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

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1 Structural Spurious Regressios ad A Hausma-type Coitegratio Test Choi, Chi-Youg, Lig Hu, ad Masao Ogaki Workig Paper No. 517 May 25 UNIVERSITY OF ROCHESTER

2 Structural Spurious Regressios ad A Hausma-type Coitegratio Test Chi-Youg Choi Departmet of Ecoomics Uiversity of New Hampshire Masao Ogaki Departmet of Ecoomics Ohio State Uiversity April, 25 Lig Hu Departmet of Ecoomics Ohio State Uiversity Abstract This paper proposes two estimators based o asymptotic theory to estimate structural parameters with spurious regressios ivolvig uit-root ostatioary variables. This approach motivates a Hausma-type test for the ull hypothesis of coitegratio for dyamic Ordiary Least Squares estimatio usig oe of our estimators for spurious regressios. We apply our estimatio ad testig methods to four applicatios: (i) log-ru moey demad i the U.S.; (ii) log-ru implicatios of the cosumptio-leisure choice; (iii) output covergece amog idustrial ad developig coutries; (iv) Purchasig Power Parity for traded ad o-traded goods. Keywords: Spurious regressio, GLS correctio method, Dyamic regressio, Test for coitegratio. JEL Classificatio Numbers: C1, C15 Correspodig Author: Masao Ogaki, Departmet of Ecoomics, Ohio State Uiversity, Columbus, Ohio ; ogaki.1@osu.edu We thak Do Adrews, Yaqi Fa, Clive Grager, Lars Hase, Kotaro Hitomi, Hide Ichimura, Lutz Kilia, Yuichi Kitamura, Yoshihiko Nishiyama, Peter Phillips, Pig Wag, Arold Zeller, ad semiar participats at the Bak of Japa, Kyoto Uiversity, Ohio State Uiversity, Uiversity of Tokyo, Natioal Chug Cheg Uiversity, Vaderbilt Uiversity, Yale Uiversity, the 24 Far Easter Meetig of the Ecoometric Society, ad the 24 Aual Meetig of the Midwest Ecoometrics Group for helpful commets. Choi gratefully ackowledges the fiacial support through Summer Fellowship from the Uiversity of New Hampshire.

3 1 Itroductio Ecoomic models ofte imply that certai variables are coitegrated. However, tests ofte fail to reject the ull hypothesis of o coitegratio for these variables. Oe possible explaatio of these test results is that the error is uit-root ostatioary due to a ostatioary measuremet error i oe variable or ostatioary omittted variables. I the uit-root literature, whe the stochastic error of a regressio is uit-root ostatioary, the regressio is techically called a spurious regressio. This is because the stadard t-test teds to be spuriously sigificat eve whe the regressor is statistically idepedet of the regressad i Ordiary Least Squares. Mote Carlo simulatios have ofte bee used to show that the spurious regressio pheomeo occurs with regressios ivolvig uit-root ostatioary variables (see, e.g., Grager ad Newbold (1974), Nelso ad Kag (1981, 1983)). Asymptotic properties of estimators ad test statistics for regressio coefficiets of these spurious regressios have bee studied by Phillips (1986, 1998) ad Durlauf ad Phillips (1988) amog others. The purpose of this paper is twofold. First, we propose a ew approach to estimatig structural parameters with spurious regressios. Whe structural parameters ca be recovered from spurious regressios, we call these structural spurious regressios. Secod, we propose a Hausma-type test for the ull hypothesis of coitegratio. This test is aturally motivated by the structural regressio approach. We also show that this test ca be used to test for coitegratio eve whe the spurious regressio is ot structural uder the alterative hypothesis. As a example of a structural spurious regressio, cosider a regressio to estimate the moey demad fuctio whe moey is measured with a ostatioary error. Currecy held by domestic ecoomic agets for legitimate trasactios is very hard to measure sice currecy is held by foreig residets ad is also used for black market trasactios. Therefore, moey may be measured with a ostatioary error. As show by Stock ad Watso (1993) amog others, uder the assumptio that the moey demad fuctio is stable i the log-ru, we have a coitegratig regressio if all variables are measured without error. If the variables are measured with statioary measuremet errors, we still have a coitegratig regressio. However, if moey is measured with a ostatioary measuremet error, we have a spurious regressio. We ca still recover structural parameters uder certai coditios. The crucial assumptio is that the ostatioary measuremet error is ot coitegrated with the regressors. Aother example of a structural spurious regressio is a regressio of moey demad with ostatioary omitted variables. Cosider the case i which the moey demad is stable i the log-ru whe a measure of shoe leather costs of holdig moey is icluded as a argumet. If a ecoometricia omits the measure of shoe leather costs from the moey demad regressio, ad if the measure is ostatioary, the regressio error is ostatioary. Shoe leather costs of holdig moey are related to the value of time, ad therefore to the real wage rate. Because the real wage rate is ostatioary i stadard dyamic stochastic geeral equilibrium models with a ostatioary techological shock, the omitted measure of the shoe leather costs is likely to be ostatioary. I this case, the moey demad regressio that omits the measure is spurious, but we ca still recover structural parameters uder certai coditios. The crucial assumptio is that the omitted variable is ot coitegrated with the regressors. Our structural spurious regressio approach is based o the Geeralized Least Squares (GLS) solutio of the spurious regressio problem aalyzed by Ogaki ad Choi (21), 1 who use a exact small 1 Aother approach would be to take the first differece to iduce statioarity ad the use istrumetal variables. This is the approach proposed by Lewbel ad Ng (25) for their Nostatioary Traslog Demad System. Our approach exploits the particular form of edogeeity assumed by may authors i the coitegratio literature ad avoids the use of istrumetal variables. Our approach yields more efficiet estimators as log as the particular form of edogeeity is correctly specified. This is importat especially whe weak istrumets cause problems. 1

