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6.4 Forecasting ARIMA Models 201
where φ ∗ 1 ,...,φ∗ p+d are the coefficients of z,...,zp+d in
φ ∗ (z) (1 − z) d φ(z).
The solution of (6.4.7) is well known from the theory of linear difference equations
(see TSTM, Section 3.6). If we assume that the zeros of φ(z) (denoted by ξ1,...,ξp)
are all distinct, then the solution is
g(h) a0 + a1h +···+adh d−1 + b1ξ −h
1
−h
+···+bpξ p , h>q− p − d, (6.4.8)
where the coefficients a1,...,ad and b1,...,bp can be determined from the p + d
equations obtained by equating the right-hand side of (6.4.8) for q − p − dq− p − d. In the case q 0, the values of g(h) in
the equations for a0,...,ad,b1,...,bp are simply the observed values g(h) Xn+h,
−p − d ≤ h ≤ 0, and the expression (6.4.6) for the mean squared error is exact.
The calculation of the forecast function is easily generalized to deal with more
complicated ARIMA processes. For example, if the observations X−13,X−12,...,Xn
are differenced at lags 12 and 1, and (1 − B) 1 − B 12 Xt is modeled as a causal
invertible ARMA(p, q) process with mean µ and max(p,q)q. (6.4.10)
To find the general solution of these inhomogeneous linear difference equations, it
suffices (see TSTM, Section 3.6) to find one particular solution of (6.4.10) and then
add to it the general solution of the same equations with the right-hand side set equal
to zero. A particular solution is easily found (by trial and error) to be
g(h) µh2
24 ,
and the general solution is therefore
g(h) µh2
24 + a0 + a1h +
11
j1
cje ij π/6 + b1ξ −h
1
+···+bpξ −h
p ,
h>q− p − 13. (6.4.11)
(The terms a0 and a1h correspond to the double root z 1 of the equation φ(z)(1 −
z)(1−z 12 ) 0, and the subsequent terms to each of the other roots, which we assume
to be distinct.) For q − p − 13