0.353$
0.99 | $2.795$ | $2.601$ | $2.867$ | $0.521$ | $-0.583$ | $2.665$ | $-0.062$ | $0.664$ | $3.389$ | $0.124$ | $8.301$ | $0.355$

Table A.6: Simple Experiments Fixed Covariance alpha. This is similar to Table
7 except that the covariance matrix used to calculate the log alpha ratio is
fixed (the covariance matrix of anomaly residuals from the 1980-1989 decade).

| Panel A: Log Fixed Cov. Mispricing Ratio, February 2010 - January 2020
---|---
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | -0.068 | 0.103 | 0.008 | 0.242 | -0.057 | 0.065 | -0.008 | 1.847 | 0.166
0.0005 | 1.976 | 1.85 | 0.719 | 0.12 | -0.09 | 1.017 | 0.176 | 4.783 | 0.14
0.001 | 2.942 | 2.924 | 1.46 | 0.143 | -0.089 | 1.944 | 0.263 | 5.903 | 0.08
0.005 | 4.422 | 4.672 | 3.03 | 0.872 | -0.087 | 3.637 | 0.54 | 7.86 | 0.066
0.01 | 4.81 | 5.155 | 3.501 | 1.267 | -0.086 | 4.118 | 0.663 | 8.433 | 0.151
0.05 | 5.314 | 5.826 | 4.158 | 1.899 | -0.089 | 4.775 | 0.869 | 9.261 | 0.376
0.1 | 5.407 | 5.973 | 4.302 | 2.046 | -0.103 | 4.916 | 0.914 | 9.446 | 0.435
0.25 | 5.47 | 6.083 | 4.411 | 2.156 | -0.132 | 5.022 | 0.941 | 9.587 | 0.478
0.5 | 5.499 | 6.127 | 4.453 | 2.193 | -0.165 | 5.064 | 0.936 | 9.644 | 0.491
0.75 | 5.513 | 6.144 | 4.468 | 2.204 | -0.188 | 5.079 | 0.924 | 9.664 | 0.493
0.9 | 5.519 | 6.149 | 4.473 | 2.207 | -0.199 | 5.085 | 0.917 | 9.671 | 0.494
0.99 | 5.522 | 6.152 | 4.475 | 2.208 | -0.204 | 5.087 | 0.914 | 9.675 | 0.494
| Panel B: Log Fixed Cov. Mispricing Ratio, February 2000 - January 2010
LF ($\theta_{t}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | 0.098 | 0.118 | 0.169 | 0.167 | 0.01 | 0.258 | -0.005 | -0.422 | 0.108
0.0005 | -0.156 | -0.125 | 0.115 | 0.447 | 0.044 | 0.106 | 0.073 | 2.428 | 0.305
0.001 | 0.035 | 0.208 | -0.054 | 0.431 | 0.065 | -0.13 | 0.113 | 3.71 | 0.357
0.005 | 1.895 | 2.524 | 0.617 | 0.204 | 0.06 | 1.173 | 0.114 | 5.885 | 0.272
0.01 | 2.583 | 3.289 | 1.374 | 0.156 | 0.044 | 2.015 | 0.063 | 6.567 | 0.195
0.05 | 3.601 | 4.426 | 2.613 | 0.437 | 0.005 | 3.26 | -0.05 | 7.759 | 0.052
0.1 | 3.834 | 4.697 | 2.904 | 0.61 | -0.013 | 3.546 | -0.073 | 8.117 | 0.028
0.25 | 4.012 | 4.912 | 3.132 | 0.779 | -0.031 | 3.771 | -0.087 | 8.45 | 0.012
0.5 | 4.083 | 5.001 | 3.225 | 0.855 | -0.041 | 3.863 | -0.095 | 8.61 | 0.007
0.75 | 4.11 | 5.034 | 3.258 | 0.883 | -0.044 | 3.897 | -0.099 | 8.674 | 0.006
0.9 | 4.119 | 5.045 | 3.27 | 0.893 | -0.046 | 3.908 | -0.101 | 8.696 | 0.005
0.99 | 4.123 | 5.05 | 3.276 | 0.897 | -0.046 | 3.914 | -0.101 | 8.707 | 0.005

Table A.7: Simple Experiments Fixed Covariance alpha. This is similar to Table
A.6, except this shows the counterfactual experiment results for statistical
arbitrageurs without a level/market term.

| Dependent variable: CRSP Fund Flows
---|---
past year average excess returns | 0.382∗∗∗
(excluding the previous month) | (0.009)
Observations | 5,451,860
$R^{2}$ | 0.002
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

Table A.8: Flows. This table shows the regression: $\text{(fund
flows)}^{j}_{t}=\underaccent{\dot}{\phi}\frac{1}{11}\sum_{\tau=t-12}^{t-1}r_{\tau}^{j}+e_{t}^{j}$,
where $\text{(fund flows)}^{j}_{t}$ is the flows of fund $j$ at time $t$,
$r_{t}^{j}$ is the excess return of fund $j$ at time $t$,
$\underaccent{\dot}{\phi}$ is the slope coefficient, and $e_{t}^{j}$ is the
residual. The data used here are the CRSP monthly fund flows data. In other
words, this shows the results of a regression of fund flows on average lagged
returns.

