Asymptotic inference for a one-dimensional simultaneous autoregressive model

A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n ², while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n ^1/2.