1. INTRODUCTION
Let yt be an observed univariate time
series generated by
where μty is
deterministic component and
vty is an unobserved
error process with initial condition
v1y =
u1y and generating mechanism
where uty is a
stationary I(0) process. (In this paper, a process is said to
be I(0) if its partial sum process converges weakly to a
Brownian motion.)
The problem of testing the null hypothesis H0 :
θ = 1 against H1 : θ < 1 has attracted
considerable attention in the literature, as has the closely related
problem of testing for parameter constancy in the
“local-level” unobserved components model. Pertinent
references include LaMotte and McWorther (1978), Nyblom and Mäkeläinen (1983), Nyblom (1986),
Nabeya and Tanaka (1988), Tanaka (1990), Kwiatkowski, Phillips, Schmidt, and Shin
(1992), Saikkonen and Luukkonen (1993a, 1993b), Choi
(1994), and Leybourne and McCabe (1994). (For a review, see Stock, 1994.) Under H0,
vty =
uty and
yt is a (trend-) stationary process,
whereas yt is an integrated process with a
random walk–type nonstationarity under the alternative
hypothesis. For this reason, tests of H0 are often
referred to as stationarity tests. The cited papers differ somewhat
with respect to the assumptions on the underlying stationary process
uty and the form of
the deterministic component
μty. On the other hand, all
previous studies (of which the author is aware) have been concerned
with the situation where yt is observed in
isolation. Specifically, all previously devised tests have exploited
only the information contained in yt when
testing H0.
In applications, it is extremely rare that individual time series are
observed in isolation. As a consequence, it seems reasonable to ask
whether more powerful stationarity tests can be obtained be utilizing
the information contained in related time series. To fix ideas, suppose
a k-vector time series xt of
covariates is observed, whose generating mechanism is
where μtx is
deterministic component and
utx is an unobserved
stationary I(0) process. Moreover, suppose the deterministic
components μty and
μtx are pth-order
polynomial trends; that is, suppose
where
are unknown parameters.
The present paper proposes two new tests that exploit the information
contained in the covariates xt when
testing the null hypothesis that yt is
(trend-) stationary. Both tests are valid under mild moment and memory
conditions on ut =
(uty,utx′)′ and enjoy optimality
properties in the special case where ut is
Gaussian white noise. The tests can be viewed as generalizations of
existing univariate stationarity tests, and the new tests dominate
their univariate counterparts in terms of asymptotic local power
whenever the zero-frequency correlation between
uty and
utx is nonzero. (When
the zero-frequency correlation equals zero, the new tests coincide with
their univariate counterparts.) In fact, substantial power gains can be
achieved if an appropriate set of covariates
xt can be found. The paper therefore
provides an affirmative answer to the question posed in the beginning
of the previous paragraph. Results complementary to those obtained here
can be found in Hansen (1995) and Elliott and
Jansson (2003). These papers demonstrate the
usefulness of covariates in the context of testing for an
autoregressive unit root.
Section 2 derives the tests and establishes their asymptotic
optimality properties in the special case where the underlying
innovation sequence is Gaussian white noise. In Section 3, the tests
are extended to accommodate general stationary errors by means of
nonparametric corrections. Section 4 shows how the tests can be applied
to test the null hypothesis that a vector integrated process is
cointegrated with a prespecified cointegration vector and presents an
empirical illustration. Finally, Section 5 offers a few concluding
remarks, and all proofs are collected in the Appendix.
2. TESTING WITH WHITE NOISE ERRORS
Let (yt,
xt′)′ be generated by
(1)–(4) and suppose
,
where
is a known, positive definite matrix (partitioned in conformity with
ut). Consider the problem of testing
This problem is that of testing whether the (permanent) component
is absent from the following permanent-transitory decomposition of
yt:
To see how the use of stationary covariates
xt facilitates the testing problem,
consider the series
,
whose permanent-transitory decomposition is
where
.
Because xt is stationary, the transformation
does not affect the permanent component. On the other hand,
Var(uty.x) =
(1 −
ρ2)Var(uty),
so the transformation reduces the variance of the transitory component
by a fraction ρ2, where
is the squared coefficient of multiple correlation computed from Σ.
The covariates xt can therefore be used to
attenuate the transitory component of yt
without affecting the permanent component. As a consequence, the use of
covariates makes it easier to detect the permanent component of
yt if it is present, thereby leading to
improvements in power relative to the case where the covariates are
ignored. The remainder of this section makes these heuristic ideas more
precise.
2.1. Point Optimal Invariant Tests
Define β = (β0y,…,
βpy,
β0x′,…,
βpx′)′ and for
any t = 1,…,T, let
where dty =
dtx = (1,…,
tp)′. Using this notation, the model
can be written as
The problem of testing H0 : θ = 1 vs.
H1 : θ < 1 is invariant under the group of
transformations of the form
.
A maximal invariant is mT =
D⊥′
vec(z1,…, zT),
where D⊥ is a matrix whose columns form an
orthonormal basis for the orthogonal complement of the column space of
(d1,…,
dT)′. For any θ*, let
where yt(θ*) satisfies
the difference equation
yt(θ*) =
Δyt +
θ*yt−1(θ*)
with initial condition y1(θ*) =
y1 and
dty(θ*)
is defined analogously. The probability density of
mT is proportional to
where, for any θ*,
By the Neyman–Pearson lemma, the test that rejects for large
values of
is the most powerful invariant test of θ = 1 against the
specific alternative θ = θ.
