The Annals of Statistics

Statistical inference for autoregressive models under heteroscedasticity of unknown form

Ke Zhu

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This paper provides an entire inference procedure for the autoregressive model under (conditional) heteroscedasticity of unknown form with a finite variance. We first establish the asymptotic normality of the weighted least absolute deviations estimator (LADE) for the model. Second, we develop the random weighting (RW) method to estimate its asymptotic covariance matrix, leading to the implementation of the Wald test. Third, we construct a portmanteau test for model checking, and use the RW method to obtain its critical values. As a special weighted LADE, the feasible adaptive LADE (ALADE) is proposed and proved to have the same efficiency as its infeasible counterpart. The importance of our entire methodology based on the feasible ALADE is illustrated by simulation results and the real data analysis on three U.S. economic data sets.

Article information

Ann. Statist., Volume 47, Number 6 (2019), 3185-3215.

Received: April 2018
Revised: August 2018
First available in Project Euclid: 31 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 62F03: Hypothesis testing 62F12: Asymptotic properties of estimators 62F35: Robustness and adaptive procedures 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Adaptive estimator autoregressive model conditional heteroscedasticity heteroscedasticity weighted least absolute deviations estimator wild bootstrap


Zhu, Ke. Statistical inference for autoregressive models under heteroscedasticity of unknown form. Ann. Statist. 47 (2019), no. 6, 3185--3215. doi:10.1214/18-AOS1775.

