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Robust Tests of Market Efficiency using
         Statistical Arbitrage

     Melvyn Teo, Yiu Kuen Tse, and Mitch Warachka




This project is funded by the Wharton-SMU Research Center, Singapore Management University
       Robust Tests of Market Efficiency using Statistical
                                            Arbitrage

                    Melvyn Teo, Yiu Kuen Tse and Mitch Warachka∗



                                               Abstract

           This paper develops robust tests of market efficiency using statistical arbitrage
        which circumvent the joint-hypotheses dilemma confounding the traditional literature.
        Hogan, Jarrow, Teo and Warachka (2004) identify statistical arbitrage opportunities
        in momentum and value strategies. However, their results are sensitive to the assump-
        tion that expected incremental trading profits are constant. We demonstrate that this
        empirical discrepancy results from a lack of statistical power in their Bonferroni test
        procedure. To resolve the disparity, we propose a Min-t test statistic for trading strate-
        gies with time-varying expected profits. This procedure is then extended to examine
        autocorrelated and non-normal trading profit innovations. Our improved statistical
        approach consistently rejects market efficiency across four different trading profit as-
        sumptions and provides a robust methodology for determining the convergence rate to
        arbitrage of a trading strategy.




  ∗
      Mitch Warachka gratefully acknowledges funding by the Wharton-SMU Research Center, Singapore
Management University.

                                                    1
   Tests of market efficiency have long been confounded by the joint-hypotheses dilemma,
which states that conclusions regarding market efficiency are conditional on specifying the
correct equilibrium model for stock returns. According to Fama (1998), this critical caveat
limits our ability to confidently reject market efficiency despite numerous empirical challenges
such as the profitability of momentum and value strategies, documented by Jegadeesh and
Titman (1993) and Lakonishok, Shleifer and Vishny (1994) respectively. Recently, Hogan,
Jarrow, Teo and Warachka (2004) (HJTW hereafter) develop a new approach for testing mar-
ket efficiency. They do so by determining whether the trading profits of persistent anomalies
constitute statistical arbitrage (SA hereafter) opportunities. This approach circumvents the
joint-hypotheses dilemma since the definition of SA is independent of any equilibrium model
and, as with arbitrage opportunities, its existence contradicts market efficiency. A detailed
summary of SA is found in the next section.
   Empirical tests for SA require assumed trading profit dynamics. The difficulty in testing
for SA stems from having sub-hypotheses for the parameters which describe the evolution
of trading profits. These sub-hypotheses arise from having several different parametric re-
strictions within a single statistical model, with each sub-hypothesis imposing a constraint
on the trading profit parameters. In contrast, the traditional market efficiency literature
involves an equilibrium model and a subsequent statistical test, with the equilibrium model
being a maintained assumption that is not explicitly tested.
   For emphasis, testing different incremental trading profit dynamics in the SA framework
is not comparable to assuming multiple models of market equilibrium. By facilitating the
estimation of more general trading profit dynamics, this paper allows researchers to choose
an appropriate process depending on the time dependence, autocorrelation and normality of
the data. In contrast, the traditional approach specifies an equilibrium model apriori, the
empirical validity of which is not tested.
   This paper offers two contributions to the SA framework for testing market efficiency.
First, a Min-t statistic is proposed which offers higher statistical power than the Bonferroni
approach adopted by HJTW. This allows us to re-examine the more general SA test with
time-varying expected trading profits. Second, while HJTW only estimate incremental trad-
ing profit models that assume innovations are serially uncorrelated and normally distributed,


                                              2
we extend our Min-t test procedure by relaxing these assumptions. This enhancement en-
ables more robust tests of market efficiency in the presence of two well known regularities in
empirical finance.
   Our first contribution provides a test procedure for the null hypothesis of no SA (NSA
hereafter) without having to invoke the Bonferroni inequality. For a null hypothesis involving
an intersection of sub-hypotheses, the Bonferroni approach controls the test’s Type I error
and provides a consistent test, with the probability of rejecting an incorrect null hypothesis
approaching one as the sample size increases towards infinity. However, the null hypothesis
of NSA is a union, not an intersection, of sub-hypotheses. This distinction is critical. With a
union, accepting even one sub-hypothesis results in accepting the null of NSA, hence market
efficiency.
   As a consequence of the null being a union of sub-hypotheses, the Bonferroni approach
does not provide a consistent statistical test and merely provides an upper bound on the Type
I error. Furthermore, its lack of consistency implies a loss of power in empirical applications,
a situation which is aggravated when more sub-hypotheses comprise the joint test. HJTW
consider two processes for incremental trading profits; a constrained mean (CM) model with
constant expected trading profits and a generalized unconstrained mean (UM) model. The
UM model allows for time-varying expected profits but requires additional sub-hypotheses
which motivates the introduction of our proposed Min-t methodology.
   We apply our test procedures to the momentum and value trading strategies of Jegadeesh
and Titman (1993) and Lakonishok, Shleifer and Vishny (1994) respectively. For the simpler
CM version of SA, conclusions resulting from our Min-t test parallel the results of HJTW who
find that approximately 50% of these strategies constitute CM SA opportunities. However,
they are unable to detect any UM SA opportunities in the same trading strategies. In
contrast, the Min-t methodology finds that the CM and UM tests produce nearly identical
results. Thus, we resolve the empirical discrepancy regarding market efficiency and trading
profit dynamics in HJTW.
   The second contribution of this paper extends the Min-t procedure to incremental trading
profits which have autocorrelated and non-normal innovations. We denote these respective
models as the CMC and UMC versions of SA. Our empirical results suggest that conclusions


                                               3
regarding market efficiency are not altered by these generalizations.
     Theoretically, our proposed Min-t statistic facilitates robust tests of market efficiency
using SA by enabling researchers to examine different trading profit processes and determine
the most appropriate model. Empirically, for momentum and value trading strategies, we
find that rejections of market efficiency are robust to different trading profit assumptions.
Therefore, the improved statistical procedures in this paper minimize the possibility that
discrepancies result from a lack of statistical power when investigating alternative trading
profit dynamics.
     Although the four (CM, UM, CMC and UMC) SA tests yield nearly identical conclusions
regarding the existence of SA, the rate at which the probability of a loss converges to zero
often varies between these formulations. Indeed, investors with limited capital who are
concerned with incurring intermediate losses benefit from estimating more elaborate trading
profit processes. Therefore, in light of the academic literature on the limits of arbitrage (see
Shleifer and Vishny (1997)), our proposed methodology is of considerable importance.
     HJTW investigate the effects of transaction costs, margin requirements, additional re-
serves for short-selling, higher borrowing rates than lending rates and the exclusion of small
stocks on conclusions regarding market efficiency. Despite these adjustments, the ability of
momentum and value trading strategies to generate SA is not seriously compromised for the
CM model. Therefore, we focus our attention in this paper on the robustness of their results
across different trading profit assumptions, rather than on replicating previous robustness
tests for the effects of market frictions.
     The organization of this paper is as follows. The next section summarizes the theoretical
underpinnings of SA. Section II introduces our statistical methodologies, while the data used
in our empirical examination are discussed in section III. Section IV presents our empirical
results. Section V concludes and offers directions for future research.



I.     Review of Statistical Arbitrage
Previous empirical tests in the market efficiency literature focus exclusively on positive ex-
pected returns. However, a positive expected return may be justified by risk premiums


                                               4
associated with an equilibrium model. Thus, the joint-hypotheses dilemma confounds tradi-
tional market efficiency tests. In contrast, as with arbitrage opportunities, the existence of
SA rejects all candidate models of market equilibrium.1


A.        Definitions and Hypotheses

A SA opportunity requires the cumulative discounted profits of a zero cost, self-financing
trading strategy to satisfy certain axioms which are restated in Definition 1 for ease of
reference.2 Specifically, cumulative discounted trading profits constitute a SA if they have a
positive expectation, a declining time-averaged variance and a probability of a loss converging
to zero. Such an opportunity is inconsistent with well functioning financial markets as shown
in HJTW in terms of the trading strategy’s Sharpe ratio as well as its contribution to expected
utility.
       Let {vi } for i = 1, . . . , n be a sequence of discounted portfolio values generated by a self-
                                                         n
financing trading strategy. We denote v(n) =              i=1   ∆vi as the trading strategy’s cumulative
discounted trading profit with the differenced terms ∆vi representing incremental discounted
trading profits.