4 sample aalysis based o the coditioal probability versio of the Gauss-Markov Theorem. We develop asymptotic theory for two estimators motivated by the GLS correctio: GLS corrected dyamic regressio ad feasible GLS (FGLS) corrected dyamic regressio estimators. These estimators will be show to be cosistet ad asymptotically ormally distributed i spurious regressios. Whe the error term is i fact statioary, ad hece the variables are coitegrated, the GLS corrected estimator is ot efficiet, but the FGLS corrected estimator, like the OLS estimator, is supercosistet. Hece the FGLS estimatio is a robust procedure with respect to the error specificatio. The FGLS corrected estimator is asymptotically equivalet to the GLS corrected estimator i spurious regressios ad it is asymptotically equivalet to the OLS estimator i coitegratig regressios. I some applicatios, it is hard to determie whether or ot the error i the regressio is statioary or uit-root ostatioary because test results are icoclusive. I such applicatios, the FGLS corrected estimator is attractive because it is cosistet i both situatios as log as the method of the dyamic regressio removes the edogeeity problem. This approach aturally motivates a Hausma-type test 2 for the ull hypothesis of coitegratio agaist the alterative hypothesis of o coitegratio (or a spurious regressio) i the dyamic OLS framework. We costruct this test by otig that while both the dyamic OLS ad GLS corrected dyamic regressio estimators are cosistet i coitegratio estimatio, the dyamic OLS estimator is more efficiet. 3 O the other had, whe the regressio is spurious oly the GLS corrected dyamic regressio estimator is cosistet. Hece, we could do a coitegratio test based o the specificatios o the error. We show that uder the ull hypothesis of coitegratio the test statistics have a usual χ 2 limit distributio, while uder the alterative hypothesis of a spurious regressio, the test statistic diverges. I some applicatios, the assumptio that the spurious regressio is structural uder the alterative hypothesis is ot very attractive. If the violatio of coitegratio arises from reasos other tha a ostatioary measuremet error, it is hard to believe that the resultig spurious regressio is structural. For this reaso, we relax the assumptio that the spurious regressio is structural ad show that the Hausma-type coitegratio test statistic still diverges uder the alterative hypothesis. Dyamic OLS is used i may applicatios of coitegratio. However, few tests for coitegratio have bee developed for dyamic OLS with the exceptio of Shi s (1994) test. As i Phillips ad Ouliaris (199), the popular Augmeted Dickey-Fuller (ADF) test for the ull hypothesis of o coitegratio was origially desiged to be applied to the residual from static OLS rather tha the residual from dyamic OLS. Because the OLS ad dyamic OLS estimates are ofte substatially differet, it is desirable to have a test for coitegratio applied to dyamic OLS. Aother aspect of our Hausma type test is that it is for the ull hypothesis of coitegratio. Ogaki ad Park (1998) argue that it is desirable to test the ull hypothesis of coitegratio rather tha that of o coitegratio i may applicatios whe ecoomic models imply coitegratio. Usig Mote Carlo experimets, we compare the fiite sample performace of the Hausma-type test with the test proposed by Shi (1994), which is a locally best ivariat test for the ull of zero variace of a radom walk compoet i the disturbaces. The experimet results show that up to the sample size of 3 the Hausma-type test is domiat i both size ad power. Whe the sample size icreases, Shi s test is better i power, but it suffers from a serious oversize problem. 2 This test ca also be called Durbi-Wu-Hausma-type test as it is closely related to ideas ad tests i Durbi (1954) ad Wu (1973) as well as a family of tests proposed by Hausma (1978). 3 After completig the first draft, it has come to our attetio that the Hausma-type test was origially proposed by Ferádez-Macho ad Mariel (1994) for the static OLS coitegratig regressio with strict exogeeity ad without ay serial correlatio. The test probably has ot bee popular because these assumptios are hard to justify i applicatios ad because the test was ot developed for dyamic regressios. 2

5 I some applicatios, it is appropriate to cosider the possibility that measuremet error is I(1) ad is ot coitegrated with the regressors. For these applicatios, the ADF test is applicable uder the ull hypothesis of a structural spurious regressio as show i Hu ad Phillips (25). For such applicatios, we recommed that both the ADF test ad Hausma-type test be applied because it is ot clear which ull hypothesis is more appropriate. We apply our estimatio ad testig methods to four applicatios: (i) log-ru moey demad i the U.S.; (ii) log-ru implicatios of the cosumptio-leisure choice; (iii) output covergece amog idustrial ad developig coutries; (iv) Purchasig Power Parity (PPP) for traded ad o-traded goods. I the first two applicatios, we focus o estimatig ukow structural parameters, while i the last two applicatios we purport to test for coitegratio with the Hausma-type coitegratio test, where we relax the assumptio that the spurious regressio uder the alterative hypothesis is structural. The rest of the paper is orgaized as follows. Sectio 2 gives ecoometric aalysis of the model, icludig asymptotic theories ad fiite sample simulatio studies. Sectio 3 presets models of ostatioary measuremet error ad ostatioary omitted variables. It the presets the empirical results of four applicatios. Sectio 4 cotais cocludig remarks. 2 The ecoometric model Cosider the regressio model where {x t } is a m-vector itegrated process geerated by The error term i (1) is assumed to be η t = y t = β x t + η t, (1) x it = v it. m k i=1 j= k γ i,j v i,t j + e t, (2) e t = ρe t 1 + u t. (3) Assumptio 1 Assume that v t = (v 1t,..., v mt ) ad u t are zero mea statioary processes with E v it α <, E u t α < for some α > 2 ad strog mixig with size α/(α 2). We also assume that the method of dyamic regressio removes the edogeeity problem, that is, E(u t v s ) = for all t, s. We call this the strict exogeeity assumptio for the dyamic regressio. The coditios o v t ad u t esure the ivariace priciples: for r [, 1, 1/2 [r v t d V (r), 1/2 [r u t d U(r), where V (r) is a m-vector Browia motio with covariace j= E(v tv t j ), ad U(r) is a Browia motio with variace j= E(u tu t j ). The fuctioal cetral limit theorem holds for weaker assumptios tha assumed here (De Jog ad Davidso (2)), but the coditios assumed above are geeral eough to iclude may statioary Gaussia or o-gaussia ARMA processes that are commoly assumed i empirical modelig. Let v t = ( x 1,t k,..., x 1,t,..., x 1,t+k,..., x m,t k,..., x m,t,..., x m,t+k ), ad γ = (γ 1, k,..., γ 1,,..., γ 1,k,..., γ m, k,..., γ m,,..., γ m,k ). We estimate the structural parameter β i the regressio y t = β x t + γ v t + e t. (4) 3

