Inference in Cointegrating Regression with Time

Inference in Cointegrating Regression with Time Series whose Roots are in the Vicinity of Unity.1 Peter C. B. Phillips Cowles Foundation for Research in Economics Yale University and University of Auckland & University of York, UK Tassos Magdalinos University of Nottingham, UK March 31, 2009

1 Phillips

acknowledges partial research support from a Kelly Fellowship and the NSF under Grant No. SES-04-142254.

Abstract Present econometric methodology of inference in cointegrating regression is extended to mildly integrated of the type introduced by Magdalinos and Phillips (2009). Conventional approaches to estimating cointegrating regressions fail to produce even asymptotically valid inference procedures when the regressors are nearly integrated, and substantial size distortions can occur in econometric testing. The new framework enables a general approach to inference that resolves this di¢ culty and is robust to the persistence characteristics of the regressors, making it suitable for general practical application. Mildly integrated instruments are employed and the resulting IV approach eliminates the endogeneity problems of conventional cointegration methods and produces robust inference. Keywords: Central limit theory, Cointegration, Robustness, Instrumentation, Mild integration, Mixed normality. JEL classi…cation: C22

1. Introduction For the last two decades, autoregressive models with roots near unity have played an important role in time series econometrics, resulting in a vast literature of theory and applications. Theoretical developments have engaged econometricians, probabilists, and statisticians, and the empirical applications extend well beyond economics and …nance into other social and business sciences like mass media communications, political science, and marketing. The theoretical work makes extensive use of functional laws for partial sums to Brownian motion, functional laws of weighted partial sums to linear di¤usions, mapping theorems for functionals of these processes, and weak convergence of discrete martingales to stochastic integrals and certain nonlinear functionals to Brownian local time. This theory provides a foundation for econometric estimation, testing, power curve evaluation and con…dence interval construction for (unit root) nonstationary time series, certain nonlinear nonstationary models and nonstationary panels. Much of the limit theory for nonstationary time series involves processes with autoregressive roots that are local to unity of the form = 1 + c=n; where n is the sample size and c is a constant. Deviations from unity of this particular form are mathematically convenient and help in characterizing local asymptotics, power functions (Phillips, 1987), point optimal asymptotic tests (Elliott, Rothenberg, Stock, 1996) and con…dence interval construction. But the speci…c O (n 1 ) rate of approach to unity has no intrinsic signi…cance or economic meaning. To accommodate greater deviations from unity within this framework, we may allow the parameter c to be large or even consider limits as c ! 1; as was done in Phillips (1987) and Chan and Wei (1988). Such analysis produces certain useful insights, but it does not resolve the di¢ culties of the discontinuity in the unit root asymptotics. In particular, it does not bridge the very di¤erent convergence rates of the stationary, unit root/local to unity, and explosive cases. Phillips and Magdalinos (2007) and Magdalinos and Phillips (2009, hereafter MP) recently explored another approach, giving a limit theory for time series with an autoregressive root of the form n = 1 + c=kn ; where (kn )n2N is a deterministic sequence increasing to in…nity at a rate slower than n. Such roots represent moderate deviations from unity in the sense that they belong to larger asymptotic neighborhoods of one than conventional local to unity roots. But in practice, for …nite n the di¤erences need not be large: e.g, when n = 100 and kn = n= ln n; = 0:95 is captured by the value c = 1:086, whereas under the standard local to unity model the same value of is captured with c = 5: When c < 0; time series generated with such roots may be regarded as mildly integrated, and when c > 0 they are mildly explosive. An interesting family of moderate deviations from unity roots occurs when we consider kn = n or n = 1 + c=n , where the exponent lies on (0; 1). The boundary value as ! 1 includes the conventional local to unity case, whereas the boundary value as ! 0 includes the stationary or explosive process, depending on the sign of c. 1

These linkages provided a mechanism for bridging the discontinuities between stationary, local to unity and explosive asymptotics in autoregressions, thereby helping to complete the passage of the limit theory as the autoregressive coe¢ cient moves through unity. Mildly integrated and mildly explosive series may also occur in multivariate regression settings in which certain subsets of the variables may move together over time. Linear regression models with mildly integrated regressors provide a framework for generalizing standard cointegrating regressions to this setting and for linking these regressions to simultaneous equations models with stationary regressors. As Elliott (1998) showed, conventional approaches to estimating cointegrating regressions do not produce valid asymptotic inference in cases where the regressors have autoregressive roots that are local to unity and there can be substantial size distortion in econometric testing. An analogous situation arises when the regressors are mildly integrated. In such cases and recognizing that roots at unity are a special (if important) case in practical applications, there is a need to develop more robust approaches to estimation and inference. The present paper contributes by tackling this problem. We present results that provide a framework of limit theory that can be used to validate inference in cointegrating models with regressors at arbitrary neighbourhoods of unity. We consider cases where the regressors are (i) exactly integrated, (ii) local to unity (in the usual sense of O (n 1 ) departures) and (iii) mildly integrated. An instrumental variables procedure is developed that addresses the need for asymptotically valid inference procedures in cointegrated systems where the regressors have roots in the vicinity of unity, but their precise integration properties are unknown. It is hoped that the resulting theory will resolve many outstanding issues of inference in cointegrated models with roots near unity and also provide some useful new practical tools for time series econometric work.

2. Cointegrated regression with arbitrary degree of persistence As is often emphasized in empirical work, economic time series seem to have autoregressive roots in the general neighborhood of unity and so insistence that roots be at unity in cointegrating regressions is likely to be too harsh a requirement in practice. Matrix cases, where the long-run autoregressive coe¢ cient matrix has the form Rn = I + C=n were considered in Phillips (1988, 1988b) and this theory has been useful in developing power functions for testing problems in cointegrated regressions (Phillips, 1988; Johansen, 1995) and in the analysis of cointegrating regressions for near integrated time series (Elliot, 1998). In this paper, we allow the regressors to have an arbitrary rate of persistence, ranging from unit root processes to processes with roots that lie close to the boundary with stationarity. A convenient general modeling framework for a multivariate cointegrated system 2

with regressors in the viccinity of unity takes the form (1) (2)

yt = Axt + u0t xt = Rn xt 1 + uxt

for each t = 1; :::; n; with some diagonal autoregressive matrix Rn satisfying kRn k 1 and Rn ! IK as n ! 1. The innovations u0t and uxt are correlated linear processes de…ned in Assumption LP below and the system is initialized at some x0 that could be any constant or a random process x0 (n) = op n( ^1)=2 with speci…ed by Assumption N below. Deterministic components may also be included in (1) - (2) and such extensions are easily accommodated, following Park and Phillips (1988). Asymptotic inference for the matrix A of cointegrating coe¢ cients depends on the degree of persistence of the nearly integrated regressor xt , i.e. the rate at which the autoregressive matrix Rn converges to the identity matrix. The e¤ect of changes in persistence on cointegration methods can be categorised by distinguishing between three classes of neighbourhoods of unity. These classes characterise the asymptotic behavior of the regressor in (2) and are listed in the following assumption. Assumption N. The autoregressive matrix in (2) satis…es the following condition: Rn = IK +

C ; for some n

>0

(3)

and some matrix C =diag(c1 ; :::; cK ); with ci 0 for all i 2 f1; :::; Kg. The regressor xt in (2) belongs to one of the following classes: (i) Integrated regressors, if C = 0 or

> 1 in (3).

