Fully Modified Estimation With Nearly Integrated Regressors

Fully Modified Estimation with Nearly K.7 Integrated Regressors Erik Hjalmarsson International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 854 January 2006

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 854 January 2006 Fully Modified Estimation with Nearly Integrated Regressors 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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/.

Fully Modi(cid:133)ed Estimation with Nearly Integrated Regressors Erik Hjalmarsson (cid:3) Division of International Finance Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA January 2006 Abstract I show that the test procedure derived by Campbell and Yogo (2005, Journal of Financial Economics, forthcoming) for regressions with nearly integrated variables can be interpreted as the natural t-test resulting from a fully modi(cid:133)ed estimation with near-unit-root regressors. This clearlyestablishesthemethodsofCampbellandYogoasanextensionofpreviousunit-rootresults. JEL classi(cid:133)cation: C22. Keywords: Fully modi(cid:133)ed estimation; Near-unit-roots; Predictive regressions. Tel.: +1-202-452-2436; fax: +1-202-263-4850; email: erik.hjalmarsson@frb.gov. The views presented in this paper (cid:3) are solely those ofthe authorand do not represent those ofthe FederalReserve Board orits sta⁄.

1 Introduction In the recent past, there has been much e⁄ort spent on the econometric analysis of forecasting regressions with nearly persistent regressors. Using Monte Carlo simulations, Mankiw and Shapiro (1986) showed in an in(cid:135)uential paper that when the regressor variables in a predictive regression are almost persistent and endogenous, test statistics will no longer have standard distributions. Since then, a much better understanding of this phenomenon has been established and several alternative methods have been proposed; e.g. Cavanagh et al. (1995), Stambaugh (1999), Jansson and Moreira (2004), Lewellen (2004), and Campbell and Yogo (2005). The issues encountered when performing inference in forecasting regressions with near-persistent variablesare,ofcourse,similartothoseinthecointegrationliteraturewheretheregressorsareassumed to follow unit-root processes. Indeed, the case with nearly persistent regressors can be seen as a generalization of the standard unit-root setup. Inthisnote,Ishowthatthee¢ cienttestforinferenceinpredictiveregressionsderivedbyCampbell and Yogo (2005) can also be seen as the natural test resulting from a generalization of fully modi(cid:133)ed estimation (Phillips and Hansen, 1990, and Phillips, 1995) to the case of near-unit-root regressors. In addition, the optimality properties of the Campbell and Yogo (2005) test-statistic can be seen as a direct analogue of the optimal inference results derived by Phillips (1991) for cointegrated unit-root systems. These results (cid:133)rmly establish the link between current work on predictive regressions and earlier work on cointegration between unit-root variables. 2 Model and assumptions Let the dependent variable be denoted y , and the corresponding vector of regressors, x , where x is t t t an m 1 vector and t=1;:::;T. The behavior of y and x are assumed to satisfy, t t (cid:2) y = (cid:11)+(cid:12)x +u ; (1) t t 1 t (cid:0) x = Ax +v ; (2) t t 1 t (cid:0) where A=I+C=T is an m m matrix. (cid:2) Assumption 1 Let w t =(u t ;(cid:15) t )0 and t = w s s t be the (cid:133)ltration generated by w t . Then F f j (cid:20) g 1