Theorem 1 characterizes the limiting distribution of
PT(θ) under a
local-to-unity reparameterization of θ and θ in
which λ = T(1 − θ) ≥ 0 and λ
= T(1 − θ) > 0 are held constant as
T increases without bound. The limiting representation of
PT(θ) involves the random
functional φP, the definition of which is
given next.
Let R ∈ [0,1), λ ≥ 0, and λ
> 0 be given. Let Σ1/2 be the (lower
triangular) Cholesky factor of the 2 × 2 matrix
and for l ∈ {0, λ}, define
,
and (V,W)′ is a Brownian motion with covariance
matrix Σ. (Here, and elsewhere, the dependence of
Ulλ and
Dl on R is suppressed.) Finally,
let R# = (1 −
R2)−1/2 and define
THEOREM 1. Let zt be generated by
(1)–(4). Suppose ut ∼
i.i.d. N(0, Σ) and suppose
λ = T(1 − θ) ≥ 0 and λ =
T(1 − θ) > 0 are fixed as T
increases without bound. Then
PT(θ)
→d
φP(λ;λ,
ρ2).
Corresponding to any invariant (possibly randomized) test of
H0 : θ = 1 there is a test function
such that H0 is rejected with probability
φT(m) whenever
mT, the maximal invariant, equals
m. For any given θ and any such
φT, the probability of rejecting
H0 is ∫ φT(m)
fT(m|θ, Σ)
dm, where
fT(·|θ, Σ) denotes
the probability density of the maximal invariant and the domain of
integration is
.
A test φT is of level α ∈ (0,1) if its
size, namely, ∫ φT(m)
fT(m|1, Σ)
dm, is less than or equal to α. Similarly, a sequence
{φT} of test functions is said to be
asymptotically of level α if
When
on the left-hand side equals limT→∞ and
the inequality is an equality, {φT} is said to
be asymptotically of size α.
The test statistic
PT(θ) is point optimal
invariant (POI) in the sense that the power
against the point alternative θ = θ is
maximized over all invariant tests of level α by the test function
1(PT(θ) >
cTP(θ,
α, Σ)), where 1(·) is the indicator function and
cTP(θ,
α, Σ) is such that the test is of size α. This optimality
result has an obvious asymptotic analogue. Let the function
cP(·,·,·) be
implicitly defined by the relation
Pr(φP(0;λ, ρ2)
> cP(λ, α,
ρ2)) = α. The statistic
PT(θ) is asymptotically
POI under local-to-unity asymptotics in the sense that
φTP(mT;λ,
α, Σ) = 1(PT(1 −
T−1λ) >
cP(λ, α,
ρ2)) maximizes
over all invariant tests asymptotically of level α; that is,
whenever {φT} is asymptotically of level
α. Moreover,
on the right-hand side equals limT→∞ and
is given by
Pr(φP(λ;λ,
ρ2) > cP(λ,
α, ρ2)).
Theorem 2 of Saikkonen and Luukkonen (1993a) obtains an upper bound on the asymptotic
power function of any location and scale invariant stationarity test in
the univariate case. Because scale invariance is not imposed, the
result stated here covers a larger class of tests than Theorem 2 of
Saikkonen and Luukkonen (1993a) even in the
univariate case. (The present paper obviates the need to impose scale
invariance by assuming that Σ is known.) Moreover, the multivariate
model studied here contains the univariate model of Saikkonen and
Luukkonen (1993a) as a special case.
The function πα(λ;ρ2) =
Pr(φP(λ;λ, ρ2) >
cP(λ, α, ρ2))
provides an upper bound on the asymptotic power function of any
invariant test asymptotically of level α. The bound is sharp in the
sense that it can be attained for any given λ by the test
φTP(mT;λ,
α, Σ). Moreover, although no test statistic attains the upper
bound uniformly in λ, it turns out that it is possible to construct
tests whose power functions are very close to the bound. The Gaussian
power envelope therefore constitutes a useful benchmark against which
the power function of any invariant test (asymptotically of level
α) can be compared.
The univariate counterpart of
PT(θ) is
for any θ*. When
,
the test that rejects for large values of
PTy(θ)
is more powerful against the specific alternative θ =
θ < 1 than any other invariant test of
H0 based solely on yt,
where invariance is with respect to transformations of the form
.
When ρ2 = 0, the time series
yt and xt are
independent. In that case, the covariates
xt carry no information about
yt, and the statistics
PT(θ) and
PTy(θ)
are equivalent. In contrast, the rejection regions of the tests based
on the statistics PT(θ)
and
PTy(θ)
differ whenever ρ2 ≠ 0. These differences persist
asymptotically, as
PTy(θ)
→d
φP(λ;λ,0) under the
assumptions of Theorem 1. Comparing
φP(λ;λ,0) and
φP(λ;λ,
ρ2), the limiting distribution of
PT(θ) is seen to depend
on the covariates xt only through the
parameter ρ2. As a consequence, the
“quality” of the covariates can be summarized by this
scalar parameter.
Figure 1 plots
π0.05(λ;ρ2) for selected values of
ρ2 in the constant mean (p = 0) case. (The
curves were generated by taking 20,000 draws from the distribution of
the discrete approximation [based on 2,000 steps] to the
limiting random variables.) The lowest curve corresponds to
ρ2 = 0 and therefore provides an upper bound on the
(local asymptotic) power function of any invariant univariate
stationarity test. An increase in the quality of the covariates (as
measured by ρ2) leads to an increase in the level of the
power envelope. Indeed, the difference between the power envelope and
its univariate counterpart is quite remarkable for most values of
ρ2. For concreteness, consider the alternative λ =
5, which corresponds to a moving average coefficient θ of 0.975
when T = 200. The univariate power envelope is 0.32, whereas
the envelopes are 0.40 and 0.58 when ρ2 equals 0.2 and
0.5, respectively. Because they are upper bounds, these power envelopes
do not by themselves illustrate the power gains attainable by feasible
tests. On the other hand, the evidence presented in Figure 1 clearly suggests that substantial power
gains can be achieved by including covariates in a stationarity test
provided an appropriate set of covariates can be found. The power
envelopes are lower in the linear trend (p = 1) case, but the
qualitative conclusion remains the same, as can be seen from Figure 2.
