PERGAMON zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA M athematical and Computer zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Estimation with Infinite MATHEMATICAL Modelling 29 (1999) 177-180 COMPUTER MODELLING for Regression Variance Errors A. THAVANESWARAN Department of Statistics, University of Manitoba Winnipeg, Manitoba, Canada R3T 2N2 S. PEIRIS School of Mathematics and Statistics, University of Sydney New South Wales, Australia Abstract-This paper addresses the problem of modelling time series with nonstationarity from a finite number of observations. Problems encountered with the time varying parameters in regression type models led to the smoothing techniques. The smoothing methods basically rely on the finiteness of the error variance, and thus, when this requirement fails, particularly when the error distribution is heavy tailed, the existing smoothing methods due to [l], are no longer optimal. In this paper, we propose a penalized minimum dispersion method for time varying parameter estimation when a regression model generated by an infinite variance stable process with characteristic exponent a E (1,2). Recursive estimates are evaluated and it is shown that these estimates for a nonstationary process with normal errors is a special case. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-Stable distribution, Penalized dispersion, Nonstationary, Recursive estimate. 1. INTRODUCTION The occurrence of sharp spikes or occasional bursts of outlying observations, as well as high tail probabilities in time series data, is usually attributed to the possibility of the underlying process having an infinite variance stable distribution. The existence of such processes has generated a great deal of interest in their modeling and inference by Autoregressive Moving Average (ARMA) processes in a wide range of fields. Particular examples include the work of Mandelbrot [2] and Fama [3] in econometrics and Stuck and Kleiner [4] in communications. Granger and Orr [5] discuss some approaches to recognizing infinite variance processes. The recent work of Kokoska and Taqqu (61, and the references therein give the complete theory of fractional innovations with infinite variance. Suppose {qt, t = 0, fl, f2,. . . } is an identically distributed independent sequence of random variables. The random variable n is said to be stable if the distribution of Y = VI+ 772 + . I - + q-, is the same as the distribution of cr,,~ + b, for constants on > 0 and zyxwvutsrqponmlkjihgfedcbaZYXW b, and integer n. Such a distribution has characteristic function @P(Y) = E(exp(ivq)) and is given by ~( )_ exp(iqj3-6]v]a(1-ir3fitan(y))), ifcrfl v- 1 exp (ivp - 61~1~ (1 - 2i0$1n(]V]))) , if o = 1, 08957177/99/g - see front matter. @ 1999 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(99)00100-4 Typeset by -Q&-W