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# On statistical arbitrage under a conditional factor model of equity returns.

Trent Spears , Stefan Zohren, Stephen Roberts University of Oxford, Oxford-Man
Institute of Quantitative Finance. Corresponding author. E-mail:
<EMAIL_ADDRESS>
###### Summary

We consider a conditional factor model for a multivariate portfolio of United
States equities in the context of analysing a statistical arbitrage trading
strategy. A state space framework underlies the factor model whereby asset
returns are assumed to be a noisy observation of a linear combination of
factor values and latent factor risk premia. Filter and state prediction
estimates for the risk premia are retrieved in an online way. Such estimates
induce filtered asset returns that can be compared to measurement
observations, with large deviations representing candidate mean reversion
trades. Further, in that the risk premia are modelled as time-varying
quantities, non-stationarity in returns is de facto captured. We study an
empirical trading strategy respectful of transaction costs, and demonstrate
performance over a long history of 29 years, for both a linear and a non-
linear state space model. Our results show that the model is competitive
relative to the results of other methods, including simple benchmarks and
other cutting-edge approaches as published in the literature. Also of note,
while strategy performance degradation is noticed through time – especially
for the most recent years – the strategy continues to offer compelling
economics, and has scope for further advancement.

Keywords: Statistical arbitrage, state space models, dynamic factor models,
machine learning.

##  1 Introduction

Statistical arbitrageurs seek trading opportunities whereby an asset price (or
return) is dislocated from a measure of fair value to which it is assumed it
will revert. Such dislocation needs to be sufficiently large so as to be
profitably traded – at least, in expectation. These strategies are documented
as originating within the financial services in the late 1970s and early
1980s, within both the hedge fund and investment banking industries [1]. To
this day, statistical arbitrage strategies are of popular practical
application, and strategy techniques and insights have inspired a deep
academic literature. A comprehensive review, for work dated up till 2016, can
be found in [2].

Indeed, much of the academic literature on statistical arbitrage ultimately
concerns estimating a fair value model for an asset spread or portfolio,
and/or modelling and analysing the stochastic dynamics of the dislocation
about fair value. This paper is mostly concerned with the former. To our
knowledge, we are the first authors to present and assess a conditional factor
model for estimating fair value in the context of statistical arbitrage. The
model assumes that future returns are predictable given (i) the asset-wise
factor exposures for the universe of assets of interest, and (ii) a latent
factor vector that is estimable by regression in the asset cross-section.
Further, we assume the dynamic nature of the factor vector, suggesting a state
space framework for the joint modelling of factors and asset returns.

Our fair value model is closest in spirit to existing work that presents
explanatory models of returns based on Principal Component Analysis (PCA) [3,
4]. In these papers, the principal components (and their loadings) that are
deemed to have sufficiently high explanatory power are an analogue to our
model’s factor exposures (and factor vectors). However the papers differ
significantly in many key workflow assumptions relative to ours, most
importantly with respect to data set construction, asset feature modelling,
and the trading strategy assumed. We posit, critically, that the details of
each of these steps can have a significant impact on the estimated performance
statistics. Hence we clearly describe our methods in the interest of ease of
interpretability and reproducibility, and present a range of performance data
based on competing sets of reasonable assumptions. Owing to transparency, we
also find two key, well-known works with which we can directly compare our
results; this is less commonly found possible for academic quantitative
finance papers that depend on strategy backtests. In any case, we further
describe these comparable papers, and other recent work in the spirit of ours,
in Section 2 below. This summary mostly covers work not contained in the
aforementioned review [2], due to their recency.

In Section 3, we present further the state space framework as it relates to
the conditional factor model. We model returns as a noisy observation of the
latent factor vectors, and describe the well-known Kalman Filter algorithm for
the case of a linear model, and the Unscented Kalman Filter algorithm for the
case the model specified is non-linear. In Section 4 we review details of the
dataset construction, and the factor dynamics model. We also discuss the
particulars of the trading strategy, that amounts to the management of a long-
short beta neutral equity portfolio. We present the specification of the
trading rule, realistic transaction cost assumptions, the hedging of market
risk, and the calculation of strategy returns and bankroll management.

Our results are presented in Section 5. We show that the conditional factor
state space model is useful for trading mid-frequency statistical arbitrage in
US equity markets. This is true with respect to a meaningful benchmark model
and a simpler ordinary least squares-based approach, and to the results given
in other well-known academic studies calculated over similar data sets. We
conclude in Section 6, and given that the model presents as a compelling
starting point for building a strategy at larger scale, we include notes for
both academic and practitioner future work.

##  2 Existing literature

We collect and describe studies relating to statistical arbitrage trading, for
work mostly post the review period of [2]. Unsurprisingly, many recent methods
for approaching the problem subsume modern machine learning techniques. These
approaches mostly concern pairs trading. For example, surrendering performance
interpretability, the work of [5] presents a spectrum of techniques –
Principal Component Analysis (PCA) for dimension reduction of raw data,
clustering as a strategy for pairs identification, and forecasting with deep
learning seeking better trade entry points. Similarly [6], but without the
lattermost step. Many authors appeal to dimension reduction when a large
universe of portfolio assets are under consideration; this often amounts to an
application of PCA. (Non-PCA based) Factor models are an alternative with a
strong theoretical grounding, and are presented (amongst many other
introductory ideas) in the well-known work of [7]. Such models are often
characterised by a collection of hand-constructed theoretically-justified
features called ‘factors’ that to correlate to price or returns time series of
the target assets. An insightful empirical demonstration of both applied PCA
and factor models can be found in [3]; also of note, the paper jointly trades
a multivariate portfolio of assets – beyond targeting pairs only. The approach
of [8] is to model returns via a statistical factor model, with an Ornstein-
Uhlenbeck process for the error term. Novelty is introduced at the level of
trade execution, formulated via a stochastic control model, though only
applied on synthetic data. Approaches to optimizing mean-reverting
multivariate portfolios, subject to either budget or leverage constraints, and
influenced by Modern Portfolio Theory, are found in [9, 10]. An approach to
multivariate statistical arbitrage modelling beyond factor models is found in
[11], that presents a framework for both linear and non-linear relationship
modelling via vine copulas.

