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ON MULTIPLE STRUCTURAL BREAKS IN DISTRIBUTION: AN EMPIRICAL CHARACTERISTIC FUNCTION APPROACH

Published online by Cambridge University Press:  25 March 2022

Zhonghao Fu ,
Yongmiao Hong  and
Xia Wang
Zhonghao Fu*
Affiliation:
Fudan University and Shanghai Institute of International Finance and Economics
Yongmiao Hong*
Affiliation:
Chinese Academy of Sciences and University of Chinese Academy of Sciences
Xia Wang*
Affiliation:
Renmin University of China
*
Address correspondence to Zhonghao Fu, School of Economics, Fudan University, Shanghai, China; e-mail: zhfu@fudan.edu.cn; Yongmiao Hong, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, China; e-mail: yh20@cornell.edu; Xia Wang, School of Economics, Renmin University of China, Beijing, China; e-mail: wxia@ruc.edu.cn.
Address correspondence to Zhonghao Fu, School of Economics, Fudan University, Shanghai, China; e-mail: zhfu@fudan.edu.cn; Yongmiao Hong, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, China; e-mail: yh20@cornell.edu; Xia Wang, School of Economics, Renmin University of China, Beijing, China; e-mail: wxia@ruc.edu.cn.
Address correspondence to Zhonghao Fu, School of Economics, Fudan University, Shanghai, China; e-mail: zhfu@fudan.edu.cn; Yongmiao Hong, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, China; e-mail: yh20@cornell.edu; Xia Wang, School of Economics, Renmin University of China, Beijing, China; e-mail: wxia@ruc.edu.cn.
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Abstract

We estimate and test for multiple structural breaks in distribution via an empirical characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup-F type test for structural breaks in distribution as well as an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge possible sources of structural breaks, which is asymptotically more powerful than the smoothed nonparametric tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power in finite samples. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our tests also indicate that the documented breaks appear to occur in variance and higher-order moments, but not so often in mean.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 3 , June 2023 , pp. 534 - 581
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Fu acknowledges financial support from the National Science Foundation of China (NSFC) (Nos. 71903032 and 72121002), Hong acknowledges financial support from the NSFC’s Fundamental Scientific Center Project (No. 71988101), and Wang acknowledges financial support from the NSFC (No. 71873151). We thank the Editor (Peter C.B. Phillips), the Co-Editor (Robert Taylor), three anonymous referees, and seminar participants in the 2019 City University of Hong Kong Workshop in Econometrics and Statistics, the International Workshop on Extreme Value Theory at Fudan University, and the 2019 Guangzhou Econometrics Workshop, for their insightful comments and suggestions. All remaining errors are solely ours.

References

REFERENCES

Andrews, D.W. & Fair, R.C. (1988) Inference in nonlinear econometric models with structural change. Review of Economic Studies 55, 615–640.CrossRefGoogle Scholar
Aston, J.A. & Kirch, C. (2012) Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis 109, 204–220.CrossRefGoogle Scholar
Aue, A., Gabrys, R., Horváth, L., & Kokoszka, P. (2009) Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis 100, 2254–2269.CrossRefGoogle Scholar
Aue, A., Rice, G., & Sönmez, O. (2018) Detecting and dating structural breaks in functional data without dimension reduction. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80, 509–529.CrossRefGoogle Scholar
Bai, J. (1994) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15, 453–472.CrossRefGoogle Scholar
Bai, J. (1996) Testing for parameter constancy in linear regressions: An empirical distribution function approach. Econometrica 64, 597–622.CrossRefGoogle Scholar
