REGRESSION THEORY FOR NEARLY COINTEGRATED TIME SERIES

Transcription

1 Econometric heory, 8, 2002, Printed in the United States of America+ DOI: 0+070S REGRESSION EORY FOR NEARLY COINEGRAED IME SERIES MICAEL JANSSON University of California, Berkeley NIELS ALDRUP University of Aarhus his paper proposes a notion of near cointegration and generalizes several existing results from the cointegration literature to the case of near cointegration+ In particular, the properties of conventional cointegration methods under near cointegration are characterized, thereby investigating the robustness of cointegration methods+ In addition, we obtain local asymptotic power functions of five cointegration tests that take cointegration as the null hypothesis+. INRODUCION In a highly influential Monte Carlo study, Granger and Newbold ~974! considered regressions of independent random walks on each other and found that the usual significance test based on the regression F-statistic tends to overreject the null+ o describe this phenomenon, the term spurious regression was coined+ he numerical findings of Granger and Newbold are given an analytical explanation by Phillips ~986!, whereas Park, Ouliaris, and Choi ~988! and Park ~990! provide further clarification+ hese authors consider regressions involving quite general integrated processes and show that the asymptotic properties of the appropriate F-statistic depend crucially on r 2, the squared multiple correlation coefficient computed from the long-run covariance matrix of the underlying innovation sequence+ If r 2, the F-statistic diverges at rate ~where is the sample size! whereas F has a nondegenerate limiting distribution, which only depends on the dimension of the system+ In other words, the regression is spurious whenever the coefficient of correlation is less than unity+ In contrast, when r 2 the series are cointegrated and F O p ~! with a complicated limiting distribution+ Conventional asymptotic results therefore depend discontinuously on r 2 + his paper is an abridged version of Chapter of the first author s Ph+D+ dissertation at University of Aarhus+ he paper has benefited from the comments of + Peter Boswijk, Bruce ansen ~the co-editor!, two anonymous referees, and seminar participants at Penn State University, inbergen Institute, University of California ~Riverside, San Diego!, the 998 European Meeting of the Econometric Society, and the 999 NOS-S conference on macroeconomic transmission mechanisms+ Address correspondence to: Michael Jansson, Department of Economics, University of California, Berkeley, 549 Evans all #3880, Berkeley, CA , USA; mjansson@econ+berkeley+edu Cambridge University Press $

2 30 MICAEL JANSSON AND NIELS ALDRUP On the other hand, it is quite obvious that the finite sample distribution of the F-statistic depends continuously on r 2 + As a consequence, there is reason to believe that spurious regression asymptotics provide a poor approximation to the finite sample behavior of the F-statistic when the processes are nearly cointegrated in the sense that r 2 is close to unity+ More generally, finite sample approximations based on spurious regression theory are likely to be of limited usefulness whenever the limiting behavior of the object of interest ~e+g+, an estimator or a test statistic! exhibits a discontinuity at r 2 and values of r 2 close to unity are of particular interest+ In contrast, a model of near cointegration in which r 2 is a sequence of parameters lying in a shrinking neighborhood of unity as tends to infinity is much more appealing in such situations+ Motivated by these considerations, the present paper introduces a model in which r 2 is a primitive parameter and uses this model to propose a notion of near cointegration+ 2 By construction, the limiting behavior of the F-statistic depends continuously on r 2 in our setup, and the model of near cointegration can therefore be used to bridge the gap between spurious regression and cointegration with respect to the limiting behavior of the F-statistic+ 3 We use the model of near cointegration to generalize several existing results from the cointegration literature to the case of near cointegration+ In particular, the robustness of cointegration methods is investigated+ We characterize the limiting behavior under near cointegration of the usual Wald statistic devised to test hypotheses on a cointegrating vector and show that the limiting distribution obtained under near cointegration stochastically dominates the x 2 distribution applicable under cointegration+ his result complements Elliott s recent study ~998!, where the implications of near integration in exactly cointegrated models are examined+ In addition to studying the robustness of cointegration methods, we characterize the behavior of five regression based cointegration tests under local alternatives and compute the corresponding local asymptotic power functions+ Among the tests under study, three are found to have virtually identical local asymptotic power properties, whereas the remaining two are significantly inferior in terms of local asymptotic power+ he paper proceeds as follows+ In Section 2, we introduce our model+ Section 3 discusses the behavior of regression estimators under near cointegration, and Section 4 contains the corresponding results for inference procedures based on these estimators+ In Section 5, we report the behavior of several cointegration tests under near cointegration+ Finally, Section 6 offers a few concluding remarks+ Proofs of all results of the paper are outlined in the Appendix+ Before we begin, a word on notation+ he inequality. 0 signifies positive definiteness when applied to square matrices, and 7{7 is the Euclidean norm+ For any symmetric A 0, A 02 ~A 02! and A 02 is the upper triangular matrix with positive diagonal elements such that A 02 A 02 A+ he symbols L, r d, and r p signify equality in law, convergence in distribution, and convergence in probability, respectively+ Finally, to simplify the notation integrals such as * 0 W~r! dr and stochastic integrals such as * 0 W~r! dw~r! are typically written as *W and *WdW, respectively+

3 D D D 2. E MODEL AND ASSUMPIONS NEARLY COINEGRAED IME SERIES 3 Suppose $z t : 0 t, % is an m-vector triangular array generated by Dz t C ~L!e t, () where each C ~L! ( ` i 0 C i L i is an m m matrix lag polynomial, whereas $e t : t % is independent and identically distributed ~i+i+d+! with E~e t! 0 and E~e t e t! 0+ riangular arrays are considered to be able to define a notion of near cointegration similar to those of anaka ~993, 996!+ Our objective is to do so by means of a parameterization of $C ~L! : % and E~e t e t! that involves minimal loss of generality for any fixed and depends on in the simplest possible fashion+ Applying the BN ~Beveridge and Nelson, 98! decomposition 4 to C ~L!, we can write z t as z t C ~!j t CD ~L!e t z 0 CD ~L!e 0, t where j t ( e s + We assume that C ~! is upper triangular ~with nonnegative diagonal elements! and E~e t e t! I m + For any fixed, these assumptions entail no loss of generality+ 5 Partition z t into m y and m x m components as z t ~ y t, x t! + Conformably, partition C ~! and C ~L! after the first row as C ~! ~C y ~!,C x ~!! and C ~L! ~ CD y ~L!, CD x ~L!!, respectively+ Assuming C y ~! and C x ~! have full row rank, C ~! can be written as C ~! v yy, 02 ~ r 2! 02 ~r V 02 xx, 02 0 V xx, v xy,!, (2) where v yy, 0 is a scalar, V xx, 0isanm x m x matrix, v xy is an m x - vector such that v xy, V xx, v xy, v yy, and 0 r + Under this parameterization, the rank of C ~! depends solely on the scalar r + Indeed, C ~! has full rank whenever 0 r, whereas C ~! has ~deficient! rank m if r + his feature is very convenient for our purposes, as it enables us to define a notion of near cointegration by modeling $r % as a sequence lying in a shrinking neighborhood of unity as increases without bound+ In recognition of the fact that our main emphasis is on the cointegration properties of $z t %, we henceforth make the simplifying assumption that r is the only parameter of the model that varies with + o make this assumption explicit, the redundant subscript will be omitted in expressions involving the parameters C x ~L!, C ~L!, v yy,, V xx,, and v xy, that do not vary with + Likewise, x t will be written as x t + Let u t C u ~L!e t, where C u ~L! ~, b! C~L! D and b V xx v xy + Define w t ~u t, Dx t! and let V ww be the long-run covariance matrix of w t, namely, V ww v uu v xu v xu V xx lim r` ( ( E~w t w s!,

