TESTING STRUCTURAL HYPOTHESES ON COINTEGRATION RELATIONS WITH SMALL SAMPLES.

ZHOU, SU

SU ZHOU [*]

This study examines the finite-sample bias of Johansen's [1991] likelihood ratio tests for structural hypotheses on cointegration relations among economic variables through the Monte Carlo experiments. It is found that the Johansen tests with small samples are biased toward rejecting the null hypotheses more often than what asymptotic theory suggests, even after the test statistics are adjusted by Sims's correction. A bootstrap method for obtaining problem-specific critical values for the tests is proposed It is shown that using the bootstrap procedure may substantially reduce the small-sample bias. An empirical application of the procedure is demonstrated. (JEL C12, C22)

I. INTRODUCTION

Since Nelson and Plosser [1982] discovered that most macroeconomic variables have a nonstationary time series structure, cointegration analysis has been substantially developed and widely applied in empirical studies in various areas of economics. Cointegration analysis offers a natural way to test for the existence of long-run relationships among nonstationary economic variables. A set of variables, [X.sub.t], is said to be cointegrated if these variables are individually nonstationary, but at least one linear combination of them, [z.sub.t] = [beta]'[X.sub.t], exists that is stationary. The vector [beta] is referred to as the cointegration vector. If [X.sub.t] is a cointegrated system, we may say that the variables in [X.sub.t] do not drift "too far" apart and there exists one or more long-run equilibrium relationships among these variables.

Cointegration analysis mainly consists of three segments: Tests for the rank of cointegration, estimation of cointegration vectors, and tests for structural hypotheses suggested by economic theory. One of the most popular methods of cointegration analysis is the multivariate maximum-likelihood approach of Johansen [1988, 1991]. The Johansen approach yields trace statistics and maximal eigenvalue statistics for identifying the number of cointegration vectors and provides consistent maximum-likelihood estimates of the coefficients of cointegration vectors. In addition, Johansen [1991] and Johansen and Juselius [1990, 1992] develop the likelihood ratio tests for structural hypotheses, that is, the hypotheses of linear restrictions on the cointegration relations, with a multivariate error correction model.

Johansen and Juselius [1990, 1992] have tabulated the asymptotic critical values for both the maximal eigenvalue and the trace tests. Also, Johansen [1991] has derived the asymptotic distributions of the test statistics for structural hypotheses. Their critical values from the asymptotic distributions have been extensively employed in the cointegration studies. Unfortunately, empirical studies of long-run (cointegration) relationships are limited by finite sample sizes. It could be misleading to draw conclusions based on a small sample study while using the asymptotic critical values, which are appropriate only for large samples.

Cheung and Lai [1993] investigate the accuracy of Johansen and Juselius's asymptotic approximations by computing the finite-sample distributions of the maximal eigenvalue and the trace statistics; they find that the asymptotic critical values are more liberal than the finite-sample critical values. That is, using the asymptotic critical values, one would reject the null of no cointegration too often when the sample size of the data used is rather small. There have been a number of other studies examining the small sample performance of several popular estimation and testing procedures designed for cointegrated systems, including the Johansen approach. [1] However, as far as I know, none of the existing studies has systematically investigated the small sample performance of any test for structural hypotheses. The present study intends to fill this gap and thus expand this body of literature.

There is an obvious reason that researchers have overlooked the small sample problem of the tests for structural hypotheses on cointegration relations. Unlike Johansen's likelihood ratio (LR) tests for cointegration rank, which have nonstandard asymptotic distributions and therefore previous results about small-sample bias of standard LR tests do not apply to those tests, the asymptotic distributions of the LR tests for structural hypotheses are shown by Johansen [1991] to be the usual [[chi].sup.2] distributions. The finite-sample bias of such LR tests has been widely acknowledged. A well- known finite-sample correction of standard LR tests was recommended by Sims [1980] and has been extensively utilized in the empirical literature and introduced in almost every textbook in multivariate time series. Yet it is demonstrated in this paper that Sims's adjustment is hardly capable of correcting the small-sample bias of the Johansen LR tests for structural hypotheses.

In this study, I examine the finite-sample bias of Johansen's [1991] LR tests for structural hypotheses on cointegration relations with and without Sims's correction. A bootstrap procedure for obtaining problem-specific critical values for the tests is proposed.

The rest of the paper is organized as follows. Section II briefly describes Johansen's tests for the rank of cointegration and estimation of cointegration vectors and then introduces the Johansen tests for structural hypotheses in a multivariate cointegrated system. Tests for both long-run exclusion and some other linear restrictions on the cointegration space are considered. Section III gives the design of the Monte Carlo experiments applied in the study for examining the small-sample bias of the LR tests for structural hypotheses. The simulation results are reported and summarized in the same section. In section IV, I analyze the effects of the presence of a constant term in the model and/or of a linear time trend in the data generating process on the finite-sample behavior of the test statistics. Section V suggests the use of the bootstrap procedure to obtain problem-specific critical values for the LR tests and investigates the performance of the bootstrap method. An example of using the bootstrap critic al values is illustrated in section VI. Section VII presents the conclusions of this study.

