2. THE UNOBSERVED COMPONENTS MODEL

Consider the unobservable components (UC) data generating process (DGP)

under the following set of assumptions (which are taken to hold throughout the paper, except where stated otherwise).

Assumption

. The term {σ_t} satisfies σ_[sT] := ω(s), where

is a nonstochastic function with a finite number of points of discontinuity; moreover, ω(·) > 0 and satisfies a (uniform) first-order Lipschitz condition except at the points of discontinuity. Similarly, except where otherwise stated, σ_η[sT] := ω_η(s), where ω_η(·) satisfies the same conditions as ω(·).

Assumption

. The irregular component {ε_t} is a zero-mean, unit variance, strictly stationary mixing process with E|ε_t|^p < ∞ for some p > 2 and with mixing coefficients {α_m} satisfying

for some r ∈ (2,4], r ≤ p. The long-run variance

is strictly positive and finite. Furthermore, {ε_t} is independent of {η_t} at all leads and lags. As is standard, we refer to {ε_t} as an I(0) process.

Assumption

. The component x_t is a p × 1 deterministic vector satisfying the condition that there exist a scaling matrix δ_T and a bounded piecewise-continuous function F(·) on [0,1] such that δ_T x_[·T] → F(·) uniformly on [0,1], with

positive definite.

From (1) and (2), observe that under Assumption

, the variance of both the irregular component, u_t := σ_tε_t, and the shocks to the level process {μ_t} are heteroskedastic. Consequently, {u_t} is I(0) provided {σ_t} is constant, whereas {μ_t} reduces to a standard random walk if {σ_ηt} is constant and vanishes from (1) when σ_ηt = 0, all t. Notice that the model considered here generalizes the UC model discussed in Kwiatkowski et al. (1992) by allowing both {σ_t} and {σ_ηt} to be potentially nonconstant over time.

Assumption

allows for a wide class of models for the variances of the errors. Models of single or multiple variance shifts satisfy Assumption

with ω(·) piecewise constant. For example, the function

gives the single break model with a variance shift at time [mT], 0 < m < 1, analyzed by Busetti and Taylor (2003) and Cavaliere (2004a). If ω(·)² is an affine function, then the unconditional variance of the errors displays a linear trend. Piecewise-affine functions are also permitted, allowing for variances that follow a broken trend. Moreover, smooth transition variance shifts also satisfy Assumption

: e.g., the function

, which corresponds to a smooth (logistic) transition from σ₀² to σ₁². The parameter m determines the transition midpoint (for t = [mT], σ_t² = 0.5(σ₀² + σ₁²)) whereas γ > 0 controls the speed of transition (the fixed change-point model follows as a limiting case for γ → ∞).

Assumption

is standard and allows for a wide variety of possible forms for the deterministic component, including the pth-order trend function x_t := (1,t,…,t^p)′, 0 ≤ p < ∞. The broken intercept and broken intercept and trend functions considered, e.g., in Busetti and Harvey (2001) are obtained by specifying

respectively, in (1), t_m^j being defined as

satisfying lim_T→∞(m/T) = μ ∈ (0,1) (see Phillips and Xiao, 1998, p. 448).

Remark 1. If ω(·) is not constant then the irregular component, {u_t}, is unconditionally heteroskedastic. Conditional heteroskedasticity is also permitted through Assumption

(see, e.g., Hansen, 1992b). Assumption

has been used extensively in the econometric literature as it allows {ε_t} to belong to a wide class of weakly dependent stationary processes. The strict stationarity assumption is made without loss of generality and may be weakened to allow for weak heterogeneity of the errors, as in, e.g., Phillips (1987).

Remark 2. The assumption of nonstochastic variance functions {ω(·),ω_η(·)} can be easily weakened simply by assuming stochastic independence between {ε_t,η_t} and {σ_t,σ_ηt}, given that the stochastic functionals {ω(·),ω_η(·)} must have sample paths satisfying the requirements of Assumption

. In the stochastic variance framework, the results given in this paper hold conditionally on a given realization of {ω(·),ω_η(·)}.

3. STATIONARITY TESTS

Kwiatkowski et al. (1992) focus on testing the I(0) null hypothesis, H₀ : σ_η² = 0, against the I(1) alternative hypothesis, H₁ : σ_η² > 0, under the ancillary assumption that σ_t = σ,σ_ηt = σ_η, all t, so that, under H₀, {y_t} reduces to the I(0) process y_t = x_t′ β + u_t, t = 1,…,T. Kwiatkowski et al. (1992) propose the test that rejects H₀ for large values of the statistic

where

, the ordinary least squares (OLS) residuals from the regression of y_t on x_t, t = 1,…,T;

is a consistent estimator of the long-run variance of {u_t} under H₀ and has the form

being a bandwidth parameter and k(·) a weighting function. Kwiatkowski et al. (1992) assume

(Bartlett weights). However, because we are dealing with mixing errors (see Assumption

), throughout the paper we will require that q_T and k(·) satisfy the following assumption (de Jong, 2000).

