Detecting Variance Change-Points for Blocked Time Series and Dependent Panel Data

This article proposes a class of weighted differences of averages (WDA) statistics to test and estimate possible change-points in variance for time series with weakly dependent blocks and dependent panel data without specific distributional assumptions. We derive the asymptotic distributions of the test statistics for testing the existence of a single variance change-point under the null and local alternatives. We also study the consistency of the change-point estimator. Within the proposed class of the WDA test statistics, a standardized WDA test is shown to have the best consistency rate and is recommended for practical use. An iterative binary searching procedure is suggested for estimating the locations of possible multiple change-points in variance, whose consistency is also established. Simulation studies are conducted to compare detection power and number of wrong rejections of the proposed procedure to that of a cumulative sum (CUSUM) based test and a likelihood ratio-based test. Finally, we apply the proposed method to a stock index dataset and an unemployment rate dataset. Supplementary materials for this article are available online.


INTRODUCTION
In statistical literature, there exist abundant studies and methods focusing on detecting change-points in mean for univariate series (e.g., Chernoff and Zacks 1964;Gardner 1969;Sen and Srivastava 1975;Worsley 1979Worsley , 1986Hawkins 1992;Siegmund and Venkatraman 1995;Loader 1996;Han and Tsung 2006). The problem of detecting variance change-points in a sequence has also received increasing attention since volatility plays an important role in finance and many economic fields. Many pioneering works developed methods to identify a single variance change-point for a univariate sequence of independent random variables (e.g., Hsu 1977;Prem and James 1981;Booth and Smith 1982). For detecting multiple variance changepoints, Inclán and Tiao (1994) proposed a multi-step iterative algorithm based on cumulative sum (CUSUM) for an independent Gaussian sequence. Chen and Gupta (1997) used the Schwarz information criterion (SIC) and adopted a binary segmentation scheme. Meanwhile, there are considerable efforts in studying variance change problems for more sophisticated univariate time series models. Among them, Tsay (1988) employed the least-square techniques and residual variance ratios to detect changes in a univariate ARMA time series. Lee and Park (2001) considered the detection in an infinite order moving average (MA) process. Kokoszka and Leipus (2000) studied CUSUM-type change-point estimator in Autoregressive Conditional Heteroscedasticity (ARCH) models. Rapach and Strauss (2008) applied CUSUM tests for detecting structural breaks in Generalized ARCH (GARCH)(1,1) models of exchange rate volatility. Xu (2013) studied the CUSUM and Lagrange Multiplier (LM) tests for detecting structural breaks in volatility models including GARCH-type models.
In this article, we propose a method for detecting variance changes in panel data. Our interest is to detect and identify common variance change-points among different individuals. Common change-points in panel data are a widely observed phenomenon. Changes in fiscal or monetary policy may affect many companies' stock volatility simultaneously. Finding common change-points in panel data is of great importance in practice (Joseph and Wolfson 1993;Bai 2010). But the research on the stability of panel models received much less attention (Horvátha and Husková 2012). Some recent development in testing change-points for panel data includes Horvátha and Husková (2012) who proposed CUSUM-type test for testing changes in mean and Xing and Ying (2012) who considered the change-point in regression coefficients. See Joseph and Wolfson (1993), Bai (2010), and Tapsoba, Lee, and Wang (2011) for more research works. However, none of the above articles consider the detection of variance change-points.
In addition to panel data, the proposed method is also applicable to blocked time series data, where we consider time series that have natural blocks. For example, an hour or a day may be considered as a natural block. Since the samples are observed in a short period (block), the observations in each block could be treated as having homogenous variance but being dependent to each other. Our interest is to test and identify the variance changes among blocks in the blocked time series data. For example, we can track the stock index collected at high frequency at each day for T days to detect the news/events that exert a large influence on the daily volatility. Similarly, to reduce the variability in production and improve quality, one can keep track of an assembly line for T days and collecting n sequencing observations each day. With such dataset, we can detect the situations that variability (variance) increases, which may serve as an alarm to the quality control manager.
