"D. S. Poskitt and A. R. Tremayne, "Testing the Specilication of a Fitted ARMA Model," Biometrika. Vol. 67, 1980, pp. 359-363.

"•See Box and Jenkins, Time Series Analysis. The first edition of the book appeared in 1970. These models were discussed earlier in M. H. Quenouille, The Analysis of Multiple Time Series (London: Charles Griffin, 1957).

applicable when the alternative is ARMA(p -I- ky, q + k- with m = max (/:,, k, as shown in Poskitt and Tremayne." Despite its limitations discussed here, there are numerous studies that quote the Box-Pierce g-statistic and argue that if it is not significant (at the 5 percent level), this is confirmatory evidence that the fitted model is adequate. The available evidence shows that the LM tests are more appropriate, and they can be implemented by some regression routines. However, until they are built into computer programs, they are not likely to be used. Currently, most time-series programs have the g-statistic built into them despite all the limitations of the g-statistics discussed here.

13.6 The Box-Jenlons Approach

The Box-Jenkins approach" is one of theThost widely used methodologies for the analysis of time-series data. It is popular because of its generality; it can handle any series, stationary or not, with or without seasonal elements, and it has well-documented computer programs. It is perhaps the last factor that contributes most to its popularity. Although Box and Jenkins have been neither the originators nor the most important contributors in the field of ARMA models," they have popularized these models and made them readily accessible to everyone, so much so that ARMA models are sometimes referred to as Box-Jenkins models.

The basic steps in the Box-Jenkins methodology are (I) differencing the series so as to achieve stationarity, (2) identification of a tentative model, (3) estimation of the model, (4) diagnostic checking (if the model is found inadequate, we go back to step 2), and (5) using the model for forecasting and control. Schematically, we can describe the steps as in Figure 13.3. We will now discuss these steps in turn.

1. Differencing to achieve stationarity: How do we conclude whether a time series is stationary or not? We can do this by studying the graph of the correlogram of the series. The correlogram of a stationary series drops off as k, the number of lags, becomes large, but this is not usually the case for a nonstationary series. Thus the common procedure is to plot the correlogram of the given series y, and successive differences , , and so on, and look at the correlograms at each stage. We keep differencing until the correlogram dampens.

2. Once we have used the differencing procedure to get a stationary time series, we examine the correlogram to decide on the appropriate orders of the AR and MA components. The cdirelogram of a MA process is zero after a point. That of an AR process declines geometrically. The corre-

Differencing the series to achieve stationarity

Identify model to be tentatively entertained

Estimate the parameters of the tentative model

Diagnostic checking. Is the model adequate?

Use the model for forecasting and control

Figure 13.3. The Box-Jenkins methodology for ARIMA models.

lograms of ARMA processes show different patterns (but all dampen after a while). Based on these, one arrives at a tentative ARMA model. This step involves more of a judgmental procedure than the use of any clear-cut rules.

3. The next step is the estimation of the tentative ARMA model identified in step 2. We have discussed in the preceding section the estimation of ARMA models.

4. The next step is diagnostic checking to check the adequacy of the tentative model. We discussed in the preceding section the Q and Q* statistics commonly used in diagnostic checking. As argued there, the Q statistics are inappropriate in autoregressive models and thus we need to replace them with some LM test statistics.

5. The final step is forecasting. We shall now discuss this problem.

Forecasting from Box-Jenkins Models

To fix ideas we shall illustrate forecasting from the ARMA(2, 2) model we have been considering. Suppose that we have estimated the model with n observations. We want to forecast :„+. This is called a A-period ahead forecast. It is

"R. M. Leuthold, A. M. A. MacCormick, A. Schmitz, and D. G. Watts, "Forecasting Daily Hog Prices and Quantities: A Study of Alternative Forecasting Techniques," Journal of the American Statistical Association, March 1970, pp. 90-107.

denoted by je„. The first subscript gives the time period when the forecast is made, and the second subscript denotes the time periods ahead for which the forecast is made. Let us start with = 1 so that we need a forecast of „+, at time period n. We have

We observe x„ and „„,. We can replace e„ and e„ , by the predicted residuals (obtaining the residuals was described in the preceding section). The only unknown is e„+. This we replace by its expected value zero. Hence

4.1 = « + 1 + + biK-\ Now let us go to = 2. We have

„+2 = , „ + + a.2X„ + e„ + 2 + Pie« + i +

We replace e„+2 and 6,,+, by zero, their expected value. „+ is not known, but we have the forecast jc„,. Thus we get

««.2 = ai4.i + 0i2X„ + p2e« We continue like this. The procedure is:

1. Write out the expression for x„+i,-

2. Replace all future values x„j (j > 0,j < k) by their forecasts.

3. Replace all e„+j (j > 0) by zero.

4. Replace all „ 0 - 0) by the predicted residuals.

An alternative procedure is to write x, in terms of all the lagged xs. For this we use the procedure outlined in the preceding section. We have

(1 + 7,L + yiL + ? + • • = e, The 7s are obtained as discussed in the preceding section. Now we have

i = + 72-,-1 + , 2 +•••] + e/+i

To get Jf,,, we replace 6,+, by zero, its expected value, and 7s by their estimates. The procedure is the same as earlier except that we do not have to deal with the predicted residuals.

Illustrative Example

As an illustrative example we consider the problem of forecasting hog marketings considered by Leuthold et al."" It is an old study, but the example illustrates how the correlogram can be used to arrive at a model that uses higher than first order differences. The data consist of 275 daily observations. The correlograms for the original data are presented in Figure 13.4. The correl-