Understanding the Sources of Momentum Profits:

Stock-Specific Component versus Common-Factor

Component∗

Qiang Kang †

University of MiamiCanlin Li ‡

University of California-Riverside

This Draft: August 2007

∗We thank Eric C. Chang, Doug Emery, Yuanfeng Hou, Chris Kirby, Ken Kopecky, A. Craig MacKinlay,Oded Palman, Yexiao Xu, and seminar participants at HKU, Temple, Tulane, UT-Dallas, Rutgers BusinessSchool, University of Miami, FMA/Asian Finance Conference, and European Finance Association Meetingfor valuable comments and suggestions. An earlier draft of the paper was completed while Kang wasaffiliated with HKU, whose hospitality is gratefully acknowledged. We also wish to thank Ken Frenchand Rob Stambaugh for providing us the Fama-French factors data and the liquidity data, respectively.The financial support from University of Hong Kong and University of Miami (Kang) and University ofCalifornia-Riverside (Li) are gratefully acknowledged. All errors, of course, remain our own responsibility.

†Mailing address: Finance Department, University of Miami, Coral Gables, FL 33124-6552. Phone:(305)284-8286. Fax: (305)284-4800. E-mail: q.kang@miami.edu.

‡Mailing address: Graduate School of Management, University of California-Riverside, 900 UniversityAvenue, Riverside, CA 92521. Phone: (909)787-2325 Fax: (909)787-3970. E-mail: canlin.li@ucr.edu.

Understanding the Sources of Momentum Profits: Stock-Specific Componentversus Common-Factor Component

Abstract

This paper examines the relative importance of the stock return’s stock-specificcomponent versus its common-factor component in explaining the momentum profits.Using a model nesting both Chordia and Shivakumar (2002) and Grundy andMartin (2001), we demonstrate that the Fama-French three-factors model leaves outimportant predictive variations in stock returns needed for Chordia and Shivakumar’sresults. In the context of a linear asset pricing model with any choice of factors, weshow that the predictive intercept and hence the predicted returns contain both astock-specific component and a common-factor component. We propose a method,which is free from the missing-factor problem in specifying the asset pricing model,to extract the stock-specific component from the predictive intercept and find that amomentum strategy based solely on this component generates significant profits. Forrobustness, we consider the Fama-French three-factors model with time-varying betasand its extended four-factors model with Pastor and Stambaugh’s (2003) liquidityfactor as the fourth factor. The stock-specific component, if not the only source,appears to be a very important source of momentum profits. In various models andsetups it always generates significant momentum profits in magnitude of over one half ofthe momentum profits in stock returns. We also explore various horizons for portfolioformation and get similar results on the significance of momentum profits based onrankings of this stock-specific component.

JEL Classification: G12, G14Keywords: Momentum, time-varying risk, time-varying risk premium, stock-specificcomponent

1 Introduction

The profitability of the momentum strategy - the strategy of buying recent winning stocksand shorting recent losing stocks - as first documented in Jegadeesh and Titman (1993)remains one of the anomalies that cannot be explained by the otherwise very successfulFama-French three-factors model, and is thus very puzzling (Fama and French, 1996).Jegadeesh and Titman (2001) show that momentum profits remain large even subsequentto the period of their 1993 study. Rouwenhorst (1998) and Griffin, Ji, and Martin (2003)report economically significant and statistically reliable momentum profits in areas outsidethe US. All these studies suggest that the momentum phenomenon is not a product of datamining or data snooping bias.

Although the momentum phenomenon has been well accepted, the source of the profitsand the interpretation of the evidence are widely debated. A variety of papers rangingfrom behavior models to rational-expectation models attempt to offer an explanation. Forthe behavioral arguments, the momentum phenomenon is often interpreted as evidencesthat investors under-react to new information. Along this line, Barberis, Shleifer andVishny (1998), Daniel, Hirshleifer and Subrahmanyam (1998), and Hong and Stein (1999)have developed behavioral models to explain the momentum. The behavioral argumentcommands support of empirical evidences that momentum profits are related to severalcharacteristics not typically associated with the priced risk in standard asset pricingmodels.1 Against the backdrop of the behavioral arguments, others have suggested thatthe profitability of momentum strategies may simply be compensation for risk. Conrad andKaul (1998) argue that the momentum profit is attributed to the cross-sectional dispersionin (unconditional) expected returns. Lewellen (2002) finds that the negative cross-serialcorrelation among stocks, not underreaction, is the main source of momentum profits.2

Using the frequency domain component method to decompose stock returns, Yao (2003)provides strong evidence that momentum is a systematic phenomenon. Theoretic modelshave been developed to link momentum to economic risk factors affecting investment lifecycles and growth rates. Berk, Green and Naik (1999) illustrate that momentum profitsarise because of persistent systematic risk in a firm’s project portfolios. Johnson (2002)posits that momentum comes from a positive relation between expected returns and firmgrowth rates.

Overall, the risk-based explanation attributes the source of momentum profits to thecommon-factor components of stock returns while the behavioral explanation is more likelyto attribute the source to the non-factor-related components (or, for simplicity, stock-specificcomponents hereinafter). To gauge the relevance of these two different stories, it is necessaryfor us to better understand which part is more important in generating the momentumprofits. However, two recent papers seem to offer contradictory evidences on the relativeimportance of these two components.

1An incomplete list of those characteristics includes: earnings momentum (Chan, Jegadeesh, andLakonishok, 1996); industry factor (Moskowitz and Grinblatt, 1999); volume and turnover (Lee andSwaminathan, 2000), analyst coverage (Hong, Lim and Stein, 2000), and 52-week high price (George andHwang, 2003).

2Both Conrad and Kaul (1998) and Lewellen (2002) employ Lo and MacKinlay’s (1990) statisticalframework to decompose the profits of an investment strategy.

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Grundy and Martin (2001) show that the momentum strategy’s profitability cannot beexplained by the Fama-French three-factors model. They argue that the gain instead reflectsthe momentum in the stock-specific components of returns. Chordia and Shivakumar (2002)report that the profits to momentum strategies are completely explained by predictivereturns using the lagged common macroeconomic variables (e.g. dividend yield, term spread,default spread, and short-term rate). The momentum profits are related to the businesscycles and mainly reflect the persistence in the time-varying expected returns. Giventhe well-documented strong predictability of the Fama-French factors by those commonmacroeconomic variables (e.g., Campbell and Shiller, 1988; Fama and French, 1988, 1989;Keim and Stambaugh, 1986), it’s natural to ask how these two seemingly “contradictory”results can coexist and whether a Fama-French-factor model with time-varying risk premiumscan generate the needed time-varying pattern in expected returns as argued in Chordiaand Shivakumar. From the viewpoint of an asset pricing model, the predictive variation ofstock returns obtains from three different sources: time-varying risk, time-varying factor riskpremiums, and time-varying stock-specific component. It is thus important to understandhow much predictability pattern each of the three sources can generate in explaining themomentum profits. This would in turn help us pin down the “right” theoretic model formomentum profits, i.e., whether we should focus our search on a behavioral model or on arisk-based model.

In this paper we apply a model that nests the two models used in Grundy andMartin (2001) and Chordia and Shivakumar (2002) to U.S. equity markets to answer thisquestion and show how these two different sets of results are reconciled with each other.We first confirm the results of Grundy and Martin and Chordia and Shivakumar usingtheir models, respectively. Similar to Grundy and Martin, we find that the Fama-Frenchthree-factors model cannot explain the momentum profits although it can explain most ofthe winner or loser return variability. And like Chordia and Shivakumar, we find thatthe momentum profits are mainly driven by the predictive variations in stock returns thatare related to the common macroeconomic variables. We then use a model similar toFerson and Harvey (1999) to reconcile the two seemingly contradictory results. The Fama-French three-factors model leaves out important predictive variations in stock returns, whichexplains the co-existence of the two different results. We further impose in Chordia andShivakumar’s (2002) predictive framework cross-sectional restrictions implied by an assetpricing model and illustrate that the stock-specific component generates the major part ofthe momentum profits. The time-varying risk premium appears to only have very limitedpower in explaining momentum profits.

In the context of a linear asset pricing model with any choice of factors and by nesting thepredictive relation as a reduced form of such model, we show that the predictive interceptand hence the predicted returns contain both a stock-specific component and a common-factor component that can not be uniquely separated from each other. The momentumprofits explained by the predicted returns can come from either component. We thenpropose a method, free of the missing-factor problem in specifying the asset pricing model,to extract the stock-specific component from the predictive intercept. The momentumstrategy based solely on this stock-specific component delivers significant profits that exhibita striking seasonality and account for more than one half of the total momentum profits

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in stock returns. Although Ferson and Harvey (1991) demonstrate that the time-varyingrisk premiums are mainly responsible for the return predictability that seems to be able toexplain the momentum profits, we find that the bulk of the predictive variations needed togenerate momentum profits actually appear to be driven by the stock-specific component.Interestingly, Griffin, Ji, and Martin (2003) use a predictive regression framework similar toChordia and Shivakumar (2002) to study the sources of momentum profits in internationalequity markets and they report that momentum profits are more related to the stock-specific(or country-specific, in their case) component.

For robustness check, we consider the Fama-French three-factors model with the time-varying betas, its extended four-factors model with Pastor and Stambaugh’s (2003) liquidityfactor as the fourth factor, and variants of the predictive regression. Overall, our resultsremain the same: the stock-specific component always generates significant momentumprofits while the common-factor component rarely does. Like Cooper, Gutierrez, andHameed (2003) and Griffin, Ji, and Martin (2003), we find that some of Chordia andShivakumar’s (2002) results are not robust to the one-month skipping commonly employedto take account of microstructure concerns. We also explore various horizons for portfolioformation and get similar results on the significance of momentum profits based on rankingsof the stock-specific component.

The remainder of the paper proceeds as follows. Section 2 features the models usedto identify the sources of momentum profits; Section 3 describes the data and presentsmomentum profits in stock returns; Section 4 analyzes the empirical evidences on the sourcesof momentum profits; Section 5 discusses the robustness analysis; and Section 6 concludes.

2 Model

2.1 Theory

Consider the following linear asset pricing model3

rit = E(rit|Zt−1) + β′i{Ft − E(Ft|Zt−1)} + εit, (1)

withE(rit|Zt−1) = αi +

∑k

βikλkt(Zt−1), (2)

andλkt(Zt−1) = E(Fkt|Zt−1). (3)

Here rit stands for stock i’s returns in excess of one-month Treasury Bill rates, Zt−1 is thevector of common information variables at t−1, which we assume to include market dividendyield DIVt−1, short-term interest rate Y LDt−1, term spread TERMt−1, and default spread

3Equation (3) says that the factor risk premiums are equal to expected factor means, which is only true ifthe factors are portfolio returns. To derive the predictive regression as in equation (5), we really don’t needequation (3). But to put the predictive regression and the Fama-French-type model into a unified framework,we do need to use this equation.

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DEFt−1, the coefficient vectors βi are the conditional betas of the return ri on the factorsFt, λkt is the risk-premium on factor k, and αi is a stock-specific return component notrelated to any systematic risks. Equation (1) identifies a stock’s systematic risk (βi) througha linear regression of the unexpected stock returns on the unanticipated parts of the riskfactors. Equation (2) links the stock’s expected returns to these systematic risks and theirassociated risk premiums as well as to the possible reward from stock-specific risks. Note intheory an asset pricing model implies that αi equals zero (i.e., nonsystematic risk should notbe rewarded). However this may not be always true in the data and therefore we include astock-specific component in the expected returns. The factor risk premium is assumed to betime-varying and modeled through a linear function of Zt−1 as in Ferson and Harvey (1991)among others:

λkt = ak0 + a′kZt−1 = ak0 + ak1DIVt−1 + ak2Y LDt−1 + ak3TERMt−1 + ak4DEFt−1 (4)

Combining equations (1), (2) and (4) gives the following predictive regression as inChordia and Shivakumar (2002):

rit = ci + δi1DIVt−1 + δi2Y LDt−1 + δi3TERMt−1 + δi4DEFt−1 (5)

+ β′i{Ft − E(Ft|Zt−1)} + εit

≡ ci + δ′iZt−1 + ξit

where ξit = β′i{Ft−E(Ft|Zt−1)}+εit , ci = αi+

∑k βikak0, δij =

∑k βikakj, j = 1, 2, 3, 4. Note

that δij are cross-sectionally restricted through δij =∑

k βikakj, j = 1, 2, 3, 4, and βik are notidentified in this predictive regression. Equation (5) can be regarded as the reduced form ofthe linear asset pricing model characterized by equation (1), equation (2) and equation (4).

For each month t, the above model is typically estimated with a history of 60-month data(the estimation period), say t − 61 to t − 2. The estimated parameters can then be used tocompute the expected/predicted returns in the portfolio formation period, say t− 7 to t− 2.Based on its expected/predicted return over the formation period, a stock is then sortedinto one of the 10 decile portfolios. Profits from the momentum strategy based on rankingof the predicted returns are defined as the profits of longing the top decile portfolio andshorting the bottom decile portfolio. Chordia and Shivakumar (2002) use equation (5), i.e.,the reduced-form linear asset pricing model, to study the relative importance of commonfactors and stock-specific information as sources of momentum profits. They find thatthe momentum strategy based on rankings of the predicted returns generates significantprofits while the momentum strategy based on rankings of the unexplained part does notgenerate significant profits at all. They claim that the momentum profits based on rankingsof past returns are attributable to cross-sectional differences in the conditionally expectedreturns that are explained/predicted by common macroeconomic variables. However, thepredictive regression in equation (5) shows that the intercept ci actually contains twocomponents: the stock-specific component αi and one piece of the common factor component(i.e., the risk premium implied by the factor model)

∑k βikak0. Thus the predicted returns

ci + δ′izt−1 contain both a stock-specific component αi and a common-factor component∑k βikak0 + δ′izt−1. A stock can be sorted into the top decile portfolio because of either a

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high αi or a high∑

k βikak0 (or∑

k βikak0 + δ′izt−1). Accordingly a trading strategy based onthe ranking of predicted returns ci + δ′izt−1 as in Chordia and Shivakumar (2002) is unableto uniquely identify the sources of momentum profits.

To extract the stock-specific component from the predictive relationship, we decomposethe model estimation period into two subperiods: the in-formation period during whichindividual stocks are ranked and momentum portfolios are formed; and the out-of-formationperiod. We assume that the systematic risks of stock i, βi, do not change from the out-of-formation period to the in-formation period but the stock-specific return component αi, if itexists, may change.4 We thus have two (potentially) different predictive intercepts betweenthe two subperiods, and we take the difference to obtain:

ci0 − ci1 = (αi0 +∑

k

βikak0) − (αi1 +∑

k

βikak0) = αi0 − αi1 (6)

where αi0 is the in-formation-period stock-specific return and αi1 is the out-of-formation-period stock-specific return. Unlike ci0 or ci1, the term ci0−ci1 only contains the stock-specificinformation.5 If only the common-factor component in the predicted returns is responsiblefor the momentum profits, a strategy based on rankings of ci0 − ci1 would generate NOmomentum profits, and the two strategies based on rankings of ci0 + δ′iz and ci1 + δ′iz wouldrender similar results. On the other hand, if the stock-specific component is also the source,a strategy based on rankings of ci0 − ci1 is expected to generate significant momentumprofits. Notice that our above discussions do not rely on the assumption of a particularfactor model as the true asset pricing model, though we do need the assumptions of a linearasset pricing model and a linear relationship between the risk premiums and the laggedcommon macroeconomic variables. Further note that equation (6) is free from the missing-factor problem that may occur in the specification of the linear asset pricing model. If,for example, there is one factor F ∗ missing from the specification and the time-varying riskpremium of F ∗ is linearly linked to the instruments via the loadings vector a∗, then, denotingthe stock loadings on the missing factor by β∗, we have

ci0 − ci1 = (αi0 +∑

k

βikak0 + β∗i a

∗0) − (αi1 +

∑k

βikak0 + β∗i a

∗0) = αi0 − αi1, (7)

which is the same as equation (6).

