2.1. Point Optimal Invariant Tests

Define β = (β₀^y,…, β_p^y, β₀^x′,…, β_p^x′)′ and for any t = 1,…,T, let

where d_t^y = d_t^x = (1,…, t^p)′. 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 m_T = D_⊥′ vec(z₁,…, z_T), where D_⊥ is a matrix whose columns form an orthonormal basis for the orthogonal complement of the column space of (d₁,…, d_T)′. For any θ*, let

where y_t(θ*) satisfies the difference equation y_t(θ*) = Δy_t + θ*y_t−1(θ*) with initial condition y₁(θ*) = y₁ and d_t^y(θ*) is defined analogously. The probability density of m_T 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 P_T(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 P_T(θ) involves the random functional φ_P, the definition of which is given next.

Let R ∈ [0,1), λ ≥ 0, and λ > 0 be given. Let Σ^1/2 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 U_l^λ and D_l on R is suppressed.) Finally, let R_# = (1 − R²)^−1/2 and define

THEOREM 1. Let z_t be generated by (1)–(4). Suppose u_t ∼ 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 P_T(θ) →_d φ_P(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 φ_T(m) whenever m_T, the maximal invariant, equals m. For any given θ and any such φ_T, the probability of rejecting H₀ is ∫ φ_T(m) f_T(m|θ, Σ) dm, where f_T(·|θ, Σ) denotes the probability density of the maximal invariant and the domain of integration is

. A test φ_T is of level α ∈ (0,1) if its size, namely, ∫ φ_T(m) f_T(m|1, Σ) dm, is less than or equal to α. Similarly, a sequence {φ_T} of test functions is said to be asymptotically of level α if

When

on the left-hand side equals lim_T→∞ and the inequality is an equality, {φ_T} is said to be asymptotically of size α.

The test statistic P_T(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(P_T(θ) > c_T^P(θ, α, Σ)), where 1(·) is the indicator function and c_T^P(θ, α, Σ) is such that the test is of size α. This optimality result has an obvious asymptotic analogue. Let the function c^P(·,·,·) be implicitly defined by the relation Pr(φ_P(0;λ, ρ²) > c^P(λ, α, ρ²)) = α. The statistic P_T(θ) is asymptotically POI under local-to-unity asymptotics in the sense that φ_T^P(m_T;λ, α, Σ) = 1(P_T(1 − T⁻¹λ) > c^P(λ, α, ρ²)) maximizes

over all invariant tests asymptotically of level α; that is,

whenever {φ_T} is asymptotically of level α. Moreover,

on the right-hand side equals lim_T→∞ and is given by Pr(φ_P(λ;λ, ρ²) > c^P(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 π^α(λ;ρ²) = Pr(φ_P(λ;λ, ρ²) > c^P(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 φ_T^P(m_T;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 P_T(θ) is

for any θ*. When

, the test that rejects for large values of P_T^y(θ) is more powerful against the specific alternative θ = θ < 1 than any other invariant test of H₀ based solely on y_t, where invariance is with respect to transformations of the form

.

When ρ² = 0, the time series y_t and x_t are independent. In that case, the covariates x_t carry no information about y_t, and the statistics P_T(θ) and P_T^y(θ) are equivalent. In contrast, the rejection regions of the tests based on the statistics P_T(θ) and P_T^y(θ) differ whenever ρ² ≠ 0. These differences persist asymptotically, as P_T^y(θ) →_d φ_P(λ;λ,0) under the assumptions of Theorem 1. Comparing φ_P(λ;λ,0) and φ_P(λ;λ, ρ²), the limiting distribution of P_T(θ) is seen to depend on the covariates x_t only through the parameter ρ². As a consequence, the “quality” of the covariates can be summarized by this scalar parameter.

Figure 1 plots π^0.05(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 P_T(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 P_T(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 P_T(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 L_T equals that of the random variable φ_L(λ;ρ²), where, for any 0 ≤ R < 1,

The test that rejects for large values of L_T is asymptotically equivalent (in an obvious sense) to the test that rejects for large values of the second-order Taylor approximation to P_T(1 − T⁻¹λ), namely,

. This observation suggests that L_T enjoys certain local optimality properties. A sequence {φ_T} 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

where l_T^(q)(m|Σ) = ∂^q log f_T(m|1 − T⁻¹λ, Σ)/∂λ^q|_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 f_T(m|1 − T⁻¹λ, Σ)·f_T(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, L_T is asymptotically LBI in the (stronger) sense defined here.²

THEOREM 2. Let z_t be generated by (1)–(4). Suppose u_t ∼ i.i.d. N(0, Σ) and suppose λ = T(1 − θ) ≥ 0 is fixed as T increases without bound. Then

(a) L_T →_d φ_L(λ;ρ²).

If {φ_T} is asymptotically of size α ∈ (0,1), then

(b)

(c)

where φ_T^L(m_T;α, Σ) = 1(L_T > c^L(α, ρ²)) and Pr(φ_L(0;ρ²) > c^L(α, ρ²)) = α.

The univariate counterpart of L_T is

where

. The statistics L_T and L_T^y are equivalent if and only if ρ² = 0. Moreover, L_T^y →_d φ_L(λ;0) under the assumptions of Theorem 2, so the difference between L_T and L_T^y 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.)

3.1. Feasible Tests

Define the matrices

where the partitioning is in conformity with u_t. Moreover, let ρ² = ω_yy⁻¹ω_xy′Ω_xx⁻¹ω_xy be the squared coefficient of multiple correlation computed from Ω, the long-run covariance matrix of u_t. (Because Ω = E(u_tu_t′) when u_t is white noise, the present definition of ρ² is consistent with that of Section 2.)

Under A1 and local-to-unity asymptotics, L_T(Ω) →_d φ_L(λ;ρ²), so an “autocorrelation robust” version of L_T 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 P_T(θ;Ω). In general, P_T(θ;Ω) suffers from “serial correlation bias” under A1. Specifically, P_T(θ;Ω) →_d φ_P(λ;λ, ρ²) + 2λω_yy.x⁻¹γ_yy.x, where γ_yy.x = γ_yy − ω_xy′Ω_xx⁻¹γ_xy. Let

The statistic Q_T(θ;Ω, Γ) coincides with P_T(θ;Ω) when u_t is white noise, because Γ = 0 in that case. More generally, Q_T(θ;Ω, Γ) corrects P_T(θ;Ω) for serial correlation bias and Q_T(θ;Ω, Γ) →_d φ_P(λ;λ, ρ²) under A1 and local-to-unity asymptotics.

In most (if not all) applications, the tests based on L_T(Ω) and Q_T(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 z_t 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,

φ_L(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 z_t 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, sup_s≥0|k(s)| < ∞, and

, where k(r) = sup_s≥r|k(s)| (for every r ≥ 0).

(ii)

, where

are positive with

and b_T⁻¹ + T^−1/2b_T = 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 z_t 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 z_t 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 A_LS (as is true under both local and fixed alternatives), whereas A2(iv) holds if the matrix A_LS 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 u_t are generated by the bivariate model

where

. Two specifications of c_yy(L) are considered:

corresponding to an AR(1) and an MA(1) model for u_t^y, 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 b_T = 1.3221·T^1/5 (following Andrews, 1991) and

, where

is computed from Andrews's (1991) equation (6.4) (with w_a = 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 y_t.

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)