Interpreting Long-Horizon Estimates in Predictive Regressions

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 928 April 2008 Interpreting Long-Horizon Estimates in Predictive Regressions 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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/.

Interpreting long-horizon estimates in predictive regressions Erik Hjalmarsson (cid:3) Division of International Finance Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA March 2008 Abstract Thispaperanalyzestheasymptoticpropertiesoflong-horizonestimatorsunderboththenullhypothesis and an alternative of predictability. Asymptotically, under the null of no predictability, the long-run estimator is an increasing deterministic function of the short-run estimate and the forecasting horizon. Under the alternative of predictability, the conditional distribution of the long-run estimator, given the short-run estimate, is no longer degenerate and the expected pattern of coe¢ cient estimates across horizons di⁄ers from that under the null. Importantly, however, under the alternative, highly endogenous regressors, such as the dividend-price ratio, tend to deviate much less than exogenous regressors, such as the short interest rate, from the pattern expected under the null, making it more di¢ cult to distinguish between the null and the alternative. JEL classi(cid:133)cation: C22, G1. Keywords: Predictive regressions; Long-horizon regressions; Stock return predictability. Thispaperhasbene(cid:133)ttedfrom commentsbyLennartHjalmarsson,RandiHjalmarssonandJonathanWright,aswellasan (cid:3) anonymousreferee. Tel.: +1-202-452-2426;fax: +1-202-263-4850;email: erik.hjalmarsson@frb.gov. Theviewsinthispaperare solely the responsibility of the author and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the FederalReserve System orofany otherperson associated with the FederalReserve System.

1 Introduction Long-runregressionsweremadepopularbyin(cid:135)uentialarticlessuchasFamaandFrench(1988)andCampbell and Shiller (1988). When attempting to predict stock returns over longer horizons, often covering several years, rather than on a month-to-month basis, the evidence in favour of predictability generally appears much stronger. For instance, the estimated regression coe¢ cients tend to increase almost linearly with the forecasting horizon. However, in a recent paper, Boudoukh, Richardson, and Whitelaw (2006) (BRW hereafter)showthatthispatternisexactlytheonetobeexpectedintheabsence ofpredictability. Althoughit is well known that the estimated slope coe¢ cient will increase with the horizon when there is predictability, BRW appear to be the (cid:133)rst ones to observe that the same is also true under the null hypothesis. They interpret their (cid:133)ndings as a strong critique of the widespread belief that long-horizon regressions provide solid evidence of predictability (e.g. Cochrane, 2001).1 The aim of the current paper is to understand in more detail the properties of long-horizon estimates under an alternative of predictability. The practical purpose of this is to establish the pattern of estimated coe¢ cientsthatmaybeexpectedbothunderthenullandthealternative,acrossdi⁄erentforecastinghorizons. Although the generally increasing pattern of coe¢ cients in long-horizon regressions is well established, the exact asymptotic sampling properties of long-run estimators under an alternative of predictability are not previously well understood. I derive the asymptotic distribution of the long-run OLS estimator, with overlapping observations, under the assumptions that the true data generating process is given by the standard linear predictive regression model and that the regressors are highly persistent variables. Under the alternative of predictability, the sampling properties of the long-run estimator are fundamentally di⁄erent than under the null hypothesis, andthelimitingdistributionishighlynon-standard. Fromapracticalperspective, thisresultisofindividual interest. It shows that con(cid:133)dence intervals for long-run estimates, based on inverting a test statistic that is valid under the null hypothesis, will not be correctly sized under the alternative, given the non-standard distribution. The theoretical results allow for an exact characterization of the conditional distribution of the long-run estimator,giventheshort-runestimate. Underthenullhypothesis,thelong-runestimatoris,asymptotically, completelydetermined once theshort-runestimate isgiven. Importantly, however, this is nottrueunderthe alternative of predictability. In fact, the degree to which the long-run estimate can vary independently of the short-run estimate is determined by the degree of endogeneity of the regressors.2 Long-run estimates for 1Earlierstudiesdiscussingotherinferentialissuesinlong-horizonregressionsincludeHansenandHodrick(1980),Richardson and Stock (1989), Richardson and Smith (1991, 1994), Hodrick (1992), Goetzman and Jorion (1993), Nelson and Kim (1993), Richardson (1993),Kirby (1997),and Valkanov (2003). 2A predictive regressorisgenerally referred to asendogenousifthe innovationsto the returnsare contemporaneously corre- 1