6 The iferece procedure about β differs accordig to differet assumptios o the error term e t i (3). Whe ρ < 1, e t is statioary, ad hece regressio (4) is a coitegratio regressio with serially correlated error. Whe ρ = 1, e t is a uit root ostatioary process ad the OLS regressio is spurious. Both models are importat i empirical studies i macroecoomics ad fiace. I the ext two sectios, we will study the asymptotic properties of differet estimatio procedures uder these two assumptios. Uder the assumptio that ρ = 1, OLS is ot cosistet while both GLS correctio ad FGLS correctio will give cosistet ad asymptotically equivalet estimators. Uder the assumptio that ρ < 1, the GLS corrected estimator is ot efficiet as it is coverget, but the FGLS estimator is coverget ad asymptotically equivalet to the OLS estimator. Therefore, FGLS is robust with respect to the error specificatios (ρ = 1 or ρ < 1). 2.1 Regressios with I(1) error I this sectio we cosider the situatio whe the error term is I(1), i.e., ρ = 1 i (3). The estimatio methods we study are dyamic OLS, GLS correctio, ad FGLS correctio The dyamic OLS spurious estimatio We start with dyamic OLS estimatio of regressio (4). Uder the assumptio of ρ = 1, this regressio is spurious sice for ay value of β the error term is always I(1). I Appedix A, we show that the DOLS estimator ˆβ dols has the followig limit distributio: [ 1 ( ˆβ dols β ) 1 [ 1 V (r)v (r) dr V (r)u(r)dr. (5) ˆγ i the estimatio is also icosistet with ˆγ γ = O p (1). As remarked i Phillips (1986, 1989), i spurious regressios the oise is as strog as the sigal. Hece, ucertaity about β persists i the limitig distributios GLS corrected estimatio Whe ρ = 1, we ca filter all variables i regressio (4) by takig the full first differece, ad use OLS to estimate y t = β x t + γ v t + u t = θ z t + u t, (6) where θ = (β, γ ) ad z t = (x t, v t). This procedure ca be viewed as GLS corrected estimatio. 4 If we let θ dgls deote the GLS corrected estimator, the we ca show that ( θdgls θ ) d N(, Ω), (7) where Ω = Q 1 ΛQ 1 with Q = E(z t z t) ad Λ is the log ru variace matrix of z t u t. Thus β i a structural spurious regressio ca be cosistetly estimated (joitly with γ) ad the estimators are asymptotically ormal. I the special case whe m = 1, {v 1t } ad {u t } are i.i.d. sequeces ad η t = e t, (7) gives that ( θdgls θ ) d N(, σ 2 u/σ 2 1v), where σ 2 u ad σ 2 1v are variaces of u t ad v 1t, respectively. 4 This is a covetioal GLS procedure whe u t is i.i.d.. Whe u t is serially correlated as i our approach, we ame this procedure GLS corrected dyamic estimatio. 4

7 2.1.3 The FGLS estimatio To use GLS to estimate a regressio with serial correlatio i empirical work, a Cochrae-Orcutt FGLS procedure is usually adopted. This procedure also works for spurious regressios as show by Phillips ad Hodgso (1994). They show that the FGLS estimator is asymptotically equivalet to that i the differeced regressio whe the error is uit-root ostatioary. I the preset paper, we will show that the FGLS correctio to the dyamic regressio provides a cosistet ad robust estimator to structural spurious regressios. Let the residual from OLS regressio (4) be deoted by ê t, ê t = y t ˆβ x t ˆγ v t. To coduct the Cochrae-Orcutt GLS estimatio, we first ru a AR(1) regressio of ê t, ê t = ˆρ ê t 1 + û t. (8) It ca be show that (ˆρ 1) = O p (1). Coduct the followig Cochrae-Orcutt trasformatio of the data: ỹ t = y t ˆρ y t 1, x t = x t ˆρ x t 1, ṽ t = v t ˆρ v t 1. (9) The cosider OLS estimatio of the regressio ỹ t = β x t + γ ṽ t + error = θ z t + error, (1) where z t = ( x t, ṽ t ). The OLS estimator of θ i (1) is computed as [ 1 [ θ fgls = z t z t z t ỹ t. (11) The limitig distributio of θ fgls ca be show to be the same as i (7). Ituitively, eve though the dyamic OLS estimator is icosistet, the residual is uit-root ostatioary because o liear combiatio of y t ad x t is statioary. Therefore, ρ approaches uity i the limit, ad z t behaves asymptotically equivalet to z t. A detailed proof of results i this sectio is give i Appedix A. 2.2 Regressios with I() error I this sectio we cosider the asymptotic distributios of the three estimators (DOLS estimator, GLS corrected estimator ad FGLS corrected estimator) uder the assumptio of coitegratio, i.e., ρ < 1 (3) The dyamic OLS estimatio Uder the assumptio of coitegratio, the DGP of y t is y t = β x t + γ v t + e t, e t = ρe t 1 + u t, ρ < 1. (12) Applyig the ivariace priciple, for r [, 1, 1/2 [r e t E(r), where E(r) is a Browia motio with variace j= E(e te t j ). The limitig distributio of the OLS estimator of β, which is asymptotically idepedet of ˆγ, is kow to be ( ˆβ dols β ) d ( 1 ) 1 ( 1 ) V (r)v (r) dr V (r)de(r). (13) 5

8 2.2.2 GLS corrected estimatio Now, if we take the full first differece as we did i the spurious regressios, the regressio becomes y t = β x t + γ v t + e t e t 1 = θ z t + e t e t 1. (14) Note that this trasformatio leads to loss i efficiecy sice the estimator β dgls is ow coverget rather tha coverget as the DOLS estimator. With some mior revisios to equatio (7), the limitig distributio of the estimator i this case ca be writte as ( θdgls θ ) d N(, Ω ), (15) where Ω = Q 1 Λ Q 1. Q is defied as followig equatio (7) ad Λ is the log ru variace matrix of vector z t e t. I the special case whe m = 1, {v 1t } ad {u t } are i.i.d. sequeces ad η t = e t, Ω = 2σ 2 e(1 ψ e )/σ 2 1v, where ψ e is the first order autocorrelatio coefficiet of {e t } The FGLS estimatio Istead of takig the full first differece, if we estimate the autoregressio coefficiet i the error ad use this estimator to filter all sequeces, we will obtai a estimator that is asymptotically equivalet to the DOLS estimator. Ituitively, i the case that the error e t = u t is serially ucorrelated, the the AR(1) coefficiet ˆρ will coverge to zero, ad hece the trasformed regressio will be asymptotically equivalet to the origial regressio. Or, if the error is statioary ad serially correlated, the the AR(1) coefficiet will be less tha uity, ad, as show i Phillips ad Park (1988), the GLS estimator ad the OLS estimator i a coitegratio regressio are asymptotically equivalet. If we coduct the Cochrae-Orcutt trasformatio (9) ad estimate β i regressio ỹ t = β x t + γ ṽt + error, (16) the Appedix B shows that the limitig distributio of β is the same as the limit of the OLS estimator give i (13). We summarize the FGLS corrected estimator i the followig propositio: Propositio 1 Suppose Assumptio 1 holds. I spurious regressios, the FGLS corrected estimator is asymptotically equivalet to the GLS corrected estimator, ad its limit distributio ca be writte as ( θfgls θ ) d N(, Ω). I coitegratio regressios, the FGLS corrected estimator is asymptotically equivalet to the DOLS estimator, ad its limit distributio ca be writte as ( ˆβ fgls β ) d ( 1 ) 1 ( 1 ) V (r)v (r) dr V (r)de(r). So FGLS is ot oly valid whe the regressio is spurious but also asymptotically efficiet whe the regressio is coitegratio. Remarks: 1. If a costat is added to (4), we show i Appedix D that our methods still apply. I particular, the GLS or FGLS corrected estimators are asymptotically equivalet to that give i (7) uder the assumptio of spurious regressios. 2. If a tred term is added to (4) (this is the case i which the determiistic coitegratio restrictio is ot satisfied i the termiology of Ogaki ad Park (1988)), the the GLS corrected estimatio leads to a sigular covariace matrix for the estimator whe ρ is less tha oe i absolute value. This is 6