Definition 1 A SA is a zero initial cost, self-financing trading strategy with cumulative
discounted trading profits v(n) such that:3

   1. v(0) = 0

   2. lim E P [v(n)] > 0
         n→∞


   3. lim P (v(n) < 0) = 0 and
         n→∞
                 P [v(n)]
   4. lim V ar    n
                            = 0 if P (v(n) < 0) > 0 ∀n < ∞.
         n→∞
   1
       SA is a sufficient but not necessary condition for market inefficiency. Furthermore, although NSA does
not imply market efficiency, we often refer to these concepts interchangeably given our empirical objective.
   2
     Bondarenko (2003) also utilizes the statistical arbitrage terminology. Besides having different axioms in
their respective definitions, the approach of HJTW is intended for applications to persistent anomalies while
Bondarenko (2003) investigates option pricing in incomplete markets.
  3
    The “if” statement in the fourth axiom is a technical condition and may be ignored when evaluating
persistent anomalies which are not standard finite time horizon arbitrage opportunities.

                                                     5
       To test for SA, we begin by assuming the following process for incremental trading profits4

                                           ∆vi = µiθ + σiλ zi ,                                      (1)

where zi are i.i.d. N (0, 1) random variables, although the assumptions of normality and
independence are subsequently relaxed. The initial quantities z0 = 0 and ∆v0 are both zero
by definition. The parameters σ and λ determine the volatility of incremental trading profits
while the parameters µ and θ specify their corresponding expectation.
       We implement two tests for SA under the assumption that trading profit innovations are
uncorrelated with a normal distribution. The model described in equation (1) represents the
UM model which allows for time-varying expected trading profits. We also consider a more
restrictive CM model that assumes constant expected trading profits by setting θ equal to
zero. Consequently, the CM version of SA has incremental trading profits evolving as

                                            ∆vi = µ + σiλ zi .                                       (2)

According to HJTW, SA opportunities exist in the UM model when the following sub-
hypotheses hold jointly:5

   1. H1: µ > 0

   2. H2: λ < 0

   3. H3: θ > max λ − 1 , −1 .
                      2


       The first sub-hypothesis tests for positive expected profits while the second implies the
trading strategy’s time-averaged variance declines over time. The third sub-hypothesis en-
sures that a potential decline in expected trading profits does not prevent convergence to
arbitrage. This restriction involves the trend in expected profits as well as volatility and
allows for negative θ values which imply expected trading profits decline over time. Eco-
nomically, the third sub-hypothesis tests for “long run” market efficiency by ascertaining
   4
       The functions iθ and iλ are justified in HJTW using a Taylor series expansion.
   5
       The third hypothesis θ > max{λ − 1 , −1} actually contains two hypotheses but the second component
                                           2

θ > −1 is a technicality (see Theorem 1 of HJTW) while the remaining three conditions have economic
interpretations.

                                                     6
whether trading profits are reduced fast enough to prevent convergence to arbitrage. For the
CM version of SA in equation (2), the third sub-hypothesis is eliminated.
     Overall, testing for SA involves a joint test of multiple sub-hypotheses, two in the CM
version and three in its extended UM counterpart. From a statistical perspective, if even
one sub-hypothesis fails to be satisfied, market efficiency is not rejected. A more detailed
discussion of the null hypothesis is provided in section II.


B.      Correlated Incremental Trading Profits

Given the manner in which financial anomaly (e.g., momentum) portfolios are typically con-
structed, it is not surprising that autocorrelation is manifested in their incremental trading
profits. In particular, the overlapping nature of the monthly holding periods likely induces
positive serial correlation. To address this statistical issue, we modify the innovations of
equation (1) to follow an MA(1) process given by

                                        zi =    i   +φ   i−1   ,                           (3)

where    i   are i.i.d. N (0, 1) random variables. We abbreviate the incremental trading profit
assumption in equation (1), modified to incorporate serially correlated innovations described
by equation (3), as the UMC model.
     As proved in HJTW, the presence of an MA(1) process does not alter the conditions for
SA, nor increase the number of sub-hypotheses. However, including the additional parameter
φ may improve the statistical efficiency of the remaining parameter estimates and avoid
inappropriate standard errors.
     The corresponding model with constant expected incremental trading profits but serially
correlated innovations is abbreviated CMC. In this version of SA, incremental trading profit
dynamics combine equation (2) with equation (3).


C.      Probability of Loss

The probability of a trading strategy generating a loss after n periods, the subject of the
third axiom in Definition 1, depends on the µ, σ, λ, θ and φ parameters as follows:


                                                7
                                                                             n    θ
                                                                     −µ      i=1 i
                       Pr{Loss after n periods} = Φ                              n          ,                 (4)
                                                                 σ(1 + φ)        i=1 i
                                                                                      2λ




where Φ(·) is the cumulative standard normal distribution function. This probability con-
verges to zero at a rate which is faster than exponential as shown in HJTW. Observe that
φ directly influences the convergence rate to arbitrage, as emphasized in section IV during
the presentation of our empirical results for momentum and value trading strategies.
       The UMC trading profit process includes all five parameters in equation (4) while the
associated loss probability for the CMC model has θ set equal to zero. By ignoring serial
correlation in trading profits, both the UM and CM models have φ = 0 with the CM model
further constraining θ to be zero. Thus, equation (4) is the most general expression for the
convergence rate to arbitrage and nests the more restrictive trading profit assumptions.
       Having specified four different processes for incremental trading profits, the next section
introduces our improved statistical methodology.



II.         Robust Tests for Statistical Arbitrage
In this section, we provide a robust methodology to test for SA. In principle, hypothesis
testing may be conducted with either SA or NSA as the null. However, the accepted paradigm
has the null hypothesis being market efficiency. Hence, we only consider NSA as the null.
       The hypothesis of market inefficiency, namely the existence of SA, consists of joint re-
strictions on the parameters underlying the evolution of trading profits. For the UM model,
the following restrictions have to be satisfied simultaneously for a SA opportunity to exist:6

   1. R1 : µ > 0         and

   2. R2 : −λ > 0            and

                         1
   3. R3 : θ − λ +       2
                             >0    and
   6
       A slight change of notation is adopted to separate H3 into two restrictions to facilitate the exposition of
the proposed test.

                                                         8
   4. R4 : θ + 1 > 0 .

      Thus, SA is defined by an intersection of sub-hypotheses. Conversely, the NSA null
hypothesis involves a union of sub-hypotheses (a consequence of DeMorgan’s Laws). In
particular, the NSA null hypothesis is written as:

       c
   1. R1 : µ ≤ 0        or

       c
   2. R2 : −λ ≤ 0           or

       c                1
   3. R3 : θ − λ +      2
                            ≤0   or

       c
   4. R4 : θ + 1 ≤ 0 .

                                             c
      Therefore, if a single sub-hypothesis Ri is accepted, then market efficiency is not rejected.
Statistically, the NSA null hypothesis presents a challenge as the Bonferroni procedure ap-
plies to an intersection, not union, of sub-hypotheses.7 In Appendix A, we examine the
Bonferroni approach in HJTW for testing a union of sub-hypotheses and highlight its lack
of power as the number of sub-hypotheses increase.
      Given the limitations of the Bonferroni approach, this section proposes a new SA testing
methodology centered on the Min-t statistic. We first consider trading profit innovations that
are assumed to be normally distributed and serially uncorrelated. In these circumstances,
critical values for the Min-t test procedure are estimated using Monte Carlo simulation.
We then allow the innovations to be non-normal as well as serially correlated and estimate
p-values for the Min-t statistics using a bootstrap procedure.


A.       Monte Carlo Procedure for Uncorrelated Normal Errors
                                                                                           1
                                                                  ˆ     θ ˆ
When each Ri is considered separately, the t-statistics t(ˆ ), t(−λ), t ˆ − λ +
                                                          µ                                2
                                                                                               and t(ˆ +1)
                                                                                                     θ
test the restrictions R1 , R2 , R3 and R4 respectively, where hats denote the MLE parameter
estimates. However, as all restrictions must be simultaneously satisfied to reject the null
  7
      See Gourieroux and Monfort (1995), Chapter 19, for an exposition of testing joint hypotheses using the
Bonferroni procedure.