1. v t =D(L)(cid:15) t = 1j=0 D j (cid:15) t (cid:0) j , and 1j=0 j jj D j jj < 1 : 2. E[w ]=0P; E u4 < ; andPE (cid:15) 4 < : t jF t (cid:0) 1 t 1 jj t jj 1 3. E[w w ]=(cid:6)=(cid:2) [((cid:3)(cid:27) ;(cid:27) );((cid:27) h ;I)]: i t t0 jF t (cid:0) 1 11 12 21 Themodeldescribedbyequations(1)and(2)andAssumption1capturestheessentialfeaturesofa predictive regression with nearly persistent regressors. It states the usual martingale di⁄erence (mds) assumptionfortheerrorsinthedependentvariablesbutallowsforalineartime-seriesstructureinthe errorsofthepredictors. Theerrortermsu andv arealsooftenhighlycorrelated. Theauto-regressive t t roots of the regressors are parametrized as being local-to-unity, which captures the near-unit-root behavior of many predictor variables, but is less restrictive than a pure unit-root assumption. The local-to-unity parameter C is generally unknown and not consistently estimable. Following Campbell and Yogo (2005), I derive the results under the assumption that C is known. Bonferroni type methods can then be used to form feasible tests, as in Cavanagh et al. (1995) and Campbell and Yogo (2005); such methods are extensively explored in these papers and will not be further discussed here. Let E t = (u t ;v t )0 be the joint innovations process. Under Assumption 1, by standard arguments, 1 [Tr]E B(r) = BM((cid:10))(r); where (cid:10) = [(! ;! );(! ;(cid:10) )]; ! = (cid:27) , ! = D(1)(cid:27) , pT t=1 t ) 11 12 21 22 11 11 21 12 ! 12 P= ! 021 , (cid:10) 22 = D(1)D(1)0, and B( (cid:1) ) = (B 1 ( (cid:1) );B 2 ( (cid:1) ))0 denotes an 1+m (cid:0) dimensional Brownian motion. Also, let (cid:3) 22 = 1k=1 E(v k v 00 ) be the one-sided long-run variance of v t . The following lemma sumsupthekeyasymptoPticresultsforthenearlyintegratedmodelinthispaper(Phillips1987,1988). Lemma 1 Under Assumption 1, as T , (a) T 1=2x J (r); (b) T 3=2 T x ! 1 (cid:0) i;[Tr] ) C (cid:0) t=1 t ) 0 1 J C (r)dr; (c) T (cid:0) 2 T t=1 x t x 0t ) 0 1 J C (r)J C (r)0dr; (d) T (cid:0) 1 T t=1 u t x 0t (cid:0) 1 ) 0 1 dB 1 P(r)J C (r)0; aRnd (e) T (cid:0) 1 T t=1 v t x 0t P (cid:0) 1 ) 0 1 dB 2 (rR)J C (r)0+(cid:3) 22 ; where J C (r)P= 0 r e(r (cid:0) s)CdB 2 R(s): P R R Analogousresultsholdforthedemeanedvariablesx =x T 1 n x ,withthelimitingprocess t t (cid:0) (cid:0) t=1 t 1 J replaced by J =J J . P C C C (cid:0) 0 C R 2

3 Fully modi(cid:133)ed estimation Let(cid:12)^ denotethestandardOLSestimateof(cid:12)inequation(1). ByLemma1andthecontinuousmapping theorem (CMT), it follows that 1 1 1 T (cid:12)^ (cid:12) dB J J J (cid:0) ; (3) (cid:0) ) 1 0C C 0C (cid:16) (cid:17) (cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19) as T . Analogous to the case with pure unit-root regressors, the OLS estimator does not have an !1 asymptotically mixed normal distribution due to the correlation between B and B , which causes B 1 2 1 and J to be correlated. Therefore, standard test procedures cannot be used. C Inthepureunit-rootcase,onepopularinferentialapproachisto(cid:147)fullymodify(cid:148)theOLSestimator as suggested by Phillips and Hansen (1990) and Phillips (1995). In the near-unit-root case, a similar method can be considered. De(cid:133)ne the quasi-di⁄erencing operator C (cid:1) x =x x x =v ; (4) C t t (cid:0) t (cid:0) 1 (cid:0) T t (cid:0) 1 t and let y+ = y !^ (cid:10)^ 1(cid:1) x and (cid:3)^+ = !^ (cid:10)^ 1(cid:3)^ ; where !^ ;(cid:10)^ 1, and (cid:3)^ are consistent t t (cid:0) 12 (cid:0)22 C t 12 (cid:0) 12 (cid:0)22 22 12 (cid:0)22 22 estimates of the respective parameters.1 The fully modi(cid:133)ed OLS estimator is now given by 1 T T (cid:0) (cid:12)^+ = y+x T(cid:3)^+ x x ; (5) t 0t (cid:0) 1(cid:0) 12 ! t (cid:0) 1 0t (cid:0) 1 ! t=1 t=1 X X where y+ = y !^ (cid:10)^ 1(cid:1) x and y = y T 1 t y . The only di⁄erence in the de(cid:133)nition of t t (cid:0) 12 (cid:0)22 C t t t (cid:0) (cid:0) t=1 t (5), totheFM-OLSestimatorforthepureunit-rootPcase, istheuseofthequasi-di⁄erencingoperator, as opposed to the standard di⁄erencing operator. Once the innovations v are obtained from quasit di⁄erencing, the modi(cid:133)cation proceeds in exactly the same manner as in the unit-root case. De(cid:133)ne ! =! ! (cid:10) 1! and the Brownian motion B =B ! (cid:10) 1B =BM(! ). 11 (cid:1) 2 11 (cid:0) 12 (cid:0)22 21 1 (cid:1) 2 1 (cid:0) 12 (cid:0)22 2 11 (cid:1) 2 The process B is now orthogonal to B and J . Using the same arguments as Phillips (1995), it 12 2 C (cid:1) follows that, as T , !1 1 1 1 1 1 T (cid:12)^+ (cid:12) dB J J J (cid:0) MN 0;! J J (cid:0) : (6) (cid:16) (cid:0) (cid:17) ) (cid:18)Z0 1 (cid:1) 2 C0 (cid:19)(cid:18)Z0 C 0C (cid:19) (cid:17) 11 (cid:1) 2 (cid:18)Z0 C 0C (cid:19) ! 1Thede(cid:133)nitionof(cid:3)^+ isslightlydi⁄erentfrom theonefoundin Phillips(1995). Thisisduetothepredictivenature 12 ofthe regression equation (1),and the martingale di⁄erence sequence assumption on ut. 3

