Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T06:35:54.542Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERAL LIMIT THEORY FOR NONLINEAR FUNCTIONALS OF NONSTATIONARY TIME SERIES

Published online by Cambridge University Press:  25 November 2024

Qiying Wang  and
Peter C. B. Phillips Open the ORCID record for Peter C. B. Phillips [Opens in a new window]
Qiying Wang*
Affiliation:
University of Sydney
Peter C. B. Phillips
Affiliation:
Yale University, University of Auckland, and Singapore Management University
*
Address correspondence to Qiying Wang, University of Sydney, Sydney, NSW, Australia; e-mail: qiying.wang@sydney.edu.au.
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.

New limit theory is provided for a wide class of sample variance and covariance functionals involving both nonstationary and stationary time series. Sample functionals of this type commonly appear in regression applications and the asymptotics are particularly relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root, or fractional processes. The limit theory is unusually general in that it covers both parametric and nonparametric regressions. Self-normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals interesting strong bimodality in the finite sample distributions of conventional self-normalized statistics similar to the bimodality that can arise in t-ratio statistics based on heavy tailed data. Bimodal behavior in these statistics is due to the presence of long memory innovations and is shown to persist for very large sample sizes even though the limit theory is Gaussian when the long memory innovations are stationary. Bimodality is shown to occur even in the limit theory when the long memory innovations are nonstationary. To address these complications, new self-normalized versions of the test statistics are introduced that deliver improved approximations that can be used for inference.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 58
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

Wang acknowledges research support from the Australian Research Council (Grant No. DP170104385). Phillips acknowledges research support from the NSF (Grant No. SES 18-50860) and a Kelly Fellowship at the University of Auckland.