State space models combine knowledge of the data process with noisy
measurement data; a focus of this work is to consider their utility for
statistical arbitrage applications. Hence our eventual contribution – a
filtering approach to multivariate statistical arbitrage, scalable to large
portfolios. Indeed, state space models for finance have been reviewed in [12,
13]. The articles include applications to stochastic volatility modelling, and
the modelling of the term structures of commodity prices and interest rates.
Applications to pairs trading are relevant, and were introduced by the seminal
work of Elliott [14], who models a latent pair spread, and noisy observations
thereof, in a state space setting. Their results are presented for synthetic
data. The work of [15, 16] extends the research by introducing time-varying
parameters to the model, hence improving model flexibility, and testing the
theory in realistic online trading settings. The Elliott model is also
generalised somewhat for richer spread dynamics in [17], which designs a novel
trading strategy based on model-estimated probabilities of large dislocations
mean reverting. Further generalisations are apparent in the recent article
[18] which presents a quasi-Monte Carlo estimation algorithm for the state
space framework presented. This allows for estimating models with richer
specifications, designed to express stylized features of financial data,
including non-Gaussianity and heteroskedastic variance.

It is natural to incorporate (dynamic) factor models into a state space
framework, as in [19], which also models stochastic volatility within the
state variable; similarly [20]. Arguments for dynamic factor modelling for
statistical arbitrage are given by [4], though the approach does not model
volatility (beyond a white noise process). PCA for factor mining, and
prediction (of note, at the level of asset prices, rather than returns) is a
heavily exploited feature of the trading strategy.

The studies related to statistical arbitrage are summarised in Figure 1 below,
and include information on the underlying market data analysed, by asset
class, time scale and date range. ‘SS’ denotes modelling in a state space
framework, ‘Target’ is given by ‘pair’ for pair-trading examples, and ‘MV’ for
multi-variate portfolios, while ‘Dim. red.’ denotes a dimension reduction
technique is a feature of the modelling approach.

Author (Year) | SS | Target | Dim. red. | Asset class | Date range | Time scale
---|---|---|---|---|---|---
[2] (2017) |  | pair |  | - | - | -
[5] (2020) |  | pair | ✓ | Commodities | 2009–2019 | intraday – 5min
[6] (2023) |  | MV | ✓ | Equities – Global | 2011–2021 | daily
[7] (2004) |  | pair | ✓ | - | - | -
[3] (2010) |  | MV | ✓ | Equities – US | 1997–2007 | daily
[8] (2019) |  | MV | ✓ | - | - | -
[9] (2018) |  | MV |  | Equities – US | 2012–2014 | daily
[10] (2019) |  | MV |  | Equities – US | 2010–2014 | daily
[11] (2016) |  | MV |  | Equities – US | 1992–2015 | daily
[14] (2005) | ✓ | pair |  | - | - | -
[15] (2011) | ✓ | pair |  | Equities – US, Commodities | 1980–2008 | daily
[16] (2010) | ✓ | pair |  | Equities – US | 1997–2005 | daily
[17] (2016) | ✓ | pair |  | Equities – US, Brazil | 2011–2013 | daily
[18] (2021) | ✓ | pair |  | Equities – US, Taiwan, Hong Kong | 2012–2019 | daily
[4] (2016) | ✓ | MV | ✓ | Equities – US | 1989–2011 | daily

Figure 1: Literature themed around statistical arbitrage, toward applications
of state space models for multivariate portfolios.

##  3 A dynamic model of returns and factors

_State space modelling_. Consider the case of discrete time increments
$k=0,1,...$. Then a linear Gaussian state space model is given by the time-
varying system of equations:

$\displaystyle\bm{x}_{k+1}$
$\displaystyle=\bm{A}_{k}\bm{x}_{k}+\bm{\epsilon}^{\bm{x}}_{k},$ (1)
$\displaystyle\bm{y}_{k}$
$\displaystyle=\bm{B}_{k}\bm{x}_{k}+\bm{\epsilon}^{\bm{y}}_{k}.$ (2)

Here $\bm{\epsilon}^{\bm{x}}_{k}$ and $\bm{\epsilon}^{\bm{y}}_{k}$ model
independent zero-mean white-noise processes with respective covariances
$\bm{\Psi}_{k}$ and $\bm{\Omega}_{k}$, assumed to be known functions of time.
The matrices $\bm{A}_{k}$ and $\bm{B}_{k}$ are also assumed to be known
functions of time. We call equations (1) and (2) the ‘state’ and ‘measurement’
equations, respectively. The state $\bm{x}_{k}$ is assumed to be a latent
random variable, of which $\bm{y}_{k}$ denotes a noisy measurement at time
$k$. Given the system parameters, probabilistic beliefs about the state can be
inferred, and these beliefs can be updated given measurement data.