Let {Y it } n i=1 be responses observed at tth time point (t = 1, . . . , T ), which are associated with a p-dimensional covariate X it through the following model: where g 1 (·) is a known function of (X it , t) up to q-dimensional unknown parameters β t and g 2 (·) is a known function of X it up to p-dimensional unknown parameters θ t (p, q fixed). The model (1.1) allows the mean and variance of Y it to be subject dependent. For the purpose of parameter identification, we require that θ t = 1. In addition, (Z 1t , . . . , Z nt ) T are the corresponding random error with mean 0 and the same unknown marginal variance σ 2 t for t = 1, . . . , T . For a given time point t, (Z 1t , . . . , Z nt ) T can be dependent; and for a given sequence i, (Z i1 , . . . , Z iT ) T can also be dependent. The model (1.1) includes many models such as Y it = Z it and Y it = g 1 (X it , t; β t ) + Z it as special cases. The model is able to incorporate the influence of the covariates on variance through a linear combination (i.e., θ T t X it ). Since the variance of Y it equals g 2 2 (θ T t X it )σ 2 t , the effect of θ T t X it on the variance depends on g 2 (·) and the impact coefficient σ 2 t . The interest of the article is to test the existence of common change-points for impact coefficients σ 2 t among n time series ({Y it } T t=1 , i = 1, 2, . . . , n) through testing the hypothesis (Gupta and Tang 1987;Chen and Gupta 1997): versus the alternative H 1 : σ 2 1 = · · · = σ 2 k 1 = σ 2 k 1 +1 = · · · = σ 2 k 2 = σ 2 k 2 +1 = · · · = σ 2 k q = σ 2 k q +1 = · · · = σ 2 T , where q is the unknown fixed number of change-points and 1 ≤ k 1 < k 2 < · · · < k q ≤ T − 1 are the unknown time positions of changepoints. If change-points exist, we will estimate the change-point locations B q = {k 1 , . . . , k q }. The above change-point testing and estimation problems are of great interests in practice. For example, Western and Bloome (2009) found that the earning insecurity (variance) of a released prisoner is related to covariates including working experience and years of schooling. In this example, one can consider θ T t X it as a summary of "working ability." The impacts of "working ability" on released prisoners' earning insecurity may return (change) to a "normal" person level after some time. An interesting question is to detect whether such change-points exist, and identify them (if exist) to find out when the prison effect will vanish.
The weighted difference of averages (WDA) statistic U k (1 ≤ k ≤ T ) for the above change-point problem can be constructed by taking the difference in averages of the sample variances s 2 t before and after time point k, where it is the sample version analog of s * 2 t = n −1 n i=1 Z 2 it based on the residualsẐ it = (Y it − g 1 (X it , t;β t ))/g 2 (θ T t X it ) withβ t ,θ t being the least-square estimators of β t and θ t for t = 1, . . . , T . The WDA statistic U k is defined by for some appropriate chosen h so that the correlation between s 2 1 and s 2 t+1 are small enough for t > h. In particular, when η = 1/2, we denote U k by U 0 k , which is called the standardized WDA (SWDA) and will be recommended in Section 3. We will show in Section 3 that the class of WDA statistics includes Inclán and Tiao's (1994) CUSUM statistic as a special case when η = 1 and covers Chen and Gupta's (1997) likelihood ratio-based test for normally distributed Z it when k and T − k are both large and η = 1/2. For testing the existence of a single variance change-point, we derive the asymptotic distributions of the test statistics based on U k in (1.3) with 1/2 ≤ η ≤ 1 for weakly dependent sequences without specific distributional assumptions. The proposed procedure for estimating change-point based on U k is also shown to be consistent. To identify multiple change-points, we adopt the iterative binary segmentation searching scheme, which has the merits of detecting multiple change-points and locating their positions simultaneously and expediently.
The contribution of the current article includes the following aspects. First, among all the WDA statistics with weights function w T (k) = {k(T − k)/T 2 } η for 1/2 ≤ η ≤ 1, SWDA is shown to have the power converging to 1 at the fastest rate for detecting a change-point with the order of change magnitude larger thanV √ log log T /T , whereV 2 = T −1 T t=1 V 2 t and V 2 t = var(s * 2 t ). Second, the proposed test and estimation procedures are shown to be applicable to a quite general dependency structure, which allows dependence across time and across individuals. Third, we show the consistency of the proposed procedure in estimating the location of the variance change-point when both n and T go to infinity. Under some mild conditions, we further show that the proposed binary segmentation method can estimate the number and the locations of the change-points consistently. This generalizes the results of Venkatraman (1992) to allow dependence.