Power envelopes: 5% level tests, constant mean (p =
0).
Power envelopes: 5% level tests, linear trend (p =
1).
2.2. Locally Best Invariant Tests
Even asymptotically, the critical region of the test based on
PT(1 −
T−1λ) depends on
λ. As a consequence, no test is asymptotically
uniformly most powerful (with respect to the class of invariant tests)
in the sense of Basawa and Scott (1983). In
such cases, tests based on weaker optimality concepts seem worth
considering. One such concept, the concept of point optimality,
justifies the test based on PT(1 −
T−1λ[dagger]),
where λ[dagger] is a prespecified
alternative against which maximal power is desired. As an alternative
to that test, the present section develops a test based on a Taylor
series expansion of PT(1 −
T−1λ) around
λ = 0. The resulting test can be implemented without
specifying an alternative in advance and enjoys certain local
optimality properties.
Using simple algebra, it can be shown that
where
.
(The dependence of
has been suppressed to achieve notational economy, and the notation
recognizes the fact that
does not depend on Σ.)
Under the assumptions of Theorem 1,
.
As a consequence, the limiting distribution of
is degenerate:
.
On the other hand, Theorem 2(a), which follows, shows that under the
same assumptions the limiting distribution of
LT equals that of the random variable
φL(λ;ρ2), where, for any 0
≤ R < 1,
The test that rejects for large values of
LT is asymptotically equivalent (in an
obvious sense) to the test that rejects for large values of the
second-order Taylor approximation to PT(1
− T−1λ), namely,
.
This observation suggests that LT enjoys
certain local optimality properties. A sequence
{φT} of tests is asymptotically locally
efficient (with respect to the class of invariant tests asymptotically
of size α) in the sense of Basawa and Scott (1983) if it maximizes
over all invariant tests asymptotically of size α. As Theorem
2(b) shows, any invariant test (asymptotically of size α) is
asymptotically locally efficient according to that definition.1
In fact, the conclusion of Theorem 2(b) holds
whenever {φT} is asymptotically of level
α.
To obtain a nontrivial characterization of local
optimality in the present context, the following alternative concept of
asymptotic local optimality is useful. Let
q* be
the smallest integer
q such that
where
lT(q)(m|Σ)
= ∂q log
fT(m|1 −
T−1λ,
Σ)/∂λq|λ=0.
An invariant test is said to be asymptotically locally best invariant
(LBI) if it maximizes
over all invariant tests asymptotically of the same size. In regular
cases where partial derivatives of ∫ log
fT(m|1 −
T−1λ,
Σ)·fT(m|1,
Σ) dm with respect to λ can be obtained by
differentiating under the integral sign, this concept of local
asymptotic optimality agrees with that of Basawa and Scott (1983) when q* = 1. The testing
problem studied here has q* = 2 and as Theorem 2(c)
shows, LT is asymptotically LBI in the
(stronger) sense defined here.2
An
alternative sufficient condition for the conclusion of Theorem 2(c) is
that {φT} is asymptotically of level α and
α ≤ Pr(φL(0;ρ2) >
E(φL(0;ρ2))).
THEOREM 2. Let zt be generated by
(1)–(4). Suppose ut ∼
i.i.d. N(0, Σ) and suppose
λ = T(1 − θ) ≥ 0 is fixed as T
increases without bound. Then
(a) LT →d
φL(λ;ρ2).
If {φT} is asymptotically of
size α ∈ (0,1), then
(b)
(c)
where
φTL(mT;α,
Σ) = 1(LT >
cL(α, ρ2)) and
Pr(φL(0;ρ2) >
cL(α, ρ2)) = α.
The univariate counterpart of LT is
where
.
The statistics LT and
LTy are equivalent if
and only if ρ2 = 0. Moreover,
LTy
→d φL(λ;0) under
the assumptions of Theorem 2, so the difference between
LT and
LTy persists
asymptotically whenever ρ2 ≠ 0. As was the case with
the power envelopes derived in the previous section, the inclusion of
covariates can have a substantial effect on the power properties of the
LBI test. (This will become apparent in Section 3.2.)
3. TESTING WITH WEAKLY DEPENDENT ERRORS
The analysis in the previous section proceeded under the restrictive
assumption that
,
where Σ is known. The optimality theory seems to depend on the
normality assumption. On the other hand, it is straightforward to
construct feasible test statistics having limiting representations of
the form φP and φL
under much less stringent assumptions on
ut. For instance, the following assumption
suffices.
A1.
,
where
has full rank, and
,
where ∥·∥ is the Euclidean norm.
3.1. Feasible Tests
Define the matrices
where the partitioning is in conformity with
ut. Moreover, let ρ2 =
ωyy−1ωxy′Ωxx−1ωxy
be the squared coefficient of multiple correlation computed from
Ω, the long-run covariance matrix of
ut. (Because Ω =
E(utut′)
when ut is white noise, the present
definition of ρ2 is consistent with that of Section
2.)
