2. THE TEST STATISTICS

Let {y_t}_t=1^T be a univariate time series generated by

where u_t is a weakly stationary zero-mean error with a strictly positive long-run variance defined by

where γ_j = E(u_tu_t−j). For simplicity, the initial condition is set to y₀ = 0. We consider a test for the null hypothesis of α = 1 against the alternative hypothesis of |α| < 1. Therefore, under the alternative hypothesis, y_t is the zero-mean stationary process with the long-run variance ω_y² = (1 − α)⁻²ω².

Throughout this paper, the long-run variance of the zero-mean series x_t is estimated by a nonparametric kernel estimator with the Bartlett kernel,

where K is the bandwidth/lag truncation parameter. As emphasized in Newey and West (1987), this choice of the kernel function ensures nonnegative estimates, and thus the long-run variance ratio test statistic defined subsequently will always be nonnegative. In addition, this long-run variance estimator is known to be consistent when bandwidth K grows at a rate slower than T^1/2, with an optimal growth rate being T^1/3 under some moment conditions (Andrews, 1991). When x_t has a nonzero mean,

where

will provide a consistent estimator. This estimator, however, becomes inconsistent if K grows too fast, for example, at the rate T. In what follows we have a convenient shortcut formula for the Bartlett kernel estimator when its bandwidth equals the sample size.

LEMMA 1. Let

. Then, (2) with K = T can be written as

This lemma generalizes equation (1) of Kiefer and Vogelsang (2002) to the case when S_T ≠ 0 and will be used to derive the main theoretical results of our paper.

The VN ratio is usually defined as the ratio of the sample variances of the first differences and the levels of a time series. The ratio is often applied to regression residuals to conduct the Durbin–Watson test for serial correlation. As a test statistic for a unit root hypothesis, however, we utilize the following generalization of the VN ratio:

where Δy_t = y_t − y_t−1 for t = 1,…,T. This ratio replaces the sample variances in the original VN ratio with the sample long-run variances. If u_t is independent and identically distributed (i.i.d.), the ratio with the choice of K = M = 1 can be used to test the null hypothesis of a unit root. But for the serially correlated u_t, it does not provide the nuisance parameter free distribution under the null. We consider the following combinations of growth rates of K and M that provide asymptotically pivotal test statistics in the presence of serially correlated error, u_t.

The choice of C0 is a combination of the bandwidth growth rates that ensures the numerator providing a consistent estimator of ω² under the null and the denominator providing a consistent estimator of the variance of y_t (or the autocovariance of order zero) under the alternative. With the choice of CC, the numerator provides a consistent estimator of ω² under the null, and the denominator provides a consistent estimator of ω_y² under the alternative. CI is the case of the denominator being an inconsistent estimator of ω_y² under the alternative, whereas the numerator is still a consistent estimator of ω² under the null. Finally, II is the combination in which both the numerator and denominator are inconsistent estimators under the null and alternative, respectively. The relationship between our assumptions on bandwidth and the asymptotic properties of the long-run variance estimators is summarized in Table 1. Note that employing a rate other than T^1/3 is also possible in C0, CC, and CI, and theoretical results will not be affected as long as it provides a consistent estimator. The T^1/3 rate is employed here simply because it is the optimal rate, and this particular rate will be used in the simulation in the next section.

Bandwidth and the long-run variance estimator

When nonzero mean stationarity or trend stationarity is allowed as an alternative hypothesis, a demeaned and detrended version of the unit root test is often employed in practice. The long-run variance ratio test can be also extended to these more empirically relevant cases. Suppose

. The demeaned and detrended test statistics are given by

and

When K = M = 1, R_μ corresponds to the test of Sargan and Bhargava (1983) and R_τ corresponds to the R₂ test proposed by Bhargava (1986). Note that R_τ is based on a detrending procedure that is efficient under the null. Schmidt and Phillips (1992) also showed that, for a Gaussian likelihood, the LM principle leads to these tests. With the choice of C0, the test is equivalent to the nonparametric extension of the VN ratio test considered by Nabeya and Tanaka (1990) and Schmidt and Phillips (1992). It is also equivalent to the MSB test considered by Stock (1994, 1999) and Perron and Ng (1996). The one-dimensional case of the cointegration tests considered by Phillips and Ouliaris (1990), Shintani (2001), and Harris and Poskitt (2004) reduces to the same unit root test under C0. The terms R and R_μ under CC are equivalent to P*(1,0) and P_μ*(1,0) of Shintani (2001), respectively. For II, R_μ is somewhat similar to the Breitung (2002) test based on the variance ratio

where

. Note that BR can be rewritten as

Applying Lemma 1 to (5) under II yields

Therefore, the first term is common between the two test statistics. Nevertheless, two statistics are different with their own limiting distribution as the remaining terms are not negligible. In the simulation, we will also consider BR for the purpose of comparison.¹

We now introduce the following assumption on the error term.