4 D D 32 MICAEL JANSSON AND NIELS ALDRUP where the partitioning is in conformity with w t + he development of formal results will proceed under the following assumptions+ A+ C ~L! C ~! C~L!~ D L!, where C~L! ( i 0 C i L i is a lag polynomial with ( ` i 0 i7cd i 7 `, and C ~! is defined as in ~2! with v yy, v xy, and V xx fixed and r 2 2 l 2 v uu+x 0v yy for some l 0, where v uu+x v uu v xu 0 and C u ~!~,0! 0+ v xu V xx A2+ $e t % is i+i+d+ with E~e t! 0 and E~e t e t! I m + A3+ z 0 C~L!e D 0 + he long-run covariance matrix of Dz t is V zz lim r` ( E~Dz t Dz s!+ In view of the relation ( V zz lim C ~!C ~! lim r` r` v yy r v xy r v xy V xx v yy v xy v xy V xx, the parameters r, v yy, V xx, and v xy in A all have natural interpretations+ Indeed, it is apparent that C ~! is parameterized directly in terms of the elements of V zz and the scalar r + In turn, r 2 is the squared coefficient of multiple correlation computed from C ~!C ~! + Under A, $z t % is nearly cointegrated in the sense that r 2 lies in a shrinking neighborhood of unity as increases+ Of course, near cointegration reduces to cointegration when l 0inA+In that case, the assumption v uu+x 0 states that the cointegration is regular in the sense of Park ~992!+ On the other hand, when l 0, the condition C u ~!~,0! 0 can be interpreted as an identification assumption+ 6 he assumption V xx 0 implies that $x t % is a noncointegrated integrated process+ Although somewhat restrictive, the assumption of noncointegrated regressors is fairly standard in the related literature, 7 so to facilitate comparisons with existing results we shall maintain this assumption throughout+ Assumption A3 is introduced to avoid any complications that might arise as a result of a nonzero mean in z t and0or a remote past initialization of the $z t % process ~e+g+, Canjels and Watson, 997!+ he parameter l introduced in A will play a prominent role in the sequel+ Because l {v yy 02 ~ r 2! 02, (3) 02 v uu+x l can be interpreted as a signal-to-noise ratio+ Specifically, the numerator in ~3! is proportional to v 02 yy ~ r 2! 02, which is the long-run standard deviation of Dy t conditional on Dx t in the case where r does not vary with + he denominator, v 02 uu+x, is the long-run standard deviation of u t conditional on Dx t + Under cointegration, the former is zero and l 0+ Under spurious regression ~when r is fixed!, on the other hand, the right-hand side of ~3! diverges+ `

5 ^ ^ ^ ^ Near cointegration corresponds to the intermediate case where the numerator and denominator of ~3! are of the same order of magnitude+ Our model can be written in triangular form as y t b x t d j t C u ~L!e t, NEARLY COINEGRAED IME SERIES 33 Dx t C x ~L!e t, (4) where d ~v 02 yy ~ r 2! 02,~r! v xy V 02 xx! + As it turns out, the presence of the additional error term d j t on the right-hand side of ~4! leads to an increase in the asymptotic variance of estimators of b+ On the other hand, because d j t is asymptotically uncorrelated with the regressor x t in ~4! in the sense that the long-run covariance between Dx t and d e t, namely, lim r` {C x ~!d, equals zero, 8 the presence of d j t does not affect the asymptotic bias of estimators of b+ Notions of near cointegration that are closely related to ours have been proposed by anaka ~993, 996!+ 9,0 he near cointegration model of anaka ~993, equation ~20!! can be written in triangular form as y t b x t d j t C u ~L!e t, Dx t C x ~L!e t, t where j t ( e s, $e t % satisfies A2, C u ~L! C u ~L! C u ~L!, and lim r` {C x ~!d 0+ he asymptotic distributions of interest are unaffected by the presence of the term C u ~L!e t, and anaka s model ~993! yields results that coincide with the results of the present paper+ anaka s near cointegration model ~996, equations ~+68! and ~+70!! can be written as y t b x t d j t C u ~L!e t, Dx t C x ~L!e t, t where j t ( e s, $e t % satisfies A2, b b b, ^ C u ~L! C u ~L! C u ~L!, and lim r` {C x ~!d 0+ he term b ~b b! is nonzero in general and gives rise to a drift term in the limiting distribution of estimators of b~such as b and b defined in Section 3! but does not affect the limiting distribution of cointegration tests, the objects of interest in anaka ~996!+ Unlike the parameterizations employed by anaka ~993, 996!, the parameterization of near cointegration proposed here is explicitly one-dimensional ~involving only the scalar parameter l! and therefore leads to simpler representations of the limiting distributions of interest, which facilitates the interpretation of the results+

6 [ 34 MICAEL JANSSON AND NIELS ALDRUP 3. BEAVIOR OF REGRESSION ESIMAORS Let a[ and b be the ordinary least squares ~OLS! estimators in the multiple regression y t a[ d t b x t u[ t ~t,+++,!, (5) where d t ~,+++,t md! for some m d + Let C diag~ 02, +++, md 02,{i mx!, where i mx is an m x -vector of ones+ As is well known ~e+g+, Phillips and Durlauf, 986!, the limiting distribution of C ~ a[,~ b b!! under cointegration is rather complicated and depends on the parameters V ww and G ww, where V ww was defined in Section 2, whereas G ww ~G {u G {x! g uu g ux g xu G xx lim r` ( A similar situation occurs under near cointegration+ C a[ b b r d Q x Q x t ( E~w t w s!+ LEMMA + Suppose $z t % is generated by () and suppose A A3 hold. hen v uu+x 02 Q x du l Q x dx V xx v xu 0 g ux, where Q x ~r! ~D~r!, X~r!!,D~r! ~,r,+++,r md!,x~r! V 02 xx V~r!, and U l ~r! l* r 0 U~s!ds U~r!, whereas V and U are independent Wiener processes of dimension m x and, respectively. In addition to ~ a[, b!, we want to study an estimator that has a compound normal limiting distribution under cointegration+ For concreteness, we consider the canonical cointegration regression ~CCR! estimator proposed by Park ~992!+ o construct this estimator, we need consistent estimators of V ww, G ww, and S ww lim r` ( E~w t w t!+ Let w[ t ~ u[ t, Dx t!, where $ u[ t % are the residuals from ~5!+ We can estimate S ww by S ww ( w t w[ t, whereas V ww and G ww can be estimated by kernel estimators of the form 6t s6 V ww ( ( k b w[ t w[ s and G ww ( t 6t s6 ( k b w[ t w[ s, where k~{! is a kernel function and $ b % is a sequence of ~possibly sampledependent! bandwidth parameters+ 2 As shown in Lemma 5 in the Appendix, the estimators V ww, G ww, and S ww are consistent under A A4, where

