Structural Spurious Regressions and A Hausman-type Cointegration Test
Choi, Chi-Young, Ling Hu, and Masao Ogaki
Working Paper No. 517
May 2005

UNIVERSITY OF

ROCHESTER

Structural Spurious Regressions and
A Hausman-type Cointegration Test∗
Chi-Young Choi
Department of Economics
University of New Hampshire

Ling Hu
Department of Economics
Ohio State University

Masao Ogaki
Department of Economics
Ohio State University
April, 2005

Abstract
This paper proposes two estimators based on asymptotic theory to estimate structural parameters with spurious regressions involving unit-root nonstationary variables. This approach motivatesa Hausman-type test for the null hypothesis of cointegration for dynamic Ordinary Least Squares
estimation using one of our estimators for spurious regressions. We apply our estimation and testing
methods to four applications: (i) long-run money demand in the U.S.; (ii) long-run implications of
the consumption-leisure choice; (iii) output convergence among industrial and developing countries;(iv) Purchasing Power Parity for traded and non-traded goods.
Keywords: Spurious regression, GLS correction method, Dynamic regression, Test for cointegration.
JEL Classification Numbers: C10, C15
Corresponding Author: Masao Ogaki, Department of Economics, Ohio State University, Columbus, Ohio 43210-1172; e-mail: ogaki.1@osu.edu∗ We thank Don Andrews, Yanqin Fan,

Clive Granger, Lars Hansen, Kotaro Hitomi, Hide Ichimura, Lutz Kilian,
Yuichi Kitamura, Yoshihiko Nishiyama, Peter Phillips, Ping Wang, Arnold Zellner, and seminar participants at the Bankof Japan, Kyoto University, Ohio State University, University of Tokyo, National Chung Cheng University, Vanderbilt
University, Yale University, the 2004 Far Eastern Meeting of the Econometric Society, and the 2004 Annual Meeting of theMidwest Econometrics Group for helpful comments. Choi gratefully acknowledges the financial support through Summer
Fellowship from the University of New Hampshire.

1

Introduction

Economic models often imply that certain variables are cointegrated. However, tests often fail to reject
the null hypothesis of no cointegration for these variables. One possible explanation of these test results
is that the error is unit-root nonstationary due to a nonstationary measurement error in one variable ornonstationary omittted variables. In the unit-root literature, when the stochastic error of a regressionis unit-root nonstationary, the regression is technically called a spurious regression. This is because thestandard t-test tends to be spuriously significant even when the regressor is statistically independent of
the regressand in Ordinary Least Squares. Monte Carlo simulations have often been used to show thatthe spurious regression phenomenon occurs with regressions involving unit-root nonstationary variables(see, e.g., Granger and Newbold (1974), Nelson and Kang (1981, 1983)). Asymptotic properties of
estimators and test statistics for regression coefficients of these spurious regressions have been studied
by Phillips (1986, 1998) and Durlauf and Phillips (1988) among others.
The purpose of this paper is twofold. First, we propose a new approach to estimating structuralparameters with spurious regressions. When structural parameters can be recovered from spurious
regressions, we call these structural spurious regressions. Second, we propose a Hausman-type test
for the null hypothesis of cointegration. This test is naturally motivated by the structural regression
approach. We also show that this test can be used to test for cointegration even when the spurious
regression is not structural under the alternative hypothesis.
As an example of a structural spurious regression, consider a regression to estimate the money
demand function when money is measured with a nonstationary error. Currency held by domestic
economic agents for legitimate transactions is very hard to measure since currency is held by foreign
residents and is also used for black market transactions. Therefore, money may be measured with anonstationary error. As shown by Stock and Watson (1993) among others, under the assumption thatthe money demand function is stable in the long-run, we have a cointegrating regression if all variables
are measured without error. If the variables are measured with stationary measurement errors, we still
have a cointegrating regression. However, if money is measured with a nonstationary measurement error,
we have a spurious regression. We can still recover structural parameters under certain conditions. The
crucial assumption is that the nonstationary measurement error is not cointegrated with the regressors.
Another example of a structural spurious regression is a regression of money demand with nonstationary omitted variables. Consider the case in which the money demand is stable in the long-run
when a measure of shoe leather costs of holding money is included as an argument. If an econometrician omits the measure of shoe leather costs from the money demand regression, and if the measure isnonstationary, the regression error is nonstationary. Shoe leather costs of holding money are related
to the value of time, and therefore to the real wage rate. Because the real wage rate is nonstationary
in standard dynamic stochastic general equilibrium models with a nonstationary technological shock,the omitted measure of the shoe leather costs is likely to be nonstationary. In this case, the money
demand regression that omits the measure is spurious, but we can still recover structural parameters
under certain conditions. The crucial assumption is that the omitted variable is not cointegrated withthe regressors.
Our structural spurious regression approach is based on the Generalized Least Squares (GLS) solution of the spurious regression problem analyzed by Ogaki and Choi (2001),1 who use an exact small
1 Another approach would be to take the first difference to induce stationarity and then use instrumental variables.

This is the approach proposed by Lewbel and Ng (2005) for their Nonstationary Translog Demand System. Our approach
exploits the particular form of endogeneity assumed by many authors in the cointegration literature and avoids the useof instrumental variables. Our approach yields more efficient estimators as long as the particular form of endogeneity iscorrectly specified. This is important especially when weak instruments cause problems.

1sample analysis based on the conditional probability version of the Gauss-Markov Theorem. We develop asymptotic theory for two estimators motivated by the GLS correction: GLS corrected dynamic
regression and feasible GLS (FGLS) corrected dynamic regression estimators. These estimators will
be shown to be consistent and asymptotically normally distributed in spurious regressions. When theerror term is in fact stationary, and hence the variables are cointegrated, the GLS corrected estimatoris not efficient, but the FGLS corrected estimator, like the OLS estimator, is superconsistent. Hencethe FGLS estimation is a robust procedure with respect to the error specification. The FGLS corrected
estimator is asymptotically equivalent to the GLS corrected estimator in spurious regressions and it isasymptotically equivalent to the OLS estimator in cointegrating regressions.
In some applications, it is hard to determine whether or not the error in the regression is stationary orunit-root nonstationary because test results are inconclusive. In such applications, the FGLS corrected
estimator is attractive because it is consistent in both situations as long as the method of the dynamic
regression removes the endogeneity problem.
This approach naturally motivates a Hausman-type test2 for the null hypothesis of cointegration
against the alternative hypothesis of no cointegration (or a spurious regression) in the dynamic OLS
framework. We construct this test by noting that while both the dynamic OLS and GLS corrected
dynamic regression estimators are consistent in cointegration estimation, the dynamic OLS estimatoris more efficient.3 On the other hand, when the regression is spurious only the GLS corrected dynamic
regression estimator is consistent. Hence, we could do a cointegration test based on the specifications
on the error. We show that under the null hypothesis of cointegration the test statistics have a usualχ2 limit distribution, while under the alternative hypothesis of a spurious regression, the test statistic
diverges.
In some applications, the assumption that the spurious regression is structural under the alternative
hypothesis is not very attractive. If the violation of cointegration arises from reasons other than anonstationary measurement error, it is hard to believe that the resulting spurious regression is structural.
For this reason, we relax the assumption that the spurious regression is structural and show that theHausman-type cointegration test statistic still diverges under the alternative hypothesis.
Dynamic OLS is used in many applications of cointegration. However, few tests for cointegration
have been developed for dynamic OLS with the exception of Shin’s (1994) test. As in Phillips andOuliaris (1990), the popular Augmented Dickey-Fuller (ADF) test for the null hypothesis of no cointegration was originally designed to be applied to the residual from static OLS rather than the residualfrom dynamic OLS. Because the OLS and dynamic OLS estimates are often substantially different, it
is desirable to have a test for cointegration applied to dynamic OLS. Another aspect of our Hausman
type test is that it is for the null hypothesis of cointegration. Ogaki and Park (1998) argue that itis desirable to test the null hypothesis of cointegration rather than that of no cointegration in many
applications when economic models imply cointegration.
Using Monte Carlo experiments, we compare the finite sample performance of the Hausman-type
test with the test proposed by Shin (1994), which is a locally best invariant test for the null of zerovariance of a random walk component in the disturbances. The experiment results show that up to thesample size of 300 the Hausman-type test is dominant in both size and power. When the sample size
increases, Shin’s test is better in power, but it suffers from a serious oversize problem.
2 This test can also be called Durbin-Wu-Hausman-type test as it is closely related to ideas and tests in Durbin (1954)
and Wu (1973) as well as a family of tests proposed by Hausman (1978).
3 After completing the first draft, it has come to our attention that the Hausman-type test was originally proposed by
Fernández-Macho and Mariel (1994) for the static OLS cointegrating regression with strict exogeneity and without any
serial correlation. The test probably has not been popular because these assumptions are hard to justify in applications
and because the test was not developed for dynamic regressions.

2

In some applications, it is appropriate to consider the possibility that measurement error is I(1)and is not cointegrated with the regressors. For these applications, the ADF test is applicable under
the null hypothesis of a structural spurious regression as shown in Hu and Phillips (2005). For suchapplications, we recommend that both the ADF test and Hausman-type test be applied because it is
not clear which null hypothesis is more appropriate.
We apply our estimation and testing methods to four applications: (i) long-run money demand in
the U.S.; (ii) long-run implications of the consumption-leisure choice; (iii) output convergence amongindustrial and developing countries; (iv) Purchasing Power Parity (PPP) for traded and non-traded
goods. In the first two applications, we focus on estimating unknown structural parameters, while in
the last two applications we purport to test for cointegration with the Hausman-type cointegration
test, where we relax the assumption that the spurious regression under the alternative hypothesis is
structural.
The rest of the paper is organized as follows. Section 2 gives econometric analysis of the model,
including asymptotic theories and finite sample simulation studies. Section 3 presents models of nonstationary measurement error and nonstationary omitted variables. It then presents the empirical resultsof four applications. Section 4 contains concluding remarks.

2

The econometric model
Consider the regression modelyt = β 0 xt + ηt ,(1)

where {xt } is an m-vector integrated process generated by∆xit = vit .
The error term in (1) is assumed to be
ηt

=

m X
k
X

γi,j vi,t−j + et ,(2)

i=1 j=−ket= ρet−1 + ut .

(3)

Assumption 1 Assume that vt = (v1t , . . . , vmt )0 and ut are zero mean stationary processes with
E|vit |α < ∞, E|ut |α < ∞ for some α > 2 and strong mixing with size −α/(α − 2). We also assume that the method of dynamic regression removes the endogeneity problem, that is, E(ut vs ) = 0 for
all t, s. We call this the strict exogeneity assumption for the dynamic regression.
P[nr]The conditions on vt and ut ensure the invariance principles: for r ∈ [0, 1], n−1/2 t=1 vt →d V (r),
P[nr]
P∞
0
n−1/2 t=1 ut →d U (r), where V (r) is an m-vector Brownian motion with covariance j=−∞ E(vt vt−j),P∞
and U (r) is a Brownian motion with variance j=−∞ E(ut ut−j ). The functional central limit theorem
holds for weaker assumptions than assumed here (De Jong and Davidson (2000)), but the conditions assumed above are general enough to include many stationary Gaussian or non-Gaussian ARMA processes
that are commonly assumed in empirical modeling.
Let vt = (∆x1,t−k , . . . , ∆x1,t , . . . , ∆x1,t+k , . . . , ∆xm,t−k , . . ., ∆xm,t , . . ., ∆xm,t+k )0 , and γ = (γ1,−k ,. . . , γ1,0 , . . . , γ1,k , . . . , γm,−k , . . . , γm,0 , . . . , γm,k )0 . We estimate the structural parameter β in the regression
yt = β 0 xt + γ 0 vt + et .
(4)3

The inference procedure about β differs according to different assumptions on the error term et in(3). When |ρ| < 1, et is stationary, and hence regression (4) is a cointegration regression with seriallycorrelated error. When ρ = 1, et is a unit root nonstationary process and the OLS regression is spurious.
Both models are important in empirical studies in macroeconomics and finance.
In the next two sections, we will study the asymptotic properties of different estimation procedures
under these two assumptions. Under the assumption that ρ = 1, OLS is not consistent while both GLS
correction and FGLS correction will give consistent and asymptotically equivalent estimators. Under√
the assumption that |ρ| < 1, the GLS corrected estimator is not efficient as it is n convergent, but
the FGLS estimator is n convergent and asymptotically equivalent to the OLS estimator. Therefore,
FGLS is robust with respect to the error specifications (ρ = 1 or |ρ| < 1).

2.1

Regressions with I(1) error

In this section we consider the situation when the error term is I(1), i.e., ρ = 1 in (3). The estimation
methods we study are dynamic OLS, GLS correction, and FGLS correction.
2.1.1

The dynamic OLS spurious estimation

We start with dynamic OLS estimation of regression (4). Under the assumption of ρ = 1, thisregression is spurious since for any value of β the error term is always I(1). In Appendix A, we showthat the DOLS estimator β̂dols has the following limit distribution:
Z 1(β̂dols − β0 ) →0−1 Z 1

V (r)V (r) dr0
V (r)U (r)dr .