                                                      9
hypothesis, the minimum of these t-statistics serves as the rejection criterion. Thus, we
define

                                                            1
                    Min-t = Min t(ˆ ), t(−λ), t ˆ − λ +
                                  µ       ˆ     θ ˆ         2
                                                                 , t (ˆ + 1) ,
                                                                      θ                     (5)

and reject the null of NSA if Min-t > tc , where the critical value tc depends on the test’s
significance level denoted α. Intuitively, the Min-t statistic evaluates the “weakest link” in
the union by focusing on the sub-hypothesis that is “closest” to being accepted.
   As the null of NSA involves a family of t-distributions, rather than a single distribution,
the probability of rejecting the null varies across different parameter values. However, the
probability of rejecting the null cannot exceed α. In other words, we require

                                Pr{Min-t > tc |µ, λ, θ, σ} ≤ α                              (6)

for all (µ, λ, θ, σ) combinations satisfying the null. Thus, two issues have to be addressed.
First, while each of the four t-statistics has an asymptotic standard normal distribution, their
joint distribution is unknown. Hence, the distribution of the Min-t statistic is intractable.
We propose to overcome this difficulty using Monte Carlo simulation. Second, to achieve
a size-α test as in equation (6), the critical value tc must be maximized over the null’s
parameter space.
   We first consider the CM model whose two SA sub-hypotheses are R1 and R2 . Obviously,
tc is maximized when (µ, λ) = (0, 0). Furthermore, as the t-statistics are scale free, we
are able to select an arbitrary value of σ when estimating tc . We assume σ = 0.01, which
approximates its sample MLE estimate in our later empirical study. To estimate tc , residuals
zi are obtained from a normal random number generator to form the incremental trading
profits ∆vi in equation (2) based on assumed model parameters (µ, λ, σ) = (0, 0, 0.01). The
MLE model parameters, their individual t-statistics and the corresponding Min-t statistic
are then computed. This procedure is repeated 10,000 times, from which tc is estimated as
the 100(1 − α) percentile of the Min-t statistics.
   Note that the distribution of Min-t is a function of the sample size n. As the series of
trading profits used in our empirical study vary from 288 to 427 observations, sample sizes
of 300 and 400 are examined. Table I summarizes the critical values for α equal to 0.10,

                                              10
0.05 and 0.01. Although the critical values for n equal to 300 and 400 are very similar, we
adopt the larger estimates 0.4754, 0.7484 and 1.2694 (CM with n = 400 in Table I) for our
empirical study.
       For the UM model, there are four inequality restrictions involving three parameters and
not all restrictions may be binding. Thus, a distribution within the null family and on the
boundary of all inequality restrictions is not available. Nonetheless, as the t-statistics that
comprise the Min-t statistic are monotonic in the underlying restrictions, it is appropriate
to focus on their boundaries. Consequently, we consider the following parameter sets where
σ is once again assumed to be 0.01 in all cases:

   1. Model A: (µ, λ) = (0, 0)

   2. Model B: (µ, λ, θ) = −1 × 10−4 , 0, − 1
                                            2


   3. Model C: (µ, λ, θ) = −1 × 10−6 , 0, − 1
                                            2


   4. Model D: (µ, λ, θ) = −1 × 10−4 , − 1 , −1
                                         2


   5. Model E: (µ, λ, θ) = −1 × 10−6 , − 1 , −1
                                         2


       Note that for Model A, the trading profit process does not depend on θ. For Model B,
the distribution is on the boundary of the R1 , R2 and R3 restrictions, while for Model D,
the boundary of R1 , R3 and R4 is relevant. As θ is not identified for µ = 0, a negative value
close to zero, µ = −1 × 10−4 , is selected implying that R1 is nearly on the boundary. We also
study distributions closer to the NSA region with µ = −1 × 10−6 , yielding Models C and
E. Thus, our Monte Carlo results are derived from five parametric specifications with the
estimated critical values summarized in Table I. Since Model E provides the largest critical
value estimates, 0.4189, 0.6007 and 0.9260 (Model E with n = 300) for α equal to 0.10, 0.05
and 0.01 respectively, they are employed in later empirical tests for SA.8 Once again, the
critical values for n equal to 300 and 400 are very similar.
   8
       We have reduced the value of µ to −1 × 10−8 and find the critical values are very stable. While our
search strategy is reasonable, we are not claiming the result is necessarily the absolute maximum over the
null’s parameter space. If the maximum is not correctly identified, the power of the test is weakened but the
Min-t procedure remains a valid size-α test.

                                                    11
B.     Bootstrap Procedure for Correlated Non-Normal Errors

The previous methodology assumes the innovations in incremental trading profits are nor-
mally distributed and serially uncorrelated. However, both assumptions have been shown to
be dubious in empirical finance. Thus, we relax these assumptions by allowing the trading
profit innovations to be non-normal and serially correlated. However, the MA(1) process
for innovations described by equation (3) introduces an unspecified nuisance parameter φ.
Therefore, searching for the maximum critical values using Monte Carlo methods becomes in-
tractable. In particular, the influence of φ on the individual components of the Min-t statistic
is unknown, offering little guidance for a search strategy. Therefore, a bootstrap procedure
estimates the required p-values and enables us to simultaneously relax the assumption of
normality.
     Brock, Lakonishok and LeBaron (1992) introduce the bootstrap technique into the em-
pirical finance literature to study technical trading rules. Since then, this procedure has
been adopted by many authors including Bessembinder and Chan (1998), Chang and Osler
(1999) as well as Sullivan, Timmermann and White (1999). Ruiz and Pascual (2002) provide
an excellent survey of the bootstrap method in empirical finance. Additional methodological
issues surrounding bootstrapping are found in Hall and Wilson (1991) as well as Horowitz
(2001).
     The steps we employ in our bootstrap procedure for the UMC model are:

  1. Estimate the parameters of the UM model with MA(1) errors using quasi-MLE and
       calculate the residuals ˆi using the following equations:
                                                           ˆ
                                                ∆vi − µ iθ
                                                      ˆ
                                           zi =
                                           ˆ         ˆ
                                                  σi
                                                   ˆ λ

       and

                                                ˆ ˆ
                                           ˆi = zi − φ ˆi−1 ,

       with the starting value of ˆ0 being zero. In addition, the Min-t statistic according to
       equation (5) is calculated.

  2. Sample with replacement a set of n residuals denoted { ∗ , . . . ,
                                                            1
                                                                          ∗
                                                                          n}   from the original set
       of residuals {ˆ1 , . . . , ˆn }.

                                               12
                                                      ∗
   3. Generate a bootstrap sample of trading profits ∆vi with the parameter values (µ, λ, θ, σ) =
          −1 × 10−6 , − 1 , −1, 0.01 and the MLE estimate φ using the equations:
                        2
                                                          ˆ

                                                   ∗       ∗    ˆ   ∗
                                                  zi =     i   +φ   i−1


         and

                                                 ∗                ∗
                                               ∆vi = µ iθ + σ iλ zi .

         Note that the (µ, λ, θ, σ) parameters for Model E of Table I are adopted as they rep-
         resent the null member with the largest critical values for normal and uncorrelated
         residuals.

                                        ∗
   4. Calculate the MLE estimates for ∆vi and hence the Min-t statistic denoted Min-t∗ .

   5. Repeat Steps 2 to 4 a total of 1,000 times. The estimated p-value of the Min-t statistic
         is given by the empirical percentage of bootstrapped Min-t∗ values that are larger than
         the sample Min-t calculated in Step 1.

       The relative frequency of trials for which the Min-t∗ statistic from the bootstrap procedure
in Steps 2 to 4 exceeds the sample Min-t statistic in Step 1 estimates the test’s p-value.
Implementing the bootstrap procedure for the CMC model follows in an identical fashion
with θ equal to zero. Note that the guidelines9 provided by Hall and Wilson (1991) and
Horowitz (2001) are adhered to in our procedure.
       As discussed in section II, with normal and uncorrelated errors, the critical Min-t values
estimated from a large scale Monte Carlo experiment are appropriate. Therefore, to validate
the bootstrapped p-values, we modify the above methodology for the CM and UM models
by constraining φ to be zero. As discussed in section IV, the resulting bootstrapped p-value
estimates for the CM and UM models agree with those from the Monte Carlo procedure that
   9
       Horowitz (2001) points out that bootstrapping should be used to estimate a test’s critical value based
on an asymptotically pivotal statistic whose asymptotic distribution under the null does not depend on any
unknown parameters. This condition is satisfied by our test as the t-statistics are asymptotically standard
normal, and thus pivotal. Furthermore, Hall and Wilson (1991) argue that the resampling in the bootstrap
process should be conducted in a manner that reflects the null.


                                                      13
implicitly assumes φ = 0 (as well as normality). This reassuring result indicates convergence
of the bootstrap procedure, and demonstrates the robustness of the Min-t statistic with
respect to the assumption of normality.