(ii) Local to unity regressors, if C < 0 and (iii) Mildly integrated regressors, if C < 0 and

= 1 in (3). 2 (0; 1) in (3).

Asymptotic inference in each of the above cases is well documented: see Park and Phillips (1988), Phillips and Hansen (1990), Johansen (1991), Saikkonen (1991) and Stock and Watson (1993) for (i), Phillips (1988) and Elliott (1998) for (ii) and Magdalinos and Phillips (2009) for (iii). The problem is that the validity of the above methods is conditional on correct speci…cation of the degree of persistence of the regressor, i.e. on a priori knowledge that the regressor xt belongs to class (i), (ii) or (iii). Elliott (1998) showed that the presence of local to unity regressors induces a size distortion in testing that can be serious when conventional cointegration estimators are used. A similar situation occurs when conventional cointegration methods are applied in the presence of mildly integrated regressors. This paper addresses the question of how to conduct inference in such ‘cointegrated’systems in a context that is of su¢ cient generality to be useful in practical 3

work. In particular, we derive a Wald test for linear restrictions on the cointegrating matrix A which is robust to the type of persistence of the regressor in (2) and follows a standard chi-square limit distribution. The key step to robustifying inference is the development of an instrumental variables (IV) procedure based on mildly integrated instruments. Given a Kz -vector of mildly integrated instruments zt = Rnz zt

1

(4)

+ uzt

with Kz

K and a known autoregressive matrix Cz Rnz = IKz + ; 2 (0; 1) ; Cz = diag(cz;1 ; :::; cz;Kz ); cz;i < 0 (5) n it is possible to remove the endogeneity that is present in conventional cointegration theory. Such instruments may be constructed from the regressors (see Section 3 below) so the method is feasible given the information contained in (1) and (2). On the other hand, if extra information is available and the instruments in (4) may be constructed exogenously relative to the original system (1)-(2), it is possible to establish a connection between cointegrated and conventional simultaneous equations systems. In order to accommodate for this possibility, it is notationaly convenient to consider the equations (1), (2) and (4) in one system and defer the construction of the instruments in (4) until the next section. The correlation structure of the innovations of the system de…ned by (1), (2) and (4) is provided by the following assumption.

Assumption LP. tation

Let ut = (u00t ; u0xt ; u0zt )0 . For each t 2 N, ut has Wold represenut = F (L) "t =

P1

1 X

Fj "t j ;

j=0

j

1 X j=0

j kFj k < 1;

where F (z) = j=0 Fj z ; F0 = Im+K+Kz ; F (1) has full rank and ("t )t2Z is a sequence of independent and identically distributed (0; ) random vectors satisfying > 0 and the moment condition E k"1 k4 < 1.

Under Assumption LP, the partial sums of ut satisfy a functional central theorem (cf. Phillips and Solo, 1992) bnsc

1 X p ut ) B (s) on DRm+K+Kz [0; 1] ; n t=1

(6)

where B denotes a Brownian motion with variance = F (1) F (1)0 > 0. The matrices Fj , F (1) and may be partitioned conformably with ut = (u00t ; u0xt ; u0zt )0 as 2 3 2 3 2 3 F0j F0 (1) 00 0x 0z 5 Fj = 4 Fxj 5 ; F (1) = 4 Fx (1) 5 ; = 4 x0 xx xz Fzj Fz (1) z0 zx zz 4

0 so that ij = FP x; zg. The one sided long run covariance i (1) Fj (1) for each i; j 2 f0; P1 0 0 matrices = 1 E u u and = t t h h=1 h=0 E ut ut h may be partitioned in a similar way to . Finally, the Brownian motion in (6) can be written as

B (s) = B0 (s)0 ; Bx (s)0 ; Bz (s)0

0

(7)

where Bi (s) is a Brownian motion with variance ii for each i 2 f0; x; zg. Inference in cointegrated systems requires estimation of the long run covariance matrices and . The following result concerns the order of consistency of the usual Newey and West (1987) estimator of : X ^n = 1 1 n h=0 M

h M +1

n X

u^t u^0t

h:

(8)

t=h+1

2.1 Lemma. Consider the estimator ^ n in (8) with M = Kn1=3 , for some …xed constant K. Under Assumption LP, ^n for any

= op n

1 2

as n ! 1

2 (2=3; 1).

3. Feasible IV inference In this section we propose a two-stage least squares procedure for the estimation of the coe¢ cient matrix A in the cointegrated system (1) - (2). The idea is to construct instruments with lower degree of persistence than the regressor in (2). These instruments will necessarily be mildly integrated processes of the form (4) with < . It turns out that this method eliminates the endogeneity present in conventional cointegration methods with unit root and local to unity regressors (see Lemma 3.2 below) and leads to inference that is robust to the degree of persistence in the regressors. Given the cointegrated system in (1) and (2), we construct instruments z~t from (2) as follows: t X t j z~t = Rnz xj (9) j=1

where Rnz is de…ned as in (5) for known Cz , and Kz = K. We begin by discussing the case where the instrument is less persistent than the regressor, i.e. the instrument autoregressive matrix Rnz satis…es the restriction < min ( ; 1) : 5

Since xt = uxt + nC xt 1 , a decomposition for z~t applies in terms of a moderately P t j integrated instrument zt = tj=1 Rnz uxj satisfying zt = Rnz zt

1

and a remainder term nt = n di¤erencing the regressor in (2):

+ uxt ; t 2 f1; :::; ng z0 = 0 (10) Pt t j j=1 Rnz Cxj 1 that arises as a result of quasiz~t = zt +

(11)

nt :

The following lemma shows that the e¤ect of quasi-di¤erencing on IV regression P sample moments is present only on the signal matrix nt=1 xt z~t0 .