_Kalman filter_. By the classic Kalman filter (KF) algorithm, given an initial
state $\bm{x}_{0}$, and the state and measurement equations above, the filter
distribution for $\bm{x}_{k}|\bm{y}_{1:k}$ at time $k>0$ can be shown to be
$\thicksim MVN(\bm{\mu}_{k}^{f},\bm{\Sigma}_{k}^{f})$ where

$\displaystyle\bm{\mu}_{k}^{f}$
$\displaystyle=\bm{\mu}_{k}^{p}+\bm{G}_{k}(\bm{y}_{k}-\bm{B}_{k}\bm{\mu}_{k}^{p}),$
(3) $\displaystyle\bm{\Sigma}_{k}^{f}$
$\displaystyle=(\bm{I}-\bm{G}_{k}\bm{B}_{k})\bm{\Sigma}_{k}^{p}.$ (4)

These equations describe the mean and covariance in terms of the ‘Kalman gain’
$\bm{G}$ given by:

$\bm{G}_{k}:=\bm{\Sigma}_{k}^{p}\bm{B}_{k}^{\prime}(\bm{B}_{k}\bm{\Sigma}_{k}^{p}\bm{B}_{k}^{\prime}+\bm{\Omega}_{k})^{-1}.$

Also, the predictive distribution for $\bm{x}_{k+1}|\bm{y}_{1:k}$ at time $k$
can be shown to be $\thicksim MVN(\bm{\mu}_{k}^{p},\bm{\Sigma}_{k}^{p})$ where

$\displaystyle\bm{\mu}_{k}^{p}$ $\displaystyle=\bm{A}_{k}\bm{\mu}_{k}^{f},$
(5) $\displaystyle\bm{\Sigma}_{k}^{p}$
$\displaystyle=\bm{A}_{k}\bm{\Sigma}_{k}^{f}\bm{A}_{k}^{\prime}+\bm{\Psi}_{k}.$
(6)

_Unscented Kalman Filter_. Generalising equations (1) and (2), we consider the
non-linear system dynamics

$\displaystyle\bm{x}_{k+1}$
$\displaystyle=f(\bm{x}_{k};\bm{A}_{k})+\bm{\epsilon}^{\bm{x}}_{k},$ (7)
$\displaystyle\bm{y}_{k}$
$\displaystyle=g(\bm{x}_{k};\bm{B}_{k})+\bm{\epsilon}^{\bm{y}}_{k}.$ (8)

Here $f$ and $g$ may be non-linear functions of the state and parameters. An
extension of the Kalman filter was developed based on the unscented transform
[21], a method for calculating statistics of a random variable that has
undergone a non-linear transformation. The idea behind the algorithm – called
the Unscented Kalman Filter (UKF) – is to first draw sigma points, which are
sample points drawn from the support of an underlying probability
distribution. The sigma points are passed through a non-linear function, so
that by the unscented transform the means and covariances of the predictive
and filter distributions are estimable; indeed, they are calculable as
functions of (weighted) sample means and (cross-)covariances of the
transformed points.

In keeping with the approach of [22], so for $N$ the state dimension, at
initialisation we define vectors of weights $\bm{w}^{c},\bm{w}^{m}$ of size
$(2N+1)$, and select a set of sigma points denoted $\bm{\mathcal{X}}$111There
are various suggestions within the literature for how one might choose the
weights and sigma points; we keep with the presentation of [22] for our
implementation.. Each sigma point is transformed on passing through $f$,
yielding the set of transformed sigma points
$\bm{\mathcal{Y}}=f(\bm{\mathcal{X}})$. Then the predictive distribution for
$\bm{x}_{k+1}|\bm{y}_{1:k}$ at time $k$ can be shown to be $\thicksim
MVN(\bm{\mu}_{k}^{p},\bm{\Sigma}_{k}^{p})$ where

$\displaystyle\bm{\mu}_{k}^{p}$
$\displaystyle=\sum_{i=0}^{2N}w^{m}_{i}\mathcal{Y}_{i,k},$ (9)
$\displaystyle\bm{\Sigma}_{k}^{p}$
$\displaystyle=\sum_{i=0}^{2N}w^{c}_{i}(\mathcal{Y}_{i,k}-\bm{\mu}_{k}^{p})(\mathcal{Y}_{i,k}-\bm{\mu}_{k}^{p})^{T}+\bm{\Psi}_{k}.$
(10)

Next we calculate $\bm{\mathcal{Z}}:=g(\bm{\mathcal{Y}})$, and
$\hat{\bm{y}}_{k}^{-}:=\sum_{i=0}^{2N}w^{m}_{i}\mathcal{Z}_{i,k}$. Then the
filter distribution for $\bm{x}_{k}|\bm{y}_{1:k}$ at time $k$ can be shown to
be $\thicksim MVN(\bm{\mu}_{k}^{f},\bm{\Sigma}_{k}^{f})$ where

$\displaystyle\bm{\mu}_{k}^{f}$
$\displaystyle=\bm{\mu}_{k}^{p}+\bm{G}_{k}(\bm{y}_{k}-\hat{\bm{y}}_{k}^{-}),$
(11) $\displaystyle\bm{\Sigma}_{k}^{f}$
$\displaystyle=\bm{\Sigma}_{k}^{p}-\bm{G}_{k}\bm{P}_{y,k}\bm{G}_{k}^{T}.$ (12)

In analogy with the Kalman gain defined above, here we have $\bm{G}$ given by:

$\displaystyle\bm{P}_{y,k}$
$\displaystyle:=\sum_{i=0}^{2N}w^{c}_{i}(\mathcal{Z}_{i,k}-\hat{\bm{y}}_{k}^{-})(\mathcal{Z}_{i,k}-\hat{\bm{y}}_{k}^{-})^{T},$
(13) $\displaystyle\bm{P}_{xy,k}$
$\displaystyle:=\sum_{i=0}^{2N}w^{c}_{i}(\mathcal{Y}_{i,k}-\bm{\mu}_{k}^{p})(\mathcal{Z}_{i,k}-\hat{\bm{y}}_{k}^{-})^{T},$
(14) $\displaystyle\bm{G}_{k}$
$\displaystyle:=\bm{P}_{xy,k}\bm{P}_{y,k}^{-1}.$ (15)

_A state space conditional factor model_. We consider a conditional factor
model, as inspired by the Arbitrage Pricing Theory (APT) of Ross [23]. For a
single-period investment setting, we assume the predictive model

$\displaystyle\bm{r}_{k+1}=\bm{X}_{k}\bm{f}_{k}+\bm{\epsilon}_{k}^{r},\qquad\bm{\epsilon}_{k}^{r}\thicksim
N(\bm{0},\bm{D}_{k}).$ (16)

Here $\bm{r}_{k+1}$ is the $n$-dimensional random vector containing the cross-
section of excess returns for $n$ equities over the next time increment
$[k,k+1]$; $\bm{X}_{k}$ is a non-random $n\times m$ matrix of observable
factor exposures estimated at $k$, for $m$ the number of explanatory asset
factors; $\bm{D}_{k}$ is a covariance matrix for the returns, assumed to be
diagonal; and $\bm{f}_{k}$ is an $m$-dimensional latent factor vector that we
estimate at each time step by cross-sectional regression.

