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**Journal of Business & Economic Statistics**

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**Convergence Rates to Purchasing Power Parity for**

**Traded and Nontraded Goods**

**Jaebeom Kim**

**To cite this article:** Jaebeom Kim (2005) Convergence Rates to Purchasing Power Parity for
Traded and Nontraded Goods, Journal of Business & Economic Statistics, 23:1, 76-86, DOI:
10.1198/073500104000000226

**To link to this article: ** http://dx.doi.org/10.1198/073500104000000226

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## Convergence Rates to Purchasing Power Parity

## for Traded and Nontraded Goods: A Structural

## Error-Correction Model Approach

**Jaebeom K**

**IM**

Department of Economics, University of St. Thomas, St. Paul, MN 55105-1096 (jkim1@stthomas.edu)

This article estimates the speed of the adjustment coefficient in structural error-correction models. We use a system method for real exchange rates of traded and nontraded goods by combining a single-equation method with Hansen and Sargent’s instrumental variables methods for linear rational expectations models. We apply these methods to a modified version of Mussa’s model. Our results show that the half-lives of purchasing power parity deviations for the rates of traded goods are less than 1 year and are shorter than those for general price and for nontraded goods in most cases, implying a faster adjustment speed to parity.

KEY WORDS: Convergence rate; Half-life; Purchasing power parity; Real exchange rate; Structural error-correction model.

1. INTRODUCTION

The half-life of purchasing power parity (PPP) deviations has been provided as a measure of the adjustment of real exchange rates. When univariate methods are applied to real exchange rates, point estimates of autoregressive coefficients typically imply very slow rates of mean reversion. Rogoff (1996) de-scribed the half-lives of PPP deviations as about 3–5 years in studies using long-horizon data and argued that this seems far too long to be explained by nominal rigidities. Studies us-ing panel data for a modern period of flexible exchange rates find only slightly shorter half-lives of 2.5 years (see, e.g., Wu 1996; Papell 1997b). However, some recent empirical litera-tures on the half-life of innovations to real exchange rates still confirm that univariate methods provide virtually no informa-tion regarding the size of the half-lives (see, e.g., Murray and Papell 2002). The present article reexamines the convergence question by contrasting the half-lives of PPP deviation across traded and nontraded goods in an exchange rate model with a system method based on Kim, Ogaki, and Yang’s (2003) struc-tural error-correction model (ECM).

Many theories expect small and short-lived deviations from PPP for traded goods and larger and more persistent deviations for nontraded goods. Because the consumption basket of the typical country involves a mix of both, estimates using the over-all consumption deflator or general price index become difficult to interpret in light of existing theories. For instance, when PPP with general price indices is tested, nontraded goods have been issued as a problem in the deviation of the real exchange rate from its PPP value, because general price indices contain prices of both traded and nontraded goods (see, e.g., Balassa 1964; Samuelson 1964; Dornbusch 1976; Stockman 1983; Backus and Smith 1993).

Another interesting fact we can find in the current literature is that the remarkable consensus of 3–5 year half-life might come from reduced-form models. Some recent studies, using producer price indices and tradable sector deflators, which ap-ply panel unit root tests to real exchange rates, report strong evidence against the unit root null and estimate the half-life of PPP deviation to be 4–5 years (see, e.g., Wu 1996; Wei and Parsley 1995; Canzoneri, Cumby, and Diba 1996). Note that

even for the rates of traded goods, this remarkable consensus of 3–5 year half-life is the same as that found for real CPI ex-change rates in many studies. However, the studies that attempt to solve the PPP puzzle of the 3–5 year half-life typically con-duct Dickey–Fuller (DF) unit root tests for the real exchange rates of traded goods’ prices and general prices, and the half-life is calculated from the coefficient of the lagged real exchange rate. Therefore, this suggests that the prices of nontraded goods in general price indices do not dictate the frequency of deviation from PPP. However, these results might come from reduced-form models.

In general, it is known that reduced-form models may not recover the structural parameters, and the speed of the adjust-ment coefficient in a structural-form model is different from that in its reduced-form model. As shown by Kim et al. (2003), because the reduced-form speed of the adjustment coefficient mixes the structural speed of the adjustment coefficient with other parameters in the system, it is not easy to interpret it. Fur-thermore, the reduced-form speed of the adjustment coefficient is a nonlinear function of the structural speed of the adjustment, the weight on the nontraded goods, and the interest elasticity of money demand, so that it may not be used to compute the half-life of PPP deviation.

In the literature of estimation of half-lives of real exchange rates, the first-order autoregressions of real exchange rates have been typically estimated by univariate methods. When a uni-variate method is combined with Hansen and Sargent’s (1982) method as in our system method, then the system method es-timator for the autoregressive coefficient is more efficient than the univariate method estimator as long as linear rational ex-pectations model is correctly specified. When the linear rational expectations model used in this article is misspecified, the sys-tem method estimator is inconsistent. However, if the model is a good approximation, then the estimator’s mean may be close to the true value and its variance may be smaller than the uni-variate estimator.

**© 2005 American Statistical Association**
**Journal of Business & Economic Statistics**
**January 2005, Vol. 23, No. 1**
**DOI 10.1198/073500104000000226**
**76**

The main purpose of the present article is to estimate half-lives of real exchange rates based on traded goods, nontraded goods, and general prices in an exchange rate model with sticky prices using a system method. The model that we use for this purpose is the structural ECM developed by Kim et al. (2003). This article extends a one-good version of the structural ECM to a two-good model for traded and nontraded goods. This is a methodological contribution made by the current study. More-over, this article examines how the adjustment speed to parity is affected when a system method based on structural ECM uses the prices of traded goods as compared with general prices and the prices of nontraded goods.

In the system method, all of the markets, including money and bond markets, are considered, and Hansen and Sargent’s (1980, 1982) method, which applies generalized method of mo-ments (GMM) to linear rational expectations models, is used. This system method is applied to a modified version of Mussa’s (1982) model with traded and nontraded goods, which may be viewed as a discrete-time version of Dornbusch’s (1976) model. The Mussa model includes a gradual adjustment equation in which the domestic price of traded goods adjusts to the long-run equilibrium level determined by PPP with rational expectations. Kim and Ogaki (2004) applied the same method developed in this article to a different dataset. They used the producer price index, the consumer price index, and gross domestic product deflators from 1973 Q1 to 2001 Q1 to construct the real ex-change rates for traded, nontraded, and general prices. They used the G7 countries in their study of bilateral exchange rates, considering each of the seven currencies alternatively as the base currency in their empirical work. Papell (1988) derived a reduced-form model from an exchange rate model that is similar to ours; however, the real exchange rate is nonstation-ary in his model, unlike in ours. Papell applied constrained maximum likelihood methods to the model and concluded that the estimation of Dornbusch’s model is much more success-ful than the single-equation method (see, e.g., Papell 1988; Kim et al. 2003).

