In this paper, we consider the role of "leads" of the first difference of integrated variables in the dynamic OLS estimation of cointegrating regression models. We demonstrate that the role of leads is related to the concept of Granger causality and that in some cases leads are unnecessary in the dynamic OLS estimation of cointegrating regression models. Based on a Monte Carlo simulation, we find that the dynamic OLS estimator without leads substantially outperforms that with leads and lags; we therefore recommend testing for Granger noncausality before estimating models.Cointegration, dynamic ordinary least squares estimator, Granger causality

Eiji Kurozumi

Kazuhiko Hayakawa

English

Research Papers in Economics

 
   
Hi-Stat  
 
 
 
Discussion Paper Series 
 
No.194 
 
 
The Role of “Leads” in the Dynamic OLS Estimation of 
Cointegrating Regression Models 
 
Kazuhiko Hayakawa 
Eiji Kurozumi 
 
December 2006 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Hitotsubashi University Research Unit 
for Statistical Analysis in Social Sciences 
A 21st-Century COE Program 
 
Institute of Economic Research   
Hitotsubashi University 
Kunitachi, Tokyo, 186-8603 Japan 
http://hi-stat.ier.hit-u.ac.jp/ The Role of \Leads" in the Dynamic OLS Estimation of
Cointegrating Regression Models
Kazuhiko Hayakawa and Eiji Kurozumiyz
Department of Economics, Hitotsubashi University,
2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan
December, 2006
Abstract
In this paper, we consider the role of \leads" of the rst dierence of integrated variables
in the dynamic OLS estimation of cointegrating regression models. We demonstrate that the
role of leads is related to the concept of Granger causality and that in some cases leads are
unnecessary in the dynamic OLS estimation of cointegrating regression models. Based on a
Monte Carlo simulation, we nd that the dynamic OLS estimator without leads substantially
outperforms that with leads and lags; we therefore recommend testing for Granger non-
causality before estimating models.
JEL classication: C13; C22
Key Words: Cointegration; dynamic ordinary least squares estimator; Granger causality
Hayakawa is a JSPS research fellow and acknowledges its nancial support.
yKurozumi's research was partially supported by the Ministry of Education, Culture, Sports, Science and
Technology under Grants-in-Aid No. 17203016 and 18730142.
zCorresponding author. Tel.: +81 42 580 8787. E-mail: kurozumi@stat.hit-u.ac.jp
11 Introduction
Since the seminal work of Engle and Granger (1987), cointegrating regressions have become
one of the standard tools in analyzing integrated (I(1)) variables. Although the ordinary least
squares (OLS) estimator is consistent in the presence of a serial correlation in the error term
and/or a correlation between the regressors and cointegration errors, it is well known that
the OLS estimator contains the so-called second-order bias. In the literature, there are three
typical estimators that deal with this problem: the fully modied OLS estimator proposed by
Phillips and Hansen (1990), Park's (1992) canonical cointegrating regression estimator, and
the dynamic OLS (DOLS) estimator of Phillips and Loretan (1991), Saikkonen (1991), and
Stock and Watson (1993). These three estimators are known to be asymptotically equivalent
and ecient. In this paper, we focus on the DOLS estimator among the three estimators
and consider the role of \leads" of the rst dierence of the integrated variables in DOLS.
We investigate the case where leads are unnecessary for the DOLS method, and by using the
Monte Carlo simulation, we demonstrate that in such a case, we can expect the improvement
of the DOLS estimator in terms of the mean squared error (MSE) by excluding leads from
the regressors.
2 Relation between Leads and Granger Causality
We consider a typical cointegrating regression model as follows:
yt =  + 0xt + u1t = 0zt + u1t (1)
xt = u2t
where  = [;0]0, zt = [1;x0
t]0, xt is an n-dimensional I(1) vector, and ut = [u1t, u0
2t]0 is a
stationary process that satises the condition of the multivariate invariance principle. Under
the regularity condition given in Saikkonen (1991), u1t is expressed as
u1t =
1 X
j= 1
0
ju2t j + vt (2)
where
P1
j= 1 kjk < 1 and vt is a stationary process such that E(u2svt) = 0 for all s and
t. See also Brillinger (1981). By inserting (2) into (1), the model can be expressed as
yt =  + 0xt +
K X
j= K
0
jxt j + _ vt (3)
where _ vt = vt +
P
jjj>K 0
j u2t j and K is known as the lead-lag truncation parameter.
Saikkonen (1991) showed that the OLS estimator of  based on (3) does not suer from the
second-order bias and is ecient in a certain class of distributions.
2Let us consider the case where
j = 0 for 8j < 0: (4)
In this case, the model becomes
yt =  + 0xt +
K X
j=0
0
jxt j + _ vt (5)
and then we do not have to include the leads of xt as regressors. We therefore expect
an improvement of the nite sample eciency by estimating (5) because we do not have to
include extra regressors. In this case, we note that condition (4) is related to the concept
of Granger causality. According to Sims (1972) and Proposition 11.3 in Hamilton (1994),
condition (4) holds if and only if u1t does not Granger-cause u2t. In other words, it is
possible to eciently estimate the cointegrating regression model without any leads of the
rst dierence of integrated variables if the past values of u1t do not help to predict u2t.
Therefore, we recommend that the null of Granger non-causality be tested before estimating
the cointegrating regression model.
Tests for Granger non-causality can be conducted by approximating the process of ut by
a nite-order vector autoregressive model: ut = 	1ut 1 + 	2ut 2 +  + 	put p + et. Let
^ ut = [^ u1t;u0
2t]0, where ^ u1t = u1t   (^    )0zt is the regression residual from (1) with ^  as the
OLS estimator of . We then estimate
^ ut = 	1^ ut 1 + 	2^ ut 2 +  + 	p^ ut p + ^ et (6)
and test the hypothesis that 	1;21 = 	2;21 =  = 	p;21 = 0 where 	j;21 is the (2;1) block
of 	j and ^ et = et   (In+1   	1L      	pLp)[z0
t(^    );0]0 and L being the lag operator.
Although ^ u1t includes an estimation error, its eect is asymptotically negligible. In fact, we
can show that for j =  p; ;p,
1
T
X
1t;t jT
^ u1t^ u1t j =
1
T
X
1t;t jT
u1tu1t j + Op

