FIGURE 21.8 Correlogram of U.S. GDP, 1970-I to 1991-IV. See Figure 21.6 for definitions.

FIGURE 21.8 Correlogram of U.S. GDP, 1970-I to 1991-IV. See Figure 21.6 for definitions.


Two practical questions may be posed here. First, how do we choose the lag length to compute the ACF? Second, how do you decide whether a correlation coefficient at a certain lag is statistically significant? The answer follows.

The Choice of Lag Length. This is basically an empirical question. A rule of thumb is to compute ACF up to one-third to one-quarter the length of the time series. Since for our economic data we have 88 quarterly observations, by this rule lags of 22 to 29 quarters will do. The best practical advice is to start with sufficiently large lags and then reduce them by some statistical criterion, such as the Akaike or Schwarz information criterion that we discussed in Chapter 13. Alternatively, one can use the following statistical tests.

Statistical Significance of Autocorrelation Coefficients

Consider, for instance, the correlogram of the GDP time series given in Figure 21.8. How do we decide whether the correlation coefficient of 0.638 at lag 10 (quarters) is statistically significant? The statistical significance of any pk can be judged by its standard error. Bartlett has shown that if a time series is purely random, that is, it exhibits white noise (see Figure 21.6), the sample autocorrelation coefficients pk are approximately19

that is, in large samples the sample autocorrelation coefficients are normally distributed with zero mean and variance equal to one over the sample size. Since we have ^ 88 observations, the variance is 1/88 = 0.01136 and the standard error is V0.01136 = 0.1066. Then following the properties of the standard normal distribution, the 95% confidence interval for any (population) pk is:

In other words,

Prob (pk - 0.2089 < pk < Pk + 0.2089) = 0.95 (21.8.7)

If the preceding interval includes the value of zero, we do not reject the hypothesis that the true pk is zero, but if this interval does not include 0, we reject the hypothesis that the true pk is zero. Applying this to the estimated value of /O10 = 0.638, the reader can verify that the 95% confidence interval for true P10 is (0.638 ± 0.2089) or (0.4291, 0.8469).20 Obviously, this inter-

19M. S. Bartlett, "On the Theoretical Specification of Sampling Properties of Autocorrelated Time Series," Journal of the Royal Statistical Society, Series B, vol. 27, 1946, pp. 27-41.


val does not include the value of zero, suggesting that we are 95% confident that the true p10 is significantly different from zero.21 As you can check, even at lag 20 the estimated p20 is statistically significant at the 5% level.

Instead of testing the statistical significance of any individual autocorrelation coefficient, we can test the joint hypothesis that all the pk up to certain lags are simultaneously equal to zero. This can be done by using the Q statistic developed by Box and Pierce, which is defined as22

where n = sample size and m = lag length. The Q statistic is often used as a test of whether a time series is white noise. In large samples, it is approximately distributed as the chi-square distribution with m df. In an application, if the computed Q exceeds the critical Q value from the chi-square distribution at the chosen level of significance, one can reject the null hypothesis that all the (true) Pk are zero; at least some of them must be nonzero.

A variant of the Box-Pierce Q statistic is the Ljung-Box (LB) statistic, which is defined as23

Although in large samples both Q and LB statistics follow the chi-square distribution with m df, the LB statistic has been found to have better (more powerful, in the statistical sense) small-sample properties than the Q statistic.24 Returning to the GDP example given in Figure 21.8, the value of the LB statistic up to lag 25 is about 891.25. The probability of obtaining such an LB value under the null hypothesis that the sum of 25 squared estimated autocorrelation coefficients is zero is practically zero, as the last column of that figures shows. Therefore, the conclusion is that the GDP time series is nonstationary, therefore reinforcing our hunch from Figure 21.1 that the GDP series may be nonstationary. In exercise 21.16 you are asked to confirm that the other four U.S. economic time series are also nonstationary.

20Our sample size of 88 observations, although not very large, is reasonably large to use the normal approximation.

^Alternatively, if you divide the estimated value of any pk by the standard error for sufficiently large n, you will obtain the standard Z value, whose probability can be easily obtained from the standard normal table. Thus for the estimated pi0 = 0.638, the Z value is 0.638/0.1066 = 5.98 (approx.). If the true p10 were in fact zero, the probability of obtaining a Z value of as much as 5.98 or greater is very small, thus rejecting the hypothesis that the true p10 is zero.

22 G. E. P. Box and D. A. Pierce, "Distribution of Residual Autocorrelations in Autoregressive Integrated Moving Average Time Series Models," Journal of the American Statistical Association, vol. 65, 1970, pp. 1509-1526.

23G. M. Ljung and G. P. E. Box, "On a Measure of Lack of Fit in Time Series Models," Bio-metrika, vol. 66, 1978, pp. 66-72.

24The Q and LB statistics may not be appropriate in every case. For a critique, see Maddala et. al., op. cit., p. 19.


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