Jackknifing Stock Return Predictions

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 932 June 2008 Jackknifing Stock Return Predictions Benjamin Chiquoine and Erik Hjalmarsson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

Jackkni(cid:133)ng Stock Return Predictions Benjamin Chiquoine Erik Hjalmarsson (cid:3) Division of International Finance Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA May 2, 2008 Abstract We show that the general bias reducing technique of jackkni(cid:133)ng can be successfully applied to stock return predictability regressions. Compared to standard OLS estimation, the jackkni(cid:133)ng procedure delivers virtually unbiased estimates with mean squared errors that generally dominate those of the OLS estimates. Thejackkni(cid:133)ngmethodisverygeneral,aswellassimpletoimplement,andcanbeappliedto models with multiple predictors and overlapping observations. Unlike most previous work on inference inpredictiveregressions,nospeci(cid:133)cassumptionsregardingthedatageneratingprocessforthepredictors arerequired. AsetofMonteCarloexperimentsshowthatthemethodworkswellin(cid:133)nitesamplesandthe empiricalsection(cid:133)ndsthatout-of-sampleforecastsbasedonthejackknifedestimatestendtooutperform those based on the plain OLS estimates. The improved forecast ability also translates into economically relevant welfare gains for an investor who uses the predictive regression, with jackknifed estimates, to time the market. JEL classi(cid:133)cation: C22, G1. Keywords: Bias correction; Jackkni(cid:133)ng; Predictive regression; Stock return predictability. Helpfulcomments have been provided by DanielBeltran,Lennart Hjalmarsson,RandiHjalmarsson,and Mike McCracken. (cid:3) Corresponding author: Erik Hjalmarsson. Tel.: +1-202-452-2426; fax: +1-202-263-4850; email: erik.hjalmarsson@frb.gov. The viewsinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasre(cid:135)ectingtheviewsoftheBoard ofGovernors ofthe FederalReserve System orofany otherperson associated with the FederalReserve System.

1 Introduction OrdinaryLeastSquares(OLS)estimationofpredictiveregressionsforstockreturnsgenerallyresultsinbiased estimates. This is true in particular when valuation ratios, such as the dividend- and earnings-price ratios, areusedaspredictorvariables. Thebiashasbeenanalyzedanddiscussedinnumerousarticlesandanumber of potential solutions have been suggested (e.g., Mankiw and Shapiro, 1986, Stambaugh, 1999, and Jansson and Moreira, 2006). However, most of the attention in the literature has been directed at constructing valid tests in the case of a single regressor that follows an auto-regressive process, and much less attention has been given to the problem of obtaining better estimators, both in the case of single or multiple predictor variables.1 Although the testing problem is arguably the more fundamental issue from a strictly statistical point of view, the estimation problem is of great interest from an economic and practical perspective. The statistical tests answer the question whether there is predictability, but the coe¢ cient estimate speaks more directly to the economic magnitude of the relationship. Since there is an emerging consensus in the (cid:133)nance profession thatstockreturnsaretosomeextentpredictable,itisofvitalinteresttodeterminetheeconomicimportance of this predictability. In addition, if forecasting regressions are to be used for out-of-sample forecasts, which is often their ultimate purpose, the point estimate obviously takes on the main role. In this paper, we propose the application of a general bias reduction technique, the jackknife, to obtain better point estimates in predictive regressions. Unlike most other methods that have been proposed, this procedure does not assume a particular data generating process for the regressor and allows for multiple predictor variables. The jackknifed estimator, which is based on a combination of OLS estimates for a small numberofsubsamples,isalsotrivialtoimplementandcouldeasilybeusedwithcommonstatisticalpackages. In relation to previous work, the current paper contributes to both the emerging literature on bias-reducing techniques in predictive regressions, such as Amihud and Hurvich (2004) and Eliasz (2005), as well as the ongoing debate on out-of-sample predictability in stock-returns, as exempli(cid:133)ed by Goyal and Welch (2003, 2007) and Campbell and Thompson (2007). In a series of Monte Carlo experiments, we show that the jackknifed estimator can reduce the bias in the estimates of the slope coe¢ cients in predictive regressions. This applies both to the standard one-regressor, one-period regression as well as to the case of multiple regressors and longer forecasting horizons. Although 1The only bias corrections, in predictive regressions, that have been used to any great extent are ad hoc corrections for the biasderviedbyStambaugh(1999),forthecaseofasingleregressorthatfollowsanAR(1)process. AmihudandHurvich(2004) provide justi(cid:133)cations for similar corrections in the case of multiple regressors. Lewellen (2004) provides a (cid:145)conservative(cid:146)bias correction, also based on a single AR(1)regressor, which is primarily useful as a tool for obtaining conservative test statistics, sinceingeneralthecorrectedestimatewillnotbeunbiasedbut,rather,underestimatethetrueparametervalue. Infact,oneof the main reasons that testing, rather than estimation, has been the main focus is that most studies on inference in predictive regressions resort to some conservative test, which does not deliver a unique estimation analogue; e.g., Cavanagh et al. (1995) and Campbelland Yogo (2006). 1

the jackknifed estimates have a larger variance than the OLS estimates, the jackknifed estimates still often outperform the OLS ones in a mean squared error sense. Thus, to the extent that it is desirable to have as small a bias as possible, for a given mean squared error, the jackknifed estimator tends to dominate the OLS estimator. In the empirical section of the paper, we consider forecasting of aggregate U.S. stock returns, using (cid:133)ve di⁄erent predictor variables: the dividend- and earnings-price ratios, the smoothed earnings-price ratio suggested by Campbell and Shiller (1988), the book-to-market ratio, and the short interest rate. Although manyotherstockreturnpredictorshavebeenproposed(see,forinstance,GoyalandWelch,2007),theabove valuationratiosareofmostinteresthere,sincetheytendtoresultinthelargestbiasesintheOLSestimates. The short interest rate is also analyzed since some recent work by Ang and Bekaert (2007) suggests that it workswellasapredictortogetherwiththedividend-priceratio,whichthusprovidesanopportunitytostudy the performance of the jackknifed estimator with multiple regressors. The in-sample results show that the jackknifed estimates, in some cases, deviate substantially from the OLSestimates. Forinstance,themagnitudeofthecoe¢ cientforthebook-to-marketratioisoftendrastically smaller when using the jackknife procedure. On average, the OLS estimates often overstate the magnitude of predictability compared to the jackknife estimates. Inordertoevaluatewhetherthesediscrepanciesinthefull-sampleestimatesactuallytranslateintobetter real time forecasting ability, we perform two di⁄erent out-of-sample exercises. First, we calculate the out-ofsampleR2sforthedi⁄erentpredictorvariables,and(cid:133)ndthattheforecastsbasedonthejackknifedestimates typically dominate those based on the OLS estimates; this is true also if one imposes some of the forecast restrictions proposed by Campbell and Thompson (2007). In a second out-of-sample exercise, we estimate the welfare gains to a mean-variance investor who uses either the OLS estimates or the jackknifed estimates toformhisportfolioweightsinordertotimethemarket. In this case, thejackknifed estimatesproduceeven clearergains,dominatingboththeportfoliochoicesbasedontheOLSestimatesaswellasthebaselinechoice based on the historical average returns. Overall, the promising results seen in the Monte Carlo simulations carry over to the real data. The rest of the paper is organized as follows. Section 2 outlines the jackknife procedure and provides an explicit example of how it works in a predictive regression. Section 3 presents the results from the Monte Carlo exercises. The empirical analysis is performed in Section 4 and Section 5 concludes. 2

