The Power of Cointegration Tests Versus Data Frequency and Time Spans.

Su Zhou [*]

Using Monte Carlo methods, this study illustrates the potential benefits of using high frequency data series to conduct cointegration analysis. The study also provides an account of why the results are different from those reported by Hakkio and Rush (1991). The simulation results show that when the studies are restricted by relatively short time spans of 30 to 50 years, increasing data frequency may yield considerable power gain and less size distortion, especially when the cointegrating residual is not nearly nonstationary, and/or when the models with nonzero lag orders are required for testing cointegration. The study may help clarify some misconceptions and misinterpretations surrounding the role of data frequency and sample size in cointegration analysis.

1. Introduction

In the empirical literature of cointegration analysis, researchers often face the limitation of using relatively short time spans of data. In many cases, this is simply due to the absence of longer spans of data. In other cases, some equilibrium relationships have to be studied for certain time periods. For instance, when the models require a flexible exchange rate or a variable price of gold, studies have to be undertaken for the flexible exchange rate period, starting from the early 1970s, or for the period since 1968 when the price of gold was allowed to fluctuate. With the limits of relatively short time spans, many researchers chose to use relatively high frequency data to conduct the studies. Such attempts have been criticized in the literature. Hakkio and Rush (1991) argue that "the frequency of observation plays a very minor role" (p. 572) in exploring a cointegration relationship, because "cointegration is a long-run property and thus we often need long spans of data to properly test it" (p. 579). H akkio and Rush's point is similar to the one made by Shiller and Perron (1985), that the length of the time series is far more important than the frequency of observation when testing for unit roots.

While those who criticize the collection of high frequency data to deal with the short time span problem advocate the use of long spans of data to test properly for cointegration, their suggestion is sometimes misinterpreted as a support for using a small number of annual data. [1] For instance, Bahmani-Oskooee (1996, p. 481) borrows Hakkio and Rush's (1991, p. 572) "testing a long-run property of the data with 120 monthly observations is no different than testing it with ten annual observations" to defend his use of annual data by saying that, using annual data of over 30 years "is as good as using quarterly or monthly data over the same period." Taylor (1995, P. 112) claims that the deficiency of using less than 50 annual observations "should be compensated by the fact that the data set spans nearly half a century."

I would like to point out that Hakkio and Rush's study has several limitations, and therefore it may not be appropriate to cite the conclusions of the study for the cases beyond its limitations. Their study only allows the cointegrating residual to be a pure first-order autoregressive (AR[1]) process and is limited to the single-equation method of cointegration tests. They show the results only for some extreme cases where the cointegrating residual for the monthly data is either very highly serially correlated (nearly nonstationary and thus all the cointegration tests would have very low test power regardless of the frequency of the data) or with a quite low coefficient of serial correlation (thus all the cointegration tests can easily reject the null of no cointegration regardless of the frequency of the data).

The present paper is motivated by seeking the answers to the following questions: (i) Does the frequency of observation play a very minor role in exploring a cointegration relationship in the cases where the cointegrating residual is not nearly nonstationary? (ii) Can the validity of the conclusions of Hakkio and Rush (1991) based on a single-equation method be extended to other popular cointegration tests and to more realistic cases, where the models with higher lag orders are required when the cointegrating residual is generated with more noise than a pure first-order autoregressive process? (iii) While testing cointegration with 120 monthly observations could be no different than testing it with 10 annual observations as both cases are subject to very low test power, does this warrant that using annual data of 30 to 40 years is as good as using quarterly or monthly data over the same period? (iv) How serious would the problem of size distortion be for the use of a small number of annual observations?

This paper examines the power of cointegration tests versus frequency of observation and time spans, as well as the small-sample size distortions of the tests, through the Monte Carlo experiments. [2] The above questions are addressed by doing the following: (i) Instead of focusing on extreme cases corresponding to the cointegrating residuals with either very high or rather low serial correlation coefficients, this study also pays attention to moderately serially correlated cointegrating residuals. (ii) Both the cointegration tests in single equations, such as the Engle-Granger (1987) tests, and those in systems of equations, such as the Johansen (1988) tests and the Horvath-Watson (1995) tests, are examined. (iii) After generating the data at the monthly frequency, two sets of quarterly and annual data are produced. One set is obtained by taking the last observation in the period. As demonstrated by Hakkio and Rush (1991), the cointegrating residual of these end-of-quarter and end-of-year data remains an AR (1) process as long as the cointegrating residual at monthly interval is generated as an AR(1) process. Another set is computed by averaging the 3 or 12 monthly observations that correspond to each quarter or each year. It can be shown that these average quarterly and annual data contain a first-order moving average (MA[1]) component. Therefore, the models with higher orders of lag length are required for testing cointegration. This allows us to examine the effects of time span and data frequency on the power of cointegration tests with different lag orders as well as the impact of under- or overparameterization on the power and empirical sizes of the tests. (iv) The simulations are first conducted with a fixed time span of 30 annual observations, 120 quarterly observations, and 360 monthly observations to illustrate the influence of different sampling frequency on the power of cointegration tests for the cointegrating residuals with different degrees of serial correlation and for the models with different la g orders. The test power and corresponding size distortions are then further analyzed for different combinations of time spans and data frequencies.

The paper is organized as follows. The next section introduces the data-generating processes. Section 3 briefly describes the cointegration tests under examination. The design of the Monte Carlo experiments applied in the study and the simulation results are reported in section 4. The last section concludes.

2. Data-Generating Processes

Following Hakkio and Rush (1991), the study starts with generating the monthly data [[X.sup.M].sub.t] by a random walk without a drift:

[[X.sup.m].sub.t] = [[X.sup.M].sub.t-1] + [[[eta].sup.M].sub.t], [[[eta].sup.M].sub.t] [sim] N(0,1).

Monthly [[Y.sup.M].sub.t] is defined as

[[Y.sup.M].sub.t] = [[X.sup.M].sub.t] + [[[epsilon].sup.M].sub.t]

where [[[epsilon].sup.M].sub.t] is an AR(1) process,

[[[epsilon].sup.M].sub.t] = [rho][[[epsilon].sup.M].sub.t-1] + [[e.sup.M].sub.t], [[e.sup.M].sub.t] [sim] N(0, [[[sigma].sup.2].sub.e]).

[[X.sup.M].sub.t] and [[Y.sup.M].sub.t] are cointegrated if [rho] [less than] 1, and are not cointegrated if [rho] = 1.

The end-of-period quarterly and annual data are

[[X.sup.end].sub.t] = [[X.sup.M].sub.t,s], [[Y.sup.end].sub.t] = [[Y.sup.M].sub.t,s], (1)

where s = 3 for quarterly data and s = 12 for annual data, hence

[[X.sup.end].sub.t] = [[X.sup.end].sub.t-1] + [[[eta].sup.end].sub.t], [[[eta].sup.end].sub.t] = [[[eta].sup.M].sub.t,1] + [[[eta].sup.M].sub.t,2] + ... [[[eta].sup.M].sub.t,s], E([[[eta].sup.end].sub.t], [[[eta].sup.end].sub.t-j] = 0 for j [neq] 0,

[[Y.sup.end].sub.t] = [[X.sup.end].sub.t] + [[[epsilon].sup.end].sub.t], [[[epsilon].sup.end].sub.t] = [[rho].sup.s][[[epsilon].sup.end].sub.t-1] + [[e.sup.end].sub.t], [[e.sup.end].sub.t] = [[e.sup.M].sub.t,s] + [rho][[e.sup.M].sub.t,s-1] + ... + [[rho].sup.s-1][[e.sup.M].sub.t,1].