The outline of the article is as follows. Section 2 presents the asymptotic distributions of the proposed test statistics for testing a single variance change-point under the null hypothesis and local alternatives, and studies the consistency of the estimator for the location of variance change-point. Section 3 proposes and recommends the SWDA statistic, where we outline the advantages of the SWDA method and compare it with the CUSUM statistic and the likelihood ratio statistic. In Section 4, we suggest an iterative searching method for estimating multiple change-points, which is shown to be consistent. Section 5 reports simulation studies designed for comparing the performance of the proposed procedure to the other two approaches. Section 6 illustrates the method using a financial time series and an unemployment rate dataset. Some conclusional remarks are made in Section 7. Details of the proofs of the main results are provided in the Appendix and a supplemental article.  (Chen and Gupta 1997),

TESTING AND ESTIMATING
where 1 ≤ k 0 ≤ T − 1 is the unknown location of the changepoint in the sequence. Clearly, a large statistic |U k | value indicates that there may exist a change-point occurring at time position k. Thus, we propose a test statistic based on U k for testing the existence of a single variance change-point, that is, where ε T is some slowly varying function of T, which will be specified shortly in Theorem 1. We incorporate dependence into our model by assuming that all the sequences {Z it } T t=1 are α-mixing, which implies that the correlation between Z it and Z it diminishes as the time lag |t − t | increases, which is practical in many real applications including financial data sequences (e.g., Connor and Korajczyk 1993). Detailed discussion on the mixing concept is provided in White (1984) and Bradley (2005). For any two σ -fields A and B , the α-mixing coefficient is defined as are independent sequences (Bradley 2005), which means that α Z (k) → 0 implies α S (k) → 0.
We now study the asymptotic distribution of U T for α-mixing sequences Z it . For simplicity, we assume that V 2 t are of the same order whose change points appear in a subset of B q . To present the asymptotic distribution for the test statistic U T , we need the following assumption: . . , n) are weakly stationary, which satisfy the moment are weakly dependent with uniformly finite dth moments but not necessary strictly stationary. We can see that d can be taken to be slightly larger than 2, when δ is close to 0. Assuming the sequence {Z 2 it − σ 2 t } T t=1 to be α-mixing allows a general dependency structure, which includes the general ARMA process (Withers 1981), Markov process (Bradley 2005), ARCH and GARCH processes (Lindner 2009). Thus, our method is applicable to detect variance change in these time series models.
With the following theorem (proved in the Appendix), we can formulate the rejection region for the null hypothesis (1.2).
where ε T is any slowly varying sequence such that ε T → ∞ as T → ∞ and ε T = o(T δ ) for arbitrary small δ > 0, and [T r] is the integer part of T r.
Remark 1. The condition T = o(n) is needed to guarantee the estimation error of β t and θ t are asymptotically ignorable. This condition is not needed if Y it = Z it in model (1.1) or the coefficients β t , θ t are not time-varying, that is, β t = β and θ t = θ for all t = 1, . . . , T .
Theorem 1 shows that √ T U k behaves like a weighted Brownian bridge under H 0 asymptotically. Note that we have to restrict the range of maximization of U k between ε T and T − ε T to ensure the uniform convergence. However, this is a minor restriction on the range of k since ε T is a slow varying function, which can be essentially written in term of log functions. For instance, we might take ε T = log{log(T )}, which is small even for large T.
An asymptotic level α test rejects the null hypothesis (1.2) when U T falls in the rejection region which could be obtained using numerical simulations. When η = 1/2, an asymptotic approximation to the distribution of M η,B can be derived, which can be approximated by a Gumbel distribution as the following (proved in the Appendix): 2 log log τ and c * (τ ) = 2 log log τ + 2 −1 log log log τ − 2 −1 log 4π . Therefore, by Theorem 1, an asymptotic α level test based on The following theorem considers the asymptotic distribution of U T under the alternative H 1 in (2.1) such that Theorem 2. Assume (A1) and the alternative H 1 in (2.1) such that (2.6) holds.