We write the state space conditional factor model

$\displaystyle\bm{f}_{k+1}$
$\displaystyle=g(\bm{f}_{k};\bm{\theta}^{f}_{k})+\bm{\epsilon}^{\bm{f}}_{k},$
(17) $\displaystyle\bm{r}_{k+1}$
$\displaystyle=\bm{X}_{k}\bm{f}_{k}+\bm{\epsilon}^{\bm{r}}_{k},$ (18)

assuming dynamics in the latent factor vector. The state equation implies the
dynamic predictability of $\bm{f}$ via some known function $g$. Recently, non-
linear models for factor evolution have been explored in [24].

Of course, a popular modern choice for $g$ might be one of the various machine
learning-based time series models. Such models could depend on a recent
history of the state variable, rather than just the current state. In the case
of the factor model above, we can expand the state space to include lags in
$f$. Consider the one dimensional case where

$f_{k}=g(f_{k-1},\dots,f_{k-M};\bm{w})+v_{k}.$

The function $g$, for example, can be estimated by a time series model, with
$v_{k}$ being its aleatoric uncertainty estimate. For this single factor case,
per [22], the state space model can be written:

Figure 2: Factor risk premia as estimated by a conditional factor model using
(i) ordinary least squares (‘OLS’); and (ii) an underlying non-linear state
space model (‘UKF’). Four factors are displayed – size, volatility, momentum
and value, for the year 1997 (which was arbitrarily chosen).

$\displaystyle\bm{f}_{k+1}$
$\displaystyle=\bm{g}(\bm{f}_{k};\bm{w})+\bm{B}\bm{\epsilon}^{\bm{f}}_{k},$
$\displaystyle\begin{bmatrix}f_{k+1}\\\ f_{k}\\\ \vdots\\\
f_{k-M+2}\end{bmatrix}$ $\displaystyle=\left[\begin{array}[]{cc}&\\\
\lx@intercol\hfil\smash{\raisebox{6.0pt}{$g(f_{k},\dots,f_{k-M+1};\bm{w})$}}\hfil\lx@intercol\\\
\begin{bmatrix}1&0&\dots&0\\\ 0&\ddots&0&\vdots\\\
0&0&1&0\end{bmatrix}&\begin{bmatrix}f_{k}\\\ \vdots\\\
f_{k-M+1}\end{bmatrix}\end{array}\right]+\begin{bmatrix}1\\\ 0\\\ \vdots\\\
0\end{bmatrix}v_{k},$ $\displaystyle r_{k+1}$
$\displaystyle=\left[\begin{array}[]{cccc}1&0&\dots&0\end{array}\right]f_{k}+\epsilon^{r}_{k}.$

We demonstrate an application of the model for four latent factors, assumed
uncorrelated, showing a sample of model output in Figure 4 above. This
compares with a predictive model for returns as a function of factors
estimated by ordinary least squares. We provide further details of the model
specification as part of the following section on experimental methods and
strategy.

##  4 Experimental methods and strategy

Our statistical arbitrage scheme has three non-trivial components – data set
construction and feature creation, data modelling, and specification of the
trading strategy. Each of these components can have a significant bearing on
performance results. In Section 3 we introduced the data model, and we present
details of the particulars here. We also address the other 2 components in
this section, commenting on the data and the trading strategy.

_Data._ The data set construction is the same as used in [25], with the
exception that the period is daily rather than bi-monthly. From that work,
recall that the data is sourced from CRSP and IBES databases, with access
granted via the Wharton Research Data Service. The raw data is daily US equity
market data for the years 1992-2021 inclusive. On each trading day, we select
the top $N$ greatest stocks across all available US equities sorted by market
capitalisation, for $N=2000$. The data set is filtered for USD denominated
common stock only – there are no closed end funds, REITs, ETFs, unit trusts,
depository receipts, warrants etc. We also collected the S&P500 (excess)
return time series, stock market capitalisations, and earnings per share,
adjusted for splits.

In that we experiment with factor models, the data set requires the creation
of our 5 chosen risk factors: market beta, size, volatility, momentum and
value. Calculating size is relatively straightforward in CRSP: the calculation
is the product of shares outstanding and the daily close price. Value was
determined by analysing IBES data, and could be merged back to the CRSP
database based on the CUSIP identifier. We note that risk factor construction
is often non-trivial. Momentum was codable using CRSP data and the definitions
of [26]; we calculated the daily compounded 12 month returns sans the most
recent 1 month of data. We created the market Beta and volatility factors
using the relevant CRSP data and the details of [27]. Indeed, over a two year
rolling window we estimate a collection of linear functions by regressing each
asset’s daily excess returns against the S&P500 excess returns. The market
Beta is the regression gradient, while the volatility factor is set to be the
regression mean-square error.

Figure 3: Suggestive neural network architecture for a non-linear latent
factor vector state evolution model. The model outputs state prediction and
aleatoric uncertainty estimates given an input historic time series.

