"Frontmatter". In: Analysis of Financial Time Series

SIMPLE ARMA MODELS 55This representation shows the dependence of the current return r t on the past returnsr t−i ,wherei > 0. The coefficients {π i } are referred to as the π-weights of an ARMAmodel. To show that the contribution of the lagged value r t−i to r t is diminishing asi increases, the π i coefficient should decay to zero as i increases. An ARMA(p, q)model that has this property is said to be invertible. For a pure AR model, θ(B) = 1so that π(B) = φ(B), which is a finite-degree polynomial. Thus, π i = 0fori >p, and the model is invertible. For other ARMA models, a sufficient condition forinvertibility is that all the zeros of the polynomial θ(B) are greater than unity inmodulus. For example, consider the MA(1) model r t = (1 − θ 1 B)a t . The zero of thefirst order polynomial 1 − θ 1 B is B = 1/θ 1 . Therefore, an MA(1) model is invertibleif | 1/θ 1 | > 1. This is equivalent to | θ 1 | < 1.From the AR representation in Eq. (2.28), an invertible ARMA(p, q) series r t is alinear combination of the current shock a t and a weighted average of the past values.The weights decay exponentially for more remote past values.MA RepresentationAgain, using the result of long division in Eq. (2.26), an ARMA(p, q) model canalso be written asr t = µ + a t + ψ 1 a t−1 + ψ 2 a t−2 +···=µ + ψ(B)a t , (2.29)where µ = E(r t ) = φ 0 /(1 − φ 1 −···−φ p ). This representation shows explicitlythe impact of the past shock a t−i (i > 0) on the current return r t . The coefficients{ψ i } are referred to as the impulse response function of the ARMA model. For aweakly stationary series, the ψ i coefficients decay exponentially as i increases. Thisis understandable as the effect of shock a t−i on the return r t should diminish overtime. Thus, for a stationary ARMA model, the shock a t−i does not have a permanentimpact on the series. If φ 0 ̸= 0, then the MA representation has a constant term,which is the mean of r t [i.e., φ 0 /(1 − φ 1 −···−φ p ].The MA representation in Eq. (2.29) is also useful in computing the variance of aforecast error. At the forecast origin h, we have the shocks a h , a h−1 ,....Therefore,the l-step ahead point forecast isand the associated forecast error isˆr h (l) = µ + ψ l a h + ψ l+1 a h−1 +···, (2.30)e h (l) = a h+l + ψ 1 a h+l−1 +···+ψ l−1 a h+1 .Consequently, the variance of l-step ahead forecast error isVar[e h (l)] =(1 + ψ 2 1 +···+ψ2 l−1 )σ 2 a , (2.31)which, as expected, is a nondecreasing function of the forecast horizon l.