Large sample estimation in nonstationary autoregressive processes with multiple observations (Q1344957)

Let \(Z_ t = (Z_{t1}, \dots, Z_{tn})^ T\) denote an \((n \times 1)\) vector of observations at time \(t=1, \dots, m\). Suppose that \(\{Z_ t\}\) satisfies the autoregressive difference equation \(Z_ t = \sum^ p_{i=1} \theta_ i Z_{t-i} + \xi_ t\), \(t = 0, \pm 1, \pm 2, \dots\). The asymptotic distributions of the least-squares estimations of the parameters are derived both when \(m \to \infty\) and \(n \to \infty\) for the two nonstationary cases, viz., (a) the explosive case and (b) the unstable case. Consequently, the authors investigate the two-dimensional process \(Z_{ts} = \sum^ p_{i=1} \theta_ i Z_{t-i,s} + \xi_{ts}\), \(t,s = 0, \pm 1, \pm 2, \dots\). Nonstandard limit distributions are obtained.