Infinite Variance Innovations Research Articles

New tests for unit roots in autoregressive processes with possibly infinite variance errors

For autoregressive processes with possibly infinite variance innovations, tests for unit roots are constructed. The limiting null distributions of the test statistics are standard normal both for finite variance innovations and for infinite variance innovations. The test statistics are the pivotal statistics of modified M-estimators in which the signs of regressors rather than the regressors themselves are used as instrumental variables in estimating unit roots. A Monte-Carlo experiment compares the proposed tests favorably with tests based on the OLSE and tests based on the M-estimators for several innovations.

Cointegrated processes with infinite variance innovations

It is widely accepted that the Gaussian assumption is too restrictive to model either financial or some important macroeconomic variables, because their distributions exhibit asymmetry and heavy tails. In this paper we develop the asymptotic theory for econometric cointegration processes under the assumption of infinite variance innovations with different distributional tail behavior. We extend some of the results of Park and Phillips which were derived under the assumption of finite variance errors.

Fractional ARIMA with stable innovations

We develop the theory of fractionally differenced ARIMA time series with stable infinite variance innovations establishing conditions for existence and invertibility. We analyze their asymptotic dependence structure by means of the codifference and the covariation, measures of dependence which are extensions of the covariance and are applicable to stochastic processes with infinite variance.

On the Noninvertible Moving Average Time Series with Infinite Variance

The limiting distribution of the least squares estimate of the derived process of a noninvertible and nearly noninvertible moving average model with infinite variance innovations is established as a functional of a Lévy process. The form of the limiting law depends on the initial value of the innovation and the stable index α. This result enables one to perform asymptotic testing for the presence of a unit root for a noninvertible moving average model through the constructed derived process under the null hypothesis. It provides not only a parallel analog of its autoregressive counterparts, but also a useful alternative to determine “over-differencing” for time series that exhibit heavy-tailed phenomena.

Limit Theory for M-Estimates in an Integrated Infinite Variance

We consider the limiting distributions of M-estimates of an “autoregressive” parameter when the observations come from an integrated linear process with infinite variance innovations. It is shown that M-estimates are, asymptotically, infinitely more efficient than the least-squares estimator (in the sense that they have a faster rate of convergence) and are conditionally asymptotically normal.

J. of time series anal.

In the first part of this paper, I am concerned with bilinear time series with innovations having infinite variance. A Least Gamma Deviation (LGD) estimation procedure is proposed for the inference of the parameters of some bilinear time series with infinite variance innovations and shown to be strongly consistent. Numerical simulations demonstrate the superiority of this new procedure to the conventional Least Squares (LS) procedure. The concepts of causality and invertibility are also briefly discussed. In the remainder of this paper, attention is focused on the derivation of the asymptotic distributions of the least squares estimates under finite variance assumptions.

A SIMPLE CONDITION FOR THE EXISTENCE OF SOME STATIONARY BILINEAR TIME SERIES

Abstract. A sufficient condition is derived for the existence of a strictly stationary solution of some bilinear time series which may have infinite variance innovations. This condition is equivalent to the condition that a polynomial of degree r has no zeros within the unit circle. In the special case when the innovations have finite variance, the computational effort involved in checking this condition is significantly reduced compared with checking the stationarity conditions given by Bhaskara Rao et al. and Liu and Brockwell which requires a knowledge of the maximum eigenvalue in the absolute value of an r2 x r2 matrix.