| Panel A: Average Learning Fraction ($\theta_{t}$), February 2010 - January
2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | 0.0013 | 0.0018 | 0.0049 | 0.0054 | 0.0075 | 0.0162 | 0.0078 | 0.0098 | 0.0103 | 0.0001 | 0.0003 | 0.0008
0.0005 | 0.0021 | 0.0023 | 0.0064 | 0.0121 | 0.0208 | 0.0288 | 0.0173 | 0.0211 | 0.0142 | 0.0005 | 0.0002 | 0.0027
0.001 | 0.0047 | 0.0035 | 0.0068 | 0.0169 | 0.032 | 0.0363 | 0.0273 | 0.0289 | 0.0156 | 0.001 | 0.0002 | 0.0046
0.005 | 0.0033 | 0.0087 | 0.0061 | 0.0377 | 0.0798 | 0.0649 | 0.0729 | 0.0599 | 0.0169 | 0.006 | 0.0001 | 0.0153
0.01 | 0.0071 | 0.0158 | 0.0065 | 0.0572 | 0.1179 | 0.0878 | 0.1114 | 0.0856 | 0.0175 | 0.0141 | 0.0002 | 0.0304
0.05 | 0.0125 | 0.0293 | 0.0094 | 0.1875 | 0.2956 | 0.2432 | 0.3028 | 0.2558 | 0.0205 | 0.1339 | 0.0001 | 0.1633
0.1 | 0.0325 | 0.0531 | 0.0093 | 0.3534 | 0.4047 | 0.3741 | 0.4203 | 0.3842 | 0.0249 | 0.3274 | 0.0001 | 0.3499
0.25 | 0.3613 | 0.3704 | 0.0185 | 0.5213 | 0.483 | 0.4693 | 0.494 | 0.4794 | 0.0686 | 0.4975 | 0.0002 | 0.4971
0.5 | 0.4914 | 0.4851 | 0.2757 | 0.5671 | 0.514 | 0.5018 | 0.528 | 0.515 | 0.4218 | 0.5392 | 0.0003 | 0.5392
0.75 | 0.5526 | 0.5113 | 0.474 | 0.5897 | 0.5303 | 0.5167 | 0.5467 | 0.5322 | 0.4858 | 0.5574 | 0.0002 | 0.5605
0.9 | 0.5492 | 0.5303 | 0.4913 | 0.599 | 0.5386 | 0.5242 | 0.5564 | 0.5412 | 0.5005 | 0.5696 | 0.0003 | 0.5708
0.99 | 0.5608 | 0.5398 | 0.4997 | 0.603 | 0.543 | 0.5283 | 0.5616 | 0.5458 | 0.5078 | 0.5731 | 0.0002 | 0.5764
| Panel B: Average Learning Fraction ($\theta_{t}$), February 2000 - January
2010
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | 0.0012 | 0.0037 | 0.0085 | 0.0028 | 0.0026 | 0.0106 | 0.0029 | 0.0047 | 0.0096 | 0.0001 | 0.0003 | 0.0006
0.0005 | 0.0022 | 0.0045 | 0.0098 | 0.0066 | 0.007 | 0.0186 | 0.0067 | 0.0103 | 0.0119 | 0.0005 | 0.0013 | 0.0019
0.001 | 0.0031 | 0.0055 | 0.0097 | 0.0093 | 0.0107 | 0.0231 | 0.011 | 0.014 | 0.0125 | 0.001 | 0.0017 | 0.0033
0.005 | 0.0043 | 0.0109 | 0.0086 | 0.0198 | 0.0269 | 0.0384 | 0.031 | 0.0284 | 0.0128 | 0.0055 | 0.0004 | 0.0111
0.01 | 0.0045 | 0.0163 | 0.009 | 0.0291 | 0.0406 | 0.0501 | 0.0482 | 0.0402 | 0.0129 | 0.012 | 0.0015 | 0.0203
0.05 | 0.0073 | 0.0234 | 0.0111 | 0.0948 | 0.1327 | 0.13 | 0.1539 | 0.1234 | 0.0144 | 0.0855 | 0.0012 | 0.0954
0.1 | 0.0198 | 0.0421 | 0.0107 | 0.1967 | 0.2443 | 0.2378 | 0.2688 | 0.2297 | 0.0164 | 0.2011 | 0.0005 | 0.2095
0.25 | 0.2012 | 0.2347 | 0.0159 | 0.4154 | 0.3858 | 0.3892 | 0.4029 | 0.3746 | 0.0373 | 0.3986 | 0.0004 | 0.3951
0.5 | 0.4517 | 0.4522 | 0.1622 | 0.5303 | 0.4721 | 0.4789 | 0.4928 | 0.4658 | 0.3145 | 0.505 | 0.0003 | 0.4979
0.75 | 0.5514 | 0.519 | 0.4534 | 0.5915 | 0.5224 | 0.5271 | 0.5487 | 0.5172 | 0.4771 | 0.5665 | 0.0015 | 0.5581
0.9 | 0.5904 | 0.5536 | 0.5096 | 0.6227 | 0.5494 | 0.5534 | 0.5788 | 0.5451 | 0.5238 | 0.5965 | 0.0002 | 0.5887
0.99 | 0.608 | 0.577 | 0.5385 | 0.6355 | 0.5644 | 0.5681 | 0.5953 | 0.56 | 0.5482 | 0.6039 | 0.0001 | 0.6057

Table A.9: Learning Fraction of Recursive Experiments. This shows the
arbitrageur fraction ($\theta_{t}$) that endogenously arises across
statistical arbitrageur model (columns) and for different initial arbitrageur
fraction (ILF) values in the recursive experiments. I just take the average
arbitrageur fraction for the February 2010 - January 2020 decade (Panel A) and
the February 2000 - January 2010 decade (Panel B).