4Empirical asset pricing models typically assume that loadings on factors are stable within a 60-monthestimation horizon (see, e.g., Fama and MacBeth, 1973; and Fama and French, 1992). Ghysels (1998, p569)argues that betas change through time very slowly. Barun, Nelson and Sunier (1995) provide evidence ofvery weak time variation in direct estimates of conditional betas with monthly data. If betas only move veryslowly, the difference in betas between the in-formation period and the out-of-formation period is likely tobe negligible. We relax this assumption and allow for time-varying betas in Section 5.1 and come up withqualitatively very similar results.

5If the asset pricing model holds, then αi0=αi1=0. If the stock-specific return does not change betweenthe two subperiods, then αi0=αi1. In either case, ci0 − ci1=0 and a sorting of stocks based on the ci0 − ci1

term can not generate cross-sectional variations in returns at all.

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2.2 Models

In light of these discussions, we allow for different intercepts ci between the in-formationperiod and the out-of-formation period in our specification of the predictive regression[Chordia and Shivakumar (2002)] and name this as Model A.

Model A: rit = ci0Dt + ci1(1 − Dt) + δ′izt−1 + ξit, (8)

where Dt = 1 for the in-formation period and Dt = 0 for the out-of-formation period. Thecommon factor component is

∑k βikλkt(Zt−1) =

∑k βikak0 + δ′izt−1 and the stock-specific

component is αi0 (note that the momentum portfolios are formed based on rankings inthe formation period). Since αi0 and

∑k βikak0 can not be separately identified from the

in-formation-period intercept ci0, the predicted returns ci0 + δ′izt−1 actually contains boththe stock-specific information and the common-factor-related information. As shown inequation (6), the component ci0−ci1 in contrast contains only the stock-specific information.

Model B is closely related to equation (8) (or equation (5)). When the time-varying riskpremiums are not explicitly modeled and the Fama-French three-factors model is assumedas the true asset pricing model so that βik can be explicitly modeled, the model used inGrundy and Martin (2001) obtains:

Model B: rit = αi0Dt + αi1(1 − Dt) +∑

k

βikλkt(Zt−1) + β′i{Ft − E(Ft|Zt−1)} + εit

= αi0Dt + αi1(1 − Dt) + β′iFt + εit, (9)

The common-factor component is β′iFt and the stock-specific component is αi0.

6 UnlikeModel A, the common-factor component in Model B is defined on the basis of the ex-postfactor risk premiums.7

Model C is a generalized version of Model B. We explicitly specify the predicted patternin rt that may be left unexplained by Model B through the predictable variations in theFama-French factor risk premiums. This specification follows Ferson and Harvey (1999) inwhich a similar model is used to show that the Fama-French three-factors model fails inexplaining the time-varying pattern in expected returns.

Model C: rit = αi0Dt + αi1(1 − Dt) + γ′izt−1 + β′

iFt + εit, (10)

The common-factor component in this model is β′iFt while the stock-specific component

is αi0 + γ′izt−1.

8 If the Fama-French model explains the conditional expected returns, the

6The term αi0 is actually a non-Fama-French-factor-related component. From this point onward, we, asin Grundy and Martin (2001), call it the stock-specific component for brevity. Such convenience is furtherused in Section 5.2.

7Strictly speaking, Model B and equation (8) are non-nested. Given equation (3), Model B requires aspecific Fama-French factor model but no specifications of the time-varying risk premiums while equation (8)requires a linear specification of the time-varying risk premiums but no specifications of an asset pricingmodel. See also Footnote 3 for more subtle differences between the two models.

8For convenience, we abuse the notation αi0 and αi1 in that they stand for the abnormal returns in ModelB but have no such asset-pricing implications in Model C.

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predictive coefficients γ equal zero (and so do the intercepts α0 and α1).

Model D is a special case of Model B when the time-varying Fama-French factor riskpremiums are explicitly modeled as linear functions of lagged macroeconomic variables. Itis also a special case of equation (8) when a particular cross-sectional restriction is imposed,that is, when the Fama-French three-factors model is used as the true asset pricing model.

Model D: rit = αi0Dt + αi1(1 − Dt) + β′iFt + εit, (11)

Fkt = ak0 + ak1DIVt−1 + ak2Y LDt−1 + ak3TERMt−1 + ak4DEFt−1 + ηkt (12)

The first equation allows us to identify βi through the factor model and the second equationmodels the time-varying risk premiums. The common-factor component is

∑k βikλkt(Zt−1) =∑

k βikF̂kt, where F̂kt is the fitted Fkt from the second equation, and the stock’s stock-specificcomponent is then the difference between the return ri and its common-factor componentβ′

iF̂t.

We use the above four models to decompose stock returns into different parts and studythe profitability of momentum strategies based on rankings of each part. This would helpus understand the sources of momentum profits in stock returns.

3 Data and Momentum Payoffs in Stock Returns

3.1 Data

The study uses all NYSE/AMEX stocks on the Center for Research in SecurityPrices (CRSP) monthly database from December 1925 through December 2002 (925 months).Fama-French three factors are used in the tests to control for risks in Models B, C and D.They include the return on CRSP value-weighted market index in excess of the one-monthTreasury bill rate (MKT RF), the small-minus-big size factor (SMB) and the high-minus-lowbook-to-market-ratio factor (HML) from July 1926 through December 2002 (918 months).9

To capture the time-varying risk premium, we use the following four macroeconomic variablesthat prior studies have found to predict market returns: the lagged values of the value-weighted market dividend yield, term premium, default premium, and short rate, all fromDecember 1926 through December 2002 (913 months). Fama and French (1989), amongothers, show that those variables are related to business conditions. The dividend yield (DIV)is defined as the total dividend payment accrued to the CRSP value-weighted market indexover the past 12 months divided by the current price level of the market index. The termpremium (TERM) is the yield spread of a ten-year Treasury bond over a three-monthTreasury bill, the default premium (DEF) is the yield spread between Moody’s Baa andAaa rated bonds, and the short rate (YLD) is the yield on the three-month Treasurybill. The dividend yield data is calculated using the CRSP data set while the other threemacroeconomic variables are obtained from the DRI database. Table 1 summarizes thosefactors and macroeconomic variables.

9The data are available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html.We thank Ken French for providing us the data.

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3.2 Momentum Profits

As favored in Jegadeesh and Titman (1993), momentum portfolios are formed based on thepast six-month returns and held for the following six months. To minimize the spuriousnegative autocorrelation due to bid-ask bounces, one month is skipped between the portfolioformation period and the portfolio holding period. Compounded returns are calculated forstocks in the portfolio formation period.10 Also, only stocks with returns throughout theentire ranking period are eligible for winner/loser selection. Specifically, for each month t,all NYSE/AMEX stocks on the monthly CRSP tape with returns for months t− 7 throught − 2 are ranked into decile portfolios according to their compounded raw returns duringthat period. Decile portfolios are formed monthly by weighting equally all firms in thatdecile ranking. Thus, P1 and P10 are equal-weighted portfolios of the 10 percent of thestocks with the lowest and highest returns over the pervious six months, respectively. Themomentum strategy longs the winner portfolio (P10) and shorts the loser (P1) and holdsthe position for the following six months (t through t + 5). To increase the power of ourtests, we follow Jegadeesh and Titman (1993) to construct overlapping portfolios. Note thatwith a six-month holding period each month’s decile portfolio return is a combination of thepast six ranking strategies, and the weights of one-sixth of the stocks change each monthwith the rest being carried over from the previous month. Each monthly cohort is assignedan equal weight in that decile portfolio. Test statistics are based on the non-overlappingportfolio returns. As a result, the sample for momentum profits covers the period of August1926 through December 2002 (917 months).

To compare with the results in Jegadeesh and Titman (1993, 2001), we also divide thewhole sample into four subperiods: 08/1926-12/1950, 01/1951-12/1964, 01/1965-12/1989,and 01/1990-12/2002. Table 2 reports the average monthly holding returns for the tendecile momentum portfolios as well as the momentum profits. Portfolio P1 consists of stockswith the lowest decile ranking period returns and P10 consists of stocks with the highestdecile ranking period returns. Portfolio P10-P1 is formed as the momentum strategy oflonging past winners and shorting the past losers. Over the full sample period, there isa clearly monotonic relation between returns and momentum ranks, consistent with theresults in Jegadeesh and Titman (1993, 2001). The overall average momentum profit is asignificant 0.76%. Like Jegadeesh and Titman (1993, 2001), there is also a strong seasonalityin momentum profits: on average, the winners outperform the losers by 1.34% per month inall non-January months, but the losers outperform the winners by a significant 5.67% permonth in January. This seasonality is likely driven by the tax-loss selling of losing stocksat calendar year-end, which subsequently rebound in January when the selling pressure isalleviated (Grinblatt and Moskowitz, 2003). Overall, the momentum strategy generatespositive profits in about 67% of the months and in about 71% of the non-January months.Only less than 20% of Januaries in this period witness a positive profit from this momentumstrategy, though.

Table 2 reveals that, except for the pre-1951 subperiod, the above return patterns existin all three subsequent subperiods. The momentum profits are insignificantly different fromzero during the period 08/1926-12/1950. The overall momentum profits in the three post-

10Using cumulative returns for ranking generates similar results.

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1950 subperiods are all significant and greater than the profits from the full sample period.In particular, the subperiod of 01/1965-12/1989 generates the largest momentum profits at1.15% per month with a t−statistics of 4.25, which is close to the corresponding momentumprofits in the original Jegadeesh and Titman (1993) sample period. The momentumprofits, though only marginally significant, continue in the more recent 1990 to 2002 period,corroborating the results in Jegadeesh and Titman (2001) and offering further evidences onthe robustness of the momentum phenomenon.11 Table 2 illustrates a striking seasonality inmomentum profits during all the four subperiods too.

4 Sources of Momentum Profits

In this section, we use Models A-D to study the sources of momentum profits. This isdone through decomposing stock returns into different parts and studying the profitabilityof momentum strategies according to rankings of each part. Based on how much each partexplains the total return momentum profits, we can identify the sources and evaluate theirrelative importance in generating the momentum profits.

For each month t, each of the four models is estimated for each NYSE/AMEX stock on themonthly CRSP tape using data from t−61 through t−2 (the estimation period). We requirea stock to have at least 36 monthly observations within that estimation period to be includedin estimation. The estimated model is then used to decompose stock returns in excess of one-month Treasury Bill rates into various components. Based on each component compoundedin the portfolio formation period (months t − 7 to t − 2) the stock is then ranked into oneof the ten deciles if it has returns throughout the entire formation period. The momentumstrategy based on rankings of that component longs the winner portfolio (P10) and shortsthe loser (P1) and holds the position for the following six months (t through t + 5). Again aone-month gap is imposed between the portfolio formation period and the portfolio holdingperiod to reduce the impact of bid-ask bounces. Within each decile portfolio returns areequally-weighted. The sample period for the such-computed momentum profits is 02/1930-12/2002 (875 months). Table 3 through Table 7, report the results of these four models,respectively. We analyze them in details in the following subsections.

4.1 Model A: Predictive Regression (Chordia andShivakumar (2002))

Table 3 documents the results of Model A which is close to the predictive regression usedin Chordia and Shivakumar (2002).12 The first row reports the average momentum profitsand the associated t−statistics for various periods. For the full sample, the momentum

11The magnitude and significance level of the momentum profits for the post-1989 period are differentbetween our study and Jegadeesh and Titman (2001). This is partially due to the inclusion in our study ofthe most recent data of 1999 through 2002 during which a market downturn and one economic recession areoverlaid. Exclusion of the four-year period generates results very similar to Jegadeesh and Titman (2001).

12Chordia and Shivakumar (2002) use the out-of-sample one-period-ahead predictions in their analysis,while we use the in-sample predictions in the analysis of Model A. The analysis based on the out-of-sampleprediction is reported in Table 11.

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profit is an average of 0.64%. Similar to Table 2, the momentum profits are significantfor the subperiod 01/1965-12/1989, insignificant for the pre-1965 subperiod, and marginallyinsignificant for the post-1989 subperiod; The subperiod of 01/1965-12/1989 has the largestmomentum profits of 1.05% per month; The momentum phenomenon exhibits seasonalityand, in particular, the momentum strategy generates a significant negative payoff in Januariesfor all periods.

As in Chordia and Shivakumar (2002), the momentum strategy based on the ranking ofthe predicted returns c0 + δ′z generates significant payoffs for the full sample and the twopost-1964 subperiods. The momentum strategy based on the ranking of unpredicted returns,measured by either the intercept c0 or the residual ε or the sum of the intercept and theresidual c0 + ε, cannot generate any significant payoffs in any period. These evidences seemto suggest that the stock’s predicted component is the source of momentum profits. Noticethat if the predicted return is truly the only source of momentum profits, a momentumstrategy based on rankings of predicted returns should generate a payoff at least as high asthe strategy based on rankings of raw returns that are noisy signals of the predictive returns.Table 3 offers results that are consistent with this intuition. Specifically, the predicted-return-based momentum payoffs are 0.78%, 1.00% and 0.94% for the 02/1930-12/2002 period, the01/1965-12/1989 subperiod and the 01/1990-12/2002 subperiod, respectively, and all aresignificant. They are higher than or close to the raw-return-based momentum payoffs in thecorresponding periods. Also, the payoffs of the predicted-return-based momentum strategyexhibit a strong seasonality throughout various periods. Interestingly, the strategy based onone part of the predicted returns δ′z does not deliver any significant payoffs at all.

The above evidences point to the predicted return as the source of the momentum profits.However, as the predicted return contains both a predicted stock-specific component and apredicted common-factor-related risk premium component (see equation (5)), we still do notknow which component is more important in explaining momentum profits. We will revisitthese results in Subsection 5.3.

4.2 Model B: Fama-French Regression (Grundy andMartin (2001))

Table 4 reports the results of Model B or the Fama-French regression. The momentumstrategy based on rankings of the Fama-French factor component β′F does not generate anysignificant payoffs for any period. In contrast, the momentum strategy based on rankingsof the stock-specific component α0 generates larger and more statistically significant payoffsthan the strategy based on the raw excess returns. Since the intercept αi0 can be interpretedas the total return adjusted for rewards to the Fama-French factor exposures, the resultssuggest that the Fama-French model can explain the variation of winer or loser returnsbut cannot explain their mean returns (Grundy and Martin, 2001). This also confirmsGrundy and Martin’s findings that the momentum strategy based on the past stock-specificreturn components is more profitable than the strategy based on the past total returns andthat the cross-sectional difference in the Fama-French factor exposures cannot explain themomentum profitability. Instead, the results from this model suggest that momentum profits

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mainly reflect the cross-sectional difference and momentum in the stock-specific componentof returns.

The results from Model A and Model B seem to contradict each other given thewell-documented predictability of the Fama-French factor risk premiums by these laggedmacroeconomic variables [e.g., Campbell and Shiller (1988), Fama and French (1988, 1989),and Keim and Stambaugh (1986)]. What might have caused this?

