Weekly FM Sales

Sales (thousands)	35 40
				Week

Fig. 9.1. The FM Sales series (thousands of units sold per week). The shaded regions show 90% prediction intervals: the light-shaded region is from the local level model without regressors; the dark-shaded region is from the local level model with regressors.

Table 9.1. Analysis of the FM sales data using the local level model with regressors.

Model	p 1	p 2	p 3	α	_ 0	R 2
Regression only	− 8.65	5.88	18.73	–	–	0.793
Local level (LL)	–	–	–	0.731	23.47	0.575
LL with regressors	− 4.80	4.65	17.29	0.471	28.47	0.808

• z ₂=1 in weeks 13–15 and 48–51 denoting high sales periods before Easterand Christmas, and z ₂ = 0 otherwise

• z ₃=1 in week 52 to denote the after-Christmas sales peak, and z ₃=0otherwise

The results are given in Table 9.1; the maximum likelihood estimates were obtained by direct maximization. The contributions of the regressor terms tend to dominate in this case, but the persistence in the series is clearly seen with the value of α = 0.471.

The local level with regressors model yields a point forecast for the next period of 31.26. The estimated standard deviation is σ ˆ = 2.39, so that a 90% one-step-ahead prediction interval would be [27.33, 35.19]. By contrast, the local level model without regressors gives a point forecast of 32.59, σ ˆ = 3.54 and the prediction interval [26.75, 38.43].

9.2 Some Examples

9.2.2 Use of a Leading Indicator

Time series regression based upon two or more variables requires some care in model development. For example, a researcher may use the trans-fer function approach of Box et al. (1994, Chaps. 10 and 11). The detailed discussion of such procedures is beyond the scope of this book, because the methodology is similar whether an ARIMA or a state space approach is employed. Accordingly, we use a single explanatory variable to illustrate ideas. Consider two series relating to US gasoline prices:

Y =US retail gas prices (the average price per gallon, in dollars)

X =The spot price of a barrel of West Texas Intermediate (WTI) oil indollars as traded at Cushing, Oklahoma

The Cushing spot price is widely used in the industry as a “marker” for pricing a number of other crude oil supplies traded in the domestic spot market at Cushing, Oklahoma. The data are monthly and cover the period January 1991 to November 2006.

The two series are plotted in Fig. 9.2 and show both marked nonstation-arity and considerably increased variability in the later years. At this point, we will examine the series through the end of 2001 to reduce the effects of

Дата добавления: 2015-10-24