_State space model specification._ The state space models described in Section
3 require the specification of the state transition function $g$, and the
state and measurement covariances, denoted $\bm{F}$ and $\bm{D}$. In the case
of the linear model, we choose $g$ to be the identity function222The choice of
$g$ is in analogy with the state space model underlying the pairs trading
application of [16]., and estimate a diagonal state covariance for
$\bm{f}_{k+1}-g(\bm{f}_{k};\bm{\theta}^{f}_{k})$ based on a rolling (lookback)
window of size 20. For the non-linear model, we estimate a multi-layer
perceptron model for $g$, that takes the previous history of $f$ on a window
of size 10, and that predicts $f$ at the next time-step, and well as the
aleatoric uncertainty of $f$, which we set as an estimate of $\bm{F}$. To be
clear, this estimate is a weighted average of the individual uncertainty
estimates collected when calculating the state update equation for each sigma
point. For reference, an image of the neural network architecture is given by
Figure 3. This model is in the spirit of [28], and in this simplified setting
we implement a multi-layer perceptron with 32 nodes followed by a drop out
layer with rate 0.1, and L2 regularisation with parameter fixed at $10^{-5}$
chosen by cross-validation on data from the earliest years. Finally, for both
the linear and the non-linear model, our estimate for $\bm{D}$ is a diagonal
covariance gleaned from estimating Eq. (16) via ordinary least squares
regression.

_Benchmark models._ We utilise a benchmark model with the trading strategy
mechanics as outlined below, but with the key difference that the spread $S$
is defined by the rolling 10 day sum total returns for each asset against the
mean total return over a rolling 10 day window. This benchmark is discussed in
[29], and is intuitive for considering reversals of returns. To be clear, this
benchmark strategy is with reference to the total return series for each stock
only; that is, it makes no reference to any underlying factors.

Figure 4:  Trades are entered when the Z score deviation of the difference
between returns and filtered returns passes a high (absolute) threshold; these
points are indicated by the large green dots. Trades are closed at the first
time the deviation has crossed the origin, as indicated by the large red dots.

_Trading strategy._ The state space model-inspired trading strategy is a
long/short strategy, hedged to be beta neutral. The strategy depends on the
spread defined by the rolling sum total returns in excess of the rolling sum
filtered total returns – as estimated using the models of Section 3 – for each
asset. That is, for each time $k$, $r_{k}$, $X_{k}$ and $X_{k-1}$ are
observed, so the filter estimate for $f_{k-1}$, denoted by
$f^{(\text{filt})}_{k-1},$ is calculable. Hence we can calculate
$r^{(\text{filt})}_{k}=X_{k}f^{(\text{filt})}_{k-1}$. The spread, or (sum)
error deviation, $\sum(r_{k}-r^{(\text{filt})}_{k})$ is the target for our
trading strategy, and we implicitly assume that it is mean-reverting.

Being arbitrary, the rolling window size for calculating the spread, denoted
$WS$, is a target parameter for performance sensitivity analysis. Given a
trading threshold $L$ ($S$), a trade is entered long (short) the first time
the Z score of spread divergences is below (above) the negative (positive)
values of $L$ ($S$). The trade is exited the first time the Z score reverts
to, or through, the level $0$, with trades ‘closed out’ on the last trading
day of the year otherwise. The quantity $L$ ($S$) should represent a
sufficiently large deviation such that expected excess returns are not
dominated by transaction costs. All position sizes long or short are set to a
constant notional value. To assist intuition, the mechanics of the strategy
are outlined diagrammatically in Figure 4 above, for the choice $L/S=1.0/1.0$,
for a target asset over the course of a year.

There are two other minor assumptions to the trading strategy that we state
for completeness. Firstly, when entering a trade we assume that we can execute
at the market close price. Secondly, at time $t$, if our database contains no
close price for a stock at $t+1$, we assume that time $t$ is the last trading
day for the stock, and that this is known to market. Hence, we close any
existing position in the stock at the time $t$ close price, and do not enter a
new trade at $t$ in the event that the trade entry criteria is met.

_Transaction costs._ In our primary experiments transaction costs are set to 5
basis points (bps) per trade entry and per trade exit, making 10 bps for a
round-trip trade in a single underlying (be it a single stock or index). This
is consistent with the works of [11, 3] and chosen for ease of comparison;
these works are important benchmarks for multivariate statistical arbitrage in
US equities. Further, we assume that we are able to borrow to short at all
times, and that the transaction costs subsume any cost of borrowing. This
deviates to some extent from reality, where not all stocks can be shorted all
the time, and where some stocks incur a ‘hard to borrow’ fee, particularly
when in high demand. Unless specified otherwise, all returns discussed for
stocks and indices are total returns, and short positions pay the risk-free
rate assuming funding is received at the time of shorting, and able to be
invested in the riskless asset. We test the robustness of our strategies to
alternative levels of transaction costs; this is discussed further in the
Results section below.

_Hedging market risk._ We hedge our trading strategy to be beta neutral. Our
simple approach is to calculate the market beta of our portfolio of long and
short positions each trade period, and to maintain a position of dollar value
$P$ in the S&P500 index, where

$P=-\beta_{port}\Pi,$

thus providing the hedge. The term $\Pi$ denotes the market dollar value of
the portfolio to be hedged (and is defined further below). A positive value of
$P$ indicates a net long postion in the index as the hedge, while a negative
value indicates a short position. The term $\beta_{port}$ denotes the
weighted-average beta of the portfolio given by

$\beta_{port}=\frac{\bm{\beta}\bm{D}^{T}}{\Pi}.$

The term $\bm{D}$ is an $N$-vector of position sizes whose $i^{th}$ entry,
consistent with the trading strategy notes above, takes the value $1$,$-1$ or
$0$ for a long, short or nil position in stock $i$. The term $\bm{\beta}$ is
an $N$-vector of corresponding stock beta estimates.

Implicit in the hedging strategy is the assumption that we can trade long or
short positions in the S&P 500 index and earn or pay the total return of the
index each period. In reality, the investor would take a position in the
futures contract written on the index, which has a myriad of practical
implications. Regardless, we do not consider these further, and so, for
example, we ignore the spot / futures basis that could induce additional
slippage.

