Unit Roots In White Noise

We show that the empirical distribution of the roots of the vector autoregression (VAR) of order p fitted to T observations of a general stationary or nonstationary process converges to the uniform distribution over the unit circle on the complex plane, when both T and p tend to infinity so that (ln T)/p → 0 and p3/T → 0. In particular, even if the process is a white noise, nearly all roots of the estimated VAR will converge by absolute value to unity. For fixed p, we derive an asymptotic approximation to the expected empirical distribution of the estimated roots as T → ∞. The approximation is concentrated in a circular region in the complex plane for various data generating processes and sample sizes.