3.1 Lemma. Consider the model (1) - (2) satisfying Assumptions N and LP and instruments z~t de…ned by (9) with 1=2 < < min ( ; 1) : The following approximations are valid as n ! 1: (i) n

1+ 2

Pn

(ii) n

(1+ )

(iii) n

(1+ )

t=1

Pn

u0t z~t0 = n

t=1

xt z~t0 = n

Pn

~t z~t0 t=1 z

=n

1+ 2

Pn

t=1

(1+ ) (1+ )

where zt is de…ned in (10).

u0t zt0 + op (1)

Pn

t=1

xt zt0

Pn

0 t=1 zt zt

n

(1+ )

+ op (1)

Pn

t=1

xt 1 x0t 1 CCz 1 + op (1)

The proof can be found in the Appendix. Lemma 3.1 reveals the asymptotic behavior of the sample moments in (i) and (iii). Since zt is a mildly integrated process satisfying (10) with Rnz = IK + Cz =n , equations (7) and (9) of Magdalinos and Phillips (2009) yield 1 n1+ as n ! 1, and, for any 1 n

1+ 2

n X

zt zt0

t=1

!p

Z

1

erCz

rCz dr xx e

(12)

0x )

= Un (1) + op (1) ;

(13)

[zt

1

F0 (1) "t ]

(14)

0

2 (1=3; 1), n X

vec (u0t zt0

t=1

where Un ( ) denotes the martingale array Un (s) =

1 n

1+ 2

bnsc X t=1

on the Skorokhod space DRmK [0; 1]. 6

P The asymptotic behavior of the sample moment nt=1 xt zt0 of part (ii) of Lemma 3.1 can be determined as follows: using the recursive property of (2) and (10) we can write xt zt0 = Rn xt 1 zt0 1 Rzn + Rn xt 1 u0xt + uxt zt0 1 Rzn + uxt u0xt : Vectorising and summing over t 2 f1; :::; ng we obtain 1X (IK 2 Rzn Rn ) vec xt 1 zt0 1 n # " t=1 n n n X X X 1 1 1 = [IK + op (1)] vec xt 1 u0xt + uxt zt0 1 + uxt u0xt n t=1 n t=1 n t=1 # " n 1X = vec xt 1 u0xt + 0xx + E (ux1 u0x1 ) + op (1) n t=1 n

(15)

by Lemma 3.1(d) of MP and the ergodic theorem. The asymptotic behavior of P n 1 nt=1 xt 1 u0xt depends on the order of persistence of xt . A standard application of the Phillips and Solo (1992) method yields 1X 1X xt 1 u0xt = xt 1 "0t Fx (1)0 + n t=1 n t=1 n

n

xx

+ op (1) as n ! 1:

(16)

If xt is mildly integrated in the sense of Assumption N(iii), the right side of (16) reduces asymptotically to the constant matrix xx by Lemma 3.3 of MP. If xt is local to unity, the martingale arrayRon the right side of (16) converges in distribution to 1 the matrix stochastic integral 0 JC (s) dBx (s)0 where Bx (s) is the Brownian motion with variance xx de…ned in (7) and Z s JC (s) = eC(s r) dBx (r) (17) 0

is the associated Ornstein-Uhlenbeck process. Since IK 2

Rzn

Rn =

1 (Cz n

IK ) IK + Op

1 n

as n ! 1

(15) and (16) imply that 1 n1+

n X t=1

xt 1 zt0

1

)

8 > > < > > :

R1

0 R1 0

Bx dBx0 +

JC dBx0 + 1 xx Cz

xx

Cz 1

under N(i)

xx

Cz 1

under N(ii) under N(iii)

as n ! 1, where N(i)-N(iii) refer to Assumption N. 7

(18)

The results of Lemma 3.1, (13) and (18) determine the individual asymptotic behavior of various sample moments of interest, but o¤er no information on the presence or lack of endogeneity which is the critical issue underlying robust inference in nearly cointegrated systems. The next lemma addresses this issue by establishing Pn 1+ Pn 0 (1+ ) asymptotic independence between n 2 ~t0 . 0x ) and n t=1 (u0t zt t=1 xt z

3.2 Lemma. Consider the model (1) - (2) satisfying Assumptions N and LP and the mildly integrated process zt de…ned in (10). The martingale array Un (s) in (14) satis…es Un (s) ) U (s) on DRmK [0; 1] as , with variance R 1 nrC! 1,rCwhere U is a Brownian motion, independent of B1xP n 0 z z e P xx e dr. Joint convergence in distribution of Un (1), n t=1 xt 1 "t and 0 n 1 0 n t=1 xt 1 xt 1 also applies as n ! 1. 3.3 Remarks.

(i) Lemma 3.2 is a direct consequence of Proposition A1 in the Appendix. It is important to note that, unlike conventional methods for removing endogeneity, the asymptoticPindependence between the martingale of the sample Pn part n 0 0 covariance matrix t=1 u0t zt and the signal matrix t=1 xt z~t holds by virtue of the reduced order of magnitude of the instrument zt and does not rely on an orthogonality correction. This is evident from equation (32) in the Appendix which shows that the conditional variance matrix of the martingale array consisting of Un (s) and the martingale part of the partial sum process of ut is asymptotically block diagonal if and only if 1 n1+ 2

bnsc X t=1

zt

1

!p 0

i.e. if and only if zt is a mildly integrated process with

2 (0; 1).