Assumption 1.

(a)

where δ and B are some positive constants.

Under Assumption 1, we have

where [Ts] signifies the integer part of Ts and W(s) denotes a standard Brownian motion defined on C[0,1]. The limiting distribution of the long-run variance ratio test is given in the following theorem.

THEOREM 1. Suppose that {y_t}_t=1^T is generated by (1) with α = 1 and Assumption 1 is satisfied. Then,

where

.

where

.

where

.

The limiting distribution of each test statistic is a function of a Brownian motion or a Brownian bridge. Evidently, this contrasts with the autocorrelation robust test in regression where only the test with an inconsistent long-run variance estimator has a nonstandard limiting distribution. Critical values for the limiting distribution of the long-run variance ratio tests with all the combinations of bandwidth growth rates are provided in Table 2. Numbers are obtained by simulation using an approximation of Brownian motion by partial sums of standard normal random variables with 10,000 steps and 10⁷ iterations. In the following section, we evaluate the finite-sample size property of each test using these asymptotic critical values. Note that the test rejects the null hypothesis for large values of the long-run variance ratio and the critical region is constructed accordingly. The consistency of the tests is also provided in the following theorem.

Critical values

THEOREM 2. Suppose that {y_t}_t=1^T is generated by (1) with |α| < 1 and Assumption 1 is satisfied. Then, for any bandwidth growth rate combinations C0, CC, CI, or II,

for any fixed constant c*.

In practice, the ordinary least squares (OLS) residuals from the regression model (1) are often used to estimate the long-run variance of u_t to ensure the consistency of the unit root test. Theorem 2, however, shows that the long-run variance estimator, using the overdifferenced series Δy_t under the fixed alternative, still provides consistency of the long-run variance ratio tests. This result is based on the fact that the long-run variance estimator based on Δy_t converges to zero from a positive value at a sufficiently slow rate under the alternative. In the simulation that follows, we focus on the case with the long-run variance estimator using Δy_t rather than using the quasi-differenced series from the OLS residuals. Nevertheless, the residual-based long-run variance ratio test seems to be a reasonable alternative to our test.

3. FINITE-SAMPLE SIZE OF THE TESTS

In this section, the finite-sample size properties of each test introduced in the previous section are investigated by a Monte Carlo simulation. We follow previous experimental studies in the unit root testing literature and consider the autoregressive (AR) and moving-average (MA) models to introduce serial correlation in the error term. In particular, our data generating process is (1) with α = 1 using the following three different error structures:

where ε_t is an i.i.d. standard normal random variable, ρ = −0.8, −0.5, 0.5, 0.8, and θ = −0.8,−0.5,0.5,0.8. Initial values y₀, u₀, and ε₀ are set to 0. In all cases we use 10,000 replications. There is fairly general agreement that the data-based bandwidth selection method in the long-run variance estimation has very important effects on improving the finite-sample performance of the semiparametric and nonparametric unit root tests (see, e.g., Stock, 1994; Phillips and Xiao, 1998). For this reason, we use the Andrews (1991) automatic bandwidth selection procedure (designed for the Bartlett kernel) to select K when tests based on C0, CC, and CI are applied to AR(1) and MA(1) errors. For the test with CC, the value of the automatic bandwidth selected for K is also used for M. For i.i.d. errors, we simply use K = 1.

Table 3 reports the rejection frequency of the standard long-run variance ratio test, R, with an asymptotic level of 5% for a sample of five different sizes, T = 25, 50, 100, 200, and 500. For each pair of bandwidth growth rates, the first column shows the empirical size when the unit root process has an i.i.d. error. With the exception of a slight underrejection for the C0/CI case when T = 25, the empirical size of the long-run variance ratio tests is very close to the asymptotic level for all combinations of bandwidth growth rates. The difference among the various choices of bandwidth, however, becomes more evident when the error terms are serially correlated.

Empirical size of the standard test with 5% level

Consistent with the finding by Schwert (1989) for semiparametric unit root tests, the long-run variance ratio tests suffer from size distortion mostly in the case of the near MA unit root (θ = −0.8). Overrejection is observed for all tests, which implies that the tests are too liberal. However, when inconsistent asymptotics are used for both the numerator and the denominator (II), the size distortion becomes smaller, and the empirical size approaches the asymptotic level as sample size increases. In contrast, the size distortion of other tests for the near MA unit root case does not disappear, even for T = 500. The size distortion appears to be largest when the test is based on CC. The problem seems to be less severe when the combination of the consistent and inconsistent estimators (CI) is employed in comparison with the conventional case (C0 and CC) when the sample size increases. For the positively correlated MA error (θ = 0.5,0.8), the tests based on CC, CI, and II have their empirical size quite close to the nominal size. In this case, only the test with C0 has a noticeable size distortion that results in the conservative test.

