1. INTRODUCTION
In various applications, e.g., analyzing financial and macroeconomical data, autocovariances often exhibit a slow decay, and a memory parameter estimator indicates the presence of strong dependence. This phenomenon can be observed because of long memory in the data and also because of nonstationarity (e.g., trend in a mean). In this case it is useful to perform the test for stationarity as a null hypothesis. Testing for stationarity against some other kind of instabilities is an important problem in number of applied fields, such as network traffic modeling, geophysics, and biomedical signal processing.
One of the oldest techniques of testing for stationarity is based on the rescaled range, or R/S, statistic of Hurst (1951), later modified by Lo (1991) (to take into account the short-range dependent errors). Bhattacharya, Gupta, and Waymire (1983) applied the R/S test for detecting the presence of a deterministic trend in the data. Kwiatkowski, Phillips, Schmidt, and Shin (1992) proposed a widely used Lagrange multiplier (KPSS) test for the null hypothesis that an observable series is stationary (or, more generally, can contain a linear trend), whereas their alternative hypothesis corresponds to the unit root (or unit root plus linear trend) model.
Tests for stationarity and other forms of stability under a wide class of alternatives, including structural changes, were considered by Ploberger, Krämer, and Kontrus (1989), Andrews (1993), Atkinson, Koopman, and Shephard (1997), and Horowitz and Spokoiny (2001), among others.
In our paper we assume that, under the null hypothesis, X1,…, Xn is a sample from a (covariance) stationary sequence, whereas the alternative is based on two main data generating schemes of nonstationarity: we suppose that X1,…, Xn are generated by the nonstationary trend model
or by the stochastic trend (unit root) model
where gn(k) is a deterministic trend and {ξk} is a second-order stationary sequence (noise). The schemes (1.1) and (1.2) cover a wide class of deterministic trends, such as multiple change points, nonparametric regression, and polynomial function, and stochastic trends.
The stationarity test considered in this paper is based on the rescaled variance, or V/S, statistic introduced by Giraitis, Kokoszka, Leipus, and Teyssière (2003), which is the centered version of the KPSS test, related to the modified R/S test of Lo (1991). Because V/S, like KPSS, is an integral-type statistic, the criterion of weak convergence in Lp space of Cremers and Kadelka (1986) can be applied to derive its limit distribution.
The classical stationarity testing literature deals mostly with independent and identically distributed (i.i.d.) or weakly dependent errors {ξk} and does not cover the case of long memory errors. In practice, however, the dependence structure of the error process is not known a priori, and the errors ξk might have short, long, or negative memory. Moreover, the tests designed for weakly dependent errors often mistake long memory for a “spurious trend” and vice versa (see Lobato and Savin, 1998). It is also difficult to distinguish small trends graphically because the paths of long memory time series contain patterns that resemble “spurious” local trends or nonstationarities.
Our test allows us to differentiate between (covariance) stationary observations that may have short, long, or negative memory and deterministic/stochastic trends. The main objective of the test is to detect “small changes,” e.g., multiple change points, whereas detection of deterministic (polynomial) and stochastic trends, which are far from the null, is easier. In the case of weakly dependent errors studied in Section 4.2, the corresponding test possesses exceptionally good power properties. The alternatives in Section 3 allow heterogeneity in the errors.
Testing for stationarity might be also useful in the context of unit root testing, because the low power of the unit root tests against the alternative of long memory fractionally integrated errors (see Diebold and Rudebusch, 1991) or stable autoregressive model with roots near unity (DeJong, Nankervis, Savin, and Whiteman, 1992) may lead to the acceptance of the false unit root hypothesis.
The test for stationarity is a pretest in the sense that the rejection of the null does not provide information on the type of alternative. Consequently, differentiation between deterministic trend and stochastic trend requires further investigation. The assumptions given in Section 3 are weak, and therefore the test might be applicable for linear and nonlinear models such as conditionally heteroskedastic time series.
The rest of the paper is organized as follows. In Section 2 we introduce the model and the testing procedure. The asymptotic results are given in Section 3. In Sections 4 and 5 we report the results of a Monte-Carlo simulation study and discuss empirical examples. Section 6 provides some results on the convergence in L2 [0,1] space by Cremers and Kadelka (1986) and discusses their applications. Section 7 contains auxiliary lemmas.
In what follows, we use C to denote a generic constant, and
denote, respectively, the convergence in probability and the convergence in distribution.
2. THE TESTING PROCEDURE
2.1. Stationarity Hypothesis and Alternatives
In this paper, we assume that a (covariance) stationary sequence {ξk,k ∈ Z} may have short, long, or negative memory.
Short memory is characterized by absolutely summable covariances γk = cov(ξk,ξ0),
(this holds as a rule for i.i.d. variables and mixing processes).
We say that a sequence {ξk} exhibits long memory if it has slowly decaying covariance
for some c ≠ 0 and 0 < d < ½.
Negative memory of {ξk} is described by the properties
for some c ≠ 0, −½ < d < 0. The notation an ∼ bn means that an /bn → 1 as n → ∞.
We shall write (γk) ∈ G(d) to denote that γk has covariance structure (2.1)–(2.3) with d ∈ (−½,½), where d = 0 corresponds to the short memory case (2.1).1
Note that the convention d = 0 indicates the short memory property (2.1) only and that (2.2) and (2.3) are not valid for d = 0, whereas 0 < d < ½ indicates (2.2) and −½ < d < 0 corresponds to (2.3).
Now we describe rigorously the hypotheses tested in our paper.