II. COINTEGRATION TESTS AND TESTS FOR STRUCTURAL HYPOTHESES

The Johansen tests are conducted through a vector error-correction (VEC) mechanism. For a vector of n variables, [X.sub.t], a VEC model can be written as

(1) [delta][X.sub.t] = [[[sigma].sup.k-1].sub.j=1] [[gamma].sub.j][delta][X.sub.t-j] + [pi][X.sub.t-1] + [micro] + [[epsilon].sub.t],

where [[epsilon].sub.t], is a vector of independent Gaussian variables with mean zero and variance matrix [sigma]. [micro] is a constant term. The hypotheses of interest involve [pi]; if the rank of [pi] is q, where q [less than or equal to] n - 1, then [pi] can be decomposed into two n x q matrices [alpha] and [beta] such that [pi] = [alpha][beta]'. The matrix [beta] consists of q linear cointegration vectors, while [alpha] can be interpreted as a matrix of vector error-correction parameters. The maximum likelihood estimates of [alpha] and [beta] are obtained by regressing [delta][X.sub.t] and [X.sub.t-1] on [delta][X.sub.t-1],..., [delta][X.sub.t-k+1] and 1. This gives residuals [R.sub.0t] and [R.sub.1t] and residual product matrices

[S.sub.ij] = [T.sup.-1] [[[sigma].sup.t].sub.t=1] [R.sub.it][R'.sub.jt], i,j = 0, 1.

One may solve the eigenvalue system

\[lambda][S.sub.11] - [S.sub.10][[S.sup.-1].sub.00][S.sub.01]\ = 0

for eigenvalues [[lambda].sub.1] [greater than] ... [greater than] [[lambda].sub.n], and eigenvectors V = ([v.sub.1],..., [v.sub.n]). The estimates of [alpha] and [beta] are given by [alpha] = [S.sub.01][beta] and [beta] = ([v.sub.1],..., [v.sub.q]), where [v.sub.1],..., [v.sub.q] are the eigenvectors associated with the q largest eigenvalues. The number of cointegration vectors could be determined using Johansen's maximal eigenvalue statistics and/or the trace statistics. The maximal eigenvalue ([[lambda].sub.max]) statistic for the null hypothesis of q cointegration vectors against the alternative of q + 1 cointegration vectors is

[[lambda].sub.max] = -T ln(1 - [[lambda].sub.q+1]),

and the trace statistic for the null hypothesis of at most q cointegration vectors is

Trace = -T [[[sigma].sup.n].sub.j=q+1] ln(1 - [[lambda].sub.j]).

If the results are consistent with the hypothesis of at least one cointegration vector, structural hypotheses, that is, the hypotheses regarding the restrictions on [beta]'s, could be tested using the maximum likelihood methodology.

Testing structural hypotheses is an important segment of cointegration analysis. Important long-run equilibrium relationships based on economic theory can be tested in a cointegration framework. Consider the following two examples:

1. Long-Run Exclusion. It occurs frequently that only a subset of the variables in [X.sub.t] can be assumed to be relevant for the long-run relations. In this case there is a need to test the hypothesis of exclusion of a subset of variables, [[beta].sub.i] = [[beta].sub.j] = ... = 0, from the long-run relations.

2. Linear Restrictions on the Cointegration Space. Economic theory often suggests certain long-run equilibrium relationships existing among a set of economic variables. These equilibrium relationships have been utilized as important building blocks in economic modeling. For instance, two long-run relationships are frequently used in monetary and international economics. One is purchasing power parity (PPP), which plays an important role in international economic modeling. The PPP condition implies a cointegration relation between the exchange rate (ER) of two currencies and the price levels ([P.sub.A] and [P.sub.B]) of the two relevant countries: [z.sub.t] = [[beta].sub.pa] ln [P.sub.At] + [[beta].sub.er] ln [ER.sub.t] + [[beta].sub.pb] ln [P.sub.Bt] with a cointegration vector [[[beta].sub.pa], [[beta].sub.er], [[beta].sub.pb]] = [-1, 1, 1]. Another is a long-run money relation linking nominal money (M), price level (P), real income (Y), and interest rate (R): [z.sub.t] = [[beta].sub.m] ln [M.sub.t]+[[beta].sub.p] ln [P.sub.t]+[[beta].sub.y] ln [Y.sub.t]+[[beta].sub.R][R.sub.t]. Monetary theory suggests long-run price homogeneity: [[[beta].sub.m], [[beta].sub.p], [[beta].sub.y], [[beta].sub.R]] = [-1, 1, [[beta].sub.y], [[beta].sub.R]], or long-run income homogeneity: [[[beta].sub.m], [[beta].sub.p], [[beta].sub.y], [[beta].sub.R]] = [-1, [[beta].sub.p], 1, [[beta].sub.R]], or both: [[[beta].sub.m], [[beta].sub.p], [[beta].sub.y], [[beta].sub.R]] = [-1, 1, 1, [[beta].sub.R]]. These relations could be examined through an analysis of the likelihood function by imposing some linear restrictions on the cointegration relations.

