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Office of Research 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. References [1] Bessembinder, Hendrik, and Kalok Chan, 1998, Market eﬃciency and the returns to technical analysis, Financial Management 21, 5-17. [2] Bondarenko, Oleg, 2003, Statistical arbitrage and securities prices, Review of Financial Studies 16, 875-919. [3] Brock, William A., Josef Lakonishok, and Blake LeBaron, 1992, Simple technical trading rules and the stochastic properties of stock returns, Journal of Finance 47, 1731-1764. [4] Chang, P.H. Kevin, and Carol L. Osler, 1999, Methodical madness: Technical analysis and the irrationality of exchange-rate forecasts, Economic Journal 109, 636-661. [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 stocks and bonds, Journal of Financial Economics 33, 3-56. [7] Gourieroux, Christian, and Alain Monfort, 1995. Statistics and Econometric Models, Volume 2 (Cambridge Press, Cambridge, UK.). [8] Hall, Peter, and Susan R. Wilson, 1991, Two guidelines for bootstrap hypothesis testing, Biometrics 47, 757-762. [9] Hogan, Stephen C., Robert A. Jarrow, Melvyn Teo, and Mitch Warachka, 2004, Testing market eﬃciency using statistical arbitrage with applications to momentum and value trading strategies, Forthcoming in Journal of Financial Economics. [10] Horowitz, Joel L., 2001, The bootstrap and hypothesis tests in Econometrics, Journal of Econometrics 100, 37-40. [11] Jegadeesh, Narasimhan, and Sheridan Titman. 1993, Returns to buying winners and selling losers: Implications for stock market eﬃciency, Journal of Finance 48, 65-91. [12] Lakonishok, Josef, Andrei Shleifer, and Robert W. Vishny, 1994, Contrarian investment, extrapolation, and risk, Journal of Finance 49, 1541-1578. [13] Ruiz, Esther, and Lorenzo Pascual, 2002, Bootstrapping ﬁnancial time series, Journal of Economic Surveys 16, 271-300. [14] Shleifer, Andrei, and Robert W. Vishny, 1997, The limits of arbitrage, Journal of Fi- nance 52, 35-55. [15] Sullivan, Ryan, Allan G. Timmermann, and Halbert L. White, 1999, Data-snooping, technical trading rule performance and the bootstrap, Journal of Finance 46, 1647-1691. 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