HYPOTHESIS H0 (Stationarity). We say that the random variables Xk, k = 1,…, n satisfy the null hypothesis H0 if
where μ is a real number and {ξk} is a stationary sequence with zero mean and covariance function (γk) ∈ G(d) with d ∈ [−a1,a2] ⊂ (−½,½), 0 ≤ a1,a2 < ½.
The alternative hypothesis to H0 includes the commonly used deterministic trend plus noise model and the unit root hypotheses. In Section 3 we consider alternatives that allow heterogeneity in errors ξk.
ALTERNATIVE HT (Deterministic trend). We say that the Xk, k = 1,…, n satisfy the alternative hypothesis HT if
where μ ∈ R, {ξk} is a covariance stationary sequence with zero mean and covariance function (γk) ∈ G(d) with |d| < ½.
ALTERNATIVE HU (Stochastic trend). We say that the Xk, k = 1,…, n satisfy alternative hypothesis HU if
where μ and the ξk are the same as under hypothesis HT. (The alternative Xk − Xk−1 = μ + ξk is also included in HU, as a special case.)
The following assumption on the trend functions gn(k) plays an important role in the proof of consistency results.
Assumption T(γ). There exist a constant γ > −½ and two functions g*,g* ∈ L2 [0,1], g* ≠ const such that
The dominated convergence theorem implies that g* ∈ L2 [0,1]; we assume that g*(t) ≠ const in L2.
Example 2.1
Assumption T(γ) is satisfied for the following classes of trends. For instance,
(a) polynomial trend
satisfies Assumption T(γ) with γ = β;
(b) nonparametric regression
where g(t),t ∈ [0,1] is a bounded function, satisfies T(γ) with γ = 0. This class of trends also covers structural breaks in the mean (single and multiple change points) of the type
with 0 < τ < 1.
Under H0, our testing procedure requires the use of an estimator
that is consistent at least at a logarithmic rate (see (3.5)), whereas under alternative HT or HU the only assumption imposed on
is the restriction
with 0 ≤ a1,a2 < ½.
To estimate d we use the local Whittle estimator
(see Künsch, 1987; Robinson, 1995a) defined as
where 0 ≤ a1,a2 < ½ and Un(d) is a local contrast function
Here λk = 2πk/n, k = 1,…, [(n − 1)/2] are the Fourier frequencies and
is the periodogram; m = mn denotes the bandwidth parameter that satisfies conditions nε ≤ m ≤ n1−ε with ε > 0.
The rate of convergence of the local Whittle estimator in the nonparametric regression model with dependent errors was discussed in Robinson (1997) and Hurvich, Lang, and Soulier (2005), among others. Other semiparametric estimation methods of the parameter d were considered by Robinson (1995b), Moulines and Soulier (1999), and Shimotsu and Phillips (2005) (also see the survey by Moulines and Soulier, 2003).
Note that the parametric testing procedure for testing fractional I(d0) against I(d1) with d1 < d0 (fractional Dickey–Fuller test) suggested by Dolado, Gonzalo, and Mayoral (2002) requires much stronger n1/2-consistency of the estimator
.
2.2. Testing the Null Hypothesis
Our test is based on the V/S statistic (see Giraitis et al., 2003). It is defined as the ratio
, where
with
,
where
are sample covariances. We assume that the bandwidth q = qn has the following property:
The V/S statistic is similar to the R/S statistic introduced by Hurst (1951) and modified by Lo (1991) to take into account short-range dependence. Compared with the R/S statistic, the numerator (2.5) is based on the sample variance of the sums Sk*, rather than their range in Lo's statistic.
Define the statistic
In Section 3 we provide a few unrestrictive additional assumptions that imply the weak convergence of the test statistic
to the random variable
where
is the fractional Brownian bridge (fBb) and WH(t), 0 ≤ t ≤ 1 denotes the fractional Brownian motion (fBm) with parameter H. Recall that fBm is a Gaussian process with zero mean and covariance
In particular, W1/2(t) is the standard Wiener process (Brownian motion).
The limit process Zd depends on the unknown memory parameter d ∈ [−a1,a2], and so does the upper α-quantile of Zd:
where FZd(x) is the distribution function of the random variable Zd. The quantile function cα(d) is computed in Section 2.3; see (2.16). Applying the test, the unknown parameter d in (2.11) is replaced by
.
TEST. We reject H0 if
Theorem 3.1, which follows, provides rigorous assumptions for the probability of rejection of the true null hypothesis to be asymptotically equal to the significance level α:
Theorems 3.2 and 3.3 discuss consistency of the test under alternatives HT or HU, as n → ∞, when the hypothesis of stationarity H0 is rejected with probability tending to 1:
In Section 3 we show that convergence (2.13) and consistency (2.14) are valid for a large class of stationary errors {ξk} including linear processes.
Remark 2.1.
(a) Under the alternatives HT or HU, the test statistic
diverges to ∞ not more slowly than (n/q)1−2a2. The power of the test increases when the minimization is carried out over the interval [−a1,a2] with a smaller a2. Covariance stationary errors ξk having the memory parameter d ∈ (a2,½) can be included in the alternative model. However, when d is close to a2, the test will detect such alternatives with low power. In applications, a1 and a2 can be chosen between 0 and 0.4.
(b) If a priori it is known that the errors ξk have short memory, which is a common case in practice, then we can test the following hypothesis against the alternatives HT and HU.
HYPOTHESIS H0*.
where μ is a real number and {ξk} is a short memory (covariance) stationary sequence.