highly endogenous regressors, such as the dividend-price ratio, are also almost completely pinned down by the short-run estimate under the alternative hypothesis. On the other hand, for near exogenous regressors suchastheshortinterestrate,thelong-runestimatorhasarelativelylargeindependentcomponentgiventhe short-run estimate.3 Given the short-run estimate, under the alternative of predictability, the implications are that: (i) the highly endogenous regressor will have a more predictable pattern for long-horizons than the near exogenous one, and (ii) this pattern will closely resemble that under the null hypothesis. These results provide an alternative interpretation of the empirical (cid:133)ndings in BRW. BRW interpret the (cid:133)ndings that the dividend-price ratio has a pattern very similar to that predicted under the null, whereas the short interest rate does not, as evidence against the predictive ability of the dividend-price ratio and in favor of the predictive ability of the short rate. Given the results in this paper, however, their (cid:133)ndings could merelyre(cid:135)ectthefactthatthereismoreindependentvariationinthelong-runestimatesforfairlyexogenous regressors. 2 Model and assumptions Letr denotetheoneperiodstockreturnfromttot+1andletr (q)= q r bethecorresponding t+1 t+q j=1 t+j q period return from t to t+q. The standard long-run forecasting regressioPn is speci(cid:133)ed as follows, (cid:0) r (q)=(cid:11) +(cid:12) x +u (q); (1) t+q q q t t+q where long-run future returns are regressed onto a one period predictor, x . Let the OLS estimator of (cid:12) in t q equation (1), using overlapping observations, be denoted by (cid:12)^ . The primary focus of interest will be the q properties of (cid:12)^ for di⁄erent values of q, and in particular the relationship between (cid:12)^ and (cid:12)^ for q >1. q 1 q Inordertoformallyanalyzethesamplingpropertiesof(cid:12)^ , thedatageneratingprocessforr andx need q t t to be explicitly speci(cid:133)ed. Following Nelson and Kim (1993) and Campbell (2001), I assume that r and x t t satisfy: r = (cid:11)+(cid:12)x +u ; (2) t+1 t t+1 x = (cid:13)+(cid:26)x +v : (3) t+1 t t+1 lated with the innovations to the regressor. When the regressor is strictly stationary, such endogeneity has no impact on the propertiesoftheestimator,butwhentheregressorispersistentinsomemanner,thepropertiesoftheestimatorwillbea⁄ected (e.g. Stambaugh,1999). 3The contemporaneous correlation between the innovations to the returns and the innovations to the regressor determines the endogeneity of a predictive regressor (see previous footnote). Campbell and Yogo (2006) show that for valuation ratios, such as the dividend-price ratio, this correlation is large and often greater than 0:9 in absolute magnitude. For interest rate variables, however, the correlation is close to zero and, from the perspective of a predictive regression, these variables are thus nearly exogenous. 2

Thus, the one-period regression in equation (1) coincides with the true data generating process and for any horizon, x will be the optimal forecaster of returns given current information at time t.4 To capture the t near persistence found in most forecasting variables, such as interest rates or valuation ratios, it is further assumed that the auto-regressive root, (cid:26), is close to one in a local sense. In particular, it is assumed that (cid:26) = 1+c=T, where c is some (cid:133)nite parameter and T is the sample size with t = 1;:::;T. This captures the near unit-root, or highly persistent, behavior of many predictor variables, but is less restrictive than a pure unit-rootassumption. Thenearunit-rootconstruction,wheretheautoregressiverootdriftsclosertounityas thesamplesizeincreases,isusedasatooltoenableanasymptoticanalysiswherethepersistenceinthedata remains large relative to the sample size, also as the sample size increases to in(cid:133)nity. That is, if (cid:26) is treated as (cid:133)xed and strictly less than unity, then as the sample size grows, the process x will behave as a strictly t stationary process asymptotically and the standard (cid:133)rst order asymptotic results will not provide a good guide to the actual small sample properties of the model. For (cid:26) = 1, the usual unit-root asymptotics apply to the model, but this is clearly a restrictive assumption for most potential predictor variables. Instead, by letting (cid:26)=1+c=T, the e⁄ects from the high persistence in the regressor will appear also in the asymptotic results, but without imposing the strict assumption of a unit root. Cavanagh et al. (1995), Lanne (2002), Valkanov (2003), Torous et al. (2004), and Campbell and Yogo (2006) all use similar models, with a near unit-root construct, to analyze the predictability of stock returns. The error processes are assumed to satisfy a martingale di⁄erence sequence with (cid:133)nite fourth order moments. That is, let w t = (u t ;v t )0 and t = w s s t be the (cid:133)ltration generated by w t . Then F f j (cid:20) g E[w ]=0, E[w w ]=(cid:6)=[((cid:27) ;(cid:27) );((cid:27) ;(cid:27) )], sup E u4 < , and sup E v4 < : t jF t (cid:0) 1 t t0 11 12 21 22 t t 1 t t 1 By standard arguments, T (cid:0) 1=2 [ t T = r 1 ]w t ) B(r)=BM((cid:6))(cid:2)(r)(cid:3); where B( (cid:1) )=(B 1 (cid:2)( (cid:1) )(cid:3);B 2 ( (cid:1) ))0 denotes a two dimensional Brownian motionPand denotes weak convergence of the associated probability measures. ) Further, as T , T 1=2x J (r)= r e(r s)cdB (s) and an analogous result holds for the demeaned !1 (cid:0) [Tr] ) c 0 (cid:0) 2 variables x = x T 1 n x , with theRlimiting process J replaced by J = J 1 J (Phillips, 1987, t t (cid:0) (cid:0) t=1 t c c c (cid:0) 0 c 1988). Let W () and WP() be the standardized Brownian motions, with unit variRance and correlation 1 2 (cid:1) (cid:1) (cid:14) = (cid:27) 12 ((cid:27) 11 (cid:27) 22 )(cid:0) 1=2, that correspond to B 1 () and B 2 (), respectively. By the properties of conditional (cid:1) (cid:1) normal distributions, W = 1 (cid:14)2W +(cid:14)W , where W is a Brownian motion with unit variance and 2 21 1 21 (cid:0) (cid:1) (cid:1) orthogonal to W . Further, lpet JW be the standardized version of J . 1 c c 4Therearetwoprimaryreasonswhytheanalysisofthelong-runregressioninequation(1)isofinterestundertheassumption thatthetruemodelisgiven bytheshort-run equation (2). First,thereisthelongstandingpopularbeliefthatpredictabilityis more evidentin thelong run,and thattheremay thereforebe powergainsto analyzing thelong-horizon regression,even ifthe short-runspeci(cid:133)cationgivenbyequation(2)iscorrect;forinstance,Campbell(2001)analyzesthepoweroflong-runtestsunder the same speci(cid:133)cation that is used in this paper. Alternatively, since there appears to be no other data generating processes thatare widely used formodeling return predictability,in the short-orlong-run,the results derived underthe data generating processgivenbyequations(2)and(3)canbeviewedasabenchmarkagainstwhichtocompareresultsfromotherspeci(cid:133)cations. 3