In this article we follow Stockman and Tesar (1995) and use the implicit deflators of nonservice consumption and service consumption classified by type and total consumption deflators to construct the real exchange rate for traded, nontraded, and general prices. The countries included in our study are Canada, France, Italy, Japan, Sweden, the United Kingdom, and the United States. Our dataset does not include Germany, because the data are not available.

For all cases, we obtain positive estimates for the structural speed of the adjustment coefficients. In particular, we estimate half-lives of deviations from PPP that are almost uniformly shorter for traded price indices than for nontraded price indices with the estimates using the overall price index lie in between. These estimates suggest that theories of international price de-termination should treat traded and nontraded goods differently to match their differential convergence rates. Moreover, it may be that convergence rates of traded goods are more plausible estimates of the impact of nominal rigidities, whereas such considerations as international factor immobility and nontraded components of goods’ prices are important for the dynamic be-havior of the overall price index.

2. AN EXCHANGE RATE MODEL WITH STICKY PRICES

2.1 The Gradual Adjustment Equation

Let *pT _{t}* be the log domestic price level of traded goods, let

*pT*∗ be the log foreign price level of traded goods, and let

_{t}*et*be the log nominal exchange rate (the price of one unit of

the foreign currency in terms of the domestic currency). We
assume that these variables are first-difference stationary. We
also assume that PPP holds in the long run, so that the real
ex-change rate,*pT _{t}* −

*pT*∗−

_{t}*et*, is stationary, or

**y**

*t*=(

*pTt*,

*et*,

*pTt*∗)′

is cointegrated with a cointegrating vector (1,−1,−1). Let

µ=*E*[*pT _{t}* −

*p*∗−

_{t}T*et*], then µcan be nonzero when different

units are used to measure prices in the two countries.

To derive the form of a structural ECM, we consider an ex-change rate model with sticky prices. We use Mussa’s (1982) model, which may be viewed as a stochastic discrete time ver-sion of Dornbusch’s (1976) model, in which the domestic price of traded goods is assumed to be sticky in the short run and to adjust gradually to its long-run equilibrium level determined by PPP with rational expectation (see, e.g., Mussa 1982).

Using Mussa’s (1982) model, the domestic price of traded goods is assumed to adjust slowly to the PPP level in the long run through

*pT _{t}*

_{+}

_{1}=

*b*(µ+

*p*∗+

_{t}T*et*−

*pTt*)

+*Et*[*pTt*+∗1+*et*+1] −(*pTt*∗+*et*), (1)

where*xt*+1=*xt*+1−*xt* for any variable*xt*;*E*[·|*It*] is the

ex-pectation operator conditional on*It*, the information available

to the economic agents at time*t*; and a positive constant*b*<1 is
the adjustment coefficient. Based on work of Mussa (1982), the
main idea behind (1) is that the price level of domestic traded
goods adjusts slowly toward its PPP level of*pT _{t}*∗+

*et*, its

long-run equilibrium level, while it adjusts instantaneously to the
ex-pected change in its PPP level. The adjustment speed is slow
when*b*is close to 0 and fast when*b*is close to 1. From (1), we
have

*pT _{t}*

_{+}

_{1}=

*d*+

*b*(

*pT*∗+

_{t}*et*−

*pTt*)+

*pTt*+∗1+

*et*+1+ε

*t*+1, (2)

where *d* =*b*µ and ε*t*+1=*Et*[*pTt*+∗1+*et*+1] −(*ptT*+∗1+*et*+1).

Hence ε*t*+1 is a one-period-ahead forecasting error, and

*E*[ε*t*+1|*It*] =0. Because*pTt*∗+*et*−*pTt* is the log real exchange

rate,*b*coincides with 1 minus the first-order autoregressive
co-efficient of the log real exchange rate. We define the half-life
of the log real exchange rate as the number of periods required
for a unit shock to dissipate by one-half in this first-order
au-toregression. Without measurement errors, the coefficient*b*can
be estimated by ordinary least squares directly from (2).
Equa-tion (2) motivates the form of the structural ECM used in this
article, and it can be referred to as the structural gradual
adjust-ment equation.

2.2 The Exchange Rate Under Rational Expectations We close the model by adding the money demand equation and the uncovered interest parity condition. The general price level is a weighted average of the prices of traded and nontraded

goods. The money demand depends on the general price level rather than on the traded good price. Let

*Pt*=(1−α)*pTt* +α*pNt* , (3)
budget shares of the traded and nontraded goods. The*mt* is the

log nominal money supply minus the log real national income,
*it*is the nominal interest rate in the domestic country, and*i*∗*t* is

the nominal interest rate in the foreign country. In (5), we are assuming that the income elasticity of money is 1. From (3), (4), (5), and (6), we obtain

Following Mussa (1982), solving (1) and (7) as a system of
stochastic difference equations for*E*[*pT _{t}*

_{+}

*|*

_{j}*It*]and

*E*[

*et*+

*j*|

*It*]for

We assume thatψ*t* is first difference stationary. Becauseδis

a positive constant smaller than 1, this implies that *Ft* is also

first-difference stationary. From (8) and (9),

*et*+*pTt*∗−*pTt* =

Because the right side of (12) is stationary (i.e., assuming that
*Et*[*Ft*] −*Et*−1[*Ft*]is stationary, which is true for a large class

of first-difference stationary variable*Ft* and information sets),

*et*+*pTt*∗−*pTt* is stationary. Thus (12) implies that (*pTt*,*et*,*pTt*∗)

is cointegrated, with the cointegrating vector(1,−1,−1).

2.3 Hansen and Sargent’s Formula

In this article we use the formula of Hansen and Sargent (1980, 1982) for linear rational expectations models to obtain a structural ECM representation from the exchange rate model. From (9), we obtain

cause this equation involves a discounted sum of expected
fu-ture values ofψ*t*, the system method using the method of

Hansen and Sargent (1982) is applicable.