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  • Amado, C. and Teräsvirta, T. (2013). Modelling volatility by variance decomposition. J. Econometrics 175 142–153.
  • Amado, C. and Teräsvirta, T. (2014). Modelling changes in the unconditional variance of long stock return series. J. Empir. Finance 25 15–35.
  • Andrews, D. W. K. (1988). Laws of large numbers for dependent nonidentically distributed random variables. Econometric Theory 4 458–467.
  • Bassett, G. Jr. and Koenker, R. (1978). Asymptotic theory of least absolute error regression. J. Amer. Statist. Assoc. 73 618–622.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327.
  • Busetti, F. and Taylor, A. M. R. (2003). Variance shifts, structural breaks, and stationarity tests. J. Bus. Econom. Statist. 21 510–531.
  • Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17–39.
  • Carroll, R. J. (1982). Adapting for heteroscedasticity in linear models. Ann. Statist. 10 1224–1233.
  • Cavaliere, G. (2004). Unit root tests under time-varying variances. Econometric Rev. 23 259–292.
  • Cavaliere, G. and Taylor, A. M. R. (2007). Testing for unit roots in time series models with non-stationary volatility. J. Econometrics 140 919–947.
  • Chen, B. and Hong, Y. (2016). Detecting for smooth structural changes in GARCH models. Econometric Theory 32 740–791.
  • Cragg, J. G. (1983). More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica 51 751–763.
  • Davidson, J. (2002). Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. J. Econometrics 106 243–269.
  • Davidson, J. (2004). Moment and memory properties of linear conditional heteroscedasticity models, and a new model. J. Bus. Econom. Statist. 22 16–29.
  • Davis, R. A. (1996). Gauss–Newton and $M$-estimation for ARMA processes with infinite variance. Stochastic Process. Appl. 63 75–95.
  • Davis, R. A. and Dunsmuir, W. T. M. (1997). Least absolute deviation estimation for regression with ARMA errors. Dedicated to Murray Rosenblatt. J. Theoret. Probab. 10 481–497.
  • Diebold, F. X. (1986). Modeling the persistence of conditional variances: A comment. Econom. Rev. 5 51–56.
  • Ding, Z., Granger, C. W. J. and Engle, R. F. (1993). A long memory property of stock market returns and a new model. J. Empir. Finance 1 83–106.
  • Drost, F. C. and Nijman, T. E. (1993). Temporal aggregation of GARCH processes. Econometrica 61 909–927.
  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007.
  • Escanciano, J. C. (2006). Goodness-of-fit tests for linear and nonlinear time series models. J. Amer. Statist. Assoc. 101 531–541.
  • Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
  • Francq, C. and Zakoïan, J.-M. (2010). GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley, Chichester.
  • Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2006). A Haar–Fisz technique for locally stationary volatility estimation. Biometrika 93 687–704.
  • Giraitis, L., Kokoszka, P. and Leipus, R. (2000). Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory 16 3–22.
  • Gonçalves, S. and Kilian, L. (2004). Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. J. Econometrics 123 89–120.
  • Gutenbrunner, C. and Jurečková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305–330.
  • Hong, Y. (1996). Consistent testing for serial correlation of unknown form. Econometrica 64 837–864.
  • Hong, Y. and Lee, Y.-J. (2005). Generalized spectral tests for conditional mean models in time series with conditional heteroscedasticity of unknown form. Rev. Econ. Stud. 72 499–541.
  • Jin, Z., Ying, Z. and Wei, L. J. (2001). A simple resampling method by perturbing the minimand. Biometrika 88 381–390.
  • Kato, K. (2009). Asymptotics for argmin processes: Convexity arguments. J. Multivariate Anal. 100 1816–1829.
  • Knight, K. (1998). Limiting distributions for $L_{1}$ regression estimators under general conditions. Ann. Statist. 26 755–770.
  • Kuersteiner, G. M. (2002). Efficient IV estimation for autoregressive models with conditional heteroskedasticity. Econometric Theory 18 547–583.
  • Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. Ann. Statist. 16 1696–1708.
  • Mikosch, T. and Stǎricǎ, C. (2004). Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Rev. Econ. Stat. 86 378–390.
  • Nicholls, D. F. and Pagan, A. R. (1983). Heteroscedasticity in models with lagged dependent variables. Econometrica 51 1233–1242.
  • Pan, J., Wang, H. and Yao, Q. (2007). Weighted least absolute deviations estimation for ARMA models with infinite variance. Econometric Theory 23 852–879.
  • Phillips, P. C. B. and Xu, K.-L. (2006). Inference in autoregression under heteroskedasticity. J. Time Series Anal. 27 289–308.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. Springer, New York.
  • Robinson, P. M. (1987). Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55 875–891.
  • Robinson, P. M. (1989). Nonparametric estimation of time-varying parameters. In Statistical Analysis and Forecasting of Economic Structural Change (P. Hackl, ed.) 253–264. Springer, Berlin.
  • Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47 67–84.
  • Robinson, P. M. (2012). Nonparametric trending regression with cross-sectional dependence. J. Econometrics 169 4–14.
  • Sensier, M. and van Dijk, D. (2004). Testing for volatility changes in U.S. macroeconomic time series. Rev. Econ. Stat. 86 833–839.
  • Shao, Q.-M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
  • Stǎricǎ, C. and Granger, C. (2005). Nonstationarities in stock returns. Rev. Econ. Stat. 87 503–522.
  • Tsay, R. S. (1988). Outliers, level shifts and variance changes in time series. J. Forecast. 7 1–20.
  • Watson, M. W. (1999). Explaining the increased variability in long-term interest rates. Econ. Q.- Fed. Reserve Bank Richmond 85 71–96.
  • Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 14 1261–1350.
  • Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154.
  • Xu, K.-L. and Phillips, P. C. B. (2008). Adaptive estimation of autoregressive models with time-varying variances. J. Econometrics 142 265–280.
  • Yao, Q. and Brockwell, P. J. (2006). Gaussian maximum likelihood estimation for ARMA models. I. Time series. J. Time Series Anal. 27 857–875.
  • Yu, K. and Jones, M. C. (2004). Likelihood-based local linear estimation of the conditional variance function. J. Amer. Statist. Assoc. 99 139–144.
  • Zhang, R. and Ling, S. (2015). Asymptotic inference for AR models with heavy-tailed G-GARCH noises. Econometric Theory 31 880–890.
  • Zhao, Q. (2001). Asymptotically efficient median regression in the presence of heteroskedasticity of unknown form. Econometric Theory 17 765–784.
  • Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.
  • Zhu, K. (2016). Bootstrapping the portmanteau tests in weak auto-regressive moving average models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 463–485.
  • Zhu, K. (2019). Supplement to “Statistical inference for autoregressive models under heteroscedasticity of unknown form.” DOI:10.1214/18-AOS1775SUPP.
  • Zhu, K. and Li, W. K. (2015). A bootstrapped spectral test for adequacy in weak ARMA models. J. Econometrics 187 113–130.
  • Zhu, K. and Ling, S. (2012). The global weighted LAD estimators for finite/infinite variance $\operatorname{ARMA}(p,q)$ models. Econometric Theory 28 1065–1086.
  • Zhu, K. and Ling, S. (2015). LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises. J. Amer. Statist. Assoc. 110 784–794.

Supplemental materials

  • Supplement to “Statistical inference for autoregressive models under heteroscedasticity of unknown form”. The supplement includes additional simulation results, applications, some technical lemmas and the remaining proofs.