Weak convergence of the sequential empirical processes of residuals in ARMA models

Bai, Jushan (1991): Weak convergence of the sequential empirical processes of residuals in ARMA models. Published in: Annals of Statistics , Vol. 22, (1994): pp. 2051-2061.

Abstract

This paper studies the weak convergence of the sequential empirical process $\hat{K}_n$ of the estimated residuals in ARMA(p,q) models when the errors are independent and identically distributed. It is shown that, under some mild conditions, $\hat{K}_n$ converges weakly to a Kiefer process. The weak convergence is discussed for both finite and infinite variance time series models. An application to a change-point problem is considered.

• Item Type: MPRA Paper
• Language: English
• Keywords: Time series models, residual analysis, sequential empirical process, weak convergence, Kiefer process, change-point problem
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Bai, J. (1991a). On the partial sums of residuals in autoregressive and moving average models. J. of Time Series Analysis. To appear.

Bai, J. (1991b). Weak convergence of sequential empirical processes of ARMA residuals. Manuscript, Department of Economics, U.C. Berkeley.

Bhansali, R.J. (1988). Consistent order determination for processes with infinite variance. {\em J. R. Stat. Society,} 50 46-60.

Bickel, P.J. and Wichura, M.J. (1971). Convergence for multiparameter stochastic processes and some applications. {\em Ann. Math. Statist.} 42 1656-1670.

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

Boldin, M.V. (1982). Estimation of the distribution of noise in an autoregression scheme. {\em Theory Probab. Appl.} 27 866-871.

Boldin, M.V. (1989). On testing hypotheses in sliding average scheme by the Kolmogorov-Smirnov and $\omega^{2}$ tests. {\em Theory Probab. Appl.,} 34, 699-704.

Brockwell, P.J. and Davis, R.A. (1986). Time Series: Theory and Method. Springer-Verlag, New York.

Chung, K.L. (1968). A Course in Probability Theory. Harcourt, Brace \& World, Inc.

Carlstein, E. (1988). Nonparametric change point estimation. {\em Ann. Statist.}, 16, 188-197.

Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and its Applications. Academic Press, San Diego.

Kanter, M. and Hannan, E.J. (1977). Autoregressive processes with infinite variance. {\em J. Applied Prob.} 14 411-415.

Koul, H. L. (1969). Asymptotic behavior of Wilcoxon type confidence regions in multiple linear regression. {\em Ann. Math. Statist.} 40, 1950-1979.

Koul, H. L. (1984). Tests of goodness-of-fit in linear regression. {\em Colloaquia Mathematica Societatis Janos Bolyai.,} 45. {\em Goodness of fit,} Debrecen, Hungary. 279-315.

Koul, H. L. (1991). A weak convergence result useful in robust autoregression. {\em Statist. Plann. \& Inference}, 29, 1291-308.

Koul, H.L. and Levental, S. (1989). Weak convergence of the residual empirical process in explosive autoregression. {\em Ann. Statist.} 17 1784-1794.

Kreiss, P. (1991). Estimation of the distribution of noise in stationary processes. {\em Metrika,} 38, 285-297.

Loynes, R.M. (1980). The empirical distribution function of residuals from generalized regression. {\em Ann. Statist.} 8 285-298.

Miller, S. M. (1989). Empirical processes based upon residuals from errors-in-variables regressions. {\em Ann. Statist.} 17 282-292.

Mukantseva, L.A. (1977). Testing normality in one-dimensional and multidimensional linear regression. {\em Theory Prob. Appl.} 22 591-602.

Picard, D. (1985). Testing and estimating change-point in time series. {\em Adv. Appl. Prob.} 17 841-867.

Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.