In this case we use the test with the critical region
See Section 4.2.
2.3. Quantiles
Let cα(d), defined by (2.11), denote the upper α-quantile of Zd. It can be easily shown that Zd has a continuous strictly increasing distribution function for all −½ < d < ½ and the quantile function has the desirable continuity property.
LEMMA 2.1. Suppose that
as n → ∞. Then
The proof of Lemma 2.1 is given in Section 7.
In Table 1 we provide the values of the quantiles cα(d) calculated using two approaches: the first is based on 10,000 Monte-Carlo simulations of the statistic
(with n = 10,000, q = n1/3) applied to the sequence FARIMA(0,d,0), whereas the second relies on the discretization of the integrals in the expression of the random variable Zd in (2.10). Both approaches give almost identical results. To obtain the values of the quantile cα(d) for all d ∈ (−½,½) we use a polynomial extrapolation. Numerical experiments show that this approximation fits very well. For α = 5% and α = 10% the standard least squares procedure gives the following formulas:
Figure 1 shows the values of quantile cα(d) for d ∈ [−0.45,0.45] and several values of α.
Quantiles cα(d).
In the case of d = 0, the distribution function of the random variable Z0 is given by the formula
and was established by Watson (1961) in the context of goodness-of-fit tests on a circle. Note that
, where FK is the asymptotic distribution function of the standard Kolmogorov statistic. Moreover, Z0 admits the representation
where the Yj are independent standard normal variables and
(for more details, see Giraitis et al., 2003).
3. ASYMPTOTIC THEORY
3.1. Asymptotic Null Distribution
To derive the asymptotic distribution of the test statistic
under H0 we have to impose additional conditions. Recall that Eξk = 0 and we assume that the autocovariance γk = cov(ξk,ξ0) has the property (γk) ∈ G(d), d ∈ [−a1,a2] ⊂ (−½,½).
In Lemma 7.3 in Section 7, we show that under the condition (γk) ∈ G(d)
and
where c is given by (2.2) and (2.3).
To derive the asymptotic distribution of the numerator Vn we introduce the following assumption.
Assumption A.1. The finite-dimensional distributions (f.d.d.'s) of the process
converge to those of the fBm process sdW1/2+d(t).
To ensure consistency of the estimator
we introduce an additional assumption.
Assumption A.2.
where
denotes sample autocovariance of {ξk} and the bandwidth parameter q = qn satisfies (2.8).
Note that Assumption A.2 and (3.1) imply convergence (3.3).
Assumption A.3. Under H0, estimator
takes values in [−a1,a2], where 0 ≤ a1,a2 < ½, and is such that
THEOREM 3.1 (Asymptotic distribution under H0). Suppose that
where μ is a real number and {ξk} is a zero mean covariance stationary sequence with (γk) ∈ G(d), d ∈ [−a1,a2] ⊂ (−½,½).
Suppose that Assumptions A.1–A.3 are satisfied. Then as n → ∞,
and
where Zd is given by (2.10) and cα(d) is the upper α-quantile of Zd defined by (2.11).
Proof of Theorem 3.1. In Lemmas 7.1 and 7.2 in Section 7 we show that under Assumptions A.1 and A.2,
and Assumption A.3 implies that
. Therefore
thus proving the convergence (3.6). We also have
because Zd has a continuous distribution function and
, by Lemma 2.1. █
The set of assumptions under H0 is unrestrictive. In Section 3.2 we show that the assumptions of Theorem 3.1 are satisfied for a linear process.
Remark 3.1.
(a) Assumption A.1 (convergence of f.d.d.'s of partial sums to those of fBm) holds for a large class of covariance stationary processes including nonlinear ARCH-type models; see Giraitis, Robinson, and Surgailis (2000).
(b) Consistency property (3.5) (Assumption A.3) of the local Whittle estimator
is valid under weak assumptions (Dalla, Giraitis, and Hidalgo, 2005) for a wide class of fourth-order stationary time series.
Assumption A.2 is also not restrictive. It means that the variance
is well approximated by the sample variance
.
Example 3.1
Let {ξk} be a fourth-order stationary process such that (γk) ∈ G(d), d ∈ (−½,½),
where κ(h,r,s) is a fourth-order cumulant defined by
and, if d ∈ (−½,0), the spectral density of {ξk} has property fξ(λ) ≤ C|λ|−2d,|λ| ≤ π. Then, under (2.8), convergence (3.3) and Assumption A.2 hold true (for the proof in the case of d ≥ 0 see Giraitis et al., 2003, Thm. 3.1; see Abadir, Distaso, and Giraitis, 2005, Thm. 2.2, in the case of d < 0).
The following proposition characterizes asymptotically negligible trends gn(k) that cannot be detected by the test.
PROPOSITION 3.1 (Negligible trend). Suppose that
where {ξk} is a zero mean covariance stationary sequence with (γk) ∈ G(d), d ∈ [−a1,a2] ⊂ (−½,½). Assume that {ξk} satisfy Assumptions A.1 and A.2,
satisfies Assumption A.3, and gn(k) is a deterministic trend such that
Then, as under
satisfies (3.6) and (3.7).
Proof of Proposition 3.1. In Lemmas 7.1 and 7.2 in Section 7, we show that, as
. The rest of the proof follows using the same argument as in the case of Theorem 3.1. █
Condition (3.12) describes negligible trends gn(k), for which the local Whittle estimator
satisfies Assumption A.3. Proof of this fact is similar as in the case of a signal plus noise model in Abadir, Distaso, and Giraitis (2004). Hurvich et al. (2005) showed consistency of
in the case of a “stronger” trend gn(k) = r(t/n) where r is a differentiable function. Such a trend does not satisfy condition (3.12) and will be detected by our test.
