The Limiting Distribution of the Residual Processes in Nonstationary Autoregressive Processes

The weak limit of the partial sums of the normalized residuals in an AR(1) process y _t = ρ y _{t −1} + e _t is shown to be a standard Brownian motion W ( x ) when |ρ| ≠ 1. However, when |ρ| = 1, the weak limit is W ( x ) plus an extra term due to estimation of ρ. Asymptotic behaviour of the partial sums is investigated with ρ = exp( c )/ n ) in the vicinity of unity, yielding a c -dependent weak limit as n ←∞, whose limit is again W ( x ) as | c | ←∞. An extension is made to nonstationary AR( p ) processes with multiple characteristic roots on the unit circle. The weak limit of the partial sums has close resemblance to that for the polynomial regression.