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198 Chapter 6 Nonstationary and Seasonal Time Series Models
6.4 Forecasting ARIMA Models
indicates that there is no appreciable measurement error due to sales and deliveries
(i.e., σ 2 V 0), and hence testing for a unit root in this case is equivalent to
testing that σ 2 U 0. Assuming that the mean is known, the unit root hypothesis
is rejected at α .05, since −.818 > −1 + 6.80/57 −.881. The evidence
against H0 is stronger using the likelihood ratio statistic. Using ITSM and entering
the MA(1) model θ −1and σ 2 2203.12, we find that −2lnL(−1, 2203.12)
604.584, while −2lnL(ˆθ, ˆσ 2 ) 597.267. Comparing the likelihood ratio statistic
λn 604.584 − 597.267 7.317 with the cutoff value cLR,.01, we reject H0 at level
α .01 and conclude that the measurement error associated with sales and deliveries
is nonzero.
In the above example it was assumed that the mean was known. In practice, these
tests should be adjusted for the fact that the mean is also being estimated.
Tanaka (1990) proposed a locally best invariant unbiased (LBIU) test for the unit
root hypothesis. It was found that the LBIU test has slightly greater power than the
likelihood ratio test for alternatives close to θ −1but has less power for alternatives
further away from −1 (see Davis et al., 1995). The LBIU test has been extended to
cover more general models by Tanaka (1990) and Tam and Reinsel (1995). Similar
extensions to tests based on the maximum likelihood estimator and the likelihood
ratio statistic have been explored in Davis, Chen, and Dunsmuir (1996).
In this section we demonstrate how the methods of Section 3.3 and 5.4 can be adapted
to forecast the future values of an ARIMA(p, d, q) process {Xt}. (The required numerical
calculations can all be carried out using the program ITSM.)
If d ≥ 1, the first and second moments EXt and E(Xt+hXt) are not determined
by the difference equations (6.1.1). We cannot expect, therefore, to determine best
linear predictors for {Xt} without further assumptions.
For example, suppose that {Yt} is a causal ARMA(p, q) process and that X0 is
any random variable. Define
Xt X0 +
t
Yj, t 1, 2,....
j1
Then {Xt,t ≥ 0} is an ARIMA(p, 1,q) process with mean EXt EX0 and autocovariances
E(Xt+hXt) − (EX0) 2 that depend on Var(X0) and Cov(X0,Yj), j
1, 2,....The best linear predictor of Xn+1 based on {1,X0,X1,...,Xn} is the same
as the best linear predictor in terms of the set {1,X0,Y1,...,Yn}, since each linear
combination of the latter is a linear combination of the former and vice versa. Hence,
using Pn to denote best linear predictor in terms of either set and using the linearity