9 because a tred term i (4) leads to a costat term i the first differeced regressio (14) ad because the log ru variace of the first differece of e t multiplied by a costat is zero. Therefore, our methods do ot apply to regressios with time treds. 3. Uder some coditios, the methods proposed i this paper also apply to other model cofiguratios, such as regressios where the regressors have drifts. These extesios will be studied i future work. 2.3 Fiite sample performace of the three estimators From the above aalysis, we show that the FGLS corrected estimatio is a robust procedure with respect to error specificatios. I this sectio, we use simulatios to study its fiite sample performaces compared to the other two estimators. I the simulatio we cosider the case whe x t is a scalar variable ad we geerate v t ad u t from two idepedet stadard ormal distributios ad let e t = ρe t 1 + u t. The structural parameter is set to β = 2, ad γ v t =.5v t. The umber of iteratios i each simulatio is 5 ad i each replicatio, 1 + observatios are geerated of which the first 1 observatios are discarded. Table 1 shows the bias ad the mea square error (MSE) of all three estimators for ρ =,.95, ad 1. Whe ρ =, the regressio is coitegratio with i.i.d. error. It is clear that the DOLS estimator is the best oe whe = 5. But whe reaches 1, the FGLS estimator becomes almost as good as the DOLS estimator. Whe ρ =.95, the regressio is coitegratio with serially correlated error. I this case, the GLS ad FGLS estimators are much better tha the DOLS estimator. Whe the sample size icreases, the FGLS estimator becomes the best oe. Fially whe ρ = 1, the regressio is spurious ad the GLS corrected estimator performs the best as expected. Figure 1 plots the three estimators whe = 1 as ρ approaches 1. The figures show that the DOLS estimator becomes flatter ad flatter as ρ 1. The GLS estimator remai largely the same for ρ close to uity. Ad the FGLS estimator becomes a bit flatter whe ρ reaches 1, but it still shows a clear peak aroud zero. From the fiite sample performace, it ca be see that the FGLS estimator is almost as good as the DOLS estimator i coitegratios ad it sigificatly outperforms the DOLS estimator i spurious regressios. The GLS estimator is the best whe ρ approaches 1, but it suffers from sigificat loss i efficiecy whe ρ is small. So we may wat to take full differece oly whe we are very sure that the error is uit root ostatioary. Otherwise the FGLS estimator is a good choice. 2.4 Hausma specificatio test for coitegratio The test statistic ad its asymptotic properties I this sectio we costruct a Hausma-type coitegratio test based o the differece of two estimators: a OLS estimator ( ˆβ dols ) ad a GLS corrected estimator ( β dgls ). This is equivalet to comparig estimators i a level regressio ad i a differeced regressio. The test is for the ull of coitegratig relatioships agaist the alterative of a spurious regressio: H : ρ < 1; agaist H A : ρ = 1. Our discussios so far suggest that uder the ull of coitegratio, both OLS ad GLS are cosistet but the OLS estimator is more efficiet. However uder the alterative of a spurious regressio, oly the GLS corrected estimator is cosistet. 7

10 dyamic OLS regressio estimator rho =.98 rho =.99 rho = GLS corrected dyamic regressio estimator rho =.98 rho =.99 rho = FGLS corrected dyamic regressio estimator rho =.98 rho =.99 rho = Figure 1: Compariso of three estimators whe = 1 ad ρ 1 8

11 Let ˆV β deote a cosistet estimator for the asymptotic variace of ( β dgls β). Uder our assumptios, it coverges to the correspodig submatrix of Ω uder the ull hypothesis ad to the correspodig submatrix of Ω uder the alterative. For example, whe m = 1, {v 1t } ad {u t } are idepedet i.i.d. ad η t = e t, take ˆV β = ( 1 ) ( ŵ2 t / 1 ) x2 t, where ŵt deotes the residuals from OLS estimatio of differeced regressio. Uder the ull of coitegratio, ˆV β 2σe(1 2 ψ e )/σ1v. 2 Uder the alterative of spurious regressio, ˆV β σu/σ 2 1v. 2 We defie the Hausma-type test statistic as: h = ( β dgls ˆβ dols ) ˆV 1 β ( β dgls ˆβ dols ). (17) Propositio 2 Suppose Assumptio 1 holds. Uder the ull hypothesis of coitegratios, h χ 2 (m). Uder the alterative of spurious regressios, h = O p (). Proof: uder the ull of coitegratio, ( βdgls ˆβ dols ) = ( β dgls β ) ( ˆβ dols β ) = ( β dgls β ) + o p (1) N(, V β ), where V β is the asymptotic variace of β dgls uder the assumptio of coitegratio. Therefore, if ˆV β is a cosistet estimator for V β, h = ( β dgls ˆβ dols ) ( ˆV β ) 1 ( β dgls ˆβ dols ) χ 2 (m). Uder the alterative of spurious regressios, ( βdgls ˆβ dols ) = ( β dgls β ) ( ˆβ dols β ) = O p (1) + O p ( ) = O p ( ). Hece, h = O p () uder the alterative. We ca exted the test to allow edogeeity uder the alterative. Cosider the followig DGP: y t = β x t + γ v t + φs t + e t, e t = ρe t 1 + u t, where {s t } satisfies the same coditios as u t ad v t, but it is correlated with {v t }. The statistic defied i (17) ca be applied to test the hypotheses: H : ρ < 1 ad φ = agaist H A : ρ = 1 ad φ. The asymptotics of h uder the ull H are the same as that uder H. Uder the alterative H A, we show i Appedix C that the DOLS estimator has the same asymptotic distributio as that uder H A ad h = O p (). Therefore, this Hausma-type test is cosistet for the ull hypothesis of coitegratios agaist the alterative of spurious regressios, regardless of whether the exogeeity assumptio holds uder the alterative. 9