4.3 Model C: Fama-French Regression with Time-Varying Alpha(Ferson and Harvey (1999))

To reconcile the different results from Model A and Model B, we follow Ferson andHarvey (1999) to explicitly specify in Model C the stock-specific component as a linearfunction of the lagged macroeconomic variables zt−1. Table 5 exhibits the payoffs of themomentum strategy based on rankings of various components of this model.

Like Model B, the strategy based on rankings of the common-factor component β′Fdoes not generate significant payoffs for all periods. Interestingly, the strategy based onrankings of either α0 or α0 + ε does not generate significant payoffs, either. In contrast,the strategy based on the time-varying alpha α0 + γ′z does generate significant payoffsin magnitudes comparable to the payoffs from the momentum strategy based on rankingsof raw excess returns r. Specifically, the average monthly returns on the time-varying-alpha-based momentum strategy (t−statistics in parentheses) are 0.66% (4.86) for the fullsample, 0.45% (2.27) for the 02/1930-12/1964 subperiod, 0.85% (4.37) for the 01/1965-12/1989 subperiod, and 0.84% (2.16) for the 01/1990-12/2002 subperiod, respectively. Themomentum strategy based on the term α0 + γ′z + β′F delivers even higher payoffs than thetime-varying-alpha-based strategy in each of the four periods. This shows that the resultsof Grundy and Martin (2001) and Chordia and Shivakumar (2002) can actually co-exist.Although the Fama-French three-factors model is good at explaining the cross-sectionaldifference in expected returns, it fails to capture important cross-sectional differences relatedto momentum profits that are in turn related to the loadings on predictive variables as firstdocumented in Ferson and Harvey (1999). This observation helps illustrate why the Fama-French component can not explain momentum profits but the predictive returns can.

4.4 Model D: Fama-French Regression with Time-Varying RiskPremium

Model D is a special case of Model B when time-varying Fama-French factor risk premiumsare explicitly modeled as linear functions of the lagged macroeconomic variables. These(ex-ante) time-varying risk premiums can be calculated from either the in-sample one-period-ahead or the out-of-sample one-period-ahead predictive regressions with parametersestimated using the data over the estimation period. Results of Model D with the in-sample-estimated risk premiums and the out-of-sample-estimated risk premiums are reported inTable 6 and Table 7, respectively.

Table 6 Panel A reports the results of Model D using the in-sample-estimated risk

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premiums F̂ . The momentum strategy based on rankings of the common-factor componentβ′F̂ generates no significant payoffs in any of the four periods. The payoffs of such momentumstrategy are insignificantly different from zero in either the non-January months or Januaries.In contrast, the momentum strategy based on rankings of the stock-specific componentr − β′F̂ delivers statistically significant payoffs for the full sample and the two post-1964subperiods. The average monthly payoffs of the stock-specific-component-based momentumstrategy are 0.50%, 0.90%, and 0.82% for the 02/1930-12/2002 period, the 01/1965-12/1989subperiod, and the 01/1990-12/2002 subperiod, respectively. The magnitude is very closeto the corresponding average momentum payoffs based on the raw excess returns, which are0.64%, 1.05%, and 0.90% per month, respectively. Moreover, the stock-specific-component-based momentum strategy produces a similar seasonality: the payoffs are significantlypositive in non-January months but significantly negative in Januaries.

Table 7 Panel A displays the results of Model D using the out-of-sample-estimated time-varying risk premiums F̂ . The momentum strategy based on rankings of the common-factor component β′F̂ now generates marginally significant payoffs in the 01/1965-12/1989period, which is 0.59% per month with t−statistics 1.98. For the full sample, the averagepayoff is 0.38% and is close to being significant (t−statistics is 1.83). The payoffs areinsignificant in the pre-1965 and post-1989 subperiods. The momentum strategy basedon rankings of the stock-specific component r − β′F̂ delivers higher and more significantpayoffs than the momentum strategy based on rankings of the common-factor component,particulary in the 01/1965-12/1989 subperiod. The average monthly payoffs are (t-statisticsin parentheses) 0.42% (1.87) and 0.82% (3.27) for the full sample and the 01/1965-12/1989 subperiod, respectively. The payoffs are statistically insignificant in the othertwo subperiods. Furthermore, the common-factor-component-based momentum payoffs donot exhibit seasonality but the stock-specific-component-based momentum payoffs shows astriking seasonality, a pattern shared by the raw-return-based momentum payoffs.

The results from Table 6 and Table 7 seem to demonstrate that the stock-specificcomponent is one important source of the momentum profits. There are mixed evidenceson whether the time-varying premium is another source of the momentum profits, though.Table 7 provides only very weak evidence to support the time-varying risk premium asanother source while Table 6 offers quite strong evidences against the time-varying riskpremium as another source.

Note that Model D can also be regarded as a special case of equation (8) by treatingthe Fama-French three-factors model as the true linear asset pricing model. The resultsof Model D imply that the time-varying risk premium story suggested by Chordia andShivakumar (2002) may no longer be robustly valid once the cross-sectional restrictions areimposed on the coefficients on predictive variables. Of course, this conclusion depends onthe assumption that the Fama-French three-factors model is the correct asset pricing model.More on the factor models are discussed in Section 5. Certainly, if we have imposed the wrongcross-sectional restrictions here, Chordia and Shivakumar’s conclusions may still hold.

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4.5 The Sources of Momentum Profits: Common-FactorComponent or Stock-Specific Component?

The empirical results so far offer strong support for the stock-specific component as the mainsource of momentum profits and only weak support for the common-factor component asanother source. However, these results depend on the validity of the Fama-French modelbeing the correct asset pricing model. To better understand Chordia and Shivakuma’s (2002)results and identify the sources of momentum profits without making a stand on a particularasset pricing model, we use equation (5) and consider the same predictive regression asChordia and Shivakumar except that we allow for different intercepts between the in-formation period and the out-of-formation period. As we have argued in Section 2.1, giventhe assumptions of a linear asset pricing model and a linear relationship between the riskpremiums and the conditioning information, the intercept in the predictive regression consistsof both a stock-specific component and a common-factor component, but the differencebetween the in-formation-period intercept c0 and the out-of-formation-period interceptc1 contains only the stock-specific component (see equation (6)). If the common-factorcomponent is the only source of momentum profits, then the two momentum strategiesbased on rankings of the two predicted returns c0 + δ′z and c1 + δ′z are expected to deliversimilar results. On the other hand, if the stock-specific component is also the source ofmomentum profits, the momentum strategy based on rankings of c0 − c1 is expected togenerate significant momentum profits.

Table 3 offers some evidences on these two conjectures. First, the average monthlypayoffs of the momentum strategy based on the predicted returns c0 + δ′z are 0.78% in thefull sample period, 1.00% in the 01/1965-12/1989 subperiod, and 0.94% in the post-1989subperiod, respectively, and all are statistically significant. In contrast, the average monthlypayoffs of the momentum strategy based on the predicted returns c1 + δ′z are insignificantlydifferent from zero in all the corresponding periods. This finding suggest that the common-factor component couldn’t be the main source of the momentum profits. Second, the averagemonthly payoffs of the momentum strategy based on rankings of c0− c1 are (with t-statisticsin parentheses) 0.41% (2.83) in the full sample period, 0.56% (3.63) in the 01/1965-12/1989subperiod, and 0.84% (3.34) in the post-1989 subperiod, respectively. For comparison thecorresponding momentum payoffs of the raw-return-based strategy are 0.64%, 1.05%, and0.90%. Clearly, the payoffs of the strategy based on rankings of c0 − c1, which containsthe stock-specific information only, are significant and they account for over 50% of themomentum profits from either the raw-return-based strategy or the predicted-return-basedstrategy. Moreover, the payoffs of the strategy based on rankings of c0 − c1 exhibit someseasonality, a pattern shared by the raw-return-based strategy and the predicted-return-based strategy. That is, the momentum payoffs are significantly positive in non-Januarymonths and negative, though not significant, in Januaries. These observations suggest thatmore than half of the explaining power of the predicted returns as the source of momentumprofits comes from the stock-specific component contained in the predicted returns.

The results of Model C provide further evidences on the above observations (see Table 5).As shown in Section 4.3, the time-varying alpha (α0 + γ′z) is the source of momentumprofits. Since it is closely related to the common macroeconomic variables, some might

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argue that this commonality in individual stock returns favors a risk-based explanation ofthe momentum phenomenon, and hence we just need to find out the responsible non-Fama-French factors to proxy for such underlying risks. However, it should be cautioned thatthe commonality in stock-specific components is also consistent with a behavioral argument.For example, this commonality could be primarily driven by investors’ (mis)interpretationsof the macroeconomic shocks. Moreover, even if the commonality reflects some (unknown)macro risks, the term (α0 + γ′z)− (α1 + γ′z)=α0 −α1 should only contain the stock-specificinformation or the information not related to the common macro-factors.13 The averagemonthly momentum payoffs based on rankings of this stock-specific component α0 − α1 are0.46% in the full sample period, 0.56% in the 01/1965-12/1989 subperiod, and 0.79% in themore recent post-1989 subperiod, respectively, and all are statistically significant. Thesepayoffs represent over half of the payoffs of the raw-return-based momentum strategy inthe corresponding periods. Such evidences substantiate the claim that the stock-specificinformation is one important source, if not the only source, of momentum profits.

5 Discussions

5.1 Time-Varying Risk

It is well documented in the literature that the risk is varying over time (e.g., Harvey, 1989;and Ferson and Harvey, 1991) and hence an asset pricing model with time-varying-beta isadvocated to capture time-varying risks. We address here the concern whether our aboveanalysis is sensitive to incorporating of time-varying risks into the models.

Table 8 reports the results of the time-varying-beta counterpart to Model B. Like theconstant Fama-French three-factors model in Table 4, the momentum strategy based onrankings of the common-factor component β′F + θ′zF does not generate any significantpayoffs across the four periods, and the momentum strategy based on rankings of the stock-specific component α0 generates significantly positive payoffs. Interestingly, for the stock-specific-component-based momentum strategy the profits in Table 8 are significantly smallerthan the corresponding-period momentum profits in Table 4; the same pattern occurs forthe payoffs of the common-factor-based momentum strategy. This evidence seems to beconsistent with Ghysels (1998) who reports that the pricing errors with constant traditionalbeta models are smaller than with time-varying beta models. If the beta risks are inherentlymisspecified, we are likely to commit larger pricing errors with a time-varying beta modelthan with a constant beta model so that the popular methods of modelling time-varyingrisks do not necessarily lead to improvements over the constant risk models.

The results of the time-varying-beta counterpart to Model C are displayed in Table 9.Similar to the constant-beta model in Table 5, the strategy based on rankings of the common-factor component β′F + θ′zF does not generate significant payoffs in either period; thestrategy based on rankings of the time-varying alpha α0 + γ′z generates significant payoffswith average monthly payoffs (and t−statistics) as follows: 0.48% (3.97) for the full sample,

13Note that only the in-formation period instruments Z are used in calculating the time-varying alpha torank stocks for portfolio formation.

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0.63% (4.03) for the 01/1965-12/1989 subperiod, and 0.77% (2.41) for the 01/1990-12/2002subperiod, respectively. The co-existence of Grundy and Martin’s (2001) and Chordia andShivakumar’s (2002) results still holds with a time-varying-beta model, which implies thateven with the time-varying betas the Fama-French three-factors model fails to capture theimportant cross-sectional differences related to momentum profits that are in turn relatedto the loadings on predictive variables (Ferson and Harvey, 1999). For every period, theaverage momentum payoffs of the time-varying-alpha-based strategy are smaller in the time-varying-beta model than in the constant-beta model, again consistent with the findingsof Ghysels (1998). Table 9 shows that the average monthly momentum payoffs (andt−statistics) based on rankings of the stock-specific component α0 − α1 are 0.39% (5.11)in the full sample period, 0.50% (4.92) in the 01/1965-12/1989 subperiod, and 0.61% (3.46)in the more recent post-1989 subperiod, respectively. These payoffs represent over half ofthe payoffs of the raw-return-based momentum strategy in the corresponding periods.

Table 6 Panel B and Table 7 Panel B present the results of the time-varying-riskversions of Model D when the in-sample and the out-of-sample one-period-ahead predictedrisk premiums are used, respectively. In either of the two panels, the strategy based onrankings of the common-factor component β′F̂ + θ′zF̂ does not generate significant payoffsfor any periods; in contrast, the strategy based on rankings of the stock-specific componentr−β′F̂−θ′zF̂ delivers more than half of the momentum profits of the raw-excess-return-basedmomentum strategy and they are significant and exhibit seasonality.

In summary, incorporating the time-varying risk into our models does not change ourconclusions. That is, the stock-specific component, not the common-factor component,appears to be the main source of momentum profits. The stock-specific component alwaysgenerates significant momentum payoffs while only in few occasions does the common-factorcomponent generate marginally significant momentum payoffs.

5.2 Liquidity Factor

Some of our above discussions are based on the assumption that the Fama-French three-factors model is the true asset pricing model. It’s natural to ask the question: What if theFama-French model is not the true asset pricing model?

Recently, Pastor and Stambaugh (2003) show that the market-wide liquidity is a pricedstate variable and that the expected stock returns are related cross-sectionally to thesensitivities of returns to the aggregate liquidity risk. They report that a liquidity riskfactor accounts for half of the momentum profits over the period under study. Grinbalttand Moskowitz (2003) note that most of the apparent momentum gains come form shortpositions in small, illiquid stocks.

To address this valid concern, we include the liquidity factor into our analysis and wethence have a four-factor asset pricing model.14 Due to the data availability, the sample istruncated to the period of January 1966 through December 1999, a total of 408 months.As a result of model estimation (requiring a minimum of 36 months of data) to decomposethe raw excess returns into explained and unexplained returns for portfolio formation, the

14We thank Rob Stambaugh for providing us the liquidity factor data.

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momentum payoffs cover the 02/1969-12/1999 period. We further divide the sample intotwo equal subsamples: 02/1969-12/1984 and 01/1985-12/1999. To compare with Jegadeeshand Titman (1993), we also separately list the subperiod 02/1969-12/1989. Table 10 reportsthe results with risks controlled for by the four-factors model with constant betas (Panel A)and the four-factors model with time-varying betas (Panel B).

The first row of Table 10 documents the average monthly momentum profits for thefour periods. The overall profits (and t−statistics) are 1.01% (3.90) in the 02/1969-12/1999period, 0.92% (2.53) in the 02/1969-12/1984 subperiod, 1.10% (2.98) in the 01/1985-12/1999subperiod, and 1.06% (3.58) in the 02/1969-12/1989 subperiod, respectively. The momentumpayoffs exhibit strong seasonality in each of the four periods: higher and significantly positivepayoffs in non-January months and significantly negative payoffs in Januaries. Panel A showsthat the momentum payoffs of the strategy based on the common-factor component β′GFare insignificantly different from zero in every period. The momentum payoffs of the strategybased on the stock-specific component α0 are significantly positive in every period and thepayoffs (and t−statistics) are 0.92% (4.94) in the 02/1969-12/1999 period, 0.65% (2.54)in the 02/1969-12/1984 subperiod, 1.20% (4.45) in the 01/1985-12/1999 subperiod, and0.84% (3.96) in the 02/1969-12/1989 subperiod, respectively. These payoffs clearly displayseasonality: significantly positive in non-January months and significantly negative inJanuaries. Panel B reports very similar results with a time-varying-beta four-factors model.The common-factor-based momentum strategy does not generate any significant payoffs, andthe stock-specific-part-based momentum strategy generates significantly positive payoffs inevery period. Again, the momentum payoffs of the strategy based on the stock-specificcomponent with the conditional four-factors model are smaller than with the unconditionalfour-factors model in each corresponding period.