The likelihood ratio (LR) test procedure for these structural hypotheses is prescribed by Johansen [1991]. The test is formulated by Johansen and Juselius [1990, 1992] as

[H.sub.2] : [beta] = H[phi]

where the matrix H of dimension n x m imposes (n - m) restrictions on i = 1, ..., q cointegration vectors. The test statistic is calculated as

Q([H.sub.2]\[H.sub.1]) = -T [[[sigma].sup.q].sub.i=1]ln[(1 - [[lambda].sub.1,i])/(1 - [[lambda].sub.2,i])],

where [[lambda].sub.1,i], and [[lambda].sub.2,i] are the ith estimated eigenvalues in the unrestricted model [H.sub.1] and the restricted model [H.sub.2], respectively. Johansen [1991] shows that the asymptotic distribution of the test statistics when there exists only one cointegration vector is [[chi].sup.2] with degrees of freedom equal to the number of restrictions (n - m).

However, empirical analyses necessarily deal with finite samples. It is widely recognized that finite-sample studies with the LR test statistics using the asymptotic critical values tend to bias the tests toward rejecting the structural hypotheses too often. Therefore, using (T - c)--where c is the number of parameters in the unrestricted equation (per equation)--instead of T in forming the LR test statistics has been recommended by almost every textbook in multivariate time series, a correction usually attributed to Sims [1980, 17]. Both the degree of the finite sample bias of the LR tests as well as the ability of Sims's adjustment in correcting the finite-sample bias are investigated by the following experiments.

III. THE MONTE CARLO SIMULATION EXPERIMENTS

I examine the finite-sample bias of the LR tests for structural hypotheses through the Monte Carlo simulations, where the simulated test statistics, with or without Sims's correction, are compared with the asymptotic critical values. The study is limited to the test statistics for one and two linear restrictions, allowing only one cointegration vector (q = 1) in the system. In the experiments, multivariate models with the dimensions n = 2, 3, 4, 5 are generated. The numbers of effective observations are T = 100 and 500 and the lag length of equation (1) is set by k = 1, 2, 4, 6. [X.sub.t] is generated by setting the initial values of [X.sub.0] equal to zero and creating T + k + 100 observations, of which the first 100 observations are discarded to minimize the effect of the initial condition. The GAUSS matrix programming language is used to write the computer programs for this study, and GAUSS's RNDN functions are used to generate pseudo-random normal innovations. Each simulation experiment consists of 20,000 replications for a sample size T, a number of variables n, and a lag length k. The data-generating processes (DGPs) for the experiments are similar to the ones utilized by Engle and Granger [1987], Gonzalo [1994], and Phillips [1994]. The true lag of the DGPs is set to be k = 1.

To test for the long-run exclusion of a variable, the following DGP(1) is used: For n [greater than] 2, [X.sub.t] is set by [X.sub.t] = [[y.sub.t], [W.sub.t]] and [W.sub.t] = [[W.sub.1t], [w.sub.2t]] with [W.sub.t] being an (n - 1)-dimensional random walk process, that is, [W.sub.t] = [W.sub.t-1] + [e.sub.wt] or [delta][W.sub.t] = [e.sub.wt], [w.sub.2t] being a random-walk variable (within [W.sub.t]) to be excluded from the cointegration relation. The cointegration equation is expressed as [beta]'[X.sub.t] = [[beta].sub.1][y.sub.t] + [[beta].sup.*][W.sub.1t] = [z.sub.t] with [beta]' = [[[beta].sub.1], [[beta].sub.2], ..., [[beta].sub.n]] = [[[beta].sub.1], [[beta].sup.*], 0] being the cointegration vector where [[beta].sup.*] = [[[beta].sub.2], ..., [[beta].sub.n-1]] and [[beta].sub.n] being restricted to be zero, [z.sub.t] = [rho][z.sub.t-1] + [e.sub.zt], [absolute val. of [rho]] [less than] 1. [u.sub.t] = [[e.sub.zt], [e.sub.wt]]' is an n x 1 vector of independently distributed normal errors of mean zero and covariance matrix [I.sub.n]. The null hypothesis of the test is [H.sub.0]: [[beta].sub.i] = 0 or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [omega] is a 1 x (n - 2) vector of zeros.