(5)0γ̂ in the estimation is also inconsistent with γ̂ − γ 0 = Op (1). As remarked in Phillips (1986, 1989),
in spurious regressions the noise is as strong as the signal. Hence, uncertainty about β persists in thelimiting distributions.
2.1.2

GLS corrected estimation

When ρ = 1, we can filter all variables in regression (4) by taking the full first difference, and use OLS
to estimate∆yt = β 0 ∆xt + γ 0 ∆vt + ut = θ0 ∆zt + ut ,(6)
where θ = (β 0 , γ 0 )0 and zt = (x0t , vt0 )0 . This procedure can be viewed as GLS corrected estimation.4
If we let θ̃dgls denote the GLS corrected estimator, then we can show that√n(θ̃dgls − θ0 ) →d N (0, Ω),(7)where Ω = Q−1 ΛQ−1 with Q = E(zt zt0 ) and Λ is the long run variance matrix of zt ut . Thus β ina structural spurious regression can be consistently estimated (jointly with γ) and the estimators are
asymptotically normal. In the special case when m = 1, {v1t } and {ut } are i.i.d. sequences and ηt = et ,(7) gives that√2n(θ̃dgls − θ0 ) →d N (0, σu2 /σ1v),2where σu2 and σ1v
are variances of ut and v1t , respectively.
4 This is a conventional GLS procedure when u is i.i.d.. When u is serially correlated as in our approach, we namettthis procedure GLS corrected dynamic estimation.

42.1.3

The FGLS estimation

To use GLS to estimate a regression with serial correlation in empirical work, a Cochrane-Orcutt FGLS
procedure is usually adopted. This procedure also works for spurious regressions as shown by Phillips
and Hodgson (1994). They show that the FGLS estimator is asymptotically equivalent to that in the
differenced regression when the error is unit-root nonstationary. In the present paper, we will show that
the FGLS correction to the dynamic regression provides a consistent and robust estimator to structural
spurious regressions.
Let the residual from OLS regression (4) be denoted by êt ,êt = yt − β̂n0 xt − γ̂ 0n vt .
To conduct the Cochrane-Orcutt GLS estimation, we first run an AR(1) regression of êt ,êt = ρ̂n êt−1 + ût .

(8)

It can be shown that n(ρ̂n − 1) = Op (1). Conduct the following Cochrane-Orcutt transformation of thedata:ỹt = yt − ρ̂n yt−1 , x̃t = xt − ρ̂n xt−1 , ṽt = vt − ρ̂n vt−1 .
(9)Then consider OLS estimation of the regressionỹt = β 0 x̃t + γ 0 ṽt + error = θ0 z̃t + error,
where z̃t = (x̃0t , ṽt 0 )0 . The OLS estimator of θ in (10) is computed as" n#−1 " n#X
X
0
θ̃fgls =z̃t z̃tz̃t ỹt .
t=1

(10)

(11)

t=1

The limiting distribution of θ̃fgls can be shown to be the same as in (7). Intuitively, even though
the dynamic OLS estimator is inconsistent, the residual is unit-root nonstationary because no linear
combination of yt and xt is stationary. Therefore, ρn approaches unity in the limit, and z̃t behaves
asymptotically equivalent to ∆zt . A detailed proof of results in this section is given in Appendix A.

2.2

Regressions with I(0) error

In this section we consider the asymptotic distributions of the three estimators (DOLS estimator,
GLS corrected estimator and FGLS corrected estimator) under the assumption of cointegration, i.e.,|ρ| < 1 (3).
2.2.1

The dynamic OLS estimation

Under the assumption of cointegration, the DGP of yt is
yt = β 0 xt + γ 0 vt + et ,et = ρet−1 + ut , |ρ| < 1.
(12)
P
[nr]
Applying the invariance principle, for r ∈ [0, 1], n−1/2 t=1 et → E(r), where E(r) is a BrownianP∞
motion with variance j=−∞ E(et et−j ). The limiting distribution of the OLS estimator of β, which is
asymptotically independent of γ̂ n , is known to beZ 1n(β̂dols − β0 ) →d0

−1 Z 1

V (r)V (r) dr
0

0

5


V (r)dE(r) .

(13)

2.2.2

GLS corrected estimation

Now, if we take the full first difference as we did in the spurious regressions, the regression becomes∆yt = β 0 ∆xt + γ 0 ∆vt + et − et−1 = θ0 ∆zt + et − et−1 .

(14)√Note that this transformation leads to loss in efficiency since the estimator β̃dgls is now n convergent
rather than n convergent as the DOLS estimator. With some minor revisions to equation (7), thelimiting distribution of the estimator in this case can be written as√n(θ̃dgls − θ0 ) →d N (0, Ω∗ ),(15)
where Ω∗ = Q−1 Λ∗ Q−1 . Q is defined as following equation (7) and Λ∗ is the long run variance matrixof vector zt ∆et . In the special case when m = 1, {v1t } and {ut } are i.i.d. sequences and ηt = et ,2Ω∗ = 2σe2 (1 − ψe )/σ1v, where ψe is the first order autocorrelation coefficient of {et }.
2.2.3

The FGLS estimation

Instead of taking the full first difference, if we estimate the autoregression coefficient in the error anduse this estimator to filter all sequences, we will obtain an estimator that is asymptotically equivalent
to the DOLS estimator. Intuitively, in the case that the error et = ut is serially uncorrelated, then theAR(1) coefficient ρ̂n will converge to zero, and hence the transformed regression will be asymptoticallyequivalent to the original regression. Or, if the error is stationary and serially correlated, then theAR(1) coefficient will be less than unity, and, as shown in Phillips and Park (1988), the GLS estimatorand the OLS estimator in a cointegration regression are asymptotically equivalent.
If we conduct the Cochrane-Orcutt transformation (9) and estimate β in regressionỹt = β̃n0 x̃t + γ̃ 0n ṽt + error,(16)

then Appendix B shows that the limiting distribution of β̃n is the same as the limit of the OLS estimatorgiven in (13). We summarize the FGLS corrected estimator in the following proposition:
Proposition 1 Suppose Assumption 1 holds. In spurious regressions, the FGLS corrected estimator is
asymptotically equivalent to the GLS corrected estimator, and its limit distribution can be written as√n(θ̃fgls − θ0 ) →d N (0, Ω).
In cointegration regressions, the FGLS corrected estimator is asymptotically equivalent to the DOLS
estimator, and its limit distribution can be written asZ 1n(β̂fgls − β0 ) →d

V (r)V (r)0 dr

0−1 Z 1
V (r)dE(r) .

0

So FGLS is not only valid when the regression is spurious but also asymptotically efficient when the
regression is cointegration.
Remarks: 1. If a constant is added to (4), we show in Appendix D that our methods still apply.
In particular, the GLS or FGLS corrected estimators are asymptotically equivalent to that given in (7)under the assumption of spurious regressions.
2. If a trend term is added to (4) (this is the case in which the deterministic cointegration restriction
is not satisfied in the terminology of Ogaki and Park (1988)), then the GLS corrected estimation leads
to a singular covariance matrix for the estimator when ρ is less than one in absolute value. This is6because a trend term in (4) leads to a constant term in the first differenced regression (14) and because
the long run variance of the first difference of et multiplied by a constant is zero. Therefore, our methods
do not apply to regressions with time trends.
3. Under some conditions, the methods proposed in this paper also apply to other model configurations, such as regressions where the regressors have drifts. These extensions will be studied in future
work.

2.3

Finite sample performance of the three estimators

From the above analysis, we show that the FGLS corrected estimation is a robust procedure withrespect to error specifications. In this section, we use simulations to study its finite sample performances
compared to the other two estimators. In the simulation we consider the case when xt is a scalar variable
and we generate vt and ut from two independent standard normal distributions and let et = ρet−1 + ut .
The structural parameter is set to β = 2, and γ 0 vt = 0.5vt . The number of iterations in each simulation
is 5000 and in each replication, 100 + n observations are generated of which the first 100 observations
are discarded.
Table 1 shows the bias and the mean square error (MSE) of all three estimators for ρ = 0, 0.95, and
1 . When ρ = 0, the regression is cointegration with i.i.d. error. It is clear that the DOLS estimator isthe best one when n = 50. But when n reaches 100, the FGLS estimator becomes almost as good as
the DOLS estimator. When ρ = 0.95, the regression is cointegration with serially correlated error. In
this case, the GLS and FGLS estimators are much better than the DOLS estimator. When the sample
size increases, the FGLS estimator becomes the best one. Finally when ρ = 1, the regression is spurious
and the GLS corrected estimator performs the best as expected.
Figure 1 plots the three estimators when n = 100 as ρ approaches 1. The figures show that the
DOLS estimator becomes flatter and flatter as ρ → 1. The GLS estimator remain largely the same forρ close to unity. And the FGLS estimator becomes a bit flatter when ρ reaches 1, but it still shows aclear peak around zero.
¿From the finite sample performance, it can be seen that the FGLS estimator is almost as good as
the DOLS estimator in cointegrations and it significantly outperforms the DOLS estimator in spurious
regressions. The GLS estimator is the best when ρ approaches 1, but it suffers from significant loss inefficiency when ρ is small. So we may want to take full difference only when we are very sure that theerror is unit root nonstationary. Otherwise the FGLS estimator is a good choice.

2.4

Hausman specification test for cointegration

2.4.1

The test statistic and its asymptotic properties

In this section we construct a Hausman-type cointegration test based on the difference of two
estimators: an OLS estimator (β̂dols ) and a GLS corrected estimator (β̃dgls ). This is equivalent tocomparing estimators in a level regression and in a differenced regression. The test is for the null of
cointegrating relationships against the alternative of a spurious regression:
H0 : |ρ| < 1; against HA : ρ = 1.
Our discussions so far suggest that under the null of cointegration, both OLS and GLS are consistent
but the OLS estimator is more efficient. However under the alternative of a spurious regression, only
the GLS corrected estimator is consistent.

7dynamic OLS regression estimator4
rho = 0.98
rho = 0.99
rho = 1

3
2
1
0
−1

−0.8

−0.6

−0.4
−0.2
0
0.2
0.4
GLS corrected dynamic regression estimator

0.6

0.8

1

4
rho = 0.98
rho = 0.99
rho = 1

3
2
1
0
−1

−0.8

−0.6

−0.4
−0.2
0
0.2
0.4
FGLS corrected dynamic regression estimator

0.6

0.8

1

4
rho = 0.98
rho = 0.99
rho = 1

3
2
1
0
−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 1: Comparison of three estimators when n = 100 and ρ → 1

8

0.8

1

√
Let V̂β denote a consistent estimator for the asymptotic variance of n(β̃dgls − β). Under ourassumptions, it converges to the corresponding submatrix of Ω∗ under the null hypothesis and to thecorresponding submatrix of Ω under the alternative. For example, when m = 1, {v1t } and {ut } are


Pn
Pn
independent i.i.d. and ηt = et , take V̂β = n1 t=1 ŵt2 / n1 t=1 ∆x2t , where ŵt denotes the residuals
2from OLS estimation of differenced regression. Under the null of cointegration, V̂β → 2σe2 (1 − ψe )/σ1v.
2
2
Under the alternative of spurious regression, V̂β → σu /σ1v .
We define the Hausman-type test statistic as:
hn = n(β̃dgls − β̂dols )0 V̂β−1 (β̃dgls − β̂dols ).

(17)

Proposition 2 Suppose Assumption 1 holds. Under the null hypothesis of cointegrations, hn → χ2 (m).
Under the alternative of spurious regressions, hn = Op (n).
Proof: under the null of cointegration,
√
=
=

√
√

n(β̃dgls − β̂dols )n(β̃dgls − β0 ) −√n(β̂dols − β0 )n(β̃dgls − β0 ) + op (1)→ N (0, Vβ ),where Vβ is the asymptotic variance of β̃dgls under the assumption of cointegration. Therefore, if V̂β isa consistent estimator for Vβ ,hn = n(β̃dgls − β̂dols )0 (V̂β )−1 (β̃dgls − β̂dols ) → χ2 (m).
Under the alternative of spurious regressions,√n(β̃dgls − β̂dols )√√n(β̃dgls − β0 ) − n(β̂dols − β0 )=√= Op (1) + Op ( n)√= Op ( n).
Hence, hn = Op (n) under the alternative.
We can extend the test to allow endogeneity under the alternative. Consider the following DGP:yt

= β 0 xt + γ 0 vt + φst + et ,et

= ρet−1 + ut ,where {st } satisfies the same conditions as ut and vt , but it is correlated with {vt }. The statistic definedin (17) can be applied to test the hypotheses:

against

H00 : |ρ| < 1and φ = 0

0
HA
:ρ=1

and φ 6= 0.