Because E([[e.sup.end].sub.t], [[e.sup.end].sub.t-j]) = 0 for j [neq] 0, [[[epsilon].sup.end].sub.t] remains and AR(1) process.

The average quarterly and annual data are

[[X.sup.av].sub.t] = [[[sigma].sup.s].sub.i=1] ([[X.sup.M].sub.t,i])/s, [[Y.sup.av].sub.t] = [[[sigma].sup.s].sub.i=1] ([[Y.sup.M].sub.t,i])/s, and [[X.sup.av].sub.t] = [[X.sup.av].sub.t-1] + [[[eta].sup.av].sub.t],

[[[eta].sup.av].sub.t] = (1/s) [[[sigma].sup.s].sub.i=1] ([[X.sup.M].sub.t,i] - [[X.sup.M].sub.t-1,i]) = (1/s){[[[[sigma].sup.s].sub.i=1] (s + 1 - i) [[[eta].sup.M].sub.t,i]] + [[[[sigma].sup.s].sub.i=2] (i - 1) [[[eta].sup.M].sub.t-1,i]]},

[[Y.sup.av].sub.t] = [[[X.sup.av].sub.t] + [[[epsilon].sup.av].sub.t] , [[[epsilon].sup.av].sub.t] = [[[sigma].sup.s].sub.i=1] ([[[epsilon].sup.M].sub.t,i])/s, (2)

which give

[[[epsilon].sup.av].sub.t] = [[rho].sup.s][[[epsilon].sup.av].sub.t-1] + [[e.sup.av].sub.t], [[e.sup.av].sub.t] = (1/s){[[[sigma].sup.s].sub.i=1][[[e.sup.M].sub.t,i]([[[sigma].sup.s-i ].sub.j=0] [[rho].sup.j])] + [[[sigma].sup.s].sub.i =2][ [[e.sup.M].sub.t-1,i]([[[sigma].sup.s-1].sub.j=s-i+1] [[rho].sup.j])]}

It can be easily shown that E([[[eta].sup.av].sub.t], [[[eta].sup.av].sub.t-1]) [neq] = 0 and E([[e.sup.av].sub.i], [[e.sup.av].sub.t-1]) [neq] 0, yet E([[[eta].sup.av].sub.t], [[[eta].sup.av].sub.t-j]) = 0 and E([[e.sup.av].sub.t], [[e.sup.av].sub.t-j]) = 0 for j [greater than] 1. This means that [[[eta].sup.av].sub.t] and [[e.sup.av].sub.t] are MA(1) processes and thus could be expressed in a typical MA(1) form

[[[eta].sup.av].sub.t] = [u.sub.t] - [[theta].sub.e][u.sub.t-1] = (1-[[theta].sub.e]L)[u.sub.t], [[e.sup.av].sub.t] = [v.sub.t] - [[theta].sub.e][v.sub.t-1] = (1- [[theta].sub.e]L)[v.sub.t],

where E([u.sub.t], [u.sub.t-j]) = 0 and E([v.sub.t], [v.sub.t-j]) = 0 for j [not equal to] 0, [[theta].sub.[eta]] and [[theta].sub.e] are the moving average parameters, and L is the lag operator. Because the first-order autocorrelation of an MA(1) process is equal to -[theta]/(1 + [[theta].sup.2]), the approximate values of [[theta].sub.[eta]] and [[theta].sub.e] can be computed by solving the following

E([[e.sup.av].sub.t],[[e.sup.av].sub.t-1])/E[([[e.sup.av].sub.t]).sup .2]= [[[sigma].sup.s].sub.i=2][([[[sigma].sup.s-i].sub.j=0][[rho].sup.j])( [[[sigma].sup.s-1].sub.j=s-i+1][[rho].sup.j])]/ [[[sigma].sup.s].sub.i=1][[([[[sigma].sup.s-i].sub.j=0][[rho].sup.j]) .sup.2]]+[[[sigma].sup.s].sub.i=2][([[[sigma].sup.s-1].sub.j=s-i+1][[ rho].sup.j]).sup.2] = -[[theta].sub.e]/1+[[[theta].sup.2].sub.e] and

E([[[eta].sup.av].sub.t],[[[eta].sup.av].sub.t-1])/E[([[[eta].sup.av] .sub.t]).sup.2] = [[[sigma].sup.s].sub.i=2][(s+1-i)(i-1)]/[[[sigma].sup.s].sub.i=1][(s+ 1-i).sup.2]+[[[sigma].sup.s].sub.i=2] ([[i-1].sup.2]) = -[[theta].sub.[eta]]/1+[[[theta].sup.2].sub.[eta]] (3)

Note that [[theta].sub.e] is a function of monthly [rho], and [[theta].sub.[eta]] = [[theta].sub.e] corresponding to [rho] = 1. The computed values of [[theta].sub.e] corresponding to different values of monthly [rho] are reported in Table 1.

As the models of cointegration tests are mostly presented in an autoregressive (AR) or a vector autoregressive (VAR) form, the existence of an MA(1) term in [[X.sup.av].sub.t] and [[[epsilon].sup.av].sub.t] may require the models to have an infinite lag structure. That is, when

[[X.sup.av].sub.t] = [[X.sup.av].sub.t-1] + (1- [[theta].sub.[eta]]L)[u.sub.t], [[[epsilon].sup.av].sub.t] = [[rho].sup.s][[[epsilon].sup.av].sub.t-1] + (1 - [[theta].sub.e]L)[v.sub.t],

let [delta][[X.sup.av].sub.t] = [[X.sup.av].sub.t-1] and [w.sub.t] = [[[epsilon].sup.av].sub.t] - [[rho].sup.s][[[epsilon].sup.av].sub.t-1], we have

[delta][[X.sup.av].sub.t]/(1 - [[theta].sub.[eta]]L) = [delta][[X.sup.av].sub.t] - [[theta].sub.[eta]][delta][[X.sup.av].sub.t-1] + [[[theta].sup.2].sub.[eta]][delta][[X.sup.av].sub.t-2] - [[[theta].sup.3].sub.[eta]][delta][[X.sup.av].sub.t-3] + ... = [u.sub.t],

[w.sub.t]/(1 - [[theta].sub.e]L) = [w.sub.t] - [[theta].sub.e][w.sub.t-1] + [[[theta].sup.2].sub.e][w.sub.t-2] - [[[theta].sup.3].sub.e][w.sub.t-3] + ... = [v.sub.t].

[delta][[X.sup.av].sub.t] and [w.sub.t], are AR processes with an infinite lag structure.

In practice, if the MA coefficient [theta] is relatively small, the study would not be hurt by the problem of under parameterization if one uses the models with finite but sufficient lag lengths. Based on the computed [theta] listed in Table 1, [[theta].sup.i] is less than 0.02 for i [greater than] 2. Therefore, AR(2) or VAR(2) models for the variables expressed in the first differences seem to be enough to capture the influence of the MA(1) term in [[X.sup.av].sub.t] and [[[epsilon].sup.av].sub.t].

3. Tests for Cointegration

Three cointegration tests are examined by this study. They are Engle and Granger's (1987) augmented Dickey-Fuller (ADF) tests, the Johansen tests (Johansen 1988 and Johansen and Juselius 1990), and the tests of Horvath and Watson (1995).