5.3 Predictive Relationship Revisited

Chordia and Shivakumar (2002) report several key results based on the out-of-sample one-period-ahead predicted returns. We use the in-sample one-period-ahead predicted returnsin Section 4.1. To resolve the potential differences that may be caused by the two types ofpredicted returns, we re-do Model A using the out-of-sample predicted returns. Specifically,for each month t, momentum portfolios are formed based on rankings of compounded out-of-sample one-period-ahead predicted and unpredicted returns over the portfolio formationperiod {t − 7, ..., t − 2} using all NYSE/AMEX stocks on the monthly CRSP tape. Themomentum strategy longs the top decile portfolio and shorts the bottom decile portfolio andholds the position for the following six months t through t+5. For each stock in each months in the portfolio formation period s ∈ {t− 7, ..., t− 2}, the out-of-sample one-period-aheadpredicted returns and unpredicted returns are obtained with the parameters estimated usingdata from s − 60 through s − 1 (a minimum of 24 months of data required) based on thefollowing model:

ris = ci + δ′izs−1 + εis, (13)

where ris stands for returns in excess of one-month Treasury Bill rates and zs−1 is the vectorof the lagged macroeconomic variables. The sample period for the momentum profits is08/1929-12/2002 (881 months). We differ from Chordia and Shivakumar in that we skip one

16

month between the portfolio formation period and the portfolio investment period to reducethe bid-ask bounce effects while they do not.

Table 11 presents the results with the out-of-sample one-period-ahead predictiverelationship. Panel A shows that the payoffs of the momentum strategy based on rankingsof the unpredicted returns, measured by either c or c + ε, are insignificantly different fromzero in the full sample period and each subperiod, which is consistent with Chordia andShivakumar’s results. The payoffs of the momentum strategy based on rankings of thepredicted return c + δ′z generates significantly positive payoffs in the full sample period andthe 01/1965-12/1989 subperiod, but they are insignificant in the 08/1929-12/1964 subperiodand the post-1989 subperiod. Though significant in the 01/1965-12/1989 subperiod, thepredicted-return-based momentum payoffs are averaged at 0.44% per month and accountsfor less than 50% of the payoff from the raw-excess-return strategy, which is 1.07% per month.The finding of almost zero profits from the predicted-return-based momentum strategy inthe post-1989 subperiod is different from Chordia and Shivakumar. These two observationssuggest that some of Chordia and Shivakumar’s results may not be robust to the one-monthskipping commonly employed to take account of microstructure concerns (see also Cooper,Gutierrez, and Hameed, 2003; and Griffin, Ji, and Martin, 2003).

We also examine one variation of the predictive regression by imposing the zero-interceptrestriction in equation (13) so that the fitted term δ′z is the so-called “Best ProportionalPredictor (BPP)”.15 The estimation of such a restricted model is biased but consistent(Goldberger 1997, pp.57&148). The BPP or the predicted return does not contain thestock-specific component and it explicitly captures the co-movement between individualstock returns and the macroeconomic variables, while the residual ε contains only thestock-specific information, although the restricted predictive relationship is lack of an asset-pricing-model interpretation. Panel B shows that the residual-based momentum strategyproduces significantly positive payoffs of 0.61% and 0.82% per month in the 01/1965-12/1989subperiod and the post-1989 subperiod, respectively, and the payoffs display a clear patternof seasonality. In contrast, the BPP-based momentum strategy produces only marginallysignificant payoffs of 0.38% in the 01/1965-12/1989 subperiod, and insignificantly negativepayoffs in the post-1989 subperiod, respectively. Again the evidences using the BPP-type predictive regression support the stock-specific component as the main source of themomentum profits.16

5.4 Other Horizons for Portfolio Formation

The in-formation-period and out-of-formation-period predictive intercepts c0 and c1 in thepredictive framework (equation (8)) are defined based on the ranking-period horizon. A six-month horizon is used in the above analysis for portfolio formation to classify individualstocks into decile portfolios for the momentum strategy. We show that the difference

15We thank Paul Lau and James Vere for helpful discussions of this.16Note that the BPP-based momentum payoffs are significantly positive in non-January months and

significantly negative in Januaries in each subperiod. This seasonality may confound the statisticalsignificance of the overall BPP-based momentum payoffs. Thus our evidences should be interpreted withsome caution.

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of the two predictive intercepts c0 − c1 contains only the stock-specific component andthe momentum strategy based solely on this stock-specific component generates significantmomentum profits. In this subsection we examine whether the above results are sensitive tothe different choices of ranking-period horizons.

Momentum profits arise in the short to intermediate horizon, which typically refers to 3-to 12-month ranking periods (Jegadeesh and Titman, 1993). Table 12 presents the profitsof the momentum strategies based on the raw excess returns r, the two predicted returnsc0 + δ′z and c1 + δ′z, and the stock-specific component c0 − c1 with ranking horizons of 3,4, 5, 9, 11, and 12 months, respectively. Interestingly, over the full sample period, exceptfor the strategy based on c1 + δ′z, the profits of all other momentum strategies (and thecorresponding t-statistics) increase steadily as the ranking horizon increases up to the 9-month horizon and then decline as the horizon extends to 12 months. This pattern alsoholds for the three subperiods, in particular for the 01/1965-12/1989 subperiod.

Except for the 3- and 4-month horizons, the momentum profits based on the two predictedreturns c0 + δ′z and c1 + δ′z behave quite differently. The momentum strategy basedon c0 + δ′z generates significantly positive profits for the whole sample and the 01/1965-12/1989 subperiod, and the profits are positive though not strongly significant for theother two subperiods. In contrast, the momentum strategy based on c1 + δ′z generatesinsignificant profits in the full sample and all the three subperiods, and they tend to benegative for horizons longer than 5 months. These observations suggest that some stock-specific component drives such differences.

A more direct evidence comes from the payoffs of the momentum strategy based onthe c0 − c1 term. For horizons longer than 5 months, the momentum strategy based onthe stock-specific component c0 − c1 delivers positive and strongly significant profits inthe full sample and the two post-1964 subperiods. In the full sample, such profits are0.603% (9-month horizon), 0.600% (11-month horizon), and 0.462% (12-month horizon),and the corresponding raw-return-based momentum profits are 0.705%, 0.583%, and 0.436%,respectively. In the 01/1965-12/1989 subperiod, such profits are 0.792% (9-month horizon),0.821% (11-month horizon), and 0.728% (12-month horizon), and the corresponding raw-return-based momentum profits are 1.119%, 1.055%, and 0.981%, respectively. In the post-1989 subperiod, the stock-specific-component-based profits are respectively 1.148% (9-monthhorizon), 0.988% (11-month horizon), and 0.752% (12-month horizon), and all are stronglysignificant; the corresponding raw-return-based profits are 0.977%, 0.737%, and 0.521%,respectively, and they are statistically insignificant. For the 5-month horizon, the momentumstrategy based on the c0 − c1 term generates significantly positive profits, about half of themagnitude of the raw-return-based momentum profits, in the two post-1964 subperiods andpositive but insignificant profits in the full sample. For horizons smaller than 5 months,though the momentum strategy based on the c0−c1 term does not generate significant profitsin some of the periods, it still produces significant and positive profits in the 01/1965-12/1989subperiod.

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6 Conclusions

We examine in this paper the sources of momentum profits in terms of the stock return’scommon-factor component versus its stock-specific component. Using a model nesting bothChordia and Shivakumar (2002) and Grundy and Martin (2001), we first confirm their results.That is, as in Chordia and Shivakumar (2002) the expected returns predicted by a set oflagged common macroeconomic variables seem to be the main source of momentum profits instock returns; yet as in Grundy and Martin (2001) the stock-specific component but not thecommon-factor component of stock returns explains the momentum profits in the frameworkof the Fama-French three-factors model with either constant or time-varying betas. We thenreconcile these two seemingly “contradictory” evidences by showing that the Fama-Frenchthree-factors model leaves out important predictive variations in stock returns responsible forChordia and Shivakumar’s (2002) results. We further impose on the coefficients of predictiveregressions the cross-sectional restrictions implied by the Fama-French three-factors model.We show that the bulk of the predictive variations needed for explaining the momentumprofits appears to mainly come from the stock-specific component rather than the (predicted)common-factor component.

In the context of a linear asset pricing model with any choice of factors and by nestingthe predictive regression as a reduced form of such model, we show that the predictiveintercept and hence the predicted returns contain both a stock-specific component and acommon-factor component. We propose a method, which is free from the missing-factorproblem in specifying the asset pricing model, to extract the stock-specific componentfrom the predictive intercept. The momentum strategy based solely on this stock-specificcomponent generates significant payoffs accounting for more than half of the momentumprofits in stock returns. Even though Ferson and Harvey (1991) demonstrate that the time-varying risk premiums are mainly responsible for the return predictability, which in turnseems to explain the momentum profits, we show that the bulk of the predictive variationsneeded for explaining the momentum profits are nevertheless driven by the predicted stock-specific component. For robustness, we include both the time-varying risk and Pastor andStambaugh’s (2003) liquidity factor, on top of the Fama-French three factors, into the assetpricing model; we examine variants of the predictive relation; and we explore various horizonsfor portfolio formation. In almost all set-ups, we get very similar results on the significance ofthe profits from the momentum strategy based on rankings of this stock-specific component.

Our empirical results on the sources of momentum profits carry implications for modellingthe momentum phenomenon in theory. The stock-specific component, if not the only source,appears to be a very important source. In various models and setups it always generatessignificant momentum profits in magnitude of over half of the momentum profits in stockreturns. The common-factor component, i.e., the time-varying risk premiums but not thetime-varying risks, may be another source but the empirical support is very weak. Similar toChordia and Shivakumar (2002) we find a striking commonality in individual stock returnsin response to the macroeconomic shocks. The commonality explains a large proportion ofthe total momentum profits and hence warrants a further study.

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[7] Conrad, Jennifer, and Gautam Kaul, 1998, An anatomy of trading strategies, Reviewof Financial Studies 11, 489-519.

[8] Cooper, Michael, Roberto C. Gutierrez, and Allaudeen Hameed, 2003, Market statesand momentum, Journal of Finance, forthcoming.

[9] Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 1998, Investorpsychology and security market under- and overreactions, Journal of Finance 53, 1839-1885.

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[11] Fama, Eugene F., and Kenneth R. French, 1989, Business conditions and expectedreturns on stocks and bonds, Journal of Financial Economics 25, 23-49.

[12] Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stockreturns, Journal of Finance 47, 427-465.

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[16] Ferson, Wayne E., and Campbell R. Harvey, 1991, The variations of economic riskpremiums, Journal of Political Economy 99, 385-415.

20

[17] Ferson, Wayne E., and Campbell R. Harvey, 1999, Conditioning variables and the cross-section of stock returns, Journal of Finance 54, 1325-1360.

[18] George, Thomas J., and Chuan-Yang Hwang, 2003, The 52-week high and momentuminvesting, Journal of Finance, forthcoming.

[19] Ghysels, Eric, 1998, On stable factor structures in the pricing of risk: Do time-varyingbetas help or hurt? Journal of Finance 53, 549-573.

[20] Goldberger, Arthur S., 1997, A course in econometrics, Harvard University Press,Cambridge, MA.

[21] Griffin, John M., Susan Ji, and J. Spencer Martin, 2002, Momentum investing andbusiness cycle risk: Evidence from pole to pole, Journal of Finance, forthcoming.

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[23] Grundy, Bruce D., and J. Spencer Martin, 2001, Understanding the nature of the risksand the source of the rewards to momentum investing, Review of Financial Studies 14,29-78.

[24] Harvey, Campbell R., 1989, Time-varying conditional covariances in tests of asset pricingmodels, Journal of Financial Economics 24, 289-317.

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[26] Hong, Harrison, Terrence Lim, and Jeremy Stein, 2000, Bad news travels slowly: Size,analyst coverage, and the profitability of momentum strategies, Journal of Finance 55,265-295.

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[28] Jegadeesh, Narasimhan, and Sheridan Titman, 2001, Profitability of momentumstrategies: An evaluation of alternative explanations, Journal of Finance 56, 699-720.

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21

[33] Lo, Andrew W., and A. Craig MacKinlay, 1990, When are contrarian profits due tostock market overreaction?, Review of Financial Studies 3, 175-205.

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[35] Pastor, Lubos, and Robert Stambaugh, 2003, Liquidity risk and expected stock returns,Journal of Political Economy 111, 642-685.

[36] Rouwenhorst, Geert K., 1998, International momentum strategies, Journal of Finance53, 267-284.

[37] Yao, Tong, 2003, Systematic momentum, working paper, University of Arizona.

22

Table 1: Summary Statistics

This table presents descriptive statistics of Fama-French factors from July 1926 throughDecember 2002 (918 months) and of macroeconomic variables from December 1926 throughDecember 2002 (913 months). MARKET RF is the monthly return on CRSP value-weighted market index in excess of the one-month Treasury bill rate (RF). SMBand HML stand for the small-minus-big size factor and the high-minus-low book-to-market-ratio factor, respectively. The factors data are obtained from French’s websiteat http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html. The dividendyield (DIV) is defined as the total dividend payment accrued to the CRSP value-weighted marketindex over the past 12 months divided by the current price level of the index. The termpremium (TERM) is the yield spread of a ten-year Treasury bond and a three-month Treasurybill, the default premium (DEF) is the yield spread of Moody’s Baa and Aaa rated bonds, and theshort rate (YLD) is the yield on the three-month Treasury bill.

MARKET RF SMB HML RF DIV TERM DEF YLD

Mean 0.623 0.210 0.403 0.311 3.968 1.464 1.137 3.892Std Dev 5.527 3.386 3.629 0.257 1.544 1.141 0.735 3.196Median 0.96 0.03 0.22 0.28 3.75 1.48 0.88 3.50Max 38.17 38.22 35.86 1.35 13.297 4.42 6.27 16.042Min -29.01 -16.26 -13.23 -0.06 1.055 -2.65 0.32 -0.091Skewness 0.224 2.180 1.841 0.970 1.033 -0.393 2.520 0.958Kurtosis 10.564 24.137 18.012 4.085 6.181 2.970 12.530 4.007

Autocorrelation:

1-month 0.104 0.068 0.183 0.972 0.978 0.954 0.976 0.9906-month -0.022 0.016 0.027 0.918 0.859 0.730 0.873 0.938

12-month 0.000 0.116 0.050 0.867 0.700 0.529 0.744 0.88824-month 0.027 0.023 0.001 0.770 0.516 0.229 0.544 0.784

Correlation with:

Market rf 1.000 0.323 0.204 -0.066 -0.101 0.052 -2.64e-4 -0.066SMB 1.000 0.088 -0.056 -0.008 0.095 0.087 -0.050HML 1.000 0.013 0.003 0.024 0.0312 -0.002RF 1.000 -0.286 -0.367 -0.076 0.979DIV 1.000 0.129 0.501 -0.289TERM 1.000 0.398 -0.373DEF 1.000 -0.087YLD 1.000

23

Tab

le2:

Raw

Mom

entu

mPor

tfol

ioan

dM

omen

tum

Str

ateg

yPay

offs

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

for

mom

entu

mpo

rtfo

lios

form

edba

sed

onpa

stsi

x-m

onth

retu

rns

and

held

for

the

follo

win

gsi

xm

onth

s.Fo

rea

chm

onth

t,al

lN

YSE

/AM

EX

stoc

kson

the

mon

thly

CR

SPta

pew

ith

retu

rns

for

mon

ths

t−

7th

roug

ht−

2ar

era

nked

into

deci

lepo

rtfo

lios

acco

rdin

gto

thei

rco

mpo

unde

dre

turn

sdu

ring

that

peri

od.