Hansen and Sargent (1982) proposed projecting the
condi-tional expectation of the discounted sum,*E*[δ*j*ψ*t*+*j*+1|*It*], onto

an information set *Ht*, which is a subset of *It*, the economic

agents’ information set. Let*E*[·|*Ht*]be the linear projection

op-erator conditional on an information set*Ht*, which is a subset

of*It*.

We take the econometrician’s information set at*t*,*Ht*, to be

the set generated by the linear functions of the current and
past values of*pT _{t}*∗. Then, replacing the economic agents’ best
forecast,

*E*[∞

_{j}_{=}

_{0}δ

*j*ψ

*t*+

*j*+1|

*It*], by the econometrician’s linear

forecast based on*Ht*in (13), we obtain

*et*+1=

Because *E*[·|*Ht*] is the linear projection operator onto *Ht*,

there exist possibly infinite-order lag polynomialsβ(*L*),γ (*L*),

andξ(*L*), such that

Then, following Hansen and Sargent (1980, app. A), we obtain
the restrictions imposed by (14) onξ(*L*),

and

*E*[ψ*t*+1|*Ht*] =γ1*pTt*∗+γ2*pTt*−∗1+ · · · +γ*p*−1*pTt*−∗*p*+2.

(20)

Here we assume thatβ(*L*)is of order*p* andγ (*L*)is of order
*p*−1 to simplify the exposition, but we do not lose generality,
because any of β*i* and γ*i* can be 0. Then (as in Hansen and

can be applied to these four equations. There exist additional
complications for obtaining data forψ*t*+1, as we discuss later.

3. STRUCTURAL MODELS AND ERROR CORRECTION MODELS

Let**y***t*be an*n*-dimensional vector of first-difference

station-ary random variables. We assume that there exist ρ linearly
independent cointegrating vectors, so that **A**′**y***t* is stationary,

where**A**′ is a (ρ×*n*) matrix of real numbers whose rows are
linearly independent cointegrating vectors. Consider a standard
ECM,

**y**_{t}_{+}_{1}=**k**+**GA**′**y***t*+**F**1**y***t*+**F**2**y***t*−1

+ · · · +**F***p***y***t*−*p*+1+ν*t*+1, (27)

where**k** is an (*n* ×1) vector,**G**is an (*n* ×ρ) matrix of real
numbers, and**v***t*is a stationary*n*-dimensional vector of random

variables with*E*[ν*t*+1|*Ht*−τ] =0.

A class of structural models can be written in the following form of a structural ECM:

**C**0**y***t*+1=**d**+**BA**′**y**+**C**1**y***t*+**C**2**y***t*−1

+ · · · +**C***p***y***t*−*p*+1+**u***t*+1, (28)

where**C***i* is an (*n*×*n*) matrix,**d**is an (*n*×1) vector, and**B**is

an (*n*×ρ) matrix of real numbers. If the deterministic
cointegra-tion restriccointegra-tion (see Ogaki and Park 1998 for this terminology)
is not satisfied, then a linear trend term must be added to (28).
Here**C0**is a nonsingular matrix of real numbers with 1’s along
its principal diagonal,**u***t*is a stationary*n*-dimensional vector of

random variables with*E*[**u***t*+1|*Ht*−τ] =0. Even though

cointe-grating vectors are not unique, we assume that there is a
nor-malization that uniquely determines**A**so that parameters in**B**

have structural meanings.

The exchange rate model can be written in the SECM
form (28) as in the system of (23)–(26). We have **y***t* =

Comparing (27) with (28), in many applications of standard
ECMs given in (27), elements in**G**are given structural
inter-pretations as parameters of the speed of adjustment toward the
long-run equilibrium represented by **A**′**y***t*. However, if we

as-sume that in (28)**C0**is nonsingular and premultiply both sides
of (28) by**C**−_{0}1, then we obtain the standard ECM (27), where

**k**=**C**−_{0}1**d**,**G**=**C**−_{0}1**B**,**F***i*=**C**−_{0}1**C***i*,andν*t*=**C**−_{0}1**u***t*. Thus the

standard ECM estimated by Engle and Granger’s (1987)
two-step method or Johansen’s (1988) maximum likelihood method
is a reduced-form model. Hence, it cannot be used to recover
structural parameters in**B**, and the impulse-response functions
based on ν*t* cannot be interpreted in a structural way unless

some restrictions are imposed on**C**0.

As in a vector autoregression, various restrictions are
possi-ble for**C**0. One example is to assume that**C**0is lower triangular.

If**C**0is lower triangular, then the first row of**G**is equal to the

first row of**B**, and structural parameters in the first row of**B**are
estimated by the standard methods used to estimate an ECM.

In the exchange rate model discussed in Section 2, *b* is
a structural parameter of interest. For the purpose of
estimat-ing *b* in the model, the restriction that**C**0 is lower triangular

is not attractive. However, as is clear from (29), the structural
ECM from the two-goods version of the exchange rate model
does not satisfy the restriction that**C**0is lower triangular for any

ordering of the variables. Even though some structural models

may be written in lower-triangular form, this example suggests that many structural models cannot be written in that particular form.

It is instructive to observe the relationship between the struc-tural ECM and the reduced-form ECM in the exchange rate model. Because

ous interactions between the domestic price and the exchange
rate affect the speed of the adjustment coefficients. The speed
of the adjustment coefficient for the domestic price is*b*in the
structural model, and*b*2*h*/(*bh*+(1−α)) in the reduced-form
model. The error correction term does not appear in the second
equation for the exchange rate in the structural ECM, whereas
it does appear with the speed of the adjustment coefficient of
*b*(1−α)/(*bh*+(1−α))in the reduced-form model.

4. THE SYSTEM METHOD

To implement the system method, we need data for ψ*t*,

which requires knowledge ofα and*h*. To computeαandα∗,
weights on the nontraded goods, we take the average budget
share over the whole sample as the weight for all periods. Even
though*h*is unknown, a cointegrating regression can be applied
to money demand if money demand is stable in the long run (as
in Stock and Watson 1993). For this purpose, we augment the
model as follows:

*mt*=*k*+*Pt*−*hit*+η*m*,*t*, (32)

whereη*m*,*t*is the money demand shock, which is assumed to be

stationary, so that money demand is stable.