For n = 2, [X.sub.t] is set by [X.sub.t] = [[y.sub.t], [w.sub.t]]' with [[beta].sub.1][y.sub.t] = [z.sub.t], [z.sub.t] = [rho][z.sub.t-1] + [e.sub.zt], [absolute val. of [rho]] [less than] 1, [delta][w.sub.t] = [e.sub.wt] and [u.sub.t] = [[e.sub.zt], [e.sub.wt]]'. That is, the null hypothesis is [H.sub.0]: [[beta].sub.2] = 0 or

[beta] = H[phi] = [1 0] [[beta].sub.1] = [[[beta].sub.1], 0]'

To test other linear restrictions on the cointegration space, we use the following DGP(2): [X.sub.t] = [[y.sub.t], [W.sub.t]]' with [W.sub.t] being an (n - 1)-dimensional random walk process, that is, [delta][W.sub.t] = [e.sub.wt]. The cointegration equation is [beta]'[X.sub.t] = [[beta].sub.1][y.sub.t] + [gamma][W.sub.t] = [z.sub.t] with [beta]' = [[[beta].sub.1], [gamma]] being the cointegration vector where [gamma] = [[[beta].sub.2], ..., [[beta].sub.n]], [z.sub.t] = [rho][z.sub.t-1] + [e.sub.zt], [absolute val. of [rho]] [less than] 1. Again, [u.sub.t] = [[e.sub.zt], [e.sub.wt]]' is an n x 1 vector of independently distributed normal errors of means zero and covariance matrix [I.sub.n].

The first linear restriction taken into consideration was the equality of two coefficients of the cointegration vector, [[beta].sub.i] = [+ or -][[beta].sub.j]. We set [H.sub.0]: [[[beta].sub.1], [[beta].sub.2], [[beta].sub.3], ..., [[beta].sub.n]] = [[[beta].sub.1], -[[beta].sub.1], [[gamma].sup.*]] where [[gamma].sup.*] = [[[beta].sub.3], ..., [[beta].sub.n]]. This is, [H.sub.0]: [[beta].sub.1] = -[[beta].sub.2] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This restriction matches the hypothesis of proportional co-movements of two variables (like the long- and short-term interest rates, consumption and income, money demand and price level, or two countries' price levels, etc.) in a cointegrated system. It is not surprising to find that the finite- sample distributions of the test statistics for such a restriction, corresponding to different model dimensions, are almost the same as those for testing the long-run exclusion of a single variable because either of the two is testing for one linear restriction. Therefore, only the results for testing the long-run exclusion are reported.

I then study the finite-sample bias problem associated with the test statistics for two linear restrictions for the models of n = 3, 4, 5. The null hypothesis is set as [H.sub.0]: [[[beta].sub.1], [[beta].sub.2], [[beta].sub.3], [[beta].sub.4],...,[[beta].sub.n]] = [[[beta].sub.1], -[[beta].sub.1], -[[beta].sub.1], [[gamma].sup.**]] where [[gamma].sup.**] = [[[beta].sub.4],..., [[beta].sub.n]]. That is, [H.sub.0]: [[beta].sub.1] = -[[beta].sub.2] = -[[beta].sub.3] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

= [[[beta].sub.1], -[[beta].sub.1], -[[beta].sub.1], [[gamma].sup.**]]'

here [omega] is a 1 x (n - 3) vector of zeros. Note that the distributions of these test statistics are not affected by the values of [beta]'s to be set under the null. The tests of two linear restrictions are applicable for testing the PPP condition as well as the long-run monetary relation, as mentioned in section II.

Table I reports the results of the Monte Carlo experiments with and without Sims's correction. Consistent with the findings in the existing literature, size distortions of the LR tests are greater when the sample size, T, is smaller. The degree of size distortion is positively related to [rho], to the model dimensions, n, and to the number of lags, k. These suggest that finite-sample studies based on the asymptotic critical values would bias the tests toward rejecting the structural hypotheses too often for high dimensional models, when the sample size is small, and/or selected lag length is long. This bias would become more severe when [rho] is closer to 1, that is, when the cointegrating residual [z.sub.t] is highly serially correlated. [2] As can be seen in Table I, Sims's correction appears to be incompetent for reducing size distortions.

IV. THE EFFECTS OF DIFFERENT TREND SPECIFICATIONS ON THE TESTS

The practice of section III is consistent with the analysis conducted by Johansen and Juselius [1990] in their study for the Finnish data, where the DGP has no linear trend but a constant term, [micro], is included in the VEC model of equation (1) to capture the impact of a possible time trend in the sample variables. I call it Case [1.sup.*]. Johansen has realized that the critical values for his popular test for cointegration rank depend on the structure of the deterministic components that prevail in the system. Three alternative deterministic-trend specifications are considered in Johansen [1988], Johansen and Juselius [1990], and Johansen [1992]. Although these alternative specifications would not affect the asymptotic distributions of the test statistics for structural hypotheses (they are still [[chi].sup.2] distributions), they might have different effects on the finite-sample distributions. Correspondingly, we investigate the three cases for the effects of different trend specifications on the finit e-sample tests for structural hypotheses.