7 NEARLY COINEGRAED IME SERIES 35 A4+ ~i! k~0!, k~{! is continuous at zero, sup s 0 6k~s!6 `, and * 0` k~r! O dr `, where k~r! O sup s r 6k~s!6 ~ for all r 0! ~ii! b a[ b, where a[ and b are positive with a[ a[ O p ~! and b 02 b o~!+ he CCR estimator ~ a[, b! is the OLS estimator obtained from the multiple regression y t a[ d t b x t u[ t ~t,+++,!, (6) where y t y t b G {x S ww w[ t [ the OLS estimator from ~5!+ v xu V xx Dx t, x t x t G {x S ww w[ t, and b is LEMMA 2+ Suppose $z t % is generated by () and suppose A A4 hold. hen a[ b C b r d Q x Q x v uu+x 02 Q l x du, where C, Q x, and U l are defined as in Lemma. he limiting distribution is compound normal: Q x Q x L N v uu+x 02 Q x du l FV 0, v uu+x Q x Q x Q x, l Q x, l Q x Q x, where Q x, l ~r! l* r Q x ~s!ds Q x ~r! and {8 FV signifies the conditional distribution relative to F V s~v~r!:0 r!. anaka ~993! obtains a result equivalent to the first half of Lemma 2+ 3 In important respects, the near cointegration case closely resembles the cointegration case+ For instance, b and b are superconsistent estimators of b+ Moreover, the limiting distribution of ~ b b! is compound normal+ he mean of this compound normal distribution is zero even under near cointegration, and the presence of d j t on the right-hand side of ~4! therefore does not introduce a bias term in the limiting distribution of b + Under cointegration ~i+e+, when l 0!, * Q x, l Q x, l * Q x Q x and the covariance matrix in the mixture representation in Lemma 2 reduces to v uu+x ~* Q x Q x! + Otherwise, if l 0, the covariance matrix is of sandwich form+ As pointed out by the co-editor, this suggests that OLS-type inference ~based on the CCR estimates! will be

8 36 MICAEL JANSSON AND NIELS ALDRUP misleading under near cointegration+ Indeed, defining Q x ~r! * r Q x ~s! ds, we have Q x, l Q x, l Q x Q x lq x ~0!Q x ~0! l 2 Q x Q x 0+ he presence of d j t on the right-hand side of ~4! therefore leads to an increase in the asymptotic variance of b, suggesting that overrejection of true null hypotheses ~on b! is likely to occur under near cointegration+ A more precise statement corroborating this conjecture will be provided in the next section+ 4. INFERENCE ON REGRESSION COEFFICIENS his section is concerned with inference on regression coefficients+ Particular attention is given to the robustness of conventional cointegration procedures under near cointegration+ Consider a general linear hypothesis of the form 0 : F b b f b, where F b is a p m x matrix of rank p and f b is a p-vector+ 4 Define G v[ uu+x ~F b where v[ uu+x v[ uu [ b f b! F b ( v xu V xx [ x t, d x t, d F b ~F b b f b!, (7) v xu and x t, d x t ~( d s!~( d s d s! d t + EOREM 3+ Suppose $z t % is generated by () and suppose A A4 hold. (a) When 0 : F b b f b is true, G r d V D p du l 2, where V p D ~r! ~* V p D V p D! 02 V p D ~r!, V p D ~r! V D ~r! ~* V D V 2D! ~* V 2D V 2D! V 2D ~r!, and V D ~r! ~V D ~r!,v 2D ~r!! V~r! ~* VD! ~* DD! D~r!, where the partitioning is after the pth row, whereas V, D, and U l are defined as in Lemma. (b) he limiting distribution in part (a) satisfies V p D du l 2 FV p L ( ~ l 2 {m i!x 2 i ~!, i where $x 2 p i ~!% i are i.i.d. x 2 ~! variates and 0 m {{{ m p are the eigenvalues of the matrix * 0 V p D ~r! V D p ~r! dr, where V p D ~r! * r V p D ~s! ds. In particular, the family $L~7* V p D du l 7 2! : l 0% is stochastically increasing in l, where L~{! denotes the probability law of the argument. Under cointegration, the limiting distribution of G is x 2 ~ p! when 0 is true+ In a recent paper, Elliott ~998! investigates the robustness of this remarkable result by considering a model in which the regressors are nearly integrated x s

9 NEARLY COINEGRAED IME SERIES 37 whereas some linear combination of the regressand and the regressor is exactly stationary+ It turns out that the x 2 result can break down when the regressors are not exactly integrated+ heorem 3 enables us to conduct a complimentary experiment: we can investigate the robustness of cointegration methods in a model where the regressors are exactly integrated whereas some linear combination of the regressand and the regressors is nearly stationary+ It follows from heorem 3~b! that tests based on the distribution applicable under cointegration ~the x 2 ~ p! distribution! are oversized ~asymptotically! under near cointegration+ hat is, lim Pr~G t! Pr V p D du l r` 2 t Pr V D p du 0 2 t Pr~x 2 ~p! t! for all t 0 whenever l 0+ For concreteness, consider the case where m d, F b I mx, and f b b 0 + In other words, consider the null hypothesis 0 : b b 0 in a regression of y t on x t and a constant+ o illustrate the magnitude of the size distortions encountered under near cointegration, we have simulated the limiting distribution of G for m x,+++,4 and for various values of l+ Specifically, we have made 20,000 draws from the distribution of the discrete approximations ~using 2,000 steps! to the limiting random variables+ Figure plots the rejection frequencies corresponding to a test with a nominal size of 5%+ he evidence presented in Figure suggests that severe size distortions can occur if conventional cointegration methods are being used when the series are nearly cointegrated rather than exactly cointegrated+ In fact, the size increases dramatically as ~the absolute value of!lincreases from 0, and substantial size distortions are encountered even for values of l in the range 5 0+ Whether or not this is a problem obviously depends on whether or not researchers can be expected to be able to detect such departures from exact cointegration+ It is therefore of interest to know whether or not tests for cointegration can be expected to reject the null hypothesis of cointegration when l is equal to 0, say+ A partial answer to this question is provided in the next section, where we illustrate how to obtain the local asymptotic power functions of tests for cointegration+ 5. LOCAL ASYMPOIC POWER OF COINEGRAION ESS During the last decade, numerous cointegration tests taking cointegration as the null hypothesis have been proposed+ hese test procedures utilize different properties of cointegrated systems, and it therefore seems desirable to investigate what, if anything, can be said about the power properties of the different tests+ In this section, we characterize the behavior of five regression based cointegration tests under local alternatives+ 5 Moreover, we obtain the correspond-