The asymptotics of hn under the null H00 are the same as that under H0 . Under the alternative0
HA , we show in Appendix C that the DOLS estimator has the same asymptotic distribution as that
under HA and hn = Op (n). Therefore, this Hausman-type test is consistent for the null hypothesisof cointegrations against the alternative of spurious regressions, regardless of whether the exogeneity
assumption holds under the alternative.
9

2.4.2Finite sample properties of the Hausman-type cointegration test

Before applying the Hausman-type cointegration test empirically, it will be instructive to examine its
finite sample properties in comparison with other comparable tests under the same null hypothesis. To
this end, we conduct a small simulation experiment based on the following dynamic regression model,yt= γ1 ∆xt+1 + βxt + γ2 ∆xt−1 + et ,

(18)

et

= ρet−1 + ut ,(19)

where γ1 = 0.3, β = 2, γ2 = −0.5, and setting ρ = 0.9 for the size performance and ρ = 1 for the power
performance. We consider sample sizes of n ∈ {50, 100, 200, 300, 500} that are commonly encounteredin empirical analysis. In the simulations pseudo random numbers are generated using the GAUSS(version 6.0) RNDNS procedures. Each simulation run is carried out with 5,000 replications. At each
replication 100 + n random numbers are generated of which the first 100 observations are discarded to
avoid a start-up effect.
Table 2 reports selected finite sample properties of the Hausman-type cointegration test together
with a residual based test under the null of cointegration due to Shin (1994, Shin’s test) who extended
the KPSS test in the parametrically corrected cointegrating regression. In the simulations the lengthsof lead and lag terms for DOLS and DGLS are chosen by the BIC rule.5 A nonparametric estimation method for long run variance estimation is employed using the QS kernel with the bandwidthof ‘integer[8(n/100)1/4 ]’. The results in Table 2 illustrate two points. First, the empirical size of theHausman-type test is close to the nominal size, in particular when sample size is relatively large, whereas
Shin’s test suffers from a serious oversize problem. Second, in terms of power, the Hausman-type test
dominates Shin’s test for moderate sample sizes that are very likely to be encountered empirically.
Shin’s test seems more powerful when n is relatively large, but only at the cost of severe size distortions. Overall, our simulation results provide evidence in favor of the Hausman-type test.

3

Empirical applications

In this section we apply the GLS-type correction methods and the Hausman-type cointegration test to
analyze four macroeconomic issues: (i) long-run money demand in the U.S.; (ii) long-run implications of
the consumption-leisure choice; (iii) output convergence among industrial and developing countries; (iv)
Purchasing Power Parity (PPP) for traded and non-traded goods. Tthe main purpose of the first two
applications is to illustrate the spurious regression approach to estimating unknown structural parameters. Identification of the structural parameters in these two applications are based on nonstationary
measurement error or nonstationary omitted variables that are explained in the following two subsections. The main purpose of the last two applications is to apply the Hausman-type cointegration test.
The alternative hypothesis is not taken as structural spurious regressions in the last two applications.

3.1

A model of nonstationary measurement error

Consider a set of variables that are cointegrated. One model of a structural spurious regression is basedon the case in which one of the variables is measured with nonstationary measurement error. Let yt0 be
the true value of yt , and assume that
5 It is an interesting research topic to investigate the performance of various lag length selection rules, but would bebeyond the scope of this paper.

10yt0 = β 0 xt + γ 00 vt + e0t(20)

is a dynamic cointegrating regression that satisfies the strict exogeneity assumption.6 Let yt be themeasured value of yt0 , and assume that the measurement error satisfies
yt − yt0 = γ m0 vt + em
t ,

(21)

where emt is I(1) and its expectation conditional on xs for all s is zero. Here, the crucial assumption for
identification is that the measurement error is not cointegrated with xt . Thenyt = β 0 xt + γ 0 vt + et(22)

where γ = γ 0 + γ m , and et is I(1) and satisfies the strict exogeneity assumption.7

3.2

A model of nonstationary omitted variables

Another case that leads to a structural spurious regression is a model of nonstationary omitted variables.
yt = β 0 xt + θ0 x0t + γ 10 vt + γ 00 vt0 + e0t(23)

where x0t is a vector of I(1) variables and vt0 is a vector of leads and lags of the first differences of x0t .
We imagine that the econometrician omits x0t from his regression. We assume that
θ0 x0t + γ 00 vt0 = γ m0 vt + em
t ,

(24)

where emt is I(1) and its expectation conditional on xs for all s is zero. Here, the crucial assumption for
identification is that the x0t is not cointegrated with xt . Thenyt = β 0 xt + γ 0 vt + et(25)

where γ = γ 1 + γ m , and et is I(1) that satisfies the strict exogeneity assumption.

3.3

U.S. money demand

The long-run money demand function has often been estimated under a cointegrating restriction among
real balances, real income, and the interest rate. The restriction is legitimate if the money demand
function is stable in the long-run and if all variables are measured without nonstationary error. Indeed,
Stock and Watson (1993) found supportive evidence of stable long-run M1 demand by estimating
cointegrating vectors. However, either if money is measured with a nonstationary measurement error or
if nonstationary omitted variables exist, then we have a spurious regression and the estimation results
based on a cointegration regression are questionable.
First, consider the model of a nonstationary measurement error described above. To be specific, we
follow Stock and Watson (1993) and assume that the dynamic regression error is stationary and thestrict exogeneity assumption holds for the dynamic regression error when money is correctly measured.
We then assume that money is measured with a multiplicative measurement error. We assume that thelog measurement error is unit-root nonstationary, and that the residuals of the projection of the log
6 Note that any variable can be chosen as the regressand in a cointegrating regression. Therefore, we choose the variablewith nonstationary measurement error as the regressand.
7 Here, we assume that the dimensions of γ 0 and γ m are the same without loss of generality, because we can add zeros
as elements of γ 0 and γ m as needed.

11measurement error on the leads and lags of the regressors in the dynamic regression satisfy the strict
exogeneity assumption. Given that a large component of the measurement error is arguably currency
held by foreign residents and black market participants, the log measurement error is likely to be very
persistent. Therefore, the assumption that the log measurement error is unit root nonstationary may
be at least a good approximation. The assumption that the measurement error is not cointegrated withthe regressors is plausible if the error is mainly due to currency held by foreign residents.
Second, consider the model of nonstationary omitted variables. A possible omitted variable is ameasure of the “shoe leather cost” that represents transaction costs. For example, in the literature ofmoney demand estimation, the real wage rate has sometimes been used as a regressor for this reason.
Because the real wage rate is I(1) in standard dynamic stochastic general equilibrium models with anI(1) technological shock, the omitted measure of the “shoe leather cost” is to be nonstationary. If thereal wage rate is the omitted variable, the assumption that it is not cointegrated with the regressorsthat include log income is not very plausible. However, it is possible that the true omitted variable
that represents the “shoe leather cost” is not the real wage rate and that it is not cointegrated with log
income.
We apply our GLS correction methods to estimate long-run income and interest elasticities of M1
demand.8 To this end, the regression equations are set up with the real money balance ( MP ) as regressandand income (y) and interest (i) as regressors. Following Stock and Watson (1993), the annual time series
for M1 deflated by the net national product price deflator is used for M
P , the real net national productfor y and the six month commercial paper rate in percentage for i. M
P and y are in logarithms while
three different regression equations are considered depending on the measures of interest. We have tried
the following three functional forms. Equation 1 has been studied by Stock and Watson (1993).
 M= α + β ln (yt ) + γit + et , (equation 1)
ln
P t
 
M
ln
= α + β ln (yt ) + γ ln(it ) + et , (equation 2)
P t

 

1 + itM= α + β ln (yt ) + γ lnln
+ et . (equation 3)
P t
it
It is worth noting that the liquidity trap is possible for the latter two functional forms. When thedata contain periods with very low nominal interest rates, the latter two functional forms may be more
appropriate.
Table 3 presents the point estimates for β (income elasticity of money demand) and γ based on thethree estimators under scrutiny: dynamic OLS, GLS corrected dynamic regression estimator, and FGLS
corrected dynamic regression estimator.9 Several features emerge from the table. First, all the estimatedcoefficients have theoretically ‘correct’ signs: positive signs for income elasticities and negative signsfor γ for the first two functional forms and positive signs for γ for the third functional form. Second,
GLS corrected estimates of the income elasticity are implausibly low for all three functional forms forlow values of k and increase to more plausible values near one as k increases. The fact that the results
become more plausible as k increases suggests that the endogeneity correction of dynamic regressions
works in this application for moderately large values of k such as 3 and 4. The results for lower valuesof k are consistent with those of low income elasticity estimates of first differenced regressions that
8 Readers are referred to Appendix F for the empirical guidelines on the use of estimation and testing techniquesdeveloped in this paper.
9 For the FGLS corrected dynamic regression estimator, the serial correlation coefficient of the error term is estimatedbefore being applied to the Cochrane-Orcutt transformation. This coefficient is assumed to be unity in the GLS corrected
dynamic regression estimator which is equivalent to regressing the first difference of variables without a constant term.

12had been used in the literature before 1980. Therefore, the estimators in the old literature of firstdifferenced regressions before cointegration became popular are likely to be downward biased becauseof the endogeneity problem. Third, all point estimates of the three estimators are very similar, andthe Hausman-type test fails to reject the null hypothesis of cointegration for large enough values of k.
Hence, there is little evidence against cointegration. However, it should be noted that a small random
walk component is very hard to detect by any test for cointegration. Therefore, it is assuring to knowthat all three estimators are similar for large enough values of k, and the estimates are robust withrespect to whether the regression error is I(0) or I(1).
We report the value of k chosen by the Bayesian Information Criterion (BIC) rule throughout ourempirical applications in order to give some guidance in interpreting results. It is beyond the scope of
this paper to study detailed analysis of how k should be chosen because this issue has not been settled
in the literature of dynamic cointegrating regressions.
Table 3 also reports the results when the ADF test is applied to the residual of OLS. The results
show evidence against the null hypothesis of structural spurious regressions and hence corroborate theresults from the Hausman-type test under the opposite null hypothesis.

3.4

Long-run implications of the consumption-leisure choice

Consider a simplified version of Cooley and Ogaki’s (1996) model of consumption and leisure in which
the representative household maximizes∞Xδ t e(t)]U = E0 [t=0where Et denotes the expectation conditioned on the information available at t. We adopt a simple
intraperiod utility function that is assumed to be time and state-separable and separable in nondurable
consumption, durable consumption, and leisuree(t) =

C(t)1−β − 1+ v(l(t))
1−βwhere v(·) represents a continuously differentiable concave function, C(t) is nondurable consumption,and l(t) is leisure.
The usual first order condition for a household that equates the real wage rate with the marginal
rate of substitution between leisure and consumption is given as:
W (t) =v 0 (l(t))C(t)−βwhere W (t) is the real wage rate. We assume that the stochastic process of leisure is (strictly) stationary
in the equilibrium as in Eichenbaum, Hansen, and Singleton (1988). An implication of the first ordercondition is that ln(W (t)) − β ln(C(t)) = ln(v 0 (l(t))) is stationary. When we assume that the log of
consumption is difference stationary, this implies that the log of the real wage rate and the log of
consumption are cointegrated with a cointegrating vector (1, −β)0 .
We first consider the model of the measurement error. We assume that (W (t)) is measured witha multiplicative measurement error, so that ln(W (t)) is measured with a measurement error (t). We
assume that this measurement error is unit-root nonstationary. One component of the measurement
error arises because it is difficult to measure fringe benefits. Therefore, we expect the measurement error
to be very persistent, and the assumption of unit-root nonstationarity may be a good approximation.
13

We also assume that the measurement error and log consumption are not cointegrated for identificaiton.
This assumption does not rule out nonlinear long-run relationships between these variables.
Consider a regressionln(W m (t)) = a + β ln(C(t)) + e(t),(26)
where W m (t) is the measured real wage rate, and e(t) = −(t) + ln(v 0 (l(t))) − a. If (t) is stationary,then e(t) is stationary, and Regression (26) is a cointegrating regression as in Cooley and Ogaki. In
this simple version, the preference parameter β is the Relative Risk Aversion (RRA) coefficient, whichis equal to the reciprocal of the intertemporal elasticity of substitution (IES). Cooley and Ogaki showthat the same regression can be used to estimate the reciprocal of the long-run IES when preferences
for consumption are subjected to time nonseparability such as habit formation. For simplicity, weinterpret β as the RRA coefficient in this paper.
If (t) is unit-root nonstationary, then Regression (26) is a spurious regression because e(t) is nonstationary in this case. Hence, the standard methods for cointegrating regressions cannot be used.
However, the preference parameter β can still be estimated by the spurious regression method. The
key assumption for identification is that the measurement error is not cointegrated with the regressor
as discussed above. As long as the log multiplicative measurement error and log consumption are notlinearly related in the long-run by a linear cointegration relationship, the assumption is satisfied. It
should be noted that either consumption or the wage rate can be used as the regressand in the case of
cointegration. However, if the wage rate is measured with nonstationary error and consumption is not,then the wage rate should be chosen as the regressand.
Another interpretation of this application is based on the nonstationary omitted variables for thisapplication. A possible omitted variable represents demographic changes. In our analysis, per capita
real consumption series was constructed by dividing constant 1982 dollar consumption series by civilian
noninstitutional adult (age 16 and over) population. If this population series does not capture the trendof all the demographic change that affects the relationship between the real wage rate and consumption,
we have a nonstationary omitted variable. As long as the omitted demographic change and log consumption are not linearly related in the long-run by a linear cointegration relationship, the assumptionthat the omitted variable is not cointegrated with the regressor is satisfied. Another possible omitted
variable is human capital if human capital affects the marginal rate of substitution between leisure andconsumption.
Table 4 presents the estimation results for the RRA coefficient (β) based on various estimators. We
used the same data set that Cooley and Ogaki used.10 The results in Table 4 illustrate several points.
First, all point estimates for β have the theoretically correct positive sign. Second, for nondurables(ND), GLS-corrected dynamic regression estimates of β are much lower than Dynamic OLS estimates
for all values of k. As a result, the Hausman-type cointegration test rejects the null hypothesis ofcointegration for all values of k at the 1 percent level. Therefore, the evidence supports the viewthat Regression (26) is a spurious regression, and the true value of the RRA coefficient is likely to bemuch lower than the dynamic OLS estimates. Both the GLS corrected and the robust FGLS corrected
dynamic regression estimation results are consistent with the view that the RRA coefficient is about
one for the value of k chosen by BIC. For nondurables plus services (NDS), GLS corrected dynamic
regression estimates of β are much lower than Dynamic OLS estimates for small values of k. As aresult, the Hausman-type cointegration test rejects the null hypothesis of cointegration at the 5 percent
level when k is 0, 1, and 2. It still rejects the null hypothesis of cointegration at the 5 percent level
when k is 3. It does not reject the null hypothesis when k is 4 or 5. According to the BIC rule, k is
chosen to be 3, and there is some evidence against cointegration. However, because the GLS corrected
10 See Cooley and Ogaki (1996, page 127) for a detailed description of the data.