| Panel A: Log GRS Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.058^{***}$ | $0.089^{***}$ | $-0.057^{***}$ | $0.157^{***}$ | $-0.037^{***}$ | $0.088^{***}$ | $0.059^{***}$ | $-0.032^{***}$ | $-0.014^{***}$
0.0005 | $0.053^{***}$ | $0.124^{***}$ | $-0.035^{***}$ | $0.339^{***}$ | $-0.036^{***}$ | $0.119^{***}$ | $0.172^{***}$ | $0.007^{***}$ | $-0.023^{***}$
0.001 | $-0.099^{***}$ | $0.124^{***}$ | $-0.016^{***}$ | $0.425^{***}$ | $-0.036^{***}$ | $0.134^{***}$ | $0.238^{***}$ | $0.314^{***}$ | $-0.026^{***}$
0.005 | $-0.048^{***}$ | $0.137^{***}$ | $0.031^{***}$ | $0.624^{***}$ | $-0.042^{***}$ | $0.136^{***}$ | $0.414^{***}$ | $0.034^{***}$ | $-0.048^{***}$
0.01 | $-0.13^{***}$ | $0.205^{***}$ | $0.052^{***}$ | $0.666^{***}$ | $-0.06^{***}$ | $0.144^{***}$ | $0.47^{***}$ | $0.068^{***}$ | $-0.06^{***}$
0.05 | $-0.033^{***}$ | $0.295^{***}$ | $0.125^{***}$ | $0.866^{***}$ | $-0.125^{***}$ | $0.199^{***}$ | $0.521^{***}$ | $0.16^{***}$ | $-0.091^{***}$
0.1 | $0.017^{***}$ | $0.371^{***}$ | $0.129^{***}$ | $0.937^{***}$ | $-0.011^{***}$ | $0.238^{***}$ | $0.577^{***}$ | $0.318^{***}$ | $-0.055^{***}$
0.25 | $0.064^{***}$ | $0.54^{***}$ | $0.198^{***}$ | $1.095^{***}$ | $0.006^{***}$ | $0.397^{***}$ | $0.642^{***}$ | $0.242^{***}$ | $-0.034^{***}$
0.5 | $-0.128^{***}$ | $0.652^{***}$ | $0.701^{***}$ | $1.154^{***}$ | $0.012^{***}$ | $0.81^{***}$ | $0.666^{***}$ | $0.126^{***}$ | $-0.026^{***}$
0.75 | $-0.056^{***}$ | $0.658^{***}$ | $0.626^{***}$ | $1.013^{***}$ | $0.017^{***}$ | $0.834^{***}$ | $0.678^{***}$ | $0.062^{***}$ | $-0.028^{***}$
0.9 | $0.034^{***}$ | $0.633^{***}$ | $0.631^{***}$ | $0.942^{***}$ | $0.019^{***}$ | $0.835^{***}$ | $0.683^{***}$ | $0.054^{***}$ | $-0.026^{***}$
0.99 | $-0.041^{***}$ | $0.578^{***}$ | $0.635^{***}$ | $0.977^{***}$ | $0.02^{***}$ | $0.837^{***}$ | $0.684^{***}$ | $0.046^{***}$ | $-0.026^{***}$
| Panel B: Log GRS Ratio, February 2000 - January 2010
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.002^{***}$ | $0.137^{***}$ | $0.314^{***}$ | $0.006^{***}$ | $0.009^{***}$ | $0.602^{***}$ | $-0.004^{***}$ | $-0.009^{***}$ | $0^{***}$
0.0005 | $-0.032^{***}$ | $0.241^{***}$ | $0.328^{***}$ | $0.035^{***}$ | $0.001^{***}$ | $0.629^{***}$ | $-0.011^{***}$ | $0.014^{***}$ | $0^{***}$
0.001 | $-0.042^{***}$ | $0.294^{***}$ | $0.31^{***}$ | $0.085^{***}$ | $0.001^{***}$ | $0.643^{***}$ | $-0.012^{***}$ | $-0.017^{***}$ | $-0.001^{***}$
0.005 | $-0.047^{***}$ | $0.273^{***}$ | $0.164^{***}$ | $0.193^{***}$ | $0.005^{***}$ | $0.494^{***}$ | $0.005^{***}$ | $0.02^{***}$ | $-0.005^{***}$
0.01 | $-0.124^{***}$ | $0.309^{***}$ | $0.177^{***}$ | $0.256^{***}$ | $0.014^{***}$ | $0.471^{***}$ | $0.021^{***}$ | $-0.029^{***}$ | $-0.005^{***}$
0.05 | $-0.016^{***}$ | $0.278^{***}$ | $0.209^{***}$ | $0.414^{***}$ | $0.011^{***}$ | $0.459^{***}$ | $0.065^{***}$ | $-0.044^{***}$ | $-0.014^{***}$
0.1 | $0.011^{***}$ | $0.254^{***}$ | $0.17^{***}$ | $0.536^{***}$ | $0.009^{***}$ | $0.441^{***}$ | $0.087^{***}$ | $-0.039^{***}$ | $-0.02^{***}$
0.25 | $-0.151^{***}$ | $0.251^{***}$ | $0.163^{***}$ | $0.621^{***}$ | $0.006^{***}$ | $0.475^{***}$ | $0.106^{***}$ | $-0.021^{***}$ | $-0.031^{***}$
0.5 | $0.019^{***}$ | $0.239^{***}$ | $0.015^{***}$ | $0.653^{***}$ | $0.005^{***}$ | $0.304^{***}$ | $0.116^{***}$ | $-0.016^{***}$ | $-0.036^{***}$
0.75 | $0.368^{***}$ | $0.147^{***}$ | $0.096^{***}$ | $0.673^{***}$ | $0^{***}$ | $0.383^{***}$ | $0.12^{***}$ | $-0.008^{***}$ | $-0.038^{***}$
0.9 | $0.996^{***}$ | $-0.004^{***}$ | $0.101^{***}$ | $0.648^{***}$ | $0^{***}$ | $0.396^{***}$ | $0.122^{***}$ | $-0.013^{***}$ | $-0.039^{***}$
0.99 | $-0.041^{***}$ | $-0.013^{***}$ | $0.104^{***}$ | $0.599^{***}$ | $0^{***}$ | $0.402^{***}$ | $0.123^{***}$ | $-0.006^{***}$ | $-0.04^{***}$
|  Note: ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