(ii) Lemma 3.2 shows that instrumentation with a mildly integrated process completely removes the endogeneity present in least squares inference of cointegrated systems of the form (1) - (2) with integrated and local to unity regressors. On the other hand, this approach introduces …rst order asymptotic bias 1 of the form n 2 0x which can be estimated by standard methods. The limit theory established in this section reveals the possibility of constructing a bias-corrected IV estimator of the coe¢ cient matrix A in (1) which is asymptotically mixed Gaussian for a very general class of persistent regressors. Using conventional regression notation, let 0 0 0 Y = [y10 ; :::; yn0 ] ; X = [x01 ; :::; x0n ] and Z~ = [~ z10 ; :::; z~n0 ] :

8

(19)

Then, the bias corrected two stage least squares estimator A~n = Y 0 Z~

n ^ 0x

X 0 Z~

1

(20)

is asymptotically mixed Gaussian, as the following theorem now shows. 3.4 Theorem. Consider the model (1) - (2) satisfying Assumptions N and LP and instruments z~t de…ned (9) with 2=3 < < min ( ; 1) : Then, the following limit theory applies for the estimator A~n in (20): Z 1 0 1+ 1 ~ ~ 2 n vec An A ) M N 0; Cz erCz xx erCz dr Cz ~ xx1 00 xx 0

as n ! 1, where ~ xx =

8
1=3 in the sense that it can be implemented using only the information provided by the statistical model (1) - (2). The instruments are constructed from the regressors in (2) without assuming any exogenous information. For precisely this reason, the method cannot accommodate mildly integrated regressors that are close to the stationary region. As earlier work in MP has shown, simultaneity in cointegrated systems with mildly integrated regressors becomes more severe as we approach the stationary region and, eventually for 2 (0; 1=3], it presents similar di¢ culties to simultaneous equations bias in that it cannot be removed without the use of exogenous instrumental variables. Classical IV procedures that utilize exogenous information and address this problem are considered in the next section. 11

(iii) The price paid for the method’s robustness is a reduction in the rate of con1+ vergence of A~n to O n 2 from O (n) in the case of applying standard cointegration methods under the knowledge that the regressors are exact unit root processes. Interestingly, A~n remains asymptotically e¢ cient in the case where the vector of regressors is mildly integrated and we instrument by a more persistent process (Theorem 3.7 with < ) and retains the OLS rate of convergence when = . In any case, a reduction in asymptotic e¢ ciency does should not a¤ect the performance of self normalized tests such as the Wald test of Theorem 3.8. (iv) A related point to the above remark is that the approach of removing endogeneity by constructing instruments with reduced rate of convergence induces …rst order asymptotic bias. As earlier work in MP has shown, cointegrated systems with mildly integrated regressors do not su¤er from endogeneity problems but the asymptotic bias becomes more severe than in the case of systems with integrated and local to unity regressors. This observation suggests that, in the present context, there is a trade-o¤ between …nite sample endogeneity and bias. This is of practical importance for the implementation of the method when choosing the rate of persistence n of the instrument. A choice of close to 1 would minimise the e¤ect of …nite sample bias but would exacerbate …nite sample endogeneity. On the other hand, the e¤ect of …nite sample bias increases the closer we choose the instrument to the stationary direction and indeed the bias becomes too severe for the method to work for 2 (0; 1=2). The restriction > 2=3 is imposed in order to ensure the inclusion of optimal bandwidths associated with non-parametric estimation of the long run covariance matrix , see Lemma 2.1.

4. Classical IV inference In this section we conduct inference on the cointegrated system (1) - (2) on the basis that additional information is available in the form of a Kz -vector of mildly integrated instruments zt ; with Kz K, satisfying (4), (5) and the following form of the classical IV long-run relevance condition. Assumption IV(i). The long run covariance matrix rank equal to K.

xz

= Fx (1) Fz (1)0 has full

The most plausible way that such instruments may be available in macroeconomic or …nancial time series is through the existence of a Kz -vector unit root process t that is correlated with the vector of regressors xt of the original system (1) - (2). In

12

this case the mildly integrated instruments zt may be constructed by di¤erencing zt =

t X

t j Rnz

t:

j;

j=1

for a chosen matrix Rnz satisfying (5). Assuming that such a unit root process t exists eliminates the e¤ects of quasi-di¤erencing (see Lemma 3.1 above) and allows greater ‡exibility in the instrumentation of cointegrated systems with mildly integrated regressors because the restriction < is no longer necessary. Classical IV inference also imposes incoherence between the innovations of (1) and those of (4). Although we will only be making use of such an orthogonality condition in order to establish a boundary with stationary simultaneous equation systems, it will be convenient for subsequent reference to state it here. Assumption IV(ii). The innovations u0t and uzt of (1) and (4) satisfy the orthogonality condition 0z = 0. We now return to our program of providing robust inference for the cointegrated system (1) - (2) by means of the mildly integrated instruments zt in (4). Under both Assumptions IV(i) and IV(ii) this can be achieved by the classical IV estimator A^IV = Y 0 PZ X (X 0 PZ X)

1

where Y and X are de…ned in (19), Z = [z10 ; :::; zn0 ]0 and PZ = Z (Z 0 Z) 1 Z 0 . Given the di¢ culty of assessing the validity of the orthogonality condition IV(ii) in practice, it is desirable to conduct inference on the basis of Assumption IV(i) only. To this end, we de…ne a bias corrected version of A^IV A~IV

1

= A^IV =

1

n ^ 0z (Z 0 Z) Z 0 X (X 0 PZ X) 1 Y 0 Z n ^ 0z (Z 0 Z) Z 0 X (X 0 PZ X)

1

:

The estimator A~IV is constructed using a similar bias correction principle to that used for the two stage least squares estimator A~n of the previous section. The di¤erence between the two estimators consists of the additional information used in the construction of the instrument matrix Z which, in the case of A~IV , avoids the restrictions and ine¢ ciency associated with quasidi¤erensing the regressors in (2). P n 0 PnAs in 0Section 3, the asymptotic behavior of the sample moments t=1 zt zt and t=1 u0t zt can be deduced directly from MP. Following the notation of MP, let Z 1 Z 1 rCz rCz e dr and Vxz = erC xz erCz dr: (24) Vzz = zz e 0

0

13

4.1 Lemma. Consider the model (1), (2) and (4) satisfying Assumptions N, LP and IV(i). Then, as n ! 1: 1+ Pn 0 2 (1=3; 1) (i) n 2 0z ) = Un (1) + op (1), for each t=1 (u0t zt P (ii) n (1+ ) nt=1 zt zt0 !p Vzz

where Un ( ) is the martingale array de…ned in (14) and Vzz is given by (24). Under Assumption IV(ii), 0z = 0, and part (i) holds for all 2 (0; 1).