Assumption

. (K₁) For all

is continuous at 0 and for almost all

, where l(x) is a nonincreasing function such that

; (K₂) q_T ↑ ∞ as T ↑ ∞, and q_T = o(T^γ), γ ≤ 1/2 − 1/r, where r is given in

.

Assumption

is sufficiently general for our purposes as it is satisfied by many of the most commonly employed kernels (see Hansen, 1992a; Jansson, 2002).

Remark 3. The

statistic maps the sequence

onto [0,1] by averaging the squared values of the sequence. Other stationarity tests can be obtained by taking different mappings. For example, the supremum of

and range of

deliver, respectively, the test of Xiao (2001) and the rescaled range (RS) test of Lo (1991), which reject H₀ for large values of the statistics

, respectively.

4. ASYMPTOTIC SIZE

Under the null hypothesis considered by Kwiatkowski et al. (1992), H₀ : σ_ηt² = σ_η² = 0, all t, if {σ_t} is constant across the sample, it is well known that (e.g., Kwiatkowski et al., 1992, pp. 164–165)

, where

, with B(·) a standard Brownian motion. For example, if x_t := (1,t,…,t^p−1)′, then F(s) := (1,s,…,s^p−1)′ and V(·) is a pth-level Brownian bridge.

Now, assume that H₀ holds but that σ_t is not necessarily constant over time; rather it satisfies Assumption

. Then, the asymptotic distribution of the

statistic assumes the form detailed in the following theorem.

THEOREM 1. Under H₀ : σ_ηt² = σ_η² = 0, all t,

, where

and where

.

Consequently, with respect to the homoskedastic case, the asymptotic distribution of the

statistic has the usual structure but with B(·) replaced by B_ω(·). It is only where ω(·) is constant throughout the sample that B_ω(·) reduces to a standard Brownian motion and, hence, that

has the standard limiting distribution.

Remark 4. The process B_ω(·) is a diffusion corresponding to the stochastic differential equation dB_ω(s) = (ω(s)/ω)dB(s) with initial condition B_ω(0) = 0. Because B_ω(·) has zero mean, variance

(where Λ_ω(·) is an increasing homeomorphism on [0,1]) and has independent increments, Corollary 29.10 of Davidson (1994) implies that B_ω(·) is distributed as B(Λ_ω(·)), and therefore at time s ∈ [0,1], B_ω(·) has the same distribution as the standard Brownian motion B(·) at time Λ_ω(s) ∈ [0,1]. That is, B_ω(·) is a Brownian motion under modification of the time domain (see, e.g., Revuz and Yor, 1991, p. 170).

Remark 5. Under the conditions of Theorem 1,

. Interestingly, in the case of no deterministic terms (i.e., x_t′ β = 0), because

(see Remark 4), it holds that

,

, and the asymptotic sizes of the

tests are not affected by variance changes that satisfy Assumption

. Simulation evidence reported in Cavaliere and Taylor (2004) suggests that this invariance property also holds reasonably well in small samples.

5. ASYMPTOTIC POWER

In this section we investigate the impact of time-varying variances in the irregular component in (1), and/or the error driving the level equation, (2), on both the consistency and local asymptotic power properties of the tests.

5.1. Consistency

It is well known (e.g., Kwiatkowski et al., 1992, eqn. (25)) that if σ_ηt² = σ_η² > 0, then

where

is a standard Brownian motion independent of B(·). Because q_T /T → 0, (4) implies that

diverges to +∞ at rate O_p(T/q_T) under the I(1) alternative. In addition to this result, note that if the {u_t} component has a time-varying variance,

is still distributed as in (4), because as T → ∞, the I(1) component {η_t} dominates.

Now, consider the general case where σ_ηt² ≠ 0 but is not necessarily constant, satisfying Assumption

. Here the following result holds.

THEOREM 2. If σ_ηt² ≠ 0, all t, the weak convergence (4) holds with W(·) replaced by W_ωη(·), where W_ωη(s) := B_ωη(s) − P_F B_ωη(s) with

.

Consequently, as in the case of constant variances, because q_T /T → 0, Theorem 2 implies that

diverges to +∞ at rate O_p(T/q_T) under global I(1) alternatives.

Remark 6. Under the conditions of Theorem 2,

and

, which imply that both

also diverge to +∞, at rate O_p((T/q_T)^1/2).