III.     Data and Terminology
Our data is identical to that used in HJTW to facilitate a comparison between their Bon-
ferroni results and those of our proposed methodology. The sample period starts in January
1965 and ends in December 2000. Monthly equity returns data are derived from the Center
for Research in Security Prices at the University of Chicago (CRSP). Our analysis covers all
stocks traded on the NYSE, AMEX and NASDAQ that are ordinary common shares (CRSP
sharecodes 10 and 11), excluding ADRs, SBIs, certificates, units, REITs, closed-end funds,
companies incorporated outside the U.S. and Americus Trust Components.
   The stock characteristics underlying the trading strategies include book-to-market eq-
uity, cash flow-to-price ratio, earnings-to-price ratio and annual sales growth. To calculate
book-to-market equity, book value per share is taken from the CRSP/COMPUSTAT price,
dividend and earnings database. We treat all negative book values as missing. We take the
sum of COMPUSTAT data item 123 (Income before extraordinary items (SCF)) and data
item 125 (Depreciation and amortization (SCF)) as cash flow. Only data 123 item is used
to calculate the cash flow if data 125 item is missing. To compute earnings, we draw on
COMPUSTAT data item 58 (Earnings per share (Basic) excluding extraordinary items) and
to compute the sales we utilize COMPUSTAT data item 12 (sales (net)). Also, all prices and
common shares outstanding numbers employed in the calculation of the ratios are computed
at the end of the year.
   To ensure that the accounting variables are known before hand and to accommodate
variation in fiscal year ends among firms, sorting on stock characteristics is performed in
July of year t using the accounting information from year t − 1. Hence, following Fama
and French (1993), to construct the book-to-market deciles from July 1st of year t to June
30th of year t + 1, the stocks are sorted into deciles based on their book-to-market equity
(BE/ME), where the book equity is in the fiscal year ending in year t − 1 and the market


                                             14
equity is calculated in December of year t − 1. Similarly, to construct the cash flow-to-price
deciles from July 1st of year t to June 30th of year t + 1, the stocks are sorted into deciles
based on their cash flow-to-price, where the cash flow is in the fiscal year ending in year t − 1
and the price is the closing price in December of year t − 1. Earnings-to-price is calculated
in a similar fashion. All portfolios are rebalanced every month as some firms disappear from
the sample over the 12-month period.
   The momentum strategies we implement are from Jegadeesh and Titman (1993). These
strategies long (short) the top (bottom) return decile based on formation and holding period
combinations of 3, 6, 9 and 12 months. The value strategies follow from Lakonishok, Shleifer
and Vishny (1994) and long (short) the top (bottom) decile of stocks based on book-to-
market, cash flow-to-price or earnings-to-price ratios of the past year along with past sales
growth.
   Once the long and short portfolio returns are generated, a self-financing condition is
enforced by investing (borrowing) trading profits (losses) generated by the various trading
strategies at the riskfree rate. Riskfree rate data are obtained from Kenneth French’s website.
   In summary, we investigate 16 momentum strategies and adopt the notational convention
of JTx y for a formation period of x months and a holding period of y months. We also
examine 12 value portfolios, denoted BMx, CPx, EPx and Sx for book-to-market, cash
flow-price, earnings-to-price and sales growth strategies respectively, where x represents the
holding period and the formation period is implicitly understood as being one year.



IV.       Empirical Results
We now discuss the empirical implications of the Min-t test for SA based on the momentum
and value strategies described in the previous section. HJTW report that roughly 50% of
the value and momentum strategies are SA opportunities for the CM model. This result
poses a serious challenge to market efficiency. However, the lack of power associated with
their Bonferroni approach results in only one trading strategy out of 28 being a UM SA
opportunity.
   With our methodology circumventing the Bonferroni inequality, we estimate the CM and


                                              15
UM incremental trading profit models with renewed confidence. Furthermore, we investigate
the sensitivity of our results to the assumptions of normality and serial independence.


A.        Uncorrelated Normal Errors

We first estimate both the CM and UM models with uncorrelated errors. Normality is
assumed with model parameters estimated using MLE. Table II summarizes the results for
the Min-t statistics. As the MLE estimates are quite similar to those of the CMC and UMC
models, we defer their reporting to subsequent tables.10
       The results reveal a strong similarity between the CM and UM models. For the momen-
tum portfolios (JTx y, for x, y = 3, 6, 9, 12), 9 of the 16 portfolios are CM SA opportunities
at the 5% significance level, while 7 of those 9 portfolios are UM SA opportunities at the 5%
level. Similarly for the value portfolios (BMx, CPx, EPx and Sx), 7 of the 12 portfolios test
positively for CM SA at the 5% significance level, while 6 of these 7 portfolios test positively
for UM SA at the 5% level. Hence, with both the CM and the UM tests, roughly 50% of the
momentum and value portfolios provide evidence of SA. Thus, the SA tests are not sensitive
to the assumption that expected incremental trading profits are constant over time. Indeed,
the Min-t methodology resolves the disparity between the CM and UM results in HJTW.
       To investigate the performance of the bootstrap procedure, Table II also records the
bootstrapped p-value estimates when the MA coefficient φ is set equal to zero. In the absence
of both serial correlation and non-normality, the Monte Carlo estimated critical values in
section II are appropriate. For ease of comparison, asterisks in Table II denote the SA test’s
statistical significance according to the bootstrapped p-values under the φ = 0 assumption.
For the UM model, there was perfect agreement between the two sets of results while for the
CM model, only 5 out of 28 portfolios displayed any discrepancies, albeit minor.11 Therefore,
besides indicating convergence of our bootstrap procedure, this robustness check confirms
the appropriateness of the uncorrelated normally distributed innovations assumption for
  10
       Allowing for serial correlation does not introduce a systematic bias in any of the parameters. Therefore,
the estimated CM and UM parameters are omitted for brevity but they are available upon request. Note
that the CM parameters are reported in HJTW.
  11
     The trading strategies are: JT9 9, BM1, BM3, CP1 and S3.



                                                        16
incremental trading profits generated by momentum and value strategies.


B.     Correlated Non-Normal Errors

Broadly speaking, Tables III and IV indicate that the CMC results are very similar to
those of the previous subsection. Indeed, 9 out of the 16 momentum portfolios and 7 of
the 12 value portfolios test positively for SA at the 5% level. Thus, the normality and
independence assumptions are only relevant for a few portfolios. For instance, the portfolio
JT3 6 tests positively for CMC SA with the bootstrap procedure but negatively for CM SA
(see Table II). As discussed in the next subsection, when deciding between the four versions
of SA, preference should be accorded to the bootstrap results given the presence of serial
correlation in trading profits.
     The estimation results of the UMC models are given in Tables V and VI for the mo-
mentum and value trading strategies. Once again, many of the portfolios remain UMC SA
opportunities after relaxing the assumptions of normality and serial independence. In par-
ticular, 3 of the 16 momentum strategies and 7 of the 12 value strategies test positively for
SA at the 5% significance level. However, three momentum strategies are “borderline” SA
opportunities at the 5% significance level. Indeed, at the 10% significance level, the results
for the CM, UM, CMC and UMC versions of the SA test are virtually indistinguishable.
At this significance level, the number of momentum and value strategies that constitute SA
varies between 8 and 10 (out of 16 and 12 respectively) across the four incremental trading
profit assumptions.
     To summarize, our empirical results demonstrate that conclusions regarding market ef-
ficiency are robust with respect to the assumed process for incremental trading profits. In
addition, our statistical procedures confirm the usefulness of the SA theory for testing market
efficiency as generalized trading profit assumptions may be estimated without compromising
statistical power.




                                             17
C.        Probability of Loss

Another advantage of the SA methodology is its ability to yield the probability of a loss at
specific time horizons. Shleifer and Vishny (1997) demonstrate the importance of capital
constraints and intermediate losses to trading decisions. Given these considerations, not all
SA opportunities are equally desirable and the convergence rates of the loss probabilities
to zero offer guidance regarding which strategies to pursue. From this perspective, the
statistical procedures provided by this paper are of considerable practical importance.
       Table VII records the number of months required for the loss probability to fall below
five and one percent for each trading strategy. The selected strategies are those which tested
positively for SA at the 10% significance level across all four models, with at least two being
significant at the 5% level. When interpreting the loss probabilities over the four models,
the appropriate version of the SA test depends on the estimated UMC parameters presented
in Tables V and VI. For example, time-varying expected profits are important when ˆ is
                                                                                 θ
                                                                                         ˆ
statistically different from zero, while accounting for serial correlation is relevant if φ is
statistically significant. In Table VII, the appropriate models are denoted in boldface.
       All else being equal, a positive estimate of φ implies the probability of a loss converges
to zero at a slower rate. The presence of positive serial correlation in the majority of the
trading strategies is confirmed by the significant φ parameters. In particular, all momentum
strategies and 3 out of 8 value strategies should be evaluated after incorporating serial
correlation.
       Conversely, even if θ is significantly negative, the rate of convergence to arbitrage may
not be reduced. Indeed, the decision to estimate the θ parameter influences the µ estimate.
For example, observe that for the JT3 9 and JT9 12 momentum strategies, the UM (UMC)
models converge considerably faster than their CM (CMC) counterparts.
       Overall, observe that the book-to-market (BM) strategies converge to arbitrage. Indeed,
after approximately 3 years, the probability of incurring a loss is below 5%. The size port-
folios S1 and S3 are also very attractive investment opportunities.12 For further illustration,
Figures 1 and 2 present loss probabilities over a ten-year time horizon for one representative
  12
       Note that the extremely slow convergence of the CP3 and CP5 value strategies is due to their large σ
estimates.