To ease the notation, de(cid:133)ne 1 1 1 1 1 1 (cid:24) dW JW JW 2 (cid:0) and (cid:24) dW JW JW 2 (cid:0) ; (4) 1 (cid:17) 1 c c 2 (cid:17) 2 c c (cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) and write (cid:24) = 1 (cid:14)2(cid:24) +(cid:14)(cid:24) ; where (cid:24) 1 dW JW 1 JW 2 (cid:0) 1 . 2 (cid:0) 2 (cid:1) 1 1 2 (cid:1) 1 (cid:17) 0 2 (cid:1) 1 c 0 c p (cid:16) R (cid:17)(cid:16) R (cid:0) (cid:1) (cid:17) 3 Asymptotic distributions under the null and the alternative ^ 3.1 The limiting distribution of (cid:12) q The foundations for the subsequent analysis is given in the following theorem, which outlines the asymptotic properties of (cid:12)^ under both the null hypothesis of no predictability and the alternative of predictability. q Theorem 1 Suppose the data are generated by equations (2) and (3). 1. Under the null hypothesis that (cid:12) =0, as T , !1 T (cid:12)^ 0 q 1 dB J 1 J2 (cid:0) 1 =q (cid:27) 11(cid:24) : (5) q(cid:0) ) 1 c c (cid:27) 1 (cid:16) (cid:17) (cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19) r 22 2. Under the alternative hypothesis that (cid:12) =0, as T , 6 !1 T (cid:12)^ (cid:12) q 1 dB J +(cid:12)(cid:17)((cid:26);q) 1 dB J 1 J2 (cid:0) 1 =q (cid:27) 11(cid:24) +(cid:12)(cid:17)((cid:26);q)(cid:24) ; (6) q(cid:0) q ) 1 c 2 c c (cid:27) 1 2 (cid:16) (cid:17) (cid:18) Z0 Z0 (cid:19)(cid:18)Z0 (cid:19) r 22 where (cid:12) q (cid:17) (cid:12)(cid:23)((cid:26);q) with (cid:23)((cid:26);q) = 1+(cid:26)+:::+(cid:26)q (cid:0) 1 = q+O T (cid:0) 1 , and (cid:17)((cid:26);q) (cid:17) q h(cid:0)= 1 1 q p(cid:0)= 1 h (cid:26)p (cid:0) h = q(q 2 (cid:0) 1) +O T (cid:0) 1 . For (cid:26)=1, it holds(cid:0)exactly that (cid:23)((cid:26);(cid:1)q)=q, and(cid:0) (cid:17)((cid:26)(cid:1);q)= q(q 2 (cid:0) 1). P P (cid:0) (cid:1) Remark 1.1 Note that the asymptotic distribution of (cid:12)^ is identical under the null and the alternative. 1 That is, for any value of (cid:12), (cid:27) T (cid:12)^ (cid:12) 11(cid:24) : (7) 1(cid:0) ) (cid:27) 1 (cid:16) (cid:17) r 22 This follows both from standard asymptotic theory applied to the one-period OLS estimator, but also from plugging in q =1 in equations (5) and (6). Remark 1.2 Under the null hypothesis, the asymptotic distribution of (cid:12)^ is identical to that of (cid:12)^ , apart q 1 from a scaling factor. This result follows from the persistent nature of the regressors; the intuition behind it is discussed in more detail in Hjalmarsson (2007). For the purposes of this paper, the implications are that the short-run and long-run estimators are perfectly correlated asymptotically. In fact, from equation (5) it 4