3.2. Example of a Linear Process
In this section we discuss the validity of Assumptions A.1–A.3 and (3.10)–(3.11) for the linear process errors
where {aj} are real weights,
, and {εj} are i.i.d. random variables with zero mean, unit variance, and finite fourth moment Eε04 < ∞. Recall that if a Gaussian process {ξk} has a spectral density then it can be written in the form (3.13).
We assume that either
or for some c1 ≠ 0, 0 < d < ½
or for some c1 ≠ 0 and negative −½ < d < 0
Because
, it is easy to check that assumption (3.14) implies
, (3.15) yields γk ∼ ck2d−1, and (3.16) implies the relations
. Thus, (γk) ∈ G(d).
Under some additional regularity assumptions on aj's, the spectral density f of {ξk} satisfies the following conditions:
where c > 0, β ∈ (0,2] and −½ < d < ½; f (λ) is differentiable in a neighborhood (0,δ) of the origin and
Remark 3.2.
(a) We use the linear model of errors ξk as an example. Robinson (1995a) has shown that if the spectral density of a linear process has properties (3.17) and (3.18) with β = 2 and the bandwidth m is such that m = o(n4/5), then the local Whittle estimator
satisfies the central limit theorem:
Although consistency of the local Whittle estimate
and Assumption A.3 remain valid for a wide class of stationary sequences (see Dalla et al., 2005), convergence (3.19) might not hold. Therefore testing procedures based on (3.19) might be not applicable for nonlinear sequences.
(b) In the case of Gaussian and linear observations, condition (γk) ∈ G(d) implies the convergence of the partial sum process to fBM; see Ibragimov and Linnik (1971, Thm. 18.6.5) and Davydov (1970).
(c) The validity of (3.10) was proved in Giraitis et al. (2003), whereas (3.11) follows from
where
.
(d) It can be shown that Assumptions A.1 and A.2 hold for linear processes (3.13) with {εk} replaced by the martingale difference sequence {εk} such that
, where
is the σ-field generated by εs,s ≤ k.
3.3. Consistency of the Test under HT
In this section we show the consistency of the test in the case of a deterministic trend.
Theorem 3.2 allows heterogeneity in the errors {ξk} and does not require consistency of the estimator
under HT. The only condition imposed on
is the restriction
.
THEOREM 3.2. Suppose
where gn(k) satisfies Assumption T(γ) with γ > −½. Assume that
and q satisfies (2.8).
Then, as n → ∞,
in the following cases:
(i) γ > 0, max1≤k<∞ Eξk2 < ∞;
(ii) γ = 0, max1≤k<∞ Eξk2 < ∞ and E(ξn)2 = o(1), where
;
(iii) γ < 0, {ξk} is a covariance stationary sequence, (γk) ∈ G(d) and d, q are such that
In particular, (3.22) holds if q = nθ,(0 < θ ≤ ½) and
Proof of Theorem 3.2. To show the consistency property (3.21) we apply the convergence
with V∞ > 0 a.s., shown in Lemma 7.1 in Section 7, and the bound
derived in Lemma 7.2. Because (3.25) implies that
, as n → ∞, then
Because
, we conclude that
in view of (3.24). █
Remark 3.3. Polynomial trends gn(k) = ckθ(1 + o(1)), θ > 0 satisfy condition T(γ) with γ = θ, whereas nonparametric regression type trends gn(k) = Gn(k/n) or single/multiple change points satisfy it with γ = 0. Such trends can be detected under unrestrictive assumptions (i) and (ii) of Theorem 3.2, which allow heterogeneity in the errors {ξk}. Assumption E(ξn)2 → 0 is obviously satisfied if {ξk} is a covariance stationary sequence such that γk → 0, as k → ∞.
Theorem 3.2 shows that decreasing trends with γ < 0, e.g., gn(k) = ckγ, can be detected if the dependence in {ξk} is not too strong, i.e., if d < ½ + γ/θ and q = nθ. For example, if {ξk} has short memory, i.e., d = 0, and q = n1/2, then the test detects a trend when
.
Under HT, partial sums of gn(k) + ξk are dominated by the trend gn(k):
Indeed, if gn(k) satisfies T(γ) with γ > −½, then by the dominated convergence theorem,
whereas
because
In the case of γ ≥ 0, (3.27) follows by assumptions max1≤k<∞ Eξk2 < ∞ and E(ξn)2 = o(1). If −½ < γ < 0, then (γk) ∈ G(d) and (3.27) follows by (3.1) and (3.22).
3.4. Consistency of the Test under HU
In this section we consider the consistency of the test under a stochastic trend alternative.
THEOREM 3.3. Suppose that
and assume that
satisfies (2.8). Then, as n → ∞,
in the following cases:
(i) μ = 0, {ξk} is a covariance stationary sequence such that (γk) ∈ G(d), d ∈ (−½,½) and {ξk} satisfies Assumption A.1;
(ii) μ ≠ 0, {ξk} satisfies assumptions max1≤k<∞ Eξk2 < ∞ and E(ξn)2 = o(1).
Proof of Theorem 3.3. In Lemmas 7.1 and 7.2 in Section 7 we show that
where θ = 1 + 2d in case (i), and θ = 2 in case (ii). Then (3.29) follows using the same argument as in the proof of Theorem 3.2. █
To examine the set of conditions of consistency in full, in the following proposition we discuss stochastic trend plus deterministic trend type alternatives. The conditions are similar to those of Theorem 3.2 under HT, where a deterministic trend dominates the noise term and heterogeneity in the errors ξk is allowed.