10 38 MICAEL JANSSON AND NIELS ALDRUP Figure. Rejection rates for G + Nominal size is 5%+ ing local asymptotic power functions and use these to address the following questions+ ~i! Does any one of these tests dominate the others in terms of local asymptotic power? ~ii! Can cointegration tests be expected to detect those departures from cointegration that seriously distort the size of conventional cointegration procedures ~cf+ Section 4!? he variable addition test proposed by Park ~990! is computed as follows+ Let k and k 2 be arbitrary nonnegative integers such that k k k 2 and for t,+++,, let r t ~t m d,+++,t m d k! ~if k! and ~if k 2! let $r 2t % be a k 2 -dimensional computer generated random walk such that $Dr 2t %;i+i+d+ N~0,I k2!+ 6 Finally, let r t ~r t, r 2t!+ Based on the multiple regressions ~6! and y t a] d t b\ x t g] r t u] t ~t,+++,!, (8) construct the statistic J ~k, k 2! v[ ~ u[ t! 2 ( ~ u] t! 2 #+ his is simply the Wald test used to test the significance of the regressor r t in ~8!+ As

11 [ O X [ X O X a consequence, J ~k, k 2! r d x 2 ~k! under the null hypothesis of cointegration ~Park, 990!+ Several cointegration tests based on partial score sums have been proposed+ We consider the test due to Shin ~994!+ 7 hat test is based on CI v uu+x 2 ( S 2 t t, where S t ( u s + 8 Evidently, CI is constructed by applying the stationarity test proposed by Kwiatkowski, Phillips, Schmidt, and Shin ~992! to the residuals $ u[ t % from ~6!+ Cointegration tests also can be based on the residuals $ SX t % from the multiple regression S y t ay S d t bx S x t SX t, NEARLY COINEGRAED IME SERIES 39 where S y t t ( y s, S d t t ( d s, and S x t t ( x s for t + Choi and Ahn ~995! propose the statistics LM v[ uu+x ~ ( t 2 S, DSX t g[ uu+x!# 2, II LM ~ v[ uu+x { 2 2 ( t 2 S,! {LM I I, and SBD v[ uu+x 2 ( S 2 t, where g[ uu+x ~, v[ xu V xx!~ G ww S ww!~, v[ xu V xx! + hese tests are intimately related to the stationarity tests of Choi and Ahn ~998!+ EOREM 4+ Suppose $z t % is generated by () and suppose A A4 hold. hen J ~k, k 2! r d R Q du l 2, CI r d U 2 l, SBD I 2 r d U l, QO, LM I r d U l, QO du l, O LM II r d U l, QO du l, O 2 U l, QO Q 2, Q 2, where R Q ~r! ~* R Q R Q! 02 R Q ~r!,r Q ~r! R~r! ~* RQ!~*QQ! Q~r!, Q ~D,V!, R ~R,R 2!, R ~r! ~r m d,+++,r m d k!, R 2 is a k 2 - dimensional Wiener process independent of Q and U l, U l ~r! U l ~r! ~*QdU l! ~*QQ! Q~r!, O and U l, Q O ~r! U l ~r! ~* QU O l! ~*QQ! Q~r!, Q~r! * r 0 Q~s! ds, whereas D, V, and U l are defined as in Lemma.

12 320 MICAEL JANSSON AND NIELS ALDRUP Figure 2. Local power of tests for cointegration; m d, m x + An expression equivalent to the representation of the limiting distribution of CI is obtained by anaka ~996, heorem +!+ 9 o obtain local asymptotic power functions, we have simulated ~the discrete time counterparts of! the limiting distributions of the J ~2,2!, 20 CI, LM I, LM II, and SBD I test statistics in the case where m d and m x + 2 As in Section 4, we have used 2,000 steps and have repeated the procedure 20,000 times+ Figure 2 shows the local asymptotic power functions of tests with size 5%+ he local asymptotic power properties of J ~2,2!, CI, and SBD I are very similar, whereas LM II and ~in particular! LM I are remarkably inferior in terms of local asymptotic power+ Because the local asymptotic power properties of J ~2,2!, CI, and SBD I are almost indistinguishable, our tentative conclusion is that the choice among these tests should be guided by finite sample considerations concerning size distortions+ Remark + Notice that LM II ~SBD I! {LM I o p ~!+ Under fixed alternatives ~i+e+, under spurious regression!, LM diverges at a faster rate than I SBD I II, and a test based on LM is therefore consistent ~Choi and Ahn, I I 995!+ In contrast, because both LM and SBD are O p ~! under near cointegration, LM II might be expected to have rather disastrous local asymptotic

13 NEARLY COINEGRAED IME SERIES 32 I power properties if the local asymptotic power of SBD is higher than the local asymptotic power of LM I + Figure 2 confirms this conjecture+ More generally, our findings illustrate the obvious, but important, point that the ~local asymptotic! power properties of a test cannot be deduced from the rate of divergence under fixed alternatives+ In the present example, e+g+, LM I and I I SBD diverge at the same rate under fixed alternatives and LM diverges faster than both of these ~Choi and Ahn, 995!+ Evidently, Figure 2 tells an entirely different story+ Remark 2+ he local asymptotic power properties of the tests depend solely on l+ In particular, our distributional results do not depend on the particular estimator used to estimate nuisance parameters such as v uu+x + In fact, the asymptotic results are the same as if these nuisance parameters were known+ As pointed out by a referee, this is somewhat unfortunate, because there is ample ~simulation! evidence documenting that the finite sample size properties of tests can be very sensitive to the choice of nuisance parameter estimation method ~see McCabe, Leybourne, and Shin, 997, and references therein!+ We share this view and encourage the reader to interpret the local asymptotic power curves presented here as approximations to the finite sample size-adjusted power curves ~as opposed to the true power curves! of the corresponding tests+ In the previous section, we argued that Wald tests based on conventional cointegration methods can encounter severe size distortions when the series are nearly cointegrated and l exceeds 5+ On the other hand, the evidence presented in Figure 2 indicates that even when l 0 the power of the tests for cointegration can be well below 50%+ his suggests that even if the departure from ~exact! cointegration is substantial ~in the sense that it severely affects the size of the conventional tests!, tests for cointegration cannot be expected to detect such departures very frequently+ herefore, whenever a researcher rejects a structural hypothesis ~on the coefficient b! using cointegration methods, the result should be interpreted carefully+ Indeed, it might be the case that the structural hypothesis is correct, whereas the ~possibly auxiliary! assumption of cointegration is not+ his of course leaves open the question of how to interpret the coefficient vector in a noncointegrated system, a question that we shall not attempt to answer here CONCLUDING REMARKS Based on a new representation, a notion of near cointegration was proposed+ he notion of near cointegration was used to generalize several existing results from the cointegration literature to the case of near cointegration+ hroughout, we have deliberately studied the properties of known inference procedures under near cointegration rather than proposed new methods+ As a result, several extensions are possible+ For instance, a companion paper by one of us ~Jansson, 200c! takes the analysis of Section 5 one step further and uses a model of

