Cointegration, variance shifts and the limiting distribution of the OLS estimator

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Economics Letters 99 (2008) 103 – 106 www.elsevier.com/locate/econbase

Cointegration, variance shifts and the limiting distribution of the OLS estimator Nikolaos Kourogenis ⁎, Nikitas Pittis Department of Banking and Financial Management, University of Piraeus, 80 M. Karaoli and A. Dimitriou str., 18534 Piraeus, Greece Received 27 October 2006; received in revised form 2 May 2007; accepted 8 June 2007 Available online 14 June 2007

Abstract This paper investigates the performance of the OLS estimator in the context of a cointegrating system, which exhibits a single variance shift. It is shown that the limiting distribution of OLS and that of the associated t-statistic depend on the time, the size and the direction of the break. © 2007 Elsevier B.V. All rights reserved. Keywords: Cointegration; Unconditional heteroscedasticity; Variance shifts JEL classification: C32

1. Introduction Standard asymptotic theory of cointegration requires the cointegration error, u1t, and the error that drives the regressor, u2t, to satisfy some regularity conditions, in order for the Functional Central Limit Theorem (FCLT) to apply (see Phillips and Durlauf, 1986). These conditions are violated when the variance of u1t and/ or the variance of u2t experience a permanent shift at some point within the sample. In such a case, the asymptotic properties of the cointegration estimators become an open issue. There are several papers in the literature that consider the issue of structural breaks in the error variance in the context of univariate autoregressive models with a unit root (see, for example, Hamori and Tokihisa, 1997; Kim et al., 2002; Burridge and Taylor, 2001; Cavaliere, 2004). More recently, Cavaliere and Taylor (2006) examine variance breaks in the context of cointegration and show that a change in the error covariance matrix is likely to produce spurious rejections of the null hypothesis of cointegration. In this paper, we adopt the triangular cointegration system, proposed by Phillips (1988, 1991), and assume that the variances of u1t and u2t are subject to a single break at a certain ⁎ Corresponding author. Tel.: +30 2104142142; fax: +30 2104142341. E-mail addresses: [email protected], [email protected] (N. Kourogenis). 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.06.004

point in the sample. For simplicity, we assume that the errors are independent both temporally and contemporaneously. This means that the simple OLS estimator is asymptotically optimal in the case of no variance shifts. Under this set of assumptions, we derive the limiting distribution of the OLS estimator of the cointegration parameter along with that of the corresponding tstatistic. It is shown that these distributions depend on the time, s, the size, a, and the direction of the break, that is whether a N 1 or 0 b a b 1. It is also shown that the variance shift does not affect the super-consistency property of OLS. 2. The model Let zt and ut be two bivariate processes, with zt = [yt, xt]T and ut = [u1t, u2t]T. We further assume that the generating mechanism for yt is given by the system: yt ¼ hxt þ u1t

ð1Þ

xt ¼ xt1 þ u2t

ð2Þ

Concerning the errors, we make the following assumption: 1. Assumption A


u1t u2t




 2 r ~ IID 0; 1 0

0 r22

 ; 1 ≤ t b ½sT 

104

N. Kourogenis, N. Pittis / Economics Letters 99 (2008) 103–106

and

 2 u1t r ~ IID 0; a  1 u2t 0

 0 ; ½sT  ≤ t ≤ T r22

where s ∈ (0, 1]. In order to describe the limiting distributions, we define the following functions: Definition. D (·), Q(·) are functions defined on the interval [0, 1], such that D ðr Þ ¼ Q rÞ ¼ 1 rðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ð 1  aÞ D ðr Þ ¼ a þ r

where for the special case of r = 0, the definition of XT (r) yields XT (r) = 0 with probability 1. From Eq. (4) we observe that the process XT (·) divided only by the standard deviation 2 does not converge to a standard Wiener process. If, however, we set ZT ðrÞ ¼

then, for every r ∈ [0, 1] we have

for r b s and

L

pffiffiffi and QðrÞ ¼ a for r ≥ s:

ZT ðrÞ Y N ð0; rÞ: Therefore, since u2t are independent we obtain

Then, we prove the following theorem: Theorem 1. If Eqs. (1), (2) and assumption A hold then   L r R 1 QðrÞDðrÞW ðrÞdW ðrÞ 2 1 1 0 T h ̂ h Y ; R1 2 2 r2 0 D ðrÞW2 ðrÞdr