PROPOSITION 3.2. Suppose that
where gn(k) is a deterministic trend, and assume that
satisfies (2.8). Then divergence (3.29) holds in the following cases:
We omit the proof of Proposition 3.2, which is similar to the proofs of Theorems 3.2 and 3.3.
4. SIMULATION STUDY
4.1. Large Samples
In this section, we present some results of simulation studies examining the finite-sample performance of the proposed test. In all examples, we assume that
is the local Whittle estimate, the minimization in (2.4) is carried out over d ∈ [−0.4, 0.4], and m = n0.7. The bandwidth q is equal to n1/3. We conducted a set of simulations to find “optimal” values of the bandwidth parameters m and q. These values seem to minimize size distortions in the interval −0.3 ≤ d ≤ 0.3. Simulations in Dalla et al. (2005) indicate that the “optimal” bandwidth for the local Whittle estimator
is n0.6 ≤ m ≤ n0.8.
We first study the empirical size of the test using stationary series generated by a FARIMA(0,d,0) model, where d ∈ (−½,½). The FARIMA(0,d,0) processes are simulated using the circulant matrix embedding method (see Bardet, Lang, Oppenheim, Philippe, and Taqqu, 2003). Figure 2 shows that the size is close to the nominal level α when the memory parameter d ∈ (−0.30, 0.30).
Empirical test size under FARIMA(0,d,0) model with i.i.d. standard normal innovations. The significance level α = 5% and 10%; the sample size n = 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/3. Estimations are based on 5,000 independent replications.
To illustrate the empirical power of the test under alternatives HT and HU, we study several nonstationary models. The first example is the standard unit root model (alternative HU)
where the noise ξk is a FARIMA(0,d,0) process. Figure 3 shows that it is more difficult to detect the nonstationarity when the parameter d is negative: in that case the power is greater than 0.75 at significance level 10% (respectively, 5%) for d > −0.25 (respectively, d ≥ −0.1).
Empirical power of the test under unit root model (4.1). The ξk are FARIMA(0,d,0) with i.i.d. standard normal innovations. The significance level α = 5% and 10%; the sample size n = 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/3. Estimations are based on 5,000 independent replications.
Figure 4 demonstrates the power of the test in the case of a linear trend model (alternative HT)
Empirical power of the test under the linear trend model (4.2) with c = 0.005. The ξk are FARIMA(0,d,0). The significance level α = 5% and 10%; the sample size n = 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/3. Estimations are based on 5,000 independent replications.
where the ξk are FARIMA(0,d,0). The power is close to 1 for any d ∈ [−0.4, 0.4] even for very small c.
Figure 5 illustrates the power of the test in the case of nonparametric regression type trend
Empirical power of the test under trend model Xk = Gn(k/n) + ξk. (a) Change point model (4.3) with Δ = 1.5; (b) model (4.4) with c = 3 and β = ½. The ξk are FARIMA(0,d,0). The significance level α = 5% and 10%; the sample size n = 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/3. Estimations are based on 5,000 independent replications.
with FARIMA(0,d,0) errors ξk. We consider two cases:
(1) the change point model
with τ = ½;
(3) the nonlinear model
The results are satisfactory when −0.4 ≤ d ≤ 0.35, but the power decreases when the memory parameter is close to ½. For a large sample size (here n = 4,096) the power becomes greater than 0.75 for any d ∈ [−0.4, 0.4].
Asymptotic properties of the test depend on the correct use of the bandwidth parameter q. Lima and Xiao (2004) suggested a data dependent bandwidth selection procedure applicable for estimation of the long-run variance s02, (3.2), of short memory time series. It would be of interest to extend their procedure to the case d ∈ [−a1,a2] and employ it in the test for stationarity.
4.2. Testing for Short Memory Stationarity
A common assumption in the econometric literature is the short memory (weak dependence) of the errors ξk:
which corresponds to d = 0. We denote such null hypotheses of short memory stationarity by H0*. We reject H0* if
here c5%(0) = 0.190,c10%(0) = 0.153 (see Table 1 in Section 2.3).
It is natural to expect that in the case d = 0 (or, more generally, when d is known) the asymptotic properties of the test are better. To illustrate this, consider the stationary short memory AR(1) model Xk = aXk−1 + εk. The empirical size of the test for different values of a is shown in Figure 6. In this context, the simulations lead to the use of the bandwidth q = n1/2. We observe the distortion of the size when the parameter a > 0.7 is close to the nonstationarity region a = 1. The test is moderately conservative (undersized) for 300 ≤ n ≤ 500 and has good size for large n > 500.
Empirical size of the test under AR(1) model Xk = aXk−1 + εk, where εk are i.i.d. N(0,1). The significance level α = 5%; the sample size n = 256, 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/2. Estimations are based on 5,000 independent replications.
Figure 7 illustrates the power of the test under three types of alternatives:
- unit root model Xk = Xk−1 + ξk with MA(1) errors ξk = aεk−1 + εk, where εk are i.i.d. N(0,1) and a varies from −1 to 0 where a = −1 with initialization X0 = 0 corresponds to a short memory stationary process,
- linear trend model Xk = ck + εk,
- the change point model Xk = Δ1{k>[nτ]} + εk,
with varying parameters a, c, and Δ. We see that in all cases the test based on Tn(0) has good power properties. Figure 7a shows that in the case of a unit root the test is powerful even for moderate sample sizes n ≥ 300; Figure 7b shows that the linear trend ck is effectively detected even for very small c ≥ 0.005 in samples n ≥ 300 and the power tends to 1 when c increases; Figure 7c shows that the small changes in the mean can be detected with probability 0.75 and magnitude of jump Δ ≥ 0.35 when n ≥ 500.
