Regression and autoregression with infinite variance

Abstract

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression Xn = a1Xn-1 + … + akXn-k + Un, the least squares estimates are shown to be consistent as long as the roots of 1 - a1x2 - a2x2 - … - akxk = 0 are outside the complex unit circle, Xn is independent of Un+j,j ≥ 1, and the Un are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.