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Model-free Asymptotically Best Forecasting of Stationary Economic Time Series

Published online by Cambridge University Press:  11 February 2009

Herman J. Bierens
Herman J. Bierens
Affiliation:
Free University, Amsterdam
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Abstract

Given observations on a stationary economic vector time series process we show that the best h-step ahead forecast (best in the sense of having minimal mean square forecast error) of one of the variables can be consistently estimated by nonparametric regression on an ARMA memory index. Our approach is based on a combination of the ARMA memory index modeling approach of Bierens [7] with a modification to time series of the nonparametric kernel regression approach of Devroye and Wagner [16]. This approach is truly model-free, as no explicit specification of the distribution of the data generating process is needed.

Type
Articles
Information
Econometric Theory , Volume 6 , Issue 3 , September 1990 , pp. 348 - 383
Copyright
Copyright © Cambridge University Press 1990

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References

1.Bierens, H.JRobust Methods and Asymptotic Theory in Nonlinear Econometrics. New York: Springer Verlag, 1981.CrossRefGoogle Scholar
2.Bierens, H.J.A uniform weak law of large numbers under ψ-mixing with application to nonlinear least squares estimation. Statistica Neerlandica 36 (1982): 81–86.CrossRefGoogle Scholar
3.Bierens, H.JUniform consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association 11 (1983): 699–707.CrossRefGoogle Scholar
4.Bierens, H.JModel specification testing of time series regressions. Journal of Econometrics 26 (1984): 323–353.CrossRefGoogle Scholar
5.Bierens, H.J. Kernel estimators of regression functions. In Bewley, T.F. (ed.), Advances in Econometrics 1985, Vol. I. New York: Cambridge University Press, 1987, 99–144.Google Scholar
6.Bierens, H.J.ARMAX model specification testing, with an application to unemployment in the Netherlands. Journal of Econometrics 35 (1987): 161–190.CrossRefGoogle Scholar
7.Bierens, H.J.ARMA memory index modeling of economic time series (with discussion). Econometric Theory 4 (1988): 35–59.CrossRefGoogle Scholar
8.Bierens, H.JReply to comments on ARMA memory index modeling of economic time series. Econometric Theory 4 (1988): 70–76.CrossRefGoogle Scholar
9.Bierens, H.J. & Hartog, J.. Nonlinear regression with discrete explanatory variables, with an application to the earnings function. Journal of Econometrics 38 (1988): 269–299.CrossRefGoogle Scholar
10.Billingsley, P.Convergence of Probability Measures. New York: John Wiley, 1968.Google Scholar
11.Box, G.E.P. & Jenkins, G.M.. Time Series Analysis, Forecasting and Control. San Francisco: Holden-Day, 1976.Google Scholar
12.Box, G.E.P. & Pierce, D.A.. Distribution of residual autocorrelations in autoregressive integrated moving average models. Journal of the American Statistical Association 72 (1970): 397–402.Google Scholar
13.Chung, K.L.A Course in Probability Theory. New York-London: Academic Press, 1974.Google Scholar
14.Collomb, G.Estimation non-parame'trique de la régression: Revue bibliographique. International Statistical Review 49 (1981): 75–93.CrossRefGoogle Scholar
15.Collomb, G.Nonparametric regression: An up-to-date bibliography. Statistics 16 (1985): 309–324.CrossRefGoogle Scholar
16.Devroye, L.P. & Wagner, T.J.. Distribution-free consistency results in non-parametric discrimination and function estimation. Annals of Statistics 8 (1980): 231–239.CrossRefGoogle Scholar
17.Dunford, N. & Schwartz, J.TLinear Operators, Part I: General Theory. New York: Wiley Interscience, 1957.Google Scholar
18.El Badawi, I., Gallant, A.R. & Souza, O.. An elasticity can be estimated consistently without a priori knowledge of functional form. Econometrica 51 (1983): 1731–1735.CrossRefGoogle Scholar
19.Fildes, R.Quantitative forecasting–the state of the art: Econometric models. Journal of the Operations Research Society 36 (1985): 549–580.Google Scholar
20.Friedman, J.H. & Stuetzle, W.. Projection pursuit regression. Journal of the American Statistical Association 76 (1981): 817–823.CrossRefGoogle Scholar
21.Fuller, W.AIntroduction to Statistical Time Series. New York: John Wiley, 1976.Google Scholar
22.Gallant, R.AOn the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics 15 (1981): 211–245.CrossRefGoogle Scholar
23.Gallant, R.AUnbiased determination of production technologies. Journal of Econometrics 20 (1982): 285–323.CrossRefGoogle Scholar
24.Gallant, R.AThe Fourier flexible form. American Journal of Agricultural Economics 66 (1984): 204–208.CrossRefGoogle Scholar
25.Gallant, R.A. & White, H.. A Unified Theory of Estimation and Inference for Nonlinear Models. Manuscript, University of California, San Diego, 1986.Google Scholar
26.Hannan, E.JNonlinear time-series regression. Journal of Applied Probability 8 (1971): 767–780.CrossRefGoogle Scholar
27.Huber, P.Projection pursuit (with discussion). Annals of Statistics 13 (1985): 435–475.Google Scholar
28.Ibragimov, I.A. & Linnik, Yu. V.. Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff, 1971.Google Scholar
29.Li, K.C.Asymptotic optimality for Cp, CL, cross-validation and generalized crossvalidation: Discrete index set. Annals of Statistics 15 (1987): 958–975.CrossRefGoogle Scholar
30.Ljung, G.M. & Box, G.E.P.. On a measure of lack of fit in times series models. Biometrica 65 (1978): 297–303.CrossRefGoogle Scholar
31.Marron, J.S.Automatic smoothing parameter selection: A review. Empirical Economics 13 (1988): 65–86.CrossRefGoogle Scholar
32.Mc.Leish, D.L.A maximal inequality and dependent strong laws. Annals of Probability 3 (1975): 829–839.Google Scholar
33.Nadaraya, E.A.On estimating regression. Theory of Probability and its Applications 9 (1964): 141–142.CrossRefGoogle Scholar
34.Parzen, E. On the estimation of a probability density function and mode. Annals of Mathematical Statistics 33 (1962): 1065–1076.CrossRefGoogle Scholar
35.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
36.Rissanen, J.Comment on: ARMA memory index modeling of economic time series. Econometric Theory 4 (1988): 61–64.Google Scholar
37.Robinson, P.M.Nonlinear regression for multiple time series. Journal of Applied Probability 9 (1972): 758–768.CrossRefGoogle Scholar
38.Robinson, P.M.Nonparametric estimators for time series. Journal of Time Series Analysis (1983): 185–207.CrossRefGoogle Scholar
39.Robinson, P.M.Kernel estimation and interpolation for time series containing missing observations. Annals of the Institute of Statistical Mathematics 36 (1984): 403–417.CrossRefGoogle Scholar
40.Shiller, R.JRational expectations and the dynamic structure of macroeconometric models: A critical review. Journal of Monetary Economics 4 (1978): 1–44.CrossRefGoogle Scholar
41.Sims, C.A.Comment on: ARMA memory index modeling of economic time series. Econometric Theory 4 (1988): 64–69.Google Scholar
42.Watson, G.S.Smooth regression analysis. Sankhya A 26 (1964): 359–372.Google Scholar
43.White, H. & Domowitz, I.. Nonlinear regression with dependent observations. Econometrica 52(1984): 143–161.CrossRefGoogle Scholar