                                                     18
momentum and value trading strategy, JT9 12 and S3, respectively.
   The entries in Table VII are computed using equation (4) which assumes normality.
Therefore, as a robustness check, distribution-free bootstrapped loss probabilities are also
computed based on 10,000 trials. This bootstrap procedure searches every generated sample
path to determine the relative frequency of incurring a loss at each monthly horizon. How-
ever, the results are nearly identical to those produced by equation (4) and are omitted for
brevity but available upon request.
   To summarize, besides offering greater power and consistency, our robust SA testing
methodology is critical for selecting the most desirable trading strategies. Despite its impor-
tance in ascertaining the loss probability, φ has no role in the NSA null hypothesis. Indeed,
φ is a nuisance parameter in the actual SA test. Nonetheless, the influence of its inclusion
on the other parameters and their standard errors is unknown apriori. Therefore, in future
applications, the contribution of φ need not be limited to the analysis of loss probabilities.



V.     Conclusion
Given the importance of market efficiency to finance, every effort should be extended to
accurately assess the validity of this fundamental tenet. Two important contributions for
testing market efficiency using statistical arbitrage are proposed in this paper.
   First, a more powerful test for statistical arbitrage that circumvents the limitations of the
Bonferroni approach is provided. Empirically, we document the importance of our robust
statistical tests using momentum and value trading strategies. Our procedures resolve the
empirical disparity in Hogan, Jarrow, Teo and Warachka (2004) by identifying statistical
arbitrage opportunities when expected incremental trading profits are time-varying. Conse-
quently, we find that conclusions regarding market efficiency are independent of whether or
not a trend in expected profits is estimated.
   Second, the proposed procedure is extended to allow for serial correlation and non-
normality in trading profit innovations. We then investigate the sensitivity of our decision to
reject market efficiency with respect to these generalizations. Relaxing the assumption that
incremental trading profits are serially uncorrelated and normally distributed does not limit


                                               19
our ability to detect statistical arbitrage opportunities. Overall, we find that rejections of
market efficiency are robust with respect to four different assumed trading profit processes.
   In summary, this paper provides improved test procedures for statistical arbitrage that
minimize the possibility of accepting market efficiency due to a lack of statistical power.
Indeed, more general trading profit dynamics may be investigated to determine the most
appropriate process. This flexibility confirms the appropriateness of the statistical arbitrage
theory in Hogan, Jarrow, Teo and Warachka (2004) as a test of market efficiency.
   In addition, we demonstrate that computing the probability of a trading strategy incur-
ring a loss is sensitive to the assumed incremental profit process. Therefore, when determin-
ing which trading strategies are closest to being standard arbitrage opportunities, estimating
parameters associated with time-varying expectations and autocorrelation is crucial. In par-
ticular, positive serial correlation in trading profit innovations reduces the convergence rate
to arbitrage.
   Promising avenues for future research include testing other persistent anomalies, such as
the abnormal returns from earnings announcements, analyst forecasts or changes in dividend
policy, for statistical arbitrage.



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 [2] Bondarenko, Oleg, 2003, Statistical arbitrage and securities prices, Review of Financial
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 [3] Brock, William A., Josef Lakonishok, and Blake LeBaron, 1992, Simple technical trading
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 [4] Chang, P.H. Kevin, and Carol L. Osler, 1999, Methodical madness: Technical analysis
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 [5] Fama, Eugene F., 1998, Market efficiency, long-term returns, and behavioral finance,
     Journal of Financial Economics 49, 283-306.

                                             20
 [6] Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on
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 [8] Hall, Peter, and Susan R. Wilson, 1991, Two guidelines for bootstrap hypothesis testing,
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 [9] Hogan, Stephen C., Robert A. Jarrow, Melvyn Teo, and Mitch Warachka, 2004, Testing
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[10] Horowitz, Joel L., 2001, The bootstrap and hypothesis tests in Econometrics, Journal
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[11] Jegadeesh, Narasimhan, and Sheridan Titman. 1993, Returns to buying winners and
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                                             21
Appendix

A      Bonferroni Approach for Multiple Hypotheses
This appendix discusses the Bonferroni approach for testing sub-hypotheses, with particular
reference to testing for SA as in HJTW.
    Let H0 be the null hypothesis consisting of K sub-hypotheses h1 , ..., hK , all of which are
required to hold under H0 . Thus, the rejection of even one sub-hypothesis rejects the null
H0 . As a consequence, H0 is the intersection of sub-hypotheses given by
                                                           K
                                                 H0 :            hi .
                                                           i=1

In the Bonferroni procedure, each sub-hypothesis hi is tested at a given level of significance
αi with a critical region denoted Ci so that Pr(Ci |H0) = αi . The critical region of the null
                                  K
hypothesis H0 is the union        i=1   Ci . Let Cic be the complement of Ci . The null hypothesis
H0 is accepted if all the sub-hypotheses are accepted. Suppressing the conditioning notation,
                                                  K
the probability of accepting H0 is Pr             i=1   Cic .
    The Bonferroni inequality states that
                                  K                        K                        K
                            Pr         Cic     ≥1−              Pr(Ci ) = 1 −             αi
                                 i=1                    i=1                         i=1

from which we obtain
                                        K                           K
                                              αi ≥ 1 − Pr                Cic    .                             (7)
                                        i=1                        i=1

              K
Therefore,    i=1   αi is an upper bound on the size of the statistical test, that is, the probability
of committing a Type I error.
    If H0 is not satisfied, then at least one sub-hypothesis, say hj , is not satisfied. As
                                                 K
                                         Pr           Ci       ≥ Pr(Cj ) ,
                                                i=1

we observe that if all the sub-tests reject their sub-hypothesis with probability one as the
                                                                        K
sample size tends to infinity, Pr(Cj ) → 1, then Pr                      i=1   Ci → 1. As a result, the Bonferroni
test is consistent.

                                                           22
   However, in the SA test conducted by HJTW, the null hypothesis of NSA is a union
of sub-hypotheses. This statement is a consequence of the fact that to reject NSA, all the
sub-hypotheses must be rejected. Rejecting one sub-hypothesis is not sufficient to reject
NSA. Thus, the null hypothesis is defined as
                                                            K
                                                 ∗
                                                H0     :         hi
                                                           i=1

                                  ∗                   K
and the probability of accepting H0 is Pr             i=1    Cic . As the probability of a union is greater
than its corresponding intersection, we have
                                         K                             K
                                    Pr         Cic     ≥ Pr                 Cic                             (8)
                                         i=1                          i=1

which, when combined with equation (7), yields the relationship
                         K                      K                                 K
                             αi ≥ 1 − Pr              Cic        ≥ 1 − Pr               Cic   .             (9)
                       i=1                      i=1                               i=1

                             K
Thus, we conclude that       i=1   αi is also an upper bound on the size of the test for the null
            ∗
hypothesis H0 defined in terms of a union. However, equation (9) implies the Bonferroni
                                  ∗
inequality is a weaker bound for H0 than for H0 . Furthermore, the Bonferroni test is gen-
                           ∗
erally not consistent for H0 , in contrast to H0 . Indeed, when K is large the actual size of
                         ∗                                       K
the Bonferroni test for H0 may be far below                      i=1   αi , resulting in a test with low power.
Conversely, the Min-t test has the correct nominal size. To the extent that searching for the
maximum probability of rejection over the parameter space H0 results in the true maximum,
the power of the test is also enhanced.




