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PARAMETRIC SPECIFICATION TEST FOR NONLINEAR AUTOREGRESSIVE MODELS

Published online by Cambridge University Press:  02 October 2014

Kun Ho Kim ,
Ting Zhang  and
Wei Biao Wu
Kun Ho Kim*
Affiliation:
Hanyang University
Ting Zhang
Affiliation:
Boston University
Wei Biao Wu
Affiliation:
University of Chicago
*
*Address correspondence to Kun Ho Kim, Department of Economics and Finance, 222 Wangsimni-ro, Seongdong-gu Seoul 133-791, South Korea; e-mail: kunhokim8@gmail.com.
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Abstract

The paper considers testing parametric assumptions on the conditional mean and variance functions for nonlinear autoregressive models. To this end, we compare the kernel density estimate of the marginal density of the process with a convolution-type density estimate. It is shown that, interestingly, the latter estimate has a parametric $\left( {\sqrt n } \right)$ rate of convergence, thus substantially improving the classical kernel density estimates whose rates of convergence are much inferior. Our results are confirmed by a simulation study for threshold autoregressive processes and autoregressive conditional heteroskedastic processes.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 5 , October 2015 , pp. 1078 - 1101
Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

Aït-Sahalia, Y. (1996) Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, 385–426.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1999) Transition densities for interest rate and other nonlinear diffusions. Journal of Finance 54, 1361–1395.CrossRefGoogle Scholar
Aït-Sahalia, Y., Fan, J., & Peng, H. (2009) Nonparametric transition-based tests for jump-diffusions. Journal of the American Statistical Association 104, 1102–1116.CrossRefGoogle Scholar
Amemiya, T. (1985) Advanced Econometrics. Harvard University Press.Google Scholar
Andel, J., Netuka, I., & Svara, K. (1984) On threshold autoregressive processes. Kibernetika 20, 89–106.Google Scholar
Bahadur, R.R. (1966) A note on quantiles in large samples. Annals of Mathematical Statistics 37, 577–580.CrossRefGoogle Scholar
Bai, J. (2003) Testing parametric conditional distribution of dynamic models. Review of Economics and Statistics 85, 531–549.CrossRefGoogle Scholar
Bandi, F.M. & Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.CrossRefGoogle Scholar
Bandi, F.M. & Phillips, P.C.B. (2007) A simple approach to the parametric estimation of potentially nonstationary diffusions. Journal of Econometrics 137, 354–395.CrossRefGoogle Scholar
Bandi, F.M. & Phillips, P.C.B. (2009) Nonstationary continuous-time processes. In Aït-Sahalia, Y. & Hansen, L.P. (eds.), Handbook of Financial Econometrics, pp. 139–202. North Holland.Google Scholar
Bhardwaj, G., Corradi, V., & Swanson, N.R. (2008) A simulation-based specification test for diffusion processes. Journal of Business and Economic Statistics 26, 176–193.CrossRefGoogle Scholar
Bickel, P. & Ritov, Y. (1988) Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhya Series A 50, 381–393.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Black, F. & Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654.CrossRefGoogle Scholar
Borkovec, M. (2000) Extremal behavior of the autoregressive process with ARCH(1) errors. Stochastic Processes and their Applications 85, 189–207.CrossRefGoogle Scholar
Chan, K.C., Karolyi, A.G., Longstaff, F.A., & Sanders, A.B. (1992) An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227.Google Scholar
Chan, N.H. & Peng, L. (2005) Weighted least absolute deviation estimation for an AR(1) process with ARCH(1) errors. Biometrika 92, 477–484.CrossRefGoogle Scholar
Chapman, D.A. & Pearson, N.D. (2000) Is the short rate drift actually nonlinear? Journal of Finance 55, 355–388.CrossRefGoogle Scholar
Chen, S.X., Gao, J.T., & Tang, C.Y. (2008) A test for model specification of diffusion processes. Annals of Statistics 36, 167–198.CrossRefGoogle Scholar
Cline, D.B.H. & Pu, H.H. (2004) Stability and the Lyapounov exponent of threshold AR-ARCH models. Annals of Applied Probability 14, 1920–1949.CrossRefGoogle Scholar
Constantinides, G.M. (1992) A theory of the nominal term structure of interest rates. Review of Financial Studies 5, 531–552.CrossRefGoogle Scholar