The DGP with alternative trend specifications can be expressed by [beta]'[X.sub.t] + [[beta].sub.0] = [z.sub.t], where [X.sub.t] = [[y.sub.t], [W.sub.t]]' with [W.sub.t] being an (n - 1) - dimensional random walk process with a drift [a.sub.w] (i.e., [delta][W.sub.t] = [a.sub.w] + [e.sub.wt] or [W.sub.t] = [a.sub.w]t + [[W.sup.*].sub.t] with [[W.sup.*].sub.t] = [[W.sup.*].sub.t-1] + [e.sub.wt]). [[beta]' [[beta].sub.0]]' is the cointegration vector with [[beta].sub.0] being a constant term and [z.sub.t] = [rho][z.sub.t-1] + [e.sub.zt] [absolute val. of [rho]] [less than] 1. The estimated VEC model is

(2) [delta][X.sub.t] = [[[sigma].sup.k-1].sub.j=1] [[gamma].sub.j] [delta][X.sub.t-j] + [pi][[X.sup.*].sub.t-1] + [micro] + [[epsilon].sub.t].

For Case [1.sup.*], [[beta].sub.0] and [a.sub.w] are zero. [micro] is unrestricted, and [[X.sup.*].sub.t-1] = [X.sub.t-1] in the VEC model. The three alternative cases are:

* Case 1 (Johansen [1988]): [[beta].sub.0] = 0 and [a.sub.w] = 0. [micro] = 0 and [[X.sup.*].sub.t-1] [X.sub.t-1] in the VEC model.

* Case 2 (Johansen and Juselius [1990]; Johansen [1992]): [a.sub.w] = 0, [micro] = 0, but [[beta].sub.0] [not equal to] 0 and [[X.sup.*].sub.t-1] = ([X'.sub.t-1], 1), in the VEC model. That is, [X.sub.t] has no time trend, but there is a constant term [[beta].sub.0] confined to the cointegration equation of the DGP.

* Case 3 (Johansen and Juselius [1990]; Johansen [1992]): [[beta].sub.0] = 0 [a.sub.w] [not equal to] 0, that is, there is a linear time trend present in the DGP. [micro]. is unrestricted, and [[X.sup.*].sub.t-1] = [X.sub.t-1] in the VEC model.

Table II shows the results of Case 1 with T = 100. It seems that Case 1 has less size distortions than Case [1.sup.*] corresponding to various n and k. In other words, when there is no linear time trend in the DGPs (for either Case 1 or Case [1.sup.*]), the absence of the constant term from the VEC models leads to less finite-sample distortion of test size. Nevertheless, both Case 1 and Case [1.sup.*] suffer large size distortions associated with small samples. [3] Again, Sims's correction is unable to reduce size distortions effectively.

The results of the Monte Carlo experiments for Cases 2 and 3 are not presented because they are very similar to those of Case [1.sup.*]. [4] This suggests that (a) having a constant term in the cointegration relation of [X.sub.t] (i.e., Case 2) yields very similar finite-sample distributions of the test statistics to those of the case allowing a constant term in the VEC model but not in the cointegration equation of the DGP (i.e., Case [1.sup.*]); and (b) having a time trend in the DGP (Case 3) or not (Case [1.sup.*]) does not have much influence on the finite-sample distributions of the LR test statistics for structural hypothesis as long as the constant term [micro] is included in the VEC model.

V. THE BOOTSTRAP SIMULATION

Because Sims's modification of the test statistics fails to correct the finite-sample bias of the LR tests for structural hypotheses, there is a need for the ways of dealing with the finite-sample bias problem. Since the finite-sample distributions of the LR test statistics are dependent on nuisance parameters (as revealed in the last two sections), tabulating critical values based on some simple DGPs would be of little use. For practical purposes, it seems appropriate to obtain problem-specific critical values for the LR tests by employing the bootstrap technique.

The bootstrap can be utilized to simulate the finite-sample distributions of the LR test statistics for structural hypotheses under the null. The procedure for the variables that may have a linear trend in the data is summarized by the following steps.

1. We apply the Johansen LR tests for structural hypotheses to actual data to obtain the test statistics, Q([H.sub.2]\[H.sub.1]), as described in section II, with the VEC model of equation (1).

2. Under the null hypothesis [H.sub.2]: [beta] = H[phi], we estimate the model

(3) [delta][X.sub.t] = [[[sigma].sup.k-1].sub.j=1] [[gamma].sub.i][delta][X.sub.t-j] + [alpha][beta]'[X.sub.t-1] + [micro] + [[epsilon].sub.t] by solving

\[lambda]H'[S.sub.11]H - H'[S.sub.10][[S.sup.-1].sub.00][S.sub.01]H\ = 0

for eigenvalues [[lambda].sub.1] [greater than] ... [greater than] [[lambda].sub.n] and corresponding eigenvectors V = ([v.sub.1],...,[v.sub.n]). The maximum likelihood estimator of [beta] is obtained by [beta]=H[phi] where [phi] = ([v.sub.1],...,[v.sub.q]). [[gamma].sub.i], [alpha], and [micro] are then calculated following Johansen and Juselius [1992, 174 and 176]. The residuals [[epsilon].sub.t], generated in equation (3), are multiplied by a scale factor of [[T/(T - c)].sup.1/2], denoted as [[epsilon].sub.t], where c represents the loss of degrees of freedom in estimating equation (3). [5]

3. We form the bootstrap disturbances, denoted as [[[epsilon].sup.*].sub.t], by randomly sampling with replacement from [[epsilon].sub.t] and define

(4) [delta][[X.sup.*].sub.t]=[[[sigma].sup.k-1].sub.j=1][[gamma].sub.j][d elta][[X.sup.*].sub.t-j]+[alpha][beta]'[[X.sup.*].sub.t-1]+[micro]+[[ [epsilon].sup.*].sub.t]

using the actual data for the first k -1 observations as initial conditions.