By redefining*mt*as*mt*−η*m*,*t*, we obtain the same equations

as those in Section 2. For the measurement ofψ*t*, note that the

ex ante foreign real interest rate can be replaced by the ex post foreign real interest rate because of the law of iterated expecta-tions. Using the money market clearing condition (32) and (11), we obtain

from the prices of traded and nontraded goods and interest rate
data without data for monetary aggregates and national income.
We have now obtained a system of four equations, (23),
(24), (25), and (26). Because *E*[*u*1,*t*|*It*] =0, we can choose

instrumental variables, **z**1,*t* for *u*1,*t* from *It* and, because

*E*[*ui*,*t*|*Ht*] =0, instrumental variables, **z***i*,*t*, for *ui*,*t* can be

se-lected from*Ht*for*i*=2,3,4.

Because the speed of adjustment,*b*, for*pT _{t}* affects the
dynam-ics of the other variables (note that only

*pT*adjusts slowly, but

_{t}*b*affects the dynamics of other variables because of interactions of

*pT*with those variables), there are cross-equation restrictions

_{t}involving*b*in many applications to the restrictions in (21).
Us-ing the moment conditions*E*[**z***i*,*t*,*ui*,*t*] =0 for*i*=1, . . . ,4, we

form a GMM estimator, imposing the restrictions from (21) and the other cross-equation restrictions implied by the model.

Given the cointegrating vector, this system method provides more efficient estimators than the single-equation method, as long as the restrictions implied by the model are true. On the other hand, the single-equation method estimators are more ro-bust, because misspecification in the other equations does not affect their consistency. The cross-equation restrictions can be tested by Wald, likelihood ratio (LR)-type, and Lagrange mul-tiplier tests in a GMM framework (see, e.g., Ogaki 1993a,b). When the restrictions are nonlinear, LR-type and Lagrange multiplier tests are known to be more reliable than Wald tests.

5. EMPIRICAL RESULTS AND CONCLUSION In this article we use quarterly data from 1974 Q1 to 2000 Q1 (from 1974 Q1 to 1998 Q2 for EMU countries, such as France and Italy). We take the exchange rates for Canada, France, Italy, Japan, Sweden, the United Kingdom, and the United States from the International Financial Statistics (IFS) CD–ROM, and study bilateral real exchange rates of traded and nontraded goods classified by type and total consumption deflators from the Quarterly National Accounts and Data Stream. We follow the method of Stockman and Tesar (1995) to construct the real exchange rate for total consumption using the general price de-flator and the real exchange rate for traded and nontraded goods using implicit deflators for nonservice consumption and service consumption. We decompose private final consumption of com-modities by type (durables, semidurables, nondurables, and ser-vices) and use services as our proxy for the consumption of nontraded goods. To estimate the interest elasticity of money demand, we use the sum of M1 and quasi-money as the mea-sure of M2 as the IFS suggests (M0 for the U.K.). The 3-month Treasury bill rates are used for the interest rate data; but for Japan, 3-month deposit rates are used, because Japanese Trea-sury bill rates are not available for an early part of the sam-ple. In addition, because many studies point out the problem of choosing of the U.S. dollar as the numeraire, other currencies are also considered as the numeraire currencies (see Papell and Theodoridis 2001; Papell 1997b for details).

Our estimation procedure has two steps. First, we estimate the monetary equilibrium equation using Park’s (1992) canon-ical cointegrating regression to obtain the interest elasticity of money demand. Second, we estimate the speed of price adjust-ment by applying GMM to the structural ECM.

Table 1 presents the results of cointegrating regression for the money demand equations and the weights on the nontraded goods,α, for each country. The null of stochastic cointegration is rejected at the 5% significance level for most countries, but is not rejected at the 10% significance level for any country except France. The deterministic cointegrating restriction is not rejected at the 5% significance level for most countries except Sweden and the U.K. In all cases, the signs of the estimates for the interest elasticity of money demand are negative, as would be expected from the economic model. To computeα, weights on the nontraded goods, we take the average budget share over the whole sample as the weight for all periods.

Table 1. Money Demand Equation andα

Country h H(0, 1) H(1, 2) H(1, 3) α

Canada 2.697 2.907 6.479 8.551 .477

(8.017) (.088) (.011) (.013)

France 8.027 .267 17.568 25.601 .416

(3.444) (.604) (.000) (.000)

Italy 6.073 .870 .820 3.030 .355

(1.909) (.351) (.365) (.219)

Japan 14.290 1.029 5.112 13.911 .516

(4.737) (.310) (.023) (.001)

Sweden 14.823 19.865 .395 1.838 .394

(3.501) (.000) (.529) (.398)

U.K. 24.021 12.365 6.024 8.558 .434

(6.546) (.004) (.014) (.013)

U.S. 10.006 .579 .382 19.640 .534

(1.471) (.446) (.536) (.000)

NOTE: Results formt=k+Pt−hit+ηm, t. For columnh, standard errors are in parentheses.

For columnsH(0, 1),H(1, 2), andH(1, 3),pvalues are in parentheses. For columnα, weight is on the nontraded goods.

Tables 2, 3, and 4 report the results of GMM estimation
us-ing the system method [(23)–(26)]. The instrumental variables
are *pT _{t}*

_{−}∗

_{3}and

*pT*

_{t}_{−}∗

_{4}, which are foreign traded goods’ prices in all cases, based on Akaike information criterion. For each country, the estimation results are reported under the assump-tion that PPP holds in the long run. (Results of unit root tests are available from the author on request.) In the system method, the structural speed of the adjustment coefficient,

*b*, appears in two equations: the gradual adjustment equation [(2) and (23)] or the Hansen–Sargent equation [(24)]. The model imposes the restriction that the coefficient

*b*in the gradual adjustment equa-tion is the same as the coefficient

*b*in the Hansen–Sargent equa-tion. We report results with and without this restriction imposed for the system method of estimation. In the case of unrestricted estimation,

*b*hs is the estimate of

*b*from the Hansen–Sargent

equation, and*b*gais the estimate of*b*from the gradual

adjust-ment equation. The restricted estimate is denoted by*b*r. We use

Table 2. System Method Results for Traded Goods Prices

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

U.K./U.S. .13 .744 2.288 .028 .221 2.226 .062 1.707

(.009) (.137) (.682) (.008) (.290) (.526) (.803) (.425)

CA/U.S. .17 .642 1.415 .003 .181 1.356 .059 .096

(.007) (.047) (.842) (.002) (.337) (.715) (.808) (.953)

SW/U.S. .14 .718 7.043 .041 .161 5.997 1.046 3.710

(.019) (.223) (.137) (.002) (.169) (.111) (.306) (.156)