2 The Jackknife Let T be the sample size available for the estimation of some parameter (cid:18). Decompose the sample into m consecutive subsamples, each with l observations, so that T = m l. The jackknife estimator, which was (cid:2) introduced by Quenoille (1956), is given by m m ^(cid:18) ^(cid:18) = ^(cid:18) i=1 li ; (1) jack m 1 T (cid:0) m2 m (cid:0) P (cid:0) where ^(cid:18) and ^(cid:18) are the estimates of (cid:18) based on the full sample and the ith subsample, respectively, using T li some given estimation method such as OLS or maximum likelihood. In the current paper, we rely only on OLS for obtaining ^(cid:18) . Under fairly general conditions, which ensure that the bias of ^(cid:18) and ^(cid:18) can be li T li expanded in powers of T 1, it can be shown that the bias of ^(cid:18) will be of an order O T 2 instead of (cid:0) jack (cid:0) O T 1 ; Phillips and Yu (2005) provide a longer discussion on this. (cid:0) (cid:1) (cid:0) (cid:0)Note(cid:1)that (cid:18) may be a single parameter, or a vector of parameters, estimated from some model using any feasible estimation method. Furthermore, (cid:18) may also represent a combination, or complicated function, of estimatedparameters. Forinstance,PhillipsandYu(2005)showhowjackkni(cid:133)ngbondoptionpricesdirectly, rather than the estimated parameters that enters the bond option formula, can help reduce the bias in the estimated option prices. The jackknife is thus a very generally applicable method. Within the context of estimating models for stock return predictability, we consider, in addition to the standard single regressor case, also a case with multiple regressors, as well as overlapping observations. Whereas the bias in the single regressor case is well analyzed, less is understood about the biases in the case of multiple regressors or the case of long-run forecasting regressions with overlapping observations. Again, in all three cases the analysis of the bias is usually restricted to the case where the regressors follow an auto-regressive process; see, for instance,AmihudandHurvich(2004)foradiscussiononsomebiasreductionmethodsforthecaseofmultiple regressors. Asimpleexamplehelpstoillustratehowthejackkni(cid:133)ngprocedurereducesthebiasinestimates. Consider the traditional predictive regression with a single regressor which follows an AR(1) process: r = (cid:22)+(cid:12)x +u ; (2) t t 1 t (cid:0) x = (cid:13)+(cid:26)x +v : (3) t t 1 t (cid:0) Supposeu t andv t arebivariatenormallydistributedwithmeanzeroandcovariancematrix (cid:27)2 u ;(cid:27) uv ; (cid:27) uv ;(cid:27)2 v 0; the correlation between u and v is denoted by (cid:14) in the simulations below. As shown in S(cid:2)(cid:0)tambaug(cid:1)h(cid:0)(1999), (cid:1)(cid:3) t t 3

the bias in the OLS estimator of (cid:12) is given by (cid:27) 1+3(cid:26) E (cid:12)^ (cid:12) = uv +O T 2 =O T 1 : (4) OLS (cid:0) (cid:0) (cid:27)2 T (cid:0) (cid:0) h i v (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) The jackknife estimator of (cid:12) for m=2, based on OLS estimation, is equal to 1 (cid:12)^ =2(cid:12)^ (cid:12)^ +(cid:12)^ ; (5) jack T (cid:0) 2 T=2;1 T=2;2 (cid:16) (cid:17) and 1 (cid:12)^ (cid:12) =2 (cid:12)^ (cid:12) (cid:12)^ (cid:12)+(cid:12)^ (cid:12) : (6) jack(cid:0) T (cid:0) (cid:0) 2 T=2;1(cid:0) T=2;2(cid:0) (cid:16) (cid:17) (cid:16) (cid:17) Taking expectations on both sides and using the expression in (4), it follows that E (cid:12)^ (cid:12) = 2 (cid:27) uv 1+3(cid:26) + (cid:27) uv 1+3(cid:26) +O T (cid:0) 2 =O T 2 : (7) h jack(cid:0) i (cid:0) (cid:27)2 v (cid:18) T (cid:19) (cid:27)2 v (cid:18) T=2 (cid:19) (cid:18) 2 (cid:19) ! (cid:0) (cid:0) (cid:1) Thus, the bias is reduced from O T 1 to O T 2 . (cid:0) (cid:0) This result would hold for any(cid:0)m, w(cid:1)hich ra(cid:0)ises t(cid:1)he question of what value m should be set to in practice. As shown by the simulations in the following section, setting m = 2 works very well and usually eliminates almost all of the bias. However, the simulations also show that an increase in m (to 3 or 4) can reduce the variance of the jackknife estimate without any substantial increase in the bias. In general, the root mean squared error is smallest for m = 4 in the simulations presented below. Phillips and Yu (2005) present results along similar lines and provide some brief theoretical arguments that support these (cid:133)ndings. In a given context, an optimal choice of m may therefore exist, although there appears to be no studies on how to choose this optimal m. The empirical section, which presents results for m = 2;3, and 4, suggests that m=3 may be the best choice on average, although the di⁄erences are generally not great between the three alternatives, and there appears to be no choice of m that strictly dominates empirically. 3 Monte Carlo Simulations Weanalyzethe(cid:133)nitesampleperformanceofthejackknifemethodbysimulatingdatafromthemodelde(cid:133)ned by equations (2) and (3). The assumption that the predictor variable follows an AR(1) process is probably the most common one in the analysis of stock-return predictability. This stems primarily from the relative ease with which the properties of estimators of (cid:12) can be analyzed in this setup, and because the model captures the most salient features of typical forecasting variables such as valuation ratios and interest rates. 4

The results from the AR(1) speci(cid:133)cation should also be qualitatively similar to those from a more general AR(p) model. In general, the jackknife procedure should help reduce bias in other setups as well, but we focus on its properties for this familiar model which is easy to parametrize in a realistic manner, such that theOLSestimatorwillbebiasedin(cid:133)nitesamples. Inadditiontoconsideringthecasewithasingleregressor, we also simulate from a model with two forecasting variables, where each of these follows an AR(1) process asspeci(cid:133)edindetailbelow. Finally,wealsoconsiderthecasewhenforecastsareformedatahorizondi⁄erent from that at which the data were sampled. 3.1 The single regressor case Equations(2)and(3)aresimulatedforthecasewhenx isascalar. Theinnovationtermsu andv aredrawn t t t from a multivariate normal distribution with unit variances. The correlation between u and v , denoted (cid:14), t t takes on three di⁄erent values: 0:9; 0:95; and 0:99. The auto-regressive root (cid:26) is set equal to either (cid:0) (cid:0) (cid:0) 0:9;0:95, or 0:999. The sample size, T, is equal to 100 or 500 observations. The parameters (cid:22);(cid:12); and (cid:13) are all set to zero, although an intercept is still estimated in the predictive regression; since the bias in the OLS estimator is not a function of the values of these parameters (e.g. Stambaugh, 1999), this standardization does not a⁄ect the results. Campbell and Yogo (2006) show that values such as these for (cid:14) and (cid:26) are often encountered empirically, when using valuation ratios as predictors. Note that, if (cid:14) = 0, so that the error terms u and v are uncorrelated, the OLS estimator is unbiased t t and equal to the full information maximum likelihood estimator. Furthermore, for (cid:26) close to zero, the OLS estimator will also be unbiased, even when (cid:14) =0. In general, the bias for the OLS estimator is thus greater 6 as (cid:26) gets closer to unity, and the closer the absolute value of (cid:14) is to one. We therefore restrict the analysis to the part of the parameter space where there actually is a bias to correct in the OLS estimator. Results for (cid:14) <0areshownsincethisistheempiricallymostrelevantcaseandthecaseof(cid:14) >0iscompletelyanalogous. The Monte Carlo simulation is conducted by generating 10;000 sample paths from equations (2) and (3), for each combination of parameter values. From each set of generated returns and regressors, the OLS estimate of (cid:12) and the jackknife estimates for m = 2;3; and 4, are calculated. The average bias and rootmean-squarederrors(RMSE)fortheseestimatorsarethencalculatedacrossthe10;000samples. Theresults are reported in Table 1, which shows the bias and the RMSE in parentheses below, for each parameter combination. An inspection of the results in Table 1 quickly reveals three distinct (cid:133)ndings: (i) the OLS estimates are upward biased for all of the parameter combinations under consideration, (ii) the jackknife estimates are virtually unbiased in all cases, and (iii) the RMSEs for the jackknife estimates are always less than or equal 5