The ADF Tests and the Johansen Tests for Cointegration

The ADF tests of Engle and Granger and the Johansen tests are the two most well known and most commonly used cointegration tests. The ADF tests focus on the ordinary least squares (OLS) residual [[epsilon].sub.1], from the single-equation cointegration regression of [Y.sub.t] on [X.sub.t]'s and apply the OLS again to get

[delta][[epsilon].sub.t] = [alpha][e.sub.t-1] + [[[sigma].sup.p].sub.i=1] [[gamma].sub.i][delta][[epsilon].sub.t-i] + [[zeta].sub.1]. (4)

The null hypothesis that [Y.sub.t] and [X.sub.t]'s are not cointegrated is tested by checking the significance of the t-statistic for [alpha] = 0.

Johansen's cointegration rank tests, based on a full information maximum likelihood (FIML) method, are considered to be superior to the regression-based single-equation methods, because they can handle the endogeneity problem of the regressors, better model the interactions between the variables, and fully capture the underlying time series properties of the variables in the system. The Johansen tests are carried out through a VAR system

[delta][Z.sub.t] = [micro] + [pi][Z.sub.t-1] + [[gamma].sub.1][delta][Z.sub.t-1] + ... + [[gamma].sub.p] [delta][Z.sub.t-p] + [[zeta].sub.t], (5)

where [Z.sub.t] is a vector containing n variables, [[zeta].sub.t] is an n-dimensional independent Gaussian disturbance with zero mean and covariance matrix [[omega].sub.[zeta]], and [micro] is a constant term. If the rank of [pi] is r, where r [less than or equal to] n - 1, then r is called the cointegration rank and [pi] can be decomposed into two n X r matrices [alpha] and [beta] such that [pi] = [alpha][beta]'. The matrix [beta] consists of r linear cointegration vectors while a can be interpreted as a matrix of vector error-correction parameters. Regressing [delta][Z.sub.t] and [Z.sub.t-1] on [delta][Z.sub.t-1], ..., [delta][Z.sub.t-p] and 1 gives residuals [R.sub.0t] and [R.sub.1t], respectively, and residual product matrices [S.sub.ij] = [T.sup.-1] [[[sigma].sup.T].sub.t=1] [R.sub.it][R.sub.jt] for i, j = 0, 1. One may then solve the eigenvalue system

/[delta][S.sub.1t] - [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 maximum eigenvalue

([[lambda].sub.max]) statistic for the null hypothesis of r cointegrating vectors against the alternative of r + 1 cointegrating vectors is

[[lambda].sub.max](r\r + 1) = -T ln(1 - [[lambda].sub.r+1])

and the trace statistic for the null hypothesis of at most r cointegrating vectors is

trace(r\n) = -T [[sigma].sup.n].sub.j=r+1] ln(1 - [[lambda].sub.j]).

The Cointegration Tests of Horvath and Watson

Horvath and Watson (1995) argue that by testing the null of no cointegration against the diffuse alternative of cointegration in general, rather than the specific alternative of certain long-run equilibrium relationship, the Johansen tests may lose power to reject the null hypothesis. Horvath and Watson propose incorporating the information about the cointegration space into the cointegration rank tests. This could be done by testing the null hypothesis of no cointegration against the composite alternative of cointegration with the cointegrating vectors prespecified based on economic theory. In so doing, the newly developed tests of Horvath and Watson gain power over the Johansen tests to reject the false null of no cointegration, when the prespecified cointegrating vectors are correctly specified.

The Horvath--Watson (HW) procedure uses the same VAR system as the Johansen procedure, Equation 5, but it imposes the prespecified cointegrating vectors [beta] on the model

[delta][Z.sub.t] = [micro] + [alpha]([beta] '[Z.sub.t-1] + [[gamma].sub.1][delta][Z.sub.t-1] + ... [[gamma].sub.p][delta][Z.sub.t-p] + [[zeta].sub.t]. (6)

The tests for cointegration are performed by computing the Wald statistic for the hypothesis that [alpha] = 0. Letting Z = [[Z.sub.1] [Z.sub.2] ... [Z.sub.T]]', [Z.sub.-1] = [[Z.sub.0] [Z.sub.1] ... [Z.sub.T-1]]', [delta]Z = Z - [z.sub.-1], [F.sub.t] = ([delta][Z.sub.t-1] [delta][Z.sub.t-2] ... [delta][Z'.sub.t-p])', F = [[F.sub.1] [F.sub.2] ... [F.sub.T]]', and [M.sub.F] = [I - F[(F'F).sub.-1]F'], the corresponding Wald test statistic is

W = [vec ([delta]Z'[M.sub.F][Z.sub.-1][beta])]'[[([beta] '[Z'.sub.-1][M.sub.F][Z.sub.-1][beta]).sup.-1] X [[[omega].sup.-1].sub.[zeta]]][vec([delta]Z'[M.sub.F][Z.sub.-1][beta] )].

The HW tests are particularly useful for investigating the long-run economic relationships in which not only the variables but also the coefficients of the variables are suggested by the economic theory, such as the purchasing power parity relation, the consumption-investment-output relations, and the term structures of interest rates, etc. A disadvantage of the HW tests is that their power gains mainly come from correctly prespecified cointegration relations. They would have power loss with incorrectly prespecified cointegrating vectors. Revealed by Horvath and Watson (1995) and also by Hoffman and Zhou (1998), the HW tests may still have higher test power over the Johansen tests when the cointegrating vectors imposed on the models are slightly incorrectly specified.

4. The Monte Carlo Experiments and Simulation Results

The design of Monte Carlo experiments adopted for this study is similar to the setting of Hakkio and Rush (1991). Monthly [X.sub.t] and [Y.sub.t] are generated by setting the initial values equal to zero and creating T + 12 observations, of which the first 12 observations are discarded to limit the effect of the initial condition. The end-of-period and average quarterly and annual data are computed by Equations 1 and 2, respectively. The simulations are first conducted with a fixed time span of 30 years with 360 monthly observations, 120 quarterly observations, and 30 annual observations. These simulations may illustrate the influence of different sampling frequencies on the power of the cointegration tests for the cointegrating residuals with different degrees of serial correlation, as well as the size distortions of different cointegration tests associated with different numbers of observations.

The attention of this study is to both highly and moderately serially correlated cointegrating residuals. Seven different values of [rho], that I used for generating monthly data, are presented in Table 1 together with the implied values of [rho] for the quarterly and annual data. They give us monthly serial correlations of 0.8, 0.9, and 0.95, and quarterly and annual serial correlations of 0.8 and 0.9. The GAUSS matrix programming language is adopted to write the computer programs for this study, and the RNDN functions of GAUSS are used to generate pseudorandom normal innovations.

The tests with the models described by Equations 4, 5, and 6 are carried out for the lag length p = 0, 2, and 4. Based on the discussions in section 2, the models with p = 0 are appropriate for monthly data (M) and the end-of-period quarterly and annual data ([Q.sub.end] and [A.sub.end], respectively), but they may suffer the problem of underparameterization for the average quarterly and annual data ([Q.sub.av] and [A.sub.av], respectively). The models with higher lag order, p [greater than or equal to] 2, would be needed for [Q.sub.av] and [A.sub.av] to capture the impact of the moving average term.

