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Additive Nonlinear ARX Time Series and Projection Estimates

Published online by Cambridge University Press:  11 February 2009

Elias Masry  and
Dag Tjøstheim
Elias Masry
Affiliation:
University of California at San Diego
Dag Tjøstheim
Affiliation:
University of Bergen
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Abstract

We propose projections as means of identifying and estimating the components (endogenous and exogenous) of an additive nonlinear ARX model. The estimates are nonparametric in nature and involve averaging of kernel-type estimates. Such estimates have recently been treated informally in a univariate time series situation. Here we extend the scope to nonlinear ARX models and present a rigorous theory, including the derivation of asymptotic normality for the projection estimates under a precise set of regularity conditions.

Type
Articles
Information
Econometric Theory , Volume 13 , Issue 2 , April 1997 , pp. 214 - 252
Copyright
Copyright © Cambridge University Press 1997

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References

REFERENCES

Auestad, B. & Tjøstheim, D. (1991) Functional identification in nonlinear time series. In Roussas, G.G. (ed.), Nonparametric Functional Estimation and Related Topics, pp. 493–507. Amsterdam: Kluwer Academic.CrossRefGoogle Scholar
Chan, K.S. & Tong, H. (1985) On the use of deterministic Lyapunov function for the ergodicity of stochastic difference equations. Advances in Applied Probability 17, 666–678.Google Scholar
Chen, R., Härdle, W., Linton, O., & Severance-Lossin, E. (1995) Nonparametric Estimation of Additive Separable Regression Models. Preprint SFB 373, Institute of Econometrics and Statistics, Humboldt University, Berlin.Google Scholar
Chen, R. & Tsay, R. (1993a) Functional coefficient autoregressive models. Journal of the American Statistical Association 88, 298–308.Google Scholar
Chen, R. & Tsay, R. (1993b) Nonlinear additive ARX models. Journal of the American Statistical Association 88, 955–967.Google Scholar
Engle, R.F., Granger, C.W.J., Rice, J., & Weiss, A. (1986) Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association 81, 310–320.Google Scholar
Fan, J., Härdle, W., & Mammen, E. (1995) Direct Estimation of Low Dimensional Components in Additive Models. Preprint SFB 373, Institute of Econometrics and Statistics, Humboldt University, Berlin.Google Scholar
Friedman, J. (1991) Multivariate adaptive regression splines (with discussion). Annals of Statistics 19, 1–141.Google Scholar
Granger, C.W.J. & Teräsvirta, T. (1993) Modelling Nonlinear Dynamic Relationships. Oxford: Oxford University Press.CrossRefGoogle Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and Its Application. New York: Academic Press.Google Scholar
Hastie, T.J. & Tibshirani, R.J. (1990) Generalized Additive Models. London: Chapman and Hall.Google Scholar
Lewis, P.A. & Stevens, J.G. (1991) Nonlinear modeling of time series using multivariate adaptive regression splines (MARS). Journal of the American Statistical Association 86, 864–877.Google Scholar
Linton, O. & Härdle, W. (1995) Estimation of Additive Regression Models with Known Links. Preprint SFB 373, Institute of Econometrics and Statistics, Humboldt University, Berlin.Google Scholar
Linton, O. & Nielsen, J.P. (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93–101.Google Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of ARCH nonlinear time series: Strong convergence and asymptotic normality, Econometric Theory 11, 258–289.Google Scholar
Newey, W.K. (1994) Kernel estimation of partial means. Econpmetric Theory 10, 253–253.Google Scholar
Pham, D.T. (1986) The mixing property of bilinear and generalized random coefficient autoregressive models. Stochastic Processes and Their Applications 23, 291–300.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1991a) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part 1: Consistency and approximations concepts. Econometric Reviews 10, 125–216.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1991b) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: Asymptotic normality. Econometric Reviews 10, 253–325.CrossRefGoogle Scholar
Ripley, B. (1987) Stochastic Simulation. New York: Wiley.Google Scholar
Robinson, P.M. (1983) Nonparametric estimators for time series. Journal of Time Series Analysis 4, 185–207.Google Scholar
Roussas, G.G. (1990) Nonparametric regression estimation under mixing conditions. Stochastic Processes and Their Application 36, 107–116.Google Scholar
Severance-Lossin, E & Sperlich, S. (1995) Estimation of Derivatives for Additive Separable Models. Preprint SFB 373, Institute of Econometrics and Statistics, Humboldt University, Berlin.Google Scholar
Stone, C.J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13, 689–705.Google Scholar
Teräsvirta, T., Tjøstheim, D., & Granger, C.W.J. (1994) Aspects of modelling nonlinear time series. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, pp. 2919–2960. Amsterdam: Elsevier.Google Scholar
Tjøstheim, D. (1990) Non-linear time series and Markov chains. Advances in Applied Probability 22, 587–611.Google Scholar
Tjøstheim, D. & Auestad, B. (1994a) Nonparametric identification of nonlinear time series: Projections. Journal of the American Statistical Association 89, 1398–1409.Google Scholar
Tjøstheim, D. & Auestad, B. (1994b) Nonparametric identification of nonlinear time series: Selecting significant lags. Journal of the American Statistical Association 89, 1410–1419.Google Scholar
Truong, Y.K. & Stone, C.J. (1992) Nonparametric function estimation involving time series. Annals of Statistics 20, 77–97.Google Scholar
Tweedie, R.L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stochastic Processes and Their Application 3, 385–403.Google Scholar
Tweedie, R.L. (1988) Invariant measures for Markov chains with no irreducibility assumptions. Journal of Applied Probability 25A (A Celebration of Applied Probability), 275–285.Google Scholar
Volkonskii, V.A. & Rozanov, Yu. A. (1959) Some limit theorems for random functions. Theory of Probability and Applications 4, 178–197.Google Scholar
White, H. (1992) Statistical Estimation with Artificial Neural Networks. Preprint, Department of Economics, University of California, San Diego.Google Scholar