to the RMSE for the OLS estimates for m=3 and 4, and fairly similar to the RMSE for the OLS estimates for m = 2. These simulation results thus suggest that the jackkni(cid:133)ng procedure reduces the bias without inducing enough variance to in(cid:135)ate the RMSE. Figure1providessomeadditionalinsightsintotheworkingsofthejackknifedestimator. Itshowsdensity plotsfortheOLSestimatoraswellasthejackknifeestimatorsform=2;3;and4. Thedensitiesareestimated with kernel methods from 100;000samples, with T = 100, (cid:26) = 0:999 and (cid:14) = 0:99. The density of the (cid:0) OLS estimate is almost completely to the right of the true value for (cid:12), and is also highly skewed towards the right. The jackknifed estimates are both more centered around the true value as well as more symmetric. For m=2, the jackknife estimator has a distribution that is centered almost exactly at the true value and is also fairly symmetric. For m=3 and 4, the densities are more peaked, re(cid:135)ecting the lower RMSEs shown in Table 1, but also slightly less centered at the true value; these densities are also somewhat more skewed. As mentioned in the previous section, these results indicate that there is a trade o⁄between bias and variance in the choice of m, and an optimal choice of m in terms of RMSE may therefore exist. However, no formal results along these lines appear to be available. In order to understand the magnitude of the bias in the OLS estimator, and the importance of the bias reduction achieved with the jackknife estimators, it is useful to consider typical values of the estimates of (cid:12) in actual data. The results in Campbell and Yogo (2006) are particularly convenient for such a comparison since they present their estimates in a standardized manner conforming with the model simulated here; that is, theyscaletheestimateof(cid:12) tocorrespondtoamodelwithunitvarianceinu andv . CampbellandYogo t t (2006) consider stock return predictability for aggregate U.S. stock returns. They show that OLS estimates of (cid:12) are typically in the range of 0:1 to 0:2 in annual data and most often in the range of 0:01 to 0:02 in monthly data. Thus, if one uses 100 years of annual data, the bias in the OLS estimate may be between 20 and 50 percent of the actual parameter value, as seen from the results in Table 1. If one relies on a shorter (in years covered) monthly series with 500 observations, the bias could easily be as large as the parameter value itself. In proportion to the size of the parameter value, the bias reduction in the jackkni(cid:133)ng procedure is therefore at least substantial and potentially huge. 3.2 Multiple regressors Although the simple forecasting regression with just one predictor is by far the most studied and commonly used in the literature, there are instances when the use of several forecasting variables may be advantageous. For instance, Ang and Beekaert (2007) argue that the dividend-price ratio works much better as a predictor when used jointly with the short rate, rather than on its own. 6

In order to evaluate the properties of the jackknife estimator in the multiple regressor case, we restrict the attention to the case with two forecasting variables and follow a similar setup to the one used in the single regressor case. In particular, it is assumed that the data is generated by a multivariate version of the model described by equations (2) and (3). The auto-regressive matrix for the two predictor variables is set to A=[(a 11 ;0);(0;a 22 )]0and the innovations u t and v t =(v 1t ;v 2t ) are again normally distributed with unit variance. The correlation vector between u and v is labeled ! and the correlation between v and v is t t uv 1t 2t labeled(cid:17), suchthatthevariance-covariancematrixforv t isequalto(cid:10) vv =[(1;(cid:17));((cid:17);1)]0. Table2showsthe results for the estimates of the two coe¢ cients, (cid:12) and (cid:12) , that correspond to the (cid:133)rst and second predictor 1 2 variable, for various values of A and di⁄erent correlations between the innovations. Results for T =100 and T =500 are presented and the results are based on 10;000 repetitions. The (cid:133)rst two columns of results in Table 2 represent perhaps the most empirically interesting case. For these results, a 11 = 0:999, a 22 = 0:95, ! uv = ( 0:9;0)0 and (cid:17) = 0:4. That is, the (cid:133)rst predictor is the most (cid:0) persistent one and is also highly endogenous, whereas the second predictor is exogenous and less persistent. This setup corresponds fairly well to the case with the dividend-price ratio and the short interest rate as predictors,sincethedividend-priceratioishighlyendogenouswhereastheshortrateisnearlyexogenous,and usually somewhat less persistent than the dividend-price ratio (Campbell and Yogo, 2006). The correlation of 0:4 between the innovations to the two regressors results in an average correlation of around 0:25 between the levels of the regressors, which is similar to the empirical correlation between the dividend-price ratio and the short interest rate observed in the data used in this paper.2 Intuitively, given the results for the single regressor case, one would expect the OLS estimate for the coe¢ cient for the (cid:133)rst regressor ((cid:12) ) to be highly biased whereas the second estimate ((cid:12) ) should perform 1 2 better,sinceitisonlyindirectlya⁄ectedbytheendogenitybiasthroughthecorrelationofthetworegressors. This intuition is bourne out to some extent by the simulation results, which show a large bias for the (cid:133)rst predictor but a smaller, although still substantial, bias for the second. The jackknife works very well for (cid:12) , resulting in almost unbiased estimates with only a small increase in RMSE for m = 2 and a signi(cid:133)cant 1 reductioninRMSEform=3;and4. Jackkni(cid:133)ngtheestimatesfor(cid:12) alsoresultsinunbiasedestimates,but 2 with a slight increase in RMSE, particularly for T =100. ThefollowingtwosetsofresultsinTable2representcaseswherebothregressorsareendogenous. Inorder fortheoverallcovariancematrixbetweenu andv tobewellde(cid:133)ned,thisforcesv andv tobefairlyhighly t t 1t 2t correlated as well. In particular, ! uv = ( 0:7; 0:7)0 and (cid:17) = 0:5 in the (cid:133)rst case and ! uv = ( 0:9; 0:9)0 (cid:0) (cid:0) (cid:0) (cid:0) 2Thecasewhentheregressorsarecompletelyorthogonaltoeachotherisobviouslymostdesirableinempiricalspeci(cid:133)cations, althoughitisnotoftenlikelytohold. Inthatcase,however,therewillbelittletonodi⁄erencebetweentheindividualcoe¢ cient estimatesfromseparateregressionsoneachregressorandtheestimatesobtainedfromamultipleregression. Thus,thereislittle point in analyzing this case,since it would merely con(cid:133)rm the results obtained in the previous section. 7

and (cid:17) = 0:8 in the second case. The peristence parameters are set to a = 0:999 and a = 0:95 in the 11 22 (cid:133)rst speci(cid:133)cation, and to a = a = 0:999 in the second one. Thus, both variables are highly endogenous 11 22 and highly co-linear. In the (cid:133)rst speci(cid:133)cation, the (cid:133)rst predictor is more persistent than the other, whereas in the latter speci(cid:133)cation both have the same persistence. These parametrizations could correspond to, for instance, various combinations of valuation ratios, which mayhave somewhatdi⁄erentdegrees of persistence and endogeneity, and may also have di⁄erent correlations with each other. As seen in Table 2, both of these parametrizations result in OLS estimates that are biased, both for (cid:12) 1 and (cid:12) . The jackkni(cid:133)ng reduces the bias substantially, although there is a little bias left in the jackknifed 2 estimates when using the (cid:133)rst parameter speci(cid:133)cation for T = 100, as seen in the middle columns of Table 2. For m = 3 and 4, the RMSE for the jackknifed estimates are similar to the OLS ones. The second speci(cid:133)cation, shown in the last two columns of Table 2, is symmetric for the two regressors and the OLS biases for the two coe¢ cients are virtually identical. For T =100; the jackknifed estimates are now virtually unbiased, with only a small increase in the RMSE relative to the OLS estimates. For T = 500, the bias is completely removed by the jackkni(cid:133)ng. In summary, the jackknife appears to work well in the multivariate case and generally results in virtually unbiased estimates. Given the multitude of possible parameter combinations that arise as soon as one leaves the simplicity of the single regressor case, the results presented in this section are far from exhaustive but hopefully capture some of the more salient features of biases in multivariate regressions. 3.3 Overlapping observations Finally, weconsidertheperformanceofthejackknifeestimatorforpredictiveregressionwithoverlappingobservations. Inferencewithoverlappingobservationsisatopicthathasalonghistoryinthe(cid:133)nanceliterature, but most of the e⁄ort has been directed at constructing valid test statistics rather than reducing the bias in OLS estimates.3 The jackknife procedure provides a simple but (cid:135)exible way of addressing the estimation problem. To keep things tractable, the single regressor case is analyzed. The data is generated in exactly the same manner as described in Section 3.1, generating sample paths from equations (2) and (3). However, instead of estimating equation (2), the sums of future q period returns are now regressed on the value of x . The t (cid:0) forecasting horizon q is set equal to 10 for T = 100 and equal to 12 for T = 500. These two cases capture common applications of long-run forecasts using a century of annual data and annual forecasts based on monthly data. 3See, for instance, Hansen and Hodrick (1980), Richardson and Stock (1989), Richardson and Smith (1991), Goetzman and Jorion (1993), Campbell (2001), Valkanov (2003), Torous et al. (2004), and Boudoukh et al. (2006). Nelson and Kim (1993) brie(cid:135)y discuss the magnitude ofthe Stambaugh (1999) bias in regressions with overlapping observations. 8