4. We then apply the Johansen LR tests to [[X.sup.*].sub.t] and obtain the test statistics, [Q.sup.*]([H.sub.2]\[H.sub.1]).

5. Repeating steps 3 and 4 N number of times and ordering the test statistics to be [[Q.sup.*].sub.1] [less than or equal to] [[Q.sup.*].sub.2] [less than or equal to] ... [less than or equal to] [[Q.sup.*].sub.N], we obtain the [delta]% critical value, which is [[Q.sup.*].sub.[delta]N]/100.

6. The null hypothesis is rejected at the [delta]% significance level if the Q([H.sub.2]\[H.sub.1]) statistic of step 1 is greater than the [delta]% bootstrap critical value.

For the variables with no linear trend in the data, one may simply remove the constant term [micro] from equations (1) and (3) and [micro] from equation (4) when conducting the bootstrap simulation following the above procedure. Besides, the residuals [[epsilon].sub.t], generated by step 2, have to be scaled and centered, that is, [[epsilon].sub.t] = [[T/(T-c)].sup.1/2]([[epsilon].sub.t]-[T.sup.-1] [[[sigma].sup.T].sub.t=1][[epsilon].sub.t]), where [T.sup.-1] [[[sigma].sup.T].sub.t=1] [[epsilon].sub.t] may be nonzero as [micro] is removed from equation (3).

To check the performance of the bootstrap procedure, we apply the procedure to the DGP utilized earlier in producing Table I and Table II for T = 100 and n = 3 to see if this procedure may help lessen the size distortion problem for Case [1.sup.*] and Case 1. The results listed in Table III are generated with 1,000 replications. For each replication, the bootstrap critical values are estimated with 400 bootstrap samples (N = 400). Compared to the results of Table I and Table II, Panel A of Table III reflects that the small-sample size distortions of the LR tests are substantially reduced as the tests are conducted with the bootstrap critical values. There exist only minor size distortions when the cointegrating residuals are moderately serially correlated (i.e., when [rho] is less than or around 0.7).

In Panel A of Table III, the estimated empirical sizes of the tests slightly vary with the lag lengths of the VEC models. I would like to point out that the results for k [greater than] 1 are attained by overparameterization by including more lags in the VEC models than are in the designed DGPs (k = 1). The results suggest that the LR tests for structural hypotheses with the bootstrap procedure are not very sensitive to overparameterization.

To further investigate the effect of lag specifications, similar to the method of Cheung and Lai [1993], I conducted the experiments based on the following DGPs for n = 3:

[[[beta].sub.1], [[beta].sub.2]][[y.sub.t], [w.sub.1t]]'

= [z.sub.t] for testing the null of [[beta].sub.3] = 0,

[[[beta].sub.1], [[beta].sub.2], [[beta].sub.3]][[y.sub.t], [w.sub.1t], [w.sub.2t]]'

= [z.sub.t] for testing the null of [[beta].sub.1]

(5) = -[[beta].sub.2] = -[[beta].sub.3],

[z.sub.t] = [rho][z.sub.t-1]+[e.sub.zt], with

[w.sub.1t] = [w.sub.1t-1] = [[phi].sub.1]([w.sub.1t-1] - [w.sub.1t-2])+[e.sub.w1t],

and

[w.sub.2t] = [w.sub.2t-1] + [[phi].sub.2]([w.sub.2t-1] - [w.sub.2t-2]) + [e.sub.w2t],

where [e.sub.w1t], [e.sub.w2t], and [e.sub.zt] are independent Gaussian zero-mean white noise innovations. Arbitrarily, the parameters [[phi].sub.1] and [[phi].sub.2] are set equal to 0.7 and 0.2, respectively. This is a model with autoregressive dependence, which implies that the appropriate lage length of the VEC model is k = 2. The estimated test sizes for T = 100 and [rho] = 0.7 are presented in Panel B of Table III. They indicate that when using the bootstrap critical values, the biases in the empirical sizes of the LR tests are reasonably small for the models specified with the right lag length (i.e., k = 2) as the cointegrating residual is no more than moderately serially correlated. The results confirm that the test sizes are not very sensitive to overparameterization (as k [greater than] 2). However, the distortion of the test size is severe in the case of underparameterization (for k = 1).