Under A1 and local-to-unity asymptotics,
LT(Ω) →d
φL(λ;ρ2), so an
“autocorrelation robust” version of
LT can be obtained by employing the
long-run covariance matrix Ω in the definition of the test
statistic. Analogously, an autocorrelation robust POI test can be based
on PT(θ;Ω). In
general, PT(θ;Ω)
suffers from “serial correlation bias” under A1.
Specifically,
PT(θ;Ω)
→d
φP(λ;λ, ρ2)
+
2λωyy.x−1γyy.x,
where γyy.x =
γyy −
ωxy′Ωxx−1γxy.
Let
The statistic
QT(θ;Ω, Γ)
coincides with
PT(θ;Ω) when
ut is white noise, because Γ = 0 in
that case. More generally,
QT(θ;Ω, Γ)
corrects PT(θ;Ω) for
serial correlation bias and
QT(θ;Ω, Γ)
→d
φP(λ;λ, ρ2)
under A1 and local-to-unity asymptotics.
In most (if not all) applications, the tests based on
LT(Ω) and
QT(θ;Ω, Γ) are
infeasible because Ω and Γ are unknown. It therefore seems
natural to consider the test statistics
,
where
are estimators of Ω and Γ, respectively.
THEOREM 3. Let zt be generated by
(1)–(4). Suppose A1 holds and suppose λ = T(1
− θ) ≥ 0 and λ = T(1 −
θ) > 0 are fixed as T increases without
bound. If
.
Conventional (possibly prewhitened) kernel estimators of Ω and
Γ (e.g., Andrews, 1991; Andrews and Monahan, 1992) meet the consistency
requirement of Theorem 3. Conditions under which VAR(1) prewhitened
kernel estimators are consistent are provided in Section 3.3.
The statistics
and
are univariate counterparts of
,
respectively. Under the assumptions of Theorem 3,
φL(λ;0). The test statistic
is well known (e.g., Kwiatkowski et al.,
1992). On the other hand, the semiparametric version
of the univariate POI test would appear to be new.
3.2. Asymptotic Power Properties
Saikkonen and Luukkonen (1993a) considered
the constant mean case and found that their test statistic
,
which corresponds to
,
has a local asymptotic power function that is almost indistinguishable
from the univariate power envelope. The choice λ = 7
produces a test that is asymptotically 0.50-optimal, level 0.05 in the
sense of Davies (1969). In other words, λ
= 7 is the alternative for which the univariate power envelope for 5%
level tests equals 0.50. In the general case, it therefore seems
natural to consider
,
where λ[dagger] is such that the test statistic
is asymptotically 0.50-optimal, level 0.05. Although computationally
feasible, such a procedure seems cumbersome in view of the fact that
the power envelope for 5% level tests depends not only on the order of
the deterministic component in the model but also on the parameter
ρ2, which measures the quality of the covariates. To
construct test statistics that are asymptotically 0.50-optimal, level
0.05 one would therefore have to use a new
λ[dagger] for each ρ2.
Fortunately, a much simpler approach yields very satisfactory results.
The approach taken here is to use the same
λ[dagger] for all values of ρ2.
The value of λ[dagger] is chosen in such a way
that the test is asymptotically 0.50-optimal, level 0.05 in the worst
case scenario ρ2 = 0, the case where the univariate test
is optimal. This approach generates a test that has excellent power
properties (relative to the power envelope) when ρ2 is
low. Moreover,
dominates its univariate counterpart for all values of
ρ2. In fact, the test has a power function that is very
close to the power envelope even for nonzero values of
ρ2.
Figure 3 illustrates this in the constant
mean case with ρ2 = 0.50. In addition to the power
envelope and the local asymptotic power of
,
Figure 3 also plots the local power function
of the LBI test
and the univariate tests
.
Comparing
,
it is seen that the inclusion of covariates can lead to huge gains in
power in cases where an appropriate set of covariates can be found. The
Pitman asymptotic relative efficiency (ARE) of
with respect to
(evaluated at power 0.50) is 1.65, implying that in large samples the
univariate test needs 65% more observations than the test using
covariates to have comparable power properties when ρ2 =
0.50. The case where covariates are included is qualitatively similar
to the univariate case in the sense that the POI test dominates the LBI
test for all but extremely small values of λ. Indeed, the
inferiority (as measured by the Pitman ARE) of the LBI test is somewhat
more pronounced when useful covariates are available.
Power curves, ρ2 = 0.5: 5% level tests, constant mean
(p = 0).
Figure 4 presents results for the linear
trend case. The statistics
use λ[dagger] = 12, the value that yields an
asymptotically 0.50-optimal, level 0.05 test in the univariate case.
All power curves lie below the curves for the constant mean case, but
the pattern is the same as in Figure 3. In
particular, the statistic
has a power function that lies close to the envelope and far above the
power functions corresponding to
.
For instance, the Pitman ARE of
with respect to
(evaluated at power 0.50) is 1.82, indicating that the inclusion of
covariates is even more beneficial in the linear trend case than in the
constant mean case.
Power curves, ρ2 = 0.5: 5% level tests, linear trend
(p = 1).
Tables 1 and 2
give various critical values for
for p ∈ {0,1}, which seem to be the cases of empirical
relevance. In the case of
,
the critical values correspond to the recommended values of
λ[dagger], namely,
λ[dagger] = 7 when p = 0 and
λ[dagger] = 12 when p = 1. The
critical values are presented for ρ2 in steps of 0.1.
The recommendation is to use the critical value corresponding to
computed from
.
Interpolation can be used to obtain critical values for values of
between those given in the tables.