The results are shown in Table 3. The bias in the OLS estimates is of an order of magnitude larger than the ones shown in Table 1. This is entirely in line with the analytical results of Bodoukh et al. (2006), who showthatoneshouldexpecttheestimate,andhencethebias,toincreasealmostlinearlywiththeforecasting horizon. The jackkni(cid:133)ng reduces the bias substantially in all cases, although not always completely. The RMSEsforthejackknifedestimatesisslightlylargerthanthosefortheOLSestimatesinsomecases,although there are also substantial reductions for some parameter combinations. It is evident that the jackknife is also applicable in long-horizon regressions. From the results presented here,itappearstobemostusefulwhentheoverlapisnottoolargerelativetothenumberofobservations;the results for q =12 and T =500 are generally stronger than those for q =10 and T =100. Overall, however, theresultsareverypromisingandthejackknifeclearlypresentsasimplewayofalleviatingestimationbiases in long-horizon regressions, an issue which is often ignored in applied work. 4 Empirical Analysis We next apply the jackknife method to real stock market data. Since the purpose of the jackknife method is to obtain better point estimates, we primarily evaluate its usefulness by an out-of-sample (OOS) forecasting exercise. However, it is also of interest to analyze the full sample point estimates, since they directly show the di⁄erences between the plain OLS estimates and the bias corrected jackknifed estimates. Asthedependentvariable,weusemonthlytotalexcessreturnsontheS&P500index,startinginFebruary 1872 and ending December 2005. Five separate forecasting variables are used: the dividend- and earningsprice ratios (D/P and E/P), the smoothed earnings-price ratio of Campbell and Shiller (1988), the bookto-market ratio (B/M), and the short term interest rate as measured by the three-month T-Bill rate. The smoothed earnings-price ratio is de(cid:133)ned as the ratio of the 10-year moving average of real earnings to the currentrealprice. Althoughmanyotherstockreturnpredictorshavebeenproposed(see,forinstance,Goyal and Welch, 2007), the above valuation ratios are of most interest here since they tend to result in the largest biases in the OLS estimates (e.g. Campbell and Yogo, 2006). The short interest rate is also analyzed, since recent work by Ang and Bekaert (2007) suggests that it works well as a predictor together with the dividend-price ratio, which provides an opportunity to study the performance of the jackknifed estimator withmultipleregressors; theshortinterestrateisgenerallynegativelyrelatedtofuturestockreturns,andwe therefore(cid:135)ipthesignonthispredictorvariableinallregressionssothattheexpectedsignisalwayspositive. All data are recorded on a monthly basis and regressions are run either at this monthly frequency or at an annualfrequency,usingoverlappingobservationsbasedontheoriginalmonthlydata. Theannualresultsthus provide an illustration of the jackknifed procedures applied to regressions with overlapping observations. In 9

all cases, excess stock returns are regressed on the lagged predictor variable(s) and an intercept, following the basic structure of equation (2). These are a subset of the same data as those used by Campbell and Thompson (2007) in their study of out-of-sample return predictability.4 The jackknifed OOS predictions can thus be directly compared to their results. In line with Campbell and Thompson, we use the level, and not logs, of the predictor variables as well as simple rather than log-returns. 4.1 In-sample results The(cid:133)rstsetofempiricalresultsisgiveninTable4andshowsthefullsampleOLSestimates,t statisticsand (cid:0) R2, along with the jackknifed estimates; the t statistics for the annual data with overlapping observations (cid:0) areformedusingNeweyandWest(1987)standarderrors. Resultsforthemonthlyandannualfrequenciesare displayed, and two di⁄erent sample periods are considered: the longest available sample for each predictor variable, as well as the forecast period used in the out-of-sample forecasts below. As is well established, predictive regressions like these tend to generate signi(cid:133)cant t statistics but fairly (cid:0) smallR2, whichincreasewiththehorizon. Inferencebasedonthet statisticsisgenerallysubjecttopitfalls, (cid:0) as documented in, for instance, Stambaugh (1999) and Campbell and Yogo (2006), and they are primarily shown here for completeness. The focus in this paper is on the point estimates in the predictive regression, whicharealsoshowninTable4. Foursetsofestimatesareshown: thestandardOLSonesandthejackknifed ones using m = 2;3; and 4 subsamples. Within the standard stock return predictability model, where the regressors follow an auto-regressive process, the OLS estimates for the valuation ratios are generally upward biased, whereas for the short interest rate the OLS estimator should be nearly unbiased. This suggests that the jackknifed estimates, which attempt to correct the OLS bias, should generally be smaller than the OLS estimates. Overall, this is the case, especially when using m > 2. This is particularly true for the book-tomarket ratio in the shorter sample, and for the coe¢ cient on the dividend-price ratio in the regressions that include the dividend-price ratio and the short rate jointly. The jackknifed estimates using m = 2 are often close to the OLS estimates, although they sometimes deviate substantially as well. Qualitatively, the results are similar for the monthly and annual data. The results in Table 4 suggests that standard OLS estimates are likely to exaggerate the size of the slope coe¢ cient in these predictive regressions. However, from these full sample estimates alone, it is di¢ cult to tell whether the jackknifed estimates are actually more accurate than the OLS estimates and we therefore turn to out-of-sample exercises to evaluate this question. 4ThedatawereobtainedfromProfessorJohnCampbell(cid:146)swebsiteandaredescribedinmoredetailinCampbellandThompson (2007). 10

4.2 Out-of-sample results In order to evaluate the OOS performance of the jackknifed estimates, we calculate an OOS R2, de(cid:133)ned as T (r r^)2 R2 =1 t=s t (cid:0) t ; (8) OS (cid:0) T (r r(cid:22))2 Pt=s t (cid:0) t P where r^ is the (cid:133)tted value from a predictive regression estimated using data up till time t 1 and r(cid:22) is the t t (cid:0) historical average return estimated using all available data up till time t 1. The out-of-sample forecasts (cid:0) begin in 1927, at which point high quality monthly CRSP data becomes available, or 20 years after the (cid:133)rst available observation for a given predictor variable, whichever comes later. Thus, s, in equation (8), represents the length of this initial (cid:145)training-sample(cid:146), which is used to obtain the estimates on which the (cid:133)rst round of forecasts is based. Note that the historical average forecast, r(cid:22), is always based on all the data back t to1872,whichpreservesitsrealworldadvantage. TheR2 statisticispositivewhentheconditionalforecast OS based on the predictive regression outperforms the historical mean. Thus, the out-of-sample R2 is positive whentherootmeansquarederroroftheconditionalforecastislessthanthatofthehistoricalmeanforecast. Giventhattheout-of-sampleR2 andacomparisonoftherootmeansquarederrorsyieldidenticalqualitative results, we focus on the out-of-sample R2 since it is measured in comparable units to the in-sample R2 and thus allows for more direct comparison. In addition to the standard forecasts based on the predictive regression and the historical mean, we also analyzethee⁄ectsofimposingsomeoftheforecastrestrictionsproposedbyCampbellandThompson(2007). Thatis,CampbellandThompsonarguethatratherthanmechanicallyforecastingstockreturnsbasedonthe estimatedforecastingequation,itisreasonabletoimposethefollowingrestrictions: ifanestimatedcoe¢ cient does not have the expected sign, it is set equal to zero, and if the forecast of the equity premium is negative, the forecast is set equal to zero. These restrictions rule out some of the perverse results that can otherwise occur in the rolling regressions that are used in the out-of-sample forecasts.5 Table 5 shows the OOS R2s for the OLS estimator and the jackknifed estimator with m=2;3; and 4, for both the restricted forecasts, which impose the Campbell and Thompson restrictions, and the unrestricted ones. For each predictor, the highest OOS R2 is shown in bold type. In general, the results show that the forecasts based on the jackknifed estimates tend to outperform the ones based on the plain OLS estimates, although there is no given value of m that consistently produces the highest OOS R2. The jackkni(cid:133)ng 5One could consider various ways of implementing the restrictions on the jackknife estimates. Here we take the simplest approach and set the parameter estimate equalto zero if it has the wrong sign. Alternatively, one could restrict the individual sub-sample estimates in the jackknife estimator to have the right sign. Since the (cid:133)rst approach immediately generalizes to the case of multiple regressors, unlike the second one which would become complicated to implement for more than one regressor, we use the (cid:133)rst approach. In all cases, the intercepts are calculated to line up with the, potentially restricted, slope coe¢ cient such that the residuals have mean zero. In the regressions with two predictor variables, each coe¢ cient is restricted separately and the intercept is again estimated to produce zero mean residuals. 11

