was paid to the time-series structure of the underlying data and the specification of the dynamic structure. Once this is done, the forecasting performance of econometric models improves considerably, and the econometric models started to do better than univariate time-series models.* Later it was argued that the appropriate comparison should be between econometric models and multivariate time-series models (not univariate time-series models). However, in the early 1970s what was being argued is that even univariate ARIMA models did better in forecasting than econometric models. At least this claim can be put to rest by taking care of the dynamics in the specification of econometric models. As for multivariate time-series models versus econometric models, there is still some controversy.

Quite apart from the forecasting aspect, the Box-Jenkins methodology involves a lot of judgmental decisions and a considerable amount of "data mining. " This data mining can be avoided if we can confine our attention to autoregressive processes only. Such processes are easier to estimate and easier to undertake specification testing. We start with a fairly high-order AR process and then simplify it to a parsimonious model by successive "testing down." This procedure has been suggested by Anderson.* For ARMA models, on the other hand, moving from the specific to the general is difficult. This is why Box and Jenkins suggest starting with a parsimonious model and building up progressively if the estimated residuals show any systematic pattern (see the earlier discussion on diagnostic checking).

The reason why one may not be able to avoid the MA part in the Box-Jenkins methodology is that the MA part can be produced by the differencing operation undertaken to remove the trend and produce a stationary series. For instance, if the true model is

y, = a + + E,

taking second differences we eliminate the trend but the residual series e, -2e, , + e,„2 is a MA(2) process and it is not even invertible.

Seasonality in the Box-Jenkins Modeling

Suppose that we have monthly data. Then observations in any month are often affected by some seasonal tendencies peculiar to that month. A classic example is the upward jump in sales in December because of Christmas shopping. The observation in December 1991 is more likely to be highly correlated with observations in December of past years than with observations in other months closer to December 1991.

Just as the Box-Jenkins method accounts for trends by using first differences, it accounts for seasonality by using the twelfth difference, that is, (Jan-

"An illustration of this comparison is D. L. Prothero and K. F. Wallis, "Modelling Macroeco-nomic Time-Series" (with discussion). Journal of the Royal Statistical Society, Series A, Vol. 139, 1976, pp. 468-500.

"See the discussion of Andersons test in Section 10.8.

13.7 «2 MEASURES IN TIME-SERIES MODELS 549

uary 1991 - January 1990), (February 1991 - February 1990), (March 1991 -March 1990), and so on. If we denote the first difference operator by A = 1 -L, the seasonal difference will be denoted by A.j = 1 - L}. With quarterly data we use the seasonal difference operator A4 = 1 - V. Note that A,2 A because \ - V (\ - LY. If the original series is y„ we get Ay, = y, - y, i, = y, - y,-4, 12 , = y, - y,-i2- If we have monthly data and we eUminate both trend and the monthly seasonal, we use both the operators A and A12- that is, we use both first differences and twelfth differences. Instead of using first difference, that is, (1 - L), we can sometimes consider a quasi-first difference, that is, (1 - aL) [e.g., (1 - 0.8L) or (1 - 0.75L)]. Similarly, for the seasonal, instead of using (1 - V), we might use (1 - 0.8F), and so on.

Chatfield" gives an example of some telephone data analyzed by Tomasek using the Box-Jenkins method. Tomasek developed the model

(1 - 0.84F)(1 - F2)(y, - 132) = (1 - 0.60F)(1 4- . ,

which when fitted to all the data, explained 99.4% of the total variation about the mean [i.e., 2 ( , " >)]• On the basis of this good fit, Tomasek recommended the use of the Box-Jenkins method for forecasting.

Chatfield argues that if one looks at the data, one finds an unusually regular seasonal pattern which itself explains 97% of the total variation. For this model once the seasonal element is accounted for, nearly any forecasting method gives good results. For instance, an adaptive forecasting equation would be

y,., = Xy, + (1 - X)y,,., 0<X<1

(i.e., the forecasts are revised on the basis of the most recent forecast errors).

The differencing operation in the Box-Jenkins procedure is also considered one of its main limitations in the treatment of series that exhibit moving seasonal and moving trend. The Box-Jenkins procedure accounts for the trend and seasonal elements through the use of differencing operators and then pays elaborate attention to the ARMA modeUng of what is left. This inordinate attention to ARMA modeling obscures some other aspects of the time series that need more attention. It has been found that other procedures that permit adaptive revision of the seasonal elements, trend term and the residual, may give better forecasts than the Box-Jenkins method.

13.7 Measures in Time-Series Models

First lets consider pure autoregressive models: AR(p). How do we choose the order p? One might think of using and other criteria discussed in Section 12.7, but most of these criteria were discussed under the assumption of non-

™C. Chatfield, The Analysis of Time Series: Theory and Practice (London: Chapman & Hall, 1975), p. 103.

-The reference to Tomasek and the data used can be found in Chatfields book, p. 103.

The numerator RSS is the residual sum of squares from the model we estimate. The denominator 2 =2 ( , ~ / the residual sum of squares from a random walk with drift, that is, y, = y,„, + p -I- e„ r = 2, . . . , 7. For most time-series data this is the "naive alternative." A model for which R}, is negative should be discarded. To adjust for degrees of freedom, we divide the numerator and denominator in Rj, by the appropriate degrees of freedom. The denominator d.f. are ( - 2). The numerator d.f. are (T - k), where is the numbers of parameters estimated. The R], adjusted for d.f. can be called R]). With seasonal data, Harvey suggests

" RSSo

H. Akaike, "Fitting Autoregressions for Prediction," Annals of the Institute of Statistical Mathematics, VoL 21, No. 2. 1969, pp. 243-247.

-A. C. Harvey, "A Unified View of Statistical Forecasting Procedure," Journal of Forecasting, Vol. 3, 1984, pp. 245-275.

stochastic regressors. One procedure that is popular for choosing the order p is Akaikes FPE criterion. This says: Choose the order p by minimizing

FPE . +

where is the estimate of = var(e,) when the order of the autoregressive process is p, that is, = RSS/(n - p - 1).

Time-series observations normally show a strong trend (upward or downward) and strong seasonal effects. Any model that is able to pick up these effects will have a high P}. The question is: How good and reliable is this? For instance. Box and Jenkins (1976, Chap. 9) have a data set consisting of 144 monthly observations on the variable y„ defined as the logarithm of the number of airline passengers carried per month. Regressing y, on time trend and seasonal dummies gives = 0.983. Is this a good model?

Harvey suggests some measures of relative Rs to judge the usefulness of a model. Note that the criterion on which the usual R is based is the residual sum of squares from the model relative to the residual sum of squares from a naive alternative (that consists of the estimation of the mean only). For instance if Sj, = 2 ( / ~ yf and RSS is the residual sum of squares from the model,

R = \ - (see Chapter 4)

Thus the "norm" 7? judges a model compared with a naive model, where only the mean is estimated. In time-series models with strong trends and seasonals, this is not a meaningful alternative. The meaningful alternative is a random walk with drift, or with seasonal data, a random walk with drift with seasonal dummies added. Harvey suggests two alternative R measures. One is