It is also found that (a) if we allow a constant term [micro] in the VEC models of equations (1) and (3), and an intercept [micro] in equation (4), the estimated test sizes appear to be nearly unaffected by whether there is a linear trend in the DGP or not; (b) when there is no linear trend in the DGP, if we allow [micro] in the VEC models of equations (1) and (3), the resampling scheme of equation (4) containing an intercept [micro] or not has little effect on the simulation results. [6]

VI. AN EXAMPLE OF STRUCTURAL HYPOTHESIS TESTING WITH THE BOOTSTRAP

In this section, the bootstrap method is applied to a data set for testing the long-run validity of PPP for France and Germany. The study uses 100 observations of quarterly data from 1974:1 through 1998:4, with observations from 1972:1 through 1973:4 available for presample lags as needed. The price variables ([P.sub.F] and [P.sub.G]) are the consumer price indexes of France and Germany, and the exchange rate (French franc price of a German mark) is the average market exchange rate, taken from the International Financial Statistics of the International Monetary Fund.

Note that the proposed bootstrap procedure has to be performed under the conditions that the rank of cointegration q has been determined, and the number of the lagged first differences in the VEC model k - 1 is properly specified. I chose the lag length of the VEC model based on the criteria that (a) additional lags are not statistically significant and (b) the residuals from equation (1) are white noise. The former is examined by the step-down testing procedure of Campbell and Perron [1991], beginning with a lag length [k.sub.max] = 8, while the latter is checked by the well-known Q test of Box and Pierce [1970] modified by Ljung and Box [1978] for small samples. The determination of the cointegration rank is on the basis of the Johansen cointegration rank tests, as described in section II. The finite-sample critical values of the cointegration rank tests are calculated following Cheung and Lai [1993, 318].

Since the price variables have an apparent time trend in the data, we allow a constant term in the VEC model and perform the tests and the bootstrap accordingly. The test statistics and the corresponding critical values are given in Table IV. The asymptotic 5% critical values of the LR tests for structural hypotheses are adjusted by Sims's correction of T/T-c). The lag length chosen by step-down testing is k = 7. The Q-statistics of Ljung and Box [1978] fail to reject the null that the residuals from the VEC models of equation (1) with k = 7 are white noise. [7] The results show that the hypothesis of no cointegration is rejected at the 5% significance level, whereas the hypothesis of one cointegration relation is not rejected. The long-run exclusion hypotheses of [[beta].sub.pf] = 0, [[beta].sub.pg] = 0 and [[beta].sub.er] = 0 are clearly rejected. We thus conclude that there exists one cointegration vector among In [P.sub.F], ln [P.sub.G], and In ER. For the hypothesis of PPP relation [[[beta].sub.pf], [[beta].sub.er], [[beta].sub.pg]] = [-1, 1, 1], using the asymptotic critical values adjusted by Sims's correction would be subject to over-rejection. When comparing the LR test statistic with our 5% bootstrap critical value, we are unable to reject the null of [[[beta].sub.pf], [[beta].sub.er], [[beta].sub.pg]] = [-1, 1, 1]. The study provides significant evidence that PPP holds well for France and Germany.

VII. CONCLUSIONS

In this study, the finite-sample bias of Johansen's [1991] LR tests for structural hypotheses (the hypotheses of linear restrictions on the cointegration relations) has been examined. The Monte Carlo experiments have been conducted to show the roles of the sample size, the lag order, and the dimension of the variable system in determining the finite-sample bias of the Johansen tests. The effects of the presence of deterministic terms in the DGPs or in the VEC model on the finite-sample behavior of the test statistics are also studied.

It is found that Johansen's tests are biased toward rejecting the null hypotheses more often than what asymptotic theory suggests, even after the test statistics are adjusted by Sims's correction. The finite-sample bias magnifies as the lag length or the dimension of the estimated system increases. Moreover, the bias problem would become more serious when the cointegrating residual is highly serially correlated.

The simulation results also indicate that the degrees of the finite-sample bias of the LR tests are very similar for the cases where there is a constant term in the VEC model regardless of the presence of a linear trend in the DGP or not (Case [1.sup.*] and Case 3) as well as for the case of a constant term confined to the cointegration equation of the DGP (Case 2). However, when the constant term is absent from both the DGP and the VEC model (Case 1), the finite-sample bias of the test statistic is somewhat smaller than those of the other three cases.

I recommend the use of the bootstrap method to obtain problem-specific critical values for the LR tests. An empirical application of the proposed bootstrap procedure is demonstrated. It is shown that using the bootstrap procedure may substantially reduce the small-sample bias. It is also illustrated that the LR tests for structural hypotheses with the bootstrap method are robust to overparameterization in the lag length of the VEC models, but not so to underparameterization when using low order VEC models.

(*.) I would like to thank Dennis Hoffman, Michael Melvin, Asatoshi Maeshiro, Mary McGarvey, and session participants at the 1998 WEA International Conference for their helpful suggestions and comments on earlier drafts of this paper. I also wish to thank an anonymous referee and the editor of Economic Inquiry for their valuable suggestions and guidance. This research is supported by a summer research grant of the College of Business of the University of Texas at San Antonio.