14 O D O D D 322 MICAEL JANSSON AND NIELS ALDRUP near cointegration to propose a new cointegration test with ~essentially! optimal local asymptotic power properties+ NOES + Earlier, Yule ~926! used the term nonsense correlation to describe a similar phenomenon+ 2+ In the aforementioned papers, r 2 is computed from a long-run covariance matrix that is itself defined by taking limits as r `+ herefore, it is not immediately obvious how to model r 2 as a sequence of parameters that lie in a shrinking neighborhood of unity as increases without bound+ By working with a representation where r 2 is a primitive parameter, we circumvent this potential problem+ 3+ A previous version of this article ~Jansson and aldrup, 2000! contains a detailed study of the F-statistic+ ` 4+ C ~L! C ~! CD ~L!~ L!, where C ~L! ( i 0 C i L i is a lag polynomial with coefficients C i ( j i C j + hese coefficients satisfy ( ` i 0 i7cd i 7 ` ~as required under As- ` sumption A, which appears later in this section! if ( ` i 0 i 2 7C i 7 `+ ` 5+ Suppose C~L! O ( i 0 C i L i is a matrix lag polynomial and suppose $ es t : t % is i+i+d+ with E~ es t! 0 and E~ es t es t! OS 0+ We can construct an orthogonal matrix O such that C~! OS 02 O is upper triangular ~with nonnegative diagonal elements!+ Given any such O, define C~L! C~L! O OS 02 O and e t O S 02 es t + By construction, C~! is upper triangular and $e t % is i+i+d+ with E~e t! 0 and E~e t e t! O OS 02 OS OS 02 O I m + 6+ For, let D ~L! r b 0 I C ~L!+ mx It is not hard to show that C u ~!~,0! 0 holds under the identification0invertibility condition inf$6z6:6d ~z!6 0%,l Notable exceptions include Park and Phillips ~989, Sec+ 5+2!, Choi ~994!, and McCabe et al+ ~997!+ See also Phillips ~995! and Chang and Phillips ~995!+ 8+ Indeed, {C x ~!d ~r! v xy O~! under A+ 9+ Alternative conditions of near cointegration have appeared in Quintos and Phillips ~993, Sec+ 5! and Phillips ~998a, p+ 025!+ he ~multivariate extension of the! notion of near cointegration introduced by Quintos and Phillips ~993! is more general than the notion suggested in the present paper+ On the other hand, the notion of near cointegration discussed in Phillips ~998a! is fundamentally different from ours, because the series $h y t % generated by equation ~5! of that paper is nearly integrated+ 0+ Details concerning the derivation of the triangular form of anaka s models are available from the authors upon request+ + Alternative estimators with identical asymptotic properties include the estimators proposed by Johansen ~988, 99!, Phillips ~99!, Phillips and ansen ~990!, Saikkonen ~99, 992!, and Stock and Watson ~993!+ 2+ For convenience, we do not make the dependence of S ww, V ww, and G ww on and b explicit+ 3+ Using anaka s ~993! notation, the representation of the limiting distribution of ~ bd j b! in heorem 6 of that paper should read ~A W j W j d A V j ~g A ~A A! A g!# + Upon subtraction of ~A A! A d, we arrive at the representation ~A W j W j ~d A ~A A! A d! A V j ~g A ~A A! A g!#,

15 K K K D K K K K K NEARLY COINEGRAED IME SERIES 323 which is equivalent to the result in Lemma 2+ he difference in the location parameter is due to the fact that anaka ~993, p+ 49! defines the population value of the regression coefficient as b ~A A! ~A A 2 A d0!, whereas our b equals ~A A! A A 2 in anaka s ~993! notation+ 4+ It is a simple matter to generalize heorem 3 to the case of nonlinear hypotheses+ o conserve space, we shall not do so here+ 5+ arris ~997! and Snell ~998! propose tests for cointegration that utilize principal component methods+ hese tests are not considered here+ 6+ his particular choice of superfluous regressors is advocated by Park ~990!+ On the other hand, little guidance on the optimal choice of k and k 2 is provided although Remark c of the paper suggests that k k 2 2 is preferable+ 7+ Closely related tests have been proposed by ansen ~992!, arris and Inder ~994!, Kuo ~998!, Leybourne and McCabe ~993!, McCabe et al+ ~997!, Quintos and Phillips ~993!, and anaka ~996!+ In Jansson and aldrup ~2000!, we also study ansen s L c test ~992!+ he local asymptotic power properties of that test are very similar to those of Shin s test ~994!, as are the local asymptotic power properties of the test due to Xiao ~999! ~Jansson, 200a!+ 8+ In its original formulation, Shin s test ~994! uses Saikkonen s estimator ~99!+ he formulation based on the CCR estimator is due to Choi and Ahn ~995!+ 9+ o see the equivalence, notice that rows through q in the expression 0 t w~s! ds w~s! wk 0 ~s! w ~s! wk ds 0 ~s! w ~s! ds ds 0 t in heorem + of anaka ~996! are identically zero+ As a consequence, the limiting distribution of anaka s S 2 ~996! depends on the vector c~j 2, J 3!0~g~!J 3! only through the scalar c0g~!+ Indeed, the variate c 2 ~t! appearing in the statement of anaka ~996! has the following simple representation: t t c w 2 ~s!ds w 2 ~s! wk g~! 0 0 ~s! w ~s! wk ds 0 ~s! w ~s! ds ds hat is, r t ~t, t 2! and r 2t is a two-dimensional random walk in ~8!+ Changing the values of k and k 2 does not seem to affect the local power of the J test much+ 2+ Results for 2 m x 4 are qualitatively similar and can be found in Jansson and aldrup ~2000!+ 22+ For recent contributions to this discussion, see Phillips ~998! and Phillips and Moon ~999!+ REFERENCES Beveridge, S+ &C+R+Nelson ~98! A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle+ Journal of Monetary Economics 7, Canjels, E+ &M+W+Watson ~997! Estimating deterministic trend in the presence of serially correlated errors+ Review of Economics and Statistics 79, Chang, Y+ &P+C+B+Phillips ~995! ime series regression with mixtures of integrated processes+ Econometric heory, Choi, I+ ~994! Spurious regressions and residual-based tests for cointegration when regressors are cointegrated+ Journal of Econometrics 60, Choi, I+ &B+C+Ahn ~995! esting for cointegration in a system of equations+ Econometric heory, Choi, I+ &B+C+Ahn ~998! esting the null of stationarity for multiple time series+ Journal of Econometrics 88, 4 77+