Dec

ilepo

rtfo

lios

are

form

edm

onth

lyby

wei

ghti

ngeq

ually

allfir

ms

inth

atde

cile

rank

ing.

P1

and

P10

are

the

equa

l-w

eigh

ted

port

folio

sof

the

10pe

rcen

tof

the

stoc

ksw

ith

the

low

est

and

the

high

est

retu

rns

over

the

perv

ious

six

mon

ths,

resp

ecti

vely

.T

hem

omen

tum

stra

tegy

long

sth

ew

inne

rpo

rtfo

lio(P

10)

and

shor

tsth

elo

ser

(P1)

and

hold

sth

epo

siti

ons

for

the

follo

win

gsi

xm

onth

st

thro

ugh

t+

5.t−

stat

isti

csar

ere

port

edin

pare

nthe

ses.

The

row

titl

ed“%

>0”

give

sth

epe

rcen

tage

ofP

10−

P1

that

are

posi

tive

.

08/1

926-

12/2

002

08/1

926-

12/1

950

01/1

951-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

P1

1.07

90.

214

10.6

471.

817

0.94

411

.608

0.61

80.

174

5.49

90.

827

-0.1

6311

.708

0.67

4-0

.390

12.3

77(3

.031

)(0

.607

)(7

.775

)(2

.012

)(1

.002

)(4

.745

)(1

.665

)(0

.488

)(3

.340

)(1

.734

)(-

0.38

5)(4

.705

)(0

.949

)(-

0.63

9)(2

.966

)P

21.

140

0.60

67.

044

1.47

60.

929

7.60

50.

882

0.56

74.

351

1.17

70.

470

8.95

10.

714

0.30

25.

241

(3.9

43)

(2.0

47)

(7.2

75)

(1.9

40)

(1.1

54)

(4.2

82)

(2.8

54)

(1.8

78)

(3.1

29)

(3.0

35)

(1.3

28)

(4.4

83)

(1.6

71)

(0.7

27)

(2.6

35)

P3

1.22

10.

782

6.08

31.

447

0.96

86.

809

1.03

90.

766

4.03

71.

308

0.71

57.

826

0.82

50.

573

3.59

3(4

.604

)(2

.863

)(6

.855

)(2

.049

)(1

.296

)(3

.887

)(3

.739

)(2

.799

)(3

.360

)(3

.673

)(2

.171

)(4

.249

)(2

.422

)(1

.658

)(2

.738

)P

41.

228

0.86

25.

280

1.37

50.

998

5.60

01.

076

0.83

23.

758

1.35

10.

837

7.01

40.

880

0.68

72.

993

(4.9

79)

(3.3

80)

(6.4

79)

(2.0

94)

(1.4

33)

(3.4

47)

(4.1

32)

(3.2

47)

(3.2

42)

(3.9

96)

(2.6

25)

(4.1

31)

(3.0

37)

(2.3

15)

(2.8

23)

P5

1.27

50.

956

4.80

51.

405

1.06

85.

179

1.16

60.

968

3.34

31.

369

0.90

76.

457

0.97

10.

831

2.51

2(5

.360

)(3

.870

)(6

.240

)(2

.210

)(1

.582

)(3

.320

)(4

.652

)(3

.855

)(3

.160

)(4

.187

)(2

.898

)(4

.065

)(3

.755

)(3

.126

)(2

.574

)P

61.

294

1.00

64.

476

1.34

11.

038

4.74

01.

250

1.07

63.

165

1.43

61.

002

6.20

20.

979

0.87

92.

079

(5.7

50)

(4.3

04)

(6.1

03)

(2.2

64)

(1.6

50)

(3.1

93)

(5.0

37)

(4.3

15)

(2.9

69)

(4.4

84)

(3.2

32)

(4.1

64)

(3.8

73)

(3.3

58)

(2.2

25)

P7

1.34

81.

091

4.19

31.

425

1.15

34.

471

1.28

71.

154

2.74

91.

462

1.05

25.

972

1.05

20.

982

1.81

3(6

.210

)(4

.851

)(5

.613

)(2

.552

)(1

.955

)(2

.875

)(5

.108

)(4

.493

)(2

.665

)(4

.520

)(3

.332

)(4

.029

)(3

.926

)(3

.525

)(1

.852

)P

81.

417

1.16

84.

171

1.47

51.

185

4.72

71.

359

1.23

52.

726

1.57

01.

184

5.82

31.

076

1.03

51.

524

(6.5

44)

(5.2

12)

(5.5

05)

(2.7

03)

(2.0

67)

(2.8

68)

(5.1

88)

(4.6

03)

(2.5

90)

(4.7

13)

(3.5

79)

(4.0

76)

(3.7

43)

(3.4

35)

(1.5

48)

P9

1.52

41.

289

4.12

21.

622

1.32

94.

906

1.38

21.

305

2.23

51.

670

1.30

25.

713

1.21

11.

172

1.64

8(6

.903

)(5

.651

)(5

.174

)(2

.997

)(2

.351

)(2

.750

)(5

.044

)(4

.624

)(2

.031

)(4

.710

)(3

.649

)(3

.984

)(3

.798

)(3

.513

)(1

.463

)P

101.

840

1.55

64.

981

1.92

81.

559

6.06

81.

554

1.49

62.

192

1.97

91.

580

6.37

61.

712

1.56

83.

297

(7.3

71)

(6.0

31)

(5.6

10)

(3.2

42)

(2.5

24)

(2.9

65)

(4.9

08)

(4.5

26)

(1.9

39)

(4.8

52)

(3.7

99)

(4.2

18)

(4.0

64)

(3.5

77)

(2.1

88)

P10

−P

10.

761

1.34

1-5

.666

0.11

10.

615

-5.5

400.

936

1.32

2-3

.308

1.15

31.

742

-5.3

321.

038

1.95

8-9

.081

(3.5

22)

(6.4

40)

(-5.

990)

(0.2

08)

(1.1

06)

(-3.

663)

(4.2

63)

(6.9

00)

(-3.

057)

(4.2

46)

(7.5

34)

(-3.

449)

(1.9

82)

(4.9

14)

(-2.

564)

%>

067

.18

71.4

619

.74

61.0

964

.31

25.0

070

.83

75.9

714

.29

70.3

374

.55

24.0

068

.59

74.1

37.

69

24

Tab

le3:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

A

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

es.

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

c 0D

t+

c 1(1

−D

t)+

δ′z t−

1+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t−

7,...,

t−

2}an

dD

τ=

0ot

herw

ise,

and

z t−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

s.E

ach

stra

tegy

desi

gnat

esw

inne

rsan

dlo

sers

asth

eto

pan

dbo

ttom

deci

les

acco

rdin

gto

diffe

rent

rank

ing

crit

eria

onco

mpo

unde

dre

turn

sw

ithi

nth

epe

riod

ofm

onth

st−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

δ′z

0.10

40.

109

0.04

70.

211

0.20

90.

234

0.00

80.

009

-0.9

330.

001

-0.1

301.

442

(0.8

5)(0

.91)

(0.0

7)(1

.03)

(1.0

2)(0

.23)

(0.0

5)(0

.58)

(-1.

04)

(3.9

e-3)

(-0.

58)

(0.7

4)c 0

+δ′

z0.

775

1.19

4-3

.901

0.55

60.

822

-2.4

650.

996

1.47

6-4

.285

0.93

91.

653

-6.9

19(4

.08)

(6.4

8)(-

4.42

)(1

.80)

(2.5

7)(-

2.36

)(3

.89)

(6.5

6)(-

2.90

)(2

.09)

(4.4

9)(-

2.43

)c 1

+δ′

z0.

114

0.41

0-3

.185

0.28

30.

499

-2.1

640.

139

0.40

7-2

.811

-0.3

880.

175

-6.5

74(0

.77)

(2.9

7)(-

3.95

)(1

.11)

(1.9

9)(-

1.72

)(0

.84)

(2.7

5)(-

2.92

)(-

1.16

)(0

.68)

(-2.

90)

c 0−

c 10.

413

0.50

0-0

.560

0.14

60.

199

-0.4

480.

563

0.68

5-0

.778

0.83

90.

955

-0.4

35(2

.83)

(3.2

8)(-

1.14

)(0

.55)

(0.7

0)(-

0.56

)(3

.63)

(4.4

3)(-

1.09

)(3

.34)

(3.7

7)(-

0.39

)c 0

0.08

90.

093

0.03

9-0

.028

-0.0

15-0

.176

0.26

20.

229

0.62

80.

068

0.12

2-0

.531

(0.8

1)(0

.88)

(0.0

6)(-

0.16

)(-

0.09

)(-

0.19

)(1

.58)

(1.4

2)(0

.69)

(0.2

8)(0

.57)

(-0.

31)

c 1-0

.070

-0.0

930.

186

-0.1

30-0

.144

0.03

60.

056

-0.0

090.

768

-0.1

50-0

.115

-0.5

42(-

0.59

)(-

0.80

)(0

.31)

(-0.

66)

(-0.

73)

(0.0

4)(0

.32)

(-0.

06)

(0.7

7)(-

0.68

)(-

0.56

)(-

0.37

)c 0

+ε

0.03

10.

071

-0.4

20-0

.133

-0.0

83-0

.705

0.26

10.

245

0.44

00.

026

0.14

9-1

.328

(0.2

6)(0

.63)

(-0.

64)

(-0.

71)

(-0.

44)

(-0.

76)

(1.5

9)(1

.54)

(0.4

8)(0

.10)

(0.6

7)(-

0.63

)c 1

+ε

-0.1

07-0

.093

-0.2

63-0

.193

-0.1

70-0

.456

0.06

20.

018

0.54

8-0

.199

-0.0

97-1

.320

(-0.

89)

(-0.

78)

(-0.

44)

(-0.

96)

(-0.

83)

(-0.

54)

(0.3

6)(0

.11)

(0.5

5)(-

0.82

)(-

0.45

)(-

0.78

)ε

-0.7

00-0

.070

-7.7

26-1

.118

-0.5

43-7

.637

-0.1

880.

400

-6.6

55-0

.560

0.30

0-1

0.01

6(-

2.57

)(-

0.27

)(-

5.86

)(-

2.40

)(-

1.19

)(-

3.34

)(-

0.57

)(1

.26)

(-5.

56)

(-0.

92)

(0.5

7)(-

2.75

)

25

Tab

le4:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

B

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

es.

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ F

t+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t−

7,...,

t−

2}an

dD

τ=

0ot

herw

ise,

and

Ft

the

vect

orof

Fam

a-Fr

ench

fact

ors.

Eac

hst

rate

gyde

sign

ates

win

ners

and

lose

rsas

the

top

and

bott

omde

cile

sac

cord

ing

todi

ffere

ntra

nkin

gcr

iter

iaon

com

poun

ded

retu

rns

wit

hin

the

peri

odof

mon

ths

t−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

β′ F

0.15

70.

135

0.40

20.

160

0.07

41.

127

0.14

50.

225

-0.7

360.

172

0.12

50.

693

(0.7

4)(0

.63)

(0.3

9)(0

.42)

(0.1

9)(0

.64)

(0.5

5)(0

.85)

(-0.

63)

(0.4

9)(0

.39)

(0.2

8)α

0+

β′ F

0.74

61.

189

-4.1

900.

481

0.78

1-2

.910

1.00

81.

498

-4.3

880.

953

1.69

0-7

.158

(3.7

7)(6

.14)

(-4.

85)

(1.4

7)(2

.28)

(-3.

11)

(3.9

2)(6

.65)

(-2.

98)

(2.0

6)(4

.49)

(-2.

43)

α0

0.66

81.

129

-4.4

710.

453

0.84

3-3

.962

0.85

61.

302

-4.0

550.

883

1.56

4-6

.600

(4.9

4)(9

.41)

(-6.

24)

(2.2

9)(4

.44)

(-4.

73)

(4.3

7)(7

.99)

(-3.

55)

(2.2

7)(5

.14)

(-2.

63)

α0

+ε

0.54

81.

126

-5.9

000.

224

0.73

4-5

.553

0.86

91.

389

-4.8

460.

802

1.67

7-8

.833

(3.4

5)(8

.32)

(-6.

44)

(0.9

2)(3

.30)

(-4.

40)

(4.1

9)(8

.24)

(-4.

11)

(1.7

5)(5

.04)

(-2.

82)

ε-0

.761

-0.0

95-8

.190

-1.2

30-0

.609

-8.2

66-0

.173

0.45

2-7

.051

-0.6

310.

237

-10.

181

(-2.

88)

(-0.

38)

(-6.

42)

(-2.

71)

(-1.

38)

(-3.

71)

(-0.

55)

(1.5

0)(-

6.28

)(-

1.07

)(0

.47)

(-2.

92)

26

Tab

le5:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

C

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

γ′ z

t−1+

β′ F

t+

ε t.

Des

crip

tion

sof

vari

able

s,m

omen

tum

stra

tegi

esan

dm

omen

tum

profi

tsar

eth

esa

me

asin

Tab

le3.

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

β′ F

0.13

10.

107

0.39

90.

159

0.07

61.

096

0.10

10.

175

-0.7

090.

115

0.06

10.

709

(0.6

4)(0

.52)

(0.4

0)(0

.43)

(0.2

0)(0

.63)

(0.4

0)(0

.69)

(-0.

61)

(0.3

5)(0

.20)

(0.3

3)α

0+

β′ F

0.01

00.

075

-0.7

18-0

.081

-0.0

03-0

.963

0.14

70.

169

-0.0

99-0

.011

0.10

3-1

.265

(0.1

1)(0

.88)

(-1.

53)

(-0.

64)

(-0.

02)

(-1.

70)

(1.0

7)(1

.24)

(-0.

14)

(-0.

05)

(0.5

2)(-

0.75

)α

1+

β′ F

-0.1

11-0

.072

-0.5

41-0

.143

-0.1

13-0

.472

-0.0

36-0

.024

-0.1

68-0

.169

-0.0

53-1

.438

(-1.

25)

(-0.

83)

(-1.

15)

(-1.

06)

(-0.

84)

(-0.

72)

(-0.

27)

(-0.

18)

(-0.

27)

(-0.

76)

(-0.

27)

(-0.

91)

α0

+γ′ z

0.65

61.

113

-4.4

480.

446

0.82

3-3

.825

0.85

21.

301

-4.0

870.

842

1.53

4-6

.772

(4.8

6)(9

.29)

(-6.

21)

(2.2

7)(4

.33)

(-4.

64)

(4.3

7)(8

.06)

(-3.

56)

(2.1

6)(5

.06)

(-2.

69)

α1

+γ′ z

-0.0

140.

277

-3.2

590.

073

0.28

5-2

.337

0.05

30.

324

-2.9

32-0

.375

0.16

4-6

.298

(-0.

13)

(2.6

6)(-

5.54

)(0

.41)

(1.6

2)(-

3.02

)(0

.41)

(2.8

0)(-

4.68

)(-

1.14

)(0

.62)

(-2.