FR/U.S. .52 .283 2.809 .192 .371 2.150 .659 .999

(.504) (.107) (.590) (.721) (.368) (.541) (.416) (.606)

IT/U.S. .84 .187 8.233 −.106 −.138 4.946 3.287 4.582

(.956) (.049) (.083) (.002) (.114) (.175) (.069) (.101)

JP/U.S. .83 .189 6.461 −.026 .205 3.242 3.219 4.227

(.412) (.022) (.167) (.001) (.273) (.355) (.072) (.120)

U.S./U.K. .17 .629 4.653 .035 .120 2.552 2.101 3.137

(.052) (.299) (.325) (.077) (.471) (.465) (.147) (.208)

CA/U.K. .17 .644 4.922 .006 .028 2.124 2.798 .231

(.042) (.263) (.295) (.012) (.285) (.546) (.094) (.891)

SW/U.K. .76 .204 3.926 .041 .103 1.304 2.622 2.155

(.319) (.022) (.416) (.001) (.003) (.728) (.105) (.340)

FR/U.K. .36 .378 1.595 .831 .042 1.063 .532 .026

(.305) (.189) (.809) (1.991) (.148) (.786) (.465) (.987)

JP/U.K. .16 .662 5.726 .104 .957 3.150 2.576 2.243

(.026) (.194) (.221) (.001) (.083) (.532) (.108) (.325)

IT/U.K. .55 .268 5.314 −.098 .044 .880 4.434 2.973

(1.038) (.182) (.256) (.012) (.063) (.830) (.035) (.226)

U.S./SW .59 .252 6.571 −.384 1.860 4.729 1.842 4.459

(.641) (.091) (.160) (.815) (.146) (.192) (.174) (.107)

U.K./SW .16 .659 3.565 .002 1.771 2.905 .660 2.384

(.032) (.231) (.468) (.001) (.464) (.406) (.416) (.303)

CA/SW .98 .163 5.822 .910 .436 3.436 2.386 3.637

(1.959) (.063) (.213) (.110) (.215) (.329) (.122) (.162)

FR/SW .98 .162 1.597 .735 .347 1.201 .396 .526

(6.127) (.195) (.809) (1.114) (.461) (.752) (.529) (.768)

JP/SW .52 .285 1.470 .187 1.023 1.064 .406 .289

(.220) (.048) (.833) (.361) (.563) (.785) (.524) (.865)

IT/SW .54 .271 3.965 .011 .624 3.264 .701 2.292

(.230) (.042) (.410) (.029) (.089) (.352) (.402) (.317)

U.S./JP .67 .227 1.266 .350 .073 1.047 .219 .283

(.120) (.012) (.867) (.801) (.082) (.789) (.639) (.868)

U.K./JP .46 .309 1.427 −.027 −.072 1.126 .301 .043

(.136) (.040) (.839) (.001) (.005) (.770) (.583) (.978)

CA/JP .71 .216 3.414 .002 .976 3.037 .377 .129

(.646) (.054) (.491) (.001) (.043) (.385) (.539) (.937)

SW/JP .74 .207 2.101 −.015 .392 1.178 .923 .328

(.526) (.038) (.717) (.012) (.105) (.758) (.336) (.848)

IT/JP .67 .226 4.809 −.153 .208 3.575 1.234 1.582

(.216) (.021) (.307) (.040) (.052) (.311) (.266) (.453)

FR/JP .57 .259 2.462 .258 .537 .711 1.751 .304

(.308) (.048) (.651) (.178) (.118) (.870) (.185) (.858)

U.S./FR .80 .195 4.154 .054 .647 2.082 2.072 .509

(1.687) (.099) (.386) (.194) (.251) (.555) (.150) (.775)

Table 2. (continued)

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

U.K./FR .14 .709 4.921 .543 .028 2.843 2.078 1.789

(.028) (.300) (.296) (1.580) (.024) (.416) (.149) (.408)

CA/FR 1.28 .126 3.722 .003 .714 2.221 1.501 1.457

(2.310) (.033) (.445) (.000) (.401) (.695) (.220) (.482)

SW/FR .91 .173 3.143 .201 .465 2.989 .154 1.610

(.481) (.019) (.534) (.008) (.488) (.392) (.694) (.447)

JP/FR 2.71 .062 7.978 .008 .394 5.627 2.351 2.788

(7.877) (.012) (.092) (.002) (.543) (.131) (.125) (.248)

IT/FR .43 .334 2.393 −.019 1.806 2.253 .140 .551

(.098) (.038) (.663) (.020) (.110) (.471) (.708) (.759)

U.S./CA .68 .225 5.651 .009 1.994 2.685 2.966 1.055

(.252) (.024) (.227) (.021) (.019) (.197) (.085) (.590)

U.K./CA .95 .166 2.987 .244 .066 1.855 1.132 .113

(2.687) (.092) (.559) (.913) (.129) (.603) (.287) (.945)

SW/CA .91 .173 3.311 .195 −.007 3.226 .085 1.105

(1.769) (.069) (.507) (79.19) (.315) (.520) (.770) (.575)

FR/CA .82 .189 .645 .150 .677 .509 .136 .091

(2.925) (.155) (.958) (.282) (.396) (.916) (.712) (.955)

JP/CA 1.80 .092 5.321 .009 .146 3.451 1.870 3.167

(8.119) (.041) (.256) (.007) (.060) (.485) (.171) (.205)

IT/CA .21 .570 4.122 −.058 .369 3.375 .747 .213

(.040) (.139) (.389) (.052) (.099) (.497) (.387) (.898)

U.S./IT .89 .177 5.272 .734 .002 3.609 1.663 .830

(1.453) (.062) (.260) (.486) (.003) (.306) (.197) (.660)

U.K./IT .86 .182 3.891 674.6 .545 2.717 1.173 1.292

(1.239) (.058) (.421) (1251) (.151) (.437) (.278) (.524)

CA/IT 1.46 .112 3.958 −.084 .097 3.886 .072 1.037

(.930) (.009) (.411) (.138) (.025) (.374) (.788) (.595)

SW/IT .42 .337 4.950 −.045 .106 4.643 .307 1.171

(.339) (.136) (.175) (.288) (.150) (.325) (.579) (.556)

JP/IT 1.32 .123 5.606 −.011 .006 4.771 .835 3.385

(5.671) (.074) (.230) (.008) (.001) (.189) (.360) (.184)

FR/IT .15 .677 4.452 .040 2.003 1.633 2.819 2.297

(.001) (.006) (.348) (.002) (.011) (.651) (.093) (.317)

NOTE: For the unrestricted estimation,bu,hsis the estimate for the speed of adjustment coefficient from the Hansen–Sargent equation, andbu,gais the estimate for the coefficient obtained from

the gradual adjustment equation. The second column reports half-life in years. In the second, third, fifth, and sixth columns, standard errors are in parentheses. In the fourth, seventh, eighth, and ninth columns,pvalues are in parentheses.