14

dynamic regression estimates get closer to dynamic OLS estimates as k increases, the evidence is notvery strong. It is likely that a small random walk component exists for the error term of the regression
for NDS making it a spurious regression. The robust FGLS corrected dynamic regression estimates areclose to both GLS corrected dynamic regression estimates and dynamic OLS estimates as long as k is
3 or greater.
Table 4 also reports ADF test results when the test is applied to the residual of static OLS. The
lag length for the ADF test has been chosen by the sequential t-test method advocated by Campbell
and Perron (1991). We do not have any evidence against the null hypothesis of structural spurious
regressions for ND. This hypothesis cannot be rejected at the 5 percent level, but can be rejected at
the 10 percent level for NDS.
Thus, we have fairly strong evidence that we have a spurious regression for ND and some evidence
that we have spurious regression for NDS. The true value of RRA is likely to be about one for both
ND and NDS.

3.5

Output convergence across national economies

In this section, we apply the techniques to reexamine a long standing issue in macroeconomics, thehypothesis of output convergence. For this application and the next, our main purpose is not to estimate
unknown structural parameters but to test the null hypothesis of cointegration with the Hausman-type
test. For this purpose, we do not need the strict exogeneity assumption under the alternative hypothesisof no cointegration (or a spurious regression).
As a key proposition of the neoclassical growth model, the convergence hypothesis has been popular in macroeconomics and has attracted considerable attention in the empirical field, particularly
during the last decade. Besides its important policy implications, the convergence hypothesis has beenused as a criterion to discern the two main growth theories, exogenous growth theory and endogenous
growth theory. However, it remains the subject of continuing debate mainly because the empiricalevidence supporting the hypothesis is mixed. Nevertheless, the established literature based on popularinternational datasets such as the Summers-Heston (1991) suggests as a stylized fact output convergence among various national economies: convergence among industrialized countries but not amongdeveloping countries and not between industrialized and developing countries.
Given that a mean stationary stochastic process of output disparities between two economies isinterpreted as supportive evidence of stochastic convergence, unit-root or cointegration testing procedures are often used by empirical researchers to evaluate the convergence hypothesis. In this vein, ourtechniques proposed here fit in the study of output convergence. We consider four developing countries(Columbia, Ecuador, Egypt, and Pakistan) along with four industrial countries (Germany, Luxemburg,
New Zealand, Switzerland). The raw data are extracted from the Penn World Tables of SummersHeston (1991) and consist of annual real GDP per capita (RGDPCH) over the period of 1950-1992.
The following two regression equations are considered with regard to the cointegration relation.

ytD= α + βytI + et ,

(27)

ytI

α + βytI + et ,

(28)

=

where ytD and ytI denote log real GDP per capita for developing and industrial countries, respectively.
Tables 5-1 and 5-2 report the results which exhibit a large variation in estimated coefficients. Recall
that our interest in this application lies in the cointegration test based on the Hausman-type test. As
can be seen from Table 5-1, irrespective of country combinations, the null hypothesis of cointegration15

can be rejected when developing countries are regressed onto industrial countries, indicating that there
is little evidence of output convergence between developing countries and industrial countries. The
picture changes dramatically when industrial countries are regressed onto industrial countries as in(28). Table 5-2 displays that the Hausman-type test fails to reject the null of cointegration in all cases
considered. Our finding is therefore consistent with the notion of convergence clubs.

3.6

PPP for traded and non-traded goods

As a major building block for many models of exchange rate determination, PPP has been one of
the most heavily studied subjects in international macroeconomics. Despite extensive research, theempirical evidence on PPP remains inconclusive, largely due to econometric challenges involved indetermining its validity. As is generally agreed, most real exchange rates show very slow convergence
which makes estimating long-run relationships difficult with existing statistical tools. The literaturesuggests a number of potential explanations for the very slow adjustment of relative price: volatility of
the nominal exchange-rate, market frictions such as trade barriers and transportation costs, imperfectcompetition in product markets, and the presence of non-traded goods in the price basket. According
to the commodity-arbitrage view of PPP, the law of one price holds only for traded goods and thedepartures from PPP are primarily attributed to the large weight placed on nontraded goods in the
CPI. This view has obtained support from many empirical studies based on disaggregated price indices.
They tend to provide ample evidence that prices for non-traded goods are much more dispersed thanfor their traded counterpart and consequently non-traded goods exhibit far larger deviations from PPP
than traded goods. Given that general price indices involve a mix of both traded and non-traded goods,
highly persistent deviations of non-traded goods from PPP can lead to the lack of conclusive evidence
on the long run PPP relationship. As in the previous application, our main purpose for this application
is not to estimate unknown structural parameters but to test the null hypothesis of cointegration with
the Hausman-type test.
Let pt and p∗t denote the logarithms of the consumer price indices in the base country and foreigncountry, respectively, and st be the logarithm of the price of the foreign country’s currency in terms of
the base country’s currency. Long-run PPP requires that a linear combination of these three variables
be stationary. To be more specific, long-run PPP is said to hold if ft = st + p∗t is cointegrated with pt
such that et ∼ I(0) in
ftT = α + βpTt + et ,ftN = α + βpN
t + et ,where the superscripts T and N denote the price levels of traded goods and non-traded goods, respectively.
Following the method of Stockman and Tesar (1995), Kim (2004) recently analyzed the real exchange rate for total consumption using the general price deflator and the real exchange rate for tradedand non-traded goods using implicit deflators for non-service consumption and service consumption,
respectively.11 We use Kim’s dataset to apply our techniques to the linear combination of sectorally
decomposed variables. Table 6 presents the results using quarterly price and exchange rate data forsix countries: Canada, France, Italy, Japan, U.K., and U.S. for the period of 1974 Q1 through 1998
Q4. With the Canadian dollar used as numeraire, Table 6 presents the estimates for β which should beclose to unity according to long-run PPP. For traded goods, estimates are above unity in most cases,
but the variation across estimates does not seem substantial, resulting in non-rejection of the null of
11 For details, see the Appendix for the description of the data. We thank JB Kim for sharing the dataset.

16

cointegration in all cases considered. By sharp contrast, the Hausman-type cointegration test rejects
the null hypothesis in every country when the price for non-traded goods is used. It is noteworthy that
there exists considerable difference between GLS-corrected estimates for β and their DOLS and FGLS
counterparts which are far greater than unity. That is, supportive evidence of PPP is found for traded
goods but not for non-traded goods, congruent with the general intuition as well as the findings byother studies in the literature such as Kakkar and Ogaki (1999) and Kim (2004).12

4

Concluding remarks and future work

In this paper, we developed two estimators to estimate structural parameters in spurious regressions:
GLS corrected dynamic regression and FGLS corrected dynamic regression estimators. A GLS corrected dynamic regression estimator is a first differenced version of a dynamic OLS regression estimator. Asymptotic theory shows that, under some regularity conditions, the endogeneity correction of
the dynamic regression works for the first differenced regressions for both cointegrating and spurious
regressions. This result is useful because it is not intuitively clear that the endogeneity correction works
even in regressions with stationary first differenced variables.
For the purpose of the estimation of structural parameters when the possibility of nonstationary
measurement error or nonstationary omitted variables cannot be ruled out, we recommend the FGLS
corrected dynamic regression estimators. They are consistent both when the error is I(0) and I(1). They
are asymptotically as efficient as dynamic OLS when the error is I(0) and as GLS corrected dynamic
regression when the error is I(1). This feature may be especially attractive when the FGLS corrected
dynamic estimator is extended to a panel data setting when some regression errors are I(0) and theothers are I(1). This extension is studied by Hu (2005).
We also developed the Hausman-type cointegration test by comparing the dynamic OLS regression
and GLS corrected dynamic regression estimators. As noted in the Introduction, this task is important
not merely because few tests for cointegration have been developed for dynamic OLS but because tests
for the null hypothesis of cointegration are useful in many applications. For this test, the spurious
regression obtained under the alternative hypothesis does not have to be structural.
We applied our estimation and testing methods to four applications: (i) long-run money demand in
the U.S.; (ii) long-run implications of the consumption-leisure choice; (iii) output convergence amongindustrial and developing countries; (iv) Purchasing Power Parity (PPP) for traded and non-traded
goods.
In the first application of estimating the money demand function, the results suggest that theendogeneity correction of the dynamic regression works with a moderately large number of leads and lags
for the GLS corrected dynamic regression estimator. The GLS corrected dynamic regression estimatesof the income elasticity of money demand are very low with low orders of leads and lags, and thenincrease to more plausible values as the order of leads and lags increases. Dynamic OLS estimates areclose to the GLS corrected dynamic regression estimates for a large enough order of leads and lags,
and we find little evidence against cointegration with the Hausman-type cointegration test. The FGLS
corrected dynamic regression estimates are very close to the GLS corrected dynamic regression and the
dynamic OLS estimates for sufficiently large order of leads and lags.
In the second application of the long-run implications of the consumption-leisure choice, we foundstrong evidence against cointegration when nondurables are used as the measure of consumption with the
12 Engel (1999) finds little evidence for long-run PPP for traded goods with his variance decomposition method. However, it should be noted that his method is designed to study variations of real exchange rates over relatively shorter
periods compared with cointegration-type methods that are designed to study long-run relationships.

17Hausman-type cointegration test. We also found some evidence against cointegration when nondurables
plus services are used as the measure of consumption. We estimate the RRA coefficient to be about
one with both GLS corrected and FGLS corrected dynamic regression estimators.
Hence, in these first two applications, the FGLS corrected dynamic regression estimator works wellin the sense that it yield estimates which are close to those of the estimator that seems to be correctly
specified. This is confirmed by our simulation results in Section 2 that the small sample efficiency
loss from using the FGLS corrected dynamic regression estimator is negligible for reasonable sample
sizes. Therefore, we recommend the robust FGLS corrected dynamic regression estimator when the
researcher is unsure about whether or not the regression error is I(0) or I(1). This is important because
it is difficult to detect a small random walk component in the error term when the error is actually I(1)and to detect a small deviation from a unit-root when the dominant autoregressive root is very close to
one when the error is actually I(0).
In the third application, we applied the Hausman-type cointegration test to log real output of pairsof countries to study output convergence across national economies. Our test results are consistent
with the stylized fact of convergence clubs in that we reject the null hypothesis of cointegration between
developing and developed countries while failing to reject the null hypothesis of cointegration between
two developed countries. Finally, we apply the Hausman-type cointegration test to study long-run PPP.
Our test results support the commodity-arbitrage view that long-run PPP holds for traded goods but
not for non-traded goods.
In future work, it will be important to study the choice of k, the number of leads and lags in theendogeneity correction. Another aspect that will be useful in empirical work is the study of possible
deterministic time trend and seasonal effects in the model.

18

Appendix
Appendix A: Proof of results in section 2.1
To show the distribution of the OLS estimator in regression (4) with ρ = 1, define X = [x1 , . . . , xn ]0 , V =[v1 , . . . , vt ]0 , and e = [e1 , . . . , en ]0 . Also define Mv = In − V(V0 V)−1 V and Mx = In − X(X0 X)−1 X.
Then the OLS estimator for β and γ can be written as"# 0
−1  0 
X X X0 VXeβ̂n − β0
=
0
0
VX VV
V0 e
γ̂ n − γ 0 0 
Xe(X0 Mv X)−1−(X0 Mv X)−1 X0 V(V0 V)−1
.
=
0
−1 0
0
−1
0
−1
V0 e
−(V V) V X(X Mv X)(V Mx V)We are mostly interested in the structural parameter β̂. Write its limit distribution asβ̂n − β0

(X0 Mv X)−1 X0 e − (X0 Mv X)−1 X0 V(V0 V)−1 V0 e 0
−1  0 −1  0   0 −1  0 
X Mv XXe
XVVVVe
1 X 0 Mv X
=
−
n2
n2
n
n2
n
n
n
 0
−1  0 
X Mv X
Xe=+ op (1)n2n2
Z 1
−1 Z 1
0→
V (r)V (r) dr
V (r)U (r)dr ≡ h1 ,=00which gives equation (5). For γ̂, we can writeγ̂ n − γ 0

= −

V0 V
n

−1 

V0 X
n



X 0 Mv X
n2

−1 

X0 e
n2




+

V 0 Mx V
n

−1 

V0 e
n


= Op (1).