1
T

;
1
T
X
1t;t jT
^ u1tu0
2t j =
1
T
X
1t;t jT
u1tu0
2t j + Op

1
T

while for j  0,
1
p
T
X
1t;t jT
^ ut j^ e0
t =
1
p
T
X
1t;t jT
ut je0
t + Op

1
p
T

:
by the asymptotic technique explained in, for example, Chapters 17{19 in Hamilton (1994).
If the evidence of Granger non-causality is observed by tests based on (6), we can expect the
nite sample eciency gain by excluding the leads of xt from (3) and estimating (5).
We may also consider verifying condition (4) by investigating whether or not the regres-
sion error from (5) is serially uncorrelated. For this purpose, the portmanteau tests are
available as explained in L utkepohl (1993).
3To demonstrate the case where the model can be expressed as (5), we consider the
following case:
u1t =
1 X
j=0
j"1t j and u2t = "2t where
1 X
j=0
jjjj < 1
and "t =
2
4 "1t
"2t
3
5  iid
2
4
0
@ 0
0
1
A;
0
@ 11 12
21 22
1
A
3
5: (7)
We then decompose "1t as
"1t = "12t + ~ "2t (8)
where "12t = "1t   12
 1
22 "2t and ~ "2t = 12
 1
22 "2t. Note that "12t is uncorrelated with all
the leads and lags of "2t and ~ "2t. Using this decomposition, u1t can be expressed as
u1t =
1 X
j=0
j"12t j +
K X
j=0
j~ "2t j +
1 X
j=K+1
j~ "2t j
= vt +
K X
j=0
0
jxt j +
1 X
j=K+1
j~ "2t j (9)
where 0
j = j12
 1
22 and vt =
P1
j=0 j"12t j. Since E("2s"12t) = 0 for all s and t, it is
evident that vt is uncorrelated with xt j for all j. The regression form in (5) is obtained
by inserting (9) into (1).
To investigate the nite sample performance of the dynamic OLS estimator without
leads, we conduct a Monte Carlo experiment. We consider the case of n = 1 and assume
that u1t follows a rst-order autoregressive model with the AR coecient , while u2t is an
iid sequence. We set T = 100, 11 = 22 = 1, 12 = 21 = 0:4;0:8, and  = 0:1;0:5;0:9.
The computation was conducted by using the GAUSS matrix language, and the number of
replications is 10,000 for all the cases. The simulation results are summarized in Table 1
(further simulation results are available from the author upon request).
For the choice of K, we use the general-to-specic method by Ng and Perron (1995) with
1% and 5% signicant levels and information criteria, i.e., the Akaike information criterion
(AIC) and the Bayesian information criterion (BIC).
From Table 1, we observe that the dynamic OLS estimator without leads substantially
outperforms that with leads and lags in all the cases. In particular, in terms of the MSE,
the MSE of the DOLS estimator without leads are approximately half of that with leads and
lags in many cases.
3 Conclusion
In this paper, we considered the role of leads of the rst dierence of the I(1) regressors in
the dynamic OLS estimation. We demonstrated that leads are not necessary in cointegrating
4regression models when the cointegrating regression error does not Granger-cause the rst
dierence of the I(1) regressors. Based on a Monte Carlo simulation, we found that the
dynamic OLS estimator without leads substantially outperforms that with leads and lags
when leads are, in fact, unnecessary.
References
[1] Brillinger, D. R.,1981, Time Series Data Analysis and Theory. (Holden-Day, San Fran-
cisco).
[2] Engle, R. F. and C. W. J. Granger, 1987, Co-integration and error correction: Repre-
sentation, estimation, and testing, Econometrica 55, 251-276.
[3] Hamilton, J. D., 1994, Time Series Analysis. (Princeton, New Jersey).
[4] L utkepohl, H., 1993, Introduction to Multiple Time Series Analysis, 2nd. (Springer-
Verlag, New York).