16 324 MICAEL JANSSON AND NIELS ALDRUP Elliott, G+ ~998! On the robustness of cointegration methods when regressors almost have unit roots+ Econometrica 66, Granger, C+W+J+ &P+Newbold ~974! Spurious regressions in econometrics+ Journal of Econometrics 2, 20+ all, P+ &C+C+eyde ~980! Martingale Limit heory and Its Application+ New York: Academic Press+ ansen, B+E+ ~992! ests for parameter instability in regressions with I~! processes+ Journal of Business and Economic Statistics 0, arris, D+ ~997! Principal components analysis of cointegrated time series+ Econometric heory 3, arris, D+ &B+Inder ~994! A test of the null hypothesis of cointegration+ In C+ argreaves ~ed+!, Nonstationary ime Series Analysis and Cointegration, pp Oxford: Oxford University Press+ Jansson, M+ ~200a! Another est of the Null ypothesis of Cointegration+ Manuscript, University of California, Berkeley+ Jansson, M+ ~200b! Consistent covariance matrix estimation for linear processes+ Econometric heory, forthcoming+ Jansson, M+ ~200c! Point Optimal ests of the Null ypothesis of Cointegration+ Manuscript, University of California, Berkeley+ Jansson, M+ &N+aldrup ~2000! Spurious Regression, Cointegration, and Near Cointegration: A Unifying Approach+ Department of Economics Working paper , University of California, San Diego+ Johansen, S+ ~988! Statistical analysis of cointegration vectors+ Journal of Economic Dynamics and Control 2, Johansen, S+ ~99! Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models+ Econometrica 59, Kuo, B+-S+ ~998! est for partial parameter instability in regressions with I~! processes+ Journal of Econometrics 86, Kwiatkowski, D+, P+C+B+ Phillips, P+ Schmidt, &Y+Shin ~992! esting the null hypothesis of stationarity against the alternative of a unit root: ow sure are we that economic time series have a unit root? Journal of Econometrics 54, Leybourne, S+J+ &B+P+M+McCabe ~993! A simple test for cointegration+ Oxford Bulletin of Economics and Statistics 55, McCabe, B+P+M+, S+J+ Leybourne, &Y+Shin ~997! A parametric approach to testing the null of cointegration+ Journal of ime Series Analysis 8, Park, J+Y+ ~990! esting for unit roots and cointegration by variable addition+ In +B+ Fomby & G+F+ Rhodes, Jr+ ~eds+!, Advances in Econometrics: Co-integration, Spurious Regression, and Unit Roots, pp Greenwich, Connecticut: JAI Press+ Park, J+Y+ ~992! Canonical cointegrating regressions+ Econometrica 60, Park, J+Y+, S+ Ouliaris, &B+Choi ~988! Spurious Regressions and ests for Cointegration+ CAE Working paper 88-07, Cornell University+ Park, J+Y+ &P+C+B+Phillips ~989! Statistical inference in regressions with integrated processes, part 2+ Econometric heory 5, Phillips, P+C+B+ ~986! Understanding spurious regressions in econometrics+ Journal of Econometrics 33, Phillips, P+C+B+ ~988a! Regression theory for near-integrated time series+ Econometrica 56, Phillips, P+C+B+ ~988b! Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations+ Econometric heory 4, Phillips, P+C+B+ ~99! Spectral regression for cointegrated time series+ In W+A+ Barnett, J+ Powell, &G+E+auchen ~eds+!, Nonparametric and Semiparametric Methods in Economics and Statistics, pp Cambridge: Cambridge University Press+

17 NEARLY COINEGRAED IME SERIES 325 Phillips, P+C+B+ ~995! Fully modified least squares and vector autoregression+ Econometrica 63, Phillips, P+C+B+ ~998! New tools for understanding spurious regressions+ Econometrica 66, Phillips, P+C+B+ &S+N+Durlauf ~986! Multiple time series regression with integrated processes+ Review of Economic Studies 53, Phillips, P+C+B+ &B+E+ansen ~990! Statistical inference in instrumental variables regression with I~! variables+ Review of Economic Studies 57, Phillips, P+C+B+ &+R+Moon ~999! Linear regression limit theory for nonstationary panel data+ Econometrica 67, Phillips, P+C+B+ &V+Solo ~992! Asymptotics for linear processes+ Annals of Statistics 20, Quintos, C+E+ &P+C+B+Phillips ~993! Parameter constancy in cointegrating regressions+ Empirical Economics 8, Saikkonen, P+ ~99! Asymptotically efficient estimation of cointegration regressions+ Econometric heory 7, 2+ Saikkonen, P+ ~992! Estimation and testing of cointegrated systems by an autoregressive approximation+ Econometric heory 8, 27+ Shin, Y+ ~994! A residual-based test of the null of cointegration against the alternative of no cointegration+ Econometric heory 0, 9 5+ Snell, A+ ~998! esting for r versus r cointegrating vectors+ Journal of Econometrics 88, 5 9+ Stock, J++ &M+W+Watson ~993! A simple estimator of cointegrating vectors in higher order integrated systems+ Econometrica 6, anaka, K+ ~993! An alternative approach to the asymptotic theory of spurious regression, cointegration, and near cointegration+ Econometric heory 9, anaka, K+ ~996! ime Series Analysis: Nonstationary and Noninvertible Distribution heory+ New York: Wiley+ Xiao, + ~999! A residual based test for the null hypothesis of cointegration+ Economics Letters 64, Yule, G+U+ ~926! Why do we sometimes get nonsense-correlations between time-series? A study in sampling and the nature of time-series+ Journal of the Royal Statistical Society 89, 63+ APPENDIX his Appendix outlines the proofs of the main results of the paper+ o facilitate the proofs, we start with two preliminary lemmas+ LEMMA 5+ Suppose $z t % is generated by () and suppose A A4 hold. hen V ww r p V ww, G ww r p G ww, and S ww r p S ww. For t, let q t ~d t, x t!, v t y t b x t,q t ~d t, x t!, and v t y t b x t. Moreover, let Y diag~ md 02,+++,t m d k 02,{i k2!, where i k2 is a k 2 -vector of ones. LEMMA 6+ Suppose $z t % is generated by () and suppose A A3 hold. hen (a) 02 C q {{} r d Q x ~{!, (b) C ( q t v t r d v 02 uu+x *Q x du l *Q x dx V xx v xu ~0, g ux!,