The simulation results for the three cointegration tests are reported in Tables 2 and 3. [3] All the results are based on 10,000 replications. The empirical sizes for 5% and 10% level tests are obtained by comparing the simulated test statistics under the null of no cointegration (i.e., setting p 1) with the 5% and 10% asymptotic critical values, respectively. The asymptotic critical values are taken from MacKinnon (1991) for the ADF tests, Osterwald-Lenum (1992) for the Johansen tests, and Horvath and Watson (1995) for the HW tests. The powers of 5% level tests, displayed in Table 2, are size adjusted. That is, the simulated test statistics are compared under the alternative of cointegration (i.e., setting p [less than] 1) with the 5% finite-sample critical values simulated under the null. [4]

The Effects of Data Frequency and Serial Correlation

The results show that the ADF tests and the HW tests have higher power over the Johansen tests, and the HW tests have the highest power among the three tests. Sampling frequencies have similar effects on the power of all three cointegration tests. For the models with p = 0, the Johansen tests have the least size distortion for [Q.sub.end] and [A.sub.end] data, while the ADF tests and the HW test tend to either over-reject or under-reject the true null for [A.sub.end]. However, for the models with p [greater than] 0, especially with p = 2, both the ADF tests and the HW tests have less size distortions, while the Johansen tests suffer large size distortions for annual data.

The results also indicate that for moderately serially correlated cointegrating residuals (i.e., monthly p = 0.9[sim]0.95 and quarterly [rho] [approx]0.73[sim]0.86), increasing data frequency from an annual to a quarterly interval may gain notable test power even when p = 0. The power gain associated with higher data frequency is substantial for the models with higher lag order and monthly p [less than]0.96.

The Impact of Under- and Overparameterization

Because of the existence of an MA term in the average quarterly and annual data but not in the end-of-period data, models for [Q.sub.av] and [A.sub.av] with low lag order are subject to underparameterization, while models for M, [Q.sub.end], and [A.sub.end] with p [greater than] 0 may endure overparameterization. Therefore, the differences between the results for the end-of-period data and those for the average data may reflect some of the effects of underparameterization, for the models with low lag order, or overparameterization, for the models with high lag order, on the cointegration tests.

As can be seen, the power of the ADF tests is somewhat lower for than for [A.sub.end] when monthly p = 0.9 [sim] 0.95 and p = 0, but there is no such difference when p [greater than or equal to] 2. The powers of the Johansen tests and the HW tests are significantly lower for [Q.sub.av] and [A.sub.av] than for [Q.sub.end] and [A.sub.end] when p = 0 and monthly p [approximate] 0.93 [sim] 0.98, but are about the same for them when p 2. The empirical sizes of the Johansen tests and the HW tests do not seem to be very different for the end-of-period data and the average data. The sizes of the ADF tests for [Q.sub.av] and [A.sub.av], compared with those for [Q.sub.end] and [A.sub.end], are distorted toward rejecting the null too rarely when p = 0 but are about the same as those for [Q.sub.end] and [A.sub.end] when p [greater than or equal to] 2. To summarize, for the data set generated for this study, underparameterization would lower the power of all three cointegration tests and cause size distortions for the AD F tests. Although overparameterization does not have much impact on the three tests, as the results from the models with p = 2 for the end-of-period data (subject to overparameterization) are similar to those for the average data (free of overparameterization), the models with high lag orders often induce tremendous power loss and large size distortions for annual data due to the high sensitivity of small sample to the loss of degrees of freedom. Thus, it is crucial to choose appropriate lag length for the cointegration tests especially when employing a small sample of annual data.

Time Spans Versus Data Frequencies

The test power and size distortions are then further analyzed for different combinations of time spans and data frequencies. The time spans run from 20 to 100 years, corresponding to quarterly data of 80 to 400 observations and monthly data of 240 to 1200 observations. The results displayed in Tables 4 and 5 are for monthly p = 0.9 and 0.95, and p = 0 and 2. The results show that the cointegration tests with monthly observations could have significantly larger test power than those with annual observations, especially when the high frequency data are moderately serially correlated or when the models with higher lag orders are required. For a sample with a fixed time span of 20 to 50 years, when monthly p = 0.95 and p = 2, using quarterly data instead of annual data may double or even triple the power of the tests, while using monthly data may further the gain of power.

The results, on one hand, confirm that the gains in power by increasing the sample spans are greater than those by increasing the observations with a fixed time span. This can be seen by the fact that the tests with 80 annual observations have much higher test power than those with 80 quarterly observations for a time span of 20 years. On the other hand, our results reflect that, when the studies are restricted by relatively short time spans, a large part of the power loss could be compensated by increasing the data frequency. For a sample with a short time span of 30 to 50 years, when monthly p = 0.95 and p = 2, using quarterly data may gain power as much as increasing the time span by 50% with annual data, while using monthly data may gain power as much as to double the length of the time span with annual data.

Again, for the models with p = 0, the Johansen tests have relatively less size distortions for [A.sub.end], while the ADF tests and the HW test have relatively large size distortion for [A.sub.end] unless the time span is 60 years or longer. For the models with p = 2, both the ADF tests and the HW tests have less size distortions, while the Johansen tests bear large size distortions for annual data unless the time span is 80 years or longer. The results signify that the use of asymptotic critical values tends to misinterpret the power performance in small samples, especially in the case of employing annual data. Thus, a proper assessment of the power performance and the acquirement of meaningful test results hinge on the use of appropriate finite-sample critical values.

Reflected by the results with respect to p = 0, underparameterization (associated with the results for [Q.sub.av] and [A.sub.av], compared to those for [Q.end] and [A.sub.end]) may lower the power of all three cointegration tests and produce size distortions for the ADF tests. The results corresponding to p = 2 illustrate that the test statistics of these cointegration tests are not very sensitive to overparameterization, as the results for [Q.sub.end] and [A.sub.end] are similar to those for [Q.sub.av] and [A.sub.av], although the loss of degrees of freedom coming from the models with high lag orders may cause significant power loss when a small sample of annual data is utilized.

5. Conclusions

Using the Monte Carlo method, this study illustrates the potential benefits of using high frequency data series to conduct cointegration analysis. The simulation results, on one hand, confirm the view that the ability of the cointegration tests to detect cointegration depends more on the time span than on the mere number of observations. On the other hand, it is found that when the studies are restricted by relatively short time spans of 30 to 50 years, increasing data frequency may yield considerable power gain and less size distortions. [5] This is particularly evident when the cointegrating residual is not nearly nonstationary, and/or when the models with higher lag orders are required for testing cointegration as the cointegrating residual is generated with more noise than a pure AR(1) process. [6] For a two-variable model with monthly p = 0.9, p = 2, and a time span of 30 years, the power of the cointegration tests investigated by this study is lower than 0.35 with annual data, but could be around 0.9 w ith quarterly data and higher than 0.98 with monthly data, with the exception of the Johansen tests whose power is relatively low. The power difference between using annual data or higher frequency data is even more dramatic for the models with higher lag orders.