On the whole, the size distortion seems to be somewhat less severe for the AR errors compared to the MA errors. The empirical size of the test with C0 is smaller than the nominal size for the entire range of AR parameters, which suggests that the test is too conservative. The largest deviation from the nominal size is observed in the test with CC when AR errors are positively correlated (ρ = 0.5,0.8). In contrast to the C0 case that underrejects for all cases, the test with CC overrejects when AR parameters are positive (ρ = 0.5,0.8), but underrejects when AR parameters are negative (ρ = −0.8,−0.5). As in the MA error results, the AR error results again favor the tests that involve inconsistent estimators, namely, the CI and II cases. When the sample size increases, both tests have a size that is very close to the asymptotic level for all values of AR parameters.

Tables 4 and 5 report the same results for R_μ and R_τ, respectively. For the i.i.d. error and positively correlated error, the size performance of the demeaned and detrended tests is very similar to that of the standard case except for a very small sample (T = 25). For the negatively correlated case, the problem of size distortion becomes more severe in general. In particular, rejection frequency increases substantially with CC for the negatively correlated MA error. When II is used, however, stability of size remains for θ = −0.5, and increases in the rejection frequency seems to be very modest, even for θ = −0.8, compared to the other choice of bandwidths. The size performance of BR test is somewhat similar to that of the demeaned test with II.

Empirical size of the demeaned test with 5% level

Empirical size of the detrended test with 5% level

In summary, consistent with the previous findings in the literature, the test based on a consistent nonparametric long-run variance estimator suffers from substantial size distortion when the errors are negatively correlated. In contrast, the empirical size of the test using a pair of inconsistent estimators seems to be very close to nominal size on the whole regardless of the serial correlation structure. Therefore, in terms of stability and accuracy of size, this choice of bandwidth growth seems to be the most effective one, with the combination of consistent and inconsistent estimators the second best.

4. POWER OF THE TESTS

In the previous section, we found that it was possible to improve the size of the long-run variance ratio test for a unit root by introducing inconsistent long-run variance estimators. This section investigates the power properties of the same tests.

First, we consider the limiting distribution under the local alternative α = 1 + T⁻¹c for a particular value of c < 0. As in the case of other unit root tests, the limiting distribution involves the functional of a diffusion

. Under this local alternative, all the asymptotic results in Theorem 1 hold by replacing W(s) with J_c(s). This can be shown by using an argument similar to that of Stock (1999) in his analysis of the local asymptotic power of the MSB test. The local asymptotic power functions of R,R_μ, and R_τ for various bandwidth growth rates based on the 5% level are plotted in Figures 1, 2, and 3, respectively. They are approximated by discrete Gaussian random walks with 500 steps with 10,000 replications. For the standard test (R) in Figure 1, the C0/CC and CI cases have similar power when c is close to zero. The difference between their power and that of the II case is evident even if c is close to zero. Although the power of the CI case is slightly below the C0/CC power function for moderate values of c, both power functions become 1.00 relatively fast for distant alternatives. In contrast, the power for the II case is considerably lower for the entire range of local alternative parameter c. It becomes only about 0.5 even if the c is as small as −24. The local asymptotic power functions of the demeaned and detrended tests (R_μ and R_τ in Figures 2 and 3) show the reduction of the local power by detrending the data compared to their corresponding power for the standard test. However, in terms of the ranking and pattern among the different choices of bandwidth, they are very similar to those of the standard tests. Figure 2 also contains the local asymptotic power of the BR test, which shows higher power relative to the II case but lower power relative to the other cases. In summary, the asymptotic power function of the tests based on the pair of inconsistent estimators (II) is well below other power functions for the entire range of c for all cases.

Asymptotic power functions: standard test.

Asymptotic power functions: demeaned test.

Asymptotic power functions: detrended test.

Second, we investigate the finite-sample power properties using a simulation design similar to the one used in the previous section. We first obtain small-sample size-adjusted critical values based on the results in Tables 3, 4, and 5. Note that the size-adjusted critical values are computed for all combinations of data generating process and sample size. We then generate the data from (1) with α = 0.9 using the same values of the error term used for the α = 1 case and apply the unit root test to the data. The frequencies of the rejection of the null hypothesis using the size-adjusted critical value are reported in Tables 6, 7, and 8. The four main findings from the tables are as follows. First, in agreement with the local asymptotic power result, the finite-sample size-adjusted power of the test with II is much lower than that of the tests with C0, CC, and CI. For example, when the error is i.i.d., the power of the standard, demeaned, and detrended tests is only 0.76, 0.53, and 0.25, respectively, even for the large sample with T = 500. For the same sample size, the power of other tests is 1.00 or at least close to 1.00. Second, the difference among the size-adjusted power of tests with C0, CC, and CI is modest compared to the much lower power of the II case. Among the group of C0, CC, and CI, the test with CC performs somewhat better than the other two when the error is negatively correlated, namely, AR(1) error with ρ = −0.8,−0.5 and MA(1) errors with θ = −0.8,−0.5. Third, with the exception of the detrended test with a large sample (T = 250, 500), the test based on CI shows reasonably good finite-sample power very close to the power of the tests based on C0 and CC. This fact is interesting given the finding of the previous section that the test with CI has a much better size property than tests based on C0 and CC. Fourth, for a large sample, the power of the BR test is higher than the power of the demeaned test with II but is lower than the power of the other demeaned tests.

Size-adjusted power of the standard test with 5% level

Size-adjusted power of the demeaned test with 5% level

Size-adjusted power of the detrended test with 5% level

The summary of this section follows. Both the local asymptotic power and the size-adjusted finite-sample power of the long-run variance ratio tests based on the pair of inconsistent estimators (II) are found to be dramatically lower than those of the test based on the pair of consistent estimators (C0 and CC). This suggests that the stability of size in the II case seems to be too costly to justify the usefulness of the inconsistent long-run variance estimator in the long-run variance ratio test for a unit root. However, at the same time, the tests based on the combination of consistent and inconsistent long-run variance estimators (CI) are found to have reasonable power and also a good size. Therefore, among the various pairs of bandwidth growth rates in the long-run variance ratio, the choice of CI may have some practical use in testing for a unit root.

5. EXTENSIONS

5.1. Response Surface Analysis

In the previous section, some specific rates of the bandwidth growth were chosen to represent the tests with consistent and inconsistent long-run variance estimators. For the case of the numerator of the test statistics, the bandwidth can be generally written as

where k > 0 and 0 < δ ≤ 1. In particular, δ = 1/3 is used in C0, CC, and CI to represent the rate for the consistent estimator. This rate is known to minimize the mean square error of the long-run variance estimator when k = 1.1447 × (f⁽¹⁾/f⁽⁰⁾)² where f⁽ⁱ⁾ is the ith derivative of the spectral density of u_t at frequency zero (Andrews, 1991). Theoretically, as long as the selected growth rate of the bandwidth provides a consistent long-run variance estimator, the limiting distribution of the test statistic under the null and its local asymptotic power do not depend on the rate or the choice of a constant k. Even so, the choice may have some effects on the small-sample performance of the test. For the case of the inconsistent long-run variance estimator in II, the choice of δ = 1 and k = 1 (K = T) is used because it has the simplest form in the sense that it does not require any truncation in the kernel estimation. Unlike the consistent case, however, even with a common growth rate T, the test statistics based on K = kT, with 0 < k < 1, will have different limiting distribution (and thus local asymptotic power) depending on the choice of a constant k. For these reasons it is of interest to see the sensitivity of the simulation results to the choice of parameters δ and k in (9). A similar argument can also be made with the choice of bandwidth M in the denominator of the test statistics. Here we conduct a simple response surface analysis of the (finite-sample) power of the test with various bandwidths that includes the cases of CC and II.

The simulation design is identical to the one used for the analysis of the finite-sample power in Section 4. For simplicity, we use the same bandwidth for the numerator and denominator (K = M) and report only the results for the demeaned test in the case of T = 100 and i.i.d. error. Both parameters, δ and k, vary from 0.05 to 1.0. Figure 4 shows the size-adjusted power of the test as a function of δ and k.

Size-adjusted power: demeaned test (T = 100).

The power turns out to be the lowest at 0.11 when both δ and k are the largest (δ = 1 and k = 1). Note that this number corresponds to the case of II with T = 100 and i.i.d. error in Table 7. For a fixed value of δ, the power increases as k decreases. Similarly, for a fixed value of k, the power increases monotonically as δ decreases. Finally, it also shows that no combination of δ and k provides a power higher than 0.52, the value obtained in Table 7 for the case of CC.