follows that, q (cid:27) 1 (cid:27) (cid:12)^ 11(cid:24) and (cid:12)^ 11(cid:24) ; (8) q (cid:24) T (cid:27) 1 1 (cid:24) T (cid:27) 1 r 22 r 22 where is used to denote an approximate distributional equivalence. Asymptotically, therefore, the condi- (cid:24) tional distribution of (cid:12)^ given (cid:12)^ satis(cid:133)es (cid:12)^ (cid:12)^ =q(cid:12)^ . Given (cid:12)^ , the estimator (cid:12)^ is thus asymptotically a q 1 q 1 1 1 q (cid:12) deterministiclinearfunctionoftheforecasting(cid:12) horizon. ThisissimilartotheresultinBRW,whichisderived (cid:12) under the assumption of a (cid:133)xed autoregressive root (cid:26) that is strictly less than unity. Remark 1.3 Under the alternative hypothesis, the long-run estimator converges to the parameter (cid:12) = q (cid:12)(cid:23)((cid:26);q), which for (cid:26)=1+c=T implies that (cid:12) =q(cid:12)+O T 1 . That is, the (cid:145)true(cid:146)parameter value, (cid:12) , as q (cid:0) q well as the estimator (cid:12)^ , grows approximately one-to-one (cid:0)with (cid:1)the forecasting horizon. q Remark 1.4 Underthealternativehypothesisofpredictability,thedistributionof(cid:12)^ isquitedi⁄erentthan q under the null hypothesis. To understand the intuition behind this result, note (cid:133)rst that the true model is given by equations (2) and (3). The long-run regression equation is thus a (cid:133)tted regression, rather than the datageneratingprocess. AsshowninAppendixA,thelong-runreturnsr (q)actuallysatisfythefollowing t+q relationship when ignoring the constant, derived from equations (2) and (3): q 1 q 1 (cid:0) (cid:0) r (q)=(cid:12) x +u (q)+(cid:12) (cid:26)p h v : (9) t+q q t t+q 0 (cid:0) 1 t+h h=1 p=h X X @ A Therearenowtwoerrorterms,theusualu t+q (q)plustheadditionalterm(cid:12) q h(cid:0)= 1 1 q p(cid:0)= 1 h (cid:26)p (cid:0) h v t+h ,which (cid:16) (cid:17) stems from the fact that at time t there is uncertainty regarding the path oPf x fPor j = 1;:::;q 1. That t+j (cid:0) is, since the true model is given by equations (2) and (3), there is uncertainty regarding both the future realizations of the returns as well as of the predictor variable when forming q period ahead forecasts. The (cid:0) (cid:133)rst error term, u (q), corresponds to the asymptotic (cid:24) term in the limiting distribution and the second t+q 1 error term, (cid:12) q h(cid:0)= 1 1 q p(cid:0)= 1 h (cid:26)p (cid:0) h v t+h , corresponds to the (cid:24) 2 term. For large q, the second error term in (9) (cid:16) (cid:17) will clearly doPminatePthe asymptotic properties since it is of an order of magnitude larger than the (cid:133)rst one, a result re(cid:135)ected in the weights on (cid:24) and (cid:24) in equation (6). However, the weight on (cid:24) in (6) also depends 1 2 2 on (cid:12). Thus for (cid:12) close to zero, (cid:24) will still be important for relatively large q. 1 Remark 1.5 Following the analysis in BRW, the results are derived under the assumption that q is (cid:133)xed as T . However, it is easy to show that the results remain similar if q increases with the sample ! 1 size T, but at a slower pace, such that q=T 0, as q;T . Under this assumption, it follows easily ! ! 1 under the null hypothesis that T (cid:12)^ 0 (cid:27)11(cid:24) . Under the alternative hypothesis, T (cid:12)^ (cid:12) = q q(cid:0) ) (cid:27)22 1 (cid:17)((cid:26);q) q(cid:0) q (cid:16) (cid:17) q (cid:16) (cid:17) 5

T (cid:12)^ q (cid:12) q +o (1) (cid:12)(cid:24) , where the (cid:133)rst term in (6) now disappears asymptotically. As discussed in q q (cid:0) q p ) 2 (cid:18) (cid:19) the previous remark, however, the (cid:133)rst term in (6) will still be important for relatively large q, provided (cid:12) is small; this can be achieved also for asymptotically large q by treating (cid:12) as small in a local sense. All the results in this paper therefore hold also under the general assumption that q grows with the sample size but ataslowerpace. AsshowninHjalmarsson(2007), asymptoticresultsderivedunderthisassumptionseemto provide good approximations of the (cid:133)nite sample properties of (cid:12)^ for forecasting horizons spanning upwards q of 15 to 20 percent of the sample size. For completeness, however, Appendix B presents the results for the case where q is asymptotically large relative to T, in a manner such that q=T = (cid:21) (0;1), as T ; i.e. 2 ! 1 when q grows at the same pace as the sample size. 3.2 Finite sample adjustments In the analysis of BRW, it follows that under the null hypothesis, (cid:12)^ (cid:23)((cid:26);q)(cid:12)^ (see equation 6 in BRW), q (cid:24) 1 rather than (cid:12)^ q(cid:12)^ , as found here, where (cid:23)((cid:26);q) is de(cid:133)ned in Theorem 1.5 However, under the current q (cid:24) 1 assumption of (cid:26) = 1+c=T, it follows that (cid:23)((cid:26);q) = q+O T 1 . Thus, for the local-to-unity speci(cid:133)cation (cid:0) of (cid:26) that is used here, (cid:23)((cid:26);q) and q are asymptotically indis(cid:0)tingu(cid:1)ishable, and replacing q by (cid:23)((cid:26);q) does not a⁄ect the asymptotic arguments but merely provides a (cid:133)nite sample adjustment. As the analysis of BRW implies, along with simulation results that are not reported here, the rate of growth of (cid:12)^ under the null q hypothesis seems to correspond best to (cid:23)((cid:26);q), rather than q, in (cid:133)nite samples. Likewise,inPart2ofTheorem1,thefactorqinfrontof(cid:24) canbereplacedby(cid:23)((cid:26);q),sincethismultiplier 1 arises in an identical manner to the one in Part 1. That is, under the alternative hypothesis, one can write, (cid:27) T (cid:12)^ (cid:12) (cid:23)((cid:26);q) 11(cid:24) +(cid:12)(cid:17)((cid:26);q)(cid:24) : (10) q(cid:0) q ) (cid:27) 1 2 (cid:16) (cid:17) r 22 In the analysis in the next section, I use these (cid:133)nite sample adjusted results. This does not qualitatively change any of the results, and for (cid:26)=1 it holds exactly that q =(cid:23)((cid:26);q). 4 The relationship between the long-run and the short-run The results in the previous section provide the necessarybuilding blocks forunderstanding the properties of, and relationship between, the long- and short-run estimators both under the null hypothesis and under the alternative of predictability. In this section, I consider the implications of these results through an informal 5Iusethe(cid:145) (cid:146)sign herebecausein theframework ofBRW,theasymptoticdistribution of(cid:12)^ isnotentirely pinned down by (cid:24) q (cid:12)^ . 1 6