Table A.10: Recursive GRS Without Market. This is similar to Table A.1, except
this shows the counterfactual experiment results for statistical arbitrageurs
without a level/market term.

| Panel A: Log Fixed Cov. Mispricing Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.006$ | $0.148$ | $0.332$ | $0.109$ | $0.057$ | $0.117$ | $0.002$ | $-0.001$ | $0.395$ | $0.$ | $0.069$ | $0.001$
0.0005 | $0.001$ | $0.185$ | $0.37$ | $0.27$ | $0.086$ | $0.193$ | $0.001$ | $0.016$ | $0.414$ | $0.001$ | $-0.078$ | $0.005$
0.001 | $-0.123$ | $0.323$ | $0.385$ | $0.338$ | $0.092$ | $0.212$ | $-0.003$ | $0.015$ | $0.428$ | $0.003$ | $-0.074$ | $0.008$
0.005 | $-0.117$ | $0.617$ | $0.364$ | $0.518$ | $0.081$ | $0.232$ | $-0.035$ | $0.002$ | $0.441$ | $0.011$ | $-0.058$ | $0.012$
0.01 | $-0.14$ | $0.611$ | $0.396$ | $0.32$ | $0.065$ | $0.233$ | $-0.06$ | $-0.005$ | $0.399$ | $0.014$ | $-0.045$ | $0.002$
0.05 | $-0.02$ | $0.84$ | $0.523$ | $0.36$ | $0.014$ | $0.225$ | $-0.121$ | $-0.038$ | $0.351$ | $-0.091$ | $-0.029$ | $-0.09$
0.1 | $-0.121$ | $0.997$ | $0.513$ | $0.32$ | $0$ | $0.226$ | $-0.135$ | $-0.041$ | $0.347$ | $-0.095$ | $-0.04$ | $-0.126$
0.25 | $-0.177$ | $0.012$ | $0.647$ | $0.383$ | $-0.027$ | $0.223$ | $-0.158$ | $-0.047$ | $0.376$ | $-0.141$ | $-0.006$ | $-0.149$
0.5 | $-0.36$ | $0.266$ | $0.753$ | $0.426$ | $-0.044$ | $0.221$ | $-0.168$ | $-0.053$ | $0.436$ | $-0.201$ | $0.007$ | $-0.158$
0.75 | $-0.109$ | $0.654$ | $0.757$ | $0.363$ | $-0.051$ | $0.222$ | $-0.171$ | $-0.055$ | $0.467$ | $-0.214$ | $-0.016$ | $-0.16$
0.9 | $-0.279$ | $0.069$ | $0.757$ | $0.445$ | $-0.053$ | $0.222$ | $-0.171$ | $-0.056$ | $0.478$ | $-0.223$ | $0.031$ | $-0.159$
0.99 | $-0.2$ | $0.634$ | $0.758$ | $0.447$ | $-0.054$ | $0.223$ | $-0.172$ | $-0.056$ | $0.483$ | $-0.202$ | $-0.016$ | $-0.162$
| Panel B: Log Fixed Cov. Mispricing Ratio, February 2000 - January 2010
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.06$ | $0.111$ | $0.048$ | $-0.178$ | $-0.085$ | $-0.081$ | $0.001$ | $-0.024$ | $0.046$ | $0$ | $0.036$ | $-0.009$
0.0005 | $-0.103$ | $0.04$ | $0.132$ | $-0.296$ | $-0.151$ | $-0.117$ | $0$ | $-0.049$ | $0.078$ | $-0.001$ | $0.95$ | $-0.025$
0.001 | $-0.092$ | $-0.047$ | $0.143$ | $-0.354$ | $-0.184$ | $-0.129$ | $-0.002$ | $-0.052$ | $0.1$ | $-0.002$ | $1.468$ | $-0.034$
0.005 | $-0.137$ | $0.061$ | $0.103$ | $-0.442$ | $-0.233$ | $-0.122$ | $-0.011$ | $-0.043$ | $0.051$ | $-0.007$ | $-0.047$ | $-0.058$
0.01 | $-0.131$ | $0.124$ | $0.126$ | $-0.446$ | $-0.243$ | $-0.114$ | $-0.021$ | $-0.039$ | $0.037$ | $-0.012$ | $-0.005$ | $-0.078$
0.05 | $-0.091$ | $0.194$ | $0.147$ | $-0.421$ | $-0.265$ | $-0.112$ | $-0.055$ | $-0.048$ | $0.042$ | $-0.043$ | $0.055$ | $-0.116$
0.1 | $-0.08$ | $0.255$ | $0.095$ | $-0.46$ | $-0.277$ | $-0.12$ | $-0.067$ | $-0.06$ | $0.034$ | $-0.09$ | $0.054$ | $-0.146$
0.25 | $-0.101$ | $0.289$ | $0.055$ | $-0.405$ | $-0.288$ | $-0.13$ | $-0.074$ | $-0.076$ | $-0.01$ | $-0.253$ | $0.055$ | $-0.161$
0.5 | $-0.244$ | $0.291$ | $0.057$ | $-0.467$ | $-0.291$ | $-0.133$ | $-0.075$ | $-0.084$ | $0.03$ | $-0.295$ | $0.057$ | $-0.152$
0.75 | $-0.213$ | $0.332$ | $0.082$ | $-0.461$ | $-0.293$ | $-0.134$ | $-0.076$ | $-0.088$ | $0.073$ | $-0.357$ | $0.949$ | $-0.152$
0.9 | $-0.22$ | $0.351$ | $0.087$ | $-0.459$ | $-0.294$ | $-0.135$ | $-0.076$ | $-0.089$ | $0.089$ | $-0.278$ | $0.072$ | $-0.16$
0.99 | $-0.2$ | $0.334$ | $0.09$ | $-0.458$ | $-0.295$ | $-0.135$ | $-0.076$ | $-0.09$ | $0.096$ | $-0.344$ | $0.107$ | $-0.165$