. By Theorem 2, the power function of the proposed test under the alternative H 1 with (2.6) is We establish the above results under condition (2.6), which may be viewed as a local alternative to H 0 . For instance, if we assume that the alternative to (2.1) is where T is a sequence of constants and σ 2 is an unknown parameter. Then we can see that if k 0 /T → r 0 and k/T → r as T → ∞, then It is noted that under alternative (2.7), the proposed test is powerful in detecting the variance changes with magnitude of order T −1/2V , which is of the order of (nT ) −1/2 ifV 2 = O(n −1 ). The location shift W η (r)| (r)|/ √ ρ attains the maximum value at r = r 0 and which attains its maximum value when r 0 = 1/2. Thus, for all the change-points with same | T |, the distribution of U T has a larger location shift from the null distribution for the change-point r 0 that appears in the middle of sequences than those change-points that appear elsewhere. This suggests that the proposed test has better power in detecting change-points in the middle than those around boundaries in sequences with the same | T |.

Estimation of the Single Variance Change Location
If the test statistic U T in (2.2) rejects the null hypothesis (1.2), indicating there may exist a change-point, then we locate the variance change-point position bŷ (2.8) For a fixed n, when T → ∞, it is known (see Hinkley 1970;Rukhin 1994) that there is no consistent estimator for the true change-point k 0 . However, when n → ∞, we can establish the consistency ofk with the following assumption, The following theorem (proved in the Appendix) shows that the WDA-based estimator is consistent for the true change-point k 0 , as n → ∞ when T is fixed.
Theorem 3. Let k 0 be the true position of change-point under the alternative hypothesis (2.1) andk be the estimate of k 0 given by (2.8). Under Assumption (A2),k is consistent to k 0 , as n → ∞.
For the case of both n and T going to infinity, the following theorem shows that the consistency property ofk still holds as long as T goes to infinity at a rate less than o Theorem 4. Let k 0 be the true position of change-point under the alternative hypothesis (2.1) andk be the estimate of k 0 given by (2.8). Assume (A1) andV → 0 as n → ∞, if T = T (n) = o(V δ−d ) for d > (2 + δ)(1 + δ) and 0 < δ ≤ 2, thenk is consistent to k 0 , as T (n) → ∞ and n → ∞.

SWDA-BASED TESTING AND ESTIMATION
Although the asymptotic distribution and consistency have been established for the WDA statistic U k with a general weight w T (k) in Section 2, in this section we recommend a testing procedure for H 0 versus H 1 in (2.1) and estimation for the variance change location through the statistic U 0 k , which has the weight function w T (k) = {k(T − K)/T 2 } 1/2 . We named U 0 k as the standardized weighted difference of averages (SWDA).
The advantages of the SWDA-based procedure lie in the following four aspects: (a) The fastest consistency rate for detecting the existence of a change-point. (b) |U 0 k | is equivalent to the log-likelihood ratio statistic LR k when Z it are iid normally distributed random variables. Under independent normal distribution assumption, the maximum likelihood estimator κ LR = arg max 1≤k≤T LR k is optimal (Rukhin 1994) in the sense that whereκ is any estimator of k 0 ; (c) its superior performance in estimating variance change-points both around the boundary and in the middle of sequences; (d) an explicit asymptotic dis-tribution function of max ε T ≤k≤T −ε T | √ T U 0 k | can be obtained as we have demonstrated in Section 2.1, which readily provides quantiles without numerical simulation. In the following, we will elaborate (a), (b) and (c), while (d) has already been shown in Section 2.1.