1 XT ðrÞ; r2 D ð r Þ

L

ZT ðrÞ Y W2 ðrÞ; ð3Þ

where W1(·), W2(·) are two independent Wiener processes.

where W2(·) is a standard Wiener process. The last result allows us to obtain the asymptotic distribution of the denominator of the OLS bias as follows:

Proof. We start from the following expression T P

h ̂ h ¼

T 1 X x2 ¼ T 2 t¼1 t

t¼1

1

XT2 ðrÞdr Z 1 Z L D2 ðrÞZT2 ðrÞdr Y r22 ¼ r22

xt u1t

t¼1 T P

Z

x2t

0

0

0

1

D2 ðrÞW22 ðrÞdr: ð5Þ

P

rT  Let us set XT ðrÞ ¼ p1ffiffiTffi ½t¼1 u2t for T1 ≤ r ≤ 1 and XT (r) = 0 for 1 0 ≤ r ≤ T . Then, for 0 b r b s we have pffiffiffiffiffiffiffiffi ½rT  pffiffi L ½rT  1 X XT ðrÞ ¼ pffiffiffiffi pffiffiffiffiffiffiffiffi u2t Y r2 rN ð0; 1Þ≡r2 N ð0; rÞ T ½rT  t¼1

½sT 1 ½rT  1 X 1 X u2t þ pffiffiffiffi u2t XT ðrÞ ¼ pffiffiffiffi T t¼1 T t¼½sT 

xt u1t ¼

T X

xt1 u1t þ

T X

t¼1

u2t u1t

ð6Þ

t¼1

For the first term of (6) we set 1 u1t for t b ½sT  and r1 wt ¼ 1 pffiffiffi u1t for t ≥ ½sT  ar 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½sT 1 X ½sT   1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ¼ u2t T ½sT   1 t¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½rT  X ½rT   ½sT  þ 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi þ u2t T ½rT   ½sT  þ 1 t¼½sT 

Then by means of wt we generate the scaled partial sum process WT(r),

pffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffi L Y r2 s N1 ð0; 1Þ þ a r  sN2 ð0; 1Þ

½rT  1 X WT ðrÞ ¼ pffiffiffiffi wt ; 0 ≤ r ≤ 1; T t¼1

pffiffiffi aN2 ð0; r  sÞÞ



 s ð 1  aÞ ¼ r2 N 0; r a þ r rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ð 1  aÞ N ð0; rÞ; ¼ r2 a þ r

T X t¼1

For s ≤ r ≤ 1 we also have

≡ r2 ðN1 ð0; sÞ þ

Concerning the numerator of the OLS bias

for which we have L

WT ðrÞ Y W1 ðrÞ; 0 ≤ r ≤ 1; ð4Þ

where W1(·) is a standard Wiener process. Since u1t and u2t are not correlated, W1(·) and W2(·) are independent. Based

N. Kourogenis, N. Pittis / Economics Letters 99 (2008) 103–106

on this and the definition of the stochastic integral, we finally obtain T 1X

T

Proof. We have T P

xt u1t t¼1 ̂ h h ¼ T P 2 xt

xt1 u1t

t¼1

T  t 
t  1
t  1 X ¼ r2 r1 Q D ZT T T T t¼1
 
 t t1  WT  WT T T


 T X t1 t1 t1 Q ¼ r2 r1 D ZT T T T t¼1
 
 t t1  WT  WT T T

 pffiffiffi ½sT   1 x½sT  pffiffiffiffi þð a  1ÞZT T T Z 1 Lr2 r1 QðrÞDðrÞW2 ðrÞdW1 ðrÞ: Y 0

u2t u1t

t¼1

p

Y 0:

T

Therefore T P

tols ¼

ð7Þ

T P

xt u1t

t¼1

□

T T P ⇒ t¼1

The asymptotic result described in (3) may be used to derive the limiting distribution of the traditional t-statistic, tols, as reported in statistical packages:

r̂1 PT

where

r̂21

¼ Z

t¼1

T

L

Y N ð0; AÞ x0 ;x1 ; N ;xT

Q ðrÞD Z ð s þ að 1  s Þ Þ 0

ðrÞW22 ðrÞdr 1 2

D 0

ðrÞW22 ðrÞdr

T

j

QðrÞDðrÞW2 ðrÞdW1 ðrÞ

0

L

x0 ;x1 ; N ;xT

Y N ð0; V Þ

ð12Þ

1

2 QðrÞDðrÞW2 ðrÞdW1 ðrÞ

0

Z ¼ r22 r21

1

0

j

3 7 5 W2 ðÞ

Q2 ðrÞD2 ðrÞW22 ðrÞdr

Finally, for σ ˆ 12 we have 2

r21̂ ¼

T T 1X 1X û ¼ T t¼1 T t¼1



 2 h h ̂ xt þ u1t

ð13Þ

1t

T T T 2 X  X X 1  h  ĥ ¼ x2t þ 2 h  h ̂ xt u1t þ u21t T t¼1 t¼1 t¼1

ð9Þ

!