Pn
Pn
Pn
−1
Alternatively, we can define that yt∗ = yt − ( t=1 yt vt0 ) ( t=1 vt vt0 ) vt , x∗t = xt − ( t=1 xt vt0 )
Pn
P
P
−1
−1
n
n
( t=1 vt vt0 ) vt , and e∗t = et − ( t=1 et vt0 ) ( t=1 vt vt0 ) vt . Note that n−1/2 x∗t = n−1/2 xt + op (1)and n−1/2 e∗t = n−1/2 et + op (1). Then regression (4) can be written asyt∗ = β 0 x∗t + e∗t .

(29)

The OLS estimator for β in regression (4) and (29) are identical. Write"β̂n − β0

n−2

=

n
X

#−1 "
x∗t x∗t

0

n−2

t=1

"
=

n−2

n
X

Z 1

#
x∗t e∗tt=1#−1 "xt x0t

n−2

t=1

→

n
X

n
X

#
xt et + op (1)t=1

V (r)V (r)0 dr−1 Z 10
V (r)U (r)dr .

0

To show the limit distribution of the GLS corrected estimator in regression (6), write√"n(θ̃dgls − θ0 ) = n

−1

n
X

#−1 "
∆zt ∆zt0

t=1

n

−1/2

n
X
t=1

19

#
∆zt ut .

(30)

For the denominator,PnnX
n−1 t=1 vt vt0
−1
0
Pn
n
∆zt ∆zt =n−1 t=1 ∆vt vt0
t=1



Pn
Σv
n−1 t=1 vt ∆vt0
Pn
→
−1
0
n
Γv,∆v
t=1 ∆vt ∆vt

Γ0v,∆v
Γ∆v,∆v


≡ Q,

(31)

where Σv is the variance matrix of {vt } and Γ is a matrix with elements computed from the autocovariances of {vt }. In the special case when m = 1, ηt = γ1,0 v1t + et ,
n−1

n
X
t=1

∆zt ∆zt0 =





Pn
Pn
2
n−1 t=1 v1t
n−1 t=1 v1t (v1t − v1,t−1 )
1
2
P
P
→
σ
n
n
1v
−1
−1
2
n
n
1 − ψ1v
t=1 v1t (v1t − v1,t−1 )
t=1 (v1t − v1,t−1 )1 − ψ1v
2(1 − ψ1v )2
where σ1v
is the variance of v1t and ψ1v is its first order autocorrelation coefficient. If we further assume
that v1t is i.i.d., the limit matrix becomes:


1 1
2
Q = σ1v.
1 2

For the numerator, the assumptions on the innovation processes ensure that CLT holds:


Pn
n
X
n−1/2 t=1 vt ut
P
n−1/2∆zt ut =→ N (0, Λ),nn−1/2 t=1 ∆vt ut
t=1

(32)

where Λ is the long run covariance matrix of the vector ∆zt ut :
" P
#
P∞
∞
0
0
E(vt vt−j
ut ut−j )E(vt ∆vt−j
ut ut−j )
j=−∞
j=−∞
P∞
.
Λ = P∞
00
j=−∞ E(∆vt vt−j ut ut−j )j=−∞ E(∆vt ∆vt−j ut ut−j )When m = 1, v1t , ut are both i.i.d. and ηt = γ1,0 v1t + et ,


1 1
2 2
Λ = σ1v σu.
1 2Hence, for the quantity defined in (30), we have the limit distribution given in (7):√n(θ̃dgls − θ0 ) → N (0, Ω)
where Ω = Q−1 ΛQ−1 .
To derive the limit distribution for the FGLS estimator, we first derive the limit distribution for ρ̂nin regression (8). Write the process of êt asêt0= yt − θ̂dols
zt= et + (θ0 − θ̂dols )0 zt= et−1 + (θ0 − θ̂dols )0 zt + ut
= êt−1 + (θ0 − θ̂dols )0 ∆zt + ut
= êt−1 + gt ,say.

¿From this expression, we can see that êt is a unit-root process with serially correlated error gt . The
OLS estimator ρ̂n can be written asPnêt−1 gt.
(33)
ρ̂n − 1 = Pt=1n2
t=1 êt−1
20

,

If we let b = (1, (β0 − β̂n )0 )0 , at = (et , x0t )0 , we can writeêt= et + (β0 − β̂n )0 xt + (γ 0 − γ̂ n )0 vt
0

(34)

0

= b at + (γ 0 − γ̂ n ) vt .
Then the denominator in (33) can be written as
n−2

n
X

ê2t = b0 n−2

t=1

n
X

at a0t b + op (1) → α0

Z 1A(r)A(r)0 dr α,0

t=1

where α = (1, −h01 )0 and A(r) = (U (r), V (r)0 )0 .
The numerator in (33) can be written asêt−1 gt=

[b0 at + (γ 0 − γ̂ n )0 vt ][(β0 − β̂n )0 vt + (γ 0 − γ̂ n )0 (vt − vt−1 ) + ut ].

The sum of this quantity is Op (n). We omit the details here since we will not make use of the distributionPnform of ρ̂n . We let c denote the limit, i.e., n−1 t=1 êt−1 gt → c.
Then,
Pn
 Z 1
−1
n−1
êt−1 gt
00n(ρ̂n − 1) = −2 Pt=1→cαA(r)A(r)
dr
α
.
(35)
d
n
2
n
0
t=1 êt−1
In fact, in our following computations, we only make use of the fact thatn(ρ̂n − 1) = Op (1).
Below we show how to derive the limit distribution for θ̃fgls . For the sequence of ỹt , we can write itasỹt= yt − ρ̂n yt−1
= θ00 zt + et − ρ̂n (θ00 zt−1 + et−1 )= θ00 (zt − ρ̂n zt−1 ) + (et − et−1 ) + (1 − ρ̂n )et−1= θ00 z̃t + ut + (1 − ρ̂n )et−1 .

Now, we can writeθ̂fgls − θ0 =" n
Xz̃t z̃t0#−1 " n
X

t=1

t=1

z̃t z̃t0 = Pnx̃ x̃0
Pnt=1 t t0
t=1 ṽt x̃t#z̃t [ut + (1 − ρ̂n )et−1 ] .

(36)

Write the denominator as
n
X
t=1


Pn
x̃t ṽt0
t=1
Pn.
0
t=1 ṽt ṽtFirst,nXx̃t x̃0t

=

t=1

n
X

(xt − ρ̂n xt−1 )(xt − ρ̂n xt−1 )0

t=1

=

n
X

[(1 − ρ̂n )xt−1 + vt ][(1 − ρ̂n )xt−1 + vt ]0

t=1=(1 − ρ̂n )2nX

xt−1 x0t−1 + (1 − ρ̂n )

t=1

n
X
t=1

21

[xt−1 vt0 + vt x0t−1 ] +

n
X
t=1

vt vt0 .

Hence,
n−1

n
X

x̃2t

t=1
2

= n(1 − ρ̂n )

n

−2

n
X

!
xt−1 x0t−1+ (1 − ρ̂n ) n

−1

t=1

= n−1

n
X

n
X

xt−1 vt0 + n−1

t=1

n
X

!
vt x0t−1

+ n−1

t=1

n
X

vt vt0t=1

vt vt0 + op (1) → Σv .

t=1

Similarly,n−1Xx̃t ṽt0

= n−1

n
X

vt (vt − vt−1 )0 + (1 − ρ̂n )n−1

t=1

n
X

0
vt vt−1t=1+(1 − ρ̂n )n−1

nX

xt−1 (vt − vt−1 )0 + (1 − ρ̂n )2 n−1

t=1

= n−1

n
X

n
X

0
xt−1 vt−1

t=1

vt (vt − vt−1 )0 + op (1) → Γv,∆v .

t=1

Finally, n

Pn
−1

Pn
1

0
t=1 ṽt ṽt = n

n

−1

0
t=1 ∆vt ∆vt + op (1) → Γ∆v,∆v . Hence,nXz̃t z̃t0 = n−1

t=1

n
X

∆zt ∆zt0 + op (1) →p Q.

(37)

t=1

Next, consider the numerator in (36)nXz̃t [ut + (1 − ρ̂n )et−1 ] =

t=1

 Pn

x̃ [u + (1 − ρ̂n )et−1 ]Pnt=1 t t.
t=1 ṽt [ut + (1 − ρ̂n )et−1 ]Pn
It is not hard to see that n−1 t=1 z̃t [ut + (1 − ρ̂n )et−1 ] →p 0. Intuitively, z̃t behaves asymptoticallylike the differenced regressors (vt0 , ∆vt0 )0 , and u and v are uncorrelated by assumption. Our remaining
task is to show that
n−1/2

n
X

z̃t [ut + (1 − ρ̂n )et−1 ] = n−1/2

t=1

n
X

∆zt ut + op (1) → N (0, Λ).

(38)

t=1

This can be shown using similar arguments in proving (37). Combining (38) with (37), we obtain thelimit distribution for θ̃dgls as given in (11).

Appendix B: Proof of results in section 2.2
To show the limit distribution of the dynamic OLS estimator in the cointegration, define


Im n
0
Hn =
.
0
Im(2k+1) n1/2

(39)

We can write"
Hn (θ̂dols −θ0 ) =n(β̂n − β0 )n1/2 (γ̂ n − γ 0 )

#


=

Pn
n−2 t=1 xt x0tPn
n−3/2 t=1 vt x0t
22

−1 

Pn
Pn
n−3/2 t=1 xt vt0
n−1 t=1 xt et
P
P
.
n
n
n−1 t=1 vt vt0
n−1/2 t=1 vt etFor the denominator,Pn
n−2 t=1 xt x0tPn
n−3/2 t=1 vt x0t

" R

Pn
1
n−3/2 t=1 xt vt0
V (r)V (r)0 dr
0
Pn
→
−1
0
n
0
t=1 vt vt

0
Γv,v

#
.

(40)

Thus, the estimator of the I(1) and I(0) components are asymptotically independent. For the numerator,
#
" R


Pn
1
n−1 t=1 xt etV(r)dE(r)
0
Pn
,
(41)
→d
n−1/2 t=1 vt et
N (0, Λv,e )where Λv,e is the long run variance of vt et . Equation (13) then follows.
To show the limit distribution for FGLS estimator in regression (16), write
n−1

= n

−1

= n−1

n
X
t=1
n
X
t=1
n
X

ê2t
e2t + 2n−1nX!
et zt0 Hn−1

Hn (θ0 − θ̂dols ) + n−10(θ0 − θ̂dols ) Hn

t=1

Hn−1

n
X

!
zt zt0 Hn−1

Hn (θ0 − θ̂dols )t=1

e2t + op (1) → σe2 .

t=1

Similarly, we can show that
n−1

n
X

êt êt−1 = n−1

t=1

n
X

et et−1 + op (1) → ψe σe2 ,t=1

where ψe is the first order autocorrelation coefficient of {et }. Then the OLS estimator isPnn−1 t=1 êt et−1Pnρ̂n =→p ψe .
n−1 t=1 ê2t
Conduct the Cochrane-Orcutt transformation (9) and estimateỹt = β 0 x̃t + γ 0 ṽt + error.
For the sequence of ỹt , we can write it asỹt = β00 x̃t + γ 00 ṽt + ẽt ,where ẽt = et − ρ̂n et−1 . Using the same weight matrix Hn , write

 
 Pn

−1 

Pn
Pn
x̃t x̃0tx̃t ṽt 0
n−1 t=1 x̃t ẽtn(β̃n − β0 )
−1
−1
t=1
t=1
P
P
P
= Hn
Hn
.
n
n
n
0
0
n−1/2 t=1 ṽt ẽtn1/2 (γ̃ n − γ 0 )t=1 ṽt ṽt
t=1 ṽt x̃t(42)

Define that Ẽ(r) = (1 − ψe )E(r) and Ṽ (r) = (1 − ψe )V (r). By Lemma 2.1 in Phillips and Ouliaris
P[nr](1990), n−1/2 x̃[nr] → Ṽ (r) and n−1/2 t=1 ẽt → Ẽ(r). Therefore we can show that
Z 1
n
X
−2
0
n
x̃t x̃t →Ṽ (r)Ṽ (r)0 dr
n−3/2
n−1

t=1
n
X

0

x̃t ṽt0→ 0ṽt ṽt 0

→ P,

t=1
n
X
t=1

23

say.

Hence, the limit of the denominator in (42) is
" n
#
"
R1
X
(1 − ψe )2 0 V (r)V (r)0 dr−1
0−1
zt zt H n →
Hn
0
t=1

0
P

#
.

Next, consider the numerator in (42). In fact, we are only interested in the first element,
n−1

n
X

Z 1
x̃t ẽt →Ṽ (r)dẼ(r) = (1 − ψe )2

V (r)dE(r).
0

0

t=1

Z 1

Therefore, we obtain the limit distribution for β̃fgls ,Z 1n(β̃fgls − β0 ) →Ṽ (r)Ṽ (r)0 dr−1 Z 1

0

0

Z 1

−1 Z 10

V (r)V (r) dr=0Ṽ (r)dẼ(r)
V (r)dE(r) ,0which is the same as the limit of β̂dols .