References

REFERENCES

Akonom, J. (1993). Comportement asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’IHP Probabilités et Statistiques , 29, 57–81.Google Scholar
Billingsley, P. (1968). Convergence of probability measures . John Wiley & Sons.Google Scholar
Billingsley, P. (1974). Conditional distributions and tightness. The Annals of Probability , 2, 480–485.CrossRefGoogle Scholar
Borodin, A. N., & Ibragimov, I. A. (1995). Limit theorems for functionals of random walks , 195. American Mathematical Society.Google Scholar
Cai, Z., Li, Q., & Park, J. Y. (2009). Functional-coefficient models for nonstationary time series data. Journal of Econometrics , 148, 101–113.CrossRefGoogle Scholar
Chan, N., & Wang, Q. (2015). Nonlinear regressions with nonstationary time series. Journal of Econometrics , 185, 182–195.CrossRefGoogle Scholar
Dong, C., & Linton, O. (2018). Additive nonparametric models with time variable and both stationary and nonstationary regressors. Journal of Econometrics , 207, 212–236.CrossRefGoogle Scholar
Duffy, J. A. (2016). A uniform law for convergence to the local times of linear fractional stable motions. Annals of Applied Probability , 26, 45–72.CrossRefGoogle Scholar
Duffy, J. A. (2020). Asymptotic theory for kernel estimators under moderate deviations from a unit root, with an application to the asymptotic size of nonparametric tests. Econometric Theory , 36, 559–582.CrossRefGoogle Scholar
Duffy, J. A., & Kasparis, I. (2021). Estimation and inference in the presence of fractional d = 1/2 and weakly nonstationary processes. Annals of Statistics , 49, 1195–1217.CrossRefGoogle Scholar
Fiorio, C. V., Hajivassiliou, V. A., & Phillips, P. C. B. (2010). Bimodal t-ratios: The impact of thick tails on inference. The Econometrics Journal , 13, 271–289.CrossRefGoogle Scholar
Gao, J., & Phillips, P. C. B. (2013). Semiparametric estimation in triangular system equations with nonstationarity. Journal of Econometrics , 176, 59–79.CrossRefGoogle Scholar
Geman, D., & Horowitz, J. (1980). Occupation densities. The Annals of Probability , 8, 1–67.CrossRefGoogle Scholar
Gozalo, P., & Linton, O. (2000). Local nonlinear least squares: Using parametric information in nonparametric regression. Journal of Econometrics , 99, 63–106.CrossRefGoogle Scholar
Hu, Z., Phillips, P. C. B., & Wang, Q. (2021). Nonlinear cointegrating power function regression with endogeneity. Econometric Theory , 37, 1173–1213.CrossRefGoogle Scholar
Jeganathan, P. (2004). Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Annals of Probability , 32, 1771–1795.CrossRefGoogle Scholar
Jeganathan, P. (2008). Limit theorems for functionals of sums that converge to fractional stable motions. Cowles Foundation Discussion Paper No. 1649. Yale University.Google Scholar
Karlsen, H. A., & Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Annals of Statistics , 29, 372–416.CrossRefGoogle Scholar
Li, D., Tjøstheim, D., & Gao, J. (2016). Estimation in nonlinear regression with Harris recurrent Markov chains. Annals of Statistics , 44, 1957–1987.CrossRefGoogle Scholar
Liang, H.-Y., Shen, Y., & Wang, Q. (2023). Functional-coefficient cointegrating regression with endogeneity. In Essays in Honor of Joon Y. Park: Econometric Theory (pp. 157–186). Emerald Publishing Limited.CrossRefGoogle Scholar
Logan, B. F., Mallows, C. L., Rice, S., & Shepp, L. A. (1973). Limit distributions of self-normalized sums. The Annals of Probability , 1, 788–809.CrossRefGoogle Scholar
Park, J. Y. (2014). Nonstationary nonlinearity: A survey on Peter Phillips’s contributions with a new perspective. Econometric Theory , 30, 894–922.CrossRefGoogle Scholar
Park, J. Y., & Phillips, P. C. B. (1999). Asymptotics for nonlinear transformations of integrated time series. Econometric Theory , 15, 269–298.CrossRefGoogle Scholar
Park, J. Y., & Phillips, P. C. B. (2000). Nonstationary binary choice. Econometrica , 68, 1249–1280.CrossRefGoogle Scholar
Park, J. Y., & Phillips, P. C. B. (2001). Nonlinear regressions with integrated time series. Econometrica , 69, 117–161.CrossRefGoogle Scholar
Petrov, V. V. (1995). Limit theorems of probability theory; sequences of independent random variables . Oxford Studies in Probability. Clarendon Press.Google Scholar
Phillips, P. C. B. (2009). Local limit theory and spurious nonparametric regression. Econometric Theory , 25, 1466–1497.CrossRefGoogle Scholar
Phillips, P. C. B., & Park, J. Y. (1998). Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion Paper, No. 1181. Yale University.Google Scholar
Phillips, P. C. B., & Wang, Y. (2023). When bias contributes to variance: True limit theory in functional coefficient cointegrating regression. Journal of Econometrics , 232, 469–489.CrossRefGoogle Scholar
Sun, Y., Cai, Z., & Li, Q. (2016). A consistent nonparametric test on semiparametric smooth coefficient models with integrated time series. Econometric Theory , 32, 988–1022.CrossRefGoogle Scholar
Sun, Y., & Li, Q. (2011). Data-driven bandwidth selection for nonstationary semiparametric models. Journal of Business & Economic Statistics , 29, 541–551.CrossRefGoogle Scholar
Tjøstheim, D. (2020). Some notes on nonlinear cointegration: A partial review with some novel perspectives. Econometric Reviews , 39, 655–673.CrossRefGoogle Scholar
Tu, Y., & Wang, Y. (2022). Spurious functional-coefficient regression models and robust inference with marginal integration. Journal of Econometrics , 229, 396–421.CrossRefGoogle Scholar
Wang, Q. (2015). Limit theorems for nonlinear cointegrating regression (Vol. 5). World Scientific.CrossRefGoogle Scholar
Wang, Q., Lin, Y.-X., & Gulati, C. M. (2003). Strong approximation for long memory processes with applications. Journal of Theoretical Probability , 16, 377–389.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2009a). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory , 25, 710–738.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2009b). Structural nonparametric cointegrating regression. Econometrica , 77, 1901–1948.Google Scholar
Wang, Q., & Phillips, P. C. B. (2011). Asymptotic theory for zero energy functionals with nonparametric regression applications. Econometric Theory , 27, 235–259.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2016). Nonparametric cointegrating regression with endogeneity and long memory. Econometric Theory , 32, 359–401.CrossRefGoogle Scholar
Wang, Q., Phillips, P. C. B., & Kasparis, I. (2021). Latent variable nonparametric cointegrating regression. Econometric Theory , 37, 138–168.CrossRefGoogle Scholar
Xiao, Z. (2009). Functional-coefficient cointegration models. Journal of Econometrics , 152, 81–92.CrossRefGoogle Scholar