18 [ S 326 MICAEL JANSSON AND NIELS ALDRUP where {{} denotes the integer part of the argument. Moreover, if A4 holds, then (c) 02 C q,{{} r d Q x ~{!, (d) 02 ( {{} v t r d v 02 uu+x U l ~{!, (e) C ( q t v t r d v 02 uu+x *Q x du l, (f) ( t 2 ~( v s!v t r d v uu+x *U l du l g uu+x (g) 02 Y r {{} r d R~{!, (h) Y ( r t v t r d v 02 uu+x *RdU l, where g uu+x ~, v xu are defined as in the text. V xx!~g ww S ww!~, v xu, V xx!, whereas C, Q x, U l, X, and R Proof of Lemma 5. For t, let u[ * t C u ~L!e t a[ d t ~ b b! x t and u ** t d j t + Because u[ t u[ * t u[ ** t, we can write w[ t as the sum of w[ * t ~ u[ * t, Dx t! and w[ ** t ~ u[ ** t,0! + Using notation typified by t 6t s6 G *,** ww ( ( k b w[ * t w[ ** s, the corresponding decomposition of G ww is G ww G *,* ww G **,** ww G *,** ww G **,* ww + Now, G *,* ww r p G ww by Corollary 4 of Jansson ~200b!, which is applicable because A5~i! of that paper holds by Lemma and Lemma 6~a! of the present paper+ o show G ww r p G ww it therefore suffices to show that G **,** ww, G *,** ww, and G **,* ww are o p ~!+ he idea of the proof is the following+ Each of the matrices G **,** ww, G *,** **,* ww, and G ww can be written as ( i 0 k~ b i!m i, where $M i : 0 i ; % is a triangular array of random matrices+ he event $7 ( i 0 k~ b i!m i 7 «% is a subset of $ ta, a#% S sup ta a S a ( k i 0 i a{b M i «+ Under A4, inf 0 Pr~ ta, a#! S can be made arbitrarily close to zero for sufficiently large 0 and appropriately selected 0 ta a `+ We therefore have ( i 0 k~ b i!m i o p ~! if sup ta a S a ( k i 0 i a{b M i o p * ~! (A.) for every 0 ta as `, where o p *~! denotes convergence to zero in outer probability+ ~o avoid measurability complications, we consider convergence in outer probability rather than convergence in probability+! he proof proceeds by applying additive decompositions to G **,** ww, G *,** ww, and G **,* ww and establishing ~A+! for each element of these decompositions+ In each instance, we make use of the fact that sup ta a as k i a{b m i o p *~! ( i 0 if $m i : 0 i ; % is a triangular array of nonnegative random variables with max 0 i E~m i! O~ 02!+

19 O S S [ S Pr * sup ta a as Indeed, by Markov s inequality and the properties of Pr *, E *, and k~{!, O we have k i ( i 0 * sup «E ta a as «E 02 ( «02 ( O i 0 02 max «a{b m i «02 ( i 0 k i k i i 0 ab 02 m i k i ab 02 max E~m i! 0 i a{b 02 m i 02 ( O i 0 k i ab 0 j E~m j! o~! for any «0, where Pr * ~{! and E * ~{! denote outer probability and outer expectation, respectively, and the last equality uses the assumption on max 0 i E~m i! along with the fact that 02 ( i 0 k~~ O ab S! i! o~! under A5 ~Jansson, 200b!+ Defining i t s and applying the decomposition w[ ** t w[ ** s ~ w[ ** t w[ ** **,** s!, G ww can be written as **,** ~ a[! **,** ~ a[!, where G **,** ww, ~a! ( G ww,2 G ww, i 0 **,** ~a! ( i 0 G ww,2 k i i a{b ( k i i a{b ( w ** s w[ ** s, ** ~ w[, s i w[ ** s! w[ ** s + By subadditivity of 7{7, 7G **,** ww, ~a!7 ( i 0 6k~~a{b! i!6{( i 7 w[ ** s w[ ** s 7, so G **,** ww, ~ a[! o p ~! if max 0 i E~( i 7 w[ ** s w[ ** s 7! O~ 02!+ Using A2 and the relation 7d 7 O~!, we have i E ( 7 w[ ** s w[ ** s 7 E ( 7 w[ ** s w[ ** s 7 7d 7 2 ( E~j s j s! max 0 i NEARLY COINEGRAED IME SERIES 327 7d 7 2 ( m{s O~!+ Next, 7G **,** ww,2 ~a!7 ( i 0 6k~~a{b! i!6{7( i ** ~ w[, s i w[ ** s! w[ ** s 7+ herefore, G **,** ww,2 ~ a[! o p ~! if max 0 i E7( i ** ~ w[, s i w[ ** s! w[ ** s 7 O~ 02!+ By the law of iterated expectations and A2, s i j s! ~j t i j t!j s j t # E~E@~j s i j s! ~j t i j t!8g max~s, t! #j s j t! E@mmax~i 6t s6,0!j s j t # m 2 max~i 6t s6,0!min~t, s! m 2 i$6t s6 i%s

20 328 MICAEL JANSSON AND NIELS ALDRUP for any s, t and i 0, where G t s~e s :s t! for any t and ${% is the indicator function+ Using this relation, the Cauchy Schwarz inequality, and 7d 7 O~!, i E ( ** ** 2 ~ w[, s i w[ ** s! w[ s i E ( ** ~ w[, s i w[ ** s! w[ s ** 2 i i ( ( ** w[, s i w[ ** s! ** ~ w[,t i w[ ** ** t! w[ s w[ ** t # i i ( ( m 2 7d 7 4 i$6t s6 i%s i ( m 2 7d 7 4 2i 2 s m 2 7d 7 4 i 2 ~ i!~ i! m 2 7d O~! for any 0 i + herefore, G **,** ww,2 ~ a[! o p ~! and G **,** ww o p ~!, as was to be shown+ Next, consider G *,** ww, which can be written as G *,** ww, ~ a[! G *,** ww,2 ~ a[!, where G *,** ww, ~a! ( G ww,2 i 0 *,** ~a! ( Because * w[, s i w s i i 0 k i i a{b ( k i i a{b ( a[ 0 b b 0 * ~ w[, s i w s i! w[ ** s, w s i w[ ** s + C 02 C d s i x s i 02 and 7AB7 7A7{7B7 for conformable A and B, an upper bound on 7G *,** ww, ~a!7 is given by C a[ b b ~ 7d 7! ( i 0 k i a{b ( i 02 C d x s i 7 02 j s 7+ By Lemma and 7d 7 O~!, 7C ~[ a,~ b b!! 7~7d 7! O p ~!, so the following condition is sufficient for *,** ~ a[! o p ~!: max 0 i i ( G ww, E~7 02 C ~d s i, x s i! j s 7! O~ 02!+

21 By A2, max t E~7 02 j t 7 2! E~7 02 j 7 2! m+ Moreover, it can be shown that max t E~ 02 C ~d t x t! 2! O ~!+ Using these relations and the Cauchy Schwarz inequality, the proof of G *,** ww, ~ a[! o p ~! is completed as follows: i 02 C d s i max 0 i E ( max 0 i i x s i 7 02 j s 7 ( ME~7 02 j s 7 2!ME~7 02 C ~d s i M max E~7 02 j t 7 2 t!m, x s i! 7 2! max E~7 02 C ~d t, x t! 7 2! O~!+ t Let ( ` k 0 C w k L k ~C u ~L!,C x ~L!! + Because w[ ** t d ( ` l 0 $l t %e t l and w t ( ` k 0 C w k e t k, G *,** 3 *,** ww,2 ~ a[! can be written as ( j G ww,2, j ~ a[!, where G ww,2, j *,** ~a! ( with M, i, *,** ( M,i,2 *,** ( M,i,3 *,** ( ` ` k 0 l 0 ( i 0 C kw ( ` i C w l i ( l 0 ` i C w l i ( l 0 k~~a{b! i!m *,**, i, j, j 3, i e s i k e s l $l s %$k l i%~d,0!, ~e s l e s l I m!$l s %~d,0!, I m $l s %~d,0!+ By the law of iterated expectations and A2, i E ( 2 ( e s i k e s l i 2 ( i ( s l t l i i ( s l t l $k l i % 2 ( 2 ( i m 2 s l $l s %$k l i % 2 E ~e s i k e t i k e s l e t l $k l i %! s i k e t i k e s l e t l $k l i%8g max~s, t! max~i k, l!!# i s l m 2 $l % NEARLY COINEGRAED IME SERIES 329 E~e s i k e s i k!e~e s l e s l!