The asymptotic behavior of the X 0 Z matrix presents two important di¤erences in comparison to the corresponding signal matrix of Section 3. First, there is no quasi-di¤erencing e¤ect as in Lemma 3.1(ii). Second, the instrument zt does not have to be less persistent than a mildly integrated regressor, i.e. if xt satis…es Assumption N(iii), the restriction < is no longer necessary. This extends the validity of IV inference over the whole range 2 (0; 1) of mildly integrated regressors and allows linkages to the boundary with the stationary simultaneous equation systems. 4.2 Lemma. ConsiderP the model (1), (2) and (4) satisfying Assumptions N, LP n 0 1 0 and IV(i). Let Jn = n t=1 xt 1 "t Fz (1) . Then, as n ! 1: P 1 (i) n 1 ( ^ ) nt=1 vec xt 1 zt0 1 = n ^ (IKKz Rzn Rn ) vec(Jn + xz )+op (1) ; (ii) n

1 ( ^ )

xz

X 0 Z ) xz , where: 8 R1 > + Bx dBz0 Cz 1 > xz 0 > > > R1 > 0 1 < xz + 0 JC dBz Cz = Vxz > > > 1 > > xz Cz > : 1 C xz

under N(i) under under under under

N(ii) N(iii) with N(iii) with N(iii) with

= >
g !p 0;

for all s 2 [0; 1] and > 0. Using the inequality (a + b)1=2 and the fact that kF (1)k kF0 (1)k, we obtain k

nt k

(28)

a1=2 + b1=2 for all a; b > 0

kF0 (1)k kzt 1 k k"t k + kF (1)k k"t k n =2 2 kF (1)k kzt 1 k + 1 k"t k n1=2 n =2 1

n1=2

16

(29)

which, in turn, implies that 1 fk

nt k > g

p kzt 1 k k"t k p + 1 > + 1 > 1 n =2 n =4 n2 4 0 0 1 k"t k p kzt 1 k > + 1 > + 1 > 1 1 2 2 n =4 n2+ 4 n2 4 2 kF (1)k

1 1

0

where that

bnsc X t=1

=

p

EFnt

because n

1

=2 kF (1)k. Using (29) and the Fnt 1 -measurability of zt

1

k

nt k

Pbnsc t=1

2

1 fk

nt k

> g

max EFnt

1 t n

1

k"t k2 1 fk

nt k

1

(30)

we obtain

> g Op (1)

kzt 1 k2 = Op (1) by MP. Therefore, max EFnt

1 t n

k"t k2 1 fk

1

nt k

> g = op (1)

(31)

is su¢ cient for (28) . To show (31), using (30) and the fact that "t is an i.i.d. sequence with E k"1 k2 < 1 we obtain 1 fk

max EFnt

1 t n

nt k > g

2

1

kzt 1 k

1

k"t k 1 fk

1

n2+ 4

nt k

>

0

2

1

+1

>

1

n2

4

kzt 1 k

0

2

+1

k"t k p > n =4

0

E k"1 k2 2 n n p o 2 + o (1) +E k"1 k 1 k"1 k > n =4

> g

max 1

1 t n

1

1

1 +4 2

n = op (1)

1 +4 2

>

max kzt 1 k >

1 t n

0

2

E k"1 k2 + o (1)

1

since equation (53) of MP shows that n 2 4 max1 t n kzt 1 k = op (1). This establishes (28). P The conditional variance of the martingale array bnsc t=1 nt is given by 2 3 Pbnsc Pbnsc 1 1 0 0 bnsc Q X 00 t=1 zt 1 zt 1 t=1 zt 1 n1+ n1+ 2 5 EFnt 1 nt 0nt = 4 P bnsc 0 bnsc 1 z Q t=1 t=1 t 1 n 1+ n

!p s

2

Vzz

00

0

;

0

17

(32)

where Q = F (1) F0 (1)0 , because we know from MP that, for all s 2 [0; 1], and n

bnsc X

1

zt zt0

t=1

bnsc X

!p sVzz and

zt

1

1

= Op n 2 +

:

t=1

P Therefore, applying Theorem 3.33 VIII of Jacod and Shiryaev (1987) to bnsc t=1 there exists a continuous Gaussian martingale (s) with quadratic variation h is = sdiag (Vzz

00 ;

nt ,

)

P such that bnsc t=1 nt ) (s) on the Skorokhod space DRmKz +m+K+Kz [0; 1]. By Levy’s characterisation of Brownian motion (e.g. Theorem 4.4 II of Jacod and Shiryaev (1987)), (s) is a Brownian motion on DRmKz +m+K+Kz [0; 1] with covariance matrix 0 0 0 diag(Vzz conformably with its covari00 ; ). Partitioning (s) = U (s) ; B (s) ance matrix we conclude that U is independent of B. Proof of part (ii). Letting bnsc

bnsc

1 X 1 X Bnx (s) = p uxj ; Wn (s) = p "j ; n j=1 n j=1 Proposition A1 (ii) implies that Bnx (s) ;

Un (s) Wn (s)

)

Bx (s) ;

U (s) W (s)

on DRK

R(m+1)Kz +K

[0; 1] :

(33)

Since xbn c can be represented as a continuous functional of Bnx ( ), viz. Z s 1 p xbnsc = Bnx (s) C eC(s r) Bnx (s) dr + op (1) ; n 0

and n 1=2 xbnsc ) JC (s) on DRK [0; 1], (33) and the continuous mapping theorem imply that 1 p xbnsc ; n

Un (s) Wn (s)

The identity

Z

vec

0

yields 2 4

) 1

JC (s) ;

xbnsc p dWn (s)0 = n

UP n (1) vec fn 1P nt=1 xt 1 "0t g vec n 1 nt=1 xt 1 x0t 1

U (s) W (s)

on DRK

Z

xbnsc p n

1

0

2

IKz

R(m+1)Kz +K

dWn (s)

R1 dUn (s) 0 n o 6 R1 xbnsc p dWn (s) 5 = 6 0 IKz n 4 R xbnsc 1 xbnsc p p ds 0 n n Z 1 = Gn (s) dVn (s) ; 3

0

18

[0; 1] : (34)

3

7 7 + op (1) 5 (35)

where xbnsc xbnsc p ; p n n

Gn (s) = diag ImKz ; IKz

xbnsc p n

0

and Vn (s) = Un (s)0 ; Wn (s)0 ; s :

Letting G (s) = diag [ImKz ; IKz

0

JC (s)] and V (s) = U (s)0 ; W (s)0 ; s ;

JC (s) ; JC (s)

joint convergence [Gn (s) ; Vn (s)] ) [G (s) ; V (s)] on the relevant Skorokhod space is guaranteed by (34). Note that Vn (s) is a semimartingale with Doob-Meyer decom0 position Vn (s) = Mn (s) + A (s) where Mn (s) = Un (s)0 ; Wn (s)0 ; 0 is a martingale and A (s) = [0; 0; s]0 has bounded variation for all s 2 [0; 1]. Thus, Theorem 2.7 of Kurtz and Protter (1991) implies that Z 1 Z 1 Gn (s) dVn (s) ) G (s) dV (s) (36) 0

0

obtains, provided that supn E [Mn ]s < 1. The later condition holds trivially, since 20 1 3 bnsc X 5 E [Mn ]s = diag 4@n 1 Ezt 1 zt0 1 A 00 ; ; 0 t=1

Pn Pbnsc 2 1 0 sup n Ez z and supn n 1 t 1 n t 1 t=1 E kzt 1 k t=1 s 2 [0; 1]. Now part (ii) can be established by combining (35) and (36).

