The chief tools in identification are the autocorrelation function (ACF), the partial autocorrelation function (PACF), and the resulting correlograms, which are simply the plots of ACFs and PACFs against the lag length.

In the previous chapter we defined the (population) ACF (pk) and the sample ACF (pk). The concept of partial autocorrelation is analogous to the


concept of partial regression coefficient. In the k-variable multiple regression model, the kth regression coefficient jk measures the rate of change in the mean value of the regressand for a unit change in the kth regressor Xk, holding the influence of all other regressors constant.

In similar fashion the partial autocorrelation pkk measures correlation between (time series) observations that are k time periods apart after controlling for correlations at intermediate lags (i.e., lag less than k). In other words, partial autocorrelation is the correlation between Yt and Yt—k after removing the effect of the intermediate Y's.7 In Section 7.11 we already introduced the concept of partial correlation in the regression context and showed its relation to simple correlations. Such partial correlations are now routinely computed by most statistical packages.

In Figure 22.2 we show the correlogram and partial correlogram of the GDP series. From this figure, two facts stand out: First, the ACF declines


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