procedureappearstobesomewhatmoreusefulonthemonthly, ratherthantheannualdata, inlinewiththe simulation results above, although the results are somewhat mixed. Qualitatively, the results are similar for boththeunrestrictedandrestrictedforecasts. Asmightbeexpectedfromthefull-samplecoe¢ cientestimates in Table 4, where the full-sample jackknifed estimates were drastically di⁄erent from the OLS estimate, the advantages of the jackkni(cid:133)ng are particularly clear for the book-to-market ratio. With regards to the choice of m, there is no value that clearly produces the best results. However, using m = 3 in the restricted forecasts consistently dominates the OLS forecasts in the monthly data and is close to, or better, in the annual data; only for the smoothed earnings price ratio in the annual data is there a material di⁄erence in favor of the OLS forecasts. In the unrestricted case, there is no m for which the jackknifed estimates consistently dominate the OLS ones for all predictor variables. This is clearly a drawback, since, asmentionedbefore, therearenoclearguidelinesforchoosingm. However, asshowninthe sectionbelow,theresultsbecomeclearerwhenoneconsiderstheimplementationofactualportfoliostrategies. In summary, the jackknifed estimator often improves upon the OLS estimator in out-of-sample forecasts. ThisseemstobeparticularlytruewhenonealsoimposestheforecastrestrictionsproposedbyCampbelland Thompson (2007), in which case the jackknifed estimator with m=3 almost completely dominates the OLS estimator. 4.3 Portfolio strategies CampbellandThompson(2007)discusshowtheOOSR2canbetranslatedintogainsineconomictermsforan investor that attempts to time the market using these predictor variables. However, practical considerations such as short selling constraints may render such theoretical relationships less accurate; a more reliable approach to gauging the economic importance of the improvement in out-of-sample forecasts is to directly simulate a portfolio choice strategy. To keep the calculations tractable, consider an investor with a singleperiod investment horizon and mean-variance preferences; that is, in each period the investor myopically chooses the optimal portfolio based on his quadratic preferences. The investor(cid:146)s utility function is the expected excess return minus ((cid:13)=2) times the portfolio variance, where (cid:13) can be viewed as the coe¢ cient of relative risk aversion. The weight on the risky asset for this investor is given by 1 E [r ] (cid:11) = t t+1 ; (9) t (cid:13) Var (r ) (cid:18) (cid:19)(cid:18) t t+1 (cid:19) where E [r ] and Var (r ) represents the expected value and variance of the excess returns over the t t+1 t t+1 next period, conditional on the information at time t. If the investor does not use the predictive regression 12

(2), it follows that 1 (cid:22) (cid:11) =(cid:11)= ; (10) t (cid:13) (cid:12)2(cid:27)2 +(cid:27)2 (cid:18) (cid:19)(cid:18) x u(cid:19) where (cid:27)2 =Var(x ) and (cid:27)2 =Var(u ). If the investor does use regression (2), x t u t 1 (cid:22)+(cid:12)x (cid:11) = t : (11) t (cid:13) (cid:27)2 (cid:18) (cid:19)(cid:18) u (cid:19) The out-of-sample economic gains of the predictive ability of equation (2) are evaluated by comparing the utilitiesfromaninvestorwhousestheweightsin(11)toonewhodisregardsthepredictabilityinreturnsand uses the weights in (10). The weights (cid:11) are calculated using only information available at time t. When the predictive regression t is not used, the weights at each time t are estimated by 1 r(cid:22) (cid:11)(cid:22) = t ; (12) t (cid:13) (cid:27)(cid:22)2 (cid:18) (cid:19)(cid:18) r(cid:19) wherer(cid:22) isthehistoricalaveragereturnestimatedusingallavailabledatauptilltimetand(cid:27)(cid:22)2 isthevariance t r of returns estimated using a (cid:133)ve year rolling window of data; i.e. (cid:27)(cid:22)2 is estimated using the last (cid:133)ve years of r data before time t. The weights based on the predictive regression are given by 1 (cid:22)^+(cid:12)^x (cid:11)^ = t ; (13) t (cid:18) (cid:13) (cid:19) (cid:27)^2 u ! where (cid:22)^ and (cid:12)^ are the estimates of the intercept and slope coe¢ cient in the predictive regression, using the datauptilltimet,and(cid:27)^2 isthevarianceoftheresiduals,againestimatedusinga(cid:133)veyearrollingwindowof u data.6 In order for the portfolio weights to be compatible with real world constraints, we impose a no short selling restriction and a maximum of 50% leverage, so that the portfolio weights are restricted to lie between 0 and 150%. Finally, the risk aversion parameter (cid:13) is set equal to three. Table 6 reports the welfare bene(cid:133)ts from using the weights (cid:11)^ , using either the OLS estimator or the t jackknifed estimators, instead of the weights (cid:11)(cid:22) . The utility di⁄erences are expressed in terms of expected t annualized returns and can thus be interpreted as the (maximum) management fee that an investor would be willing to pay a portfolio manager that exploits the predictive ability of equation (2). As in Table 5, we consider both the forecasts that impose the Campbell and Thompson restrictions and those that do not. Qualitatively, the results in Table 6 tell the same story as those in Table 5. The portfolio strategies based 6The use of a (cid:133)ve-year window to estimate the variance of the (unexpected) returns conforms with the approach taken by Campbelland Thompson (2007). Itcan bejusti(cid:133)ed by thefactthatitiseasierto calculatethevarianceofreturns,asopposed to the expected value,overshortertime horizons,and there is a large literature that shows that variances change overtime. 13

on the jackknifed estimates tend to outperform those based on the OLS estimates, and, importantly, o⁄er welfare gains over the strategies based purely on historical average returns. Again, the jackknifed estimator appears to work best for the monthly data. The portfolio results in Table 6 provides even stronger support of the bene(cid:133)t of the jackknifed estimates than the OOS R2s reported in Table 5. In the monthly data, the results for the OLS portfolio weights are dominated by the jackknife weights, for any m, in almost all cases. This is true both for the restricted and unrestricted forecasts. If one were to choose a single m for all predictors, m=3 would appear to be the best choice; in the monthly data, it dominates the OLS results in all cases. Compared to the OLS weights, the utility gains from using the jackknife procedure are relatively large, often between 50 and 100 basis points. Although this may not sound that large in absolute terms, the gains from using the predictive regression (with OLS estimates) in the (cid:133)rst place, instead of the historical average return, are typically no larger than 50-60 basis points. In fact, the welfare gains from the OLS weights are quite often negative, whereas the jackknife weights, especially for m = 3, are almost always positive. The welfaregainsfromthejackknifeweightsarealsosimilartothosereportedbyCampbellandThompson(2007) basedontheircompletelyrestrictedforecastswherethecoe¢ cientinthepredictiveregressionistotallypinned down by theoretical arguments and not estimated at all. The results here thus suggest that improving the estimationprocedurescanleadtoatleastasbiganimprovementastheimpositionoftheoreticalconstraints. These result also add further evidence to the case that returns are predictable out-of-sample, in contrast to the conclusions of Goyal and Welch (2003, 2007). 5 Conclusion A simple bias reducing method, the jackknife, is proposed for predictive regressions of stock returns. Unlike most previous work on inference in stock return predictability regressions, this paper puts the focus on obtaining good point estimates rather than correctly sized tests, a task which has become increasingly more important as the focus in the literature has shifted towards out-of-sample forecasts and practical portfolio choice based on return forecasts. In addition, the jackknife is a general method that does not rely on speci(cid:133)c assumptions on the data generating process. Monte Carlo simulations show that the jackknife method works well in (cid:133)nite samples and virtually eliminates the bias in OLS estimates of predictive regressions. Most importantly, it also works well on actual stock returns data, and leads to substantial improvements in out-of-sample forecasts. This is illustrated not onlybypurelystatisticalmeasuressuchasout-of-sampleR2, butalsothroughsimulatedportfoliostrategies, whichoftenperformsigni(cid:133)cantlybetterwhentheforecastsarebasedonthejackknifedestimatesratherthan 14