C < 1 for all

Proposition A2. Under Assumptions N and LP, sup E 1 t n

t X

2 t j Rnz xj

= O n(

_ )+2( ^ )

j=1

as n ! 1:

Proof. For the sake of clarity, we di¤erentiate between the Euclidain matrix norm kM kE = (trM 0 M )1=2 and the spectral norm kM k de…ned as the square root of the maximal eigenvalue of M 0 M . Recall that xj =

Rnj x0

+

j X

Rnj k uxk

k=1

and denote the autocovariance matrix of uxt by E (xj x0i ) =

j i X X

Rnj

k=1 l=1

19

k

ux

ux

(k

(h) = E uxt u0xt l) Rni l ;

h

: Since

using the trace version of the Cauchy Schwarz inequality (see e.g. Abadir and Magnus, 2005) we obtain E

t X

(

2 t j Rnz xj

= tr

j=1

t X

)

t j t i Rnz Rnz E (xj x0i )

i;j=1

=

t X

j i X X

2t tr Rnz

j i

Rnj+i

k l

(k

ux

l)

i;j=1 k=1 l=1

j i t X X X

2t Rnz

j i

2t Rnz

j i

Rnj+i

k l E

i;j=1 k=1 l=1

j t X i X X

Rnj+i

E

k

k l E

i;j=1 k=1 l=1

K

j t X i X X

i;j=1 k=1 l=1

kRnz k2t

j i

(k

l)kE

k

ux

(k

k

ux

ux

kRn kj+i

k l

l)kE (k

l)kE (37)

where the last inequality follows from the fact that Rn and Rnz are diagonal K K matrices, so kRnm kE K 1=2 kRn km for any m 2 Z and the same inequality holds for Rnz . In order to estimate the right side (37) we make use of the bounds sup 1 t n

t X j=1

kRn kt

j

=O n

^1

and

sup 1 t n

t X j=1

kRnz kt

j

(38)

=O n

and also of the fact that Assumption LP guarantees summability of k ux (h)kE over h 2 Z. We consider …rst the case < . Using the fact that kRn ki l 1, for all l i, (37) yields sup E 1 t n

t X

K

t X j=1

sup 1 t n

j=1

by (38). When E

2 t j Rnz xj

= O n , letting j 2

t j Rnz xj

K

+2

t X i=1

t i

kRnz k

!2

sup 1 j n

j X k=1

kRn k

j k

!

1 X

l= 1

k

k = m in (37) we obtain that j 1 i t X X X

i;j=1 m=0 l=1

kRnz k2t

20

j i

kRn km+i l k

ux

(j

m

l)kE :

ux

(l)kE

Using the fact that kRnz kt sup E 1 t n

t X

j

1 for all j

2

t j Rnz xj

K

sup 1 t n

j=1

1 X

j= 1

= O n

t X

k

i=1

ux

t,

kRnz kt

i

!

1 X

m=0

kRn km

!

sup 1 i n

i X l=1

kRn ki

(j)kE

+2

and the result follows. Proof of Lemma 2.1. It can be shown by standard methods that, under Assumption LP, M 1 ^n = Op max ; n1=2 M for any bandwidth parameter M increasing to 1 with n. Therefore, the requirement 1 n 2 ^n = op (1) yields the restriction ) ( 1 M n 2 max ; !0 n =2 M which is satis…ed for all

2 (2=3; 1) :

Proof of Lemma 3.1. We start by noting that the remainder term nt

t 1 X t j R Cxj = n j=1 nz

1

in (11) satis…es the recursive formula nt

= Rnz

n;t 1

+

C xt n

(39)

1

and the uniform bounds 1

sup k k = Op n n =2 1 t n nt 1 sup E k nt k2 = O n ( n 1 t n

(40)

2

)

(41)

:

(41) follows immediately from Proposition A2. (40) is a direct consequence of the fact that supt2[0;1] xbntc = Op n( ^1)=2 and (38): 1 n

=2

sup k

1 t n

nt k

kCk n =2 n

sup kxj 1 k sup

1 j n

1 t n

21

t X j=1

kRnz kt

j

= Op

1 n(

)=2

:

l

!

For part (i), using (11) and the BN decomposition u0t = F0 (1) "t ~"0t we obtain ! n n n n X X 1 F0 (1) X 1 X 0 0 0 u0t z~t u0t zt = 1+ "t nt ~"0t 0nt : (42) 1+ 1+ 2 2 2 n n n t=1 t=1 t=1 t=1 1+ Pn Since nt is (x0 ; :::; xt 1 )-measurable, n 2 "t ) is a martingale array t=1 ( nt with E

1 n

1+ 2

n X

2

(

n E k"1 k2 X = Ek n1+ t=1

"t )

nt

t=1

nt k

2

Ek n

sup1

2

E k"1 k

nt k

t n

2

= o (1)

by (41). For the second term of (42), summation by parts and (39) yield n X

1 n

1+ 2

~"t

0 nt

=

t=1

= =

1 n

~"n

1+ 2

1 n

n X

1 n

1+ 2

1+3 2

~"0 0 0 nt Cz

~"t

~"t

n X

0 n;1

+

t=1

n X

1 n

0 n;n+1

0 0 nt Cz

t=1 n X

1 n

kCz k

n

sup k

=2

1 t n

1

= Op

n(

Op

)=2

n n

1 2

n1=2

n1=2 n X

1 !

1

1

+ op nt k

~"t x0t C 0 + op

t=1

t=1

1

0 nt+1

~"t

n

1 + 2

t=1

= Op

k~"t k + op

n(

1

n2 + )=2

!