95)

α0

+γ′ z

+β′ F

0.74

11.

188

-4.2

470.

474

0.77

8-2

.970

1.01

21.

507

-4.4

340.

938

1.68

0-7

.229

(3.7

4)(6

.11)

(-4.

92)

(1.4

4)(2

.26)

(-3.

20)

(3.9

3)(6

.69)

(-3.

00)

(2.0

2)(4

.46)

(-2.

45)

α1

+γ′ z

+β′ F

0.01

50.

266

-2.7

870.

047

0.19

4-1

.614

0.12

60.

412

-3.0

21-0

.287

0.17

8-5

.405

(0.1

1)(2

.08)

(-4.

30)

(0.2

1)(0

.87)

(-1.

59)

(0.7

9)(2

.80)

(-3.

71)

(-0.

92)

(0.6

5)(-

3.15

)

α0−

α1

0.45

90.

561

-0.6

760.

263

0.35

6-0

.791

0.55

90.

662

-0.5

670.

792

0.91

7-0

.583

(5.2

8)(6

.58)

(-1.

52)

(1.8

3)(2

.48)

(-1.

15)

(4.7

4)(5

.96)

(-0.

83)

(4.1

6)(5

.11)

(-0.

52)

α0

0.00

60.

086

-0.8

89-0

.083

0.02

3-1

.288

0.15

10.

176

-0.1

31-0

.032

0.08

3-1

.301

(0.0

7)(1

.03)

(-1.

80)

(-0.

65)

(0.1

8)(-

2.08

)(1

.11)

(1.3

1)(-

0.18

)(-

0.14

)(0

.42)

(-0.

73)

α1

-0.1

01-0

.060

-0.5

66-0

.109

-0.0

66-0

.606

-0.0

42-0

.038

-0.0

84-0

.194

-0.0

85-1

.388

(-1.

14)

(-0.

68)

(-1.

22)

(-0.

81)

(-0.

48)

(-0.

95)

(-0.

31)

(-0.

28)

(-0.

13)

(-0.

88)

(-0.

44)

(-0.

87)

α0

+ε

-0.0

230.

080

-1.1

71-0

.136

-0.0

05-1

.621

0.15

80.

197

-0.2

76-0

.064

0.08

6-1

.713

(-0.

25)

(0.9

4)(-

2.18

)(-

1.03

)(-

0.04

)(-

2.36

)(1

.16)

(1.4

8)(-

0.38

)(-

0.26

)(0

.43)

(-0.

87)

α1

+ε

-0.1

29-0

.062

-0.8

79-0

.163

-0.0

92-0

.963

-0.0

33-0

.015

-0.2

23-0

.224

-0.0

70-1

.921

(-1.

42)

(-0.

70)

(-1.

78)

(-1.

19)

(-0.

67)

(-1.

51)

(-0.

24)

(-0.

11)

(-0.

34)

(-0.

95)

(-0.

35)

(-1.

06)

ε-0

.762

-0.0

89-8

.263

-1.2

23-0

.595

-8.3

33-0

.187

0.44

3-7

.122

-0.6

280.

249

-10.

277

(-2.

88)

(-0.

35)

(-6.

56)

(-2.

70)

(-1.

35)

(-3.

78)

(-0.

59)

(1.4

6)(-

6.26

)(-

1.07

)(0

.49)

(-3.

04)

27

Tab

le6:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

D

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esw

ith

pred

icte

dri

skpr

emiu

mco

mpu

ted

in-s

ampl

e.Fo

rea

chm

onth

t,th

efo

llow

ing

mod

elis

esti

mat

edfo

rea

chN

YSE

/AM

EX

stoc

kon

the

mon

thly

CR

SPta

peus

ing

data

from

t−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ F

t+

θ′z t−

1F

t+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t

−7,

...,

t−

2}an

dD

τ=

0ot

herw

ise,

z t−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

san

dF

tth

eve

ctor

ofFa

ma-

Fren

chfa

ctor

s.T

hein

-sam

ple

one-

peri

od-a

head

pred

icte

dva

lue

F̂tfo

rth

epe

riod

oft−

61th

roug

ht−

2is

obta

ined

from

the

follo

win

gm

odel

(am

inim

umof

36m

onth

sof

data

requ

ired

):F

t=

d0D

t+

d1(1

−D

t)+

a′ z

t−1+

η t.

Usi

ngth

ees

tim

ated

beta

san

dF̂

t,fa

ctor

-an

dno

n-fa

ctor

-rel

ated

retu

rns

for

the

peri

odof

mon

ths

t−

7th

roug

ht−

2ar

eca

lcul

ated

and

com

poun

ded

acco

rdin

gly.

Eac

hst

rate

gyde

sign

ates

win

ners

and

lose

rsas

the

top

and

bott

omde

cile

sac

cord

ing

toth

eco

mpo

unde

dre

turn

sdu

ring

the

peri

odof

mon

ths

t−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

Pan

elA

:C

onst

ant

Bet

as–

θ=

0β′ F̂

0.23

10.

176

0.84

40.

301

0.20

11.

444

0.19

90.

234

-0.1

840.

103

-0.0

011.

250

(1.0

9)(0

.84)

(0.8

1)(0

.80)

(0.5

3)(0

.79)

(0.7

5)(0

.87)

(-0.

16)

(0.2

9)(-

3.6e

-3)

(0.5

3)

r−

β′ F̂

0.50

31.

098

-6.1

290.

101

0.62

6-5

.840

0.89

91.

437

-5.0

210.

824

1.71

8-9

.012

(2.9

0)(7

.22)

(-6.

36)

(0.3

6)(2

.39)

(-4.

34)

(4.2

2)(8

.38)

(-4.

11)

(1.7

4)(5

.02)

(-2.

75)

Pan

elB

:T

ime-

vary

ing

Bet

asβ′ F̂

-0.0

130.

023

-0.4

080.

054

0.11

6-0

.641

-0.0

81-0

.064

-0.2

63-0

.061

-0.0

60-0

.079

(-0.

15)

(0.2

8)(-

0.99

)(0

.36)

(0.7

9)(-

0.84

)(-

0.92

)(-

0.76

)(-

0.50

)(-

0.41

)(-

0.39

)(-

0.15

)β′ F̂

+θ′

zF̂

0.12

80.

070

0.77

50.

226

0.15

51.

028

0.10

40.

081

0.36

3-0

.088

-0.1

780.

906

(1.0

4)(0

.59)

(1.1

5)(1

.01)

(0.7

1)(0

.83)

(0.7

9)(0

.60)

(0.7

0)(-

0.40

)(-

0.94

)(0

.55)

r−

β′ F̂

−θ′

zF̂

0.38

20.

875

-5.1

230.

047

0.46

4-4

.679

0.63

61.

094

-4.3

970.

790

1.56

1-7

.683

(2.3

9)(6

.08)

(-5.

95)

(0.1

8)(1

.78)

(-4.

00)

(3.4

2)(7

.40)

(-3.

90)

(1.9

6)(5

.69)

(-2.

57)

28

Tab

le7:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

D1

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esba

sed

onM

odel

Dw

ith

pred

icte

dri

skpr

emiu

mco

mpu

ted

out-

of-s

ampl

e.T

heou

t-of

-sam

ple

one-

peri

od-a

head

pred

icte

dva

lue

F̂t

isca

lcul

ated

base

don

the

esti

mat

esfr

omth

eon

e-pe

riod

-ahe

adpr

edic

tive

regr

essi

onof

Fs

agai

nst

z s−

1us

ing

data

from

mon

ths

s−

60th

roug

hs−

1(a

min

imum

of36

mon

ths

ofda

tare

quir

ed),

whe

rez s

−1

isth

eve

ctor

ofth

ela

gged

mac

roec

onom

icva

riab

les

and

Fs

the

vect

orof

Fam

a-Fr

ench

fact

ors.

The

nth

efo

llow

ing

mod

elis

used

toes

tim

ate

beta

sfo

rea

chN

YSE

/AM

EX

stoc

kon

the

mon

thly

CR

SPta

peus

ing

data

from

t−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ F

t+

θ′z t−

1F

t+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t

−7,

...,

t−

2}an

dD

τ=

0ot

herw

ise.

Usi

ngth

ees

tim

ated

beta

san

dF̂

t,fa

ctor

-an

dno

n-fa

ctor

-rel

ated

retu

rns

for

the

peri

odof

mon

ths

t−

7th

roug

ht−

2ar

eca

lcul

ated

and

acco

rdin

gly

com

poun

ded.

Eac

hst

rate

gyde

sign

ates

win

ners

and

lose

rsas

the

top

and

bott

omde

cile

sac

cord

ing

toth

eco

mpo

unde

dre

turn

sdu

ring

the

peri

odof

mon

ths

t−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

Onl

yst

ocks

wit

hre

turn

sth

roug

hout

the

enti

rera

nkin

gpe

riod

are

elig

ible

for

win

ner/

lose

rse

lect

ion.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

Pan

elA

:C

onst

ant

Bet

as–

θ=

0β′ F̂

0.37

80.

322

1.00

90.

408

0.33

51.

239

0.59

00.

434

2.29

9-0

.105

0.07

3-2

.071

(1.8

3)(1

.63)

(0.8

2)(1

.16)

(0.9

9)(0

.60)

(1.9

8)(1

.54)

(1.2

9)(-

0.30

)(0

.23)

(-0.

90)

r−

β′ F̂

0.41

90.

995

-6.0

10-0

.041

0.40

6-5

.098

0.82

41.

421

-5.7

430.

872

1.76

2-8

.909

(1.8

7)(4

.61)

(-6.

04)

(-0.

11)

(1.0

4)(-

3.75

)(3

.27)

(6.7

4)(-

4.10

)(1

.66)

(4.1

3)(-

2.73

)

Pan

elB

:T

ime-

vary

ing

Bet

asβ′ F̂

0.08

00.

124

-0.4

120.

038

0.09

2-0

.564

0.17

40.

160

0.32

10.

011

0.14

1-1

.423

(0.9

2)(1

.50)

(-0.

82)

(0.2

5)(0

.63)

(-0.

62)

(1.5

7)(1

.50)

(0.5

1)(0

.08)

(1.0

2)(-

1.83

)β′ F̂

+θ′

zF̂

0.07

70.

093

-0.1

020.

154

0.18

5-0

.201

0.14

50.

056

1.11

9-0

.261

-0.0

85-2

.194

(0.6

2)(0

.80)

(-0.

13)

(0.7

0)(0

.89)

(-0.

14)

(0.9

5)(0

.39)

(1.2

7)(-

1.25

)(-

0.44

)(-

1.80

)

r−

β′ F̂

−θ′

zF̂

0.35

70.

809

-4.6

79-0

.014

0.30

3-3

.597

0.59

81.

080

-4.6

980.

892

1.65

2-7

.473

(1.8

8)(4

.45)

(-5.

04)

(-0.

04)

(0.9

3)(-

2.77

)(2

.80)

(5.9

1)(-

4.04

)(1

.89)

(4.4

2)(-

2.36

)

29

Tab

le8:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

Bw

ith

Tim

e-Var

yin

gB

etas

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esfo

rth

em

odel

Bw

ith

tim

e-va

ryin

gbe

tas.

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ F

t+

θ′z t−

1F

t+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t

−7,

...,

t−

2}an

dD

τ=

0ot

herw

ise,

z t−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

san

dF

tth

eve

ctor

ofFa

ma-

Fren

chfa

ctor

s.E

ach

stra

tegy

desi

gnat

esw

inne

rsan

dlo

sers

asth

eto

pan

dbo

ttom

deci

les

acco

rdin

gto

diffe

rent

rank

ing

crit

eria

onco

mpo

unde

dre

turn

sw

ithi

nth

epe

riod

ofm

onth

st−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

β′ F

-0.1

37-0

.005

-1.6

17-0

.213

-0.0

83-1

.678

-0.0

330.

102

-1.5

15-0

.136

0.00

2-1

.652

(-1.

66)

(-0.

06)

(-3.

58)

(-1.

40)

(-0.

56)

(-2.

10)

(-0.

36)

(1.1

7)(-

3.65

)(-

0.96

)(0

.02)

(-1.

38)

α0

+β′ F

-0.0

350.

183

-2.4

72-0

.160

0.05

2-2

.562

0.11

00.

338

-2.4

010.

021

0.23

9-2

.375

(-0.

37)

(2.0

3)(-

5.19

)(-

0.94

)(0

.31)

(-3.

38)

(1.0

0)(3

.43)

(-4.

58)

(0.1

3)(2

.14)

(-1.

59)

θ′zF

-0.1

110.

013

-1.4

85-0

.247

-0.1

34-1

.530

0.02

50.

145

-1.2

86-0

.005

0.15

3-1

.749

(-1.

71)

(0.2

1)(-

3.97

)(-

2.37

)(-

1.36

)(-

2.53

)(0

.27)

(1.6

4)(-

2.55

)(-

0.04

)(1

.30)

(-1.

77)

α0

+θ′

zF

0.02

90.

219

-2.0

90-0

.122

0.04

8-2

.043

0.18

90.

394

-2.0

610.

124

0.34

2-2

.270

(0.3

9)(3

.35)

(-4.

52)

(-1.

04)

(0.4

4)(-

2.95

)(1

.74)

(4.0

8)(-

3.39

)(0

.73)

(2.9

0)(-

1.55

)β′ F

+θ′

zF

0.03

60.

011

0.32

00.

050

0.00

10.

603

0.09

40.

095

0.07

7-0

.109

-0.1

230.

043

(0.3

0)(0

.09)

(0.5

0)(0

.22)

(0.0

1)(0

.50)

(0.7

2)(0

.71)

(0.1

4)(-

0.52

)(-

0.63

)(0

.03)

α0

+β′ F

+θ′

zF

0.70

21.

181

-4.6

470.

372

0.71

1-3

.463

1.02

81.

543

-4.6

270.

958

1.75

3-7

.780

(3.3

6)(5

.76)

(-5.

22)

(1.0

6)(1

.93)

(-3.

61)

(3.9

6)(6

.84)

(-3.

10)

(1.9

8)(4

.52)

(-2.

50)

α0

0.49

50.

876

-3.7

560.

290

0.59

1-3

.129

0.61

90.

993

-3.4

990.

809

1.41

7-5

.887

(3.9

7)(7

.70)

(-5.

80)

(1.4

8)(3

.05)

(-4.

03)

(3.7

0)(7

.08)

(-3.

62)

(2.4

2)(5

.77)

(-2.

55)

α0

+ε

0.44

10.

881

-4.4

700.

184

0.54

6-3

.919

0.63

31.

049

-3.9

370.

762

1.46

2-6

.936

(3.1

9)(7

.13)

(-5.

91)

(0.8

3)(2

.54)

(-3.

89)

(3.6

6)(7

.32)

(-4.

04)

(2.0

6)(5

.69)

(-2.

59)

ε-0

.752

-0.1

08-7

.939

-1.1

96-0

.611

-7.8

23-0

.175

0.44

6-7

.005

-0.6

680.

184

-10.

038

(-2.

96)

(-0.

45)

(-6.

39)

(-2.

76)

(-1.

46)

(-3.

55)

(-0.

57)

(1.5

3)(-

6.54

)(-

1.17

)(0

.37)

(-3.

07)

30

Tab

le9:

Pro

fits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

Cw

ith

Tim

e-Var

yin

gB

etas

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esfo

rth

eM

odel

Cw

ith

tim

e-va

ryin

gbe

tas.