These conclusions are less pessimistic than those of Hakkio and Rush (1991, pp. 572 and 579) that "the frequency of observation plays a very minor role" in exploring a cointegration relationship, and "rejecting noncointegration may be a fairly strong conclusion." The power gain from using high frequency data may also suggest that, when testing a time series model for cointegration, if one of the variables in the available data set has lower frequency than the others, it is not necessarily fruitless for researchers to seek the possibility of benefiting from linear interpolation, or other methods, to fill in the values of a low frequency data series in order to use the information contained in other higher frequency series. [7]

The study may help clarify some misconceptions and misinterpretations surrounding the role of data frequency and sample size in cointegration analysis. Whereas the statement that "testing a long-run property of the data with 120 monthly observations is no different than testing it with ten annual observations" (Hakkio and Rush 1991, p. 572) could be a legitimate statement, it may simply reflect that both cases are subject to very low test power and cannot be discriminated one from the other. It does not warrant that using annual data of 30 to 50 years is just as good as using quarterly or monthly data over the same period. The evidence presented in this study discourages the use of annual data of less than 50 years to test for cointegration with high lag order models. The results indicate that using a small sample of 30 to 50 annual observations, instead of more observations of higher frequency data, may not only result in significant loss of the test power but also very likely experience the problem of size distortion. In addition, the power of the tests with a small number of annual observations is very sensitive to the lag length of the models, and is more easily effected by the problem of underparameterization.

Although the above conclusions basically hold for all three cointegration tests in the study, it is found that the use of a small sample of annual data is particularly inappropriate for the application of the Johansen cointegration rank tests with higher lag order models, even if the data set spans half a century. The test results would suffer lower test power and larger size distortions compared with those of the ADF tests and the 11W tests. Therefore, when someone employs a small number of annual observations to carry out the Johansen cointegration tests with a lagged VAR model, one would expect a low probability of acquiring meaningful results. This is because, if one rejects the null hypothesis and concludes the existence of a cointegration relationship among the variables in the model by comparing the test statistics with the asymptotic critical values, the conclusion would be subject to over-rejection due to the problem of size distortion. [8] On the other hand, when one uses the appropriate finite-sa mple critical values that are usually much greater than the asymptotic critical values, there is rarely a chance of rejecting the null (even if it is false) as the size-adjusted power of the Johansen tests is very low for a small sample.

(*.) Division of Economics and Finance, College of Business, University of Texas at San Antonio, San Antonio, TX 78249-0633; E-mail [e-mail].

The author thanks two anonymous referees for their helpful comments and editorial suggestions on this paper. Financial support provided by a summer research grant from the College of Business of the University of Texas at San Antonio is gratefully acknowledged. The usual caveat applies.

(1.) See Bahmani-Oskooee (1996), Masih and Masih (1996), and Taylor (1995) for examples.

(2.) The size distortions of the tests have not much to do with the data frequency, or the time span, as long as the number of sample observations and the lag lengths of the models stay the same. That is, applying a model with fixed lag length to a sample of 80 quarterly observations, or 80 annual observations, yields similar size distortions. The finite-sample size distortions of the ADF tests and the Johansen tests have been analyzed by Cheung and Lai (1993, 1995). There are several reasons that this paper also examines size distortions. In some existing studies of cointegration, when authors defend their use of small samples of annual data by arguing that increasing data frequency would not have much power gain, they often ignore the problem of size distortions associated with small samples. I would like to demonstrate the seriousness of the size distortion problem rather than simply citing the available studies that may not exactly address my concerns. Besides, the finite-sample study of the Horvath-Watso n (1995) tests is not available.

(3.) Because the results for the [[lambda].sub.max] statistic and the trace statistic of the Johansen tests are very similar, I only report the power and empirical sizes of the tests for the statistics. Those for the trace statistics are not listed but are available from the author upon request. Because different values of [[[sigma].sup.2].sub.e], the relative variance of [[eta].sub.1], to [e.sub.1], do not have much effect on the results, only the results for [[[sigma].sup.2].sub.e] = 1.0 are reported.

(4.) revealed size distortions associated with small samples (reported in Table 3) show the significance of using size-adjusted power to compare the test performance.

(5.) Note that the complication of the presence of seasonal factors in the quarterly and monthly data and regime shifts in the relatively long span of annual data are not taken into consideration in this study. Besides, with actual economic and financial data, as we sample more frequently, we obtain new information on short cycle events. In other words, shorter cycle events add new sources of noise to the series. This will impact the size and power of the cointegration tests. However, the current Monte Carlo method does not allow this to happen when the sampling of data for simulations is going from high frequency to low frequency by dropping or aggregating monthly observations to obtain quarterly or annual data based on the same sample. I thank an anonymous referee for pointing out this shortcoming.

(6.) These are more general and more realistic cases than those studied by Hakkio and Rush (1991).

(7.) Detailed investigation of this issue is beyond the scope of the present study. Smith (1998) uses Monte Carlo simulation techniques to examine the effects of linearly interpolating some of the variables in the framework of the Johansen cointegration estimation and testing methodology. He found that linear interpolation does not introduce any bias into the estimates of the cointegrating vectors. Although the greater the number of variables that are interpolated, and the smaller the sample size, the more that bias in the cointegration rank test becomes a problem, linear interpolation of one annual variable to match a group of quarterly variables does not seriously bias the rank test statistics even with a sample as short as 20 years.

(8.) A number of existing cointegration studies, including those listed in footnote 1, apply She Johansen tests to a small sample of annual data using the asymptotic critical values. They may have been subject to this problem.

References

Bahmani-Oskooee, Mohsen. 1996. Decline of the Iranian rial during the postrevelutionary period: A productivity approach. Journal of Developing Areas 30:477-92.

Cheung, Yin-Wong, and Kon S. Lai. 1993. Finite-sample sizes of Johansen's likelihood ratio tests for cointegration. Oxford Bulletin of Economics and Statistics 55:313-28.

Cheung, Yin-Wong, and Kon S. Lai. 1995. Lag order and critical values of the augmented Dickey-Fuller test. Journal of Business and Economic Statistics 13:277-SO.

Engle, Robert F., and Clive W. J. Granger. 1987. Co-integration and error correction: Representation, estimation and testing. Econometrica 55:251-76.

Hakkio, Craig S., and Mark Rush. 1991. Cointegration: How short is the long-run? Journal of International Money and Finance 10:571-81.

Hoffman, Dennis, and Su Zhou. 1998. Testing for cointegration in models with alternative deterministic trend specifications: Pre-specifying portions of the cointegration space. Unpublished paper, Arizona State University and the University of Texas at San Antonio.

Horvath, Michael T. K., and Mark W. Watson. 1995. Testing for cointegration when some of the cointegrating vectors are prespecified. Econometric Theory 11:984-1014.

Johansen, Soren. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231-54.

Johansen, Soren, and Katarina Juselius. 1990. Maximum likelihood estimation and inference on cointegration--With applications to the demand for money. Oxford Bulletin of Economics and Statistics 52:169-210.

MacKinnon, James G. 1991. Critical values for cointegration tests. In Long-run economic relationships: Readings in cointegration, edited by Engle, Robert F. and Clive W. J. Granger, New York: Oxford University Press, pp. 267-76.

Masih, Abul M., and Rumi Masih. 1996. Empirical tests to discern the dynamic causal chain in macroeconomic activity: New evidence from Thailand and Malaysia based on a multivariate cointegration/vector error-correction modeling approach. Journal of Policy Modeling 18:531-60.

Osterwald-Lenem, Michael. 1992. A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank tests statistics. Oxford Bulletin of Economics and Statistics 54:313-28.

Shiller, Robert J., and Pierre Perron. 1985. Testing the random walk hypothesis: Power versus frequency of observation. Economics Letters 18:381-6.