Percentiles of
and
(1
− 7/T), constant mean case (p = 0)
Percentiles of
and
(1
− 12/T), linear trend case (p = 1)
In general, point optimal and locally optimal tests may fail to be
consistent in curved statistical models (van
Garderen, 2000). In view of the following fixed parameter
result, the tests based on
are consistent if
are well behaved under fixed alternatives.
THEOREM 4. Let zt be generated by
(1)–(4). Suppose A1 holds and suppose θ < 1 and
λ = T(1 − θ) > 0 are
fixed as T increases without bound. If
,
then
for any c ∈ R.
3.3. Covariance Matrix Estimation
Under fairly general conditions, the requirements of Theorems 3 and 4
are met by VAR(1) prewhitened kernel estimators with plug-in
bandwidths. These estimators are defined as follows.
For t = 2,…,T, let
,
where
is a (k + 1) × (k + 1) matrix and
.
Define
where k(·) is a kernel and
is a sequence of (possibly sample-dependent) bandwidth parameters. The
proposed estimators of Ω and Γ are
respectively. Consider the following assumption.
A2.
(i) k(0) = 1, k(·) is continuous at zero,
sups≥0|k(s)|
< ∞, and
,
where k(r) =
sups≥r|k(s)|
(for every r ≥ 0).
(ii)
,
where
are positive with
and bT−1 +
T−1/2bT =
o(1).
(iii)
for some A such that (I − A) is
nonsingular.
(iv) The matrix A in (iii) is block upper triangular.
Assumption A2(i) is discussed in Jansson (2002), whereas Assumptions A2(ii) and (iii) are
adapted from Andrews and Monahan (1992).
Assumption A2(iv) is helpful when studying the behavior of
under fixed alternatives. When
are standard kernel estimators and A2(iii) and (iv) are trivially
satisfied. A nondegenerate prewhitening matrix satisfying A2(iii) is
discussed subsequently.
LEMMA 5. Let zt be generated by
(1)–(4). Suppose A1 and A2(i)–(iii) hold and suppose λ
= T(1 − θ) ≥ 0 and λ =
T(1 − θ) > 0 are fixed as T
increases without bound. Then
.
LEMMA 6. Let zt be generated by
(1)–(4). Suppose A1 and A2 hold and suppose θ < 1 and
λ = T(1 − θ) > 0 are
fixed as T increases without bound. Then
.
Under local alternatives (i.e., under the assumptions of Theorem 3
and Lemma 5), A2(iii) is satisfied by the least squares estimator
On the other hand, standard cointegration arguments can be used to
show that the first column of
converges at rate T to first unit vector in
under fixed alternatives (i.e., under the assumptions of Theorem 4 and
Lemma 6). As a consequence,
violates A2(iii) under fixed alternatives.
An estimator
satisfying A2(iii) under both local and fixed alternatives can be
obtained by modifying
as follows. Let
be the Jordan decomposition of
.
Define
,
where
is a Jordan matrix obtained from
by dividing the diagonal elements of each Jordan block by
max(1,|μ|/0.97), where μ is the eigenvalue
(real or complex) associated with the Jordan block and
|·| denotes absolute value. This adjustment
preserves the eigenvectors of
and bounds the eigenvalues of
away from unity. By construction,
whenever the eigenvalues of
do not exceed 0.97. More generally, the properties of
are easily deduced once the properties of
have been established. In particular,
satisfies A2(iii) whenever
for some ALS (as is true under both local
and fixed alternatives), whereas A2(iv) holds if the matrix
ALS is block upper triangular (as is the
case under fixed alternatives). Lemmas 5 and 6 therefore demonstrate
the plausibility of the high-level assumptions on
made in Theorems 3 and 4, respectively.
3.4. Finite Sample Properties
To investigate the finite sample properties of the test statistics
introduced in Section 3.1, a small Monte Carlo experiment is conducted.
Samples of size T = 200 are generated according to
(1)–(4). The errors ut are generated
by the bivariate model
where
.
Two specifications of cyy(L) are
considered:
corresponding to an AR(1) and an MA(1) model for
uty, respectively. In
both cases,
In particular, the parameter ρ in (8) is the correlation
coefficient computed from Ω.
The parameters Ω and Γ are estimated using VAR(1)
prewhitened kernel estimators. Specifically,
are constructed using the quadratic spectral kernel (which clearly
satisfies Assumption A2(i)) along with a plug-in bandwidth. The value
of the plug-in bandwidth is obtained by setting
bT =
1.3221·T1/5 (following Andrews, 1991) and
,
where
is computed from Andrews's (1991)
equation (6.4) (with wa = 1 for all
a). Because
is imposed, A2(ii) is automatically satisfied. In particular, the condition
controls the behavior of the estimated bandwidth under fixed
alternatives, thereby circumventing the problems discussed by Choi
(1994). Finally, the matrix
used in the prewhitening procedure was computed by modifying the
ordinary least squares (OLS) estimator in the manner described in
Section 3.3.
Tables 3 and 4
and 5 and 6 summarize the results for the constant mean and linear
trend cases, respectively. The tables report the observed
rejection rates of 5% level tests implemented using critical values
based on the estimate
computed from
.
As was the case with the asymptotic analysis of Section 3.2, the
simulation evidence is favorable to the tests developed in this paper.
The rejection rates of the new tests are quite similar to those of
their univariate counterparts under the null hypothesis. No noticeable
loss in power is observed in the case where the covariates are
uninformative (when ρ2 = 0), whereas substantial power
gains are achieved in the cases where the covariates do carry
information about yt.