We first study the consistency rate of the test proposed in Section 2.1. Following the discussion after Theorem 2, we note that, under the alternative (2.7), the asymptotic power of the test is bounded above and below by where M η,B and B η,α are defined below Equation (2.4). By the asymptotic result in (2.5), M 1/2,B is of order √ log log(T ) and B 1/2,α is also of order √ log log(T ). Because M η,B and B η,α are of order √ log log(T ) for all 1/2 ≤ η ≤ 1. Hence, if | T | → ∞ at an order that is larger than √ log log(T ), the proposed tests are consistent and the power is dominated by the term {r 0 (1 − r 0 )} η | T |/ √ ρ. Thus to compare the consistency rate of the power function β( , W η ) for different η, we need only compare the coefficients {r 0 (1 − r 0 )} η . As required by Theorem 1, we only consider 1/2 ≤ η ≤ 1. Because {r 0 (1 − r 0 )} η attains the maximum value at η = 1/2 for 1/2 ≤ η ≤ 1 and fixed 0 < r 0 < 1, the power function β( , W η ) converges to 1 at the fastest rate at η = 1/2. Thus, the consistency rate of the test achieves its maximum at η = 1/2. We next show the equivalence between |U 0 k | and the loglikelihood ratio statistic LR k under iid normality assumption. Assume that Z 1t , Z 2t , . . . , Z nt are identically and independently generated from N(μ t , σ 2 t ) distribution. It follows that ns * 2 t /σ 2 t has a χ 2 n distribution. It can be shown (see the Appendix) that the log-likelihood ratio test statistic LR k for testing (1.2) versus (2.1) with variance change-point at time k is Using a second-order Taylor's expansion, Inclán and Tiao (1994) showed that, for large k and T − k, LR k ≈ Applying the following relation one can show that LR k ≈ l T |U 0 k | 2 for a constant l T = T 3V 2 /4C 2 T . Thus LR k is equivalent to |U 0 k | for large k and T − k under independent normality assumption. Chen and Gupta (1997) proposed a Schwarz information criterion (SIC) based on the likelihood ratio test statistics LR k , where they re-jected H 0 if 2 max 1≤k≤(T −1) LR k ≥ log(T ) + c α for some critical value c α obtained from a Gumbel distribution.
Since |U 0 k | is equivalent to LR k for a single variance change model as specified in (2.1) under iid normality, we expect that SWDA-based estimatork and the maximum likelihood estimation (MLE)κ LR will have similar performance under iid normality assumption. For illustrative purpose, we conducted some preliminary numerical comparisons. Figure 1 shows the comparison results of change-point estimation based on 100 simulations by generating n = 30 independent normally distributed data series with length T = 100 and true underlying variance jumping from 1 to 2 at time point k 0 , for k 0 = 10, 50, or 90. We can see that their performances are close to each other as expected. More comprehensive simulation studies in Section 5 also demonstrate such phenomena for abrupt changes under normal distribution, while for the nonnormal distributions, SWDA-based estimator outperformedκ LR .
Among the WDA procedures with 0 ≤ η ≤ 1, the SWDA with η = 1/2 is more stable than other weights in estimating change-points appearing both in the middle and around the boundaries. To appreciate this, note that for each r ∈ (0, 1) and (1), as shown in the proof of Theorem 1. We observe that (3.2) If 0 ≤ η < 1/2, we have W η (r)/ √ r(1 − r) → ∞ for r → 0 or r → 1. Thus, for 0 ≤ η < 1/2, √ T U [T r] → ∞ in probability when r is close to the boundary 0 or 1. So the values of √ T U [T r] for r's that are close to the boundaries tend to be much larger than other places even when there is no change-point near the boundaries. As a result, if 0 ≤ η < 1/2, by (2.8), we have an estimatork that tends to be closer to both boundaries than the true position. On the other hand, if 1/2 < η ≤ 1, we have an estimatork that biases toward the middle of the sequence. This could be explained by the relationship between U k and U 0 k as the following: . . , [T /2]} and decreasing in {[T /2] + 1, . . . , T }, the relative ratios U k /U 0 k take the highest value on k's around [T /2] but the smallest values on k's near the boundaries 1 or T. When T is large and k is close to 1 or T, the relative ratio between U k and U 0 k is close to 0. In other words, relative to U 0 k , U k attenuates the variances-average differences around the boundaries. Thus, U k is more likely to make errors than U 0 k in estimating the variance change location when the true variance change is around the boundaries.