12 2.4.2 Fiite sample properties of the Hausma-type coitegratio test Before applyig the Hausma-type coitegratio test empirically, it will be istructive to examie its fiite sample properties i compariso with other comparable tests uder the same ull hypothesis. To this ed, we coduct a small simulatio experimet based o the followig dyamic regressio model, y t = γ 1 x t+1 + βx t + γ 2 x t 1 + e t, (18) e t = ρe t 1 + u t, (19) where γ 1 =.3, β = 2, γ 2 =.5, ad settig ρ =.9 for the size performace ad ρ = 1 for the power performace. We cosider sample sizes of {5, 1, 2, 3, 5} that are commoly ecoutered i empirical aalysis. I the simulatios pseudo radom umbers are geerated usig the GAUSS (versio 6.) RNDNS procedures. Each simulatio ru is carried out with 5, replicatios. At each replicatio 1 + radom umbers are geerated of which the first 1 observatios are discarded to avoid a start-up effect. Table 2 reports selected fiite sample properties of the Hausma-type coitegratio test together with a residual based test uder the ull of coitegratio due to Shi (1994, Shi s test) who exteded the KPSS test i the parametrically corrected coitegratig regressio. I the simulatios the legths of lead ad lag terms for DOLS ad DGLS are chose by the BIC rule. 5 A oparametric estimatio method for log ru variace estimatio is employed usig the QS kerel with the badwidth of iteger[8(/1) 1/4. The results i Table 2 illustrate two poits. First, the empirical size of the Hausma-type test is close to the omial size, i particular whe sample size is relatively large, whereas Shi s test suffers from a serious oversize problem. Secod, i terms of power, the Hausma-type test domiates Shi s test for moderate sample sizes that are very likely to be ecoutered empirically. Shi s test seems more powerful whe is relatively large, but oly at the cost of severe size distortios. Overall, our simulatio results provide evidece i favor of the Hausma-type test. 3 Empirical applicatios I this sectio we apply the GLS-type correctio methods ad the Hausma-type coitegratio test to aalyze four macroecoomic issues: (i) log-ru moey demad i the U.S.; (ii) log-ru implicatios of the cosumptio-leisure choice; (iii) output covergece amog idustrial ad developig coutries; (iv) Purchasig Power Parity (PPP) for traded ad o-traded goods. Tthe mai purpose of the first two applicatios is to illustrate the spurious regressio approach to estimatig ukow structural parameters. Idetificatio of the structural parameters i these two applicatios are based o ostatioary measuremet error or ostatioary omitted variables that are explaied i the followig two subsectios. The mai purpose of the last two applicatios is to apply the Hausma-type coitegratio test. The alterative hypothesis is ot take as structural spurious regressios i the last two applicatios. 3.1 A model of ostatioary measuremet error Cosider a set of variables that are coitegrated. Oe model of a structural spurious regressio is based o the case i which oe of the variables is measured with ostatioary measuremet error. Let y t be the true value of y t, ad assume that 5 It is a iterestig research topic to ivestigate the performace of various lag legth selectio rules, but would be beyod the scope of this paper. 1

13 y t = β x t + γ v t + e t (2) is a dyamic coitegratig regressio that satisfies the strict exogeeity assumptio. 6 measured value of yt, ad assume that the measuremet error satisfies Let y t be the y t y t = γ m v t + e m t, (21) where e m t is I(1) ad its expectatio coditioal o x s for all s is zero. Here, the crucial assumptio for idetificatio is that the measuremet error is ot coitegrated with x t. The y t = β x t + γ v t + e t (22) where γ = γ + γ m, ad e t is I(1) ad satisfies the strict exogeeity assumptio A model of ostatioary omitted variables Aother case that leads to a structural spurious regressio is a model of ostatioary omitted variables. y t = β x t + θ x t + γ 1 v t + γ v t + e t (23) where x t is a vector of I(1) variables ad v t is a vector of leads ad lags of the first differeces of x t. We imagie that the ecoometricia omits x t from his regressio. We assume that θ x t + γ v t = γ m v t + e m t, (24) where e m t is I(1) ad its expectatio coditioal o x s for all s is zero. Here, the crucial assumptio for idetificatio is that the x t is ot coitegrated with x t. The y t = β x t + γ v t + e t (25) where γ = γ 1 + γ m, ad e t is I(1) that satisfies the strict exogeeity assumptio. 3.3 U.S. moey demad The log-ru moey demad fuctio has ofte bee estimated uder a coitegratig restrictio amog real balaces, real icome, ad the iterest rate. The restrictio is legitimate if the moey demad fuctio is stable i the log-ru ad if all variables are measured without ostatioary error. Ideed, Stock ad Watso (1993) foud supportive evidece of stable log-ru M1 demad by estimatig coitegratig vectors. However, either if moey is measured with a ostatioary measuremet error or if ostatioary omitted variables exist, the we have a spurious regressio ad the estimatio results based o a coitegratio regressio are questioable. First, cosider the model of a ostatioary measuremet error described above. To be specific, we follow Stock ad Watso (1993) ad assume that the dyamic regressio error is statioary ad the strict exogeeity assumptio holds for the dyamic regressio error whe moey is correctly measured. We the assume that moey is measured with a multiplicative measuremet error. We assume that the log measuremet error is uit-root ostatioary, ad that the residuals of the projectio of the log 6 Note that ay variable ca be chose as the regressad i a coitegratig regressio. Therefore, we choose the variable with ostatioary measuremet error as the regressad. 7 Here, we assume that the dimesios of γ ad γ m are the same without loss of geerality, because we ca add zeros as elemets of γ ad γ m as eeded. 11

