A stationary AR(2) with pseudo-cyclical behavior

Question

Consider the AR(2) process Xt = 0.8\,X{t-1} - 0.5\,X{t-2} + \varepsilont with white-noise shocks. Is it stationary? Answer yes or no.

Accepted Answer

Check the three conditions with \phi1 = 0.8, \phi2 = -0.5: 1. \phi1 + \phi2 = 0.8 - 0.5 = 0.3 < 1. OK. 2. \phi2 - \phi1 = -0.5 - 0.8 = -1.3 < 1. OK. 3. -0.5 < 1. OK. All three hold, so the process is stationary. There is an extra wrinkle worth noting. The discriminant \phi1^2 + 4\phi2 = 0.64 + 4(-0.5) = 0.64 - 2 = -1.36 is negative, which means the characteristic roots are complex. Complex roots make the autocorrelation function decay in a damped, oscillating wave rather than a smooth geometric fall, so the series shows pseudo-cyclical behavior: it swings up and down with a preferred period, …

A stationary AR(2) with pseudo-cyclical behavior

Your answer

More Statistics questions