Table A.11: Recursive Learning Experiments Fixed Covariance alpha Measure.
This is similar to Table A.6, except these are the results for the recursive
experiments rather than the simple experiments.

| Panel A: Log Fixed Cov. Mispricing Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.051$ | $-0.099$ | $0.163$ | $0.089$ | $0.002$ | $0.188$ | $0.019$ | $-0.014$ | $0.002$
0.0005 | $-0.085$ | $-0.069$ | $0.191$ | $0.239$ | $0.001$ | $0.217$ | $0.047$ | $-0.039$ | $0.006$
0.001 | $-0.082$ | $-0.055$ | $0.206$ | $0.324$ | $0.001$ | $0.241$ | $0.069$ | $-0.096$ | $0.008$
0.005 | $-0.081$ | $-0.048$ | $0.192$ | $0.506$ | $-0.011$ | $0.213$ | $0.15$ | $-0.027$ | $0.01$
0.01 | $-0.017$ | $-0.075$ | $0.219$ | $0.335$ | $-0.06$ | $0.184$ | $0.185$ | $-0.018$ | $0$
0.05 | $-0.032$ | $-0.032$ | $0.324$ | $0.293$ | $-0.158$ | $0.222$ | $0.19$ | $-0.019$ | $-0.1$
0.1 | $-0.214$ | $0.04$ | $0.288$ | $0.326$ | $-0.135$ | $0.213$ | $0.186$ | $-0.036$ | $-0.136$
0.25 | $-0.134$ | $0.072$ | $0.419$ | $0.366$ | $-0.159$ | $0.408$ | $0.19$ | $-0.026$ | $-0.149$
0.5 | $-0.133$ | $0.033$ | $0.682$ | $0.451$ | $-0.169$ | $0.541$ | $0.189$ | $-0.016$ | $-0.158$
0.75 | $1.748$ | $0.02$ | $0.69$ | $0.33$ | $-0.171$ | $0.581$ | $0.191$ | $-0.019$ | $-0.171$
0.9 | $5.956$ | $0.011$ | $0.697$ | $0.313$ | $-0.171$ | $0.595$ | $0.192$ | $-0.019$ | $-0.172$
0.99 | $-0.17$ | $0.018$ | $0.699$ | $0.336$ | $-0.172$ | $0.6$ | $0.192$ | $-0.014$ | $-0.172$
| Panel B: Log Fixed Cov. Mispricing Ratio, February 2000 - January 2010
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | GKX | KNS | KPS | NN | RF
0.0001 | $-0.095$ | $-0.083$ | $0.079$ | $-0.169$ | $0.001$ | $0.032$ | $-0.037$ | $-0.207$ | $-0.008$
0.0005 | $-0.132$ | $-0.014$ | $0.179$ | $-0.288$ | $-0.003$ | $0.104$ | $-0.091$ | $-0.232$ | $-0.02$
0.001 | $-0.155$ | $0.035$ | $0.194$ | $-0.347$ | $-0.005$ | $0.139$ | $-0.123$ | $-0.07$ | $-0.027$
0.005 | $-0.146$ | $0.11$ | $0.144$ | $-0.437$ | $-0.02$ | $0.097$ | $-0.207$ | $-0.293$ | $-0.054$
0.01 | $-0.181$ | $0.121$ | $0.162$ | $-0.442$ | $-0.021$ | $0.093$ | $-0.244$ | $-0.24$ | $-0.068$
0.05 | $-0.058$ | $0.078$ | $0.175$ | $-0.469$ | $-0.056$ | $0.121$ | $-0.315$ | $-0.235$ | $-0.112$
0.1 | $-0.085$ | $0.036$ | $0.12$ | $-0.468$ | $-0.066$ | $0.113$ | $-0.327$ | $-0.232$ | $-0.135$
0.25 | $-0.113$ | $-0.102$ | $0.097$ | $-0.468$ | $-0.074$ | $0.106$ | $-0.327$ | $-0.198$ | $-0.167$
0.5 | $-0.224$ | $-0.172$ | $0.049$ | $-0.462$ | $-0.075$ | $0.119$ | $-0.325$ | $-0.238$ | $-0.17$
0.75 | $0.356$ | $-0.187$ | $0.086$ | $-0.441$ | $-0.076$ | $0.158$ | $-0.324$ | $-0.216$ | $-0.172$
0.9 | $0.475$ | $-0.223$ | $0.081$ | $-0.448$ | $-0.076$ | $0.167$ | $-0.324$ | $-0.212$ | $-0.173$
0.99 | $4.503$ | $-0.215$ | $0.079$ | $-0.454$ | $-0.076$ | $0.172$ | $-0.323$ | $-0.212$ | $-0.173$

