Variance Homogeneity Test Based on Cumulative Wavelet Coefﬁcients

The aim of this paper is to address the problem of variance break detection in time series in wavelet domain. The maximal overlapped discrete wavelet transform (MODWT) decomposes the series variance across scales into components known as the wavelet variances. We introduce all scale wavelet coefficients based test statistic that allows detect ing a break in the homogeneity of the variance of a series through changes in the mean of wavelet variances. The statistic make s use of the traditional CUSUM (cumulative sum) based test designed to test for a break in the mean and constructed using cumulative sums of the square of wavelet coefficients. Under moments and mixing conditions, the test statistic satisfies the functional central limit theorem (FCLT) for a broad class of time series models. The overall performance of our test statistic is compared to the traditional Inclan [8] test statistic. The effectiveness of our statistic is supported by good performances reported in simulations and is as reliable as the traditional statistic . Our method provid es a nonparametric test procedure that can be applied to a large class of linear and non linear models. We illustrate the practical use of our test procedure with the quarterly percentage changes in the Americans personal savings data set over the period 1970-2016. Both statistics detect a break in the variance in the second quarter of 2001. Over-laped Discrete Wavelet Transform, Test for Homogeneity of Variance, Time Series


Introduction
The test problem that addresses the issue of change in the variance homogeneity of time series has received considerable attention in the literature. Most of the available procedure have developed tests that make use of cumulative sums of squares (CUSUMs) principale to test for breaks in the variance. From a methodological point of view, the test statistics developed for variance change detection in the case of independent and identically distributed (i.i.d) random variables setting as in [7] and [8] may not work in the time series setup and suitable modification were needed to account for the temporal dependence in the data. Many different approaches for addressing such test problem have been developed; see, for example [4], [7], [15], and [13]. The properties of the classical CUSUMs statistic in the variance break detection in general setting were examined by several authors. We can cite for example [6] and [9] for ARCH models, [11] for standardized residuals from an estimated GARCH model, [15] for a class of nonstationary and nonparametric time series models, [4] for ongoing monotring process and [10] for detecting changes in general autocovariance structure in nonstationary time series using locally stationary wavelet process. Our approach is inspired by the application of the discrete wavelet transform in [3] for testing homogeneity of variance in a time series with long memory structure. Here we limit ourselves to test statistics based on the cumulative sums of squares of wavelet coefficients and take a different approach to investigate the null hypotheses of no break in the variance. We are motivated by the fact that the sample variance of a time series can be decomposed into components known as the wavelet variances each of which is associated with a particular scales as in [17]. This decomposition can be achieved through the use of the maximal-overlap discrete wavelet transform (MODWT) also called the undecimated or shift invariant discret wavelet transform, and has been discussed in the wavelet literature, see [1][2][3], and [12]. In a similar manner the variance σ 2 X of X t is decomposed into variances of wavelets and scaling coefficients as given by equation (6) in [16]. In theory any changes in either the variance of wavelets or scaling coefficients would directly results in the shift of σ 2 X . On the other hand and in constract to the scaling coefficients the wavelet coefficients demean and remove trends from a time series. These attractive characteristics give us a motivation in this work to explore the cumulative sums of the squared wavelet coefficients which is a time dependent series designed to track the build up over time of the sample variance across scales up to a fixed level. This paper is orgonized as follows. Section 1 provides an introduction to our test problem and a brief review of the MODWT tranform in the literature. Section 2 introduces the definition of the statistics designed to test for a break detection in the variance. In section 3 we derive the asymptotic distribution of the test statistic. Monte Carlo experiments are presented and discussed in section 4. In section 5 we apply the test procedure to the real data series US personnal savings. Section 6 provides some conclusions. Let X = (X 0 , · · · , X N −1 ) be an observed time series for which we want to test for homogeneity of variance. Assume that X is regarded under H 0 as a realization of a second order stationary discrete time stochastic process with constant variance. The null hypothesis of our test problem would be which we whish to test against the alternative hypothesis the times of variance change k 0 is unknown.