Empirical power of the test under unit root model Xk = Xk−1 + ξk, where (a) ξk = aεk−1 + εk with a varying from −1 to 0; (b) linear trend model Xk = ck + εk with c varying from 0.0005 to 0.005; (c) change point model Xk = Δ1{k>[n/2]} + εk with Δ varying from 0.1 to 1. The innovations εk are i.i.d. N(0,1). Significance level α = 5%; the sample size n = 256, 512, 1,024 in (a) and n = 256, 512, 1,024, 4,096 in (b) and (c). The bandwidths are m = n0.7, q = n1/2. Estimations are based on 5,000 independent replications.
Remark 4.1. Under hypothesis H0* (d = 0), the interval [−a1,a2] of the hypothetical memory parameter becomes a single point a1 = a2 = 0, and stationary sequences {Xk} with positive memory parameter d > 0 or negative d < 0 are not covered by H0*. Note that, for 0 < d < ½, Tn(0) ≥ OP((n/q)d) → ∞ (see the proof of Theorem 3.1), and thus H0* is rejected as n → ∞. Formally, the stationary long memory case d > 0 could be considered as an alternative, but the power of the test for such an alternative will be low if d approaches 0 (see Figure 8). For negative −½ < d < 0 it holds that Tn(0) = OP((n/q)d) → 0, and in such a case the test accepts H0*.
Empirical power of the test under FARIMA(0,d,0), d varying from 0 to 0.4. Innovations εk are i.i.d. N(0,1). Significance level α = 5%; the sample size n = 256, 512, 1,024, 4,096. The bandwidths are m = n0.7, q = n1/2. Estimations are based on 5,000 independent replications.
5. EMPIRICAL APPLICATION
In this section we apply the theory to the data set of S&P 500 daily returns. Figure 9 presents the Standard and Poor's 500 Index daily closing values from year 1928 to 1993 and their log returns.
S&P 500 Index closing values and their log returns.
It is well known that, for many financial data, the returns are almost uncorrelated, whereas their transformations such as squares or absolute returns can be significantly correlated; see, e.g., Ding and Granger (1996). This indicates that the returns do not behave as a sequence of independent random variables, and hence it is useful to examine the dependence structure of the transformed returns. The slow decay of the sample autocovariance of |rk| and rk2 shows that the presence of long memory or nonstationarity is rather likely. The graphs of the corresponding autocorrelations are shown in Figure 10, and the statistical results are presented in Table 2. Recall that
estimates the memory parameter only in a stationary case, whereas under nonstationarity, positive values of
do not indicate the presence of long memory.
Autocorrelations of S&P 500 daily returns, squares of returns and absolute values of returns (from low to high).
Quantiles cα(d) for the random variable Zd with α = 5% and 10%
Values of the test statistic and quantiles for the S&P 500 daily returns, squares, absolute values, and third powers of returns, 1/3/28–5/28/93
Values of the test statistic and quantiles for the S&P 500 daily returns, squares, absolute values, and third powers of returns, 9/6/88–5/28/93
The short or long memory dependence structure of rt,|rt|,rt2 is confirmed by empirical studies; see, e.g., Ding and Granger (1996). Theoretical autoregressive conditionally heteroskedastic (ARCH) type models for rt satisfy Assumption A.1 of convergence of f.d.d.s (see Giraitis et al., 2003; Giraitis et al., 2000); consistency of the local Whittle estimator of parameter d is valid for a wide class of stationary time series under weak assumptions. Therefore we can expect our testing procedure to be applicable.
The results confirm the covariance stationarity for returns rk, whereas for squares and absolute values H0 is rejected. It is interesting to point out that, similarly to rk, for an (odd) third power rk3 the stationarity hypothesis H0 is not rejected and the small value of
does not indicate the presence of the long memory. Note, however, that for large values of
the size of the test can be significantly distorted (the null hypothesis is rejected too often), especially in the case of small samples; see Figure 2. Therefore rejection of H0 for absolute returns
could be “spurious” and should be taken with caution, unless we do not take into consideration very large sample size (n = 17,496).
Our statistical procedure shows that the absolute powers of the returns, |rk|δ,δ ≥ 1 do not follow a covariance stationary model. This finding might indicate the phenomenon of “spurious” long memory in powers |rk|δ. See, e.g., Lobato and Savin (1998) and Mikosch and Stărică (2004), where these authors provide arguments indicating that such an effect can be observed because of structural changes in data, monotonic trends, nonexistence of higher order moments, etc. In the case of our statistic,
, spurious inference can happen because of the strong bias of the statistic
; see also Abadir et al. (2005). The complete explanation of the spurious long memory phenomenon remains an interesting open problem in the field.
As expected, our test shows that the initial series of prices pk (Figure 9) and log-prices log pk do not possess the stationarity property. Application of the test shows that, in the case of
, and H0 is rejected; similarly in the case of
, and H0 is rejected too. (Because the stationarity of rk = log pk − log pk−1 is not rejected, log pk is a nice empirical example of unit root series.)