analysis. For ease of notation, it is assumed that (cid:27) =(cid:27) =1. 11 22 Under both the null and the alternative, the short-run estimator satis(cid:133)es, T (cid:12)^ (cid:12) (cid:24) , and one 1(cid:0) 1 ) 1 (cid:16) (cid:17) can write informally, 1 (cid:12)^ (cid:12)+ (cid:24) : (11) 1 (cid:24) T 1 Similarly, under the null with (cid:12) =0, the long-run estimator satis(cid:133)es 1 (cid:12)^ (cid:23)((cid:26);q) (cid:24) =(cid:23)((cid:26);q)(cid:12)^ : (12) q (cid:24) T 1 1 Thus, as noted above, under the null-hypothesis, (cid:12)^ and (cid:12)^ are perfectly asymptotically correlated. 1 q Under the alternative of predictability, (cid:23)((cid:26);q) (cid:17)((cid:26);q) (cid:17)((cid:26);q) (cid:12)^ (cid:12) + (cid:24) +(cid:12) (cid:24) =(cid:23)((cid:26);q)(cid:12)^ +(cid:12) 1 (cid:14)2(cid:24) +(cid:14)(cid:24) : (13) q (cid:24) q T 1 T 2 1 T (cid:0) 2 (cid:1) 1 1 (cid:16)p (cid:17) The distribution of (cid:12)^ is now a function of (cid:12)^ , as well as an additional term. Note, however, that given (cid:12)^ , q 1 1 the random variable (cid:24) is (cid:133)xed, and the only independent information in (cid:12)^ , given (cid:12)^ , derives from the (cid:24) 1 q 1 21 (cid:1) variable. To better understand the properties of (cid:12)^ under the alternative hypothesis, it is useful to consider the q two special cases of (cid:14) = 0 and (cid:14) close to 1. The case of (cid:14) close to 1 will be symmetrical to that of (cid:14) close (cid:0) to 1, but the latter is much more common in stock return applications. To more easily understand the (cid:0) variation in (cid:12)^ , Figure 1 shows the density plots for (cid:24) , for di⁄erent values of (cid:14), and (cid:24) for the case of c=0 q 1 2 ((cid:26)=1); the density of (cid:24) is identical to that of (cid:24) . 21 2 (cid:1) As is seen in Figure 1, (cid:24) , and hence (cid:24) , is almost always negative, a fact which will be used in the 2 21 (cid:1) discussion below. To see this analytically, consider the case when c = 0 and note that one can then write 1 (cid:24) = 1 W (1)2 1 W (1) 1 W (r)dr 1 W2 (cid:0) . Since W (1)2 is distributed as a (cid:31)2 variable, 2 2 2 (cid:0) (cid:0) 2 0 2 0 2 2 1 (cid:16) (cid:16) (cid:17) (cid:17)(cid:16) (cid:17) there is an approximately two-thiRrds probabilityRthat the (cid:133)rst term in the numerator will be negative. The secondtermwillalsotendtobenegative,sincethecorrelationbetweenW (1)andW (r)ispositive. Further, 2 2 when W (1)2 is large, the denominator will also be large, skewing the distribution further to the left. The 2 ratiowillthereforebenegativemostofthetimeandhaveanegativemean. Asimilarargumentcanbemade for c=0. 6 4.1 The case of (cid:14) = 0 When (cid:14) =0, (cid:17)((cid:26);q) (cid:12)^ (cid:23)((cid:26);q)(cid:12)^ +(cid:12) (cid:24) ; (14) q (cid:24) 1 T 2 (cid:1) 1 7

where the second term is stochastically independent of (cid:12)^ . Since E[(cid:24) ] < 0, as is apparent from Figure 1 1 21 (cid:1) and the discussion above, (cid:12)^ will tend to be below the curve (cid:12)^ (cid:23)((cid:26);q); the 5th, 50th and 95th percentile of q 1 (cid:24) , for c = 0, are given by 14:05, 4:37, and 0:13, respectively. To further understand the relationship 2 (cid:1) 1 (cid:0) (cid:0) (cid:0) between (cid:12)^ and (cid:12)^ in this case, consider a simple example. Suppose T =500; q =50, c=0, and (cid:12)^ =0:015, 1 q 1 which is a typical estimate of the short-run slope parameter in a regression with monthly standardized data such that (cid:27) = (cid:27) = 1 (e.g. Campbell and Yogo, 2006). If the true value of (cid:12) is equal to zero, then 11 22 asymptotically, (cid:12)^ = (cid:23)((cid:26);q)(cid:12)^ = 0:75. On the other hand, if (cid:12) = 0:015, so that the short-run estimate is q 1 equal to the true value, then conditional on (cid:12)^ , the 5th, 50th and 95th percentiles of (cid:12)^ are equal to 0:234, 1 q 0:589, and 0:745, respectively, based on equation (14) and the percentiles of (cid:24) . 21 (cid:1) 4.2 The case of (cid:14) 1 (cid:25) (cid:0) As (cid:14) 1; #(cid:0) (cid:17)((cid:26);q) (cid:12)^ (cid:23)((cid:26);q)(cid:12)^ (cid:12) (cid:24) : (15) q (cid:24) 1(cid:0) T 1 For (cid:14) close to minus one, (cid:24) is almost always positive, and (cid:12)^ will tend to be smaller than (cid:23)((cid:26);q)(cid:12)^ . Note 1 q 1 also, that once (cid:12)^ is determined, there is no additional variance left in the estimator (cid:12)^ . That is, since 1 q (cid:12)^ (cid:12)+(cid:24) =T, for a given (cid:12)^ and (cid:12), (cid:24) is pinned down, and hence (cid:12)^ as well. 1 (cid:24) 1 1 1 q Consider a similar thought experiment to that above. Again, suppose T = 500; q = 50, c = 0, and (cid:12)^ = 0:015. If (cid:12) = 0, then (cid:12)^ = (cid:24) =T, which implies that (cid:24) = 7:5 and (cid:12)^ = q(cid:12)^ = 0:75: Now, if (cid:12) = 0:01, 1 1 1 1 q 1 then (cid:12)^ = (cid:12) +(cid:24) =T implies that (cid:24) = 2:5, and (cid:12)^ = 0:689. If (cid:12) = 0:015, then (cid:24) = 0, and (cid:12)^ = 0:75: To 1 1 1 q 1 q the extent that (cid:12) is greater than or equal to zero, a large negative correlation (cid:14)severely limits the range of probable values that (cid:12)^ can attain once (cid:12)^ is (cid:133)xed. (The 5th, 50th and 95th percentile of (cid:24) for (cid:14) = 0:99 q 1 1 (cid:0) and c=0, are given by 0:10, 4:32, and 14:03, respectively.) 4.3 Implied long-horizon estimates Thus, when the predictor is exogenous, so that (cid:14) = 0, somewhat substantial deviations in (cid:12)^ from that q predictedunderthenullarepossibleandtosomeextentexpected. Whentheregressorsarehighlyendogenous, and (cid:14) is close to minus one, the range of possibilities also under the alternative is more restricted and large deviations from that predicted under the null are not likely.6 Figure 2 further illustrates this last point. Using equations (14) and (15), it plots potential outcomes of 6Apartfromincreasingslopecoe¢ cients,increasingR2shavealsobeenusedasanargumentinfavoroflong-runpredictability. InAppendixC,I(cid:133)rstreplicateBRW(cid:146)s(cid:133)ndingthatR2increaseswiththeforecasthorizonunderthenullhypothesis. Inaddition, Ishow thattheasymptoticpropertiesofR2 underthealternativehypothesisarenotafunction ofthedegreeofendogeneity of theregressorandthatR2 stillincreasesalmostlinearlywiththeforecastinghorizon. Thus,unlikefor(cid:12)^ ,thereisnosystematic q di⁄erence in the asymptotic properties ofR2 forexogenous and endogenous regressors underthe alternative hypothesis. 8