Table A.12: Recursive Fixed Covariance alpha Measure Without Market. This
table is similar to Table A.11, except this shows the counterfactual
experiment results for statistical arbitrageurs without a level/market term.

| Panel A: Log GRS Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.122^{***}$ | $0.335^{***}$ | $0.068^{***}$ | $0.173^{***}$ | $-0.266^{***}$ | $0.833^{***}$ | $-0.028^{***}$ | $-0.254^{***}$ | $0.397^{***}$ | $0.^{***}$ | $0.052^{***}$ | $-0.011^{***}$
0.0005 | $0.058^{***}$ | $0.488^{***}$ | $0.106^{***}$ | $0.348^{***}$ | $-0.25^{***}$ | $1.003^{***}$ | $-0.022^{***}$ | $-0.229^{***}$ | $0.428^{***}$ | $-0.001^{***}$ | $-0.083^{***}$ | $-0.016^{***}$
0.001 | $-0.086^{***}$ | $0.597^{***}$ | $0.12^{***}$ | $0.386^{***}$ | $-0.232^{***}$ | $1.06^{***}$ | $-0.02^{***}$ | $-0.212^{***}$ | $0.44^{***}$ | $0.^{***}$ | $0.082^{***}$ | $-0.017^{***}$
0.005 | $-0.129^{***}$ | $0.663^{***}$ | $0.147^{***}$ | $0.539^{***}$ | $-0.179^{***}$ | $1.182^{***}$ | $-0.016^{***}$ | $-0.167^{***}$ | $0.436^{***}$ | $0.^{***}$ | $-0.033^{***}$ | $-0.016^{***}$
0.01 | $0.171^{***}$ | $0.521^{***}$ | $0.189^{***}$ | $0.619^{***}$ | $-0.156^{***}$ | $1.241^{***}$ | $-0.014^{***}$ | $-0.143^{***}$ | $0.408^{***}$ | $0.001^{***}$ | $-0.045^{***}$ | $-0.02^{***}$
0.05 | $-0.061^{***}$ | $1.564^{***}$ | $0.308^{***}$ | $0.803^{***}$ | $-0.149^{***}$ | $1.461^{***}$ | $-0.012^{***}$ | $-0.085^{***}$ | $0.4^{***}$ | $0.001^{***}$ | $0.06^{***}$ | $-0.032^{***}$
0.1 | $-0.038^{***}$ | $0.637^{***}$ | $0.309^{***}$ | $0.888^{***}$ | $-0.167^{***}$ | $1.606^{***}$ | $-0.011^{***}$ | $-0.08^{***}$ | $0.424^{***}$ | $-0.083^{***}$ | $0.016^{***}$ | $-0.029^{***}$
0.25 | $0.088^{***}$ | $0.187^{***}$ | $0.389^{***}$ | $1.068^{***}$ | $-2.033$ | $-2.26$ | $-0.011^{***}$ | $-0.368^{***}$ | $0.413^{***}$ | $0.003^{***}$ | $-0.001^{***}$ | $-0.027^{***}$
0.5 | $0.098^{***}$ | $0.733^{***}$ | $0.573^{***}$ | $1.151^{***}$ | $-2.044$ | $-1.991^{*}$ | $-0.011^{***}$ | $-1.99^{*}$ | $0.374^{***}$ | $0^{***}$ | $-0.014^{***}$ | $-0.035^{***}$
0.75 | $0.213^{***}$ | $-0.166^{***}$ | $0.667^{***}$ | $1.136^{***}$ | $-1.9^{**}$ | $-2.051$ | $-0.011^{***}$ | $-0.17^{***}$ | $0.406^{***}$ | $0.003^{***}$ | $0^{***}$ | $-0.026^{***}$
0.9 | $0.134^{***}$ | $-0.073^{***}$ | $0.696^{***}$ | $1.011^{***}$ | $-2.039$ | $-2.051$ | $-0.011^{***}$ | $-2.078$ | $-0.332^{***}$ | $-0.001^{***}$ | $0^{***}$ | $-0.027^{***}$
0.99 | $0.013^{***}$ | $0.687^{***}$ | $-0.038^{***}$ | $1.013^{***}$ | $-2.047$ | $-2.082$ | $-0.011^{***}$ | $-1.968^{*}$ | $-2.273$ | $-0.002^{***}$ | $0^{***}$ | $-0.029^{***}$
|  Note: ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01
| Panel B: Log Fixed Cov. Mispricing, February 2010 - January 2020
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.007$ | $0.125$ | $0.287$ | $0.077$ | $0.041$ | $0.055$ | $-0.006$ | $-0.025$ | $0.344$ | $0.$ | $0.055$ | $0$
0.0005 | $-0.06$ | $0.189$ | $0.314$ | $0.196$ | $0.057$ | $0.113$ | $-0.01$ | $-0.024$ | $0.356$ | $0.$ | $-0.104$ | $-0.001$
0.001 | $-0.043$ | $0.288$ | $0.328$ | $0.247$ | $0.058$ | $0.128$ | $-0.013$ | $-0.031$ | $0.37$ | $0$ | $-0.076$ | $-0.002$
0.005 | $0.$ | $0.591$ | $0.311$ | $0.375$ | $0.035$ | $0.147$ | $-0.023$ | $-0.055$ | $0.381$ | $-0.002$ | $-0.059$ | $-0.009$
0.01 | $-0.062$ | $0.376$ | $0.338$ | $0.309$ | $0.013$ | $0.153$ | $-0.03$ | $-0.067$ | $0.338$ | $-0.005$ | $-0.044$ | $-0.017$
0.05 | $-0.102$ | $1.003$ | $0.451$ | $0.246$ | $-0.099$ | $0.19$ | $-0.07$ | $-0.13$ | $0.291$ | $-0.039$ | $-0.025$ | $-0.053$
0.1 | $-0.078$ | $0.497$ | $0.44$ | $0.33$ | $-0.131$ | $0.296$ | $-0.098$ | $-0.146$ | $0.284$ | $-0.077$ | $-0.028$ | $-0.079$
0.25 | $-0.255$ | $0.092$ | $0.543$ | $0.379$ | $45.874$ | $42.723$ | $-0.129$ | $-0.588$ | $0.3$ | $-0.105$ | $-0.003$ | $-0.11$
0.5 | $-0.181$ | $1.691$ | $0.636$ | $0.41$ | $43.701$ | $43.976$ | $-0.141$ | $48.358$ | $0.329$ | $-0.119$ | $0.007$ | $-0.129$
0.75 | $-0.333$ | $0.826$ | $0.682$ | $0.45$ | $45.484$ | $44.473$ | $-0.145$ | $-0.133$ | $0.352$ | $-0.122$ | $0.003$ | $-0.128$
0.9 | $-0.222$ | $1.507$ | $0.771$ | $0.43$ | $45.32$ | $43.958$ | $-0.146$ | $48.249$ | $-0.526$ | $-0.122$ | $0.003$ | $-0.131$
0.99 | $-0.215$ | $1.882$ | $-0.002$ | $0.432$ | $45.108$ | $48.318$ | $-0.147$ | $44.294$ | $46.031$ | $-0.122$ | $0.003$ | $-0.132$