The MODWT transform
The maximal-overlap discrete wavelet transform (MODWT) is variant of the standard discrete wavelet transform (DWT) and has been discussed in the wavelet literature see [1][2][3]. For the class of discrete compactly supported Daubechies wavelets, we denote {h j,l andg j,l , l = 0, · · · , L j − 1} the level j of a wavelet and scaling filters of length L j = (2 j − 1)(L − 1) + 1. These filters are discussed and given in more details in [3]. The stochastic processes resulting from applying these filters to {X t } are respectively given by the level j wavelet and scaling coefficients The MODWT wavelet and scaling coefficient based on a finite sample size and run up to a maximum level J are given for t = 0, 1, · · · N − 1 by obtained as a result of circularly filtering X 0 , X 1 , · · · , X N −1 with the filters {h j,l } and {g j,l } where X t mod N = X t if t ≥ 0 and X t mod N = Y N −|t| if t < 0. Note that in order to obtain the time in wavelet coefficients aligned with time is X t we circularly shift these coefficients by an amount dictated by the phase properties of the filter been used as in [3] page 198.

Cumulative wavelet variance
The wavelet transform decomposes the variance σ 2 X into finite number of cumulative wavelet variances across scales and a scaling variance.
where σ 2 Wj = V ar(W j,t ) and σ 2 Vj = V ar(V j,t ) Any break in the variance of X t would result in a break of the variances of wavelet and scaling coefficients, for more details see page 190 of [17]. Therefore a break in the variance of either W j,t or V j,t would be regarded as directly related to a break in the variance of the original series. Because of the attractive properties of the wavelet coefficients, a statistical test based on these coefficients would provide an alternative method to variance break detection in time series. Given a time series X t , t = 0, · · · , N − 1 and the transform W j,t , we then form the cumulative wavelet variance time series The mean of C J,t is regarded as the overall contribution to the true variance of X t by wavelet coefficients under the null hypothesis. We will refer to the quantity C J,t as the cumulative sums of the squared wavelet coefficients or simply the cumulative wavelet variance. From (7) it follows that any changes occurring in the variance of wavelet coefficients W j,t is equivalent to a shift in the mean of C J,t . Therefore our test problem for homogeneity of variance of X t will be regarded as a test problem of detecting change in the mean of C J,t . On the other hand the contribution to the variance of X t by the scaling coefficients is given by V ar( V J,t ), and any break of this variance would reveal as well a break in σ 2 X . Both series C J,t and V 2 J,t , t = 0, 1, · · · are positive, but the series C J,t is a nondecreasing sequence and can easily track the manner in which the sample variance of X t is building. The mean focus of this work will be to construct a wavelet coefficients based statistic to test the hypothesis of homogeneity of variance of X t .

Variance change point statistics
In order to test for a change in the mean a class of commonly used test statistics is based on the CUSUM process. In our setting the wavelet based test statistic is given by These scale related statistics are evaluated up to the physical scale τ J = 2 J δ where δ is the sampling time between observations which equals 1 in our case. Note that this statistics contains affected coefficients in the modwt given by W 2 j,t for t < L j − 1. We can easily show that in term of probability their effect on our statistics is negligible The standard test statistic for change in the variance proposed in [8] is equivalent to Under the assumption that X t , t = 1, · · · , N is a sequence of independent and identically distributed (i.i.d) normal random variables. This assumption was later relaxed to allow for various form of dependence and heterogeneity in X t , see [7], [10] and [13]. The limiting distribution of this statistic depend on the long-run variance σ 2 = ∞ k=−∞ γ k where γ k is the k-th order autocovariance of the series X 2 t which should be estimated. Under general dependence we suggest to use the Heteroskedasticity and Autocorrelation Consistent (HAC) estimator and use the popular Bartlett kernel.σ 2 = γ 0 + 2 q k=1 w k (q)γ k where w k (q) = 1 − k/(q + 1) are the Bartlett weights andγ k are the sample autocovariances of the series X 2 t . In the case of i.i.d normal random variables N (0, σ 2 ), all the autocovariances are such thatγ k = 0 for k = 0 and the estimator reduces toσ 2 = 2σ 4 . In the next section we establish the functional central limit theorem (FCLT) and derive the asymptotic distribution of the statistic T C,J for a broad class of processes under moments and mixing conditions. In the following, {B 0 (r); 0 ≤ r ≤ 1} is the standard Wiener process (Brownian motion) and {B(r); 0 ≤ r ≤ 1} is the Brownian bridge B(r) = B 0 (r) − rB 0 (1). By ⇒ we denote the weak convergence of random variables

Asymptotic properties
Let M n m (X) denote the σ-algebra generated by {X m , · · · , X n }. The process {X t } is said to be strong mixing according to Rosenblat[14], if the mixing coefficient Then under the null H 0 and assumptions (A1), (A2) and (A3) where α Z J (k) and α Wj (k) are respectively the mixing coefficients for Z J,t and W j,t .
Proof: see Appendix.