_Strategy returns._ We assume a 100% leverage portfolio so that for each
dollar invested in the portfolio, a position of up to a dollar’s worth long
and / or a dollar’s worth short can be entered. Therefore, the market cost for
$l$ longs and $s$ shorts is $\Pi=\max(l,s)$. We calculate (excess, unhedged)
portfolio returns for a time period $[k,k+dt]$ thus:

$\Delta\Pi_{k}/\Pi_{k}=\frac{\sum_{i}(R^{l}_{i,k}-r_{k}dt)+\sum_{j}(R^{s}_{j,k}-r_{k}dt)}{\Pi_{k}}+\frac{r_{k}dtS_{k}}{\Pi_{k}}-\frac{TC\cdot\bm{1}\cdot(\Delta\bm{D}_{k})^{|\cdot|}}{\Pi_{k}}.$

Here $R^{l}_{i,k}$ ($R^{s}_{j,k}$) denotes the return over the period for a
long (short) position held, $r_{k}$ denotes the (annualised) risk-free rate of
return, $TC$ denotes the transaction cost, and $\bm{X}^{|\cdot|}$ defines an
operator returning a vector of elementwise absolute values of the input
$\bm{X}$. Hence the terms on the right-hand side of the summation account for,
respectively, the excess return of the long and short positions, the risk free
rate earned on the proceeds of the short sales deposited in a money account,
and the transaction costs for changes to the portfolio holdings333The
calculation is similar to that found in [3], and is equal when $\Pi=s$.
Otherwise, our return is slightly more conservative for the case $\Pi=l$,
though this has little impact on the final value calculated..

##  5 Results

In this section, we present performance statistics for the state space factor
models of Section 3, relative to our benchmarks and the long-only market
portfolio. For readers inclined, it is possible to replicate the results given
the descriptions of the data and methodology as outlined in Section 4.

We find that the multivariate statistical arbitrage strategy has shown
reasonable performance through time, which is consistent with the expectations
for a mid-frequency strategy with reasonable capacity. Performance is best
during the first half of our 29 year sample, and we conjecture that the
impressive returns have diminished due to increased market competition and
efficiency through time. Also, we find sufficient differences between strategy
performance that supports the use of a state space conditional factor model
for multi-variate statistical arbitrage, instead of other less complex -
albeit relatively powerful - models. Despite all strategies showing relative
underperformance in the most recent years, practitioners may find latitude for
improvement by further modifying or optimizing components of the strategy; we
briefly describe some preliminary ideas for research direction at the end of
this section.

###  5.1 Strategy performance

_The earlier years: 1993–2007._ Given the data set and assumed fair value
models, per the methods described in Section 4 there is an implied sensitivity
of any estimated performance statistics to the key assumptions underlying the
trading strategy. Not the least, this includes the setting of parameters
denoting (i) the trade entry threshold Z score; (ii) the size of the returns
window; and (iii) the level of transaction costs. We set the parameters of (i)
and (ii) based on strategy performance over the ‘earlier years’ of 1993-2007.
We assume a transaction cost level of 5 bps for each trade in all results that
follow, unless otherwise specified. Choosing the parameters in this way also
allows for a reasonable performance comparison between our strategy relative
to results given by other well known academic works. Finally, the parameter
choices are maintained when we assess strategy aggregate performance, and
performance conditioned on later years only.

For various combinations of Z score-based entry thresholds for the long and
short legs (denoted L and S), and various window sizes (denoted WS), we show
average Sharpe ratio and maximum drawdown statistics in Figure 5, on the next
page. Drawdowns are calculated on a 252 day rolling window. We find utility in
solving for the trade entry parameters for the long and short legs separately,
making no assumption that the same cut off should be optimal for both. We test
the entry level to balance entering a sufficient number of trades, but not
entering too early (or late) such that the trade is not held too long (or
missed entirely). As expected, we find firstly that as the threshold
decreases, the number of trades increases to the extent that small thresholds
have bad economics owing to large turnover and cumulative transaction costs.
Window size is similarly tested, for sizes of 1, 5 and 10, corresponding to
daily, weekly and fortnightly rolling windows.

Per Figure 5, we notice two clear clusters of data, whereby (i) one exhibits
the highest average Sharpe ratio, but also exhibits amongst the worst for
maximum drawdowns; and (ii) the other is a cluster with good average Sharpe
ratio, but amongst the strategies with the better of the maximum drawdowns. It
is reasonable to suggest that different investors might have different
preferences for either cluster, so we extend our study for recent years with
respect to them. The first cluster is characterised by models with a window
size of 5, and long/short entry thresholds in {0.5,1.0,1.5,2.0 : $L<S$}. The
second cluster is the same except that the window size is 10. For both
clusters, the state space models tend to dominate both of the benchmark models
with respect to both statistics of interest. However, the performance between
the state space models is less noticeable, though is such that the KF tends to
have the higher Sharpe, for worse drawdowns, relative to the UKF, and for all
choices of cutoff. Regardless of an investors preference, these differences
are relatively small for this experiment.

Figure 5: Average Sharpe ratio over the 15 years from 1993-2007 versus maximum
drawdown for a collection of models underlying the trading strategy.

Figure 6: Tabulated average annual Sharpe ratios by date range, for our
conditional factor model (hedged UKF, L/S = 1.5/2.0, WS=5) and two leading
external methods. Note I: All methods assume transaction costs of 5bps per
trade. Note II: * results mean ‘excluding 1992 due to lack of data’.