Table 3. System Method Results for General Prices

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

U.K./U.S. 1.60 .103 2.200 .001 −.492 .882 1.318 .491

(19.128) (.141) (.699) (.023) (1.596) (.829) (.250) (.782)

CA/U.S. 2.89 .058 1.824 −.571 .042 1.312 .512 1.099

(9.806) (.012) (.768) (.014) (.070) (.726) (.474) (.577)

SW/U.S. 1.87 .089 5.057 .069 .277 2.321 2.736 1.670

(8.472) (.039) (.282) (.004) (.298) (.676) (.098) (.433)

FR/U.S. .69 .222 5.426 2.382 .326 3.424 2.002 .464

(.178) (.016) (.246) (2.338) (.330) (.330) (.157) (.792)

IT/U.S. 2.66 .063 5.325 .001 −.077 3.086 2.239 .225

(42.765) (.068) (.255) (.020) (.073) (.378) (.134) (.893)

JP/U.S. 5.01 .034 1.211 −.021 −.087 .539 .672 .284

(12.978) (.003) (.876) (.001) (.054) (.910) (.412) (.867)

U.S./U.K. 1.56 .105 4.629 −3.935 1.470 3.079 1.550 1.337

(1.566) (.012) (.327) (.194) (.575) (.379) (.213) (.512)

CA/U.K. 1.45 .113 4.648 .003 .027 2.135 2.513 .861

(6.372) (.063) (.325) (.005) (.286) (.544) (.112) (.650)

SW/U.K. 2.35 .071 3.465 .068 .009 2.295 1.170 1.457

(15.877) (.037) (.483) (.001) (.021) (.513) (.279) (.482)

FR/U.K. 1.62 .101 3.298 5.999 .001 1.457 1.841 .086

(2.432) (.017) (.509) (2.744) (.008) (.692) (.174) (.957)

IT/U.K. 1.33 .122 4.443 .162 .011 .886 3.556 1.579

(4.877) (.062) (.349) (.004) (.015) (.828) (.059) (.454)

JP/U.K. 2.82 .059 2.688 −.042 1.950 1.206 1.482 .485

(35.622) (.048) (.611) (.333) (.086) (.751) (.223) (.784)

U.S./SW .97 .164 1.126 .023 1.999 1.115 .011 .260

(1.235) (.041) (.890) (.020) (.139) (.773) (.916) (.878)

U.K./SW .73 .213 9.252 .039 .248 6.178 3.074 4.268

(.852) (.067) (.055) (.003) (.098) (.103) (.079) (.118)

CA/SW 1.32 .123 .825 .235 .012 .296 .529 .377

(1.384) (.018) (.935) (.082) (.011) (.960) (.467) (.828)

FR/SW 1.30 .125 6.488 .063 .344 4.542 1.946 .205

(3.355) (.046) (.166) (.027) (.456) (.208) (.163) (.902)

Table 3. (continued)

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

JP/SW 2.84 .059 2.699 .041 .388 1.657 1.042 1.787

(5.261) (.007) (.609) (.002) (.278) (.646) (.307) (.409)

IT/SW 1.47 .111 3.866 .071 .591 3.574 .292 .639

(3.404) (.032) (.424) (.649) (.265) (.311) (.588) (.726)

U.S./JP 1.42 .115 6.100 .135 .136 4.143 1.957 3.297

(5.328) (.056) (.192) (1.016) (.256) (.386) (.161) (.192)

U.K./JP .49 .297 6.814 .008 1.371 4.822 1.992 2.898

(.198) (.059) (.146) (.017) (.202) (.306) (.158) (.234)

CA/JP .64 .236 .960 .136 .074 .881 .079 .239

(.217) (.024) (.916) (.728) (.023) (.830) (.778) (.887)

SW/JP 3.33 .051 9.696 .268 .411 6.573 3.123 5.031

(3.322) (.003) (.046) (.089) (.130) (.086) (.077) (.080)

FR/JP .75 .205 3.414 .184 .003 1.961 1.453 1.828

(.346) (.024) (.491) (.056) (.002) (.580) (.228) (.401)

IT/JP 1.30 .124 4.934 .183 .092 3.225 1.708 3.832

(1.568) (.021) (.294) (.013) (.023) (.358) (.191) (.147)

U.S./FR 2.83 .059 3.801 −.391 .912 2.198 1.603 1.424

(28.222) (.037) (.434) (1.457) (.277) (.532) (.205) (.490)

U.K./FR 1.49 .109 4.889 .060 −.129 3.191 1.698 1.471

(40.672) (.366) (.299) (.008) (.168) (.362) (.192) (.479)

CA/FR 1.28 .126 3.722 .026 .721 2.134 1.588 1.997

(2.310) (.033) (.445) (.082) (.607) (.711) (.207) (.368)

SW/FR 2.38 .070 2.927 .044 .691 2.063 .864 1.345

(7.549) (.017) (.570) (.003) (.618) (.559) (.352) (.510)

JP/FR 3.78 .045 4.117 .052 .718 3.687 .430 1.604

(60.828) (.034) (.390) (.002) (.236) (.297) (.511) (.448)

IT/FR 1.05 .152 4.365 .025 1.258 1.434 2.931 2.648

(1.623) (.042) (.358) (.019) (1.130) (.697) (.086) (.266)

U.S./CA 1.34 .121 2.059 .068 .927 1.117 .942 1.244

(.379) (.005) (.725) (.088) (.355) (.657) (.331) (.536)

U.K./CA 1.73 .095 6.447 .061 .149 3.961 2.486 1.930

(10.039) (.058) (.168) (.026) (.097) (.411) (.114) (.381)

SW/CA 1.88 .088 1.209 .081 .059 .728 .481 .109

(3.805) (.017) (.877) (.033) (.031) (.866) (.487) (.946)

FR/CA 1.90 .087 4.707 .184 .033 3.297 1.410 3.680

(3.805) (.050) (.319) (.526) (.004) (.347) (.235) (.158)