Appendix C: Proof of results in section 2.4
In the extended test, where we allow endogeneity under the alternative, the regression can be writtenas:yt = β 0 xt + γ 0 vt + (φst + et ).
Define s = [φs1 + e1 , . . . , φsn + en ]0 . Note that n−1/2 s = n−1/2 e + op (1). Similar as in Appendix A,
the OLS estimators for β under the alternative of a spurious regression can be written asβ̂n − β0

(X0 Mv X)−1 X0 s − (X0 Mv X)−1 X0 V(V0 V)−1 V0 s 0−1  0 
−1  0   0 −1  0 VVVeXe
1 X 0 Mv X
XV
X Mv X=+ op (1)
−
n2
n2
n
n2
n
n
n
 0
−1  0 
X Mv X
Xe=+ op (1)
2
n
n2
Z 1
−1 Z 1
0→
V (r)V (r) dr
V (r)U (r)dr .
=00

Due to endogeneity, the estimator in the differenced regression is not consistent either. The es0
timators (β̃dgls− β00 , γ̃ 0dgls − γ 00 )0 → Q−1 φ(E(vt ∆st )0 , E(∆vt ∆st )0 )0 . Let β̄ denote the limit of β̃dgls ,then√n(β̃dgls − β̂dols )√=n(β̃dgls − β̄) − n(β̂dols − β̄)√= Op (1) + Op ( n)√= Op ( n).
√Finally, in the differenced regression, the variance estimate still converges. Therefore, the Hausmantype test statistic is of order n under the alternative of spurious regressions no matter whether exogeneity
holds or not.
24

Appendix D: Regression with a constantIn this section, we consider the model with a constant term, e.g.
yt = δ + β 0 xt + γ 0 vt + et .

(43)

As in Appendix A, we can also estimate the parameter δ and β in regressionyt∗ = δ + β 0 x∗t + e∗t .

(44)

The OLS estimator in regression (44) can be written as
"
#
"
#−1 

Pn
Pn
0
n−1/2 (δ̂n − δ0 )1
n−3/2 t=1 x∗tn−3/2 t=1 e∗t
P
Pn
Pn
=
0
n
n−2 t=1 x∗t e∗tn−3/2 t=1 x∗t n−2 t=1 x∗t x∗tβ̂n − β0
−1  −3/2 Pn


Pn
n
1
n−3/2 t=1 x0t
t=1 etP
P
P+ op (1)=nnnn−2 t=1 xt etn−3/2 t=1 xt n−2 t=1 xt x0t
R
R

−1 

1
V (r)dr
U (r)drR
R
R→.
V (r)drV (r)V (r)0 dr
V (r)U (r)drTherefore, the OLS estimator of the intercept diverges in the spurious regression. Similarly, we canshow that γ̂ − γ 0 in regression (43) is also Op (1). Note that the estimators δ̂, β̂ and γ̂ estimated in thisway are numerically identical to that estimated in regression (43).
If we do GLS correction or the differenced regression, the constant is canceled so the limit results
are the same as given by (7). Finally, consider the Cochrane-Orcutt FGLS estimation. Let êt denote
the OLS residual from (43),êt = yt − δ̂n − β̂n0 xt − γ̂ 0n vt .
Then do another OLS estimation ofêt = ρ̂n êt−1 + ût .
Write
êt

= yt − δ̂n − β̂n0 xt − γ̂ 0n vt= et−1 + (δ0 − δ̂n ) + (β0 − β̂n )0 xt + (γ 0 − γ̂)0 vt + ut
= êt−1 + [(β0 − β̂n )0 vt + (γ 0 − γ̂ n )0 (vt − vt−1 ) + ut ]= êt−1 + gt ,which takes the same form as in the previous section where no constant is included. Hence we still havePnêt−1 gtρ̂n − 1 = Pt=1.
n2
t=1 êt−1
Write the process of êt as:êt = yt − δ̂n − β̂n0 xt − γ̂ 0n vt = (β0 − β̂n )0 (xt − x̄) + (γ 0 − γ̂ n )0 (vt − v̄) + (et − ē).

(45)

Comparing equation (45) with (34), the only difference is that all terms in (45) are subtracted by theirsample means. This leads to demeaned Brownian motions instead of standard Brownian motions in thelimit of the distribution of ρ̂n . Using similar methods as in Appendix A, we can show thatn(ρ̂n − 1) = Op (1).
25

Next, conduct the Cochrane-Orcutt transformation as in (9), and consider the OLS estimator in the
regressionỹt = β̃n0 x̃t + γ̃ 0n ṽt + error = θ̃0 z̃t + error.
Write the estimatorθ̃fgls =" nXz̃t z̃t0#−1 " nX

t=1#
z̃t ỹt .

t=1

For ỹt , writeỹt= yt − ρ̂n yt−1=

(1 − ρ̂n )δ0 + β00 x̃t + γ 00 ṽt + et + (1 − ρ̂n )et−1= θ00 z̃t + (1 − ρ̂n )δ0 + et + (1 − ρ̂n )et−1 .
Thus we can writeθ̂fgls − θ0 =" n
Xz̃t z̃t0#−1 " nX

t=1#
z̃t [(1 − ρ̂n )δ0 + et + (1 − ρ̂n )et−1 ] .

(46)

t=1

The only difference between (46) and (36) is that (46) has one additional term (1 − ρ̂n )δ0 . However,since δ0 is a constant and 1 − ρ̂n = Op (n−1 ), this term vanishes in the limit. Therefore, using the
Cochrane-Orcutt transformation, the limit distribution of the estimators is the same regardless ofwhether we have a constant in the data generating process of the data. In either case, we have the same
result as given by (7).
Above are discussions under the assumption of spurious regressions. Under the assumption of
cointegration, the OLS estimator β̂ is still n convergent. By similar arguments, we can also show that
the FGLS estimators are still asymptotically equivalent to the OLS estimators in the cointegration.
Thus Proposition 1 holds when there is a constant in the DGP of yt . Also, the GLS corrected estimator
has the same property as when there is no constant since this constant is removed in differenced
regression. Therefore, the Hausman-type cointegration test statistic has the same asymptotics as given
in Proposition 2.

Appendix E: Data descriptions
In the first two empirical analysis we use the same data set as in Stock and Watson (1993, page 817)
for the U.S. money demand, and the data set of Cooley and Ogaki (1996, page 127) for the long-run
intertemporal elasticity of substitution. Readers are referred to the original work for further details on
data.
Per capita output series are extracted from the Penn World Tables of Robert Summers and Alan
Heston (1991). They are annual data on real GDP per capita (RGDPCH) for four developing countries(Columbia, Ecuador, Egypt, and Pakistan) along with four industrial countries (Germany, Luxemburg,
New Zealand, and Switzerland) over the period of 1950-1992.
In the PPP application we borrow the dataset from Kim (2004) who constructed the real exchange
rate for total consumption using the general price deflator and the real exchange rate for traded and nontraded goods using implicit deflators for non-service consumption and service consumption, respectively.
Data are quarterly observations spanning from 1974 Q1 to 1998 Q4. The exchange rates for Canada,
France, Italy, Japan, the United Kingdom, and the United States are taken from the International
Financial Statistics (IFS) CD-ROM, and bilateral real exchange rates of traded and non-traded goods
classified by type and total consumption deflators from the Quarterly National Accounts and Data
Stream are studied.
26

Appendix F: Guidelines on empirical application
Procedures for GLS- and FGLS- corrected estimations
Step 1 Choose a length (p) of lead and lag terms using popular lag selection rules such as AIC, BIC, ortheir modified versions due to Ng and Perron (2001). Given that the lead and lag length selection
issue has not been settled in the dynamic OLS literature, we recommend to report results fromdifferent order together with BIC as a rough guideline. To correct for the endogeneity problem
Instrumental Variable (IV) approach can be also applied. The IV approach is appealing as it does
not involve choosing proper length of leads and lags, but the downside must be that it is not easy
to find good instruments in practice.
yt =

p
X

γk ∆xt+k + βxt +

k=1

p
X

φk ∆xt−k + et ,k=1

Step 2 (GLS-corrected estimation) Filter all variables in the above equation by taking full difference
p−1
X

2

∆yt

=

γk ∆ xt+k + β∆xt +∆yt= θ0 ∆zt + ∆et

k=1

p−1
X

φk ∆2 xt−k + ∆et ,k=1

Step 2’ (FGLS-corrected estimation) Retrieve the OLS residuals such thatêt = yt −pXγ̂k ∆xt+k + β̂xt +

k=1

p
X

φ̂k ∆xt−k ,k=1

and obtain ρ̂ from regressing êt onto êt−1 . After ‘n0 iterations ρ̂n can be obtained. The variables
are transformed such that ỹt = yt − ρ̂n yt−1 , x̃t = xt − ρ̂n xt−1 , and ∆x̃t+k = ∆xt+k − ρ̂n ∆xt+k−1 .
Step 3 Apply OLS to estimate θ = {γ1 , · · · , γp−1 , β, φ1 , · · · , φp−1 }. The obtained estimates are the(F)GLS corrected estimates of θ.
The Hausman-type cointegration test
Step 1 Obtain the DOLS and GLS-corrected estimates for the parameters. We recommend to report
the results from different order of lead and lag terms together with the one chosen by the BIC rule
as a guideline. When selecting lead and lag lengths through the BIC rule, it is recommended tochoose the lengths for DOLS and DGLS separately. That is, θ̂dols is obtained using the BIC lag
length from DOLS regression equation while θ̃dgls using the BIC lag length selected from DGLS
regression equation as described above.
√
Step 2 Compute V̂β , a consistent estimate for the long run variance matrix of n(θ̃dgls − θ) using
Heteroskedasticity and autocorrelation consistent (HAC) estimator. In the empirical part ofthis paper we adopted the long run variance estimator from Andrews and Monahan (1992) witha quadratic spectral (QS) kernel using prewhitening. Readers are also referred to the recent
study by Sul, Phillips, and Choi (2005) who propose a recursive demeaning and recursive Cauchy
estimation to reduce the small sample bias in prewhitening coefficient estimates as well as a samplesize-dependent boundary condition rule that substantially enhances power without compromising
size.
27

Step 3 Construct the test statistic
hn = n(β̃dgls − β̂dols )0 V̂β−1 (β̃dgls − β̂dols ) → χ2 (m)28Table 1: The bias and square root of the mean square error of three estimatorsρρ=0

ρ = 0.95

ρ=1nn = 50n = 100
n = 500
n = 50n = 100
n = 500
n = 50n = 100
n = 500

DOLS estimator
Bias
Square root of MSE-0.0000
0.0290-0.0002
0.0169-0.0000
0.0040-0.00070.35550.00700.24150.00060.0773-0.0347
1.4769-0.01131.2809-0.0086
0.9895

GLS corrected estimator
Bias
Square root of MSE
0.00310.21560.00040.14480.00100.06340.00190.2104-0.00100.1472-0.0007
0.06370.00240.21030.00280.14390.00030.0643

FGLS corrected estimator
Bias
Square root of MSE-0.0000
0.0296-0.00020.0170-0.0000
0.00400.00000.21620.00360.1375-0.0000
0.0475-0.0095
0.59680.00310.3635-0.00110.1167

Table 2: Finite sample performance of the Hausman-type cointegration test

T
50
100
200
300
500

Hausman-type Test
power
size (5%)
0.621
0.114
0.688
0.072
0.754
0.050
0.783
0.039
0.816
0.040

Shin’s Test
power size (5%)
0.249
0.141
0.402
0.171
0.652
0.199
0.775
0.184
0.882
0.181

Note: The Hausman-type cointegration test is stipulated in section 2.4. Nonparametric estimator of long run variance isused based on the QS kernel with the bandwidth of ‘integer[8(T /100)1/4 ]’.