5.2. Asymptotic Local Power

We now focus attention on the limiting behavior of the

statistic under the local alternative (see also Busetti and Taylor, 2003, p. 513):

where c ≥ 0 is a noncentrality parameter and λ_εω/ω_η > 0 is a scale factor that simplifies the representation of the asymptotic distributions. Notice that ω/ω_η = 1 if σ_t = σ_ηt, t = 1,…,T; i.e., if the pattern of time variation is common to the variances of the irregular component in (1) and the error driving the level in (2). Moreover, where σ_t = σ and σ_ηt = σ_η, t = 1,…,T, H_c reduces to the local alternative considered by, inter alia, Stock (1994, p. 2799).

The following theorem details the large-sample behavior of

under H_c.

THEOREM 3. Under H_c of (5),

where the (independent) processes V_ω(·) and W_ωη(·) are as previously defined.

Remark 7. Notice from (6) that the asymptotic local power of

is affected by heteroskedasticity in both the irregular component and the errors driving the level process. Moreover, because the limiting processes relating to these components enter the asymptotic distribution in different forms (W_ωη(·) is integrated whereas V_ω(·) is not), it is anticipated that heteroskedasticity will have different effects in these two cases.

Remark 8. Under the homoskedastic condition that σ_t² = σ² and σ_ηt² = σ_η², for all t, the local alternative simplifies to H_c : σ_η² = (c²/T²)σ²λ_ε², and the right member of (6) reduces to

(cf. Busetti and Taylor, 2003, p. 513).

Remark 9. Under the conditions of Theorem 3,

.

7. BOOTSTRAP PROCEDURES

In this section we show that the size biases caused by time-varying second moments can be corrected by properly adapting the heteroskedastic fixed regressor bootstrap of Hansen (2000) to the present framework. Interestingly, the heteroskedastic bootstrap allows us to retrieve asymptotically correct p-values even in the presence of autocorrelated errors. The rationale behind this result is that whereas the asymptotic null distribution of the

statistic is affected by the heteroskedasticity function ω(·) it is not affected by the short memory properties of the I(0) component {ε_t} (see Theorem 1). We outline the bootstrap procedure for the

-based procedure, although the

- and

-based procedures may be bootstrapped in an entirely analogous fashion.

Let

denote the limiting null distribution of

(Theorem 1) and its cumulative distribution function (c.d.f.), respectively. Let

denote the residuals obtained by regressing y_t on x_t and let {z_t}_t=1^T denote an independent N(0,1) sequence. The bootstrap sample is defined as

, and the bootstrap statistic is given by

with

denoting the residuals obtained from the regression of y_t^b on x_t, t = 1,…,T. The bootstrap p-value is

, where G_T^b(·) denotes the c.d.f. of

.

The usefulness of the heteroskedastic bootstrap in the present framework is given in Theorem 4, which shows (i) that the bootstrap allows us to retrieve the correct asymptotic null distribution and hence that the p-values based on

are asymptotically pivotal and (ii) that a test based on the bootstrap p-values is consistent.

THEOREM 4. (i) Under the conditions of Theorem 1,

, where

denotes weak convergence in probability (see Giné and Zinn, 1990). (ii) Under the conditions of Theorem 2,

.

In practice, G_T^b(·) is not known but can be approximated in the usual way through numerical simulation by generating N (conditionally) independent bootstrap statistics,

, computed as before but from

, with {{z_n,t}_t=1^T}_n=1^N a doubly independent N(0,1) sequence. The simulated bootstrap p-value is then computed as

and is such that

.

In Table 4 we report results for the bootstrapped KPSS testing procedure, outlined before, applied to data generated according to Case (a) of Section 6.1. The results are therefore directly comparable with those given for Case (a) in Table 1 for the

test. Results are reported only for this case because this was the form of heteroskedasticity that effected the most significant size distortions in the original tests. The reported results are for experiments run over N = 1,000 bootstrap replications. Benchmark entries for the case where the errors are homoskedastic are also reported in the column labeled “IID.”

Empirical size of bootstrap KPSS tests: Heteroskedastic errors, Case (a)

A comparison of the results in Tables 1 and 4 shows that the bootstrap performs very well in practice with empirical sizes much closer to the nominal level than for the standard

test. Some oversizing, associated with early negative and late positive variance breaks, is still seen for T = 50 but is much reduced relative to that seen for the standard

test and is largely eliminated for T = 250. The undersizing seen in the standard

test for early positive and late negative breaks is eliminated by the bootstrap. Although not reported here, qualitatively similar improvements (available on request) were seen for bootstrapped implementations of the

tests and for data generated under Cases (b) and (c).