JP/CA 2.05 .081 5.686 .011 .258 3.587 2.099 2.916

(9.251) (.032) (.224) (.006) (.100) (.464) (.147) (.232)

IT/CA .26 .481 3.729 .004 1.847 1.713 2.015 .817

(.078) (.128) (.443) (.003) (.522) (.633) (.155) (.664)

U.S./IT 1.01 .158 3.817 .361 .015 3.632 .184 .910

(1.328) (.039) (.431) (.491) (.127) (.303) (.667) (.634)

U.K./IT .91 .173 4.447 .053 .114 1.931 2.516 .031

(1.441) (.057) (.348) (.003) (.037) (.586) (.112) (.984)

CA/IT 1.23 .131 4.006 −.468 .116 3.744 .261 2.447

(.876) (.014) (.405) (.816) (.034) (.290) (.608) (.294)

SW/IT .64 .237 4.573 −.024 .106 4.065 .508 1.691

(.787) (.090) (.333) (.018) (.154) (.167) (.475) (.429)

JP/IT 2.90 .058 5.907 7.342 .165 5.116 .791 .817

(17.872) (.022) (.206) (91.63) (.044) (.163) (.373) (.664)

FR/IT 1.75 .094 4.449 .005 2.001 2.107 2.341 1.813

(1.26) (.007) (.348) (.003) (.008) (.550) (.125) (.404)

NOTE: For the unrestricted estimation,bu,hsis the estimate for the speed of adjustment coefficient from the Hansen–Sargent equation, andbu,gais the estimate for the coefficient obtained from

the gradual adjustment equation. The second column reports half-life in years. In the second, third, fifth, and sixth columns, standard errors are in parentheses. In the fourth, seventh, eighth, and ninth columns,pvalues are in parentheses.

Table 4. System Method Results for Nontraded Good Prices

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

U.K./U.S. 3.35 .050 1.371 .715 1.184 1.127 .244 .097

(10.775) (.009) (.849) (.936) (1.213) (.770) (.621) (.952)

CA/U.S. 13.78 .013 2.553 .291 .037 .195 2.358 1.396

(182.83) (.002) (.635) (.009) (.043) (.978) (.124) (.497)

SW/U.S. 2.70 .062 1.679 .035 .644 1.179 .500 1.217

(24.461) (.037) (.794) (.070) (.356) (.436) (.479) (.544)

FR/U.S. 2.41 .069 2.470 4.545 .173 2.045 .425 .150

(25.916) (.055) (.649) (.971) (.113) (.563) (.514) (.927)

IT/U.S. 8.16 .021 5.125 9.134 −.031 3.632 1.493 .714

(489.40) (.027) (.274) (13.21) (.046) (.304) (.221) (.699)

JP/U.S. 6.55 .026 .833 .013 .053 .647 .186 .498

(48.714) (.005) (.934) (.007) (.059) (.885) (.666) (.779)

U.S./U.K. 4.27 .039 8.605 .091 .646 6.363 2.242 .938

(8.795) (.003) (.072) (.001) (.201) (.095) (.134) (.625)

Table 4. (continued)

Currencies Half-life br Jr bu,hs bu,ga Ju LR LR1

CA/U.K. 6.17 .028 7.162 .191 .252 5.846 1.316 1.446

(96.937) (.012) (.128) (.238) (.095) (.119) (.251) (.380)

SW/U.K. 6.10 .028 2.117 .042 .059 1.038 1.079 1.935

(46.905) (.006) (.714) (.012) (.011) (.792) (.298) (.380)

FR/U.K. 1.79 .092 7.483 .520 .084 4.870 2.613 4.737

(13.784) (.072) (.112) (.341) (.115) (.148) (.105) (.093)

IT/U.K. .30 .432 6.427 .186 .343 6.107 .320 2.640

(.236) (.247) (.169) (.719) (.087) (.106) (.571) (.267)

JP/U.K. 4.74 .036 7.533 .013 .073 4.441 3.092 3.024

(190.41) (.054) (.110) (.143) (.001) (.215) (.078) (.220)

U.S./SW .89 .176 3.828 .292 .234 2.219 1.609 .140

(.563) (.024) (.429) (.109) (.206) (.528) (.204) (.932)

U.K./SW 15.52 .011 4.914 .017 .746 3.814 1.100 .762

(398.73) (.003) (.296) (.336) (.328) (.282) (.294) (.683)

CA/SW 1.57 .105 3.672 .076 .621 1.304 2.368 .495

(12.079) (.094) (.452) (.042) (.959) (.860) (.123) (.780)

FR/SW 1.55 .106 7.571 .082 .009 6.662 .909 1.131

(5.877) (.047) (.109) (.058) (.013) (.083) (.340) (.568)

JP/SW 6.01 .028 6.649 .296 .145 6.287 .362 1.902

(297.07) (.041) (.156) (.234) (.676) (.098) (.547) (.386)

IT/SW 1.56 .105 1.296 .103 .242 1.137 .159 .179

(.253) (.002) (.860) (.005) (.145) (.768) (.690) (.914)

U.S./JP 3.05 .055 3.004 .766 .037 2.558 .446 1.065

(3.998) (.004) (.557) (.301) (.006) (.464) (.504) (.587)

U.K./JP 1.15 .139 .498 .026 .138 .392 .106 .357

(3.588) (.069) (.974) (.007) (.016) (.941) (.744) (.836)

CA/JP 4.75 .036 1.662 .011 2.023 1.404 .258 .193

(62.943) (.018) (.798) (.001) (2.447) (.704) (.611) (.908)

SW/JP 2.02 .082 .399 .041 .054 .138 .261 .014

(2.051) (.007) (.983) (.078) (.028) (.986) (.609) (.993)

FR/JP 5.01 .034 1.969 .098 −.054 1.924 .045 .733

(23.653) (.005) (.742) (.013) (.139) (.588) (.832) (.693)

IT/JP 3.37 .050 2.716 .413 −.105 2.242 .474 .805

(223.42) (.174) (.606) (.762) (.194) (.523) (.490) (.668)

U.S./FR 8.66 .019 4.732 .021 .593 1.433 3.299 3.152

(1906.5) (.088) (.316) (.024) (.602) (.697) (.069) (.206)

U.K./FR 8.08 .021 4.742 .001 .061 1.867 2.875 1.049

(156.75) (.009) (.315) (.000) (.214) (.760) (.089) (.591)

