[논문]Bootstrap-Based Test for Volatility Shifts in GARCH against Long-Range Dependence

Wang, Yu; Park, Cheolwoo; Lee, Taewook

doi:10.5351/csam.2015.22.5.495

Bootstrap-Based Test for Volatility Shifts in GARCH against Long-Range Dependence 원문보기

Communications for statistical applications and methods = 한국통계학회논문집, v.22 no.5, 2015년, pp.495 - 506

Wang, Yu (Department of Statistics, University of Georgia) , Park, Cheolwoo (Department of Statistics, University of Georgia) , Lee, Taewook (Department of Statistics, Hankuk University of Foreign Studies)

Abstract ▼ AI-Helper

Volatility is a variation measure in finance for returns of a financial instrument over time. GARCH models have been a popular tool to analyze volatility of financial time series data since Bollerslev (1986) and it is said that volatility is highly persistent when the sum of the estimated coefficients of the squared lagged returns and the lagged conditional variance terms in GARCH models is close to 1. Regarding persistence, numerous methods have been proposed to test if such persistency is due to volatility shifts in the market or natural fluctuation explained by stationary long-range dependence (LRD). Recently, Lee et al. (2015) proposed a residual-based cumulative sum (CUSUM) test statistic to test volatility shifts in GARCH models against LRD. We propose a bootstrap-based approach for the residual-based test and compare the sizes and powers of our bootstrap-based CUSUM test with the one in Lee et al. (2015) through simulation studies.

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문제 정의

This paper focuses on the persistence in volatility of financial time series. Numerous methods have been proposed to test if such persistency is due to volatility shifts in the market or natural fluctuation explained by stationary long-range dependence.

가설 설정

H₁ : The observed data # follow the LRD model.
H₁ : not H₀.

제안 방법

(2015) proposed the test statistics for volatility shifts against LRD based on the CUSUM statistic using residual series from the fitted GARCH models. In this paper, we extended it to the bootstrap methods and compared the empirical sizes and powers of the test statistics of Lee et al. (2015) and our bootstrap tests. We concluded that our bootstrap methods are more promising than the asymptotic ones for volatility shifts against LRD in small sample sizes, since empirical sizes are closer to the nominal significant level and empirical powers are higher.
(2015) proposed a residual-based cumulative sum (CUSUM) test statistic to test volatility shifts in GARCH models against LRD. In this paper, we propose a bootstrap-based approach for the residual-based test and compare the sizes and powers of our bootstrap-based CUSUM test with the one in Lee et al. (2015) through simulation studies.
(2015) adopted the residualbased CUSUM test. The test uses the standardized residuals from an estimated GARCH model that imitate the innovation series in GARCH models and reduce the dependence among observations. Unlike the statistic proposed by Lee et al.

대상 데이터

45. All results are based on 1000 replications on sample sizes 500, 750, 1000 and 2000 with significant level 5%, and the bootstrap replication is set to be B = 100.

이론/모형

Efron (1979) introduced the bootstrap method to study the distributions of estimators (sampling distribution) and test statistics by resampling the finite sample data. In this paper, we compare the size and power of the tests presented in Section 2 using the asymptotic and bootstrap-based methods.

성능/효과

This difference is barely observed as the sample size increases. In conclusion, bootstrap tests are shown to be more preferable for one volatility shift against LRD when sample sizes are small.

참고문헌 (17)

Andreou, E. and Ghysels, E. (2002). Detecting multiple breaks in financial market volatility dynamics, Journal of Applied Econometrics, 17, 579-600.

상세보기
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상세보기
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상세보기
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Berkes, I., Horvath, L., Kokoszka, P. and Shao, Q.-M. (2006). On discriminating between long-range dependence and changes in mean, Annals of Statistics, 34, 1140-1165.

상세보기
Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327.

상세보기
Ding, Z., Granger, C. W. and Engle, R. F. (1993). A long memory property of stock market returns and a new model, Journal of Empirical Finance, 1, 83-106.

상세보기
Efron, B. (1979). Bootstrap methods: Another look at the jackknife, Annals of Statistics, 7, 1-26.

상세보기
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica: Journal of the Econometric Society, 50, 987-1007.

상세보기
Jach, A. and Kokoszka, P. (2008). Wavelet-domain test for long-range dependence in the presence of a trend, Statistics, 42, 101-113.

상세보기
Kokoszka, P. and Leipus, R. (2000). Change-point estimation in ARCH models, Bernoulli, 6, 513-539.

상세보기
Kuswanto, H. (2011). A new simple test against spurious long memory using temporal aggregation, Journal of Statistical Computation and Simulation, 81, 1297-1311.

상세보기
Lee, S., Tokutsu, Y. and Maekawa, K. (2004). The cusum test for parameter change in regression models with ARCH errors, Journal of the Japan Statistical Society, 34, 173-188.

상세보기
Lee, T., Kim, M. and Baek, C. (2015). Tests for volatility shifts in GARCH against long-range dependence, Journal of Time Series Analysis, 36, 127-153.

상세보기
Qu, Z. (2011). A test against spurious long memory, Journal of Business and Economic Statistics, 29, 423-438.

상세보기
Resnick, S. I. (1992). Adventures in Stochastic Processes, Birkhauser, Boston.
Zhang, A., Gabrys, R. and Kokoszka, P. (2007). Discriminating between long memory and volatility shifts, Austrian Journal of Statistics, 36, 253-275.

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DOI : 10.5351/CSAM.2015.22.5.495
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