One can argue that the S&P data from 1928 to 1993 cover too long a span to yield stationary returns and that the test rejects stationarity in |rt|,rt2 because of structural shifts in the economy. To avoid this kind of instability we also applied the test to the sample containing the last 1,200 observations from the period 9/6/88–5/28/93 excluding the October 1987 crash. In this case the test yields the same decision on the stationarity as in the full sample case; see Table 3. The test shows that the rejection of the null does not seem to be related to the length of the series, whereas the slow decay of autocovariances of |rt| and rt2 might indicate the presence of nonstationarity (of a different type than that one described by HT and HU) rather than long memory.
6. WEAK CONVERGENCE IN L2 [0,1] SPACE
Most of the classical weak convergence studies of partial sum processes, empirical processes, quantile processes, etc., are conducted in the space D[0,1] using the Skorokhod topology framework. However, the investigation of the asymptotic properties of the KPSS, V/S, and many classical integral-type statistics, such as the Cramér–von Mises ω2 statistic, the Watson statistic, and the Anderson–Darling statistic, require weaker, only L2 or Lp, topology on the space of paths (see Billingsley, 1968, Prob. 2, p. 123). Note that the L2 [0,1] topology is not applicable for the maximum type Kolmogorov–Smirnov, Kuiper, R/S, and other statistics where convergence in D[0,1] or C[0,1] should be used.
In this section we formulate the criteria of weak convergence in the space L2 [0,1], which presents a useful sufficient condition for the convergence of integrals and sums
and other functionals of Xn(t), 0 ≤ t ≤ 1 and follows from the weak convergence criterion in the space Lp [0,1], p ≥ 1 of Cremers and Kadelka (1986). Sufficient conditions given in Theorem 6.1 are less restrictive and easier to verify than those in the space D[0,1].
THEOREM 6.1. Assume that {Xn(t),t ∈ [0,1]}, n ≥ 0 is a sequence of real-valued measurable processes with paths from the space L2 [0,1], i.e.,
Suppose that there exist a process X∞(·) and a nonnegative function f satisfying
such that
(i) f.d.d.s of the processes Xn(·) converge to those of the process X∞(·);
(ii) for any t ∈ [0,1] and n ≥ 1
(iii) for any t,
Then Xn weakly converges to X∞ in space L2 [0,1]; i.e., for any continuous real-valued bounded functional g : L2 [0,1] → R, it holds that, as n → ∞,
In particular, for any continuous function F : L2 [0,1] → L2 [0,1], F(Xn(·)) weakly converges to F(X∞(·)).
Following Cremers and Kadelka (1986), we call such a convergence a weak convergence in L2 [0,1] and write
. An analogous result can be derived for the spaces Lp [0,1], 1 ≤ p < ∞ (see Cremers and Kadelka, 1986).
COROLLARY 6.1. Theorem 6.1 implies the convergence
for continuous functionals F : L2 [0,1] → R of paths Xn(·) such as
For more continuous functionals in Lp [0,1] and related discussion see Oliveira and Suquet (1998).
The convergence of the sums
follows under additional unrestrictive regularity assumptions. The criterion of the convergence of general transformations
was studied by Park and Phillips (1999) and Pötscher (2004). Phillips and Solo (1992) provided some major examples where functional laws fail but convergence in L2 applies.
Remark 6.1. The assumption of measurability of the processes Xn(t) is not restrictive and is needed to assure that integral (6.1) is correctly defined. In fact, to ensure the measurability it is enough to consider the stochastically continuous processes. (Recall that random process X is stochastically continuous if for any t and for any ε > 0, P{|X(t) − X(s)| > ε} → 0 as s → t.)2
Indeed, because [0,1] is a compact and R is locally compact, stochastic continuity of processes (see Gikhman and Skorokhod, 1980, Ch. 3, Sect. 3, Thm. 1) implies that there exists a stochastically equivalent
measurable process
such that for any n and t it holds that
. It remains to observe that the f.d.d.s of the two stochastically equivalent processes
coincide.
which tends to zero if and only if ρX(τ) → 1 as τ → 0, i.e., ρX(τ) is continuous at τ = 0. The stochastic continuity property is satisfied by the partial sum process
(as soon as the ξj are nondegenerate random variables) and by fBm with parameter H > 0.
The numerator Vn, (2.5), appearing in the test statistic (2.9), is a function of the form (6.4). Lemma 7.1 in Section 7 shows that the limit of Vn exists under unrestrictive assumptions.
7. LEMMAS
Under Assumption T(γ), the dominated convergence theorem implies that for any 0 ≤ t ≤ 1, as n → ∞,
LEMMA 7.1. Suppose that Vn is given by (2.5). Then
holds with
where sdWd+1/2(t) is defined in Assumption A.1 and G∞(t) in (7.1).
Proof of Lemma 7.1. We show first that convergence (7.2) holds in the case of Theorem 3.1, Proposition 3.1, and Theorem 3.2.
Recall that by definition
Then
Thus
Setting
we can write
where
which together with (7.5) implies
where
Hence, (7.2) follows if we show that
and
We prove that Yn(t) satisfies conditions (i), (ii), and (iii) of Theorem 6.1, which implies convergence (7.6), in view of (6.2) and (6.4) of Corollary 6.1, whereas the proof of (7.7) is straightforward.
Theorem 3.1 is a special case of Proposition 3.1. In Proposition 3.1 we have that Xk = μ + gn(k) + ξk, where
, and therefore we can write
Then with θ = d,
which implies that
In view of (7.9), to prove (7.6), it suffices to show that Xn,ξ(t) (0 ≤ t ≤ 1) satisfies conditions (i), (ii), and (iii) of Theorem 6.1 with θ = d and X∞(t) = sdWd+1/2(t). Condition (i) follows by Assumption A.1; convergence (iii) and the bound (ii) follow from
which holds by (7.19) of Lemma 7.3, which follows.