[5] Ng, S. and P. Perron, 1995, Unit root tests in ARMA models with data-dependent
methods for the selection of the truncation Lag, Journal of the American Statistical
Society 90, 268-281.
[6] Park, J. Y., 1992, Canonical cointegrating regressions, Econometrica 60, 119-143.
[7] 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.
[8] Phillips, P. C. B. and M. Loretan, 1991, Estimating long-run economic equilibria,
Review of Economic Studies 58, 407-436.
[9] Saikkonen, P., 1991, Asymptotically ecient estimation of cointegration regressions,
Econometric Theory 7, 1-21.
[10] Sims, C. A., 1972, Money, income, and causality, The American Economic Review 62,
540-552.
[11] Stock, J. H. and M. W. Watson, 1993, A simple estimator of cointegrating vectors in
higher order integrated systems, Econometrica 61, 783-820.
5Table 1: Simulation Results
T = 100; 12 = 0:4 GS001 GS005 AIC BIC
 L&L Lags L&L Lags L&L Lags L&L Lags
BIAS 0.00231 0.00191 0.00138 0.00076 0.00156 0.00062 0.00229 0.00174
0.1 Std. Dev. 0.05592 0.03787 0.07115 0.04301 0.04820 0.03975 0.03448 0.03486
MSE 0.00313 0.00144 0.00506 0.00185 0.00233 0.00158 0.00119 0.00122
BIAS 0.01305 0.01239 0.00525 0.00482 0.00678 0.00378 0.01690 0.01178
0.5 Std. Dev. 0.09897 0.06823 0.12299 0.07647 0.10103 0.07482 0.06346 0.06415
MSE 0.00997 0.00481 0.01515 0.00587 0.01025 0.00561 0.00431 0.00425
BIAS 0.09278 0.09473 0.07786 0.07110 0.08037 0.06462 0.11620 0.09531
0.9 Std. Dev. 0.39449 0.27095 0.46348 0.29522 0.46084 0.30779 0.29436 0.27233
MSE 0.16423 0.08239 0.22087 0.09221 0.21883 0.09891 0.10015 0.08324
T = 100; 12 = 0:8 GS001 GS005 AIC BIC
 L&L Lags L&L Lags L&L Lags L&L Lags
BIAS 0.00244 0.00268 0.00043 0.00083 0.00158 0.00068 0.00385 0.00269
0.1 Std. Dev. 0.03526 0.02498 0.04597 0.02889 0.03121 0.02612 0.02297 0.02324
MSE 0.00125 0.00063 0.00211 0.00084 0.00098 0.00068 0.00054 0.00055
BIAS 0.00913 0.00916 0.00312 0.00352 0.00460 0.00215 0.01402 0.00865
0.5 Std. Dev. 0.06438 0.04777 0.07958 0.05145 0.06938 0.05101 0.04811 0.04557
MSE 0.00423 0.00237 0.00634 0.00266 0.00483 0.00261 0.00251 0.00215
BIAS 0.14712 0.14187 0.12359 0.11137 0.12629 0.10198 0.18267 0.13415
0.9 Std. Dev. 0.30634 0.23202 0.32937 0.22818 0.33710 0.22879 0.27612 0.23326
MSE 0.11549 0.07396 0.12376 0.06447 0.12959 0.06275 0.10961 0.07240
Note: \GS001", \GS005", \AIC", \BIC" denote the dynamic OLS estimator with K chosen by the general to specic
approach with 1% and 5% signicant levels, AIC, and BIC, respectively. \L&L" denotes the dynamic OLS estimator with
leads and lags, and \Lags" denotes the dynamic OLS estimator without leads.
6

A simple estimator of cointegrating vectors in higher order integrated systems,

Asymptotically ecient estimation of cointegration regressions,

Canonical cointegrating regressions,

Co-integration and error correction: Representation, estimation, and testing,

Estimating long-run economic equilibria,

Introduction to Multiple Time Series Analysis,

Money, income, and causality,

Statistical inference in instrumental variables regression with I(1) processes,

Time Series Analysis.

Time Series Data Analysis and Theory. (Holden-Day,

Unit root tests in ARMA models with data-dependent methods for the selection of the truncation Lag,

The Role of "Leads" in the Dynamic OLS Estimation of Cointegrating Regression Models

http://hi-stat.ier.hit-u.ac.jp/research/discussion/2006/pdf/D06-194.pdf

The Role of "Leads" in the Dynamic OLS Estimation of Cointegrating Regression Models

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics