1. INTRODUCTION
Conventionally, the autocorrelation robust inference relies on the
consistent estimation of the long-run variance of the data. In the
regression context, such an estimator based on the nonparametric kernel
method is often referred to as the heteroskedasticity autocorrelation
consistent (HAC) estimator and is frequently employed to construct
standard errors or the Wald-type test statistics in the presence of serial
correlation of unknown form (see, e.g., Newey and West,
1987; Andrews, 1991). HAC estimation,
however, is known to suffer from small-sample bias that results in size
distortion of the test statistics. Kiefer, Vogelsang, and Bunzel (2000) have recently proposed autocorrelation robust
test statistics standardized by an inconsistent long-run variance
estimator instead of a consistent estimator. Their alternative asymptotic
approximation to the distribution of the test statistic incorporates the
randomness of the (inconsistent) long-run variance estimator and is
considered to have some advantages in improving the size properties
compared to the conventional approach.
Since the influential paper of Phillips (1987), nonparametric long-run variance estimation has
also played an important role in the unit root/cointegration
literature. The nonparametric or semiparametric unit root test designed to
incorporate general serial correlation, however, is known to suffer from
some size distortion for the same reason as the test with HAC estimation
in the standard regression model. Therefore, it seems reasonable to
investigate whether the inconsistent estimation of the long-run variance
provides a useful alternative approach in the unit root test in addition
to in the tests in the regression model. In this paper, we conduct
theoretical and simulation analyses on the effect of consistent and
inconsistent long-run variance estimation in testing for a unit root. In
particular, we focus on a class of nonparametric tests based on the
generalization of the von Neumann (VN) ratio. This class of the unit root
test includes the test considered by Sargan and Bhargava (1983) and Bhargava (1986),
the class of the locally best invariant (LBI) test considered by Nabeya
and Tanaka (1990), the Lagrange multiplier (LM)
test of Schmidt and Phillips (1992), the
modified Sargan–Bhargava (MSB) test considered by Stock (1994, 1999) and Perron and
Ng (1996), and a nonparametric unit root test of
Breitung (2002). Its multivariate extension
includes the cointegration tests considered by Phillips and Ouliaris
(1990), Shintani (2001), and Harris and Poskitt (2004).
The main reason for the choice of the VN ratio test in our analysis,
rather than the more commonly used nonparametric variation of the
Dickey–Fuller type test proposed by Phillips (1987) and Phillips and Perron (1988), is its convenience in considering the
properties of the long-run variance estimation under the null and
alternative hypotheses. In a typical regression framework, the true
long-run variance used to standardize the Wald test statistic is common
under both null and alternative hypotheses. In the test for a unit root,
this is not the case. To be more specific, estimation of a positive
long-run variance of the first differenced observation (or the error term)
is often required for the test statistic to have a nuisance parameter free
limiting distribution under the null hypothesis of a unit root. Under the
alternative hypothesis of a stationary root, however, the long-run
variance of the same variable becomes zero because of overdifferencing. In
contrast, the long-run variance of the variable in level is positive and
finite under the alternative, whereas the corresponding long-run variance
cannot be defined under the null hypothesis. The unit root test statistic
we consider is constructed using the ratio of the long-run variance
estimator of the first differenced series to that of the series in levels.
Because the growth rate of the bandwidth in the kernel estimator is the
key to distinguishing the consistent estimator from the inconsistent
estimator, the various combinations of the bandwidths in the numerator and
denominator in the long-run variance ratio offer a systematic way to
investigate the effect of new approach under both the null and alternative
hypotheses.
The remainder of the paper is organized as follows. Section 2
introduces the long-run variance ratio test for a unit root and derives
its limiting distribution under different assumptions on the growth rate
of bandwidths. The finite-sample size properties of each test are
investigated by a Monte Carlo simulation in Section 3. The power of the
test is studied in Section 4. Some extensions, including the analysis of
cointegration, are considered in Section 5. Finally, concluding remarks
are made in Section 6. Throughout this paper, we use the symbols
to signify weak convergence and convergence in probability, respectively.
All the limits are taken as the sample size T → ∞.
2. THE TEST STATISTICS
Let
{yt}t=1T
be a univariate time series generated by
where ut is a weakly stationary zero-mean
error with a strictly positive long-run variance defined by
where γj =
E(utut−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, yt is the
zero-mean stationary process with the long-run variance
ωy2 = (1 −
α)−2ω2.
Throughout this paper, the long-run variance of the zero-mean series
xt 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
T1/2, with an optimal growth rate being
T1/3 under some moment conditions (Andrews, 1991). When xt
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 ST
≠ 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 Δyt =
yt −
yt−1 for t =
1,…,T. This ratio replaces the sample variances in the
original VN ratio with the sample long-run variances. If
ut 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 ut, 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, ut.
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 yt
(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 ωy2 under the
alternative. CI is the case of the denominator being an inconsistent
estimator of ωy2 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 T1/3 is also
possible in C0, CC, and CI, and theoretical results will not be affected
as long as it provides a consistent estimator. The
T1/3 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, Rμ
corresponds to the test of Sargan and Bhargava (1983) and Rτ corresponds to
the R2 test proposed by Bhargava (1986). Note that Rτ 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
Rμ under CC are equivalent to P*(1,0) and
Pμ*(1,0) of Shintani (2001), respectively. For II,
Rμ 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
Recently, Müller (2005) also considered the effect of an inconsistent
long-run variance estimator in his analysis of the test of stationarity.
We do not consider his test here because its null hypothesis is not a unit
root.
We now introduce the following assumption on the error term.
Assumption 1.
(a)
where δ and B are some positive constants.
(b) εt is i.i.d. with zero mean,
variance σ2, and finite fourth cumulants, and
εs = 0 for s ≤ 0.
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
{yt}t=1T
is generated by (1) with α = 1 and Assumption 1
is satisfied. Then,
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.
THEOREM 2. Suppose that
{yt}t=1T
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
ut to ensure the consistency of the unit root
test. Theorem 2, however, shows that the long-run variance estimator,
using the overdifferenced series Δyt
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 Δyt 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 Δyt
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 εt 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 Rμ and
Rτ, 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
Jc(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,Rμ, and
Rτ 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 (Rμ and
Rτ 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 ut 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 xt, a multivariate generalization of
(2). The test statistic for the null hypothesis of r
cointegration in an n × 1 vector of time series
yt 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
yt =
(y1t,y2t,y3t,y4t,y5t)′
from a vector autoregressive (VAR) model,
where A(L) = diag[1 − αL,1
− αL,1 − αL,1 − L,1
− L] with |α| ≤ 1 and
εt =
(ε1t,ε2t,ε3t,ε4t,ε5t)′
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 εt, 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 Byt 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.
6. CONCLUSION
In this paper, we investigated the properties of the long-run variance
ratio tests for a unit root, a generalization of a test based on the VN
ratio. Our main interest was in evaluating the effect of introducing
inconsistent long-run variance estimation on the size and power of the
test for a unit root.
Based on the results of the Monte Carlo simulation designed to
evaluate the finite-sample property, the unit root tests with an
inconsistent long-run variance estimator were found to have much less size
distortion compared to the tests with a conventional consistent long-run
variance estimator. This finite-sample size improvement, however, came at
the cost of a loss in power. The finite-sample power, in addition to the
local asymptotic power, of the tests with an inconsistent long-run
variance estimator was shown to be much lower than in tests with a
consistent estimator. A further simulation experiment was conducted by
sequentially applying a multivariate version of the long-run variance
ratio test to determine the cointegrating rank. When an inconsistent
long-run variance estimator was used, the frequency of selecting the true
cointegrating rank was found to be dramatically lower than in the
conventional test, reflecting the low power to reject the null hypothesis
with a value that was less than the true cointegrating rank.
These findings resemble the case of the autocorrelation robust test in
the standard regression context, where the test with a better size
property, proposed by Kiefer et al. (2000), has
a lower power compared to the test based on the conventional HAC
asymptotics. However, in the context of the long-run variance ratio test
for a unit root, our simulation result showed that a test that combines
consistent and inconsistent estimators improves the power while
maintaining the good size property of the test based only on inconsistent
estimators. Alternatively, although not pursued in this paper, (i) the
introduction of generalized least squares detrending (Elliott, Rothenberg, and Stock, 1996), (ii) the use of
an AR spectral density estimator instead of a kernel-based estimator
(Berk, 1974; Perron and Ng,
1996), and (iii) the investigation of the role of higher order
terms using an asymptotic expansion (Jansson,
2004) seem to comprise a promising direction in which to extend the
analysis of the effect of introducing an inconsistent long-run variance
estimator on the size and power of the long-run variance ratio test.
APPENDIX: Proofs
Proof of Lemma 1. By using a similar argument as in the
derivation of equation (1) of Kiefer and Vogelsang (2002), we have
where 1A is an indicator function that
takes one when A is true and zero otherwise. █
Proof of Theorem 1. In the proof, the limits on integrals
over the unit interval are omitted. For example,
is written as ∫W2. For the proof of part (a), we
first investigate the asymptotic properties of the numerator of the
long-run variance ratio. Under Assumption 1, we have
.
For the case of the inconsistent rate K = T, applying
Lemma 1 to
yields
where the joint weak convergence results
follow from Lemma 2.1 of Park and Phillips (1988). For the asymptotic properties of the
denominator, first note that
.
For
with M/T → 0, we have
where the second equality follows from
for any j = 1,…,M − 1 because
for any j = 1,…,M − 1. Finally, for
,
applying Lemma 1 to
yields
where the joint weak convergence results for the I(2) process
are from Lemma 2.1 of Park and Phillips (1989)
and
.
The required results for part (a) can be obtained by combining the
appropriate results under C0, CC, CI, and II, because the convergence of
the numerator and the denominator holds jointly and the nuisance parameter
ω2 cancels out for any combination.
For the proof of part (b), let us first note that the numerator of the
demeaned test statistic Rμ is identical to that of
the standard test statistic R and thus its asymptotic property is
already provided in the proof of part (a). For the denominator of
Rμ with M = T, the formula in
Lemma 1 simplifies to
because ST = 0 for the demeaned series
xt = yt −
y. The rest of the proof is similar to part (a) with the
standard Brownian motion replaced by the demeaned Brownian motion. The
proof of (c) is entirely analogous to that of parts (a) and (b) except for
the use of the demeaned Brownian bridge. █
Proof of Theorem 2. We show only the consistency of the
R test because the results for Rμ and
Rτ can be obtained using a similar argument. Let
us write
and maintain the assumption on the initial value y0 =
0. Then, under the fixed alternative, the limiting behavior of the
numerator for the overdifferenced series when K/T
→ 0 is given by
where
.
For the inconsistent rate with K = T, (A.1) in the
proof of Theorem 1 can be also used to obtain
For the denominator, because yt is
stationary, we have
.
Finally, for the inconsistent rate with M = T, applying
Lemma 1 to xt =
yt yields
because
is an I(1) process. Combining the all the results yields
Therefore, all the test statistics diverge in the positive direction
as required. █