Corradi, V. & Swanson, N.R. (2006) Bootstrap conditional distribution tests in the presence of dynamic misspecification. Journal of Econometrics 133, 779–806.CrossRefGoogle Scholar
Courtadon, G. (1982) The pricing of options on default-free bonds. Journal of Finance and Quantitative Analysis 17, 75–100.CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53, 385–403.CrossRefGoogle Scholar
Duffie, D. & Kan, R. (1996) A yield factor model of interest rates. Mathematical Finance 6, 379–406.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Fan, J.Q. & Yao, Q.W. (2005) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer.Google Scholar
Freedman, D.A. (1975) On tail probabilities for martingales. Annals of Probability 3, 100–118.CrossRefGoogle Scholar
Frees, E.W. (1994) Estimating densities of functions of observations. Journal of the American Statistical Association 89, 517–525.CrossRefGoogle Scholar
Gao, J. & King, M. (2004) Adaptive testing in continuous-time diffusion models. Econometric Theory 20, 844–882.CrossRefGoogle Scholar
Giné, E. & Mason, D.M. (2007) On local U-statistic processes and the estimation of densities of functions of several sample variables. Annals of Statistics 35, 1105–1145.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Hall, P. & Yao, Q.W. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.CrossRefGoogle Scholar
Hannan, E.J. (1973) Central limit theorems for time series regression. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 26, 157–170.CrossRefGoogle Scholar
Hong, Y. & Li, H. (2005) Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18, 37–84.CrossRefGoogle Scholar
Jones, D.A. (1976) Nonlinear autoregressive processes. Proceedings of the Royal Society A 360, 71–95.Google Scholar
Klimko, L.A. & Nelson, P.I. (1978) On conditional least squares estimation for stochastic processes. Annals of Statistics 6, 629–642.CrossRefGoogle Scholar
Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society, Series B 66, 63–78.CrossRefGoogle Scholar
Ling, S. (2007) A double AR(p) model: Structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Liu, W.D. & Wu, W.B. (2010) Simultaneous nonparametric inference of time series. Annals of Statistics 38, 2388–2421.CrossRefGoogle Scholar
Pritsker, M. (1998) Nonparametric density estimation and tests of continuous time interest rate models. Review of Financial Studies 11, 449–487.CrossRefGoogle Scholar
Robinson, P.M. (1983) Nonparametric estimators for time series. Journal of Time Series Analysis 4, 185–207.CrossRefGoogle Scholar
Saavedra, A. & Cao, R. (2000) On the estimation of the marginal density of a moving average process. Canadian Journal of Statistics 28, 799–815.CrossRefGoogle Scholar
Schick, A. & Wefelmeyer, W. (2004) Root n consistent density estimators for sums of independent random variables. Journal of Nonparametric Statistics 16, 925–935.CrossRefGoogle Scholar
Schick, A. & Wefelmeyer, W. (2007) Uniformly root-n consistent density estimators for weakly dependent invertible linear processes. Annals of Statistics 35, 815–843.CrossRefGoogle Scholar
Scott, D.W. (1992) Multivariate Density Estimation. Theory, Practice and Visualization. Wiley.CrossRefGoogle Scholar
Shao, X.F. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 1773–1801.CrossRefGoogle Scholar
Silverman, B.W. (1986) Density Estimation. Chapman and Hall.Google Scholar
Stanton, R. (1997) A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance 52, 1973–2002.CrossRefGoogle Scholar
Tong, H. (1990) Nonlinear Time Series: A Dynamical System Approach. Oxford University Press.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.CrossRefGoogle Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences of the United States of America 102, 14150–14154.CrossRefGoogle Scholar
Wu, W.B. & Shao, X.F. (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436.CrossRefGoogle Scholar
Zhao, Z.B. (2008) Parametric and nonparametric models and methods in financial econometrics. Statistics Surveys 2, 1–42.CrossRefGoogle Scholar
Zhao, Z.B. (2011) Nonparametric model validations for hidden Markov models with applications in financial econometrics. Journal of Econometrics 162, 225–239.CrossRefGoogle ScholarPubMed