Bai, J. (1997) Estimation of a change point in multiple regression models. Review of Economics and Statistics 79, 551–563.CrossRefGoogle Scholar
Bai, J. & Ng, S. (2002) Determining the number of factors in approximate factor models. Econometrica 70, 191–221.CrossRefGoogle Scholar
Bai, J. & Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78.CrossRefGoogle Scholar
Berkes, I., Gabrys, R., Horváth, L., & Kokoszka, P. (2009) Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71, 927–946.CrossRefGoogle Scholar
Bierens, H.J. & Wang, L. (2012) Integrated conditional moment tests for parametric conditional distributions. Econometric Theory 28, 328–362.CrossRefGoogle Scholar
Boldea, O. & Hall, A.R. (2013) Estimation and inference in unstable nonlinear least squares models. Journal of Econometrics 172, 158–167.CrossRefGoogle Scholar
Boothe, P. & Glassman, D. (1987) The statistical distribution of exchange rates: Empirical evidence and economic implications. Journal of International Economics 22, 297–319.CrossRefGoogle Scholar
Busetti, F. & Harvey, A. (2011) When is a copula constant? A test for changing relationships. Journal of Financial Econometrics 9, 106–131.CrossRefGoogle Scholar
Chacko, G. & Viceira, L.M. (2003) Spectral GMM estimation of continuous-time processes. Journal of Econometrics 116, 259–292.CrossRefGoogle Scholar
Chen, B. (2015) Modeling and testing smooth structural changes with endogenous regressors. Journal of Econometrics 185, 196–215.CrossRefGoogle Scholar
Chen, B. & Hong, Y. (2012) Testing for smooth structural changes in time series models via nonparametric regression. Econometrica 80, 1157–1183.Google Scholar
Chow, G.C. (1960) Tests of equality between sets of coefficients in two linear regressions. Econometrica 28, 591–605.CrossRefGoogle Scholar
Cowell, F.A. (1985) Measures of distributional change: An axiomatic approach. The Review of Economic Studies 52, 135–151.CrossRefGoogle Scholar
Delgado, M.A. & Hidalgo, J. (2000) Nonparametric inference on structural breaks. Journal of Econometrics 96, 113–144.CrossRefGoogle Scholar
Diebold, F.X., Gunther, T.A., & Tay, A.S. (1998) Evaluating density forecasts with applications to financial risk management. International Economic Review 39, 863–883.CrossRefGoogle Scholar
Diebold, F.X., Hahn, J., & Tay, A.S. (1999) Multivariate density forecast evaluation and calibration in financial risk management: High-frequency returns on foreign exchange. Review of Economics and Statistics 81, 661–673.CrossRefGoogle Scholar
Dümbgen, L. (1991) The asymptotic behavior of some nonparametric change-point estimators. Annals of Statistics 19, 1471–1495.CrossRefGoogle Scholar
Epps, T. (1987) Testing that a stationary time series is Gaussian. Annals of Statistics 15, 1683–1698.CrossRefGoogle Scholar
Epps, T. (1988) Testing that a Gaussian process is stationary. Annals of Statistics 16, 1667–1683.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (1999) Root-n-consistent estimation of partially linear time series models. Journal of Nonparametric Statistics 11, 251–269.CrossRefGoogle Scholar
Fengler, M.R., Mammen, E., & Vogt, M. (2015) Specification and structural break tests for additive models with applications to realized variance data. Journal of Econometrics 188, 196–218.CrossRefGoogle Scholar
Feuerverger, A. & Mureika, R.A. (1977) The empirical characteristic function and its applications. Annals of Statistics 5, 88–97.CrossRefGoogle Scholar
Fu, Z. & Hong, Y. (2019) A model-free consistent test for structural change in regression possibly with endogeneity. Journal of Econometrics 211, 206–242.CrossRefGoogle Scholar
Galí, J. & Gertler, M. (1999) Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics 44, 195–222.CrossRefGoogle Scholar
González-Rivera, G. & Sun, Y. (2017) Density forecast evaluation in unstable environments. International Journal of Forecasting 33, 416–432.CrossRefGoogle Scholar