the OLS ones. 15

References [1] Amihud,Y.,andHurvich,C.,2004.PredictiveRegressions: AReduced-BiasEstimationMethod,Journal of Financial and Quantitative Analysis 39, 813-841. [2] Ang, A., and G. Bekaert, 2007. Stock Return Predictability: Is it There? Review of Financial Studies 20, 651-707. [3] Boudoukh J., M. Richardson, and R.F. Whitelaw, 2006. The Myth of Long-Horizon Predictability, Review of Financial Studies, forthcoming. [4] Campbell, J.Y., 2001. Why long horizons? A study of power against persistent alternatives, Journal of Empirical Finance 8, 459-491. [5] Campbell,J.Y.,andR.Shiller,1988.StockPrices,Earnings,andExpectedDividends,JournalofFinance 43, 661-676. [6] Campbell., J.Y., and S.B. Thompson, 2007. Predicting the Equity Premium Out of Sample: Can Anything Beat the Historical Average?, Review of Financial Studies, forthcoming. [7] Campbell, J.Y., and M. Yogo, 2006. E¢ cient Tests of Stock Return Predictability, Journal of Financial Economics 81, 27-60. [8] Cavanagh, C., G. Elliot, and J. Stock, 1995. Inference in Models with Nearly Integrated Regressors, Econometric Theory 11, 1131-1147. [9] Eliasz, P., 2005. Optimal Median Unbiased Estimation of Coe¢ cients on Highly Persistent Regressors, Mimeo, Princeton University. [10] GoetzmanW.N.,andP.Jorion,1993.Testingthepredictivepowerofdividendyields,JournalofFinance 48, 663-679. [11] Goyal A., and I. Welch, 2003. Predicting the Equity Premium with Dividend Ratios, Management Science 49, 639-654. [12] Goyal,A.,andI.Welch,2007.AComprehensiveLookattheEmpiricalPerformanceofEquityPremium Prediction, Review of Financial Studies, forthcoming. [13] Hansen,L.P.,andR.J.Hodrick,1980.Forwardexchangeratesasoptimalpredictorsoffuturespotrates: An Econometric Analysis, Journal of Political Economy 88, 829-853. 16

[14] Jansson, M., and M.J. Moreira, 2006. Optimal Inference in Regression Models with Nearly Integrated Regressors, Econometrica 74, 681-714. [15] Lewellen, J., 2004. Predicting Returns with Financial Ratios, Journal of Financial Economics, 74, 209- 235. [16] Mankiw, N.G., and M.D. Shapiro, 1986. Do We Reject Too Often? Small Sample Properties of Tests of Rational Expectations Models, Economics Letters 20, 139-145. [17] Nelson, C.R., and M.J. Kim, 1993. Predictable stock returns: the role of small sample bias, Journal of Finance 48, 641-661. [18] Newey,W.,andK.West,1987.ASimple,PositiveSemi-De(cid:133)nite,HeteroskedasticityandAutocorrelation Consistent Covariance Matrix, Econometrica 55, 703-708. [19] Paye, B.S., and A. Timmermann, 2006. Instability of Return Prediction Models, Journal of Empirical Finance 13, 274-315. [20] Phillips, P.C.B., and J. Yu, 2005. Jackkni(cid:133)ng Bond Option Prices, Review of Financial Studies 18, 707-742. [21] Quenouille, M. H., 1956. Notes on Bias in Estimation, Biometrika 43, 353(cid:150)360. [22] Richardson,M.,andT.Smith,1991.Testsof(cid:133)nancialmodelsinthepresenceofoverlappingobservations, Review of Financial Studies 4, 227-254. [23] Richardson, M., and J.H. Stock, 1989. Drawing inferences from statistics based on multiyear asset returns, Journal of Financial Economics 25, 323-348. [24] Stambaugh, R., 1999. Predictive Regressions, Journal of Financial Economics 54, 375-421. [25] Torous, W., R.Valkanov, andS.Yan, 2004.Onpredictingstockreturnswithnearlyintegratedexplanatory variables, Journal of Business 77, 937-966. [26] Valkanov, R., 2003. Long-horizon regressions: theoretical results and applications, Journal of Financial Economics 68, 201-232. 17

SLO eht rof )sesehtnerap ni( rorre derauqs naem toor dna saib naem eht swohs elbat ehT .esac rosserger elgnis eht rof stluser olraC etnoM :1 elbaT eht ot snoitavonni eht neewteb noitalerroc eht ,(cid:14) fo seulav gnire⁄id ehT .selpmasbus 4 dna ;3;2 = m htiw srotamitse definkkcaj eht dna rotamitse fo tes hcae evoba nevig era )(cid:26)( toor evisserger-otua eht fo eulav eht dna )T( ezis elpmas ehT .wor pot eht ni nevig era ,rosserger eht dna snruter .snoititeper 000;01 no desab era stluser llA .stluser 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 999:0=(cid:26);001= T 59:0=(cid:26);001= T 9:0=(cid:26);001= T 350:0 150:0 840:0 640:0 440:0 240:0 140:0 040:0 830:0 SLO )860:0( )760:0( )560:0( )960:0( )860:0( )660:0( )270:0( )070:0( )960:0( 200:0 300:0 300:0 200:0 200:0 200:0 200:0 200:0 100:0 2=m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )960:0( )760:0( )660:0( )370:0( )370:0( )170:0( )670:0( )570:0( )470:0( 300:0 400:0 300:0 200:0 200:0 100:0 200:0 300:0 200:0 3=m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )750:0( )650:0( )650:0( )460:0( )460:0( )160:0( )070:0( )960:0( )860:0( 400:0 400:0 400:0 100:0 100:0 000:0 200:0 200:0 200:0 4=m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )350:0( )250:0( )250:0( )160:0( )060:0( )850:0( )760:0( )660:0( )560:0( 999:0=(cid:26);005= T 59:0=(cid:26);005= T 9:0=(cid:26);005= T 110:0 010:0 010:0 800:0 800:0 800:0 800:0 700:0 700:0 SLO )410:0( )410:0( )310:0( )810:0( )810:0( )810:0( )220:0( )220:0( )220:0( 000:0 000:0 000:0 100:0 100:0 000:0 000:0 000:0 000:0 2=m (cid:0) (cid:0) )410:0( )410:0( )410:0( )810:0( )810:0( )810:0( )320:0( )220:0( )220:0( 000:0 000:0 000:0 100:0 100:0 000:0 000:0 000:0 000:0 3=m (cid:0) (cid:0) )210:0( )210:0( )110:0( )710:0( )810:0( )710:0( )220:0( )220:0( )120:0( 000:0 000:0 000:0 100:0 100:0 000:0 000:0 000:0 000:0 4=m (cid:0) (cid:0) )110:0( )110:0( )110:0( )710:0( )710:0( )710:0( )220:0( )220:0( )120:0( 18

eht rof )sesehtnerap ni( rorre derauqs naem toor dna saib naem eht swohs elbat ehT .esac rosserger elpitlum eht rof stluser olraC etnoM :2 elbaT owt htiw noisserger evitciderp a ni stneic ¢eoc epols owt eht rof ,selpmasbus 4 dna ;3;2 = m htiw srotamitse definkkcaj eht dna rotamitse SLO yb nevig xirtam evisserger-otua eht htiw ,srosserger owt eht rof stoor evisserger-otua eht fo eulav eht setacidni wor pot ehT .selbairav rotciderp ehT .srosserger owt eht dna snruter eht ot snoitavonni eht neewteb , vu ! ,rotcev noitalerroc eht setacidni wor dnoces ehT .0]) 22 a;0(;)0; 11 a([ = A ro 001 rehtie ot lauqe si ,T ,ezis elpmas ehT .srosserger owt eht fo sessecorp noitavonni eht rof , (cid:10) xirtam ecnairavoc-ecnairav eht sevig wor driht vv .snoititeper 000;01 no desab era stluser llA .stluser fo tes hcae evoba detacidni dna ,005 999:0= a;999:0= a 59:0= a;999:0= a 59:0= a;999:0= a 22 11 22 11 22 11 0)9:0 ;9:0 (= vu ! 0)7:0 ;7:0 (= vu ! 0)0;9:0 (= vu ! (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 0])1;8:0(;)8:0;1([= vv (cid:10) 0])1;5:0(;)5:0;1([= vv (cid:10) 0])1;4:0(;)4:0;1([= vv (cid:10) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2 1 2 1 2 1 001= T 630:0 530:0 620:0 830:0 820:0 460:0 SLO (cid:0) )390:0( )390:0( )860:0( )260:0( )860:0( )480:0( 200:0 200:0 300:0 500:0 100:0 400:0 2=m (cid:0) (cid:0) (cid:0) )321:0( )321:0( )580:0( )570:0( )580:0( )580:0( 300:0 200:0 300:0 500:0 100:0 400:0 3=m (cid:0) (cid:0) (cid:0) )501:0( )601:0( )670:0( )460:0( )570:0( )270:0( 400:0 300:0 200:0 700:0 200:0 200:0 4=m (cid:0) (cid:0) (cid:0) )001:0( )001:0( )170:0( )950:0( )170:0( )660:0( 005= T 700:0 700:0 400:0 800:0 500:0 110:0 SLO (cid:0) )910:0( )910:0( )810:0( )210:0( )810:0( )510:0( 000:0 000:0 100:0 000:0 000:0 100:0 2=m (cid:0) (cid:0) )520:0( )520:0( )020:0( )310:0( )020:0( )510:0( 000:0 000:0 100:0 000:0 000:0 200:0 3=m (cid:0) (cid:0) )220:0( )220:0( )910:0( )110:0( )910:0( )310:0( 000:0 000:0 100:0 000:0 000:0 200:0 4=m (cid:0) (cid:0) )020:0( )120:0( )810:0( )110:0( )910:0( )210:0( 19

