Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T18:24:11.180Z Has data issue: false hasContentIssue false
  • English
  • Français

NONSTATIONARY NONLINEARITY: A SURVEY ON PETER PHILLIPS’S CONTRIBUTIONS WITH A NEW PERSPECTIVE

Published online by Cambridge University Press:  11 April 2014

Joon Y. Park
Joon Y. Park*
Affiliation:
Department of Economics, Indiana University and Sungkyunkwan University
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we provide a survey of Peter Phillips’s works on the econometrics for models with nonstationary nonlinearity, and some of the extensions that were made possible due to his original contributions. Parametric and nonparametric models are considered in both discrete time and continuous time setups. Although some of the asymptotics in the paper are applicable more generally for a wide variety of nonstationary models, we mainly analyze models with nonstationary processes that allow for the functional limit theory with limit processes having well defined local times.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 4 , August 2014 , pp. 894 - 922
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aït-Sahalia, Y. & Park, J.Y. (2012) Stationarity-based specification tests for diffusions when the process is nonstationary. Journal of Econometrics 169, 279–292.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Park, J.Y. (2013) Bandwidth Selection and Asymptotic Properties of Local Nonparametric Estimators in Possibly Nonstationary Continuous-time Models. Working paper, Indiana University.Google Scholar
Akonom, J. (1993) Comportement asymptotique du temps d’occupation du processus des sommes partielles. Annales de l’Institut Henri Poincaré 29, 57–81.Google Scholar
Astrauskas, A. (1983) Limit theorems for sums of linearly generated random variables. Lithuanian Mathematical Journal 23, 127–134.Google Scholar
Avram, F. & Taqqu, M.S. (1992) Weak convergence of sums of moving averages in the α-stable domain of attraction. Annals of Probability 20, 483–503.Google Scholar
Bandi, F.M. & Corradi, V. (2011) Nonparametric Nonstationarity Tests. Working paper, University of Warwick.Google Scholar
Bandi, F.M. & Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.Google Scholar
Bandi, F.M. & Phillips, P.C.B. (2010) Nonstationary continuous-time processes. In Aït-Sahalia, Y. & Hansen, L.P. (eds.), Handbook of Financial Econometrics, pp. 139–201. North Holland.Google Scholar
Beran, J. (1994) Statistics for Long-Memory Processes. Chapman & Hall.Google Scholar
Borodin, A. & Ibragimov, I. (1995) Limit Theorems for Functionals of Random Walks. Proceedings of the Steklov Institute of Mathematics, 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.Google Scholar
Chang, Y. (2002) Nonlinear IV unit root tests in panels with cross-sectional dependency. Journal of Econometrics 110, 261–292.Google Scholar
Chang, Y., Jiang, B., & Park, J.Y. (2012) Nonstationary regression with logistic transition. Econometrics Journal 15, 255–287.Google Scholar
Chang, Y. & Nguyen, C.M. (2012) Residual based tests for cointegration in dependent panels. Journal of Econometrics 167, 504–520.Google Scholar
Chang, Y. & Park, J.Y. (2003) Index models with integrated time series. Journal of Econometrics 114, 73–106.Google Scholar
Chang, Y. & Park, J.Y. (2011) Endogeneity in nonlinear regressions with integrated time series. Econometric Reviews 30, 51–87.Google Scholar
Chang, Y., Park, J.Y., & Phillips, P.C.B. (2001) Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometrics Journal 4, 1–36.Google Scholar
Chang, Y. & Song, W. (2009) Testing for unit roots in small panels with short-run and long-run cross-sectional dependencies. Review of Economic Studies 73, 903–935.Google Scholar
Choi, I. & Saikkonen, P. (2010) Tests for nonlinear cointegration. Econometric Theory 26, 682–709.CrossRefGoogle Scholar
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009a) Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25, 1869–1892.Google Scholar
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009b) Specification testing in nonlinear and nonstationary time series autoregression. Annals of Statistics 37, 3893–3923.Google Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Han, H. & Park, J.Y. (2008) Time series properties of arch processes with persistent covariates. Journal of Econometrics 146, 275–292.Google Scholar
Han, H. & Park, J.Y. (2012) ARCH/GARCH with persistent covariate: Asymptotic theory of MLE. Journal of Econometrics 167, 95–112.Google Scholar
Hansen, B. (1991) GARCH(1,1) processes are near epoch dependent. Economics Letters 36,181–186.CrossRefGoogle Scholar
Hong, S.H. & Phillips, P.C.B. (2010) Testing linearity in cointegrating relations with an application to purchasing power parity. Journal of Business & Economic Statistics 28, 96–114.CrossRefGoogle Scholar
Hu, L. & Phillips, P.C.B. (2004a) Dynamics of the federal funds target rate: A nonstationary discrete choice approach. Journal of Applied Econometrics 19, 851–867.CrossRefGoogle Scholar
Hu, L. & Phillips, P.C.B. (2004b) Nonstationary discrete choice. Journal of Econometrics 120,103–138.CrossRefGoogle Scholar
Ibragimov, I. & Linnik, Y.V. (1971) Independent and Stationary Sequences of Random Variables. Walters-Noordhoff.Google Scholar
Ibragimov, R. & Phillips, P.C.B. (2008) Regression asymptotics using martingale convergence methods. Econometric Theory 24, 888–947.CrossRefGoogle Scholar