29

Table 3: Application to Long Run U.S. Money Demand

Estimator

DOLS

GLScorrected

FGLScorrected
AR(1)FGLScorrected
AR(2)HausmanTest
ADF-based Test

k
0
1
2
3
4
BIC
[lag]
0
1
2
3
4
BIC
[lag]
0
1
2
3
4
BIC
[lag]
0
1
2
3
4
BIC
[lag]
0
1
2
3
4
0

Equation 1β̂γ̂
0.944 (0.054)-0.090 (0.015)
0.958 (0.048)-0.096 (0.014)
0.970 (0.051)-0.101 (0.014)
0.975 (0.055)-0.104 (0.015)
0.967 (0.054)-0.108 (0.015)

Equation 2β̂γ̂
0.889 (0.057)-0.308 (0.058)
0.884 (0.046)-0.313 (0.045)
0.879 (0.044)-0.320 (0.043)
0.871 (0.036)-0.328 (0.035)
0.855 (0.029)-0.334 (0.028)[3]
0.407 (0.081)
0.654 (0.119)
0.837 (0.134)
0.856 (0.145)
0.962 (0.161)[5]-0.014 (0.004)-0.025 (0.010)-0.050 (0.013)-0.067 (0.017)-0.086 (0.022)

0.419 (0.079)
0.685 (0.115)
0.848 (0.130)
0.884 (0.140)
0.898 (0.151)-0.083 (0.023)-0.065 (0.009)-0.081 (0.010)-0.096 (0.011)-0.108 (0.011)

0.893 (0.049)
0.872 (0.035)
0.901 (0.036)
0.905 (0.029)
0.886 (0.025)-0.083 (0.023)-0.069 (0.009)-0.086 (0.010)-0.100 (0.011)-0.109 (0.010)

0.893 (0.049)
0.872 (0.038)
0.894 (0.033)
0.907 (0.029)
0.889 (0.026)[2]
0.942 (0.052)
0.888 (0.040)
0.940 (0.045)
0.980 (0.050)
1.010 (0.045)

ln
ln
ln

M 
 M tP M tPP

0.388 (0.078)
0.643 (0.115)
0.787 (0.133)
0.816 (0.146)
0.811 (0.153)-0.290 (0.079)-0.278 (0.030)-0.309 (0.031)-0.330 (0.026)-0.333 (0.023)

0.858 (0.071)
0.815 (0.045)
0.840 (0.054)
0.851 (0.046)
0.833 (0.051)-0.290 (0.079)-0.276 (0.031)-0.312 (0.029)-0.332 (0.026)-0.335 (0.023)

0.858 (0.071)
0.809 (0.049)
0.839 (0.049)
0.853 (0.050)
0.827 (0.061)

α + β ln (yt ) + γit + et ,=α + β ln (yt ) + γ ln(it ) + et ,=α + β ln (yt ) + γ ln

it0.850 (0.435)
0.722 (0.118)
0.830 (0.131)
0.903 (0.128)
0.856 (0.142)

[5]
2.270
0.113
0.012
0.010
0.188
-2.726†(equation 1)

 1 + it t

0.850 (0.435)
0.744 (0.115)
0.797 (0.128)
0.912 (0.124)
0.895 (0.133)
[5]

[5]
2.643
0.041
0.007
0.092
0.246
-3.128‡=0.300 (0.082)
0.506 (0.148)
0.620 (0.161)
0.725 (0.185)
0.654 (0.195)
[5]

[5]

[3]
4.147
0.691
0.549
0.001
0.485
-3.768‡Note:[5]-0.086 (0.022)-0.177 (0.046)-0.248 (0.053)-0.289 (0.061)-0.283 (0.067)

[2]

[4]
0.942 (0.052)
0.900 (0.039)
0.948 (0.042)
0.991 (0.044)
1.012 (0.042)

Equation 3β̂γ̂
0.850 (0.085)
0.906 (0.280)
0.843 (0.066)
0.915 (0.216)
0.837 (0.072)
0.941 (0.233)
0.832 (0.062)
0.975 (0.205)
0.824 (0.065)
0.995 (0.215)(equation 2)+ et .

(equation 3)‘GLS-corrected (FGLS-corrected)’ denotes the GLS (FGLS) corrected dynamic regression estimator. Figures in the parenthesis
represent standard errors. ‘k’ denotes the maximum length of leads and lags. In FGLS corrected estimation, the serial correlation
coefficient in error term is estimated before being applied to the Cochrane-Orcutt transformation, whereas it is assumed to beunity in GLS corrected estimation which is analogous to regressing the first difference of variables without a constant term.

Hausman test represents the Hausman-type cointegration test as stipulated in section 2.4. The test statistic is constructed asvar(β̃dgls )cov(β̃dgls , γ̃dgls )(Γ̂dgls − Γ̃dols )Σ(Γ̂dgls − Γ̃dols )0 → χ2 (2) where Γ = [β, γ] and Σ =. The critical valuescov(β̃dgls , γ̃dgls )var(γ̃dgls )of χ2 (2) are 4.61, 5.99 and 9.21 for 10%, 5%, and 1% significance levels. The critical values of the ADF-based tests are -2.88 and-2.57 for 5% and 10% significance levels. ‡(†) represents that the null hypothesis can be rejected at 5% (10%).

30

Table 4: Application to Preference Parameter (β) Estimation
Estimator

ND

NDS

k

DOLS

0
1
2
3
4
5
6
7
BIC [lag]
0
1
2
3
4
5
6
7
BIC [lag]

1.865 (0.218)
1.865 (0.192)
1.870 (0.181)
1.873 (0.193)
1.877 (0.204)
1.880 (0.196)
1.888 (0.202)
1.891 (0.200)[0]1.102 (0.052)
1.103 (0.052)
1.102 (0.042)
1.100 (0.041)
1.099 (0.036)
1.095 (0.033)
1.091 (0.030)
1.088 (0.029)[0]Note: Results for W (t) =v 0 (l(t)).
C(t)−βGLScorrected
0.222 (0.061)
0.628 (0.091)
0.720 (0.102)
0.850 (0.110)
0.963 (0.117)
1.041 (0.123)
1.129 (0.125)
1.185 (0.129)[3]0.480 (0.066)
0.796 (0.080)
0.855 (0.084)
0.924 (0.085)
0.967 (0.087)
0.995 (0.088)
1.019 (0.089)
1.031 (0.091)
[3]

FGLS
(AR(1))
1.874 (0.151)
0.983 (0.087)
1.160 (0.089)
1.199 (0.097)
1.207 (0.107)
1.391 (0.104)
1.437 (0.109)
1.533 (0.111)[4]
1.106 (0.095)
0.912 (0.041)
0.952 (0.034)
0.978 (0.032)
0.998 (0.030)
1.018 (0.028)
1.031 (0.025)
1.040 (0.023)
[3]

FGLS
(AR(2))
1.874 (0.151)
0.983 (0.087)
1.160 (0.089)
1.199 (0.097)
1.207 (0.107)
1.391 (0.104)
1.488 (0.109)
1.558 (0.114)[4]
1.106 (0.095)
0.932 (0.035)
0.911 (0.038)
0.975 (0.031)
1.000 (0.029)
1.009 (0.029)
1.025 (0.025)
1.042 (0.024)[3]

Hausmantest
140.35‡102.46‡60.38‡
41.53‡
37.01‡
34.68‡
32.63‡
35.05‡

ADF-based
test-2.30215.93‡
8.22‡
3.88†
1.60
0.98
0.58
0.44
0.77

-2.867†‘GLS-corrected (FGLS-corrected)’ denotes the GLS (FGLS) corrected dynamic

regression estimator. Figures in parenthesis represent standard errors. ‘k’ denotes the maximum length of leads and lags.
In FGLS corrected estimation, the serial correlation coefficient in the error term is estimated before being applied to the
Cochrane-Orcutt transformation, whereas it is assumed to be unity in GLS corrected estimation which is analogous toregressing the first difference of variables without constant term. Hausman test represents the Hausman-type cointegration
test as stipulated in section 2.4. The test statistic is constructed as(β̂dgls −β̃dols )2 )V ar(β̃dgls )→ χ2 (1). The critical values of χ2 (1)are 2.71, 3.84 and 6.63 for ten, five, and one percent significance levels. The critical values of the ADF-based tests are-2.88 and -2.57 for 5% and 10% significance levels. ‡(†) represents that the null hypothesis can be rejected at 5% (10%)
significance level.

31

Table 5-1: Application to Output Convergence (Regressand: developing countries; Regressor:
industrial countries)
Regressand

Regressor

GER

LUX

COL
NZL

SWI

GER

LUX

ECU
NZL

SWI

k

DOLS

0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC

0.680 (0.156)
0.755 (0.179)
0.876 (0.245)
0.963 (0.235)
1.105 (0.267)[0]0.923 (0.144)
0.953 (0.208)
0.993 (0.136)
1.035 (0.118)
1.087 (0.093)[1]
1.218 (0.454)
1.213 (0.368)
1.203 (0.309)
1.218 (0.330)
1.178 (0.328)[0]0.972 (0.349)
0.967 (0.343)
0.948 (0.327)
0.901 (0.291)
0.926 (0.437)[0]0.784 (0.311)
0.816 (0.400)
0.873 (0.583)
0.913 (0.596)
1.041 (0.743)[0]1.078 (0.697)
1.158 (0.661)
1.261 (0.410)
1.347 (0.696)
1.438 (0.518)[0]1.496 (0.418)
1.499 (0.446)
1.487 (0.536)
1.520 (0.573)
1.535 (0.631)[0]1.155 (0.439)
1.139 (0.405)
1.086 (0.400)
1.022 (0.340)
1.061 (0.378)[0]

GLScorrected
0.464 (0.091)
0.501 (0.111)
0.490 (0.130)
0.409 (0.154)
0.607 (0.169)[0]0.322 (0.097)
0.642 (0.151)
0.633 (0.167)
0.657 (0.193)
0.739 (0.223)[1]
0.363 (0.117)
0.650 (0.199)
0.600 (0.241)
0.748 (0.261)
0.788 (0.294)[0]0.493 (0.121)
0.663 (0.162)
0.662 (0.189)
0.567 (0.212)
0.576 (0.210)[0]0.344 (0.159)
0.472 (0.191)
0.472 (0.231)
0.357 (0.273)
0.501 (0.322)[0]0.129 (0.157)
0.393 (0.263)
0.397 (0.291)
0.734 (0.301)
0.848 (0.343)[0]0.299 (0.182)
0.608 (0.331)
0.662 (0.387)
0.795 (0.409)
0.928 (0.462)[0]0.287 (0.202)
0.368 (0.294)
0.418 (0.326)
0.470 (0.352)
0.563 (0.352)[0]FGLScorrected
0.635 (0.030)
0.963 (0.109)
1.087 (0.121)
0.979 (0.058)
1.029 (0.067)[3]
0.915 (0.009)
0.669 (0.143)
0.892 (0.103)
1.030 (0.095)
1.119 (0.075)[1]
1.219 (0.053)
0.931 (0.270)
1.425 (0.221)
1.608 (0.235)
1.801 (0.191)[1]
0.969 (0.021)
0.875 (0.284)
1.350 (0.248)
1.435 (0.175)
1.491 (0.212)[1]0.779 (0.009)
0.742 (0.413)
0.956 (0.503)
1.259 (0.289)
1.245 (0.252)[1]
1.067 (0.034)
0.085 (0.293)
0.631 (0.246)
0.573 (0.407)
1.076 (0.362)[1]
1.505 (0.091)
0.988 (0.366)
1.087 (0.436)
1.367 (0.465)
1.934 (0.486)[1]
1.176 (0.052)
0.819 (0.321)
1.139 (0.294)
1.354 (0.235)
1.332 (0.246)[1]

HausmanTest
3.668†
9.261‡
11.294‡13.996‡11.654‡4.715‡
6.544‡
4.494‡
3.094†
3.012†
13.819‡9.067‡
3.104†
2.205
0.109
4.219‡
3.127†
4.359‡
6.130‡
5.398‡
5.410‡
3.464†
4.720‡
5.272‡
3.305†
8.272‡
11.803‡4.465‡
3.543†
1.487
5.338‡
5.694‡
3.645†
3.419†
0.065
10.777‡8.719‡
5.337‡
4.840‡
0.492

Note: See the notes in Table 2. Annual data covering 1950-1992 are used for four developing countries (COL: Columbia;
ECU: Ecuador; EGT: Egypt; PAK: Pakistan) and four industrial countries (GER: Germany; LUX: Luxemburg; NZL:New Zealand; SWI: Switzerland). The regression equation is ln(yDEV ) = α + βln(yIN D ) + e.

32

Table 5-1: Continued–
Regressand

Regressor

GER

LUX

EGT

NZL

SWI

GER

LUX

PAK
NZL

SWI

k

DOLS

0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC

0.916 (0.108)
1.011 (0.145)
1.125 (0.256)
1.160 (0.342)
1.233 (0.449)[1]
1.228 (0.270)
1.313 (0.362)
1.390 (0.141)
1.436 (0.144)
1.476 (0.100)[5]1.674 (0.317)
1.737 (0.301)
1.794 (0.388)
1.837 (0.456)
1.852 (0.594)[3]
1.344 (0.243)
1.422 (0.266)
1.498 (0.387)
1.537 (0.483)
1.756 (0.453)[2]0.746 (0.112)
0.858 (0.144)
0.981 (0.193)
1.020 (0.269)
0.999 (0.339)[0]0.980 (0.233)
1.014 (0.257)
1.055 (0.167)
1.059 (0.196)
1.038 (0.169)[0]1.354 (0.318)
1.377 (0.268)
1.382 (0.273)
1.447 (0.311)
1.538 (0.582)[4]
1.121 (0.185)
1.177 (0.176)
1.254 (0.169)
1.333 (0.202)
1.542 (0.170)[0]

GLScorrected
0.372 (0.133)0.550 (0.143)
0.663 (0.165)
0.691 (0.192)
0.767 (0.223)[1]
0.059 (0.136)
0.568 (0.198)
0.611 (0.229)
0.831 (0.264)
1.025 (0.287)[1]
0.299 (0.155)
0.829 (0.257)
1.090 (0.298)
1.355 (0.324)
1.473 (0.359)[1]
0.388 (0.169)
0.666 (0.229)
0.991 (0.235)
1.080 (0.268)
1.176 (0.297)[2]
0.328 (0.155)
0.454 (0.177)
0.526 (0.213)
0.727 (0.241)
0.771 (0.283)[0]0.267 (0.148)
0.315 (0.245)
0.330 (0.282)
0.617 (0.321)
0.809 (0.355)[0]0.294 (0.177)
0.675 (0.314)
0.598 (0.381)
0.936 (0.409)
1.409 (0.384)[0]0.557 (0.182)
0.655 (0.260)
0.722 (0.304)
1.036 (0.312)
1.208 (0.331)[0]

FGLScorrected
0.851 (0.009)
1.224 (0.094)
1.278 (0.140)
1.263 (0.143)
1.227 (0.151)[1]
1.203 (0.056)
0.596 (0.199)
1.239 (0.107)
1.280 (0.122)
1.396 (0.091)[1]
1.658 (0.115)
1.666 (0.260)
1.653 (0.331)
1.940 (0.412)
2.719 (0.497)[4]
1.311 (0.075)
1.464 (0.246)
1.571 (0.372)
1.636 (0.462)
2.005 (0.254)[2]0.696 (0.007)
0.930 (0.096)
0.994 (0.103)
1.066 (0.131)
0.985 (0.157)[1]
0.972 (0.031)
0.649 (0.188)
0.935 (0.127)
0.932 (0.164)
0.937 (0.140)[1]
1.347 (0.079)
1.392 (0.258)
1.453 (0.251)
1.454 (0.303)
1.701 (0.633)[4]
1.088 (0.048)
1.320 (0.172)
1.413 (0.143)
1.579 (0.170)
1.692 (0.133)[3]

Hausman
Test
5.032‡
6.349‡
6.886‡
7.081‡
3.663†
16.382‡15.242‡5.224‡
3.104†
0.951
11.892‡
6.955‡
2.523
2.061
0.203
13.502‡5.930‡
3.723†
5.455‡
4.184‡
4.188‡
6.657†
2.638
1.212
0.127
7.201‡
7.329‡
4.463‡
0.864
0.001
2.775†
2.642
2.090
0.161
1.559
3.709†
3.034†
0.767
1.055
0.982

Note: Refer to the notes in Table 2. Annual data covering 1950-1992 are used for four developing countries (COL:
Columbia; ECU: Ecuador; EGT: Egypt; PAK: Pakistan) and four industrial countries (GER: Germany; LUX: Luxemburg;
NZL: New Zealand; SWI: Switzerland). The regression equation is ln(yDEV ) = α + βln(yIN D ) + e.