Smith, Scott F. 1998. Cointegration tests when data are linearly interpolated. Unpublished paper, State University of New York at Albany.

Taylor, Mark P. 1995. Modeling the demand for U.K. broad money. The Review of Economics and Statistics 75:112-7.

                      Values for [rho] and [theta]
                   Implied Values for [b]
[[rho].sub.M] [a]   Quarterly [rho]        Annual [rho]
0.8000                   0.5120               0.0687
0.9000                   0.7290               0.2824
0.9283                   0.8000               0.4096
0.9500                   0.8574               0.5404
0.9655                   0.9655               0.6561
0.9816                   0.9457               0.8000
0.9913                   0.9740               0.9000
1.0000                   1.0000               1.0000
[[rho].sub.M] [a]  Quarterly [theta]  Annual [theta]
0.8000                  -0.2138          -0.1882
0.9000                  -0.2192          -0.2422
0.9283                  -0.2200          -0.2530
0.9500                  -0.2204          -0.2584
0.9655                  -0.2206          -0.2620
0.9816                  -0.2207          -0.2640
0.9913                  -0.2208          -0.2646
1.0000                  -0.2208          -0.2647

(a.)[[rho].sub.M] IS monthly [rho].

(b.)Implied values for quarterly [rho] and annual [rho] are [([[rho].sub.M]).sup.3] and [([[rho].sub.M]).sup.12], respectively. Implied values for [theta] are calculated based on Equation 3 with s = 3 for quarterly [theta] and s = 12 for annual [theta].

                  Power of 5% Level Tests with a Fixed
                       Time Span of 30 Years [a]
        Fre-         ADF                  Johansen
p       quency       p = 0  p = 2  p = 4  p = 0     p = 2  p = 4
0.8000  M            1.00   1.00   1.00   1.00      1.00   0.999
0.5120  [Q.sub.end]  1.00   0.999  0.951  1.00      0.969  0.733
0.0687  [A.sub.end]  0.982  0.452  0.219  0.983     0.152  0.046
0.5120  [Q.sub.av])  1.00   0.997  0.931  0.998     0.944  0.687
0.0687  [A.sub.av])  0.976  0.427  0.212  0.665     0.112  0.042
0.9000  M            0.995  0.984  0.954  0.957     0.878  0.794
0.7290  [Q.sub.end]  0.990  0.883  0.741  0.915     0.649  0.406
0.2824  [A.sub.end]  0.856  0.349  0.193  0.608     0.115  0.045
0.7290  [Q.sub.av]   0.969  0.866  0.709  0.622     0.597  0.383
0.2824  [AV.sub.av]  0.783  0.331  0.185  0.282     0.092  0.043
0.9283  M            0.895  0.850  0.788  0.698     0.591  0.523
0.8000  [Q.sub.end]  0.867  0.686  0.556  0.633     0.423  0.271
0.4096  [A.sub.end]  0.690  0.283  0.173  0.402     0.085  0.044
0.8000  [Q.sub.av]   0.782  0.670  0.535  0.306     0.383  0.260
0.4096  [A.sub.av]   0.570  0.270  0.165  0.160     0.079  0.044
0.9500  M            0.589  0.560  0.510  0.362     0.312  0.279
0.8574  [Q.sub.end]  0.561  0.443  0.373  0.329     0.242  0.167
0.5404  [A.sub.emd]  0.449  0.216  0.147  0.231     0.079  0.046
0.8574  [Q.sub.av]   0.469  0.438  0.355  0.145     0.222  0.165
0.5404  [A.sub.av]   0.339  0.211  0.143  0.088     0.067  0.045
0.9655  M            0.312  0.302  0.283  0.183     0.163  0.152
0.9000  [Q.sub.end]  0.303  0.258  0.230  0.168     0.139  0.105
0.6561  [A.sub.end]  0.268  0.160  0.123  0.133     0.064  0.046
0.9000  [Q.sub.av]   0.236  0.261  0.224  0.081     0.130  0.106
0.6561  [A.sub.av]   0.194  0.161  0.122  0.057     0.061  0.046
0.9816  M            0.122  0.129  0.126  0.086     0.078  0.076
0.9457  [Q.sub.end]  0.125  0.118  0.114  0.081     0.074  0.064
0.8000  [A.sub.end]  0.121  0.101  0.091  0.070     0.055  0.048
0.9457  [Q.sub.av]   0.099  0.122  0.112  0.048     0.072  0.063
0.8000  [A.sub.av]   0.087  0.104  0.094  0.043     0.053  0.048
0.9913  M            0.071  0.073  0.071  0.062     0.057  0.057
0.9740  [Q.sub.end]  0.071  0.067  0.070  0.057     0.055  0.052
0.9000  [A.sub.end]  0.072  0.067  0.072  0.056     0.051  0.049
0.9740  [Q.sub.av]   0.057  0.069  0.070  0.044     0.054  0.051
0.9000  [A.sub.av]   0.055  0.070  0.089  0.039     0.052  0.048
        HW
p       p = 0  p = 2  p = 4
0.8000  1.00   1.00   1.00
0.5120  1.00   0.999  0.952
0.0687  0.989  0.356  0.099
0.5120  1.00   0.997  0.927
0.0687  0.926  0.288  0.088
0.9000  0.998  0.994  0.976
0.7290  0.997  0.919  0.743
0.2824  0.895  0.275  0.096
0.7290  0.932  0.885  0.706
0.2824  0.622  0.214  0.089
0.9283  0.950  0.911  0.850
0.8000  0.926  0.753  0.568
0.4096  0.738  0.225  0.086
0.8000  0.684  0.707  0.538
0.4096  0.408  0.176  0.078
0.9500  0.693  0.654  0.589
0.8574  0.666  0.511  0.364
0.5404  0.502  0.172  0.081
0.8574  0.376  0.468  0.358
0.5404  0.232  0.142  0.072
0.9655  0.393  0.383  0.342
0.9000  0.381  0.306  0.242
0.6561  0.296  0.134  0.070
0.9000  0.196  0.280  0.226
0.6561  0.139  0.112  0.067
0.9816  0.150  0.153  0.145
0.9457  0.152  0.135  0.119
0.8000  0.130  0.085  0.059
0.9457  0.085  0.172  0.111
0.8000  0.069  0.078  0.060
0.9913  0.075  0.080  0.074
0.9740  0.077  0.073  0.071
0.9000  0.071  0.063  0.054
0.9740  0.052  0.070  0.068
0.9000  0.050  0.060  0.053

(a.)The simulation results are based on 10,000 replications. M represents monthly data of 360 observations. [Q.sub.end] and [Q.sub.av] are for end-of-period and average quarterly data of 120 observations, respectively. [A.sub.end] and [A.sub.av] are for end-of-period and average annual data of 30 observations, respectively. p is the number of lags used in the model. ADF; Johansen, and HW represent the augmented Dickey-Fuller tests, the Johansen tests, and the Horvath-Watson tests, respectively. All tests are size adjusted so that each test has the same rejection frequency of 5% when the null hypothesis is true. Also see notes to Table 1.