Monte Carlo rejection rates (AR(1) model, 5% level tests, constant
mean, T = 200)
Monte Carlo rejection rates (MA(1) model, 5% level tests, constant
mean, T = 200)
Monte Carlo rejection rates (AR(1) model, 5% level tests, linear
trend, T = 200)
Monte Carlo rejection rates (MA(1) model, 5% level tests, linear
trend, T = 200)
In addition to documenting the superiority of the new tests, the
simulation evidence also points out some problems with the small sample
properties of the new tests and their univariate counterparts.
Rejection rates under the null tend to fall far short of the nominal
level in the MA(1) model with |b| ≥ 0.5,
which leads to an unnecessary reduction in power when asymptotic
critical values are used. Likewise, power is very low in the AR(1)
model with a = 0.8, especially so for the point optimal tests.
Moreover, the pattern exhibited by the rejection rates in the AR(1)
model with a = 0.8 is rather peculiar. In part, the latter
phenomenon appears to be due to imprecision of the estimates of Ω
and Γ, because simulation results (not reported here) show that the
power of the infeasible tests using the true values of Ω and Γ
is monotonic in θ. It follows from Theorem 4 that the low power in
the AR(1) model with a = 0.8 is a finite sample phenomenon. In
an attempt to quantify the effect of a change in the sample size for
moderate values of T, Tables 7 and
8 investigate the power against the (fixed)
alternative θ = 0.9 for T ∈ {200,300,400,500} in the
AR(1) model with a = 0.8. As the sample size increases, power
increases in all cases but remains disappointingly low in the case of
the point optimal test. Indeed, even in samples of size T =
500 the point optimal test fails to dominate the locally optimal test.
As a consequence, the locally optimal test is likely to be superior to
the point optimal test in cases where the time series is believed to be
highly persistent under the null hypothesis.
Monte Carlo rejection rates (AR(1) model, a = 0.8, θ =
0.9, 5% level tests, constant mean)
Monte Carlo rejection rates (AR(1) model, a = 0.8, θ =
0.9, 5% level tests, linear trend)
4. COINTEGRATION TESTING WITH A
PRESPECIFIED COINTEGRATION VECTOR
An example of the applicability of the tests proposed in this paper
can be obtained from the theory of cointegrated time series. Suppose
(Yt,
Xt′)′ is a (k +
1)-vector integrated process generated by the cointegrated system
where Yt is a scalar,
Xt is a k-vector,
μtY and
μtX are deterministic
components, and
(utY,utX′)′ satisfies A1. Setting
yt = Yt
− ψ′Xt,
μty =
μtY −
ψ′μtX,
xt = ΔXt,
and μtx =
μtX, the cointegration
model reduces to (1)–(4) with
(uty,utx′)′ =
(utY,utX′)′ and θ = 1. In
this context, the null hypothesis θ = 1 is the hypothesis that
(Yt,
Xt′)′ is cointegrated with
cointegrating vector (1, −ψ′)′, whereas the
alternative θ < 1 is the hypothesis that
(Yt,
Xt′)′ is not cointegrated.
In many applications, the (potentially) cointegrating vector (1,
−ψ′)′ is known a priori from economic theory
(e.g., Horvath and Watson, 1995; Zivot, 2000).3
The
stationarity tests considered here cannot be used to test the null
hypothesis of cointegration if the (potentially) cointegrating vector
is unknown. For that testing problem, Shin (1994), Choi and Ahn (1995), and Nyblom and Harvey (2000) propose consistent tests, whereas Jansson
(2003) derives a Gaussian power envelope and
develops (nearly) efficient tests.
In such cases, the null
hypothesis that (
Yt,
Xt′)′ is cointegrated with
cointegrating vector (1, −ψ′)′ is invariably
tested by applying a univariate stationarity test to the series
Yt −
ψ′
Xt, thereby discarding the
potentially useful information contained in the series
Δ
Xt. As indicated by the results of
the previous sections, this empirical practice may lead to a dramatic
and unnecessary reduction in power in situations where the
zero-frequency correlation between Δ
Xt
and
Yt −
ψ′
Xt is nonzero. In economic
applications, such nonzero correlations are the rule rather than the
exception.
4 When interpreted as tests of the null hypothesis
of cointegration with a prespecified cointegrating vector, the
stationarity tests proposed in the present paper therefore seem much
more attractive than their univariate counterparts currently used in
empirical work.
As an illustration, the tests are used to examine the relevance of
long-run purchasing power parity (PPP). Specifically, the bilateral
intercountry relationship between the United States, the domestic
country, and the United Kingdom, the foreign country, is considered.
The aim is to test the following version of the PPP hypothesis (e.g.,
Froot and Rogoff, 1995):
where st is the logarithm of domestic
currency price of a unit of foreign exchange,
ptD and
ptF are the logarithms
of the price indices in the domestic and foreign countries, and
ut is a stationary error term capturing
deviations from PPP. In this setup, a rejection of the null hypothesis
of cointegration is interpreted as evidence against long-run PPP. Upon
imposing the symmetry and proportionality restriction
ψD = −ψF = 1,
the problem reduces to that of testing whether the real exchange rate
st −
ptD +
ptF is
(trend-)stationary. The data consist of st
− ptD +
ptF and
(ΔptD,
ΔptF), where the
inflation rates
ΔptD and
ΔptF serve as
covariates.