Table A.13: Recursive Learning Experiments with Fixed Incumbent Capital. This
table shows results for experiments where the incumbent institutions have
$A_{t}$ dollars of capital to manage instead of $(1-\theta_{t})A_{t}$ dollars.
In other words, statistical arbitrageurs raise funds to manage capital from
outside of the stock market. Panel A shows the log ratio of the GRS test $F$
statistics of the counterfactual experiment anomaly portfolio outcomes to the
baseline GRS test statistic. Panel B shows a similar log GRS ratio, except
that the anomalies are weighted with the inverse of the residual covariance
matrix from the pre-arbitrageur period (January 1980 - December 1989). This
latter alpha measure is not a statistical test. Positive values implies that
alpha increased relative to baseline, while negative values implies that alpha
decreased. Small values (close to zero) are approximately percentage changes.
ILF ($\theta_{0}$) represents the initial arbitrageur fraction used in the
counterfactual experiments (the arbitrageur fraction as of January 1990 when
the statistical arbitrageur starts managing capital counterfactually). Each
column represents a different statistical arbitrageur model, where Table 2
describes the initialisms of these models. Stars are given in Panel A to
represent whether the alpha remaining counterfactually is different from zero
(alpha remained statistically significant), not whether baseline and
counterfactual alpha are statistically different from one another. The
critical values for the log GRS ratio are $-2.008$, $-1.937$, and $-1.805$ for
10%, 5%, and 1% significance respectively for the 2010 - 2020 decade.