(cid:12)^ , given (cid:12)^ =0:015, for four di⁄erent values of the true (cid:12) =0;0:005;0:010;0:015. The same parameters as q 1 in the examples above are used, with T =500 and c=0. For (cid:14) 1, once (cid:12)^ and (cid:12) are (cid:133)xed, the outcome (cid:25)(cid:0) 1 of (cid:12)^ is fully determined and there is thus no range of possibilities. For (cid:14) =0, there is independent variation q left in (cid:12)^ , given (cid:12)^ and (cid:12), in the form of (cid:24) . The graphs for (cid:14) = 0 show the lower bound of (cid:12)^ , based q 1 21 q (cid:1) on the 5th percentile of (cid:24) . The upper bound, based on the 95th percentile of (cid:24) is virtually identical to 21 21 (cid:1) (cid:1) (cid:23)((cid:26);q)(cid:12)^ , since the 95th percentile of (cid:24) is almost equal to zero. 1 21 (cid:1) The graphs clearly demonstrate the limited range of plausible outcomes for (cid:12)^ given a typical one-period q estimate of (cid:12)^ , when the regressor is highly endogenous. Indeed, when the estimate (cid:12)^ is in fact identical to 1 1 the true (cid:12), the outcome is observationally equivalent to that under the null hypothesis. When the predictor is exogenous, the range of outcomes is obviously much larger, and there is a fair chance of detecting patterns that deviate substantially from those expected under the null. In their empirical analysis, BRW show that the coe¢ cients for the dividend-price ratio, which is highly endogenous, are nearly linear in the forecasting horizon whereas those for the short interest rate, which is nearly exogenous, grow at a much slower pace. In light of Figure 2, these (cid:133)ndings are suggestive of predictive ability in the short interest rate, but can say little or nothing regarding the predictive ability of the dividend-price ratio.7 5 Summary and conclusion To sum up, under the null hypothesis, the long-run estimator is asymptotically completely determined by the one-period estimate and the persistence in the regressor. Under the alternative hypothesis, the degree to which the long-run estimates can vary independently of the one period ones is determined by the degree of endogeneity in the regressors. Nearly exogenous predictors, such as the short interest rate, allow for more independent variation than highly endogenous predictors such as the earnings-price ratio. Unfortunately, long-run estimates therefore provide additional information in cases where short-run inference is relatively straightforward but adds little in the case of endogenous regressors where short-run inference is fraught with di¢ culties (i.e. Stambaugh, 1999, and Campbell and Yogo, 2006). Finally, it is worth pointing out, that the asymptotic framework used in this paper delivers an asymptotically degenerate distribution of (cid:12)^ given (cid:12)^ , under the null hypothesis. This prevents the construction q 1 7The results in this paper are all based on the assumption that the regressor follows a near unit-root process. For a (cid:133)xed autoregressive root (cid:26)strictly less than unity, the e⁄ects arising from endogeneity would not appear in the asymptotic analysis, although one could perhaps obtain some similar results using the (cid:133)nite sample bias derived in Stambaugh (1999). In practice, of course, the near unit-root construction is designed to asymptotically capture the (cid:133)nite sample bias that arises from highly persistent and endogenous regressors, which are not necessarily unit-root processes. For regressors with very low persistence, i.e. (cid:26) << 1, there will be no e⁄ects from endogeneity and there should thus be no systematic di⁄erence between endogenous and exogenous regressors in the behavior of either the short-run or the long-run estimators. However, most relevant predictors ofstock returns tend to be highly persistent. 9