To demonstrate the above point, we compare the SWDA estimator with the estimator proposed by Inclán and Tiao (1994) and a WDA estimator using (2.8) with w T (k) = 1. Recall that Inclán and Tiao's (1994) method is equivalent to U k with weight w T (k) = k(T − k)/T 2 and the weight for SWDA is w T (k) = k(T − k)/T 2 . We generated independent Gaussian data sequences and t 5 distributed data sequences with one variance change-point, and searched for the change-point viâ k = arg max 1≤k≤(T −1) |U k |. Figures 2 and 3 show the histograms ofk of 100 runs for the three different weights, with T = 100, n = 30, with variance changing from 1 to 2, and true changepoint k 0 = 10, 50, and 90. We can see from both figures that the one with weight w T (k) = 1 tended to make errors biased toward the two ends and the one with weight w T (k) = k(T − k)/T 2 tended to make errors biased toward the middle. The SWDAbased estimator performed stably no matter whether k 0 was at the ends of a sequence or in the middle.

ITERATIVE SEARCH FOR MULTIPLE VARIANCE CHANGE-POINTS
If we consider detecting multiple variance change-points in sequences, we can employ an iterative search scheme. We adopt the easy-to-implement binary segmentation method, which is based on successive application of U k to pieces of the subseries that are split consecutively after a possible change-point is found (Chen and Gupta 1997).
Let s 2 [t 1 : t 2 ] represent the series s 2 t 1 , s 2 t 1 +1 , . . ., s 2 t 2 , t 1 < t 2 and let U k (s 2 [t 1 : t 2 ]) indicate the range over which the weighted differences of averages are obtained. The search algorithm can be briefly described as follows. First, calculate U k (s 2 [1 : T ]) and U T . If U T < U * T ,α n , then accept the null hypothesis and stop; otherwise, consider there is a change-point occurring at time position k 1 , where k 1 is the point at which max k |U k | is obtained, and divide the sequence into two subsequences. Second, for the two subsequences, calculate U k (s 2 [1 : k 1 ]) and U k (s 2 [k 1 + 1 : T ]), and repeat the first step for each subsequence to detect possible change-points. Continue the process until no more changes can be found in any of the subsequences. This binary segmentation method is efficient for searching multiple changes, since when we determine there is no single change-point in a subsequence, we do not need to continue testing for the data in that subsequence, which significantly cuts down the operations for locating change-points (Yang and Kuo 2001). For such a binary searching scheme, each time we need only to test and estimate the position of a single change-point for a subsequence, and repeat the process for each subsequence until the null hypothesis is accepted. Therefore, for each step, we need only to test the no-change hypothesis H 0 defined by (1.2) against the single-change hypothesis H 1 defined by (2.1). If U T exceeds a critical value U * T ,α n , we conclude that there exists a change-point for this sequence, and we search it via (2.8).
Let 1 = k 0 < k 1 < k 2 < · · · < k q ≤ k q+1 = T be the locations of change-points and assume the following conditions are satisfied.
The next theorem shows that the binary segmentation method can consistently estimate the number of change-points and their locations. The proof of Theorem 5 is given in the supplemental article.
In particular, if T 3 = o(n), then the proposed binary segmentation procedure consistently estimates the number and the locations of the change-points.
Theorem 5 is a generalization of the consistency result proved in Venkatraman (1992). We extend the result to allow a general weight function w T (k) and a general dependency structure. In our setup, we let sample size n go to infinity so that the consistency estimation of change-points is feasible. In the same spirit, Baltagi, Kao, and Liu (2012) discussed the benefit of using panel data when the instrument variables are weak. Figure 3. Histograms of change-point estimators from U k statistic with weights of 1, k(T − k)/T 2 , and k(T − k)/T 2 for t 5 data sequences. The true change-point position k 0 is 10, 50, and 90 for the three rows, respectively.