1 n1=2

showing part (i) for > 1=2. For part (ii), using the recursive formulae (2) and (39) we obtain xt

0 nt

= Rn xt

1

0 n;t 1 Rnz

+

1 Rn xt 1 x0t 1 C + uxt n

0 n;t 1 Rnz

+

1 uxt x0t 1 C: n

Vectorising and summing along t 2 f1; :::; ng we obtain [IK 2

Rnz

Rn ]

n X

n;t 1

xt

1

t=1

=

x0

n;0

+ (Rnz

xn +

n;n

IK )

n X

n;t 1

C n

n Rn X

=

n

n Rn X

(xt

1

uxt +

xt 1 ) + op n

t=1

22

1

xt 1 )

t=1

C

t=1

C

(xt

1+ 2

n

n IK X

(xt

uxt )

1

t=1

+ Op n 1

;

P P 1+ since nt=1 n;t 1 uxt = op n 2 from the proof of part (i) and nt=1 (xt Op (n) for any > 0 (see equation (10) of MP). Since Rnz

IK 2

Rn =

1 n

Cz 1

normalising the above expression by n n X

1 n1+

xt

n;t 1

;

n

yields

1

=

1

1

IK + O

uxt ) =

1

Cz C

1

IK

n1+

t=1

n X

(xt

xt 1 ) + op (1) :

1

t=1

The result now follows by undoing the vectorisation and using (11). For part (iii), (11) yields 1 n1+

n X

z~t z~t0

t=1

n X

zt zt0

=

t=1

n X

1 n1+

0 nt

nt

+

t=1

n X

1 n1+

t=1

sup1

n X

0 nt

zt

+

t=1

k

nt k

t nk n =2

2

+

nt k

2

n1+ 2

+

n X

0 nt zt

t=1

n X t=1

k

sup1

nt k kzt k t nk n =2

nt k

2 n1+ 2

= op (1) by (40) since the Lyapounov inequality gives ! n 1 X 1 E kz k sup E kzt k2 t 1+ =2 n n 1 t n t=1 Proof of Theorem 3.4. For any n

1+ 2

A~n

A

= =

1 n

1+ 2

1 n

U00 Z~

1+ 2

(U00 Z

= Un (1) For n

1

1 n1+

t=1

1=2

= O (1) :

2 (2=3; 1), Lemma 2.1 and (13) yield 1

n ^ 0x n

n1+ 1

0x )

n1+

1

X 0 Z~ 1

X Z~ 0

+ Op n

1 2

^ 0x

1

X Z~ 0

+ op (1) :

Z~ 0 X, Lemma 3.1(ii) shows that n Ln =

n X

1 n1+

n X

1

Z~ 0 X = n

xt 1 x0t 1 CCz 1

t=1

23

1

Z 0X

Ln where

0x

kzt k

1 0 and the asymptotic behavior of n by (18). Under Assumption N(i), R 1 Z0 X is given 1 Ln = op (1). Under N(ii), Ln ) 0 JC JC dsCCz , giving

1

n1+

Z

~0

ZX )

JC dBx0

0

=

Z

1

xx

+

Z

+

Cz

xx

1

1

JC JC0 dsCCz 1

0

1

JC dJC0 Cz 1 :

0

R1 Under N(iii), equation (7) in MP gives Ln !p 0 erC xx erC drCCz 1 . Thus, in all of the above cases, n 1 Z~ 0 X ) ~ xx Cz 1 . Now, by Lemma 3.1(ii), (15), (16) and Lemma 3.2, n 1 Z~ 0 X and Un (1) converge jointly in distribution and are asymptotically independent. Thus, " # 1 1+ 1 0 n 2 vec A~n A = Z~ X Im Un (1) + op (1) n1+ has the required mixed Gaussian limit distribution. Proof of Lemma 3.5. For part (i), we can use (21) to write !0 ( n ) n n t X X X X 1 1 1 t j u0t Rnz xj Cz Rnz1 u0t z~t0 u0t x0t = 1+ 1+ n 2 2 n n t=1 t=1 t=1 j=1 n X

1

n = for any

> 1=2, because

Pn

t=1

t 1 (Rnz

1

n

1+ 2

+

n X

1+ 2

t=1

1 n

1+ 2

t 1 u0t x00 Rnz + Op

+

n X

u0t

t=1

j=1

u0t ) = Op n u0t x0t = Op

t=1

t 1 X =2

n1=2 n + =2 !0

t j Rnz xj

, x0 = op n

Cz Rnz1 + op (1) =2

and

n1=2 n =2+

by MP. For the leading term in the above display, using the BN decomposition we obtain " t 1 ! # " t 1 ! # n n X X X X 1 1 t j t j Rnz xj ~"0t + op (1) Rnz xj u0t = 1+ 1+ + 2 n n 2 + t=1 t=1 j=1 j=1 (43)

24

because, for the martingale part of the BN decomposition, Proposition A2 yields 0 1 " t 1 # 2 ! 2 n t 1 X X X 1 1 t j t j Rnz xj F0 (1) "t = O @ +2 max E Rnz xj A 1+ + 1 t n n n 2 t=1 j=1 j=1 L2

1

= O

:

n

Applying the summation by parts formula, the right side of (43) becomes ( n 1 ! ) n X X 1 n+1 j Rnz xj ~"0n (Rnz Im ) (xt ~"0t ) 1+ + n 2 t=1 j=1 " t 1 ! # n X 1 Cz X t j + 1+ + Rnz xj ~"0t : n 2 n t=1 j=1

(44)

By MP, the second term of (44) has order Op n ( + =2 1=2) . By Proposition A2, the …rst term of (44) has order Op n ((1+ )=2) . For the third term of (44), using the Cauchy Schwarz inequality and Proposition A2 and letting = E k~"01 k2 we obtain # " t 1 ! ! ! n n t 1 X X X X 1 1 t j t j Rnz xj ~"0t E Rnz xj k~"0t k 1+ 1+ +2 +2 2 n 2 n t=1 t=1 j=1 j=1 L1 8 ! 2 91=2 n < t 1 = 1=2 X X t j E Rnz xj 1+ ; n 2 +2 t=1 : j=1 8 ! 2 91=2 t 1 = 1=2 1=2 < X n t j max E R x nz j ; n 2 +2 :1 t n j=1

= O

n

1=2+ =2

n3

=o

=2

1

n

1=2

:

This shows part (i) for any > 1=2. For part (ii), we make repeated use of the decomposition (21) and Proposition A2. By (21), ! " n !0 # n n n t X X X X 1 1 1 X t j 1 0 0 0 t 1 xt z~t xt xt = xt x0 Rnz + xt Rnz xj Cz Rnz : n1+ n1+ t=1 n t=1 t=1 t=1 j=1 (45)