22 330 MICAEL JANSSON AND NIELS ALDRUP *,** for any i, l 0+ he conclusion G ww,2, ~ a[! o p ~! now follows because max E~7M *,**, i, 7! 0 i 7d 7 ( ` ` k 0 l 0 ( ` 7C k w 7 max 0 i i E ( 7d 7 ( 7C w k 7 ( m 02 $l % k 0 ` l 0 ` ~ 7d 7!m 02 ( 7C w k 7 O~ 02!, k 0 e s i k e s l $l s %$k l i % where the second inequality uses the Cauchy Schwarz inequality and the previous display, whereas the last equality uses 7d 7 O~! and the fact that ( ` k 0 7C w k 7 ` under A+ Next, 7M *,**,i,2 7 7d 7( ` l 0 7C w l i 77 ( i ~e s l e s l I m!$l s %7 for any i 0 and i E ( ~e s l e s l i l I m!$l s % E ( ~e s e s I m! for any i, l 0+ By A2, $vec~e t e t I m! : t % is a uniformly integrable martingale difference sequence, so sup max l 0 0 i sup l 0 i E ( l E ( ~e s l e s l I m!$l s % ~e s e s I m! o ~! by an argument analogous to the proof of heorem 2+22 of all and eyde ~980!+ In particular, max E~7M *,**, i,2 7! 0 i 7d 7 ( 7d 7 sup l 0 o ~!, ` 7C w l i 7 max l 0 0 i max 0 i i E ( i E ( ~e s l e s l ~e s l e s l I m!$l s % ` I m!$l s % ( l 0 7C l w 7

23 *,** *,** so G ww,2,2 ~ a[! o p ~!+ Finally, G ww,2,3 ~ a[! o~! because ` i max 7M *,**, i,3 7 7d 7 max 7C 0 i 0 i ( w l i 7 ( $l s % l 0 7d 7 max 0 i ( ` ` 7C l i l 0 7d 7 ( 7C w l 7 O~!+ l 0 NEARLY COINEGRAED IME SERIES 33 w 7 he proof of G **,* ww o p ~! is analogous to that of G *,** ww o p ~! and is omitted to conserve space+ he proof of S ww r p S ww is a special case ~with S 0 i 0 replaced by S i 0 throughout! of the proof of G ww r p G ww + Finally, V ww G ww G ww S ww is consistent for V ww G ww G ww S ww, because G ww r p G ww and S ww r p S ww + Proof of Lemma 6. Let $w t %, $j t %, and $r 2t % be defined as in the text+ By a multivariate analogue of Phillips and Solo ~992, heorem 3+4!, we have {{} w t ( 0 I 02 m 0 V~{! j 0 I k2 U~{! R 02 2 ~{!, r 2,{{} rd Vww (A.2) where U, V, and R 2 are independent Wiener processes of dimension, m x, and k 2, respectively, and V 02 ww v uu+x v xu V xx 02 0 V xx + Moreover, ( t ( ( r 2t w t r d 0 w s w t r d V ww 02 0 R 2 d by Phillips ~988b!, whereas V V U d U V 02 ww G ww, (A.3) V U V 02 ww, (A.4) ( w t w t r p S ww (A.5) in view of the law of large numbers+ Part ~a! is standard+ For t, v t u t d j t, where u t, d, and j t are defined as in Section 2+ By ~0!, ~!, and part ~a!, C ( q t u t r d v uu+x 02 Q x du Q x dx V xx v xu ~0, g ux! +

24 332 MICAEL JANSSON AND NIELS ALDRUP Moreover, C ( q t j t d r d lv uu+x 02 Q x U, by ~A+2!, part ~a!, the relation lim r` d ~lv 02 uu+x,0!, and the continuous mapping theorem ~CM!+ Because *Q x du l *Q x du l*q x U, part ~b! is obtained by combining the preceding displays+ Part ~c! is standard+ Define q t ~d t, x t! and v t u t d j t, where x t x t G {x S ww w t and u t u t v xu V xx Dx t ~, v xu V xx!w t + Let ~c! ~h! denote the counterparts of parts ~c! ~h! in which q t and v t are replaced with q t and v t, respectively+ Part ~c! follows from part ~a!+ By ~0!, lim r` d ~lv 02 uu+x,0!, and CM, {{} 02 ( u t 02 ( {{} {{} 02 ( d j t ~d! 302 ( ~, v xu V xx!w t r d v 02 uu+x U~{!, {{} j t r d v 02 uu+x l 0 Combining these results, part ~d! is obtained+ Next, ~diag~ 02,+++, m d 02!! ( d t v t r d v uu+x 02 DdU l by ~a!, ~d!, and CM, whereas ( x t u t ( x t w t G {x S ww ( and ( x t j t d V xx 2 ( { v xu r d v uu+x 02 XdU U~t! dt+ w t w t x t j t ~d! r d v 02 uu+x l XU in view of ~A+3!, ~A+5!, ~c!, ~A+2!, and CM+ Part ~e! is established by using these results and the relation C ( q t v t ~diag~ 02,+++, md 02!! ( d t v t + ( x t u t ( x t j t d

25 [ NEARLY COINEGRAED IME SERIES 333 Because ( t 2 ( ( v s v t t 2 ( u s u t d ( t 2 ( j s u t ( t 2 ( u s j t d, part ~f! can be obtained by combining the following results, each of which is obtained in standard fashion using ~A+2!, ~A+3!, ~A+5!, and CM: ( t 2 ( u s u t ( t 2 t ( u s ut ( u t u t t 2 r d v uu+x UdU g uu+x, d ( ( t 2 t 2 ( ( j s ut r d v uu+x l UdU, P u s j t d r d v uu+x l U l U, where U~r! P * r 0 U~t! dt+ Parts ~g! and ~h! are proved in the same way as ~a! and ~e!, respectively+ Now, x t x t ~ G {x S ww G {x S ww!w t G {x S ww ~ w[ t w t! and v t v t ~ b b! G {x S ww w[ t ~ [ v xu V xx v xu V xx!dx t + By Lemma 5, G {x S ww r p G {x S ww and v[ xu V xx r p v xu V xx + Furthermore, b b O p ~! and max t 7 w[ t w t 7 O p ~ 02! by Lemma, the proof of which only uses ~a! ~b! of the present lemma+ Using these facts, the proof of ~d! is completed as follows: {{} 02 ( ~v t v t! {{} ~ b b! G {x S ww 02 ( w t ~ [ v xu V xx v xu V xx! 02 x {{} r d 0+ Analogously, the proof of ~e! ~h! can be completed by using elementary manipulations to show that C ( ~q t v t q t v t! r d 0, ( t 2 ~( v s!v t ( t 2 ~( r d 0, etc+ v s!v t