p

Y r21 ðs þ að1  sÞÞ≡r21 D2 ð1Þ

2 2

xt u1t

1

Hence, from (5), (11), (12) and (13), we have that for large enough sample

and

1

2

A¼

û21t

j

Z

L

Y r2 r1

2
Z 6 V ¼ E 4 r2 r1

Theorem 2. If Eqs. (1), (2) and assumption A hold, then

t¼1

x0 ;x1 ; N ;xT

since W1(·) and W2(·) are independent, where the variance V depends on W2(·). For the derivation of V we employ Ito's isometry to obtain

Remark 2. Observe that in (3) the trivial case of a = 1 or of s = 1 agrees with the relevant literature (see Park and Phillips (1988), Phillips (1991), White (2001), etc.).

tols ¼

j

ð11Þ

From (6), (7) and (8) we have

Remark 1. Note that the superconsistency of the OLS estimator of θ under a variance shift is maintained. Moreover, Eq. (3) depicts the nonlinear effect of the time, the magnitude and the direction of the variance shift on the limiting distribution of T(θˆ − θ).

sffiffiffiffiffiffiffiffiffiffiffi! T

P x2t h ̂ h

xt u1t sffiffiffiffiffiffiffiffiffiffiffi T P r̂1 x2t t¼1

t¼1

ð8Þ

Applying (7) and (8) to (6) we obtain (3).

ð10Þ

t¼1

Finally, for the last term of Eq. (6) we have that the stochastic sequence u2tu1t has zero mean and uniformly bounded variance. Therefore, T P

105

:

6 6 E6 6 4

T P

xt u1t t¼1 sffiffiffiffiffiffiffiffiffiffiffi T P r̂1 x2t t¼1

j

2

x0 ;x1 ; N ;xT

3

Z 1 7 Q2 ðrÞD2 ðrÞW22 ðrÞdr 7 0 7→ Z 1 7 5 ðs þ að1  sÞÞ D2 ðrÞW 2 ðrÞdr 0

2

106

N. Kourogenis, N. Pittis / Economics Letters 99 (2008) 103–106

Table 1 Empirical sizes of tols (T = 100, nominal size = 0.05) S α

0.1

0.15

0.2

0.25

0.3

0.5

0.7

0.75

0.8

0.85

0.9

0.01 10

0.0915 0.0613

0.0922 0.0668

0.0881 0.0716

0.0864 0.0788

0.0824 0.0871

0.0637 0.1082

0.0554 0.1319

0.0538 0.1348

0.0475 0.1325

0.0515 0.1296

0.0522 0.1247

last PThe T xt u1t qt¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffi P

r1̂

T

x2 t¼1 t

j

result combined with the fact that the distribution of tends to be normal (see (12)) completes the

x0 ;x1 ; N ;xT

proof.

□

3. Concluding remarks In this paper, we have analyzed the effects of a single break in the error covariance matrix of a cointegrating system on the limiting distribution of the OLS estimator and that of the associated t-statistic. We have shown that the variances of these distributions are affected by the time, the size, and the direction of the break. Moreover, a small Monte Carlo study (see Table 1) suggests that the size distortions of the t-test on the cointegration parameter are maximized when a common increase in the elements of the error covariance matrix occurs a few periods before the end of the sample. For example, when σ12 =σ22 = 1, T = 100 and a = 10, the empirical size of the t-test for the null hypothesis θ = 1 ranges from 6.13% for s = 0.1 to 13.48% for s = 0.75. On the other hand, in the case of a variance decrease, the size distortions are milder and are maximized near the beginning of the sample. For example, for σ12 =σ22 = 1, T = 100 and a = 0.01, the empirical size of the t-test attains its maximum value of 9.22% for s = 0.15. Acknowledgement We are grateful to an anonymous referee for his comments and suggestions. The usual disclaimer applies.

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