of formal, and asymptotically meaningful, tests on the joint distribution of (cid:12)^ and (cid:12)^ under the null. BRW 1 q devise a joint test of (cid:12) = (cid:12) = ::: = (cid:12) = 0 under the assumption of a (cid:133)xed (cid:26) strictly less than one. 1 2 q However, under these assumptions, the standard asymptotic distribution of the test is not likely to be well satis(cid:133)ed due to the standard complications in inference with endogenous and persistent variables. Monte Carlo simulations not reported in this paper con(cid:133)rm that for a large negative (cid:14), the BRW test will tend to severelyoverreject. ItisalsointerestingtonotethattheWaldstatisticofBRWisscaledby(1 (cid:26))(cid:0) 1,which (cid:0) diverges to in(cid:133)nity as (cid:26) 1. The degenerate case encountered in the current asymptotics thus follows as a ! limiting case in their analysis. The construction of an asymptotically valid and correctly sized joint test of (cid:12) and (cid:12) for (cid:26) close to unity is thus left unresolved. 1 q A Proof of Theorem 1 Proof. For ease of notation the case with no intercept is treated. The results generalize immediately to regressions with (cid:133)tted intercepts by replacing all variables by their demeaned versions. Part 1 is proved in Hjalmarsson (2007), but is repeated here for completeness. 1. Under the null hypothesis, 1 1 T (cid:12)^ 0 = 1 T (cid:0) q u (q)x 1 T (cid:0) q x2 (cid:0) = 1 T (cid:0) q q u x 1 T (cid:0) q x2 (cid:0) : q q(cid:0) qT t+q t ! T2 t ! 0qT t+j t 1 T2 t ! (cid:16) (cid:17) X t=1 X t=1 X t=1 X j=1 X t=1 @ A By standard arguments, q 1 T T t=(cid:0)1 q q j=1 u t+j x t = q 1 T T t=(cid:0)1 q(u t+1 x t +:::+u t+q x t ) ) 0 1 dB 1 J c ; as T ! 1 , 1 since for any h>0, 1 T Pu xP 1 dB J . TherPefore, T (cid:12)^ 0 q 1 dB JR 1 J2 (cid:0) : T t=1 t+h t ) 0 1 c q(cid:0) ) 0 1 c 0 c (cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) 2. By summing upPon both sides inRequation (2), R R r (q)=(cid:12)(x +x +:::+x )+u (q) t+q t t+1 t+q 1 t+q (cid:0) q = (cid:12) x +(cid:26)x +:::+(cid:26)q 1x +v +((cid:26)v +v )+:::+ (cid:26)q pv +u (q) t t (cid:0) t t+1 t+1 t+2 (cid:0) t+p 1 t+q (cid:0) ! p=2 (cid:0) (cid:1) X q 1 q 1 (cid:0) (cid:0) = (cid:12) x +(cid:12) (cid:26)p h v +u (q); q t 0 (cid:0) 1 t+h t+q h=1 p=h X X @ A where (cid:12) =(cid:12) 1+(cid:26)+:::+(cid:26)q 1 =(cid:12)(cid:23)((cid:26);q). Thus, q (cid:0) (cid:0) (cid:1) q 1 q 1 q 1 q 1 (cid:0) (cid:0) (cid:0) (cid:0) r (q)=(cid:12) x +(cid:12) (cid:26)p h v +u (q)=(cid:12)(cid:23)((cid:26);q)x +(cid:12) (cid:26)p h v +u (q); t+q q t 0 (cid:0) 1 t+h t+q t 0 (cid:0) 1 t+h t+q h=1 p=h h=1 p=h X X X X @ A @ A 10

