This paper develops the empirical likelihood (EL) inference procedure for parameters in autoregressive models with the error variances scaled by an unknown nonparametric time-varying function. Compared with existing methods based on non-parametric and semi-parametric estimation, the proposed test statistic avoids estimating the variance function, while maintaining the asymptotic chi-square distribution under the null. Simulation studies demonstrate that the proposed EL procedure (a) is more stable, i.e., depending less on the change points in the error variances, and (b) gets closer to the desired confidence level, than the traditional test statistic.

References

• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327. https://doi.org/10.1016/0304-4076(86)90063-1
• Cavaliere, G. (2004). Unit root tests under time-varying variance shifts. Econometric Reviews, 23, 259–292. https://doi.org/10.1081/ETC-200028215
• Cavaliere, G., & Taylor, A. M. R. (2007). Testing for unit roots in time series models with nonstationary volatility. Journal of Econometrics, 140(2), 919–947. https://doi.org/10.1016/j.jeconom.2006.07.019
• Chan, N. H., & Ling, S. Q. (2006). Empirical likelihood for Garch models. Econometric Theory, 3, 403–428. https://doi.org/10.1017/S0266466606060208
• Chen, S. X., & Ingrid, V. K. (2009). A review on empirical likelihood methods for regression. Test, 18(3), 415–447. https://doi.org/10.1007/s11749-009-0159-5
• Chen, S. X., & Qin, Y. (2000). Empirical likelihood confidence intervals for local linear smoothers. Biometrika, 87, 946–953. https://doi.org/10.1093/biomet/87.4.946
• DiCiccio, T., Hall, P., & Romano, J. (1991). Empirical likelihood is Bartlett-Correctable. Annals of Statistics, 19(2), 1053–1061. https://doi.org/10.1214/aos/1176348137
• Drees, H., & Starica, C. (2002). A simple non-stationary model for stock returns (Working paper). Chalmers University of Technology.
• Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987–1008. https://doi.org/10.2307/1912773
• Engle, R. F., & Rangel, J. G. (2008). The spline-GARCH model for low-frequency volatility and its global macroeconomic causes. The Review of Financial Studies, 21(3), 1187–1222. https://doi.org/10.1093/rfs/hhn004
• Han, Y., Jin, Y. H., & Chen, M. (2013). Empirical likelihood-based subset selection for partially linear autoregressive models. Acta Mathematicae Applicatae Sinica, English Series, 29(4), 793–808. https://doi.org/10.1007/s10255-013-0256-9
• Hansen, B. E. (1995). Regression with nonstationary volatility. Econometrica, 63, 1113–1132. https://doi.org/10.2307/2171723
• Kolaczyk, E. D. (1994). Empirical likelihood for generalized linear models. Statistica Sinica, 4, 199–218. https://doi.org/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A4n111.pdf
• Li, G., & Wang, Q. H. (2003). Empirical likelihood regression analysis for right censored data. Statistica Sinica, 13, 51–68. https://doi.org/https://www.jstor.org/stable/24307094?seq=1
• Lu, X. W. (2009). Empirical likelihood for heteroscedastic partially linear models. Journal of Multivariate Analysis, 100, 387–396. https://doi.org/10.1016/j.jmva.2008.05.006
• Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and Its Applications, 9(1), 141–142. https://doi.org/10.1137/1109020
• Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237–249. https://doi.org/10.1093/biomet/75.2.237
• Owen, A. B. (1990). Empirical likelihood ratio confidence regions. Annals of Statistics, 18, 90–120. https://doi.org/10.1214/aos/1176347494
• Owen, A. B. (1991). Empirical likelihood for linear models. Annals of Statistics, 19(4), 1725–1747. https://doi.org/10.1214/aos/1176348368
• Owen, A. B. (2001). Empirical likelihood. Chapman and Hall.
• Phillips, P. C. B., & Xu, K. L. (2006). Inference in autoregression under heteroskedasticity. Journal of Time Series Analysis, 27, 289–308. https://doi.org/10.1111/jtsa.2006.27.issue-2
• Polzehl, J., & Spokoiny, V. (2006). Varying coefficient GARCH versus local constant volatility modeling: Comparison of predictive power (Working paper). Weierstrass Institute for Applied Analysis and Stochastics.
• Qiu, J., & Wu, L. (2015). A moving blocks empirical likelihood method for longitudinal data. Biometrics, 71, 616–624. https://doi.org/10.1111/biom.12317
• Qin, G., & Jing, B. Y. (2001). Empirical likelihood for censored linear regression. Scandinavian Journal of Statistics, 28, 661–673. https://doi.org/10.1111/sjos.2001.28.issue-4
• Qin, J., & Lawless, J. (1994). Empirical likelihood and general estimating equations. Annals of Statistics, 22, 300–325. https://doi.org/10.1214/aos/1176325370
• Shi, J., & Lau, T. S. (2000). Empirical likelihood for partially linear models. Journal of Multivariate Analysis, 72(1), 132–148. https://doi.org/10.1006/jmva.1999.1866
• Starica, C. (2003). Is GARCH (1,1) as good a model as the Nobel prize accolades would imply (Working paper). Chalmers University of Technology.

• Xu, K. L., & Phillips, P. C. B. (2008). Adaptive estimation of autroregressive models with time-varying variances. Journal of Econometrics, 142, 265–280. https://doi.org/10.1016/j.jeconom.2007.06.001
• Variyath, A. M., & Chen, J. H. (2010). Abraham B. Empirical likelihood based variable selection. Journal of Statistical Planning and Inference, 140, 971–981. https://doi.org/10.1016/j.jspi.2009.09.025
• Watson, G. S. (1964). Smooth regression analysis. Sankhya Series A, 26, 359–372.
• Wong, W. H. (1983). On the consistency of cross validation in kernel nonparametric regression. Annals of Statistics, 11, 1136–1141. https://doi.org/10.1214/aos/1176346327
• Zhou, M., & Li, G. (2008). Empirical likelihood analysis of the Buckley-James estimator. Journal of Multivariate Analysis, 99, 649–664. https://doi.org/10.1016/j.jmva.2007.02.007

To cite this article: Yu Han & Chunming Zhang (2021): Empirical likelihood inference in
autoregressive models with time-varying variances, Statistical Theory and Related Fields, DOI:
10.1080/24754269.2021.1913977
To link to this article: https://doi.org/10.1080/24754269.2021.1913977