               Empirical Size for 5% and 10% Level Tests
                 with a Fixed Time Span of 30 Years [a]
                 ADF                  Johansen                HW
Frequency        p = O  p = 2  p = 4  p = O     p = 2  p = 4  p = 0
5% level tests
 M               0.052  0.047  0.044  0.049     0.053  0.053  0.039
 [Q.sub.end]     0.055  0.049  0.041  0.051     0.058  0.070  0.033
 [A.sub.end]     0.081  0.054  0.028  0.059     0.124  0.276  0.017
 [Q.sub.av]      0.018  0.045  0.040  0.046     0.058  0.073  0.037
 [A.sub.av]      0.032  0.046  0.028  0.063     0.138  0.310  0.024
10% level tests
 M               0.097  0.094  0.089  0.101     0.106  0.108  0.088
 [Q.sub.end]     0.105  0.093  0.081  0.102     0.121  0.136  0.081
 [A.sub.end]     0.137  0.093  0.053  0.114     0.208  0.393  0.056
 [Q.sub.av]      0.041  0.091  0.081  0.085     0.119  0.136  0.078
 [A.sub.av]      0.060  0.086  0.052  0.112     0.236  0.431  0.062
Frequency        p = 2  p = 4
5% level tests
 M               0.039  0.041
 [Q.sub.end]     0.039  0.046
 [A.sub.end]     0.037  0.088
 [Q.sub.av]      0.040  0.050
 [A.sub.av]      0.043  0.099
10% level tests
 M               0.090  0.091
 [Q.sub.end]     0.091  0.102
 [A.sub.end]     0.095  0.175
 [Q.sub.av]      0.093  0.105
 [A.sub.av]      0.105  0.193
(a.)See notes to Table 2.
                 Power of 5% Level Tests with Different
                             Time Spans [a]
                              ADF
Span                           M    [Q.sub.end]  [A.sub.end]
p = 0, [[rho].sub.M] = 0.9
   20                        0.846     0.795        0.524
   30                        0.995     0.990        0.856
   40                        1.00      1.00         0.981
   50                        1.00      1.00         0.998
   60                        1.00      1.00         1.00
   80                        1.00      1.00         1.00
  100                        1.00      1.00         1.00
p = 2, [[rho].sub.M] = 0.9
   20                        0.759     0.559        0.178
   30                        0.984     0.883        0.349
   40                        1.00      0.987        0.566
   50                        1.00      0.999        0.754
   60                        1.00      1.00         0.892
   80                        1.00      1.00         0.987
  100                        1.00      1.00         0.999
p = 0, [[rho].sub.M] = 0.95
   20                        0.292     0.286        0.233
   30                        0.589     0.561        0.449
   40                        0.839     0.807        0.682
   50                        0.963     0.952        0.869
   60                        0.996     0.993        0.960
   80                        1.00      0.999        0.998
  100                        1.00      1.00         1.00
                                                     Johansen
Span                         [Q.sub.av]  [A.sub.av]     M
p = 0, [[rho].sub.M] = 0.9
   20                          0.676       0.436      0.616
   30                          0.969       0.783      0.957
   40                          0.999       0.956      0.998
   50                          1.00        0.995      1.00
   60                          1.00        1.00       1.00
   80                          1.00        1.00       1.00
  100                          1.00        1.00       1.00
p = 2, [[rho].sub.M] = 0.9
   20                          0.532       0.165      0.501
   30                          0.866       0.331      0.878
   40                          0.979       0.524      0.991
   50                          0.999       0.693      1.00
   60                          1.00        0.873      1.00
   80                          1.00        0.977      1.00
  100                          1.00        0.997      1.00
p = 0, [[rho].sub.M] = 0.95
   20                          0.211       0.174      0.170
   30                          0.469       0.339      0.362
   40                          0.710       0.550      0.593
   50                          0.899       0.762      0.821
   60                          0.974       0.906      0.947
   80                          0.999       0.991      0.999
  100                          1.00        1.00       1.00
Span                         [Q.sub.end]  [A.sub.end]  [Q.sub.av]
p = 0, [[rho].sub.M] = 0.9
   20                           0.543        0.253       0.261
   30                           0.915        0.608       0.622
   40                           0.995        0.880       0.903
   50                           1.00         0.979       0.990
   60                           1.00         0.998       1.00
   80                           1.00         1.00        1.00
  100                           1.00         1.00        1.00
p = 2, [[rho].sub.M] = 0.9
   20                           0.288        0.055       0.263
   30                           0.649        0.115       0.597
   40                           0.897        0.234       0.851
   50                           0.986        0.397       0.980
   60                           0.999        0.595       0.997
   80                           1.00         0.871       1.00
  100                           1.00         0.981       1.00
p = 0, [[rho].sub.M] = 0.95
   20                           0.156        0.109       0.078
   30                           0.329        0.231       0.145
   40                           0.553        0.404       0.244
   50                           0.792        0.624       0.428
   60                           0.931        0.807       0.641
   80                           0.997        0.977       0.920
  100                           1.00         0.998       0.974
                                          HW
Span                         [A.sub.av]    M    [Q.sub.end]
p = 0, [[rho].sub.M] = 0.9
   20                          0.114     0.912     0.863
   30                          0.282     0.998     0.997
   40                          0.541     1.00      1.00
   50                          0.794     1.00      1.00
   60                          0.933     1.00      1.00
   80                          0.998     1.00      1.00
  100                          1.00      1.00      1.00
p = 2, [[rho].sub.M] = 0.9
   20                          0.055     0.829     0.587
   30                          0.092     0.994     0.919
   40                          0.200     1.00      0.993
   50                          0.329     1.00      1.00
   60                          0.493     1.00      1.00
   80                          0.805     1.00      1.00
  100                          0.955     1.00      1.00
p = 0, [[rho].sub.M] = 0.95
   20                          0.058     0.371     0.350
   30                          0.088     0.693     0.666
   40                          0.148     0.910     0.884
   50                          0.253     0.987     0.981
   60                          0.394     0.999     0.998
   80                          0.721     1.00      1.00
  100                          0.929     1.00      1.00
Span                         [A.sub.end]  [Q.sub.av]  [A.sub.av]
p = 0, [[rho].sub.M] = 0.9
   20                           0.558       0.582       0.283
   30                           0.895       0.932       0.622
   40                           0.990       0.996       0.879
   50                           1.00        1.00        0.976
   60                           1.00        1.00        0.998
   80                           1.00        1.00        1.00
  100                           1.00        1.00        1.00
p = 2, [[rho].sub.M] = 0.9
   20                           0.112       0.535       0.099
   30                           0.275       0.885       0.214
   40                           0.512       0.990       0.453
   50                           0.740       1.00        0.648
   60                           0.901       1.00        0.846
   80                           0.989       1.00        0.976
  100                           0.999       1.00        0.981
p = 0, [[rho].sub.M] = 0.95
   20                           0.248       0.182       0.114
   30                           0.502       0.376       0.232
   40                           0.760       0.597       0.408
   50                           0.917       0.821       0.616
   60                           0.980       0.942       0.796
   80                           1.00        0.998       0.970
  100                           1.00        1.00        0.998
p = 2,
[[rho].sub.M] = 0.95
   20                 0.278  0.235  0.125  0.229  0.116  0.149  0.117
   30                 0.560  0.443  0.216  0.438  0.211  0.312  0.242
   40                 0.792  0.672  0.358  0.649  0.323  0.528  0.395
   50                 0.939  0.868  0.496  0.845  0.447  0.762  0.622
   60                 0.990  0.960  0.663  0.948  0.596  0.911  0.799
   80                 1.00   0.999  0.884  0.997  0.847  0.995  0.972
  100                 1.00   1.00   0.977  1.00   0.955  1.00   0.998
p = 2,
[[rho].sub.M] = 0.95
   20                 0.054  0.112  0.055  0.338  0.249  0.087  0.233
   30                 0.079  0.222  0.067  0.654  0.511  0.172  0.468
   40                 0.138  0.364  0.120  0.868  0.749  0.315  0.724
   50                 0.212  0.600  0.189  0.973  0.911  0.490  0.891
   60                 0.338  0.774  0.275  0.997  0.980  0.681  0.969
   80                 0.591  0.963  0.523  1.00   0.999  0.904  0.999
  100                 0.827  0.997  0.766  1.00   1.00   0.985  1.00
p = 2,
[[rho].sub.M] = 0.95
   20                 0.079
   30                 0.142
   40                 0.286
   50                 0.424
   60                 0.615
   80                 0.852
  100                 0.966