Proposition 1
Assume that conditions (A1)-(A3) are satisfied. Then under the null H 0 the statistics is such that T C,J ⇒ sup

|B(r)| as N → ∞
The limiting distribution of the statistic T C,J depend on the long-run variances σ 2 Z J which is unknown and is estimated bŷ whereγ Z,k are the sample autocovariances of the series Z J,t . Note that as an alternative toγ Z,k we can use the nonparametric estimator 2πf Z J (0), where f Z J (0) is the multitaper estimator of the spectrum of Z J,t at zero frequency. On the other hand it has been shown in [11] and [15], that under fairly mild regularity conditions that the statistic T N /σ converges weakly in distribution to sup As suggested early the test problem of break detection in X t can be formulated in term of the statistic T C,J which aims to test for a break in the mean of the cumulative wavelet variance C J,t . Therefore any mean break detection computed by the statistic T C,J will be regarded as a break in the variance of X t . In order to identify potential occurrence of change in variance for a given sample of time series X 0 , · · · , X n−1 , the statistic T C,J is computed at accumulated scales of the MODWT transform. Both statistics are computed and compared to the same critical value. For instance, for a level of significance α = 5% for our test, the 95% quantile value is q α = 1.358. A large value of any statistic would strongly indicates the existence of variance change. .

Simulations
We considere four different simulated models, a Gaussian X t = t model, a Moving average M A(1) model, an autoregressive AR(1) model, and the nonlinear model X t = 2 t −1 as given in the example below. For each model the variance is set to change at time k 0 from σ 2 1 to σ 2 2 . The MODWT transform was run up to level J = 4 with the Least Asymetric wavelet filter LA(8) of length L = 8. The choice of the wavelet filter was arbitrary with moderate length. For each model and for each sample size, the statistics T N and T C,J are then calculated and compared to the critical value q α . This process of generating data sets, running the above procedure separately for each model and then testing at significance level α = 5% was repeated a total of n = 1000. The rejection rate of the statistic T C,J are tabulated from level J = 1 of the MODWT up to level J = 4. At each level J we have a test statistic T C,J , and if we assume that at that level we reject H 0 , then we have a strong evidence to claim that there is a break in the variance of X t at unknown change point. For example table 1 provide the simulation setting correspending to all four models.

Example
We consider three linear models t (Gaussian) and M A(1), an AR (1), and a nonlinear model X t each with variance break occuring respectively at times k 0 = 200 for N = 400 and k 0 = 400 for N = 800 In order to assess the performance of our statistic we computed the rejection rate of the statistic T C,J for accumulated level of scales J = 1, 2, 3, 4 separetly for all models and for two different sample sizes. The results are summurized in tables 2, and show that the overall performance in term of rejection rate of the statistic T C,J for all models are as good as the statistic T N except for the AR(1) model at level j = 3. A large value of the scale related statistic T C,J at some given level J should be interpreted as evidence against the H 0 . For example for the AR(1) model, at level j = 3 the rejection rate was 11.8% and 12.0% which are very low for both sample sizes. But at levels J = 1 the rejection rate is very high. Therefore caution should be taken when a low values of the statistic show up at some scales. The important result we derive from these simulations is that our test problem for shift in the variance of X t is tranformed into a wavelet based test for shift in the mean of cumulative wavelet variances.   Figure 3 shows that the series T C,1 (k) asso- Table 1. Simulated models

Conclusions
The wavelet based statistic T C,J gives us a scale by scale evaluation test of the homogeneity of the variance of X t using the classical CUSUM based test designed to detect a change in the mean of a time series which correspond to the cumulative wavelet coefficients C J,t in our sitting. The over all performance in term of rejection rate of the statistic based on our simulation are as good as the tests in Inclan [8] like tests. This is comfirmed by the processing of real data given by the US personal savings series and support the practical utility of this statistic. The graph of the series X t in the top panel of figure 2 depicts a high variability after the year 2000 and this is detected by both statistics. Table 3 shows that the scale related statistics T C,J are doing quite well except for J = 1 which just barely fail to reject the H 0 . With regard to the identification of the change point both statistic estimate agree that the change occurs over the second quarter of the year 2001. Our wavelet based statistic is effective and this is mainly due to the attractive property of wavelets coefficients which demean and detrend time series. Its important to note here that because the wavelets demean the series, we can safely exclude any changes in the variance due to the mean. This study shows that our nonparametric method provide a reliable statistic that can be applied to a broad class of time series data.