Empirical Sizes
To illustrate that the asymptotic null distribution of the SWDA-based test and compare it with Inclán and Tiao's and Chen and Gupta's methods in finite sample cases, we simulated dependent samples from the following model: where ρ = 0.3, θ 1t = 0.1, ε it and Z i1 were independent standard normal, t 5 or centralized χ 2 3 distributed, and ε it were independent of Z i(t−1) , for i = 1, . . . , n, and t = 2, . . . , T . Here all the σ 2 t are equal, that is, σ 2 t = σ 2 1 = var(Z i1 ), for t = 1, 2, . . . , T . We ran simulations with n = 100, T = 100, 300, 500, ε T = log log(T ), and 10,000 replications to calculate the empirical size. The least-square estimates of θ 1t were obtained using the cross-section data at time t. The sizes of the proposed test are calculated as the proportions of test statistics max ε T ≤k≤T −ε T | √ T U 0 k | that are larger than U * T ,α among 10,000 replicates under different significance levels α = 0.01, 0.05, and 0.1. The values of empirical sizes are listed in Table 1. From the table, we can see that for both Gaussian and non-Gaussian distributions, the sizes from the SWDA-based test are close to the specified nominal levels, while the sizes from Inclán and Tiao's and Chen and Gupta's methods are not well controlled for non-Gaussian distributions.

Empirical Power
The simulation studies are designed to evaluate the performance of the proposed ISWDA (iterated standardized weighted differences of averages) method and compare it with Inclán and Tiao's CUSUM method and Chen and Gupta's SIC method. We conducted simulations for two variance change patterns under normal, t 5 , and centralized χ 2 3 distributions. We generated data Y it from the regression model specified in (5.1) with predictor X it , which was generated from standard normal. The random error Z it were generated from the AR(1) model specified in (5.1), in which ρ = 0.30, ε it and Z i1 were independently standard normal, t 5 or centralized χ 2 3 distributed, and ε it were independent of Z i(t−1) , for i = 1, . . . , 100, and t = 2, . . . , 120. The average number of wrong rejections and the empirical power for each change-point are reported based on nominal level α = 0.05 and 10,000 simulation iterations. Table 2 studies the cases of multiple abrupt variance changes. Specifically, the variance σ 2 is the variance of Z i1 , and the variance σ 2 jumps to 2σ 2 and 4σ 2 at time positions 30 and 60, then drops to 2σ 2 at time position 100. From the table, we can see that ISWDA procedure makes the least number of wrong rejections and good detection power for both normal and nonnormal cases, suggesting that the method is valid to apply to both Gaussian and non-Gaussian sequences. However, we observed that Inclán-Tiao's and Chen-Gupta's methods make many wrong rejections under the nonnormal cases, especially Inclán-Tiao's method. Table 3 studies the cases of gradual variance changes, which are common in real world. Such changes usually depart from the original stage persistently with small values. We explored the performances of the three methods using the same model as in Table 2. Each row in Table 3 shows the simulation result for the case that the variance σ 2 changes to 2σ 2 , 3σ 2 , 4σ 2 , 5σ 2 , and 6σ 2 at time positions 60, 61, 62, 63, and 64, then stays at 6σ 2 . From the table, we can conclude that, first, when the distribution shape departs away from normal distribution, Inclán-Tiao's and Chen-Gupta's methods make many wrong rejections, again indicating that these procedures are not suitable for non-Gaussian sequences. Second, our procedure not only makes the least number of Type I errors, but also makes less Type II errors.

EMPIRICAL EXAMPLES
To demonstrate the application of the proposed variance change detection method, we apply it to two real datasets, the Shanghai Composite Index (SCI) dataset and the World Unemployment Rate (WUR) dataset. The SCI can be considered as a high-frequency data while the WUR is a panel data.

The SCI Dataset
The SCI dataset was collected every 5 min for 100 days from July 5th to December 1st in 2010. The sequence has 48 observations each day. We first calculated the series of returns by taking the difference of the logarithm of the stock index, then applied the ISWDA method to the calculated variance sequence.
