Correlogram and partial correlogram, first differences of GDP, United States, 1970-I to 1991-IV.

8It is hard to tell whether the variance of this series is stationary, especially around 1979-1980. The oil embargo of 1979 and a significant change in the Fed's monetary policy in 1979 may have something to do with our difficulty.

844 PART FOUR: SIMULTANEOUS-EQUATION MODELS

approximate 95% confidence limits for pk are -0.2089 and +0.2089. (Note: As discussed in Chapter 21, these confidence limits are asymptotic and so can be considered approximate.) But at all other lags, they are not statistically different from zero. This is also true of the partial autocorrelations, pkk.

Now how do the correlograms given in Figure 22.3 enable us to find the ARMA pattern of the GDP time series? (Note: We will consider only the first-differenced GDP series because it is stationary.) One way of accomplishing this is to consider the ACF and PACF and the associated correlograms of a selected number of ARMA processes, such as AR(1), AR(2), MA(1), MA(2), ARMA(1, 1), ARIMA(2, 2), and so on. Since each of these stochastic processes exhibits typical patterns of ACF and PACF, if the time series under study fits one of these patterns we can identify the time series with that process. Of course, we will have to apply diagnostic tests to find out if the chosen ARMA model is reasonably accurate.

To study the properties of the various standard ARIMA processes would consume a lot of space. What we plan to do is to give general guidelines (see Table 22.1); the references can give the details of the various stochastic processes.

Notice that the ACFs and PACFs of AR(p) and MA(g) processes have opposite patterns; in the AR(p) case the AC declines geometrically or exponentially but the PACF cuts off after a certain number of lags, whereas the opposite happens to an MA(g) process.

Geometrically, these patterns are shown in Figure 22.4.

A Warning. Since in practice we do not observe the theoretical ACFs and PACFs and rely on their sample counterparts, the estimated ACFs and PACFs will not match exactly their theoretical counterparts. What we are looking for is the resemblance between theoretical and sample ACFs and PACFs so that they can point us in the right direction in constructing ARIMA models. And that is why ARIMA modeling requires a great deal of skill, which of course comes from practice.

ARIMA Identification of U.S. GDP. Returning to the correlogram and partial correlogram of the stationary (after first-differencing) U.S. GDP for 1970-I to 1991-IV given in Figure 22.3, what do we see?

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