Hall, A.R., Han, S., & Boldea, O. (2012) Inference regarding multiple structural changes in linear models with endogenous regressors. Journal of Econometrics 170, 281–302.CrossRefGoogle ScholarPubMed
Hansen, B.E. (2001) The new econometrics of structural change: Dating breaks in US labor productivity. Journal of Economic Perspectives 15, 117–128.CrossRefGoogle Scholar
Hlávka, Z., Hušková, M., Kirch, C., & Meintanis, S.G. (2012) Monitoring changes in the error distribution of autoregressive models based on Fourier methods. TEST 21, 605–634.CrossRefGoogle Scholar
Hlávka, Z., Hušková, M. and Meintanis, S.G. (2017) Change point detection with multivariate observations based on characteristic functions. In D. Ferger, W. González Manteiga, T. Schmidt, JL Wang (eds.), From Statistics to Mathematical Finance , pp. 273–290. Springer.CrossRefGoogle Scholar
Hlávka, Z., Hušková, M. & Meintanis, S.G. (2020) Change-point methods for multivariate time-series: Paired vectorial observations. Statistical Papers 61, 1351–1383.CrossRefGoogle Scholar
Hong, Y. (1999) Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. Journal of the American Statistical Association 94, 1201–1220.CrossRefGoogle Scholar
Hong, Y. & Lee, T.-H. (2003) Diagnostic checking for the adequacy of nonlinear time series models. Econometric Theory 19, 1065–1121.CrossRefGoogle Scholar
Hong, Y., Wang, X., & Wang, S. (2017) Testing strict stationarity with applications to macroeconomic time series. International Economic Review 58, 1227–1277.CrossRefGoogle Scholar
Hörmann, S. & Kokoszka, P. (2010) Weakly dependent functional data. Annals of Statistics 38, 1845–1884.CrossRefGoogle Scholar
Hsieh, D.A. (1988) The statistical properties of daily foreign exchange rates: 1974–1983. Journal of International Economics 24, 129–145.CrossRefGoogle Scholar
Hušková, M. & Meintanis, S.G. (2006) Change point analysis based on empirical characteristic functions. Metrika International Journal for Theoretical and Applied Statistics 63, 145–168.Google Scholar
Hušková, M. & Meintanis, S.G. (2008) Tests for the multivariate k-sample problem based on the empirical characteristic function. Journal of Nonparametric Statistics 20, 263–277.CrossRefGoogle Scholar
Inoue, A. (2001) Testing for distributional change in time series. Econometric Theory 17, 156–187.CrossRefGoogle Scholar
Jiang, G.J. & Knight, J.L. (2002) Estimation of continuous-time processes via the empirical characteristic function. Journal of Business and Economic Statistics 20, 198–212.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2018) New distribution theory for the estimation of structural break point in mean. Journal of Econometrics 205, 156–176.CrossRefGoogle Scholar
Kapetanios, G. (2009) Testing for strict stationarity in financial variables. Journal of Banking and Finance 33, 2346–2362.CrossRefGoogle Scholar
Knight, J.L. & Yu, J. (2002) Empirical characteristic function in time series estimation. Econometric Theory 18, 691–721.CrossRefGoogle Scholar
Koedijk, C.G., Schafgans, M., & de Vries, C.G. (1990) The tail index of exchange rate returns. Journal of International Economics 29, 93–108.CrossRefGoogle Scholar
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. Annals of Statistics 17, 1217–1241.CrossRefGoogle Scholar
Lin, C.-H. & Kao, T.-C. (2008) Multiple structural changes in the tail behavior: Evidence from stock index futures returns. Nonlinear Analysis: Real World Applications 9, 1702–1713.Google Scholar
Liu, C.Y. & He, J. (1991) A variance-ratio test of random walks in foreign exchange rates. The Journal of Finance 46, 773–785.CrossRefGoogle Scholar
Liu, J., Wu, S., & Zidek, J.V. (1997) On segmented multivariate regression. Statistica Sinica 7, 497–525.Google Scholar
Loader, C.R. (1996) Change point estimation using nonparametric regression. Annals of Statistics 24, 1667–1678.CrossRefGoogle Scholar