14 measuremet error o the leads ad lags of the regressors i the dyamic regressio satisfy the strict exogeeity assumptio. Give that a large compoet of the measuremet error is arguably currecy held by foreig residets ad black market participats, the log measuremet error is likely to be very persistet. Therefore, the assumptio that the log measuremet error is uit root ostatioary may be at least a good approximatio. The assumptio that the measuremet error is ot coitegrated with the regressors is plausible if the error is maily due to currecy held by foreig residets. Secod, cosider the model of ostatioary omitted variables. A possible omitted variable is a measure of the shoe leather cost that represets trasactio costs. For example, i the literature of moey demad estimatio, the real wage rate has sometimes bee used as a regressor for this reaso. Because the real wage rate is I(1) i stadard dyamic stochastic geeral equilibrium models with a I(1) techological shock, the omitted measure of the shoe leather cost is to be ostatioary. If the real wage rate is the omitted variable, the assumptio that it is ot coitegrated with the regressors that iclude log icome is ot very plausible. However, it is possible that the true omitted variable that represets the shoe leather cost is ot the real wage rate ad that it is ot coitegrated with log icome. We apply our GLS correctio methods to estimate log-ru icome ad iterest elasticities of M1 demad. 8 To this ed, the regressio equatios are set up with the real moey balace ( M P ) as regressad ad icome (y) ad iterest (i) as regressors. Followig Stock ad Watso (1993), the aual time series for M1 deflated by the et atioal product price deflator is used for M P, the real et atioal product M for y ad the six moth commercial paper rate i percetage for i. P ad y are i logarithms while three differet regressio equatios are cosidered depedig o the measures of iterest. We have tried the followig three fuctioal forms. Equatio 1 has bee studied by Stock ad Watso (1993). ( ) M l = α + β l (y t ) + γi t + e t, (equatio 1) P t ( ) M l = α + β l (y t ) + γ l(i t ) + e t, (equatio 2) l P ) ( M P t t [ 1 + it = α + β l (y t ) + γ l + e t. (equatio 3) i t It is worth otig that the liquidity trap is possible for the latter two fuctioal forms. Whe the data cotai periods with very low omial iterest rates, the latter two fuctioal forms may be more appropriate. Table 3 presets the poit estimates for β (icome elasticity of moey demad) ad γ based o the three estimators uder scrutiy: dyamic OLS, GLS corrected dyamic regressio estimator, ad FGLS corrected dyamic regressio estimator. 9 Several features emerge from the table. First, all the estimated coefficiets have theoretically correct sigs: positive sigs for icome elasticities ad egative sigs for γ for the first two fuctioal forms ad positive sigs for γ for the third fuctioal form. Secod, GLS corrected estimates of the icome elasticity are implausibly low for all three fuctioal forms for low values of k ad icrease to more plausible values ear oe as k icreases. The fact that the results become more plausible as k icreases suggests that the edogeeity correctio of dyamic regressios works i this applicatio for moderately large values of k such as 3 ad 4. The results for lower values of k are cosistet with those of low icome elasticity estimates of first differeced regressios that 8 Readers are referred to Appedix F for the empirical guidelies o the use of estimatio ad testig techiques developed i this paper. 9 For the FGLS corrected dyamic regressio estimator, the serial correlatio coefficiet of the error term is estimated before beig applied to the Cochrae-Orcutt trasformatio. This coefficiet is assumed to be uity i the GLS corrected dyamic regressio estimator which is equivalet to regressig the first differece of variables without a costat term. 12

15 had bee used i the literature before 198. Therefore, the estimators i the old literature of first differeced regressios before coitegratio became popular are likely to be dowward biased because of the edogeeity problem. Third, all poit estimates of the three estimators are very similar, ad the Hausma-type test fails to reject the ull hypothesis of coitegratio for large eough values of k. Hece, there is little evidece agaist coitegratio. However, it should be oted that a small radom walk compoet is very hard to detect by ay test for coitegratio. Therefore, it is assurig to kow that all three estimators are similar for large eough values of k, ad the estimates are robust with respect to whether the regressio error is I() or I(1). We report the value of k chose by the Bayesia Iformatio Criterio (BIC) rule throughout our empirical applicatios i order to give some guidace i iterpretig results. It is beyod the scope of this paper to study detailed aalysis of how k should be chose because this issue has ot bee settled i the literature of dyamic coitegratig regressios. Table 3 also reports the results whe the ADF test is applied to the residual of OLS. The results show evidece agaist the ull hypothesis of structural spurious regressios ad hece corroborate the results from the Hausma-type test uder the opposite ull hypothesis. 3.4 Log-ru implicatios of the cosumptio-leisure choice Cosider a simplified versio of Cooley ad Ogaki s (1996) model of cosumptio ad leisure i which the represetative household maximizes U = E [ δ t e(t) t= where E t deotes the expectatio coditioed o the iformatio available at t. We adopt a simple itraperiod utility fuctio that is assumed to be time ad state-separable ad separable i odurable cosumptio, durable cosumptio, ad leisure e(t) = C(t)1 β 1 1 β + v(l(t)) where v( ) represets a cotiuously differetiable cocave fuctio, C(t) is odurable cosumptio, ad l(t) is leisure. The usual first order coditio for a household that equates the real wage rate with the margial rate of substitutio betwee leisure ad cosumptio is give as: W (t) = v (l(t)) C(t) β where W (t) is the real wage rate. We assume that the stochastic process of leisure is (strictly) statioary i the equilibrium as i Eichebaum, Hase, ad Sigleto (1988). A implicatio of the first order coditio is that l(w (t)) β l(c(t)) = l(v (l(t))) is statioary. Whe we assume that the log of cosumptio is differece statioary, this implies that the log of the real wage rate ad the log of cosumptio are coitegrated with a coitegratig vector (1, β). We first cosider the model of the measuremet error. We assume that (W (t)) is measured with a multiplicative measuremet error, so that l(w (t)) is measured with a measuremet error ɛ(t). We assume that this measuremet error is uit-root ostatioary. Oe compoet of the measuremet error arises because it is difficult to measure frige beefits. Therefore, we expect the measuremet error to be very persistet, ad the assumptio of uit-root ostatioarity may be a good approximatio. 13

