Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T05:43:25.279Z Has data issue: false hasContentIssue false
  • English
  • Français

INFERENCE IN MILDLY EXPLOSIVE AUTOREGRESSIONS UNDER UNCONDITIONAL HETEROSKEDASTICITY

Published online by Cambridge University Press:  18 September 2024

Xuewen Yu Open the ORCID record for Xuewen Yu [Opens in a new window]  and
Mohitosh Kejriwal Open the ORCID record for Mohitosh Kejriwal [Opens in a new window]
Xuewen Yu*
Affiliation:
Fudan University
Mohitosh Kejriwal
Affiliation:
Purdue University
*
Address correspondence to Xuewen Yu, Department of Applied Economics, School of Management, Fudan University, 670 Guoshun Road, Shanghai, China; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Mildly explosive autoregressions have been extensively employed in recent theoretical and applied econometric work to model the phenomenon of asset market bubbles. An important issue in this context concerns the construction of confidence intervals for the autoregressive parameter that represents the degree of explosiveness. Existing studies rely on intervals that are justified only under conditional homoskedasticity/heteroskedasticity. This paper studies the problem of constructing asymptotically valid confidence intervals in a mildly explosive autoregression where the innovations are allowed to be unconditionally heteroskedastic. The assumed variance process is general and can accommodate both deterministic and stochastic volatility specifications commonly adopted in the literature. Within this framework, we show that the standard heteroskedasticity- and autocorrelation-consistent estimate of the long-run variance converges in distribution to a nonstandard random variable that depends on nuisance parameters. Notwithstanding this result, the corresponding t-statistic is shown to still possess a standard normal limit distribution. To improve the quality of inference in small samples, we propose a dependent wild bootstrap-t procedure and establish its asymptotic validity under relatively weak conditions. Monte Carlo simulations demonstrate that our recommended approach performs favorably in finite samples relative to existing methods across a wide range of volatility specifications. Applications to international stock price indices and U.S. house prices illustrate the relevance of the advocated method in practice.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 36
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

We are grateful to Peter Phillips (the Editor), Xu Cheng (the Co-Editor), four anonymous referees, Yong Bao, Giuseppe Cavaliere, Prosper Dovonon, Pierre Perron, Yundong Tu, and Alan Wan, as well as seminar participants at City University of Hong Kong, Concordia University, Fudan University, Nanyang Technological University, Peking University, University of Exeter, University of Illinois Urbana–Champaign, and University of Science and Technology of China, for their comments that improved the paper substantially. Yu is supported by Shanghai Sailing Program No. 23YF1402000 and the National Natural Science Foundation of China (Grant No. 72303040). Any errors are our own.