and T (cid:12)^ (cid:12) = (cid:12) q (cid:0) 1 q (cid:0) 1 (cid:26)p h 1 T v x + 1 T u (q)x 1 T x2 (cid:0) 1 : q(cid:0) q 0 0 (cid:0) 1T t+h t T t+q t 1 T2 t ! (cid:16) (cid:17) h X =1 p X =h X t=1 X t=1 X t=1 @ @ A A Observethat T 1 T t=1 v t+h x t ) 0 1 dB 2 J c forallh. Let(cid:17)((cid:26);q)= q h(cid:0)= 1 1 p q (cid:0)= 1 h (cid:26)p (cid:0) handitfollowsthat,asT ! 1 , q h(cid:0)= 1 1 q p(cid:0)= P1 h (cid:26)p (cid:0) h T 1 T t=1 Rv t+h x t ) (cid:17)((cid:26);q) 0 1 dB 2 J c . ByPthe resPults in Part 1., T 1 T t=1 u t+q (q)x t ) q 1 PdB J (cid:16) ,Pas T , (cid:17) andPthe desired result folloRws. Note that, P 0 1 c !1 R q 1q 1 q 1q 1 (cid:0) (cid:0) c p h (cid:0) (cid:0) c(p h) 1 (cid:17)((cid:26);q)= 1+ (cid:0) = 1+ (cid:0) +O T (cid:0) 2 = q(q 1)+O T (cid:0) 1 : T T 2 (cid:0) h X =1p X =h(cid:16) (cid:17) h X =1p X =h(cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) B Results for the case when q=T = (cid:21) (0;1) as T 2 ! 1 Some of the literature on long-horizon regressions has analyzed the case where q is asymptotically large relative to T, such that q=T =(cid:21) (0;1), as T . In the context of this study, such asymptotics are less 2 !1 usefulbecausethelong-runOLSestimatorwillnot convergetoaproperlyde(cid:133)nedlong-runcoe¢ cient. Since the current focus is on the distribution of the long-run estimator conditional on the short-run estimator, it makes more sense to consider the case when the long-run estimator does converge. Nevertheless, it is still interesting to see if any of the results derived in the main text continue to hold under this assumption. Again treating the case without an intercept, Valkanov (2003) shows that under the null hypothesis, 1 with q=T = (cid:21) as T ! 1 ; (cid:12)^ q ) 0 1 (cid:0) (cid:21) B 1 (r;(cid:21))J c (r) 0 1 (cid:0) (cid:21) J c 2 (cid:0) , where B 1 (r;(cid:21)) (cid:17) B 1 (r+(cid:21)) (cid:0) B 1 (r). Under the alternative, (cid:12)^ T q ) (cid:12) (cid:16) 0 R1 (cid:0) (cid:21) J c (r;(cid:21))J c (r) (cid:17)(cid:16) 0 1R (cid:0) (cid:21) J c 2 (cid:0) (cid:17)1 , where J c (r;(cid:21)) (cid:17) r r+(cid:21) J c (r): Thus, (cid:12)^ q (cid:16) (cid:17)(cid:16) (cid:17) no longer converges to a constanRt and it is therefore noRt surprising that the strong coRnnection between the asymptotic distributions for the short-run and long-run estimators is no longer apparent. Indeed, given the highly non-standard limiting distributions, it is di¢ cult to get a grasp of the properties of (cid:12)^ . A very rough q approximation, however, can provide some guidelines. r+(cid:21) r+(cid:21) Note that B (r+(cid:21)) B (r) = dB (s) (cid:21)dB (r) and J (r) (cid:21)J (r). Under the null 1 (cid:0) 1 r 1 (cid:25) 1 r c (cid:25) c 1 hypothesis, it follows that (cid:12)^ q (cid:24) (cid:21) R 0 1 (cid:0) (cid:21) dB 1 J c 0 1 (cid:0) (cid:21) J c 2 (cid:0) (cid:25) (cid:21)TR(cid:12)^ 1 = q(cid:12)^ 1 , and under the alternative hypothesis, (cid:12)^ T(cid:21)(cid:12) =q(cid:12). Thus, a (cid:16) lRso for q=T = (cid:17)(cid:16) (cid:21)R, it wou (cid:17) ld appear that (cid:12)^ will grow with the forecasting q (cid:24) q horizon. Under the null hypothesis, there is still some indication of the relationship between (cid:12)^ and (cid:12)^ , q 1 but under the alternative hypothesis the more subtle connections between the short-run and the long-run estimator are no longer evident, as might be expected given the lack of a consistent long-run estimator. 11

C Asymptotic properties of R2 LetR2 bethecoe¢ cientofdeterminationfromtheq periodregressioninequation(1). Usingthesamearguq (cid:0) 2 1 mentsasintheproofofTheorem1,itfollowsthatunderthenullhypothesis,TR2 1 1 dB J 1 J2 (cid:0) . 1 ) (cid:27)11 0 1 c 0 c Similarly, TR2 q 1 dB J 2 1 J2 (cid:0) 1 ; using the result that 1 T u2 (q) (cid:16) R (cid:27) , w (cid:17) hic (cid:16) hRis de (cid:17) q ) (cid:27)11 0 1 c 0 c qT t=1 t+q ! p 11 rived in Hjalmarsson ( (cid:16) 20R07). Thu (cid:17) s, (cid:16) aRsymp (cid:17) totically, under the null hypothPesis, R2 R2 =qR2. q 1 1 Becausex isanear-integratedregressor,itfollowseasilythatR2 1asT (cid:12) . Thisisnotveryuseful t q ! p !(cid:12)1 from a practical perspective and it is more interesting to analyze the properties under a local alternative, (cid:12) = b=pT. Under this alternative, using similar arguments as before, R2 1 (cid:27)11 ; and R2 1 ) (cid:0) (cid:27)11+b2( 0 1J c 2) q ) 1 (cid:0) (cid:27)11+qb (cid:27) 2 1 ( 1 0 1J c 2) : Standardizing so that (cid:27) 11 = 1, and using the approximation that (1+Rx)(cid:0) 1 (cid:25) 1 (cid:0) x, it follows thatRR2 b2 1 J2 and R2 qb2 1 J2 qR2. 1 (cid:25) 0 c q (cid:25) 0 c (cid:25) 1 (cid:16) (cid:17) (cid:16) (cid:17) R R 12

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Figure 1: Density plots of (cid:24) and (cid:24) for c = 0. The graphs show the densities for the Brownian functionals 1 2 (cid:24) ,fordi⁄erentvaluesof(cid:14),and(cid:24) ,obtainedbykernelestimationofsimulateddatausing100;000repetitions 1 2 and a sample size of 500 in each repetition. The shape of the density of (cid:24) is identical to that of (cid:24) : 21 2 (cid:1) 15

Figure 2: Possible outcomes of (cid:12)^ given (cid:12)^ and (cid:12). The graphs show potential outcomes of (cid:12)^ , given that q 1 q (cid:12)^ = 0:015, for (cid:12) = 0;0:005;0:01;0:015. The left hand side panel shows the case for exogenous regressors 1 with (cid:14) =0 and the right hand side shows the case with highly endogenous regressors. The plots are formed using equations (14) and (15), letting T =500 and c=0. For (cid:14) 1, once (cid:12)^ and (cid:12) are (cid:133)xed, the outcome of (cid:12)^ is fully determined and there is thus no range of possibili (cid:25) tie (cid:0) s. The gra 1 phs for (cid:14) = 0 show the lower q bound of (cid:12)^ , based on the 5th percentile of (cid:24) . The upper bound, based on the 95th percentile of (cid:24) , is q 21 21 virtually identical to (cid:23)((cid:26);q)(cid:12)^ ; i.e. the line for (cid:1) (cid:12) =0. (cid:1) 1 16