33

Table 5-2: Application to Output Convergence (Regressand: industrial countries; Regressor:
industrial countries)
Regressand

Regressor

LUX

GER

NZL

SWI

GER

LUX

NZL

SWI

k

DOLS

0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC

1.291 (0.246)
1.302 (0.207)
1.312 (0.174)
1.294 (0.222)
1.293 (0.262)[4]
1.829 (0.125)
1.802 (0.165)
1.775 (0.105)
1.760 (0.060)
1.692 (0.076)[5]
1.494 (0.087)
1.482 (0.127)
1.452 (0.127)
1.413 (0.118)
1.369 (0.184)[4]
0.726 (0.239)
0.800 (0.168)
0.925 (0.191)
1.030 (0.265)
1.104 (0.328)[0]
1.289 (0.633)
1.256 (0.852)
1.227 (0.555)
1.255 (0.886)
1.188 (0.676)[3]
1.040 (0.499)
1.016 (0.642)
1.006 (0.461)
0.984 (0.616)
0.932 (0.900)[1]

GLScorrected
0.623 (0.112)
1.018 (0.162)
1.094 (0.174)
1.040 (0.184)
1.074 (0.178)[4]0.494 (0.159)
1.089 (0.240)
1.268 (0.285)
1.480 (0.301)
1.671 (0.174)[5]0.811 (0.147)
1.260 (0.150)
1.326 (0.169)
1.339 (0.171)
1.347 (0.148)[4]0.702 (0.126)
0.581 (0.136)
0.536 (0.163)
0.723 (0.185)
0.838 (0.209)[1]
0.412 (0.176)
0.796 (0.294)
0.797 (0.354)
1.098 (0.362)
1.277 (0.383)[1]
0.603 (0.184)
0.898 (0.240)
0.771 (0.273)
0.875 (0.309)
0.967 (0.336)[1]

FGLScorrected
1.330 (0.044)
0.719 (0.079)
0.849 (0.077)
0.706 (0.101)
0.694 (0.119)[1]
1.889 (0.078)
1.465 (0.165)
1.658 (0.096)
1.643 (0.056)
1.746 (0.079)[5]
1.531 (0.064)
1.305 (0.121)
1.379 (0.107)
1.331 (0.090)
1.421 (0.134)[4]0.678 (0.045)
0.909 (0.092)
0.938 (0.088)
1.039 (0.111)
1.000 (0.133)[1]1.290 (0.040)
0.445 (0.447)
1.231 (0.503)
0.464 (0.814)
2.311 (0.608)[1]
1.035 (0.007)
0.932 (0.480)
1.443 (0.469)
1.578 (0.626)
1.766 (0.845)[1]

Hausman
Test
2.970†
1.219
2.314
2.046
0.224
8.602‡
4.908‡
3.161†
0.029
0.318
2.006
0.631
0.223
0.023
0.329
1.914
4.105‡
2.793†
1.971
3.667†
3.015†
1.272
0.201
0.067
0.094
0.219
0.617
0.126
0.009
0.157

Note: Refer to the notes in Table 2. Annual data covering 1950-1992 are used for four developing countries (COL:
Columbia; ECU: Ecuador; EGT: Egypt; PAK: Pakistan) and four industrial countries (GER: Germany; LUX: Luxemburg;
NZL: New Zealand; SWI: Switzerland). The regression equation is ln(yIN D ) = α + βln(yIN D ) + e.

34

Table 5-2: Continued–
Regressand

Regressor

GER

NZL

LUX

SWI

GER

SWI

LUX

NZL

k

DOLS

0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC
0
1
2
3
4
BIC

0.509 (0.044)
0.498 (0.060)
0.498 (0.075)
0.520 (0.096)
0.517 (0.142)[5]0.676 (0.232)
0.703 (0.233)
0.731 (0.133)
0.734 (0.209)
0.745 (0.184)[5]
0.797 (0.043)
0.801 (0.050)
0.778 (0.047)
0.763 (0.057)
0.844 (0.080)[0]0.634 (0.045)
0.649 (0.079)
0.671 (0.119)
0.676 (0.179)
0.709 (0.278)[1]
0.831 (0.340)
0.850 (0.244)
0.857 (0.214)
0.845 (0.250)
0.824 (0.268)[0]1.206 (0.073)
1.219 (0.098)
1.224 (0.095)
1.233 (0.088)
1.234 (0.116)[1]

GLScorrected
0.395 (0.127)
0.449 (0.149)
0.459 (0.175)
0.285 (0.177)
0.315 (0.170)[5]0.292 (0.125)
0.552 (0.202)
0.576 (0.224)
0.483 (0.206)
0.628 (0.228)[5]0.613 (0.145)
0.742 (0.201)
0.817 (0.224)
0.681 (0.227)
0.812 (0.238)[0]0.531 (0.097)
0.623 (0.100)
0.650 (0.114)
0.572 (0.130)
0.672 (0.149)[1]0.350 (0.107)
0.724 (0.162)
0.746 (0.183)
0.702 (0.214)
0.611 (0.238)[1]
0.503 (0.119)
0.991 (0.187)
1.005 (0.232)
1.056 (0.260)
1.194 (0.272)[1]

FGLScorrected
0.501 (0.046)
0.507 (0.047)
0.488 (0.055)
0.511 (0.060)
0.452 (0.081)[5]0.671 (0.044)
0.314 (0.161)
0.485 (0.105)
0.496 (0.164)
0.597 (0.144)[5]0.783 (0.013)
0.791 (0.050)
0.769 (0.046)
0.789 (0.055)
0.832 (0.080)[3]0.639 (0.045)
0.653 (0.063)
0.665 (0.078)
0.672 (0.091)
0.696 (0.122)[1]
0.848 (0.022)
0.421 (0.142)
0.469 (0.137)
0.423 (0.169)
0.442 (0.182)[1]1.233 (0.014)
1.148 (0.106)
1.171 (0.095)
1.180 (0.091)
1.263 (0.124)[1]

Hausman
Test
0.130
0.053
2.174
2.115
0.106
0.605
0.948
2.393
0.389
0.014
0.155
0.060
0.203
0.039
0.068
0.077
0.042
0.406
0.078
4.580
0.934
0.626
0.606
1.101
0.133
1.700
1.219
0.634
0.036
0.051

Note: Refer to the notes in Table 2. Annual data covering 1950-1992 are used for four developing countries (COL:
Columbia; ECU: Ecuador; EGT: Egypt; PAK: Pakistan) and four industrial countries (GER: Germany; LUX: Luxemburg;
NZL: New Zealand; SWI: Switzerland). The regression equation is ln(yIN D ) = α + βln(yIN D ) + e.

35

Table 6: Application to PPP for traded and non-traded goods
Estimator

k
0
1
2

DOLS
3
4
BIC
0
1
2
GLScorrected

3
4
BIC
0
1
2

FGLScorrected3
4Hausman
Test

BIC
0
1
2
3
4

FRA
1.149
(0.312)1.179(0.424)
1.195(0.515)1.186(0.524)1.195(0.502)[0]0.833(0.393)0.984(0.454)1.259(0.469)1.374(0.501)1.549(0.520)[0]
1.156(0.259)1.229(0.329)1.178(0.355)1.152(0.339)1.252(0.333)
[1]
0.182
0.022
0.171
0.553
0.173

Traded Goods
ITA
JPN
U.K.
1.379
1.558
1.306
(0.165)(0.326)(0.201)
1.456
1.485
1.439
(0.217)(0.459)(0.262)
1.511
1.442
1.533
(0.277)(0.646)(0.342)
1.531
1.390
1.571
(0.308)(0.561)(0.392)
1.553
1.388
1.613
(0.353)(0.471)(0.412)
[0]
[0]
[2]
1.114
1.086
1.030
(0.381)(0.411)(0.365)
1.086
1.221
1.324
(0.436)(0.477)(0.419)
1.333
1.415
1.516
(0.454)(0.477)(0.430)
1.391
1.246
1.560
(0.485)(0.504)(0.460)
1.661
1.248
1.600
(0.511)(0.535)(0.492)
[0]
[0]
[0]
1.358
1.605
1.269
(1.163)(0.783)(0.125)
1.456
1.607
1.248
(0.149)(0.339)(0.162)
1.487
1.717
1.323
(0.181)(0.396)(0.217)
1.463
1.625
1.376
(0.204)(0.318)(0.250)
1.603
1.506
1.423
(0.245)(0.277)(0.272)
[1]
[2]
[1]
0.883
0.256
0.080
0.209
0.002
0.002
0.109
0.051
0.001
0.058
0.039
0.001
0.008
0.016
0.011

U.S.
1.053
(0.198)1.078(0.276)1.092(0.328)1.102(0.381)
1.109(0.401)[0]0.919(0.140)1.027(0.161)1.103(0.167)1.158(0.176)
1.184(0.188)[0]1.049(0.054)0.766(0.242)0.802(0.266)0.822(0.301)0.826(0.284)
[1]
0.132
0.006
0.134
0.222
0.007

FRA
1.872
(0.165)1.887(0.151)1.898(0.161)1.888(0.165)1.871(0.159)[0]0.375(0.178)0.864(0.315)1.127(0.369)1.255(0.422)1.577(0.456)[0]1.868(0.232)1.909(0.141)
1.932(0.152)1.920(0.157)1.922(0.153)[2]
12.667‡
5.097‡
2.317
0.424
0.096

Non-traded Goods
ITA
JPN
U.K.
2.142
2.357
2.059
(0.241)(0.299)(0.278)
2.175
2.376
2.066
(0.222)(0.299)(0.290)
2.216
2.399
2.067
(0.237)(0.331)(0.272)
2.250
2.397
2.054
(0.247)(0.346)(0.261)
2.287
2.402
2.042
(0.252)(0.343)(0.233)
[0]
[5]
[1]
0.448
0.372
0.351
(0.176)(0.198)(0.171)
0.900
0.888
0.988
(0.309)(0.352)(0.295)
1.104
1.046
1.001
(0.368)(0.408)(0.355)
1.171
1.192
1.088
(0.422)(0.456)(0.407)
1.411
1.379
1.166
(0.468)(0.503)(0.457)
[0]
[0]
[1]
2.129
2.354
2.063
(1.223)(0.760)(0.156)
2.005
1.947
1.671
(0.201)(0.257)(0.226)
2.102
2.095
1.864
(0.218)(0.286)(0.217)
2.158
2.173
1.983
(0.233)(0.311)(0.214)
2.285
2.202
2.052
(0.240)(0.310)(0.189)[1][5][1]
20.200‡16.524‡13.972‡10.603‡9.597‡
8.118‡
6.604‡
6.494‡
4.912‡
3.363†
3.738†
3.019†
1.934
0.874
1.813

U.S.
1.711
(0.443)1.728(0.441)1.748(0.473)1.762(0.511)1.763(0.484)[0]0.159(0.080)0.397(0.142)0.569(0.166)0.751(0.185)0.785(0.208)[0]1.712(0.073)0.291(0.159)0.357(0.191)
0.392(0.216)0.631(0.234)[3]
18.459‡17.424‡
16.549‡14.485‡12.988‡N
N
Note: Results are for ftT = α + βpT
t + et and ft = α + βpt + et using Canada as a base country. Figures in parenthesis

represent standard errors. ‘k’ denotes the maximum length of leads and lags. Hausman test represents the Hausman-type(β̂dgls −β̃dols )2 )→ χ2 (1). The critical
V ar(β̃dgls )2values of χ (1) are 2.71, 3.84 and 6.63 for ten, five, and one percent significance level. ‡(†) represents that the nullcointegration test as stipulated in section 2.4. The test statistic is constructed as

hypothesis of β̂dgls = β̃dols can be rejected at 5% (10%) significance level.

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38