References

REFERENCES

Anderson, T. W. (1959). On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics , 30, 676–687.CrossRefGoogle Scholar
Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica , 59, 817–858.CrossRefGoogle Scholar
Andrews, D. W. K., Cheng, X., & Guggenberger, P. (2020). Generic results for establishing the asymptotic size of confidence sets and tests. Journal of Econometrics , 218, 496–531.CrossRefGoogle Scholar
Andrews, D. W. K., & Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica , 60, 953–966.CrossRefGoogle Scholar
Arvanitis, S., & Magdalinos, T. (2018). Mildly explosive autoregression under stationary conditional heteroskedasticity. Journal of Time Series Analysis , 39, 892–908.CrossRefGoogle Scholar
Astill, S., Harvey, D. I., Leybourne, S. J., Sollis, R., & Taylor, A. M. R. (2018). Real-time monitoring for explosive financial bubbles. Journal of Time Series Analysis , 39, 863–891.CrossRefGoogle Scholar
Basawa, I. V., Mallik, A. K., McCormick, W. P., & Taylor, R. L. (1989). Bootstrapping explosive autoregressive processes. Annals of Statistics , 17, 1479–1486.CrossRefGoogle Scholar
Beare, B. K. (2018). Unit root testing with unstable volatility. Journal of Time Series Analysis , 39, 816–835.CrossRefGoogle Scholar
Boswijk, H. P., & Zu, Y. (2018). Adaptive wild bootstrap tests for a unit root with non-stationary volatility. Econometrics Journal , 21, 87–113.CrossRefGoogle Scholar
Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli , 3, 123–148.CrossRefGoogle Scholar
Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Annals of Statistics , 14, 1171–1179.CrossRefGoogle Scholar
Cavaliere, G. (2005). Unit root tests under time-varying variances. Econometric Reviews , 23, 259–292.CrossRefGoogle Scholar
Cavaliere, G., Nielsen, H. B., & Rahbek, A. (2020). Bootstrapping noncausal autoregressions: With applications to explosive bubble modeling. Journal of Business and Economic Statistics , 38, 55–67.CrossRefGoogle Scholar
Cavaliere, G., & Taylor, A. M. R. (2007a). Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics , 140, 919–947.CrossRefGoogle Scholar
Cavaliere, G., & Taylor, A. M. R. (2007b). Time-transformed unit root tests for models with non-stationary volatility. Journal of Time Series Analysis , 29, 300–330.CrossRefGoogle Scholar
Cavaliere, G., & Taylor, A. M. R. (2008). Bootstrap unit root tests for time series with non-stationary volatility. Econometric Theory , 24, 43–71.CrossRefGoogle Scholar
Cavaliere, G., & Taylor, A. M. R. (2009). Heteroskedastic time series with a unit root. Econometric Theory , 25, 1228–1276.CrossRefGoogle Scholar
Chan, N. H., Li, D., & Peng, L. (2012). Toward a unified interval estimation of autoregressions. Econometric Theory , 28, 705–717.CrossRefGoogle Scholar
Davidson, J., & De Jong, R. M. (2000). The functional central limit theorem and weak convergence to stochastic integrals II: Fractionally integrated processes. Econometric Theory , 16, 643–666.CrossRefGoogle Scholar
Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility. Journal of Finance , 48, 1749–1778.CrossRefGoogle Scholar
Fei, Y. (2018). Limit theory for mildly integrated process with intercept. Economics Letters , 163, 98–101.CrossRefGoogle Scholar
Georgiev, I. (2008). Asymptotics for cointegrated processes with infrequent stochastic level shifts and outliers. Econometric Theory , 24, 587–615.CrossRefGoogle Scholar
Gonçalves, S., & Kilian, L. (2004). Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics , 123, 89–120.CrossRefGoogle Scholar
Guo, G., Sun, Y., & Wang, S. (2019). Testing for moderate explosiveness. Econometrics Journal , 22, 73–95.CrossRefGoogle Scholar
Hall, P. (1992). The bootstrap and Edgeworth expansion . Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Hansen, B. E. (1995). Regression with nonstationary volatility. Econometrica , 63, 1113–1132.CrossRefGoogle Scholar
Harvey, D. I., Leybourne, S. J., Sollis, R., & Taylor, A. M. R. (2016). Tests for explosive financial bubbles in the presence of non-stationary volatility. Journal of Empirical Finance , 38, 548–574.CrossRefGoogle Scholar
Harvey, D. I., Leybourne, S. J., & Zu, Y. (2019). Testing explosive bubbles with time-varying volatility. Econometric Reviews , 38, 1131–1151.CrossRefGoogle Scholar
Harvey, D. I., Leybourne, S. J., & Zu, Y. (2020). Sign-based unit root tests for explosive financial bubbles in the presence of deterministically time-varying volatility. Econometric Theory , 36, 122–169.CrossRefGoogle Scholar