16 We also assume that the measuremet error ad log cosumptio are ot coitegrated for idetificaito. This assumptio does ot rule out oliear log-ru relatioships betwee these variables. Cosider a regressio l(w m (t)) = a + β l(c(t)) + e(t), (26) where W m (t) is the measured real wage rate, ad e(t) = ɛ(t) + l(v (l(t))) a. If ɛ(t) is statioary, the e(t) is statioary, ad Regressio (26) is a coitegratig regressio as i Cooley ad Ogaki. I this simple versio, the preferece parameter β is the Relative Risk Aversio (RRA) coefficiet, which is equal to the reciprocal of the itertemporal elasticity of substitutio (IES). Cooley ad Ogaki show that the same regressio ca be used to estimate the reciprocal of the log-ru IES whe prefereces for cosumptio are subjected to time oseparability such as habit formatio. For simplicity, we iterpret β as the RRA coefficiet i this paper. If ɛ(t) is uit-root ostatioary, the Regressio (26) is a spurious regressio because e(t) is ostatioary i this case. Hece, the stadard methods for coitegratig regressios caot be used. However, the preferece parameter β ca still be estimated by the spurious regressio method. The key assumptio for idetificatio is that the measuremet error is ot coitegrated with the regressor as discussed above. As log as the log multiplicative measuremet error ad log cosumptio are ot liearly related i the log-ru by a liear coitegratio relatioship, the assumptio is satisfied. It should be oted that either cosumptio or the wage rate ca be used as the regressad i the case of coitegratio. However, if the wage rate is measured with ostatioary error ad cosumptio is ot, the the wage rate should be chose as the regressad. Aother iterpretatio of this applicatio is based o the ostatioary omitted variables for this applicatio. A possible omitted variable represets demographic chages. I our aalysis, per capita real cosumptio series was costructed by dividig costat 1982 dollar cosumptio series by civilia oistitutioal adult (age 16 ad over) populatio. If this populatio series does ot capture the tred of all the demographic chage that affects the relatioship betwee the real wage rate ad cosumptio, we have a ostatioary omitted variable. As log as the omitted demographic chage ad log cosumptio are ot liearly related i the log-ru by a liear coitegratio relatioship, the assumptio that the omitted variable is ot coitegrated with the regressor is satisfied. Aother possible omitted variable is huma capital if huma capital affects the margial rate of substitutio betwee leisure ad cosumptio. Table 4 presets the estimatio results for the RRA coefficiet (β) based o various estimators. We used the same data set that Cooley ad Ogaki used. 1 The results i Table 4 illustrate several poits. First, all poit estimates for β have the theoretically correct positive sig. Secod, for odurables (ND), GLS-corrected dyamic regressio estimates of β are much lower tha Dyamic OLS estimates for all values of k. As a result, the Hausma-type coitegratio test rejects the ull hypothesis of coitegratio for all values of k at the 1 percet level. Therefore, the evidece supports the view that Regressio (26) is a spurious regressio, ad the true value of the RRA coefficiet is likely to be much lower tha the dyamic OLS estimates. Both the GLS corrected ad the robust FGLS corrected dyamic regressio estimatio results are cosistet with the view that the RRA coefficiet is about oe for the value of k chose by BIC. For odurables plus services (NDS), GLS corrected dyamic regressio estimates of β are much lower tha Dyamic OLS estimates for small values of k. As a result, the Hausma-type coitegratio test rejects the ull hypothesis of coitegratio at the 5 percet level whe k is, 1, ad 2. It still rejects the ull hypothesis of coitegratio at the 5 percet level whe k is 3. It does ot reject the ull hypothesis whe k is 4 or 5. Accordig to the BIC rule, k is chose to be 3, ad there is some evidece agaist coitegratio. However, because the GLS corrected 1 See Cooley ad Ogaki (1996, page 127) for a detailed descriptio of the data. 14

17 dyamic regressio estimates get closer to dyamic OLS estimates as k icreases, the evidece is ot very strog. It is likely that a small radom walk compoet exists for the error term of the regressio for NDS makig it a spurious regressio. The robust FGLS corrected dyamic regressio estimates are close to both GLS corrected dyamic regressio estimates ad dyamic OLS estimates as log as k is 3 or greater. Table 4 also reports ADF test results whe the test is applied to the residual of static OLS. The lag legth for the ADF test has bee chose by the sequetial t-test method advocated by Campbell ad Perro (1991). We do ot have ay evidece agaist the ull hypothesis of structural spurious regressios for ND. This hypothesis caot be rejected at the 5 percet level, but ca be rejected at the 1 percet level for NDS. Thus, we have fairly strog evidece that we have a spurious regressio for ND ad some evidece that we have spurious regressio for NDS. The true value of RRA is likely to be about oe for both ND ad NDS. 3.5 Output covergece across atioal ecoomies I this sectio, we apply the techiques to reexamie a log stadig issue i macroecoomics, the hypothesis of output covergece. For this applicatio ad the ext, our mai purpose is ot to estimate ukow structural parameters but to test the ull hypothesis of coitegratio with the Hausma-type test. For this purpose, we do ot eed the strict exogeeity assumptio uder the alterative hypothesis of o coitegratio (or a spurious regressio). As a key propositio of the eoclassical growth model, the covergece hypothesis has bee popular i macroecoomics ad has attracted cosiderable attetio i the empirical field, particularly durig the last decade. Besides its importat policy implicatios, the covergece hypothesis has bee used as a criterio to discer the two mai growth theories, exogeous growth theory ad edogeous growth theory. However, it remais the subject of cotiuig debate maily because the empirical evidece supportig the hypothesis is mixed. Nevertheless, the established literature based o popular iteratioal datasets such as the Summers-Hesto (1991) suggests as a stylized fact output covergece amog various atioal ecoomies: covergece amog idustrialized coutries but ot amog developig coutries ad ot betwee idustrialized ad developig coutries. Give that a mea statioary stochastic process of output disparities betwee two ecoomies is iterpreted as supportive evidece of stochastic covergece, uit-root or coitegratio testig procedures are ofte used by empirical researchers to evaluate the covergece hypothesis. I this vei, our techiques proposed here fit i the study of output covergece. We cosider four developig coutries (Columbia, Ecuador, Egypt, ad Pakista) alog with four idustrial coutries (Germay, Luxemburg, New Zealad, Switzerlad). The raw data are extracted from the Pe World Tables of Summers- Hesto (1991) ad cosist of aual real GDP per capita (RGDPCH) over the period of The followig two regressio equatios are cosidered with regard to the coitegratio relatio. y D t = α + βy I t + e t, (27) y I t = α + βy I t + e t, (28) where y D t ad y I t deote log real GDP per capita for developig ad idustrial coutries, respectively. Tables 5-1 ad 5-2 report the results which exhibit a large variatio i estimated coefficiets. Recall that our iterest i this applicatio lies i the coitegratio test based o the Hausma-type test. As ca be see from Table 5-1, irrespective of coutry combiatios, the ull hypothesis of coitegratio 15