                                                       23
                                 Probability of a Loss for JT9 12 Trading Strategy
                  0.5
                                                                                            CM
                                                                                            UM
                 0.45                                                                       CMC
                                                                                            UMC

                  0.4


                 0.35


                  0.3
   Probability




                 0.25


                  0.2


                 0.15


                  0.1


                 0.05


                   0
                        0   20         40             60                 80          100          120
                                                     Months


Figure 1: The trading strategy JT9 12 denotes a Jegadeesh and Titman (1993) momentum portfolio
with a formation period of 9 months and a holding period of 12 months. Plotted above, for the JT9 12
trading strategy, are loss probabilities derived from parameter estimates of the CM (constrained mean)
and UM (unconstrained mean) models along with their counterparts CMC (constrained mean with
correlation) and UMC (unconstrained mean with correlation) that allow for non-normal correlated
innovations in trading profits. The probability of a loss is computed according to equation (4). Observe
that the unconstrained mean tests have faster convergence rates to zero despite a negative value for θ
(rate of change in mean). This result is a consequence of the larger parameter estimates for µ (mean) in
the unconstrained models. Also observe that the convergence rates are slower for the CMC and UMC
trading profit models with serially correlated innovations. This result is expected when the MA(1)
parameter φ is positive while the remaining estimates of µ, θ and λ (rate of change in volatility) are
almost identical to the case with uncorrelated trading profit increments. As seen in equation (4), the
denominator (variance) is larger when the parameter φ is positive, which reduces the rate of convergence
to arbitrage.
                                   Probability of a Loss for S3 Trading Strategy
                 0.45
                                                                                            CM
                                                                                            UM
                  0.4                                                                       CMC
                                                                                            UMC


                 0.35


                  0.3
   Probability




                 0.25


                  0.2


                 0.15


                  0.1


                 0.05


                   0
                        0   20        40              60                 80        100            120
                                                     Months


Figure 2: The sales growth portfolio of Lakonishok, Shleifer and Vishny (1994) denoted S3 has a three
year holding period and a one year formation period. Plotted above, for the S3 trading strategy, are loss
probabilities derived from parameter estimates of the CM (constrained mean) and UM (unconstrained
mean) models along with their counterparts CMC and UMC that allow for non-normal correlated
innovations in trading profits. The probability of a loss is computed according to equation (4). Observe
that the unconstrained mean tests have faster convergence rates to zero despite a negative value for θ
(rate of change in mean). This result is a consequence of the larger parameter estimates for µ (mean) in
the unconstrained models. Also observe that the convergence rates are slower for the CMC and UMC
trading profit models with serially correlated innovations. This result is expected when the MA(1)
parameter φ is positive while the remaining estimates of µ, θ and λ (rate of change in volatility) are
almost identical to the case with uncorrelated trading profit increments. As seen in equation (4), the
denominator (variance) is larger when the parameter φ is positive, which reduces the rate of convergence
to arbitrage.
Table I: Critical Values of the Min-t Statistic for CM and UM Models

The UM (unconstrained mean) version of statistical arbitrage has time-varying expected trading
profits while the CM (constrained mean) model has constant expected trading profits. Both the UM
and CM models have uncorrelated trading profit innovations as described in equations (1) and (2)
respectively. The parameters which describe the evolution of incremental trading profits are µ (mean),
θ (rate of change in mean) and λ (rate of change in volatility) as well as σ (volatility) whose value
is 0.01 without loss of generality. The various model-dependent parameter specifications lie on (or
very near) the boundaries of the no statistical arbitrage null hypothesis. For a given sample size, the
critical values presented below are estimated from Monte Carlo samples of 10,000 trials with details in
section II. Model E with n=300 produces the largest critical values for the UM version of statistical
arbitrage. Therefore, these critical values are adopted in later hypothesis testing and highlighted in
bold below. For the CM test, the sample size n=400 yields the largest critical values and serves an
identical purpose. Note that Model A does not require the θ parameter to be specified since having µ
equal zero eliminates θ from the incremental trading profit process. However, the Min-t statistic for
Model A is comprised of four individual t-statistics versus two for the CM version which explains the
disparity in their critical values.

                            Parameters          Sample size       Significance level α
              Model         µ      λ       θ        n           10%        5%         1%


            CM              0         0    -        300        0.4552    0.7639    1.2513
                            0         0    -        400       0.4754    0.7484    1.2694

            UM     A        0         0    -        300        0.2729    0.4386    0.7302
                                                    400        0.2753    0.4416    0.7468

                                           1
                   B    −1 × 10−4     0   –2        300        0.0546    0.2917    0.6299
                                                    400        0.0893    0.3036    0.6129

                                           1
                   C    −1 × 10−6     0   –2        300        0.0615    0.2951    0.6592
                                                    400        0.0777    0.2986    0.6317

                   D    −1 × 10−4    –1
                                      2   –1        300        0.3951    0.5855    0.9151
                                                    400        0.4003    0.5904    0.9092

                   E    −1 × 10−6    –1
                                      2   –1        300       0.4189    0.6007    0.9260
                                                    400        0.3874    0.5711    0.8570
Table II: Min-t Statistics and Bootstrapped p-values for the CM and UM Models

Momentum portfolios of Jegadeesh and Titman (1993) are abbreviated JTx y to denote a for-
mation period of x months and a holding period of y months. Value portfolios of Lakonishok, Shleifer
and Vishny (1994) are abbreviated BMx, CPx, EPx and Sx to represent book-to-market, cash flow-
price, earnings-to-price and sales growth strategies respectively, with x denoting the holding period and
the formation period implicitly understood as being one year. The UM (unconstrained mean) version of
statistical arbitrage has time-varying expected trading profits while the CM (constrained mean) model
has constant expected trading profits. Both the UM and CM models have uncorrelated trading profit
innovations as described in equations (1) and (2) respectively. The sample Min-t statistics are recorded
below with their significance at the 10%, 5% and 1% levels denoted by one, two and three asterisks
according to the critical values in Table 1. In particular, the corresponding critical values for the CM
test at the 10%, 5% and 1% levels are 0.4754, 0.7484 and 1.2694 while their UM counterparts are
0.4189, 0.6007 and 0.9260. Furthermore, as a robustness check, bootstrapped p-values under the φ = 0
constraint are computed and displayed below with their statistical significance indicated by asterisks
in parentheses. Thus, a p-value larger than 5% but smaller than 10% is denoted by one asterisk and
so forth. For emphasis, asterisks in parentheses denote the significance associated with bootstrapped
p-values while those without parentheses in the Min-t column are generated by Monte Carlo simulation.
Observe that the Monte Carlo and bootstrap procedures produce nearly identical results. The Min-t
test statistics are defined as Min t(ˆ ), t(−λ) and Min t(ˆ ), t(−λ), t(ˆ − λ + 1 ), t (ˆ + 1) for the
                                       µ      ˆ                 µ       ˆ    θ ˆ 2        θ
respective CM and UM versions of statistical arbitrage.

                            Sample Size         CM Model                     UM Model
                Portfolio       n            Min-t  p-value              Min-t   p-value


                JT3   3        427        –1.7851      0.875           –0.7214     0.631
                JT3   6        424         0.6734*     0.067 (*)       –0.2027     0.622
                JT3   9        421         1.9282***   0.003 (***)      0.6753**   0.043 (**)
                JT3   12       418         2.6645***   0.000 (***)      0.7341**   0.020 (**)

                JT6   3        424         0.2914      0.163           –1.0223     0.704
                JT6   6        421         3.0628***   0.000 (***)      0.3588     0.121
                JT6   9        418         1.8040***   0.002 (***)      0.8790**   0.015 (**)
                JT6   12       415         3.4718***   0.000 (***)      0.8571**   0.014 (**)

                JT9   3        421         0.9223**    0.038   (**)    0.4553*     0.100   (*)
                JT9   6        418         1.4484***   0.010   (***)   0.9143**    0.013   (**)
                JT9   9        415         1.1695**    0.009   (***)   1.1018***   0.005   (***)
                JT9   12       412         3.0312***   0.000   (***)   2.1918***   0.000   (***)

                JT12   3       418         0.4137      0.144            0.1899     0.206
                JT12   6       415         0.2542      0.165            0.1213     0.200
                JT12   9       412        –0.9525      0.632           –1.0555     0.707
                JT12   12      409         0.0115      0.250           –1.1042     0.678

                BM1            414          0.6338*    0.043 (**)       0.5256*    0.055 (*)
                BM3            372         0.8037**    0.059 (*)       0.8192**    0.016 (**)
                BM5            324         1.4265***   0.007 (***)     1.2461***   0.001 (***)

                CP1            414         0.6893*     0.109           –0.1216     0.199
                CP3            372         1.6981***   0.004 (***)      0.6513**   0.043 (**)
                CP5            324         1.8835***   0.003 (***)      0.4232*    0.097 (*)

                EP1            414        –4.9990      0.999           –4.7874     0.975
                EP3            372        –1.1115      0.681           –1.1128     0.719
                EP5            324         0.6190*     0.082 (*)        0.2480     0.166

                S1             378         1.5366***   0.002 (***)     1.4009***   0.000 (***)
                S3             336         1.3647***   0.013 (**)      1.3913***   0.001 (***)
                S5             288         2.8911***   0.000 (***)     0.9687***   0.007 (***)
Table III: CMC model with MA(1) Errors for Momentum Trading Strategies

The CMC (constrained mean with correlation) model has correlated innovations in trading prof-
its described by a MA(1) process and expected trading profits that are constant over time, as described
in equations (2) and (3). Momentum portfolios of Jegadeesh and Titman (1993) are abbreviated
JTx y to denote a formation period of x months and a holding period of y months. For each
momentum trading strategy, the first row records the MLE parameter estimates of the CM model with
MA(1) errors, while the second row provides their t-statistics. The Min-t test statistic is defined as
        µ      ˆ
Min{t(ˆ ), t(−λ)}. Trading strategies capable of producing statistical arbitrage at the 10%, 5% and
1% significance levels are denoted with *, ** and *** asterisks respectively. Thus, a p-value less than
or equal to 1% is denoted by three asterisks while those larger than 1% (5%) but less than or equal to
5% (10%) are denoted by two (one) asterisks.