5.2. Cointegration

The long-run variance ratio test for a unit root can be generalized to test for cointegration (or cointegrating rank) in a multivariate system. Let

be the long-run variance covariance matrix of an n-dimensional vector x_t, a multivariate generalization of (2). The test statistic for the null hypothesis of r cointegration in an n × 1 vector of time series y_t then can be constructed by using the sum of the (n − r) smallest eigenvalues of

. With a bandwidth growth rate that provides a consistent long-run covariance matrix estimator, both theoretical and small-sample properties of the test have already been considered in several studies, including Shintani (2001) and Harris and Poskitt (2004). For example, in the case of the demeaned test for no cointegration (r = 0), the statistic under the null weakly converges to a random variable

where

is n-vector standard Brownian motion. Limiting distribution of the test statistic based on the inconsistent long-run covariance matrix estimator can also be obtained as a multivariate generalization of Theorem 1. Again, for the demeaned test, random variables in the limit are

and

respectively, for CI and II, where

. In this section, we investigate how the findings for the univariate case obtained in the previous sections can be generalized to the multivariate case. In particular, we are interested in the effect of introducing the inconsistent estimator on the determination of the cointegrating rank in finite samples.

We follow Harris and Poskitt (2004) in simulation design and generate five-dimensional vector series y_t = (y_1t,y_2t,y_3t,y_4t,y_5t)′ from a vector autoregressive (VAR) model,

where A(L) = diag[1 − αL,1 − αL,1 − αL,1 − L,1 − L] with |α| ≤ 1 and ε_t = (ε_1t,ε_2t,ε_3t,ε_4t,ε_5t)′ is an i.i.d. multivariate normal random variable. An equicorrelation matrix with correlation coefficient 0.8 is used as a variance covariance matrix of ε_t, and a vector of 0's is used as an initial value y₀. Because all the test statistics we consider here are invariant to any transformation of the form By_t where B is any 5 × 5 nonsingular matrix, the simulation results based on (10) cover a fairly general case of cointegration, including a linear transformation considered by Harris and Poskitt (2004). However, in general, the bandwidth selected by the automatic procedure is not invariant to a linear transformation. For this reason, we employ here a fixed rule for selecting bandwidth in long-run covariance matrix estimation. In particular, we follow Schwert (1989) and use K = [4(T/100)^1/3] where [x] is an integer part of x. The sample size T varies from 25 to 500 as in the univariate case. The cointegrating rank is determined by successively testing from the hypothesis of r = 0 to the hypothesis of r = 4 in case each hypothesis is not rejected. The asymptotic critical values for the multivariate version of the demeaned tests based on C0, CC, CI, II, and BR are obtained by approximating the limiting distribution using a multivariate normal random variable with 1,000 steps and 10⁵ iterations.

Table 9 reports the relative frequencies of selected cointegrating ranks by sequentially applying the asymptotic 5% level tests in 10,000 replications. The left side of the table shows the result with α = 1, the case of no cointegration (r = 0). The right side of the table shows the result with α = 0.8, the case when the cointegrating rank is three (r = 3). The asymptotic theory predicts that the probability of selecting the true cointegrating rank converges to 95% whereas the probability of selecting the smaller rank converges to zero.

Cointegrating rank selection using the demeaned test with 5% level

When α = 1, all the tests, except the one based on CC, select the true cointegrating rank (r = 0) with a very high frequency for all the sample sizes under consideration. Among all the tests, the one based on II stands out as the frequency of selecting rank zero is closest to the theoretical value 95%, even if the sample size is as small as T = 25 or 50. The BR test shows the second-best finite-sample performance. The test based on CC often selects r = 1 for the sample sizes of T = 25 and 50, but the frequency of selecting r = 0 dramatically increases for a larger sample size. The notable difference in the finite-sample performance reveals the fact that the good size property of the test based on an inconsistent estimator (II) and size distortion in the test based on a consistent estimator (CC) observed in the univariate case are also present in the five-dimensional case.

When α = 0.8, no test selects the true cointegrating rank (r = 3) when the sample size is less than T = 100. The frequencies of selecting r = 3 increase as the sample size increases. In particular, the tests based on C0, CC, and CI perform reasonably well when the sample size becomes as large as T = 500. Among these three tests, the one based on CC dominates the other two and selects the true rank more than three times as often as other tests when T = 250. In contrast, the test based on II underestimates r for almost all the cases. This poor performance of the test based on II can be explained by its low power to reject the null when the value of r in the hypothesis is smaller than its true value, which occurs in the early stage of the sequential cointegrating rank selection procedure. The BR test performs better than the II case but is dominated by other tests.

From this additional experiment on cointegration, we find that the better finite-sample property associated with lower finite-sample power in the unit root test using an inconsistent long-run variance estimator can be well generalized to the multivariate case.