18 ca be rejected whe developig coutries are regressed oto idustrial coutries, idicatig that there is little evidece of output covergece betwee developig coutries ad idustrial coutries. The picture chages dramatically whe idustrial coutries are regressed oto idustrial coutries as i (28). Table 5-2 displays that the Hausma-type test fails to reject the ull of coitegratio i all cases cosidered. Our fidig is therefore cosistet with the otio of covergece clubs. 3.6 PPP for traded ad o-traded goods As a major buildig block for may models of exchage rate determiatio, PPP has bee oe of the most heavily studied subjects i iteratioal macroecoomics. Despite extesive research, the empirical evidece o PPP remais icoclusive, largely due to ecoometric challeges ivolved i determiig its validity. As is geerally agreed, most real exchage rates show very slow covergece which makes estimatig log-ru relatioships difficult with existig statistical tools. The literature suggests a umber of potetial explaatios for the very slow adjustmet of relative price: volatility of the omial exchage-rate, market frictios such as trade barriers ad trasportatio costs, imperfect competitio i product markets, ad the presece of o-traded goods i the price basket. Accordig to the commodity-arbitrage view of PPP, the law of oe price holds oly for traded goods ad the departures from PPP are primarily attributed to the large weight placed o otraded goods i the CPI. This view has obtaied support from may empirical studies based o disaggregated price idices. They ted to provide ample evidece that prices for o-traded goods are much more dispersed tha for their traded couterpart ad cosequetly o-traded goods exhibit far larger deviatios from PPP tha traded goods. Give that geeral price idices ivolve a mix of both traded ad o-traded goods, highly persistet deviatios of o-traded goods from PPP ca lead to the lack of coclusive evidece o the log ru PPP relatioship. As i the previous applicatio, our mai purpose for this applicatio is ot to estimate ukow structural parameters but to test the ull hypothesis of coitegratio with the Hausma-type test. Let p t ad p t deote the logarithms of the cosumer price idices i the base coutry ad foreig coutry, respectively, ad s t be the logarithm of the price of the foreig coutry s currecy i terms of the base coutry s currecy. Log-ru PPP requires that a liear combiatio of these three variables be statioary. To be more specific, log-ru PPP is said to hold if f t = s t + p t is coitegrated with p t such that e t I() i f T t = α + βp T t + e t, f N t = α + βp N t + e t, where the superscripts T ad N deote the price levels of traded goods ad o-traded goods, respectively. Followig the method of Stockma ad Tesar (1995), Kim (24) recetly aalyzed the real exchage rate for total cosumptio usig the geeral price deflator ad the real exchage rate for traded ad o-traded goods usig implicit deflators for o-service cosumptio ad service cosumptio, respectively. 11 We use Kim s dataset to apply our techiques to the liear combiatio of sectorally decomposed variables. Table 6 presets the results usig quarterly price ad exchage rate data for six coutries: Caada, Frace, Italy, Japa, U.K., ad U.S. for the period of 1974 Q1 through 1998 Q4. With the Caadia dollar used as umeraire, Table 6 presets the estimates for β which should be close to uity accordig to log-ru PPP. For traded goods, estimates are above uity i most cases, but the variatio across estimates does ot seem substatial, resultig i o-rejectio of the ull of 11 For details, see the Appedix for the descriptio of the data. We thak JB Kim for sharig the dataset. 16

19 coitegratio i all cases cosidered. By sharp cotrast, the Hausma-type coitegratio test rejects the ull hypothesis i every coutry whe the price for o-traded goods is used. It is oteworthy that there exists cosiderable differece betwee GLS-corrected estimates for β ad their DOLS ad FGLS couterparts which are far greater tha uity. That is, supportive evidece of PPP is foud for traded goods but ot for o-traded goods, cogruet with the geeral ituitio as well as the fidigs by other studies i the literature such as Kakkar ad Ogaki (1999) ad Kim (24) Cocludig remarks ad future work I this paper, we developed two estimators to estimate structural parameters i spurious regressios: GLS corrected dyamic regressio ad FGLS corrected dyamic regressio estimators. A GLS corrected dyamic regressio estimator is a first differeced versio of a dyamic OLS regressio estimator. Asymptotic theory shows that, uder some regularity coditios, the edogeeity correctio of the dyamic regressio works for the first differeced regressios for both coitegratig ad spurious regressios. This result is useful because it is ot ituitively clear that the edogeeity correctio works eve i regressios with statioary first differeced variables. For the purpose of the estimatio of structural parameters whe the possibility of ostatioary measuremet error or ostatioary omitted variables caot be ruled out, we recommed the FGLS corrected dyamic regressio estimators. They are cosistet both whe the error is I() ad I(1). They are asymptotically as efficiet as dyamic OLS whe the error is I() ad as GLS corrected dyamic regressio whe the error is I(1). This feature may be especially attractive whe the FGLS corrected dyamic estimator is exteded to a pael data settig whe some regressio errors are I() ad the others are I(1). This extesio is studied by Hu (25). We also developed the Hausma-type coitegratio test by comparig the dyamic OLS regressio ad GLS corrected dyamic regressio estimators. As oted i the Itroductio, this task is importat ot merely because few tests for coitegratio have bee developed for dyamic OLS but because tests for the ull hypothesis of coitegratio are useful i may applicatios. For this test, the spurious regressio obtaied uder the alterative hypothesis does ot have to be structural. We applied our estimatio ad testig methods to four applicatios: (i) log-ru moey demad i the U.S.; (ii) log-ru implicatios of the cosumptio-leisure choice; (iii) output covergece amog idustrial ad developig coutries; (iv) Purchasig Power Parity (PPP) for traded ad o-traded goods. I the first applicatio of estimatig the moey demad fuctio, the results suggest that the edogeeity correctio of the dyamic regressio works with a moderately large umber of leads ad lags for the GLS corrected dyamic regressio estimator. The GLS corrected dyamic regressio estimates of the icome elasticity of moey demad are very low with low orders of leads ad lags, ad the icrease to more plausible values as the order of leads ad lags icreases. Dyamic OLS estimates are close to the GLS corrected dyamic regressio estimates for a large eough order of leads ad lags, ad we fid little evidece agaist coitegratio with the Hausma-type coitegratio test. The FGLS corrected dyamic regressio estimates are very close to the GLS corrected dyamic regressio ad the dyamic OLS estimates for sufficietly large order of leads ad lags. I the secod applicatio of the log-ru implicatios of the cosumptio-leisure choice, we foud strog evidece agaist coitegratio whe odurables are used as the measure of cosumptio with the 12 Egel (1999) fids little evidece for log-ru PPP for traded goods with his variace decompositio method. However, it should be oted that his method is desiged to study variatios of real exchage rates over relatively shorter periods compared with coitegratio-type methods that are desiged to study log-ru relatioships. 17