Jeganathan, P. (2004) Convergence of functional of sums of r.v.s to local times of fractional stable motions. Annals of Probability 32, 1771–1795.Google Scholar
Jeganathan, P. (2008) Limit Theorems for Functionals of Sums that Converge to Fractional Brownian and Stable Motions. Working paper, Indian Statistical Institute.Google Scholar
Jeong, M. & Park, J.Y. (2010) Asymptotics for Maximum Likelihood Estimators of Diffusion Models. Working paper, Indiana University.Google Scholar
Jeong, M. & Park, J.Y. (2013) Asymptotics for Maximum Likelihood Estimators of Jump Diffusion Models. Working paper, Indiana University.Google Scholar
de Jong, R.M. (2004) Addendum to ‘Asymptotics for nonlinear transformations of integrated time series’. Econometric Theory 20, 627–635.CrossRefGoogle Scholar
de Jong, R.M. & Wang, C.H. (2005) Further results on the asymptotics for nonlinear transformations of integrated time series. Econometric Theory 21, 413–430.Google Scholar
Kanaya, S. (2011) A Nonparametric Test for Stationarity in Continuous-time Markov Processes. Working paper, University of Oxford.Google Scholar
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2007) Nonparametric estimation in a nonlinear cointegration type model. Annals of Statistics 35, 252–299.Google Scholar
Karlsen, H.A. & Tjøstheim, D. (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372–416.Google Scholar
Kasahara, Y. & Maejima, M. (1988) Weighted sums of i.i.d. random variables attracted to integrals of stable processes. Probability Theory and Related Fields 78, 75–96.Google Scholar
Kasparis, I. & Phillips, P.C.B. (2012) Dynamic misspecification in nonparametric cointegrating regression. Journal of Econometrics 168, 270–284.Google Scholar
Kim, J. & Park, J.Y. (2013a) Asymptotics for Recurrent Diffusions with Application to High Frequency Regression Working paper, Indiana University.Google Scholar
Kim, J. & Park, J.Y. (2013b) Mean Reversion and Unit Root Properties of Diffusion Models. Working paper, Indiana University.Google Scholar
McLeish, D. (1975) Invariance principles for dependent variables. Probability Theory and Related Fields 32, 165–178.Google Scholar
McLeish, D. (1977) On the invariance principle for nonstationary mixingales. Annals of Probability 5, 616–621.Google Scholar
Merlevéde, F., Peligrad, M., & Utev, S. (2006) Recent advances in invariance principles for stationary sequences. Probability Surveys 3, 1–36.CrossRefGoogle Scholar
Miller, J.I. & Park, J.Y. (2010) Nonlinearity, nonstationarity, and thick tails: How they interact to generate persistency in memory. Journal of Econometrics 155, 83–89.Google Scholar
Myklebust, T., Karlsen, H.A., & Tjøstheim, D. (2012) Null recurrent unit root processes. Econometric Theory 28, 1–41.Google Scholar
Park, J.Y. (2002) Nonlinear nonstationary heterskedasticity. Journal of Econometrics 110, 383–415.CrossRefGoogle Scholar
Park, J.Y. (2006) Spatial Analysis of Time Series. Working paper, Texas A& M University.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.Google Scholar
Park, J.Y. & Phillips, P.C.B. (2000) Nonstationary binary choice. Econometrica 68, 1249–1280.Google Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 1452–1498.Google Scholar
Phillips, P.C.B. (1982) On the consistency of nonlinear FIML. Econometrica 50, 1307–1324.Google Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory of autoregression. Biometrika 74,535–548.Google Scholar
Phillips, P.C.B. (1996) Econometric model determination. Econometrica 64, 763–812.Google Scholar
Phillips, P.C.B. (2001) Descriptive econometrics for non-stationary time series with empirical applications. Journal of Applied Econometrics 16, 389–413.Google Scholar
Phillips, P.C.B. (2005) Econometric analysis of Fisher’s equation. American Journal of Economics and Sociology 64, 125–168.CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557–614.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–495.Google Scholar
Phillips, P.C.B., Jin, S., & Hu, L. (2007) Nonstationary discrete choice: A corrigendum and addendum. Journal of Econometrics 141, 1115–1130.CrossRefGoogle Scholar
Phillips, P.C.B. & Park, J.Y. (1998) Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion paper 1181.Google Scholar
Phillips, P.C.B., Park, J.Y., & Chang, Y. (2004) Nonlinear instrumental variable estimation of anautoregression. Journal of Econometrics 118, 219–246.Google Scholar
Phillips, P.C.B. & Solo, V. ( 1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Phillips, P.C.B. & Wickens, M.R. (1978) Exercises in Econometrics. Ballinger and Phillip Allan.Google Scholar
Revuz, D. & Yor, M. (2005) Continuous Martingales and Brownian Motion, 3rd ed.Springer-Verlag.Google Scholar
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 301–340.CrossRefGoogle Scholar
Shi, X. & Phillips, P.C.B. (2012) Nonlinear cointegrating regression under weak identification. Econometric Theory 28, 509–547.Google Scholar
Wang, Q. & Phillips, P.C.B. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.Google 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.Google Scholar
Wang, Q. & Phillips, P.C.B. (2012) A specification test for nonlinear nonstationary models. Annals of Statistics 40, 727–758.CrossRefGoogle Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems fordependent heterogeneous processes. Econometric Theory 4, 210–230.Google Scholar
Zhang, Y., Su, L., & Phillips, P.C.B. (2012) Asymptotics for linear processes. Econometrics Journal 15, 56–100.Google Scholar