On the First-Order Autoregressive Process with Infinite Variance

Abstract

For a first-order autoregressive process Y t = $Y t _ x + e ti where the e/s are i.i.d. and belong to the domain of attraction of a stable law, the strong con-' sistency of the ordinary least-squares estimator b n of 0 is obtained for 0 = 1 , and the limiting distribution of b n is established as a functional of a Levy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the e/s are heavy-tailed. INTRODUCTIONConsider the first-order autoregressive process (AR (1)) where $ is the unknown parameter to be estimated from the data {Y t } and {e,} is the random disturbances. Customarily, /? is estimated by its leastsquares estimate /> -[ V y 2 11 V y i Y°nWhen the disturbances f e r } are independent and identically distributed (i.i.d.) or form a martingale difference sequence with (2 + b) moments 9 b > 0, the asymptotic properties of b n such as strong consistency and weak convergence are well-studied both for |/?| < 1 and |/S| = 1. See for example, Lai and Wei [14] and the references therein. Recently, there has been considerable interest in the asymptotic properties of b n when the e/s are heavy-tailed. Heavy-tailed distributions are useful in modeling certain economic variables and stock price changes. Some references are Fama [6], Mandelbrot [15], McCullough [16], and DuMouchel [5]. When the e/s are i.i.d. with common distribution function F which belongs to the domain of attraction of a stable law with index a < 2, Hannan and Kanter [9] and Knight [12] establish the strong consistency of the leastsquares estimates b n of fi for a stationary AR(/?) model. This includes (1) 'Research supported in part by the National Science Foundation through Grant DMS-8503358 and ECE-8513980. We wish to thank the editor and two referees for helpful discussions and comments on an earlier draft. 354

Unit Root Inference for Non-Stationary Linear Processes Driven by Infinite Variance Innovations

“…The asymptotic null distributions of conventional regression-based unit root statistics, such as the ordinary least squares (OLS) based statistics of Dickey and Fuller (1979), Said and Dickey (1987) and the semi-parametric statistic of Phillips (1987), di¤er from the …nite variance case when the innovations lie in the domain of attraction of a stable law, such that they have in…nite variance [IV]; see, in particular, Chan and Tran (1989), Phillips (1990), Samarakoon and Knight (2009), Rachev et al (1998) and Caner (1998). Indeed, in such cases these limiting null distributions are no longer pivotal, depending on the so-called tail index of the stable law and on the relative weights of the left and right tails of the stable distribution.…”

“…Our …rst contribution is to extend upon the work of Chan and Tran (1989) and Samarakoon and Knight (2009) to derive the limiting null distribution of the usual OLS-based augmented Dickey-Fuller [ADF] test in the case where the shocks follow a linear process which is driven by IV innovations. We show that, provided the lag length used in the ADF regression satis…es the standard o(T 1=3 ) rate condition, these distributions are free of serial correlation nuisance parameters and coincide with those given in Chan and Tran (1989) and Samarakoon and Knight (2009) who assume a …rst-order and …nite-order autoregressive process respectively. The required rate condition therefore coincides with that required for analogous results to hold in the case where the innovations have …nite variance; cf.…”

Testing for bubbles and change-points

“…Horváth and Kokoszka (2003) and Jach and Kokoszka (2004) made the mild hypothesis that the E(e t ) = 0 and that the {e t } are in the domain of attraction of an α-stable law with α ∈ (1, 2), while Kokoszka and Parfionovas (2004) considered the more general case α ∈ (1, 2] which then includes the Gaussian case α = 2. Chan and Tran (1989) have shown that under the null hypothesis H 0…”

Tests for cointegration with infinite variance errors

“…As a special case of our results, we can also analyze whether Phillips and Perron (1988) unit root tests are robust to infinite variance errors (p"1 in Tables 5 and 6). Even though the limit theory for the infinite variance case was developed by Chan and Tran (1989) and Phillips (1990) the issue of robustness has not been explored. From Tables 5 and 6 we can see that at the 5% nominal level the actual size ranges between 6% and 9% and 5% and 6% for the demeaned Z K R and Z K M tests, respectively.…”