Strategy | Date range | Avg ann Sharpe | TR SPX | Notes (Models for cited work)
---|---|---|---|---
|  | [3, 11] | Ours |  |
Ours | 1993-2021 | - | 1.29 | 0.87 | -
[3] | 1997-2007 | 1.44 | 1.61 | 0.47 | PCA based
| 2003-2007 | 0.90 | 1.01 | 0.78 | PCA based
| 1997-2007 | 1.10 | 1.61 | 0.47 | ETF based
| 2003-2007 | 1.51 | 1.01 | 0.78 | ETF based – incorporating additional volume feature
[11] | 1992-2015 | 0.75-1.12 | 1.67* | 0.73* | Vine based. Range across 4 alt. trade selection methods.

One also notices from Figure 5 that the strategies that have a lower entry
threshold for the longs, and relatively larger entry threshold for the shorts,
tend to exhibit the better performance statistics. It is the case that the
longs outperform the shorts for this data sample444Recently, other authors
have analysed long-short equity factor portfolios, albeit in an alternate
strategy setting, and made the same outperformance observation [30]., with the
shorts also more likely to have a negative year of performance. The effect of
increasing an entry threshold is to reduce the number of trades entered on
that long or short leg; this reduces relatively positive or negative returns,
hence the observation. We leave the question of why the longs may more often
outperform the shorts as future work, and include no assumption as to what leg
may outperform for any period within our backtest; this could possibly be a
source of improvement of the strategy alpha.

Our results can be reasonably compared with other well-known results from the
literature. Our approach has a similar data set to the others (daily US
equities data, albeit with different universe sizes), but most notably
different models for fair-value – in [3] factors are modelled based on
Principal Component Analysis, with an additional ETF-based model, and in [11]
asset relationships are modelled with Vine copulas. Our trading strategy is
similar to [3], though we do not refine a trade exit threshold; we have less
in common with [11], who refine their trading rules with many more steps
relative to our approach. Importantly, across all papers, the transaction cost
assumption is equal. For the papers cited, it is easiest to compare average
annual Sharpe ratios over the date ranges given in the cited paper. We
tabulate the results of this comparison in Figure 6 above, and also include
the total (excess) return of the S&P 500 over the same period. Our results are
similar in magnitude to [3], and we conjecture that daily US equity mean
reversion could be captured by strategies using a variety of underlying fair
value models over the considered date range. Superficially, there appears to
be some outperformance in our approach, though this claim would benefit from
comparison given outcomes of a controlled experiment. On the other hand,
though the average Sharpe ratios appear similar, our methods do show
sufficient variation in Sharpe ratio year-on-year. For example, the PCA based
method shows severe performance degradation, and negative Sharpes, in later
years. On the other hand, our approach had no negative years between 1997 and
2007. The outperformance of our approach is more striking against [11], and
with many degrees of freedom in the formulation of the copula-based strategy
it is hard to conjecture exactly why this may be the case.

_Aggregate performance._ In Figure 7 below, we depict the annual Sharpe ratio
distribution over 29 years by way of box and whisker plots, for the 4
underlying models. Further, we show the performance across 4 sets of parameter
choices, which include window sizes of 5 and 10, and long/short entry
thresholds of 0.5/2.0 and 1.5/2.0. We also show a plot for the Sharpe ratio of
total (excess) return of the S&P 500.

Figure 7: Sharpe ratio distribution over 29 years for 4 models underlying the
trading strategy outlined in Section 3, grouped by long/short entry Z score
criteria, and lookback window size for cumulative returns. The performance of
the S&P 500 with respect to total (excess) return is also included for
reference.

|  | Transaction costs (bps)
---|---|---
|  | 0 | 5 | 10 | 15
| 25th | 1.11 | 0.39 | -0.08 | -0.76
WS 5 | 50th | 1.39 | 0.99 | 0.57 | -0.09
| 75th | 2.75 | 1.91 | 1.14 | 0.70
| 25th | 0.80 | 0.35 | -0.10 | -0.66
WS 10 | 50th | 1.09 | 0.59 | 0.26 | -0.17
| 75th | 1.90 | 1.47 | 1.05 | 0.83

Figure 8: Sharpe ratio percentiles by transaction cost and window size levels
for UKF 0.5/2.0.

Modelwise, the KF and the UKF outperform relative to OLS and BM across all but
one groupwise comparisons of 1st, 2nd and 3rd quantiles, and averages, of
annual Sharpe ratios. On the other hand, any UKF under/outperformance over the
KF is not apparent, so that the additional computational complexity of the UKF
is not justified for this particular application. If stronger non-linear
relationships between variables were observed, or expected, given additional
training data, the UKF may still find use. The OLS approach induces good
results, with the (U)KF appearing to be a worthwhile refinement, that is, the
introduction of uncertainty quantification has a benefit. The nature of the
outperformance appears to vary depending on the parameter settings of the
trading strategy.

In Figure 8, we show the results of an exploration of robustness of our
strategy to the transaction cost assumption. As mentioned, we implemented our
strategies assuming a 5 bps (10 bps round-trip) transaction cost for each
asset traded. We show Sharpe ratio percentile levels for transaction costs of
0,5,10 and 15 bps for the UKF 0.5/2.0 across window sizes. The yellow
highlighted data are for the 5 bps transaction cost level. Clearly this is a
sweet spot – and while the quoted performance can sustain a slightly higher

Figure 9: Top: Percentage of the stock universe (N=2000) invested in, long or
short, at a given time. Bottom: Strategy rolling window (252 day) drawdown
versus the long only market portfolio; transaction cost = 5bps.

transaction cost assumption, there is an obvious performance decay such that
at 15 bps of transaction costs, the median annual Sharpe ratio over the sample
realises negative. Finally, we should mention that the cost of short selling
may have been significantly more than estimated in this work, if the capacity
to borrow to sell was even available, particularly so for the earlier years of
study. On the other hand, practitioners may have strategies for cheapening
transaction costs when trading. Given these latter points, and other cited
academic studies, we think our primary transaction cost assumption is
reasonable.

