Time Series 3

Last time we discussed two main categories of linear models, and their combination. Here w t denotes a white noise: a stationary process with E w t = 0, E w 2 t = σ 2 , and E w t w s = 0 for s = t. MA(q) x t = µ + w t + q j=1 θ j w t−j = µ + w t + θ 1 w t−1 + · · · + θ q w t−q. x t is stationary with well-defined variance and covariances for any values of the coefficients θ 0 ,. .. , θ q , if q is finite. In the limit q → ∞, we require the rather weak condition that |θ j | exists. We are respecting the standard normalization condition that θ 0 = 1. AR(p) x t = c + w t + p j=1 φ j x t−j = c + w t + φ 1 x t−1 + · · · + φ p x t−p. The process is stationary if and only if the characteristic polynomial P (z) = 1 − φ 1 z − · · · − φ p z p has all its roots outside the unit disk. Again we set φ 0 = 1. ARMA(p,q) It is straightforward to combine the above as x t = c + p j=1 φ j x t−j + q j=0 θ j w t−j