                                          Parameters
                  Portfolio       µ        λ         σ         φ      Min-t    p-value


                  JT3 3       –0.0024    0.0148   0.0249    0.4108   –1.2642   0.755
                              –1.2642    0.2005   2.8295   10.0934
                  JT3 6        0.0016   –0.0510   0.0194    0.6234    0.9498   0.036 (**)
                               1.3333   –0.9498   3.7308   16.0257
                  JT3 9        0.0022   –0.0848   0.0178    0.5692    1.2219   0.016 (**)
                               2.4444   –1.2219   3.0690   19.4266
                  JT3 12       0.0033   –0.1271   0.0174    0.5777    2.4397   0.001 (***)
                               4.7143   –2.4397   3.7021   21.4758

                  JT6 3        0.0022   –0.0045   0.0300    0.4234    0.0661   0.210
                               1.0476   –0.0661   3.1250   13.4841
                  JT6 6        0.0043   –0.1445   0.0303    0.6778    3.0394   0.000 (***)
                               3.5833   –3.0394   4.2676   17.0730
                  JT6 9        0.0055   –0.0895   0.0193    0.6217    1.2000   0.015 (**)
                               5.5000   –1.2000   2.8806   19.9904
                  JT6 12       0.0042   –0.2012   0.0240    0.7217    3.4051   0.000 (***)
                               6.0000   –3.4051   3.2877   25.3228

                  JT9 3        0.0036   –0.0181   0.0313    0.4038    0.2870   0.170
                               1.8947   –0.2870   3.3298   13.1104
                  JT9 6        0.0064   –0.0467   0.0197    0.6089    1.1173   0.023 (**)
                               5.3333   –1.1173   4.6905   18.1220
                  JT9 9        0.0053   –0.0130   0.0128    0.5665    0.2305   0.146
                               5.8889   –0.2305   3.8788   20.3777
                  JT9 12       0.0030   –0.1397   0.0166    0.6422    3.1645   0.000 (***)
                               4.2857   –3.1645   4.3684   29.7315

                  JT12 3       0.0060   –0.0106   0.0293    0.3938    0.1974   0.183
                               3.1579   –0.1974   4.1268   13.5793
                  JT12 6       0.0056   –0.0440   0.0178    0.6537    1.1369   0.015 (**)
                               4.6667   –1.1369   5.2353   18.3109
                  JT12 9       0.0038    0.0304   0.0095    0.5834   –0.5147   0.395
                               4.7500    0.5147   3.8000   22.7004
                  JT12 12      0.0015   –0.0255   0.0083    0.7004    0.6602   0.056 (*)
                               2.5000   –0.6602   5.1875   27.1473
Table IV: CMC model with MA(1) Errors for Value Trading Strategies

The CMC (constrained mean with correlation) model has correlated innovations in trading prof-
its described by a MA(1) process and expected trading profits that are constant over time, as
described in equations (2) and (3). Value portfolios of Lakonishok, Shleifer and Vishny (1994) are
abbreviated BMx, CPx, EPx and Sx to represent book-to-market, cash flow-price, earnings-to-price
and sales growth strategies respectively, with x denoting the holding period and the formation period
implicitly understood as being one year. For each value trading strategy, the first row records the
MLE parameter estimates of the CM model with MA(1) errors, while the second row provides their
                                                             µ      ˆ
t-statistics. The Min-t test statistic is defined as Min{t(ˆ ), t(−λ)}. Trading strategies capable of
producing statistical arbitrage at the 10%, 5% and 1% significance levels are denoted with *, ** and
*** asterisks respectively. Thus, a p-value less than or equal to 1% is denoted by three asterisks while
those larger than 1% (5%) but less than or equal to 5% (10%) are denoted by two (one) asterisks.

                                           Parameters
                    Portfolio      µ        λ         σ        φ      Min-t    p-value


                    BM1         0.0140   –0.0328   0.0521   0.0984    0.5426   0.080 (*)
                                5.8333   –0.5426   3.3613   1.6131
                    BM3         0.0115   –0.0432   0.0411   0.1508    0.7758   0.029 (**)
                                5.7500   –0.7758   3.4831   2.7874
                    BM5         0.0106   –0.0727   0.0401   0.2031    1.3879   0.006 (***)
                                5.5789   –1.3879   3.7477   3.4897

                    CP1         0.0020   –0.2402   0.2413   0.2023    0.5538   0.124
                                0.5538   –4.7470   4.7594   2.6973
                    CP3         0.0036   –0.2320   0.1344   0.0850    1.5699   0.011 (**)
                                1.5699   –3.8992   3.8074   1.6315
                    CP5         0.0037   –0.2794   0.1372   0.1038    1.7301   0.008 (***)
                                1.7301   –4.1577   3.7901   1.7214

                    EP1         0.0022    0.2229   0.0154   0.2208   –4.6552   1.000
                                0.8148    4.6552   4.8125   3.5215
                    EP3         0.0013    0.0728   0.0260   0.1369   –1.0645   0.638
                                0.5909    1.0645   3.2500   2.5119
                    EP5         0.0011   –0.0523   0.0389   0.1946    0.5421   0.119
                                0.5421   –0.8288   3.6019   3.0889

                    S1          0.0085   –0.0860   0.0432   0.0618    1.5198   0.003 (***)
                                5.5641   –1.5198   3.2271   0.9306
                    S3          0.0056   –0.0999   0.0366   0.0975    1.4440   0.006 (***)
                                4.3077   –1.4440   2.8154   1.4444
                    S5          0.0046   –0.1606   0.0435   0.1494    3.2354   0.000 (***)
                                3.2857   –3.2354   4.1429   2.4058
Table V: UMC models with MA(1) Errors for Momentum Trading Strategies

The UMC (unconstrained mean with correlation) model has correlated innovations in trading
profits described by a MA(1) process and expected trading profits that are time-varying, as described
in equations (1) and (3). Momentum portfolios of Jegadeesh and Titman (1993) are abbreviated
JTx y to denote a formation period of x months and a holding period of y months. For each
momentum trading strategy, the first row records the MLE parameter estimates of the UM model with
MA(1) errors, while the second row provides their t-statistics. The Min-t test statistic is defined as
Min t(ˆ ), t(−λ), t(ˆ − λ + 1 ), t (ˆ + 1) . Trading strategies capable of producing statistical arbitrage
        µ       ˆ    θ ˆ 2          θ
at the 10%, 5% and 1% significance levels are denoted with *, ** and *** asterisks respectively. Thus,
a p-value less than or equal to 1% is denoted by three asterisks while those larger than 1% (5%) but
less than or equal to 5% (10%) are denoted by two (one) asterisks.

