On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations

Let xₜ(t = 1, 2, ⋯) be defined recursively by {equation*}{1.1}x_t = axₜ₋₁ + u_t, t = 1, 2, ⋯,{equation*} where x₀ is a constant, ε uₜ = 0, ε u²ₜ = σ² and ε uₜuₛ = 0, t ≠ s. (ε denotes mathematical expectation.) An estimate of α based on x₁, ⋯, xT (which is the maximum likelihood estimate of α if the u's are normally distributed) is {equation*}{1.2} α = (∑^Tₜ₌₁ x_txₜ₋₁)/(∑^Tₜ₌₁ x^2ₜ₋₁).{equation*} If |α| < 1, √ T ( α - α) has a limiting normal distribution with mean 0 under fairly general conditions such as independence of the u's and uniformly bounded moments of the u's of order 4 + ε, for some ε > 0. (See [2], Chapter II, for example.) If |α| > 1, White [3] has shown ( α - α)|α|T/(α² - 1) has a limiting Cauchy distribution under the assumption that x₀ = 0 and the u's are normally distributed; he has also found the distribution when x₀ ≠ 0. His results can be easily modified and restated in the following form (ΣTₜ₌₁ x²ₜ₋₁)1/2( α - α) has a limiting normal distribution if the u's are normally distributed and if |α| ≠ 1. Peculiarly, for |α| = 1 this statistic has a limiting distribution which is not normal (and is not even symmetric for x₀ = 0). One purpose of this paper is to characterize the limiting distributions for |α| > 1 when the u's are not necessarily normally distributed; it will be shown that for |α| > 1 the results depend on the distribution of the u's. Central limit theorems are not applicable. Secondly, the limiting distribution for |α| < 1 will be shown to hold under the assumption that the u's are independently, identically distributed with finite variance. This was conjectured by White.