The bottom panel of Figure 9 shows rolling 252 day drawdown for 2 UKF based
strategies differing on window size. In the years up till 2007 the strategies
largely avoid the worst of the drawdowns realised by the market portfolio,
which has drawdowns touching -10% or worse on 3 occasions. In these cases, our
strategies realise drawdowns between -7 and -5 %, with the exception of the
strategy with window size of 5 that realises a -9.5% drawdown on one of the
three occasions. Post-2007, the two severe market drawdowns of 2008 and 2020,
both being worse than -20%, are matched by strategy drawdowns about -16 and -
12%, respectively. The first of these drawdowns, within the years of the Great
Financial Crisis, was to the chagrin of many real-world strategies, such that
we find it less of a concern that our strategy did not register greater
outperformance during this period. For intuition, we also depict the
proportion of assets under consideration that are invested in for all times,
in the top panel of Figure 9. We notice the obvious empirical fact that with a
larger window size, more assets are held. There is an annual seasonality
apparent, whereby at the end of each year positions are closed out, and at the
start of the year there are spikes for new trades re-entered, as expceted by
the definition of our trading strategy.

It is reasonable to assume that the beta neutral statistical arbitrage
strategy has zero or low (absolute) correlation of returns to the long only
market portfolio. Hence, in theory, a blended investment taking a position in
the strategy and the long only portfolio could show an outperformance (with
respect to Sharpe ratio)

Figure 10: Sharpe ratio distribution over 29 years for a UKF model-based
trading strategy as outlined in Section 3, the S&P 500 with respect to total
(excess) return, and a portfolio of a weighted combination of the two.

relative to both [31]. We implement such a blended strategy by weighting the
statistical arbitrage strategy and the long only portfolio as

$w_{i}=\frac{S_{i}}{S_{L}+S_{LS}},\qquad
S_{i}:=\frac{\mu_{i}}{\sigma^{2}_{i}},\qquad i\in\\{\text{L, LS}\\},$

where the weights $w$ depend on the mean $\mu$ and variance $\sigma^{2}$ of
the long portfolio $L$ or long-short portfolio $LS$ returns, optimizing for
maximum Sharpe ratio [31, 28]. The true values of the mean and variance
parameters are unknown, and for demonstration purposes we take the
straightforward approach of setting the means and variances equal in the first
year, and equal to their historical values on a rolling window of size (up to)
10 otherwise. We find that the results are not particularly sensitive to
(reasonable) values of the choice of rolling window size. Images summarising
the Sharpe ratio performance are given in Figure 10. Of note, given the right-
hand side chart, we notice considerable performance improvement with respect
to the blended strategy for the first, second and third quartiles, supporting
the theoretical result, and offering a practical hint for deploying such a
strategy.

_Recent performance: 2008–2022._ We notice from the left-hand side image of
Figure 10 that the strategy has not performed so well in the most recent
years, particularly for the years 2015-2019. We conjecture that this could be
due to a lower volatility market regime during this time, increased
sophistication of market participants and with it competition for alpha /
eroded edge, or otherwise due to limitations derived from some underlying
assumption of our strategy. Overcoming these challenges is likely of interest
to the motivated academic or practitioner, and we offer some avenues for
future work in the following conclusion.

##  6 Conclusion

We study multivariate statistical arbitrage under a conditional factor model
augmented in a state space framework for US equities trading. Our study shows
the modelling approach can yield compelling performance statistics over a 29
year period, an an absolute basis and relative to a reasonable benchmark, and
relative to a model estimating returns as a function of factors via ordinary
least squares. However, for our experimental set up, we did not find that the
non-linear state space model was justified relative to the linear case; though
it was no worse, any outperformance benefit does not contrast well with the
increased modelling complexity introduced. When we compare our results to
existing attempts in the literature, we find that our approach compares well.
We also show empirical evidence that blending our investment capital between
the long-short strategy and the long-only market portfolio as a function of
each strategies estimated mean and variance yields a strategy with improved
annual Sharpe ratio, consistent with theory. All results are with respect to
reasonable level of transaction costs.

There is the scope for academic and practioner work on the strategy presented
in this paper, particularly with respect to exploring the potential for
improvement on the period of strategy underperformance in recent years. We
conclude with some final remarks:

  * •

Regarding the data underlying the quantitative strategy, the addition of
alternative data beyond factor data and returns time series may improve the
predictive power of the model. With respect to the dataset we constructed, and
by the recent work of [32], it is the case that estimating submodels over
clusters of data – for example, stocks grouped by sector – could lead to
significant performance improvement.

  * •

The recent work of [24, 33] consider alternative specifications of dynamic
factor models for investing, including those with non-linearities at the level
of the measurement equation. Such expressions could be handled by our UKF-
based framework, and present as a reasonable target for improving performance.

  * •

Recall our brief discussion on related literature in Section 2, whereby almost
all work cited made use of data on a daily time scale. The literature would be
bolstered by analysis completed on alternative time frames, particularly on
intraday time scales. This is arguably harder, not the least in terms of data
set curation. On the other hand, the recent work of [34] provides evidence of
intraday return predictability as a function of factors using machine
learning; this could support refining reactive entry and exit rules within the
trading strategy.

  * •

Modelling techniques subsuming regime filters could support our statistical
arbitrage trading strategy, with recent evidence to this end for a pairs
trading application given in [35].

  * •

Finally, and requiring more expert / domain knowledge, it would be interesting
to understand trades entered that are directly mappable to a market phenomena,
such as a stock going ex-dividend, or being influenced by index rebalancing.
This is interesting in the context of determining whether the quantitative
strategy’s profitable trades are explainable in terms of a well-known market
effect, or else is supportive of the underlying factor-based theory.

## Acknowledgements

The authors would like to thank the Oxford-Man Institute of Quantitative
Finance for its generous support. SR would like to further thank the Royal
Academy of Engineering.

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