(a.)The simulation results are based on 10,000 replications. M represents monthly data of 12 X s observations. s is the span of the data. [Q.sub.end] and [Q.sub.av] are for end-of-period and average quarterly data of 4 X s observations, respectively. [A.sub.end] and [A.sub.av] are for end-of-period and average annual data of s observations, respectively. p is the number of lags used in the model. Also see notes to Tables 1 and 2.

                  Empirical Sizes for 5% and 10% Level
                  Tests with Different Time Spans [a]
                         ADF
Span                      M    [Q.sub.end]  [A.sub.end]  [Q.sub.av]
p = 0, 5% level tests
  20                    0.057     0.065        0.105       0.023
  30                    0.052     0.055        0.081       0.018
  40                    0.052     0.058        0.077       0.018
  50                    0.055     0.055        0.071       0.019
  60                    0.049     0.052        0.065       0.018
  80                    0.053     0.055        0.064       0.020
 100                    0.049     0.050        0.060       0.015
p = 2, 5% level tests
  20                    0.050     0.050        0.054       0.046
  30                    0.047     0.049        0.054       0.045
  40                    0.051     0.050        0.053       0.047
  50                    0.050     0.051        0.052       0.049
  60                    0.048     0.048        0.051       0.047
  80                    0.051     0.049        0.052       0.049
 100                    0.047     0.047        0.049       0.045
p = 0, 10% level tests
  20                    0.107     0.118        0.165       0.051
  30                    0.097     0.105        0.137       0.041
  40                    0.099     0.105        0.131       0.042
  50                    0.102     0.105        0.124       0.041
  60                    0.100     0.102        0.118       0.039
  80                    0.102     0.104        0.116       0.042
 100                    0.101     0.104        0.111       0.039
                                    Johansen
Span                    [A.sub.av]     M      [Q.sub.end]  [A.sub.end]
p = 0, 5% level tests
  20                      0.048      0.052       0.054        0.070
  30                      0.032      0.049       0.051        0.059
  40                      0.028      0.054       0.056        0.062
  50                      0.025      0.053       0.053        0.056
  60                      0.022      0.049       0.050        0.054
  80                      0.022      0.055       0.054        0.056
 100                      0.018      0.054       0.055        0.059
p = 2, 5% level tests
  20                      0.051      0.056       0.070        0.218
  30                      0.046      0.053       0.058        0.124
  40                      0.047      0.056       0.062        0.099
  50                      0.048      0.055       0.056        0.084
  60                      0.049      0.051       0.056        0.079
  80                      0.046      0.055       0.059        0.071
 100                      0.048      0.054       0.059        0.068
p = 0, 10% level tests
  20                      0.085      0.107       0.111        0.133
  30                      0.060      0.101       0.102        0.114
  40                      0.058      0.105       0.106        0.123
  50                      0.051      0.105       0.103        0.113
  60                      0.045      0.102       0.104        0.108
  80                      0.046      0.108       0.108        0.113
 100                      0.039      0.110       0.108        0.110
                                                 HW
Span                    [Q.sub.av]  [A.sub.av]    M    [Q.sub.end]
p = 0, 5% level tests
  20                      0.047       0.070     0.041     0.034
  30                      0.046       0.063     0.039     0.033
  40                      0.047       0.062     0.041     0.039
  50                      0.046       0.056     0.041     0.038
  60                      0.046       0.054     0.039     0.038
  80                      0.047       0.056     0.046     0.044
 100                      0.045       0.059     0.042     0.041
p = 2, 5% level tests
  20                      0.070       0.246     0.043     0.042
  30                      0.058       0.138     0.039     0.039
  40                      0.064       0.103     0.043     0.043
  50                      0.056       0.087     0.042     0.043
  60                      0.054       0.080     0.043     0.041
  80                      0.056       0.072     0.046     0.043
 100                      0.057       0.065     0.042     0.041
p = 0, 10% level tests
  20                      0.090       0.130     0.086     0.078
  30                      0.085       0.112     0.088     0.081
  40                      0.091       0.109     0.090     0.086
  50                      0.086       0.100     0.090     0.086
  60                      0.084       0.096     0.090     0.086
  80                      0.088       0.094     0.093     0.089
 100                      0.089       0.097     0.092     0.091
Span                    [A.sub.end]  [Q.sub.av]  [A.sub.av]
p = 0, 5% level tests
  20                       0.009       0.034       0.013
  30                       0.017       0.037       0.024
  40                       0.027       0.039       0.028
  50                       0.029       0.040       0.032
  60                       0.029       0.039       0.033
  80                       0.034       0.042       0.038
 100                       0.035       0.039       0.038
p = 2, 5% level tests
  20                       0.031       0.043       0.033
  30                       0.037       0.040       0.043
  40                       0.038       0.043       0.039
  50                       0.041       0.045       0.042
  60                       0.038       0.042       0.038
  80                       0.041       0.043       0.041
 100                       0.038       0.042       0.039
p = 0, 10% level tests
  20                       0.040       0.074       0.047
  30                       0.056       0.078       0.062
  40                       0.065       0.082       0.068
  50                       0.070       0.081       0.073
  60                       0.073       0.082       0.076
  80                       0.078       0.080       0.077
 100                       0.081       0.080       0.076
p = 2, 10% level tests
  20                    0.103  0.096  0.094  0.093  0.089  0.113
  30                    0.094  0.093  0.093  0.091  0.086  0.106
  40                    0.097  0.098  0.096  0.085  0.091  0.111
  50                    0.097  0.096  0.098  0.093  0.089  0.105
  60                    0.095  0.095  0.094  0.092  0.091  0.103
  80                    0.101  0.099  0.099  0.093  0.092  0.111
 100                    0.098  0.096  0.098  0.094  0.093  0.109
p = 2, 10% level tests
  20                    0.130  0.323  0.160  0.360  0.092  0.094
  30                    0.121  0.208  0.119  0.236  0.090  0.091
  40                    0.121  0.176  0.118  0.179  0.093  0.090
  50                    0.114  0.156  0.111  0.156  0.092  0.094
  60                    0.111  0.144  0.108  0.149  0.090  0.091
  80                    0.114  0.138  0.111  0.137  0.093  0.089
 100                    0.112  0.130  0.109  0.128  0.093  0.092
p = 2, 10% level tests
  20                    0.095  0.095  0.112
  30                    0.095  0.093  0.105
  40                    0.091  0.091  0.098
  50                    0.092  0.092  0.098
  60                    0.093  0.093  0.093
  80                    0.090  0.090  0.091
 100                    0.091  0.091  0.090
(a.) See notes to Table 4.