CA/FR 3.87 .044 7.569 −.380 .036 6.557 1.012 4.019

(24.109) (.013) (.109) (1.204) (.015) (.087) (.314) (.134)

SW/FR .38 .369 2.557 .405 .036 1.357 1.200 .135

(.766) (.434) (.634) (.047) (.001) (.715) (.273) (.934)

JP/FR 2.23 .074 5.043 .325 .153 2.677 2.366 2.078

(25.921) (.070) (.283) (.617) (.075) (.444) (.124) (.353)

IT/FR .88 .178 2.474 .115 .621 1.282 1.192 .104

(2.116) (.092) (.649) (.178) (.063) (.733) (.274) (.949)

U.S./CA 5.65 .030 1.326 .034 .014 1.104 .222 .309

(64.298) (.011) (.857) (.029) (.017) (.776) (.637) (.856)

U.K./CA 4.58 .037 6.508 .695 .088 4.633 1.875 1.409

(37.200) (.017) (.164) (.745) (.068) (.200) (.171) (.494)

SW/CA 8.53 .020 2.087 .784 .125 2.075 .012 1.356

(72.450) (.003) (.719) (.762) (.128) (.556) (.912) (.507)

FR/CA 4.51 .038 2.858 .018 .830 2.594 .264 .927

(91.602) (.030) (.582) (.007) (.323) (.458) (.607) (.629)

JP/CA 4.67 .036 .721 .032 .009 .558 .163 .159

(82.941) (.024) (.949) (.131) (.021) (.905) (.686) (.923)

IT/CA 1.44 .113 6.603 .009 .669 3.017 3.586 4.895

(4.221) (.042) (.158) (.002) (.265) (.388) (.058) (.086)

U.S./IT 6.57 .026 3.910 .641 .001 1.319 2.591 .902

(1175.2) (.124) (.482) (.084) (.002) (.724) (.724) (.636)

U.K./IT 1.68 .098 5.348 .236 .017 2.960 2.658 2.035

(8.528) (.054) (.253) (.491) (.512) (.441) (.103) (.361)

CA/IT 2.952 .057 1.988 .282 .011 1.973 .014 .273

(11.143) (.013) (.737) (.036) (.026) (.577) (.903) (.872)

SW/IT 2.663 .063 3.890 .395 .101 2.433 1.456 .680

(32.702) (.052) (.421) (.516) (.180) (.487) (.227) (.711)

JP/IT .95 .166 .975 .831 .055 .538 .418 .885

(3.534) (.122) (.913) (1.148) (.072) (.910) (.517) (.642)

FR/IT .91 .173 .640 .034 .553 .330 .310 .210

(1.820) (.072) (.887) (.158) (.344) (.987) (.577) (.900)

NOTE: For the unrestricted estimation,bu,hsis the estimate for the speed of adjustment coefficient from the Hansen–Sargent equation, andbu,gais the estimate for the coefficient obtained from

the gradual adjustment equation. The second column reports half-life in years. In the second, third, fifth, and sixth columns, standard errors are in parentheses. In the fourth, seventh, eighth, and ninth columns,pvalues are in parentheses.

the LR-type test statistic (see, e.g., Ogaki 1993a for an explana-tion of this test) to test the restricexplana-tion. In all cases, this restric-tion is not rejected at the 5% level. Furthermore, for the test of the Hansen–Sargent restrictions, we also report the LR-type test statistic, denoted by LR1. For all cases, the null hypothesis is not rejected, which is evidence in favor of the Hansen–Sargent restrictions.

To obtain the half-life of each estimate for*b*, we rearrange the
structural ECM equation as an AR(1) process. Because 1−*b*is
the autoregression coefficient for the first-order autoregression
representation as in (23), and because our data are quarterly,
the half-life is calculated as.25 ln(.5)/ln(1−*b*). All estimates
for the structural speed of the adjustment coefficient have the
theoretically correct positive sign. Furthermore, most of them
are significant at the 5% significance level.

The results in Tables 2, 3, and 4 show that the estimated half-lives for the rates of the traded goods range from .17 to .96 year, based on the median values. Most half-life estimates are much shorter than the consensus of 3–5 years stated by Rogoff (1996) and others (see Murray and Papell 2002). For general prices rates, the estimated half-lives are around 1.30–1.88 years, based on the median value and they are also shorter than 3–5 years. For nontraded goods prices, the adjustment speeds to PPP are much slower then those for traded goods prices and general prices, and most half-life estimates fall in the 3–5 year range.

In most cases, the point estimate for the half-life of the gen-eral price rate is larger than that of the traded goods rate and is smaller than that of the nontraded goods rate for each pair of countries. Similarly, in most cases, the same is true for standard error as well. This result indicates that a sharper estimation of the half-life is possible when we use price indices with large traded goods price components together with a system method for each country.

All of the European and other real exchange rates for traded goods show that their half-lives tend to be shorter than their general prices and nontraded goods rates and are shorter than 1 year. For each country, the standard error for the half-life of the total consumption deflator-based real exchange rate is larger than that of the traded goods-based real exchange rate, and is smaller than that of the nontraded goods-based real ex-change rate. These traded goods rates are among the most likely to exhibit evidence of short-run and long-run PPP, because trade between European countries as well as major trading partners has relatively low transaction costs and relatively stable nontar-iff barriers to trade. This result is interesting, because it con-firms that traded goods prices tend to adjust faster than general prices and nontraded goods prices, implying shorter half-lives for traded goods rates than for general prices and nontraded goods rates. This result is also consistent with the results re-garding long-run PPP of Kim (1990), Ito (1997), Kakkar and Ogaki (1999), Kim (2004), and Kim and Ogaki (2004).

6. ACKNOWLEDGMENTS

The author thanks an associate editor, two anonymous ref-erees, and seminar participants at the Ohio State University, the KAEA International Economics Conference, the Midwest Econometrics Conference at University of Chicago, the Mid-west Economics Conference at Cleveland, and the MidMid-west

Macroeconomics Conference at Atlanta for their helpful com-ments and discussions. Special thanks are due to Masao Ogaki, G. S. Maddala, Nelson Mark, Mario J. Crucini, and Paul Evans. The author acknowledges the financial support from the Charles A. Dice Fellowship and a PEGS research grant from the Ohio State University, and the Research Assistant Grant (RAG) from University of St. Thomas.

*[Received February 2003. Revised October 2003.]*

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