Jansson, M. (2002). Consistent covariance matrix estimation for linear processes. Econometric Theory , 18, 1449–1459.CrossRefGoogle Scholar
Kunsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Annals of Statistics , 17, 1217–1241.CrossRefGoogle Scholar
Kurozumi, E., Skrobotov, A., & Tsarev, A. (2023). Time-transformed test for the explosive bubbles under non-stationary volatility. Journal of Financial Econometrics , 21, 1282–1307.CrossRefGoogle Scholar
Lee, J. H. (2018). Limit theory for explosive autoregression under conditional heteroskedasticity. Journal of Statistical Planning and Inference , 196, 30–55.CrossRefGoogle Scholar
Liu, R. Y. (1988). Bootstrap procedures under some non-iid models. Annals of Statistics , 16, 1696–1708.CrossRefGoogle Scholar
Liu, X., & Peng, L. (2019). Asymptotic theory and unified confidence region for an autoregressive model. Journal of Time Series Analysis , 40, 43–65.CrossRefGoogle Scholar
Lui, Y. L., Xiao, W., & Yu, J. (2021). Mildly explosive autoregression with anti-persistent errors. Oxford Bulletin of Economics and Statistics , 83, 518–539.CrossRefGoogle Scholar
Magdalinos, T. (2012). Mildly explosive autoregression under weak and strong dependence. Journal of Econometrics , 169, 179–187.CrossRefGoogle Scholar
Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica , 55, 703–708.CrossRefGoogle Scholar
Phillips, P. C. B. (1987). Towards a unified asymptotic theory for autoregression. Biometrika , 74, 535–547.CrossRefGoogle Scholar
Phillips, P. C. B. (2005). HAC estimation by automated regression. Econometric Theory , 21, 116–142.CrossRefGoogle Scholar
Phillips, P. C. B. (2023). Estimation and inference with near unit roots. Econometric Theory , 39, 221–263.CrossRefGoogle Scholar
Phillips, P. C. B., & Magdalinos, T. (2005). Limit theory for moderate deviations from a unit root under weak dependence . Cowles Foundation for Research in Economics, Yale University.Google Scholar
Phillips, P. C. B., & Magdalinos, T. (2007a). Limit theory for moderate deviations from a unit root. Journal of Econometrics , 136, 115–130.CrossRefGoogle Scholar
Phillips, P. C. B., & Magdalinos, T. (2007b). Limit theory for moderate deviations from a unit root under weak dependence. In Phillips, G. D. A., & Tzavalis, E. (Eds.), The refinement of econometric estimation and test procedures: Finite sample and asymptotic analysis (123–162). Cambridge University Press.CrossRefGoogle Scholar
Phillips, P. C. B., & Shi, S. (2020). Real time monitoring of asset markets: Bubbles and crises. Handbook of Statistics , 42, 61–80.CrossRefGoogle Scholar
Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review , 56, 1043–1078.CrossRefGoogle Scholar
Phillips, P. C. B., & Solo, V. (1992). Asymptotics for linear processes. Annals of Statistics , 20, 971–1001.CrossRefGoogle Scholar
Phillips, P. C. B., Wu, Y., & Yu, J. (2011). Explosive behavior in the 1990s Nasdaq: When did exuberance escalate asset values? International Economic Review , 52, 201–226.CrossRefGoogle Scholar
Phillips, P. C. B., & Xu, K. L. (2006). Inference in autoregression under heteroskedasticity. Journal of Time Series Analysis , 27, 289–308.CrossRefGoogle Scholar
Rho, Y., & Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory , 35, 142–166.CrossRefGoogle Scholar
Sensier, M., & van Dijk, D. (2004). Testing for volatility changes in US macroeconomic time series. Review of Economics and Statistics , 86, 833–839.CrossRefGoogle Scholar
Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association , 105, 218–235.CrossRefGoogle Scholar
Skrobotov, A. (2023). Testing for explosive bubbles: A review. Dependence Modeling , 11, Article 20220152.CrossRefGoogle Scholar
Smeekes, S., & Urbain, J. P. (2014). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing [Working paper]. Maastricht University.Google Scholar
Wang, X., & Yu, J. (2015). Limit theory for an explosive autoregressive process. Economics Letters , 126, 176–180.CrossRefGoogle Scholar
Wu, C. F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Annals of Statistics , 14, 1261–1295.Google Scholar
Xu, K. L. (2008). Bootstrapping autoregression under non-stationary volatility. Econometrics Journal , 11, 1–26.CrossRefGoogle Scholar
Xu, K. L., & Phillips, P. C. B. (2008). Adaptive estimation of autoregressive models with time-varying variances. Journal of Econometrics , 142, 265–280.CrossRefGoogle Scholar
Supplementary material: File

Yu and Kejriwal supplementary material

Yu and Kejriwal supplementary material
Download Yu and Kejriwal supplementary material(File)
File 458.6 KB