| Panel A: Log GRS Ratio, February 2010 - January 2020
---|---
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.029^{***}$ | $0.134^{***}$ | $-0.27^{***}$ | $-0.05^{***}$ | $-0.202^{***}$ | $0.383^{***}$ | $-0.025^{***}$ | $-0.159^{***}$ | $-0.078^{***}$ | $-0.001^{***}$ | $0.22^{***}$ | $-0.009^{***}$
0.0005 | $0.043^{***}$ | $0.313^{***}$ | $-0.269^{***}$ | $0.14^{***}$ | $-0.191^{***}$ | $0.569^{***}$ | $-0.026^{***}$ | $-0.138^{***}$ | $-0.069^{***}$ | $0.006^{***}$ | $0.049^{***}$ | $-0.016^{***}$
0.001 | $-0.09^{***}$ | $0.511^{***}$ | $-0.289^{***}$ | $0.211^{***}$ | $-0.171^{***}$ | $0.633^{***}$ | $-0.028^{***}$ | $-0.122^{***}$ | $-0.083^{***}$ | $0.006^{***}$ | $0.273^{***}$ | $-0.025^{***}$
0.005 | $0.027^{***}$ | $0.746^{***}$ | $-0.298^{***}$ | $0.384^{***}$ | $-0.118^{***}$ | $0.745^{***}$ | $-0.041^{***}$ | $-0.078^{***}$ | $-0.15^{***}$ | $-0.049^{***}$ | $-0.154^{***}$ | $-0.043^{***}$
0.01 | $0.029^{***}$ | $0.629^{***}$ | $-0.294^{***}$ | $0.501^{***}$ | $-0.099^{***}$ | $0.782^{***}$ | $-0.047^{***}$ | $-0.057^{***}$ | $-0.168^{***}$ | $-0.01^{***}$ | $-0.091^{***}$ | $-0.05^{***}$
0.05 | $0.053^{***}$ | $0.895^{***}$ | $-0.293^{***}$ | $0.718^{***}$ | $-0.06^{***}$ | $0.844^{***}$ | $-0.044^{***}$ | $-0.011^{***}$ | $-0.154^{***}$ | $-0.343^{***}$ | $-0.033^{***}$ | $-0.089^{***}$
0.1 | $0.135^{***}$ | $1.^{***}$ | $-0.297^{***}$ | $0.843^{***}$ | $-0.042^{***}$ | $0.863^{***}$ | $-0.03^{***}$ | $0.008^{***}$ | $-0.129^{***}$ | $0.365^{***}$ | $-0.093^{***}$ | $-0.063^{***}$
0.25 | $-0.121^{***}$ | $1.014^{***}$ | $-0.306^{***}$ | $0.923^{***}$ | $-0.031^{***}$ | $0.877^{***}$ | $-0.022^{***}$ | $0.021^{***}$ | $-0.015^{***}$ | $0.196^{***}$ | $-0.093^{***}$ | $-0.049^{***}$
0.5 | $0.067^{***}$ | $1.352^{***}$ | $0.119^{***}$ | $0.942^{***}$ | $-0.027^{***}$ | $0.886^{***}$ | $-0.019^{***}$ | $0.025^{***}$ | $0.756^{***}$ | $0.309^{***}$ | $-0.011^{***}$ | $-0.044^{***}$
0.75 | $0.151^{***}$ | $0.584^{***}$ | $0.511^{***}$ | $0.91^{***}$ | $-0.025^{***}$ | $0.891^{***}$ | $-0.017^{***}$ | $0.027^{***}$ | $0.686^{***}$ | $0.683^{***}$ | $-0.093^{***}$ | $-0.061^{***}$
0.9 | $0.056^{***}$ | $0.724^{***}$ | $0.496^{***}$ | $0.99^{***}$ | $-0.024^{***}$ | $0.893^{***}$ | $-0.016^{***}$ | $0.028^{***}$ | $0.681^{***}$ | $0.574^{***}$ | $-0.06^{***}$ | $-0.057^{***}$
0.99 | $-0.04^{***}$ | $0.917^{***}$ | $0.49^{***}$ | $1.011^{***}$ | $-0.023^{***}$ | $0.894^{***}$ | $-0.015^{***}$ | $0.028^{***}$ | $0.679^{***}$ | $0.733^{***}$ | $-0.149^{***}$ | $-0.055^{***}$
|  Note: ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01
| Panel B: Log Fixed Cov. Mispricing, February 2010 - January 2020
ILF ($\theta_{0}$) | BPZF | BPZL | BSV | DGU | FF3 | FF6 | GKX | HXZ | KNS | KPS | NN | RF
0.0001 | $-0.172$ | $0.083$ | $0.288$ | $0.062$ | $-0.012$ | $0.258$ | $0.002$ | $0.006$ | $0.413$ | $0.002$ | $-0.338$ | $0.003$
0.0005 | $-0.053$ | $0.137$ | $0.293$ | $0.111$ | $0.018$ | $0.283$ | $0.005$ | $0.022$ | $0.45$ | $0.004$ | $-0.267$ | $0.012$
0.001 | $-0.065$ | $0.301$ | $0.293$ | $0.134$ | $0.026$ | $0.289$ | $0$ | $0.021$ | $0.455$ | $0.006$ | $-0.271$ | $0.018$
0.005 | $-0.366$ | $0.368$ | $0.28$ | $0.142$ | $0.015$ | $0.28$ | $-0.043$ | $0.008$ | $0.438$ | $0.034$ | $-0.467$ | $0.043$
0.01 | $0.$ | $0.371$ | $0.28$ | $0.128$ | $-0.001$ | $0.267$ | $-0.07$ | $0.001$ | $0.43$ | $0.067$ | $-0.595$ | $0.044$
0.05 | $-0.072$ | $0.397$ | $0.299$ | $-0.044$ | $-0.028$ | $0.225$ | $-0.089$ | $-0.016$ | $0.428$ | $0.011$ | $-0.014$ | $-0.076$
0.1 | $-0.653$ | $0.438$ | $0.296$ | $-0.067$ | $-0.02$ | $0.234$ | $-0.061$ | $0.012$ | $0.44$ | $0.281$ | $-0.036$ | $-0.072$
0.25 | $-0.394$ | $0.238$ | $0.352$ | $-0.067$ | $-0.029$ | $0.238$ | $-0.054$ | $0.03$ | $0.519$ | $0.165$ | $-0.164$ | $-0.035$
0.5 | $-0.22$ | $0.112$ | $0.456$ | $-0.076$ | $-0.037$ | $0.234$ | $-0.051$ | $0.032$ | $0.547$ | $0.332$ | $-0.02$ | $-0.03$
0.75 | $11.868$ | $-0.066$ | $0.139$ | $-0.083$ | $-0.04$ | $0.233$ | $-0.048$ | $0.033$ | $0.504$ | $0.579$ | $-0.106$ | $-0.035$
0.9 | $-0.047$ | $-0.017$ | $0.11$ | $-0.07$ | $-0.04$ | $0.233$ | $-0.047$ | $0.034$ | $0.496$ | $0.542$ | $0.001$ | $-0.028$
0.99 | $-0.318$ | $0.105$ | $0.098$ | $-0.063$ | $-0.04$ | $0.233$ | $-0.046$ | $0.035$ | $0.493$ | $0.528$ | $5.718$ | $-0.026$

Table A.14: Recursive Learning Experiments with Exact Learning This table is
similar to Table A.13, except this table shows the results for recursive
experiments where instead of fixed incumbent capital, $Z_{t}$ is filled with
$\dot{z}_{i,k,t}$ values rather than the $\underaccent{\dot}{z}_{i,k,t}$
values. This means that the linear statistical arbitrageur models—BPZL, BSV,
DGU, FF3, FF6, HXZ, and KNS models—are all exact demand functions from the
models instead of demand functions that come from first order approximations
of their portfolio weights.