                                              Parameters
              Portfolio       µ         λ           θ         σ         φ      Min-t    p-value


              JT3 3       –0.0004    0.0187      0.3871    0.0244    0.4101   –0.5737   0.423
                          –0.5737    0.2483      1.2250    2.7416   10.0024
              JT3 6        0.0173   –0.0443     –0.4916    0.0187    0.6224    0.3119   0.117
                           2.0353   –0.8055     –2.9089    3.5962   15.8776
              JT3 9        0.0155   –0.0765     –0.4039    0.0170    0.5705    0.6174   0.079 (*)
                           1.2500   –1.0066     –1.7738    2.7869   19.4048
              JT3 12       0.0122   –0.1183     –0.2570    0.0166    0.5784    0.8034   0.029 (**)
                           0.8034   –1.8989     –0.9809    3.0741   21.5019

              JT6 3        0.0392    0.0082     –0.6053    0.0281    0.4222   –0.6653   0.399
                           3.2131    0.1105     –3.2968    2.8100   13.3608
              JT6 6        0.0158   –0.1385     –0.2581    0.0294    0.6791    0.4629   0.080 (*)
                           0.4629   –2.4175     –0.5649    3.5071   16.9156
              JT6 9        0.0087   –0.0877     –0.0900    0.0192    0.6219    0.6608   0.056 (*)
                           0.6608   –1.0908     –0.3006    2.6667   20.1262
              JT6 12       0.0105   –0.1967     –0.1749    0.0234    0.7217    0.6649   0.063 (*)
                           0.6649   –2.8969     –0.5961    2.8193   25.4120

              JT9 3        0.0279   –0.0094     –0.4484    0.0299    0.4050    0.1393   0.187
                           1.2624   –0.1393     –1.5667    3.1474   13.1494
              JT9 6        0.0072   –0.0464     –0.0213    0.0196    0.6089    0.7225   0.054 (*)
                           0.7225   –1.0474     –0.0790    4.4545   18.1761
              JT9 9        0.0099   –0.0090     –0.1234    0.0126    0.5670    0.1455   0.169
                           1.0645   –0.1455     –0.6361    3.5000   20.6934
              JT9 12       0.0156   –0.1299     –0.3192    0.0157    0.6419    2.4348   0.000 (***)
                           2.4348   –2.5876     –3.4323    3.8293   29.3105

              JT12 3       0.0262   –0.0018     –0.3076    0.0280    0.3930    0.0315   0.198
                           0.8213   –0.0315     –1.0455    3.7333   13.4589
              JT12 6       0.0133   –0.0392     –0.1748    0.0173    0.6519    0.9170   0.028 (**)
                           0.9170   –0.9515     –0.7804    4.8056   18.1588
              JT12 9       0.0140    0.0412     –0.2695    0.0090    0.5823   –0.6656   0.447
                           2.5455    0.6656     –2.7956    3.6000   22.8353
              JT12 12      0.0155    0.0119     –0.4631    0.0068    0.6929   –0.3092   0.273
                           4.5588    0.3092     –5.6545    5.2308   26.7529
Table VI: UMC models with MA(1) Errors for Value Trading Strategies

The UMC (unconstrained mean with correlation) model has correlated innovations in trading
profits described by a MA(1) process and expected trading profits that are time-varying, as described
in equations (1) and (3). Value portfolios of Lakonishok, Shleifer and Vishny (1994) are abbreviated
BMx, CPx, EPx and Sx to represent book-to-market, cash flow-price, earnings-to-price and sales
growth strategies respectively, with x denoting the holding period and the formation period implicitly
understood as being one year. For each value trading strategy, the first row records the MLE parameter
estimates of the UM model with MA(1) errors, while the second row provides their t-statistics. The
Min-t test statistic is defined as Min t(ˆ ), t(−λ), t(ˆ − λ + 1 ), t (ˆ + 1) . Trading strategies capable
                                          µ       ˆ    θ ˆ 2          θ
of producing statistical arbitrage at the 10%, 5% and 1% significance levels are denoted with *, ** and
*** asterisks respectively. Thus, a p-value less than or equal to 1% is denoted by three asterisks while
those larger than 1% (5%) but less than or equal to 5% (10%) are denoted by two (one) asterisks.

                                               Parameters
               Portfolio       µ         λ           θ         σ        φ      Min-t    p-value


               BM1          0.0061   –0.0283      0.1619    0.0509   0.0973    0.4498   0.038 (**)
                            0.9839   –0.4498      0.8603    3.2215   1.5873
               BM3          0.0073   –0.0431      0.0899    0.0411   0.1501    0.7932   0.010 (***)
                            0.9481   –0.7932      0.4466    3.5739   2.7541
               BM5          0.0071   –0.0690      0.0830    0.0394   0.2026    1.2055   0.003 (***)
                            1.2055   –1.2545      0.5217    3.5818   3.4632

               CP1          0.1434   –0.2371     –0.7599    0.2106   0.1999   –0.1019   0.131
                            1.0429   –4.7802     –3.3970    4.3693   2.6268
               CP3          0.0191   –0.2325     –0.3170    0.1347   0.0843    0.6052   0.023 (**)
                            0.6052   –3.9274     –1.0060    3.8376   1.6088
               CP5          0.0026   –0.2793      0.0654    0.1370   0.1038    0.3872   0.051 (*)
                            0.3872   –4.1874      0.1303    3.8375   1.7214

               EP1          0.0314    0.2419     –1.3434    0.0139   0.2162   –4.4626   0.853
                           24.1538    4.4626     –1.4623    4.0882   3.4209
               EP3          0.0073    0.0731     –0.3548    0.0260   0.1364   –1.0653   0.530
                            0.8111    1.0653     –1.0648    3.2500   2.4982
               EP5          0.0029   –0.0523     –0.1987    0.0389   0.1946    0.2116   0.108
                            0.2116   –0.8288     –0.2072    3.6019   3.0889

               S1           0.0090   –0.0863     –0.0121    0.0432   0.0618    1.2807   0.000 (***)
                            1.2815   –1.4847     –0.0822    3.1478   0.9316
               S3           0.0125   –0.1015     –0.1611    0.0369   0.0963    1.3966   0.002 (***)
                            1.3966   –1.4668     –1.1242    2.8168   1.4288
               S5           0.0091   –0.1606     –0.1381    0.0435   0.1485    0.8269   0.010 (***)
                            0.8269   –3.2576     –0.5575    4.1827   2.3913
Table VII: Number of Months Until Loss Probability Declines Below One and Five Percent

The UM (unconstrained mean) version of statistical arbitrage has time-varying expected trading
profits while the CM (constrained mean) model has constant expected trading profits. Both the UM
and CM models have uncorrelated trading profit innovations as described in equations (1) and (2)
respectively. In contrast, their UMC and CMC counterparts allow for serial correlation in trading
profits through the addition of an MA(1) process given in equation (3). Momentum portfolios of
Jegadeesh and Titman (1993) are abbreviated JTx y to denote a formation period of x months and a
holding period of y months. Value portfolios of Lakonishok, Shleifer and Vishny (1994) are abbreviated
BMx, CPx, EPx and Sx to represent book-to-market, cash flow-price, earnings-to-price and sales
growth strategies respectively, with x denoting the holding period and the formation period implicitly
understood as being one year. Trading strategies that imply the existence of statistical arbitrage
at the 10% significance level for all four (CM, UM, CMC and UMC) trading profit models, with at
least two being significant at the 5% level, are examined below in terms of their loss probabilities.
An asterisk denotes trading strategies that constitute statistical arbitrage opportunities at the 5%
level for all four models. The number of months before the probability of a loss declines below 5%
and 1% are reported below, along with the average p-value over the four test procedures. Observe
that the unconstrained version of statistical arbitrage often implies faster convergence to arbitrage,
even for negative values of θ (rate of change in mean) while tests which account for autocorrelation
always experience slower convergence. These two results may be explained by the larger estimates
for µ (mean) in the unconstrained tests and the positive MA(1) parameters φ for serial correlation in
trading profit innovations. Numbers in boldface denote the appropriate trading profit model given the
statistical significance of the θ and φ estimates presented in Tables V and VI.

                                      Number of Months before     Number of Months before
                           Average   Loss Probability below 5%   Loss Probability below 1%
               Portfolio   p-value   CM UM CMC             UMC   CM UM CMC             UMC


               JT3 9       0.0353    162    52    211      48    287   163    383     284

               JT3 12*     0.0125     60    26     81      23    104    55    141      57

               JT6 9       0.0220     38    32     52      36     69    60     94      71

               JT6 12      0.0193     52    30     74      37     87    57    123      73

               JT9 6       0.0250     36    38     51      46     66    70     96      88

               JT9 12*     0.0000     63     9      88     15    109    25     151     40

               BM1         0.0540    33     58      38     66    63     96      73    109

               BM3         0.0285     29    43     37      52     55    74     70      89

               BM5*        0.0043     27    39     39      50     48    66     71      85

               CP3*        0.0202    416   310     467    372    673   713     752    858

               CP5         0.0398    320   328     362    375    505   497     568    568

               S1*         0.0013    41     42      49     47    73     77      88     86

               S3*         0.0055    62     32      73     40    110    70     131     86

               S5*         0.0043     83    54    104      73    142   107    176     142

				
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