rorre derauqs naem toor dna saib naem eht swohs elbat ehT .snoitavresbo gnippalrevo htiw snoisserger noziroh-gnol rof stluser olraC etnoM :3 elbaT noziroh-gnol a ni tneic ¢eoc epols eht rof ,selpmasbus 4 dna ;3;2 = m htiw srotamitse definkkcaj eht dna rotamitse SLO eht rof )sesehtnerap ni( eht ,(cid:14) fo seulav gnire⁄id ehT .noisserger eht ni desu si rosserger elgnis A .q noziroh tsacerof dna snoitavresbo gnippalrevo htiw noisserger evitciderp eht dna ,q noziroh tsacerof eht ,T ezis elpmas ehT .wor pot eht ni nevig era ,rosserger eht dna snruter eht ot snoitavonni eht neewteb noitalerroc .snoititeper 000;01 no desab era stluser llA .stluser fo tes hcae evoba nevig era ,(cid:26) toor evisserger-otua eht fo eulav 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) 99:0 = (cid:14) 59:0 = (cid:14) 09:0 = (cid:14) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 999:0=(cid:26);01= q;001= T 59:0=(cid:26);01= q;001= T 9:0=(cid:26);01= q;001= T 534:0 914:0 104:0 863:0 163:0 933:0 613:0 703:0 482:0 SLO )325:0( )315:0( )694:0( )594:0( )294:0( )084:0( )284:0( )184:0( )964:0( 171:0 571:0 661:0 480:0 890:0 680:0 050:0 250:0 340:0 2=m )515:0( )125:0( )605:0( )455:0( )055:0( )945:0( )775:0( )775:0( )375:0( 832:0 922:0 322:0 931:0 641:0 331:0 780:0 680:0 670:0 3=m )264:0( )464:0( )454:0( )674:0( )084:0( )084:0( )605:0( )015:0( )705:0( 682:0 772:0 762:0 681:0 881:0 371:0 121:0 021:0 601:0 4=m )364:0( )564:0( )454:0( )754:0( )164:0( )064:0( )574:0( )284:0( )084:0( 999:0=(cid:26);21= q;005= T 59:0=(cid:26);21= q;005= T 9:0=(cid:26);21= q;005= T 321:0 121:0 311:0 480:0 080:0 870:0 370:0 760:0 360:0 SLO )751:0( )551:0( )741:0( )181:0( )181:0( )081:0( )802:0( )802:0( )602:0( 310:0 610:0 310:0 200:0 200:0 200:0 100:0 100:0 100:0 2=m (cid:0) (cid:0) (cid:0) (cid:0) )451:0( )251:0( )941:0( )891:0( )991:0( )691:0( )622:0( )822:0( )622:0( 910:0 220:0 810:0 000:0 100:0 200:0 200:0 200:0 100:0 3=m (cid:0) (cid:0) (cid:0) )031:0( )031:0( )521:0( )881:0( )881:0( )781:0( )812:0( )122:0( )812:0( 420:0 720:0 320:0 100:0 000:0 400:0 200:0 100:0 100:0 4=m (cid:0) (cid:0) )121:0( )221:0( )811:0( )381:0( )481:0( )281:0( )512:0( )812:0( )512:0( 20

fo snoisserger evitciderp ni stneic ¢eoc epols eht fo setamitse tniop definkkcaj dna SLO eht swohs elbat ehT .stluser laciripme elpmas-nI :4 elbaT era )tnecrep ni desserpxe( 2R dna scitsitats t SLO eht ,noitidda nI .nmuloc tsr(cid:133) eht ni detacidni selbairav rotciderp eht gnisu ,snruter kcots ssecxe (cid:0) eht rehtie dna ,snoitavresbo ylhtnom lanigiro eht no desab ,atad gnippalrevo launna ro ylhtnom rehtie gnisu ,nwohs era stluser fo stes ruoF .nwohs eht ni nmuloc tsr(cid:133) ehT .sesicrexe elpmas-fo-tuo tneuqesbus eht ni desu elpmas tsacerof eht ro elbairav rotciderp hcae rof elpmas lluf elbaliava tsegnol dne selpmas lla ;elpmas eht fo etad trats eht swohs nmuloc dnoces eht dna ,noisserger evitciderp eht ni desu )s(elbairav rotciderp eht setacidni elbat fo tneic ¢eoc epols eht rof ,selpmasbus 4 dna ;3;2=m htiw ,setamitse tniop definkkcaj dna SLO eht wohs snmuloc ruof txen ehT .5002 rebmeceD ni dnoces eht rof tneic ¢eoc epols eht fo setamitse eht wohs snmuloc ruof txen ehT .noisserger gnitsacerof eht ni rotciderp )ylno yllacipyt dna( tsr(cid:133) eht wohssnmuloceerhtlan(cid:133)ehT .yltniojdedulcnietarlliB-Tehtdnaoitarecirp-dnedividehthtobhtiwnoissergerehtnielbacilppaylnosisiht ;rosserger era snoitavresbo gnippalrevo htiw atad launna eht rof scitsitats t ehT .tnecrep ni 2R SLO eht dna stneic ¢eoc epols owt eht rof scitsitats t SLO eht (cid:0) (cid:0) .srorre dradnats )7891( tseW dna yeweN gnisu detaluclac )%( 2R t t (cid:12)^ (cid:12)^ (cid:12)^ (cid:12)^ (cid:12)^ (cid:12)^ (cid:12)^ (cid:12)^ snigeB elpmaS )s(rotciderP SLO SLO;2 SLO;1 4=m;2 3=m;2 2=m;2 SLO;2 4=m;1 3=m;1 2=m;1 SLO;1 elpmaS lluF ,ylhtnoM 73.0 20.1 48.1 30.2 39.0 99.1 2m2781 P/D 42.0 37.1 31.1 23.1 40.1 50.1 2m2781 P/E 65.0 77.1 83.1 32.1 43.1 94.1 2m1881 P/E dehtoomS 91.1 82.1 21.0 51.0 42.0 12.0 6m6291 M/B 83.0 88.1 22.1 47.0 10.1 73.1 1m0291 etar lliB-T 12.1 80.2 97.1 10.1 83.0 75.0- 56.1 31.1 63.1 35.0 78.1 1m0291 etar lliB-T dna P/D elpmaS tsaceroF ,ylhtnoM 21.1 52.1 00.3 86.2 22.4 39.3 1m7291 P/D 17.0 82.2 26.1 10.2 60.2 60.2 1m7291 P/E 53.1 58.1 26.2 67.2 75.2 20.3 1m7291 P/E dehtoomS 16.0 69.1 50.0 10.0 10.0 81.0 6m6491 M/B 78.0 64.2 37.1 05.1 88.0 35.1 1m0491 etar lliB-T 65.1 31.2 33.2 32.1 82.0 22.0- 53.1 12.0 47.1- 01.0 19.2 1m0491 etar lliB-T dna P/D elpmaS lluF ,launnA 41.5 14.2 23.2 35.2 14.1 55.2 2m2781 P/D 03.4 67.2 15.1 07.1 94.1 25.1 2m2781 P/E 98.6 53.2 64.1 84.1 56.1 77.1 2m1881 P/E dehtoomS 17.31 50.4 51.0 51.0 52.0 32.0 6m6291 M/B 19.1 57.1 89.0 36.0 33.1 99.0 1m0291 etar lliB-T 10.61 23.2 57.3 53.1 05.0 87.0 23.1 61.2 92.2 83.2 66.2 1m0291 etar lliB-T dna P/D elpmaS tsaceroF ,launnA 98.01 42.3 11.3 74.2 33.4 79.3 1m7291 P/D 87.6 21.3 67.1 79.1 70.2 50.2 1m7291 P/E 75.31 52.3 47.2 58.2 66.2 90.3 1m7291 P/E dehtoomS 62.8 90.2 01.0 50.0 40.0 22.0 6m6491 M/B 62.4 81.2 43.1 61.1 35.0 21.1 1m0491 etar lliB-T 82.41 28.1 78.2 49.0 55.0 75.0 78.0 81.1 05.0 61.2 07.3 1m0491 etar lliB-T dna P/D 21