Under the assumptions of Theorem 3.2, Xk = μ + gn(k) + ξk, where trend gn(k) satisfies Assumption T(γ) with γ > −½. Then, setting θ = ½ + γ, the trend gn(k) dominates the errors ξk in decomposition (7.8):
because EXn,ξ2(t) → 0 by (3.27) and Xn,g(t) satisfies (7.1). Thus Yn(t) satisfies conditions (i), (ii), and (iii) with X∞(t) = G∞(t).
It remains to show convergence (7.2) in the case of Theorem 3.3. As before, Vn can be written in the form (7.5), where
By definition of the stochastic trend (3.28),
Setting
we can write
Together with (7.5) this yields
We shall show that
(a) the f.d.d.'s of the process Yn*(t), 0 ≤ t ≤ 1, as n → ∞, converge to those of the process X∞(t) given in (7.3),
(b) for any 0 < t ≤ 1, E(Yn*(t))2 → X∞2(t), and
(c) sup0≤t≤1 E(Yn*(t))2 ≤ C < ∞, uniformly in n ≥ 1.
(a)–(c) imply that the process {Yn*(t)} satisfies the assumptions of Theorem 6.1 with the limit process X∞(t) given in (7.3), and therefore (7.6) holds by (6.2) and (6.5) of Corollary 6.1.
In the case of μ = 0 we assume that {ξk} satisfies Assumption A.1 and (γk) ∈ G(d), d ∈ (−½,½). Then (a) holds with θ = 1 + d by Assumption A.1, and (b) and (c) follow from (7.10).
In the case μ ≠ 0 we set
. Then, as n → ∞,
because
, by the assumptions of Theorem 3.3(ii), which implies the validity of (a)–(c) with X∞(t) = μt. █
LEMMA 7.2. Suppose that
is given by (2.6). Then, under the assumptions of Theorem 3.1 and Proposition 3.1, as n → ∞,
where s d 2 is given by (3.1) and (3.2).
Under the assumptions of Theorems 3.2 and 3.3,
uniformly in n, q, where
Proof of Lemma 7.2. It suffices to show convergence (7.11) under the assumptions of Proposition 3.1 because Theorem 3.1 is a special case of Proposition 3.1.
Denote
where
with
. Then we can write
where
because under the conditions of Lemma 7.2,
. Similarly
In Proposition 3.1 we assume that
, which implies that
whereas Assumption A.2 together with convergence (7.19) of Lemma 7.3 implies that
to prove (7.11).
Assume that the assumptions of Theorem 3.2 hold. Then
where by (7.15),
because under Assumption T(γ) the dominated convergence theorem implies that
. If γ ≥ 0, then (7.12) follows from (7.16), (7.17), and the estimate
If −½ < γ < 0, then under the assumption (γk) ∈ G(d) relation (7.20) of Lemma 7.3 implies that
, which together with (7.16) and (7.17) implies
because q2d = o(qn2γ), by assumption (3.22).
Suppose that the assumptions of Theorem 3.3 hold. Then
which yields
By definition of the stochastic trend (3.28),
If μ = 0, then
by (7.19) of Lemma 7.3, and therefore
to prove (7.12). If μ ≠ 0 then
Hence by (7.18),
to prove (7.12). █
LEMMA 7.3. Let {ξk} be a covariance stationary zero mean sequence with (γk) ∈ G(d), d ∈ (−½,½) and let q satisfy (2.8). Then, as n → ∞,
where s d 2 is defined in (3.2) and
Proof of Lemma 7.3. Suppose that d > 0. Then, by assumption (2.2), as n → ∞,
In the case d ≤ 0, we can write
If d = 0 then, using assumption (2.1), it follows that
.
If d < 0, then assumption (2.3) and (7.21) imply
where, as n → ∞,
and therefore
to complete the proof of (7.19).
It remains to show (7.20). We have
where
because, by (7.19), E(ξn)2 ≤ Cn−1+2d. Hence
By (7.19), the first term satisfies
. If d ≠ 0, then |γj| = O(| j|−1+2d), and the second term can be estimated by
In the case d = 0,
Finally, the third term can be bounded by qn2d−1 = (q/n)1−2dq2d = o(q2d), to prove (7.20). █
Proof of Lemma 2.1. Convergence of the quantile (2.15) follows if we show that the mapping ν [map ] cα(ν), ν ∈ [−a1,a2] is continuous. Then,
where
is a distribution function of the estimator
, to prove (2.15).
Decompose this mapping as
denotes the distribution of random variable X). To show that the mapping
(
is continuous, it is enough to note that the quantile of a continuous strictly monotonic distribution function is a continuous functional in uniform metric, i.e., if cαF denotes the quantile of a distribution F, then cαFn → cαF whenever supt|Fn(t) − F(t)| → 0. Set Fn = FZνn, F = FZν and observe that, because F(t) is continuous,
if and only if supt|Fn(t) − F(t)| → 0.
Now we prove that
i.e.,
is continuous. Set
. Then
and
because from the definition of the fBm, using straightforward calculations, it follows that g(y) = E(Yy(t) − Yν(t))2, y ∈ [−a1,a2] is a continuous bounded function, g(ν) = 0, and
uniformly in νn,ν. This proves (7.22). █