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ F

t+

γ′ z

t−1+

θ′z t−

1F

t+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t

−7,

...,

t−

2}an

dD

τ=

0ot

herw

ise,

z t−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

san

dF

tth

eve

ctor

ofFa

ma-

Fren

chfa

ctor

s.E

ach

stra

tegy

desi

gnat

esw

inne

rsan

dlo

sers

asth

eto

pan

dbo

ttom

deci

les

acco

rdin

gto

diffe

rent

rank

ing

crit

eria

onco

mpo

unde

dre

turn

sw

ithi

nth

epe

riod

ofm

onth

st−

7th

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

640

1.19

3-5

.528

0.25

20.

669

-4.4

671.

047

1.60

8-5

.123

0.89

81.

805

-9.0

81(2

.89)

(5.5

6)(-

5.70

)(0

.68)

(1.7

3)(-

4.08

)(3

.92)

(7.0

2)(-

3.34

)(1

.71)

(4.4

0)(-

2.64

)

β′ F

-0.0

490.

071

-1.3

87-0

.130

-0.0

09-1

.492

0.08

60.

196

-1.1

23-0

.091

0.04

8-1

.616

(-0.

61)

(0.9

2)(-

3.21

)(-

0.88

)(-

0.07

)(-

2.00

)(1

.00)

(2.3

9)(-

2.58

)(-

0.65

)(0

.46)

(-1.

39)

α0

+β′ F

0.01

80.

093

-0.8

23-0

.109

-0.0

40-0

.882

0.18

70.

278

-0.8

150.

033

0.09

8-0

.687

(0.2

6)(1

.46)

(-2.

02)

(-0.

91)

(-0.

35)

(-1.

25)

(2.2

7)(3

.54)

(-1.

87)

(0.2

6)(0

.98)

(-0.

65)

α1

+β′ F

-0.0

720.

000

-0.8

72-0

.162

-0.0

84-1

.038

0.06

70.

147

-0.8

14-0

.097

-0.0

56-0

.549

(-1.

04)

(3e-

5)(-

2.12

)(-

1.30

)(-

0.71

)(-

1.37

)(0

.82)

(1.8

5)(-

1.96

)(-

0.85

)(-

0.57

)(-

0.64

)β′ F

+θ′

zF

0.00

4-0

.002

0.07

40.

035

0.01

90.

216

0.01

40.

017

-0.0

13-0

.097

-0.0

94-0

.127

(0.0

4)(-

0.02

)(0

.12)

(0.1

6)(0

.09)

(0.1

9)(0

.12)

(0.1

3)(-

0.02

)(-

0.52

)(-

0.53

)(-

0.11

)α

0+

β′ F

+θ′

zF

0.04

60.

014

0.40

0-0

.150

-0.1

50-0

.155

0.25

60.

228

0.57

00.

170

0.04

71.

525

(0.5

2)(0

.17)

(0.7

6)(-

0.98

)(-

1.04

)(-

0.16

)(2

.25)

(2.0

1)(1

.01)

(0.9

6)(0

.28)

(1.6

5)α

1+

β′ F

+θ′

zF

-0.0

56-0

.095

0.38

1-0

.220

-0.2

29-0

.128

0.10

70.

069

0.53

20.

073

-0.0

501.

420

(-0.

65)

(-1.

15)

(0.7

9)(-

1.51

)(-

1.64

)(-

0.14

)(0

.94)

(0.6

0)(0

.95)

(0.4

3)(-

0.30

)(1

.79)

To

beco

ntin

ued.

31

Tab

le9

conti

nued

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

γ′ z

0.08

70.

108

-0.1

440.

179

0.19

8-0

.035

0.02

10.

050

-0.2

96-0

.033

-0.0

24-0

.138

(0.9

5)(1

.22)

(-0.

28)

(1.1

4)(1

.32)

(-0.

04)

(0.1

8)(0

.42)

(-0.

47)

(-0.

19)

(-0.

14)

(-0.

18)

α0

+γ′ z

0.48

30.

851

-3.6

220.

268

0.56

6-3

.106

0.63

30.

995

-3.3

440.

772

1.34

2-5

.505

(3.9

7)(7

.61)

(-5.

89)

(1.3

9)(2

.96)

(-3.

84)

(4.0

3)(7

.42)

(-3.

94)

(2.4

1)(5

.53)

(-2.

57)

α1

+γ′ z

-0.0

680.

172

-2.7

49-0

.064

0.13

4-2

.293

-0.0

120.

240

-2.7

78-0

.189

0.14

7-3

.887

(-0.

67)

(1.7

5)(-

5.59

)(-

0.36

)(0

.77)

(-2.

94)

(-0.

10)

(2.2

9)(-

5.43

)(-

0.75

)(0

.67)

(-2.

51)

α0

+β′ F

+γ′ z

0.05

40.

247

-2.1

05-0

.090

0.09

4-2

.172

0.25

30.

442

-1.8

270.

055

0.28

4-2

.462

(0.6

0)(2

.88)

(-4.

58)

(-0.

56)

(0.5

9)(-

3.22

)(2

.48)

(4.7

4)(-

3.54

)(0

.32)

(2.6

1)(-

1.53

)α

1+

β′ F

+γ′ z

-0.0

590.

108

-1.9

20-0

.170

-0.0

10-1

.973

0.10

60.

276

-1.7

58-0

.077

0.10

6-2

.091

(-0.

70)

(1.3

5)(-

4.39

)(-

1.12

)(-

0.07

)(-

2.81

)(1

.15)

(3.1

9)(-

4.21

)(-

0.48

)(0

.95)

(-1.

47)

α0

+β′ F

+γ′ z

+θ′

zF

0.70

61.

188

-4.6

690.

378

0.71

6-3

.452

1.02

41.

541

-4.6

710.

977

1.78

0-7

.850

(3.3

7)(5

.78)

(-5.

22)

(1.0

7)(1

.94)

(-3.

56)

(3.9

4)(6

.84)

(-3.

13)

(2.0

1)(4

.57)

(-2.

51)

α1

+β′ F

+γ′ z

+θ′

zF

-0.0

440.

159

-2.3

09-0

.040

0.13

3-1

.997

-0.0

010.

221

-2.4

40-0

.139

0.10

9-2

.871

(-0.

43)

(1.5

7)(-

5.04

)(-

0.22

)(0

.75)

(-2.

55)

(-0.

01)

(1.9

4)(-

4.55

)(-

0.59

)(0

.49)

(-2.

51)

α0−

α1

0.39

30.

484

-0.6

220.

233

0.28

2-0

.321

0.50

20.

587

-0.4

350.

614

0.83

0-1

.766

(5.1

1)(6

.61)

(-1.

41)

(1.8

4)(2

.26)

(-0.

48)

(4.9

2)(6

.33)

(-0.

66)

(3.4

6)(5

.49)

(-1.

50)

α0

0.04

40.

028

0.22

6-0

.144

-0.1

12-0

.507

0.24

50.

228

0.43

20.

164

0.02

01.

745

(0.5

2)(0

.35)

(0.4

5)(-

1.01

)(-

0.84

)(-

0.57

)(2

.24)

(2.1

0)(0

.76)

(0.9

1)(0

.12)

(1.7

4)α

1-0

.058

-0.0

800.

181

-0.2

23-0

.200

-0.5

560.

105

0.07

40.

449

0.08

5-0

.052

1.59

4(-

0.69

)(-

0.97

)(0

.38)

(-1.

59)

(-1.

44)

(-0.

66)

(0.9

2)(0

.65)

(0.8

0)(0

.50)

(-0.

31)

(1.9

8)α

0+

ε0.

027

0.02

10.

093

-0.1

76-0

.139

-0.6

060.

251

0.24

20.

340

0.14

10.

022

1.44

5(0

.31)

(0.2

5)(0

.18)

(-1.

20)

(-1.

00)

(-0.

66)

(2.2

8)(2

.24)

(0.5

9)(0

.79)

(0.1

3)(1

.39)

α1

+ε

-0.0

74-0

.083

0.02

0-0

.255

-0.2

20-0

.653

0.10

50.

083

0.33

90.

066

-0.0

341.

168

(-0.

86)

(-0.

99)

(0.0

4)(-

1.73

)(-

1.55

)(-

0.76

)(0

.92)

(0.7

3)(0

.60)

(0.3

9)(-

0.20

)(1

.32)

ε-0

.738

-0.1

01-7

.840

-1.1

75-0

.588

-7.8

17-0

.182

0.42

5-6

.869

-0.6

340.

196

-9.7

67(-

2.93

)(-

0.42

)(-

6.35

)(-

2.73

)(-

1.42

)(-

3.52

)(-

0.60

)(1

.47)

(-6.

54)

(-1.

13)

(0.4

0)(-

3.11

)

32

Tab

le10

:P

rofits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Model

Bw

ith

the

Liq

uid

ity

Fac

tor

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esfo

ra

four

-fac

tor

exte

nsio

nof

Mod

elB

.For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

α0D

t+

α1(1

−D

t)+

β′ G

Ft+

θ′z t−

1G

Ft+

ε t,

whe

reG

Ftis

the

vect

orof

the

thre

eFa

ma-

Fren

chfa

ctor

san

dth

eliq

uidi

tyfa

ctor

.D

escr

ipti

ons

ofot

her

vari

able

s,m

omen

tum

stra

tegi

esan

dm

omen

tum

profi

tsar

eth

esa

me

asin

Tab

le3.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/196

9-12

/199

9(3

71m

onth

s).

02/1

969-

12/1

999

02/1

969-

12/1

984

01/1

985-

12/1

999

02/1

969-

12/1

989

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r1.

009

1.65

4-6

.329

0.92

01.

511

-6.0

151.

103

1.80

7-6

.642

1.05

81.

638

-5.6

40(3

.90)

(7.5

8)(-

4.25

)(2

.53)

(4.8

7)(-

2.69

)(2

.98)

(5.8

8)(-

3.25

)(3

.58)

(6.5

1)(-

3.27

)

Pan

elA

:C

onst

ant

Bet

as–

θ=

0β′ G

F-0

.044

0.05

6-1

.186

0.12

30.

234

-1.1

76-0

.221

-0.1

33-1

.195

0.00

10.

130

-1.4

85(-

0.23

)(0

.30)

(-1.

12)

(0.3

9)(0

.75)

(-0.

70)

(-1.

04)

(-0.

67)

(-0.

88)

(4.6

e-3)

(0.5

2)(-

1.13

)α

0+

β′ G

F0.

992

1.54

3-5

.278

0.85

71.

386

-5.3

511.

135

1.71

1-5

.204

0.99

51.

509

-4.9

44(4

.18)

(7.4

6)(-

4.04

)(2

.46)

(4.5

9)(-

2.49

)(3

.54)

(6.0

8)(-

3.32

)(3

.52)

(6.1

5)(-

2.98

)α

00.

917

1.35

0-4

.005

0.64

91.

061

-4.1

851.

202

1.66

0-3

.825

0.83

91.

221

-3.5

73(4

.94)

(8.5

3)(-

3.66

)(2

.54)

(4.9

4)(-

2.56

)(4

.45)

(7.1

5)(-

2.53

)(3

.96)

(6.7

4)(-

2.73

)α

0+

ε0.

898

1.43

4-5

.199

0.68

81.

168

-4.9

441.

121

1.71

9-5

.454

0.88

41.

341

-4.3

92(4

.29)

(8.4

9)(-

3.99

)(2

.55)

(5.2

3)(-

2.94

)(3

.48)

(6.7

6)(-

2.66

)(3

.94)

(7.1

6)(-

3.25

)ε

-0.1

100.

573

-7.8

820.

114

0.73

4-7

.162

-0.3

480.

403

-8.6

020.

285

0.90

4-6

.867

(-0.

36)

(2.1

1)(-

4.81

)(0

.28)

(1.9

4)(-

4.21

)(-

0.74

)(1

.02)

(-3.

01)

(0.8

6)(2

.90)

(-5.

34)

Pan

elB

:T

ime-

vary

ing

Bet

asβ′ G

F0.

051

0.16

6-1

.250

0.14

70.

231

-0.8

41-0

.050

0.09

6-1

.659

0.19

20.

290

-0.9

42(0

.70)

(2.4

2)(-

3.09

)(1

.58)

(2.5

7)(-

1.71

)(-

0.44

)(0

.92)

(-2.

58)

(2.1

3)(3

.24)

(-2.

48)

β′ G

F+

θ′zG

F-0

.083

-0.0

12-0

.890

0.04

80.

156

-1.2

17-0

.222

-0.1

91-0

.564

0.01

00.

120

-1.2

59(-

0.70

)(-

0.10

)(-

1.41

)(0

.28)

(0.9

1)(-

1.39

)(-

1.38

)(-

1.23

)(-

0.61

)(0

.07)

(0.8

4)(-

1.78

)α

0+

β′ G

F+

θ′zG

F1.

015

1.59

5-5

.578

0.90

01.

449

-5.5

511.

137

1.75

0-5

.604

1.03

01.

566

-5.1

65(4

.17)

(7.5

6)(-

4.19

)(2

.54)

(4.7

4)(-

2.55

)(3

.42)

(6.0

3)(-

3.46

)(3

.58)

(6.3

1)(-

3.08

)

α0

0.75

51.

100

-3.1

710.

531

0.80

0-2

.616

0.99

21.

421

-3.7

260.

670

0.94

2-2

.467

(5.0

6)(9

.01)

(-3.

25)

(2.8

4)(4

.84)

(-2.

33)

(4.2

3)(7

.99)

(-2.

29)

(4.2

3)(6

.66)

(-2.

78)

α0

+ε

0.73

41.

124

-3.6

970.

529

0.82

4-2

.925

0.95

21.

445

-4.4

700.

674

0.97

8-2

.845

(4.5

7)(8

.89)

(-3.

36)

(2.7

8)(4

.94)

(-2.

58)

(3.6

3)(7

.64)

(-2.

34)

(4.1

4)(6

.87)

(-3.

14)

ε-0

.142

0.52

6-7

.731

0.10

10.

695

-6.8

69-0

.399

0.34

5-8

.593

0.25

80.

865

-6.7

54(-

0.49

)(2

.02)

(-5.

14)

(0.2

7)(1

.96)

(-4.

40)

(-0.

89)

(0.9

0)(-

3.29

)(0

.82)

(2.9

4)(-

5.72

)

33

Tab

le11

:P

rofits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Pre

dic

ted

vers

us

Unpre

dic

ted

Ret

urn

s

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

esw

ith

pred

icte

dre

turn

sco

mpu

ted

out-

of-

sam

ple.

For

each

mon

tht,

allN

YSE

/AM

EX

stoc

kson

the

mon

thly

CR

SPta

pear

era

nked

into

deci

lepo

rtfo

lios

acco

rdin

gto

diffe

rent

com

pone

nts

com

poun

ded

over

the

form

atio

npe

riod

ofm

onth

st−

7th

roug

ht−

2.Fo

rea

chst

ock

inea

chm

onth

sin

the

form

atio

npe

riod

s∈{t−

7,...,

t−2}

,eac

hco

mpo

nent

isob

tain

edfr

omth

efo

llow

ing

one-

peri

od-a

head

out-

of-s

ampl

epr

edic

tion

wit

hth

epa

ram

eter

ses

tim

ated

usin

gda

tafr

oms−

60th

roug

hs−

1(a

min

imum

of24

mon

ths

ofda

tare

quir

ed):

r s=

c+

δ′z s

−1+

ε s,

whe

rer s

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes

and

z s−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

s.E

ach

stra

tegy

desi

gnat

esw

inne

rsan

dlo

sers

asth

eto

pan

dbo

ttom

deci

les.

Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is08

/192

9-12

/200

2(8

81m

onth

s).

08/1

929-

12/2

002

08/1

929-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

r0.

670

1.25

1-5

.756

0.28

90.

754

-4.8

851.

073

1.64

0-5

.156

0.93

31.

859

-9.2

57(3

.01)

(5.8

3)(-

5.85

)(0

.78)

(1.9

6)(-

4.32

)(4

.00)

(7.1

9)(-

3.29

)(1

.76)

(4.5

3)(-

2.66

)

Pan

elA

:U

nres

tric

ted

Reg

ress

ions

δ′z

0.07

00.

108

-0.3

540.

150

0.17

3-0

.109

0.04

60.

142

-1.0

13-0

.102

-0.1

340.

250

(0.5

7)(0

.92)

(0-0

.52)

(0.7

4)(0

.87)

(-0.

10)

(0.2

7)(0

.89)

(-1.

01)

(-0.

39)

(-0.

58)

(0.1

3)c+

δ′z

0.34

70.

713

-3.7

040.

399

0.65

8-2

.487

0.44

00.

762

-3.0

950.

024

0.76

8-8

.151

(2.3

1)(5

.33)

(-4.

03)

(1.7

0)(2

.95)

(-1.

78)

(2.3

6)(4

.38)

(-3.

39)

(0.0

6)(2

.55)

(-2.

83)

c-0

.011

-0.0

300.

193

-0.0

78-0

.118

0.37

50.

050

0.02

10.

365

0.05

20.

114

-0.6

26(-

0.09

)(-

0.24

)(0

.31)

(-0.

37)

(-0.

56)

(0.4

3)(0

.29)

(0.1

3)(0

.34)

(0.2

0)(0

.49)

(-0.

35)

c+

ε0.

039

0.07

7-0

.383

-0.1

34-0

.112

-0.3

740.

200

0.21

9-0

.009

0.20

20.

323

-1.1

25(0

.32)

(0.6

4)(-

0.61

)(-

0.65

)(-

0.54

)(-

0.43

)(1

.22)

(1.4

1)(-

0.01

)(0

.74)

(1.3

5)(-

0.58

)ε

0.18

80.

436

-2.5

65-0

.244

-0.0

53-2

.378

0.55

40.

829

-2.4

710.

660

1.01

5-3

.248

(1.1

0)(2

.57)

(-3.

22)

(-0.

79)

(-0.

17)

(-1.

75)

(3.0

8)(5

.08)

(-2.

38)

(2.0

2)(3

.34)

(-1.

85)

Pan

elB

:R

estr

icte

dR

egre

ssio

ns–

c=

0δ′

z0.

233

0.61

9-4

.042

0.25

70.

542

-2.9

170.

377

0.70

9-3

.276

-0.1

130.

654

-8.5

43(1

.54)

(4.5

4)(-

4.61

)(1

.09)

(2.4

3)(-

2.14

)(1

.87)

(3.7

5)(-

3.26

)(-

0.28

)(2

.14)

(-3.

56)

ε0.

298

0.59

6-2

.992

-0.1

140.

124

-2.7

660.

613

0.90

6-2

.608

0.81

71.

285

-4.3

36(1

.74)

(3.5

4)(-

3.65

)(-

0.38

)(0

.41)

(-2.

19)

(3.3

3)(5

.36)

(-2.

60)

(2.1

9)(4

.11)

(-1.

71)

34

Tab

le12

:P

rofits

(=P

10−

P1)

ofA

lter

nat

ive

Mom

entu

mStr

ateg

ies:

Var

ious

Ran

kin

gH

oriz

ons

Thi

sta

ble

repo

rts

the

mon

thly

retu

rns

inpe

rcen

tage

ofth

eal

tern

ativ

em

omen

tum

stra

tegi

es.

For

each

mon

tht,

the

follo

win

gm

odel

ises

tim

ated

for

each

NY

SE/A

ME

Xst

ock

onth

em

onth

lyC

RSP

tape

usin

gda

tafr

omt−

61th

roug

ht−

2(a

min

imum

of36

mon

ths

ofda

tare

quir

ed):

r t=

c 0D

t+

c 1(1

−D

t)+

δ′z t−

1+

ε t,

whe

rer t

stan

dsfo

rre

turn

sin

exce

ssof

one-

mon

thTre

asur

yB

illra

tes,

Dτ

isth

edu

mm

yva

riab

lew

ith

Dτ=

1fo

rτ∈{t

−L

,...,t−

2}an

dD

τ=

0ot

herw

ise,

and

z t−

1is

the

vect

orof

the

lagg

edm

acro

econ

omic

vari

able

s.E

ach

stra

tegy

desi

gnat

esw

inne

rsan

dlo

sers

asth

eto

pan

dbo

ttom

deci

les

acco

rdin

gto

diffe

rent

rank

ing

crit

eria

onco

mpo

unde

dre

turn

sw

ithi

nth

epe

riod

ofm

onth

st−

Lth

roug

ht−

2.Por

tfol

ios

are

form

edm

onth

ly.

The

mom

entu

mst

rate

gylo

ngs

the

win

ner

port

folio

(P10

)an

dsh

orts

the

lose

r(P

1)an

dho

lds

the

posi

tion

sfo

rth

efo

llow

ing

six

mon

ths.

t−st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hesa

mpl

epe

riod

for

the

mom

entu

mpr

ofits

is02

/193

0-12

/200

2(8

75m

onth

s).

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Pan

elA

:3-

mon

thho

rizo

nr

0.29

50.

795

-5.2

83-0

.064

0.33

7-4

.607

0.74

21.

224

-4.5

690.

400

1.20

3-8

.427

(1.6

6)(4

.80)

(-5.

86)

(-0.

22)

(1.1

5)(-

4.19

)(3

.35)

(6.7

0)(-

3.35

)(0

.88)

(3.5

5)(-

2.69

)c 0

+δ′

z0.

359

0.78

5-4

.388

0.07

50.

389

-3.4

810.

712

1.15

5-4

.166

0.44

21.

136

-7.1

87(2

.29)

(5.3

8)(-

5.48

)(0

.31)

(1.5

7)(-

3.93

)(3

.31)

(6.4

4)(-

3.12

)(1

.08)

(3.5

9)(-

2.60

)c 1

+δ′

z0.

387

0.67

2-2

.792

0.43

00.

611

-1.6

200.

469

0.70

1-2

.085

0.11

60.

783

-7.2

16(2

.75)

(5.3

3)(-

3.11

)(1

.84)

(2.8

0)(-

1.11

)(2

.88)

(4.4

0)(-

3.02

)(0

.32)

(3.0

9)(-

2.67

)c 0

−c 1

0.04

50.

209

-1.7

90-0

.169

-0.0

41-1

.626

0.27

10.

474

-1.9

570.

185

0.37

4-1

.899

(0.3

6)(1

.73)

(-2.

99)

(-0.

77)

(-0.

19)

(-1.

56)

(1.9

2)(3

.72)

(-2.

30)

(0.8

0)(1

.64)

(-1.

84)

Pan

elB

:4-

mon

thho

rizo

nr

0.39

70.

932

-5.5

780.

037

0.45

7-4

.716

0.82

31.

352

-4.9

970.

541

1.40

3-8

.947

(2.0

0)(4

.98)

(-5.

91)

(0.1

1)(1

.37)

(-4.

31)

(3.3

5)(6

.57)

(-3.

41)

(1.1

0)(3

.79)

(-2.

67)

c 0+

δ′z

0.46

60.

919

-4.5

830.

208

0.53

5-3

.505

0.78

01.

258

-4.4

850.

559

1.30

0-7

.591

(2.7

4)(5

.76)

(-5.

35)

(0.7

8)(1

.96)

(-3.

71)

(3.2

8)(6

.27)

(-3.

07)

(1.2

9)(3

.88)

(-2.

62)

c 1+

δ′z

0.32

20.

592

-2.6

930.

387

0.56

3-1

.611

0.41

00.

631

-2.0

28-0

.021

0.59

5-6

.805

(2.2

7)(4

.59)

(-3.

05)

(1.6

1)(2

.48)

(-1.

12)

(2.5

4)(4

.07)

(-2.

62)

(-0.

06)

(2.3

5)(-

2.65

)c 0

−c 1

0.12

60.

282

-1.6

16-0

.113

0.00

8-1

.483

0.34

40.

531

-1.7

060.

350

0.54

4-1

.790

(0.9

5)(2

.11)

(-2.

72)

(-0.

48)

(0.0

3)(-

1.46

)(2

.27)

(3.7

3)(-

2.01

)(1

.39)

(2.1

9)(-

1.52

)

To

beco

ntin

ued.

35

Tab

le12

conti

nued

02/1

930-

12/2

002

02/1

930-

12/1

964

01/1

965-

12/1

989

01/1

990-

12/2

002

Ran

king

Bas

edon

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Ove

rall

Non

-Jan

Jan

Pan

elC

:5-

mon

thho

rizo

nr

0.49

71.

061

-5.7

970.

122

0.55

8-4

.805

0.91

21.

477

-5.2

960.

704

1.61

8-9

.354

(2.3

6)(5

.28)

(-6.

00)

(0.3

5)(1

.55)

(-4.

44)

(3.5

2)(6

.74)

(-3.

50)

(1.3

7)(4

.13)

(-2.

70)

c 0+

δ′z

0.61

11.

071

-4.5

180.

376

0.69

4-3

.231

0.88

21.

383

-4.6

300.

723

1.48

6-7

.666

(3.4

0)(6

.27)

(-5.

22)

(1.3

3)(2

.38)

(-3.

56)

(3.5

2)(6

.45)

(-3.

11)

(1.5

9)(4

.15)

(-2.

58)

c 1+

δ′z

0.25

90.

524

-2.6

950.

387

0.56

3-1

.611

0.31

10.

542

-2.2

27-0

.185

0.38

3-6

.434

(1.8

0)(3

.92)

(-3.

16)

(1.5

6)(2

.37)

(-1.

15)

(1.9

2)(3

.59)

(-2.

57)

(-0.

54)

(1.4

7)(-

2.74

)c 0

−c 1

0.20

80.

359

-1.4

76-0

.062

0.05

2-1

.362

0.41

70.

604

-1.6

380.

532

0.71

3-1

.461

(1.4

5)(2

.45)

(-2.

62)

(-0.

24)

(0.1

9)(-

1.48

)(2

.64)

(4.0

3)(-

1.94

)(2

.05)

(2.7

9)(-

1.16

)

Pan

elD

:9-

mon

thho

rizo

nr

0.70

51.

334

-6.3

100.

308

0.81

0-5

.379

1.11

91.

740

-5.7

100.

977

1.96

6-9

.898

(2.9

1)(5

.65)

(-6.

30)

(0.7

4)(1

.88)

(-4.

12)

(3.9

3)(7

.09)

(-3.

67)

(1.8

2)(4

.58)

(-3.

14)

c 0+

δ′z

0.92

01.

363

-4.0

220.

708

0.99

8-2

.576

1.10

41.

615

-4.5

201.

137

1.86

2-6

.845

(4.4

7)(6

.76)

(-4.

37)

(2.0

4)(2

.80)

(-1.

95)

(4.0

8)(6

.72)

(-3.

01)

(2.5

8)(4

.88)

(-2.

95)

c 1+

δ′z

-0.1

190.

187

-3.5

350.

167

0.36

8-2

.109

-0.2

280.

066

-3.4

71-0

.678

-0.0

68-7

.388

(-0.

80)

(1.3

5)(-

4.24

)(0

.66)

(1.4

5)(-

1.73

)(-

1.30

)(0

.45)

(-2.

88)

(-2.

05)

(-0.

27)

(-3.

38)

c 0−

c 10.

603

0.64

00.

193

0.26

40.

287

0.01

10.

792

0.87

2-0

.083

1.14

81.

143

1.20

4(4

.14)

(4.2

4)(0

.35)

(0.9

9)(1

.02)

(0.0

1)(5

.02)

(5.6

5)(-

0.10

)(4

.96)

(4.7

5)(1

.37)

Pan

elE

:11

-mon

thho

rizo

nr

0.58

31.

276

-7.1

420.

186

0.75

8-6

.287

1.05

81.

732

-6.3

560.

737

1.79

4-1

0.89

2(2

.30)

(5.2

5)(-

6.45

)(0

.43)

(1.7

1)(-

4.00

)(3

.54)

(6.8

2)(-

3.83

)(1

.32)

(4.0

5)(-

3.37

)c 0

+δ′

z0.

842

1.32

0-4

.492

0.64

10.

965

-3.0

261.

069

1.61

6-4

.949

0.94

41.

707

-7.4

48(4

.04)

(6.5

2)(-

4.74

)(1

.85)

(2.7

3)(-

2.22

)(3

.80)

(6.5

4)(-

3.12

)(2

.06)

(4.2

6)(-

3.27

)c 1

+δ′

z-0

.261

0.03

8-3

.595

-0.0

310.

137

-1.9

33-0

.327

-0.0

28-3

.614

-0.7

50-0

.100

-7.9

07(-

1.77

)(0

.28)

(-4.

05)

(-0.

12)

(0.5

5)(-

1.53

)(-

1.89

)(-

0.20

)(-

2.92

)(-

2.18

)(-

0.42

)(-

3.15

)c 0

−c 1

0.60

00.

642

0.13

20.

293

0.33

0-0

.126

0.82

10.

910

-0.0

970.

988

0.96

51.

245

(4.2

4)(4

.48)

(0.2

1)(1

.15)

(1.2

6)(-

0.12

)(5

.10)

(5.9

3)(-

0.10

)(4

.11)

(3.9

7)(1

.10)

Pan

elF:12

-mon

thho

rizo

nr

0.43

61.

160

-7.6

440.

014

0.59

8-6

.602

0.98

11.

701

-6.9

380.

521

1.63

4-1

1.72

3(1

.67)

(4.6

7)(-

6.56

)(0

.03)

(1.3

2)(-

3.96

)(3

.22)

(6.6

9)(-

4.10

)(0

.90)

(3.6

0)(-

3.41

)c 0

+δ′

z0.

711

1.22

1-4

.980

0.52

60.

861

-3.2

710.

960

1.55

1-5

.539

0.72

61.

553

-8.3

71(3

.37)

(6.0

1)(-

5.14

)(1

.51)

(2.4

3)(-

2.37

)(3

.36)

(6.2

7)(-

3.45

)(1

.53)

(3.8

0)(-

3.58

)c 1

+δ′

z-0

.186

0.05

7-2

.890

0.05

70.

160

-1.1

00-0

.306

-0.0

61-2

.997

-0.6

080.

007

-7.3

65(-

1.25

)(0

.42)

(-3.

18)

(0.2

2)(0

.63)

(-0.

86)

(-1.

78)

(-0.

45)

(-2.

27)

(-1.

82)

(0.0

3)(-

2.93

)c 0

−c 1

0.46

20.

568

-0.7

160.

164

0.25

6-0

.873

0.72

80.

880

-0.9

380.

752

0.80

90.

124

(3.2

8)(4

.01)

(-1.

11)

(0.6

5)(0

.98)

(-0.

84)

(4.4

5)(5

.96)

(-0.

88)

(3.0

6)(3

.30)

(0.1

0)

36