The tests are implemented using quarterly data from the Global
Financial Database (GFD). The exchange rate data is from GFD series
__GBP_D, and the price series are consumer price indices. Prices for
the United States and the United Kingdom are from GFD series CPUSAM and
CPGBRM, respectively. When implementing the tests, the nuisance
parameters are estimated in the same way as in the Monte Carlo
experiment of Section 3.4. The linear trend version of the test
statistics is used. In other words, p = 1 is imposed.5
Empirical tests of long-run PPP are typically
conducted using the constant mean versions of the univariate
stationarity tests. The reasons for not imposing β1 = 0
in (9) are twofold. First, as pointed out to the author by Maurice
Obstfeld, the presence of a deterministic trend component in (9) cannot
be ruled out on theoretical grounds. Indeed, a simple
Harrod–Balassa–Samuelson model (e.g., Obstfeld and Rogoff,
1996, Chap. 4) in which the differential
between productivity growth in tradables and nontradables differs
between the home and foreign countries might produce a nonzero
β1 in (9). Second, the real exchange rate appears to
have a nonconstant mean, suggesting that β1 should be
unrestricted in (9).
Two sample periods are considered. One
sample period, covering the period from January 1900 through January
2001, spans the twentieth century, whereas the other sample period,
covering January 1974 through January 2001, corresponds to the period
of the recent float.
Table 9 summarizes the
results.
In agreement with other studies (e.g., Culver and
Papell, 1999; Kuo and Mikkola, 1999),
the tests fail to reject the null hypothesis of stationarity when the
covariates are ignored. The tests using covariates, in contrast,
provide mixed evidence regarding the validity of long-run PPP. The
locally optimal test based on
rejects the null at the 5% level in both cases, whereas the point
optimal test based on
fails to reject in both cases. To the extent that the stationary
component of st −
ptD +
ptF might be well
approximated by a highly persistent autoregressive process (e.g., Engel, 2000; Kuo and Mikkola,
1999), the fact that
fails to reject is to be expected in view of the simulation results
reported in Section 3.4. The estimates
are large, suggesting that substantial power gains are achieved by using
covariates, which in turn might explain why the
test reaches different conclusions than the univariate tests.
APPENDIX
The proofs of Theorems 1–4 make use of Lemma 7, which shows how
functional laws for sample moments of the transformed data
zt(θ) and
dt(θ) can be deduced from
functional laws for zt and
dt. Because these preliminary results
might be of independent interest, they are presented in greater
generality than needed for the proofs of Theorems 1–4.
In Lemma 7 and elsewhere in the Appendix, [lfloor ]·[rfloor ] denotes
the integer part of the argument, and all functions are understood to
be CADLAG functions defined on the unit interval (equipped with the
Skorohod topology).
LEMMA 7. Let {FTt : 0 ≤ t
≤ T,T ≥ 1} and
{(gTt′,
hTt′)′ : 1 ≤ t
≤ T,T ≥ 1} be triangular arrays of (vector)
random variables with FT0 = 0 for all
T. Let l > 0 be given and define
FTt(l) =
ΔFTt + (1 −
T−1l)FT,
t−1(l),
gTt(l) =
ΔgTt + (1 −
T−1l)gT,
t−1(l), and
hTt(l) =
ΔhTt + (1 −
T−1l)hT,
t−1(l) with initial conditions
FT0(l) =
FT0,
gT1(l) =
gT1, and
hT1(l) =
hT1.
(a) Suppose
where F and G are continuous. Then
jointly with (A.1), where
.
(b) Suppose
jointly with (A.1), where H, ΓFH,
and ΓGH are continuous and H is a
semimartingale. Then
jointly with (A.1)–(A.3), where
.
Proof of Lemma 7. For t = 0,…,T,
FTt(l) can expressed as
This relation can be restated as follows:
Now, limT→∞
sup0≤r≤1|(1 −
T−1l)[lfloor ]Tr[rfloor ]
− exp(−lr)| = 0 and
FT, [lfloor ]T·[rfloor ]
→d F(·), where F is
continuous, so
by the continuous mapping theorem.
Next, using summation by parts,
for t = 1,…,T, where
and GTt(l) =
ΔGTt + (1 −
T−1l)GT,
t−1(l) with initial conditions
GT0(l) =
GT0 = 0. A second application of the proof
of FT,
[lfloor ]T·[rfloor ](l)
→d Fl(·)
yields GT,
[lfloor ]T·[rfloor ](l)
→d Gl(·).
Moreover, using Billingsley (1999, Theorem 13.4),
max1≤t≤T∥GTt(l)
− GT,
t−1(l)∥ →d
0, so
as claimed.
Finally, using (GT,
[lfloor ]T·[rfloor ], gT,
[lfloor ]T·[rfloor ] − gT,
[lfloor ]T·[rfloor ](l))
→d (G(·),
lGl(·)), the continuous mapping
theorem (CMT), and the relation
,
The proof of part (a) is completed by noting that the convergence
results in the preceding displays hold jointly with (A.1).
Using the assumption on
,
part (a), and CMT,
Next,
where the equalities follow from summation by parts and integration
by parts, respectively.
This result, part (a), and CMT can be used to show that
Similar reasoning yields
The convergence results in the preceding displays hold jointly with
(A.1)–(A.3). █
Proof of Theorems 1 and 2. The proof proceeds under the assumptions
of Theorem 3, strengthening A1 only when necessary. Define Ω and
Γ as in Section 3. Let
Because limT→∞
max0≤i≤p
sup0≤r≤1|T−i[lfloor ]Tr[rfloor ]i
− ri| = 0 and
,
where
it follows from Lemma 7 that
where dt[dagger](l)
= dt(1 −
T−1l)·Ω−1/2′,
and Dly(r)
and Dx(r) are defined as in the
text.
Standard weak convergence results (e.g., Phillips
and Solo, 1992; Phillips, 1988; Hansen, 1992) for linear processes can be used to
show that the following hold jointly:
where
is a Brownian motion with covariance matrix
.