Loretan, M. & Phillips, P.C.B. (1994) Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets. Journal of Empirical Finance 1, 211–248.CrossRefGoogle Scholar
Manner, H., Stark, F., & Wied, D. (2019) Testing for structural breaks in factor copula models. Journal of Econometrics 208, 324–345.CrossRefGoogle Scholar
Müller, H.-G. (1992) Change-points in nonparametric regression analysis. Annals of Statistics 20, 737–761.CrossRefGoogle Scholar
Pinkse, J. (1998) A consistent nonparametric test for serial independence. Journal of Econometrics 84, 205–231.CrossRefGoogle Scholar
Politis, D.N. & White, H. (2004) Automatic block-length selection for the dependent bootstrap. Econometric Reviews 23, 53–70.CrossRefGoogle Scholar
Qu, Z. (2008) Testing for structural change in regression quantiles. Journal of Econometrics 146, 170–184.CrossRefGoogle Scholar
Qu, Z. & Perron, P. (2007) Estimating and testing structural changes in multivariate regressions. Econometrica 75, 459–502.CrossRefGoogle Scholar
Quintos, C., Fan, Z., & Phillips, P.C.B. (2001) Structural change tests in tail behaviour and the Asian crisis. Review of Economic Studies 68, 633–663.CrossRefGoogle Scholar
Rime, D., Sarno, L., & Sojli, E. (2010) Exchange rate forecasting, order flow and macroeconomic information. Journal of International Economics 80, 72–88.CrossRefGoogle Scholar
Rossi, B. (2013) Advances in forecasting under instability. In G. Elliott, A. Timmermann (eds.), Handbook of Economic Forecasting , vol. 2, no. 2, pp. 1203–1324. Elsevier.Google Scholar
Rossi, B. & Sekhposyan, T. (2013) Conditional predictive density evaluation in the presence of instabilities. Journal of Econometrics 177, 199–212.CrossRefGoogle Scholar
Selk, L. & Neumeyer, N. (2013) Testing for a change of the innovation distribution in nonparametric autoregression: the sequential empirical process approach. Scandinavian Journal of Statistics 40, 770–788.CrossRefGoogle Scholar
Shephard, N.G. (1991) From characteristic function to distribution function: a simple framework for the theory. Econometric Theory 7, 519–529.CrossRefGoogle Scholar
Singleton, K.J. (2001) Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics 102, 111–141.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1996) Evidence on structural instability in macroeconomic time series relations. Journal of Business and Economic Statistics 14, 11–30.Google Scholar
Su, L. & White, H. (2007) A consistent characteristic function-based test for conditional independence. Journal of Econometrics 141, 807–834.CrossRefGoogle Scholar
Su, L. & Xiao, Z. (2008) Testing structural change in time-series nonparametric regression models. Statistics and Its Interface 1, 347.CrossRefGoogle Scholar
Vogt, M. (2015) Testing for structural change in time-varying nonparametric regression models. Econometric Theory 31, 811–859.CrossRefGoogle Scholar
Wang, X. & Hong, Y. (2018) Characteristic function based testing for conditional independence: A nonparametric regression approach. Econometric Theory 34, 815–849.CrossRefGoogle Scholar
Wu, J. & Chu, C. (1993) Kernel-type estimators of jump points and values of a regression function. Annals of Statistics 21, 1545–1566.CrossRefGoogle Scholar
Yakir, B. (1997) A note on optimal detection of a change in distribution. Annals of Statistics 25, 2117–2126.CrossRefGoogle Scholar
Zhang, C., Osborn, D.R., & Kim, D.H. (2008) The new Keynesian Phillips curve: From sticky inflation to sticky prices. Journal of Money, Credit and Banking 40, 667–699.CrossRefGoogle Scholar
Zhou, Y., Fu, L., & Zhang, B. (2017) Two non parametric methods for change-point detection in distribution. Communications in Statistics-Theory and Methods 46, 2801–2815.CrossRefGoogle Scholar
Zou, C., Yin, G., Feng, L., & Wang, Z. (2014) Nonparametric maximum likelihood approach to multiple change-point problems. Annals of Statistics 42, 970–1002.CrossRefGoogle Scholar
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