Zhou: Associate Professor, Division of Economics and Finance, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, Tex. 78249-0633. Phone 1-210-458-4315, Fax 1-210-458-5837, E-mail szhou@lonestar.jpl.utsa.edu

(1.) See Gonzalo [1994], Phillips [1994], and Haug [1996].

(2.) For a better understanding of [rho], it is worth pointing out that the value of [rho] is inversely related to that of [alpha], the error-correction parameter. Consider a two-variable system, [X.sub.t] = [[y.sub.t], [w.sub.t]]', where (a) [w.sub.t] = [w.sub.t-1] + [e.sub.wt], a random walk process, (b) [y.sub.t] = [beta][w.sub.t] + [z.sub.t], the cointegration relation, and (c) [z.sub.t] = [rho][z.sub.t-1] + [rho][e.sub.zt]. One may rewrite (b) as [y.sub.t] - [y.sub.t-1] = [beta]([w.sub.t] - [w.sub.t-1]) + [z.sub.t] - ([y.sub.t-1] - [beta][w.sub.t-1]), that is [delta][y.sub.t] = [beta][e.sub.wt] + [z.sub.t] - [z.sub.t-1] = [beta][e.sub.wt] + [rho][z.sub.t-1] + [e.sub.zt] - [z.sub.t-1] = -(1 - [rho])[z.sub.t-1] + [beta][e.sub.wt] + [e.sub.zt], where [alpha] = 1 - [rho] reflects the speed of adjustment of [y.sub.t] to past deviations ([z.sub.t-1]) from the long-run equilibrium.

(3.) Note that all results except those for k = 1, reported in Table I and Table II, correspond to over-parameterized models. Cheung and Lai [1993] show that the Johansen cointegration tests are rather sensitive to underparameterization in the lag length but not so to overparameterization. Following their procedure, I investigate the effect of lag specification on the tests for structural hypotheses for both Case 1 and Case [1.sup.*] and obtain results that are consistent with their conclusion. The results are not reported but are available on request.

(4.) The results of Case 2 are obtained by setting [[beta].sub.0] = 0.5. It is found that the finite-sample distribution of the test statistic for structural hypothesis is invariant to the value of [[beta].sub.0]. For Case 3, the results vary with different values of [a.sub.w], but the effects of different [a.sub.w] values are rather small.

(5.) This scale factor adjusts the variance of [[epsilon].sub.t] to account for the loss of c degrees of freedom in estimating equation (3).

(6.) The details of these results are not reported. They are available from the author on request.

(7.) These test statistics are not reported but are available from the author on request.

REFERENCES

Box, George E. P., and David A. Pierce. "Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models." Journal of the American Statistical Association, December 1990, 1509-26.

Campbell, John Y., and Pierre Perron. "Pitfalls and Opportunities: What Macroeconomists Should Know about Unit Roots." NBER Macroeconomics Annual, 1991, 141-201.

Cheung, Yin-Wong, and Kon S. Lai. "Finite-Sample Sizes of Johansen's Likelihood Ratio Tests for Cointegration." Oxford Bulletin of Economics and Statistics, August 1993, 313-28.

Engle, Robert F., and Clive W. J. Granger. "Cointegration and Error Correction: Representation, Estimation and Testing." Econometrica, March 1987, 251-76.

Gonzalo, Jesus. "Five Alternative Methods of Estimating Long-run Equilibrium Relationships." Journal of Econometrics, January/February 1994, 203-33.

Haug, Alfred A. "Tests for Cointegration: A Monte Carlo Comparison." Journal of Econometrics, 71, March/April 1996, 89-116.

Johansen, Soren. "Statistical Analysis of Cointegration Vectors." Journal of Economic Dynamics and Control, June/September 1988, 231-54.

-----. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models." Econometrica, November 1991, 1551-80.

-----. "Determination of Cointegration Rank in the Presence of a Linear Trend." Oxford Bulletin of Economics and Statistics, August 1992, 383-97.

Johansen, Soren, and Katarina Juselius. "Maximum Likelihood Estimation and Inference on Cointegration--with Applications to the Demand for Money." Oxford Bulletin of Economics and Statistics, May 1990, 169-210.

-----. "Testing Structural Hypotheses in a Multivariate Cointegration Analysis of PPP and the UIP for UK." Journal of Econometrics, July/September 1992, 211-44.

Ljung, G. M., and George E. P. Box. "On a Measure of Lack of Fit in Time Series Models." Biometrika, August 1978, 297-303.

Nelson, Charles R., and Charles I. Plosser. "Trends and Random Walks in Macroeconomic Time Series." Journal of Monetary Economics, September 1982, 139-62.

Phillips, Peter C. B. "Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models." Econometrica, January 1994, 73-93.

Sims, Christopher A. "Macroeconomics and Reality." Econometrica, January 1980, 1-48.