By (A.7), Lemma 7, and the relation
,
simple algebra yields
where vt[dagger](l)
= Ω−1/2(zt(1
− T−1l) −
dt(1 −
T−1l)′β) and
Vlλ is defined in terms of
V as in the text. Similarly, using (A.7), (A.8), and Lemma 7,
the following results can be verified:
where ρ# = (1 −
ρ2)−1/2, ρ =
(ωyy−1ωxy′Ωxx−1ωxy)1/2,
and γyy.x =
γyy −
ωxy′Ωxx−1γxy.
The limiting distributions of
PT(θ;Ω) and
LT(Ω) do not depend on k,
the dimension of xt. The remainder of the
proof proceeds under the assumption that k = 1 and δ =
∥δ∥ = ρ, because these assumptions simplify the algebra
without leading to a loss of generality. When k = 1 and δ
= ρ, the processes
coincide with the processes Dl,
Ul and W defined in the text
(with R = ρ). Now,
where
.
By the algebra of OLS, (A.6), and (A.9),
for l ∈ {0, λ}. Using this along with
(A.10) and (A.11) and the relation
it follows that
Because γyy.x = 0 and Σ = Ω
under the assumptions of Theorem 1, the proof of that theorem is now
complete.
Next, LT(Ω) can be written as
LT*(Ω) +
LT**(Ω), where
.
When
coincide with Σ* and Σ** defined in the text.
The result LT(Ω)
→d
φL(λ;ρ2) now follows from
simple algebra and the fact that
under the assumptions of Theorem 3, where
is defined as in the text (with R = ρ). In particular,
Theorem 2(a) follows because Σ = Ω under the assumptions of
Theorem 2.
Under the assumptions of Theorem 2, integrals such as
can be differentiated with respect to λ by differentiating under
the integral sign. As a consequence,
where Var0(·) denotes the variance under
H0. The first inequality uses
|φT| ≤ 1 and the modulus
inequality for integrals, the second inequality uses the
Cauchy–Schwarz inequality, and the last equality uses ∫
l(1)(m|Σ)
fT(m|1, Σ) dm
= 0 and the fact that
l(1)(mT|Σ)
differs from
by an additive constant. Using the fact that
ut is Gaussian white noise, it is easy to
show that
.
Therefore, the
of the left-hand side of the preceding display is zero, as claimed in
Theorem 2(b).
For any T, let
,
where
is such that
By the Neyman–Pearson lemma and the fact that
l(2)(mT|Σ)
−
2T−1l(1)(mT|Σ)
differs from 2LT by an additive
constant,
Moreover, for any sequence {ηT} of bounded
functions,
where the second equality uses ∫
l(1)(m|Σ)2fT(m|1,
Σ) dm = o(1). Combining the preceding displays,
it follows that
The proof of 2(c) can be completed by showing that
which, because
is bounded, holds if
where E0(·) denotes expectation under
H0. Now, using
E0(l(1)(mT|Σ))
= 0 and
and the fact that
l(2)(mT|Σ)
−
2T−1l(1)(mT|Σ)
differs from 2LT by an additive
constant,
where LTμ =
LT −
E0(LT). Using this
relation and
,
Because {φT} is asymptotically of level
α, it can be shown (using Theorem 2(a)) that
.
Therefore,
.
Moreover, {LTμ} is uniformly
integrable under H0, so
as was to be shown. █
Proof of Theorem 3. The proof of Theorems 1 and 2(a) carries over
to the case where Ω and Γ are replaced with consistent estimators
if the following analogues of equations (A.6) and (A.9)–(A.11)
can be established:
where
.
Now,
where the first inequality uses the triangle inequality, the first
equality uses the relation
and (A.6), the second inequality uses the properties of
∥·∥, and the last equality uses (A.6) and the assumption
.
Similar reasoning establishes (A.13)–(A.15). █
Proof of Theorem 4. By the properties of seemingly unrelated
regressions,
does not depend on
:
because dty(1) =
dty =
dtx. Partition
after the first row as
.
Under the assumptions of Theorem 4, it follows from standard results
for linear processes that
and
where
W is a Wiener process, and Dy
is defined as in the text. By (A.17) and CMT,
where
.
For any
,
and the first inequality uses the fact that
is positive definite, whereas the second inequality uses
,
(A.16), (A.18), and the portmanteau theorem (e.g., Billingsley, 1999).
Next, consider
Now,
where
.
Partition
after the first row as
.
The series
satisfies the difference
with initial condition
.
As a consequence,
and the last equality uses
,
whereas the convergence result follows from (A.17), Lemma 7, and CMT.
Now,
By the portmanteau theorem and the fact that the function
Kλ(·,·) is positive
definite in the sense that
for any nonzero, continuous function f (·),
for any
Proof of Lemma 5. Let
utPW =
ut −
Aut−1, where A is the
matrix appearing in A2(iii). The equations defining
are sample counterparts of the relations
Because
under A2(iii), it therefore suffices to show that
.
Let
,
where
.
Let
.
Using notation typified by
can be written as
.
Now,
by Corollary 4 of Jansson (2002). The proof of
is completed by using the relation
and straightforward, but tedious, bounding arguments to show that
.
Indeed, the proof of Lemma 5 of Jansson and Haldrup (2002) carries over to the present case. The details
are omitted for brevity.
Proceeding in analogous fashion, it can be shown that
.
█
Proof of Lemma 6. In view of A2(iii) and (iv), it suffices to show
that
,
where
are defined in the obvious way. Now,
because
.
Moreover,
where the second inequality uses the Cauchy–Schwarz inequality
and the last equality uses
(Jansson, 2002) and
.
Similar reasoning can be used to show that
.
█