snruter kcots ssecxe fo stsacerof eht morf tluser taht )tnecrep ni desserpxe( 2R elpmas-fo-tuo eht swohs elbat ehT .stluser elpmas-fo-tuO :5 elbaT htiw ,setamitse definkkcaj eht ro setamitse SLO eht rehtie gnisu demrof era stsacerof ehT .nmuloc tsr(cid:133) eht ni detacidni selbairav rotciderp eht gnisu htob rof stluseR .)7002( nospmohT dna llebpmaC yb dednemmocer stsacerof eht no snoitcirtser eht gnisopmi tuohtiw ro htiw dna ,4 dna ;3;2 = m 2R elpmas-fo-tuo tsehgih eht ,stsacerof fo stes detcirtser dna detcirtsernu eht htob rof dna ,wor hcae roF .nwohs era atad launna dna ylhtnom eht eht wohs snmuloc owt gniwollof eht dna ,no desab era stsacerof eht taht )s(elbairav rotciderp eht setacidni nmuloc tsr(cid:133) ehT .epyt dlob ni nwohs si eerht dna owt snmuloc neewteb ecnere⁄id ehT .ylevitcepser ,nigeb stsacerof elpmas-fo-tuo eht hcihw ta etad eht dna snigeb elpmas eht hcihw ta etad 2R elpmas-fo-tuo eht wohs snmuloc ruof gniwollof ehT .tsacerof tsr(cid:133) eht rof setamitse laitini eht mrof ot desu si taht(cid:146)elpmas-gniniart(cid:145) eht stneserper stluser gnidnosperroc eht wohs snmuloc ruof tsal eht dna ,snoitcirtser nospmohT dna llebpmaC eht esopmi ton od taht stsacerof detcirtsernu eht rof .ecalp ni snoitcirtser nospmohT dna llebpmaC eht htiw detcirtseR detcirtsernU 2R 2R 2R 2R 2R 2R 2R 2R snigeB tsaceroF snigeB elpmaS )s(rotciderP 4=m 3=m 2=m SLO 4=m 3=m 2=m SLO ylhtnoM 63.0 33.0 44.0 61.0 45.0- 26.0- 13.0- 66.0- 1m7291 2m2781 P/D 83.0 73.0 64.0 42.0 92.0 13.0 93.0 21.0 1m7291 2m2781 P/E 14.0 65.0 47.0 44.0 01.0 71.0 76.0 23.0 1m7291 2m1881 P/E dehtoomS 74.0 87.0 31.0 10.0- 24.0 27.0 83.0- 44.0- 6m6491 6m6291 M/B 01.0- 48.0 57.31- 85.0 03.4- 92.0- 47.31- 45.0 1m0491 1m0291 etar lliB-T 22.0 90.1 12.9- 71.0 00.3- 22.1- 05.01- 21.0 1m0491 1m0291 etar lliB-T dna P/D launnA 72.5 97.4 37.7 36.5 92.5 27.4 96.7 35.5 1m7291 2m2781 P/D 79.3 27.4 42.5 49.4 29.3 56.4 32.5 39.4 1m7291 2m2781 P/E 16.2 32.4 07.5 58.7 10.2 51.3 76.5 98.7 1m7291 2m1881 P/E dehtoomS 18.3 38.5 16.3- 93.1 49.2 74.4 08.01- 83.3- 6m6491 6m6291 M/B 04.7 54.9 43.0 74.7 89.0- 02.8 42.2- 45.5 1m0491 1m0291 etar lliB-T 64.9 04.01 49.21 78.7 13.11 42.9 59.1 48.8 1m0491 1m0291 etar lliB-T dna P/D 22

eht sesu ohw rotsevni na rof ,snruter detcepxe dezilaunna tnecrep ni desserpxe ,sniag ytilitu eht swohs elbat ehT .stluser eciohc oiloftroP :6 elbaT htiw secnereferp ecnairav-naem sah rotsevni eht ;tekram eht emit ot ,naem lacirotsih eht fo daetsni ,nmuloc tsr(cid:133) eht ni detacidni selbairav rotciderp setamitse SLO eht rehtie gnisu demrof ,snruter kcots ssecxe eht fo stsacerof no desab era sthgiew oiloftrop ehT .eerht ot lauqe noisreva ksir evitaler dna llebpmaC yb dednemmocer stsacerof eht no snoitcirtser eht gnisopmi tuohtiw ro htiw dna ,4 dna ;3;2 = m htiw ,setamitse definkkcaj eht ro ,stsaceroffostesdetcirtserdnadetcirtsernuehthtobrofdna,worhcaeroF .nwohseraatadlaunnadnaylhtnomehthtobrofstluseR .)7002(nospmohT gniwollof eht dna ,no desab era stsacerof eht taht )s(elbairav rotciderp eht setacidni nmuloc tsr(cid:133) ehT .epyt dlob ni nwohs si niag ytilitu tsehgih eht neewteb ecnere⁄id ehT .ylevitcepser ,nigeb stsacerof elpmas-fo-tuo eht hcihw ta etad eht dna snigeb elpmas eht hcihw ta etad eht wohs snmuloc owt snmuloc ruof gniwollof ehT .tsacerof tsr(cid:133) eht rof setamitse laitini eht mrof ot desu si taht(cid:146)elpmas-gniniart(cid:145) eht stneserper eerht dna owt snmuloc ,snoitcirtser nospmohT dna llebpmaC eht esopmi ton od taht stsacerof detcirtsernu eht no desab snoisiced oiloftrop eht morf sniag ytilitu eht wohs .ecalp ni snoitcirtser nospmohT dna llebpmaC eht htiw stluser gnidnosperroc eht wohs snmuloc ruof tsal eht dna detcirtseR detcirtsernU 4=m 3=m 2=m SLO 4=m 3=m 2=m SLO snigeB tsaceroF snigeB elpmaS )s(rotciderP ylhtnoM 30.0 80.0 05.0 34.0- 90.0- 60.0- 14.0 25.0- 1m7291 2m2781 P/D 17.0 36.0 56.0 73.0 65.0 15.0 35.0 32.0 1m7291 2m2781 P/E 80.0 83.0 70.0- 62.0- 90.0- 80.0 11.0- 03.0- 1m7291 2m1881 P/E dehtoomS 02.0 83.0 46.0- 17.0- 02.0 93.0 46.0- 07.0- 6m6491 6m6291 M/B 57.1 22.2 11.2 76.1 55.1 51.2 21.2 86.1 1m0491 1m0291 etar lliB-T 74.2 98.0 05.0 56.0- 10.1 77.0 29.0 56.0- 1m0491 1m0291 etar lliB-T dna P/D launnA 53.0- 03.0- 82.0 55.0- 53.0- 03.0- 03.0 45.0- 1m7291 2m2781 P/D 24.0 64.0 85.0 26.0 24.0 64.0 85.0 26.0 1m7291 2m2781 P/E 33.0 30.0 41.0 25.0 61.0 62.0- 41.0 25.0 1m7291 2m1881 P/E dehtoomS 64.0- 30.0- 36.1- 26.0- 54.0- 20.0- 46.1- 75.0- 6m6491 6m6291 M/B 65.1 98.1 14.1 35.1 25.1 59.1 24.1 55.1 1m0491 1m0291 etar lliB-T 32.0 13.0- 15.0- 60.0- 70.2 59.0 33.1 00.0 1m0491 1m0291 etar lliB-T dna P/D 23

Figure 1: Density plots for the OLS and jackknife estimates, based on 100,000 simulations for T = 100, (cid:26) = 0:999 and (cid:14) = 0:99. The graphs shows the kernel density estimates of the bias in the OLS and (cid:0) jackknifed estimates, with m=2;3; and 4. The vertical solid line indicates a zero bias. 24