The ISWDA detected 3 changes at time positions 1, 38, and 64, while Inclán-Tiao's method located 8 changes at time positions 10, 38,62,67,68,71,86,90, and Chen-Gupta's method selected 10 changes at time positions 1, 38, 53, 56, 62, 64, 67, 71, 86, and 90. The corresponding dates for the three change-points detected by ISWDA are: July 5th, August 25th, and October  12th, which implies that there were variance jumps from these time points to their next corresponding days. Since volatility reflects the variation of expectation or emotion from people about the market, there may exist several events taking place around these time points, which exert a large influence on the overall stock market. Looking back into political and economic records, we do find three big events occurring on these time points, which may help to provide an insight for the occurrences of the changes in daily realized volatilities. First, on July 5th and July 6th, the Chinese state council held the conference about western development, which was important to accelerate western's as well as the whole nation's economic development. Second, to promote the central region's sound and rapid development, on August 25th, National Development and Reform Commission successively announced two documents requiring the central regions (Shanxi, Anhui, Jiangxi, Henan, Hubei, and Hunan provinces) to actively implement plans for agriculture, transportation, equipment manufacturing, energy, high-tech, etc. Third, on October 12th, the People's Bank of China raised the deposit reserve requirement ratio for six banks, which reflected the concerns about the excess liquidity.

The WUR Dataset
The unemployment rate is the ratio of the number of unemployed individuals over the total number of individuals in the labor market. Unemployment is a serious social problem that affects workers, their families, and the country. And unemployment rate is often used to measure the health of an economic system of a country. The unemployment rate is believed to be highly related to the gross domestic product (GDP) growth rate (Collins 2009). With the increase of the level of economic globalization, it is interesting to know whether the impact of the GDP growth rate on the variation of unemployment rate among countries will be reduced. This question may be answered by using the proposed method. We downloaded the unemployment rate and the GDP growth dataset from the world bank website (http://data.worldbank.org/).
The dataset we analyzed contains the annual unemployment rate and GDP growth for more than 200 countries from 1991 to 2012. Due to missing values, we retained data for 175 countries that have complete observations. Let Y it and X it be, respectively, the unemployment rate and GDP growth for the ith country at year t (i = 1, . . . , 175; t = 1991, . . . , 2012). For each time t, we assume the following model: where θ 2 0t + θ 2 1t = 1 and Z it is the random error that has mean 0 and variance σ 2 t . The goal of this analysis is to test if all σ 2 t are equal to each other and detect them if the change-points exist. We first applied the least-square method at each time point t to obtain estimatorsβ t0 andβ t1 by minimizing i (Y it − β t0 − β t1 X it ) 2 . Then obtaining the least-square estimate for θ 0t , θ 1t , and σ 2 t by minimizing To demonstrate the possible change-points, the least-square estimatesσ 2 t for σ 2 t are plotted in Figure 4. We then applied the proposed change-points detection method and the methods proposed by Inclán-Tiao and Chen-Gupta to the residualsẐ it =ˆ it /(θ 0t +θ 1t X it ) from the above model. The ISWDA detected five changes at year 1996, 1997, 1999, 2005, and 2006, while Inclán-Tiao's method located three changes at year 1999, 2005, and 2006, and Chen-Gupta's method selected two changes at year 2005 and 2006. These change points may be clearly observed in Figure 4. The variance estimates after 2006 are smaller than the variance estimates between 1991 and 2005, which could be understood as that the impact of GDP growth on the variation of unemployment rate becomes smaller.

CONCLUSION
In this article, we discuss the hypothesis testing and estimation problems of detecting variance change-points for sequences of weakly dependent data with dependent blocks without any specific distributional assumptions. We propose a class of WDAbased test statistics for testing a single variance change-point and the asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses. In particular, we recommend the SWDA-based procedure. The SWDA-based procedure (a) has the fastest consistency rate in detecting the existence of a change-point among the proposed class of WDA statistics; (b) is equivalent to the likelihood ratio test when the data are coming from independent normal distribution and hence it is optimal in the sense of Rukhin (1994); (b) is stable in identifying the variance change-point both in the boundaries and in the middle of sequences; and (c) is convenient to implement since the critical value for the hypothesis testing could be obtained from the explicit asymptotic null distribution of the test statistic. A binary segmentation method is proposed to estimate the locations of multiple change-points whose consistency is also established. Simulation studies and real data analyses demonstrated that the proposed method performed well under various circumstances. Finally note that, although we recommend the SWDA within the class of WDA statistics, whether the weight function family of the form {r(1 − r)} η is better than other weight families remains an open question, which needs future study to clarify.
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