The corresponding test-statistics will now have standard distributions asymptotically. For instance, the t test of the null hypothesis (cid:12) =(cid:12)0 satis(cid:133)es (cid:0) k k (cid:12)^+ (cid:12)0 t+ = k (cid:0) k N(0;1) (7) 1 ) !^ a T x x (cid:0) a 112 0 t=1 t 1 0t 1 r (cid:1) (cid:0) (cid:0) (cid:16) (cid:17) P under the null, as T . Here a is an m 1 vector with the k(cid:146)th component equal to one and zero !1 (cid:2) elsewhere. The t+ statistic is identical to the Q statistic of Campbell and Yogo (2005). Whereas Campbell (cid:0) (cid:0) and Yogo (2005) attack the problem from a test point-of-view, the derivation in this paper starts with the estimation problem and delivers the test-statistic as an immediate consequence. However, presenting the derivation in this manner makes clear that this approach is a generalization of fully modi(cid:133)ed estimation. Inaddition,ifAssumption1isreplacedbythestrongerconditionthatbothu andv aremartingale t t di⁄erence sequences, it is easy to show that OLS estimation of the augmented regression y =(cid:11)+(cid:12)x +(cid:13)(cid:1) x +u (8) t t 1 C t tv (cid:0) (cid:1) yields an estimator of (cid:12) with an asymptotic distribution identical to that of (cid:12)^+ . This is, of course, a straightforward extension of the results in Phillips (1991) for unit-root regressors. Moreover, in the unit-root case, Phillips (1991) shows that the OLS estimator of (cid:12) in equation (8) is identical to the gaussian full system maximum likelihood estimator of (cid:12). The optimality properties of Campbell and Yogo(cid:146)s (2005) Q test is thus a direct extension of the optimality results developed in Phillips (1991). (cid:0) References Campbell, J.Y., and M. Yogo, 2005. E¢ cient Tests of Stock Return Predictability, forthcoming Journal of Financial Economics. Cavanagh,C.,G.Ellliot,andJ.Stock,1995. Inferenceinmodelswithnearlyintegratedregressors, Econometric Theory 11, 1131-1147. Jansson, M., and M.J. Moreira, 2004. Optimal Inference in Regression Models with Nearly Inte- 4

grated Regressors, NBER Working Paper T0303. Lewellen, J., 2004. Predicting returns with (cid:133)nancial ratios, Journal of Financial Economics, 74, 209-235. Mankiw, N.G., and M.D. Shapiro, 1986. Do we reject too often? Small sample properties of tests of rational expectations models, Economic Letters 20, 139-145. Phillips, P.C.B, 1987. Towards a Uni(cid:133)ed Asymptotic Theory of Autoregression, Biometrika 74, 535-547. Phillips, P.C.B, 1988. Regression Theory for Near-Integrated Time Series, Econometrica 56, 1021- 1043. Phillips, P.C.B, 1991. Optimal Inference in Cointegrated Systems, Econometrica 59, 283-306. Phillips, P.C.B, 1995. Fully Modi(cid:133)ed Least Squares and Vector Autoregression, Econometrica 63, 1023-1078. Phillips, P.C.B, and B. Hansen, 1990. Statistical Inference in Instrumental Variables Regression with I(1) Processes, Review of Economic Studies 57, 99-125. Stambaugh, R., 1999. Predictive regressions, Journal of Financial Economics 54, 375-421. 5

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