The …rst term on the right of (45) has order op n 1 n1+

n X t=1

t 1 xt x00 Rnz

kx0 k 1 n =2 n1+

=2

sup kxt k

1 t n

25

(1

n X t=1

)

since

kRnz kt

1

=

kx0 k Op n =2

1 n1

:

For the second term in (45), we know from MP that sup1 t n E kxt k2 = O (n ). Thus, the Cauchy Schwarz inequality and Proposition A2 yield " t ! # ! n n t X X X X 1 1 t j t j Rnz xj xt E Rnz xj kxt k 1+ + n1+ + t=1 n t=1 j=1 j=1 L1 8 0 191=2 2 t = =2 < X O n t j @ A sup E R x nz j :1 t n ; n + j=1

1

= O

n(

:

)=2

Since > , this implies that the right side P of (45) converges to 0 in probabiln 1 ity. ItPremains to show the result for n ~t z~t0 : Given the derivations for t=1 z n n 1 ~t0 and (21), it is su¢ cient to prove that t=1 xt z n t kx0 k X X t j Rnz xj kRnz kt = op (1) 1+ + n t=1 j=1

and n t X X

1 n1+

+2

t=1

(46)

2 t j Rnz xj

= op (1) :

(47)

j=1

The left side of the (47) has L1 norm of order O n ( ) by Proposition A2. For (46), kx0 k =n =2 = op (1) and 8 9 ! 2 1=2 Pn n t t t < = X X X 1 t t j t j t=1 kRnz k E R x sup kR k E R x nz nz j nz j ; n1+ =2+ n1+ =2+ 1 t n: t=1

j=1

j=1

= O

since


1=2. For the third term of (53),

~"0t u0xt

t=1

~"0t u0xt + Op n1

+ Op n

n 1 X ~"0t zt0 1 Cz + Op n + n t=1

=2

=2

t=1

P P by MP. Since = > 1=2, and nt=1 fu0t u0xt E (u0t u0xt )g and nt=1 f~"0t u0xt both have rate Op n 1=2 by the stationary ergodic CLT, (53) implies that 1 n

1+ 2

n X

vec (u0t z~t0

0x ) =

t=1

F0 (1)] X

0x g

n

[IK n

1+ 2

(~ zt

1

"t ) + op (1) :

t=1

A standard martingale CLT implies that the right side of the above expression converges in distribution to a ! ( ) n X N 0; plimn!1 n (1+ ) z~t 1 z~t0 1 00 t=1

which, in view of part (ii), yields the required limit distribution. Proof of Lemma 4.2. Part (i) can be deduced by an identical method to that used in establishing (15) (16): " n # n X 1X 1 (IKKz Rzn Rn ) vec xt 1 zt0 1 = vec xt 1 "0t Fz (1)0 + xz + op (1) n t=1 n t=1 but now we also have to account for the possibility that IKKz

Rzn

Rn =

1 (Cz n

IK ) IK + Op 29

. When 1

n

> ;

as n ! 1

and the lemma follows by Pstandard unit root asymptotics and, in the 1 > > case, from the fact that n 1 nt=1 xt 1 "0t = op (1). It remains to show the lemma when < 1. When = , IKKz Rzn Rn ! (Cz IK + IKz C), so 1 n1+

n X

vec xt 1 zt0

1

=

t=1

=

(Cz Z

1

IK + IKz erCz

C)

erC drvec

1

vec

xz

xz

+ op (1)

= vecVxz :

0

When

> , n (IKKz 1 n1+

n X

Rzn

Rn ) !

vec xt 1 zt0

1

C so

IKz

=

IKz

C

1

vec

xz

+ op (1)

t=1

= vecC

1

xz :

7. References Abadir, K.M. and J. R. Magnus (2005). Matrix Algebra. Econometric Exercises, vol.1. Cambridge University Press. Chan, N. H. and C. Z. Wei (1988). “Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes”, Annals of Statistics 16, 367-401. Elliott, G. (1998). On the Robustness of Cointegration Methods When Regressors Almost Have Unit Roots.”Econometrica 66, 149-158. Elliott G., J. Rothenberg and J. H. Stock (1996). E¢ cient tests for an autoregressive unit root, Econometrica 64, 813-836. Hall, P. and C.C. Heyde (1980). Martingale Limit Theory and its Application. Academic Press. Kurtz, T.G. and P. Protter (1991). “Weak limit theorems for stochastic integrals and stochastic di¤erential equations”. Annals of Probability 19, 1035-1070. Jacod, J. and A.N. Shiryaev (1987). Limit Theorems for Stochastic Processes. Springer-Verlag. Johansen,S. (1991). “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models,”Econometrica, 59, 1551-80. Magdalinos, T. and P. C. B Phillips (2009). “Limit Theory for Cointegrated Systems with Moderately Integrated and Moderately Explosive Regressors”, Econometric Theory, forthcoming. 30

Newey, W.K. and K.D. West (1987). “A simple, positive semi-de…nite, heteroskedasticity and autocorrelation consistent covariance matrix”. Econometrica, 55, 703–708. Park, J. Y. and P.C.B. Phillips (1989).“Statistical Inference in Regressions with Integrated Processes: Part 1,”Econometric Theory 4, 468-497. Phillips, P. C. B. (1987). “Towards a uni…ed asymptotic theory for autoregression,” Biometrika 74, 535–547. Phillips, P. C. B. (1988a). “Regression Theory for Near-Integrated Time Series,” Econometrica, 56, 1021-1044. Phillips, P. C. B. (1988b). “Multiple regression with integrated processes.” In N. U. Prabhu, (ed.), Statistical Inference from Stochastic Processes, Contemporary Mathematics 80, 79–106. Phillips, P. C. B. and B. E. Hansen (1990). “Statistical inference in instrumental variables regression with I(1) processes,” Review of Economic Studies 57, 99– 125. Phillips, P. C. B. and T. Magdalinos (2007), “Limit theory for Moderate deviations from a unit root.”Journal of Econometrics, 136, 115-130. Phillips, P. C. B. and V. Solo (1992), “Asymptotics for Linear Processes,” Annals of Statistics 20, 971–1001. Saikkonen, P. (1991). “Asymptotically E¢ cient Estimation of Cointegration Regressions,”Econometric Theory 7, 1-21. Stock, J.H. and M.W. Watson (1993). “A simple estimator of cointegrating vectors in higher order integrated systems”, Econometrica 61, 783-821.

31