Serial Correlation

The multiple linear regression model of equation (5.1) assumes that the observations are not correlated with one another. While this is certainly believable if one has drawn a random sample, it’s less likely if one has drawn observations sequentially in time. Time series observations, which are drawn at regular intervals, usually embody a structure where time is an important component. If you are unable to completely model this structure in the regression function itself, then the remainder spills over into the unobserved component of the statistical model (its error) and this causes the errors be correlated with one another.

One way to think about it is that the errors will be serially correlated when omitted effects last more than one time period. This means that when the effects of an economic ‘shock’ last more than a single time period, the unmodeled components (errors) will be correlated with one another. A natural consequence of this is that the more frequently a process is sampled (other things being equal), the more likely it is to be autocorrelated. From a practical standpoint, monthly observations are more likely to be autocorrelated than quarterly observations, and quarterly more likely than yearly ones. Once again, ignoring this correlation makes least squares inefficient at best and the usual measures of precision (standard errors) inconsistent.

9.4.1 Подпись:
Serial Correlation in a Time-Series

To gain some visual evidence of autocorrelation you can plot the series against its lagged values. If there is serial correlation, you should see some sort of positive or negative relationship between the series. Below (Figure 9.8) is the plot of Real GDP growth against its lagged value. A least squares fit is plotted to show the general orientation of the linear relationship. The series itself certainly appears to be serially correlated.

Подпись: rk Подпись: ET=fc+i(yt- y)(yt-k - y) L(y< - y)2 Подпись: (9.5)

Other evidence can be obtained by looking at the correlogram. A correlogram is simply a plot of a series’ sample autocorrelations. The kth order sample autocorrelation for a series y is the correlation between observations that are k periods apart. The formula is

In gretl the correlogram plots a number of these against lags. The syntax to plot 12 autocorrelations of the series g is

corrgm g 12

Подпись: which yields the plot in Figure 9.9.The correlogram is the plot at the top and the partial auto-

image302

Figure 9.8: This plot shows the relationship between GDP growth vs. lagged growth.

correlations are printed in the bottom panel. Approximate 95% confidence intervals are plotted to indicate which are statistically significant at 5%.

Approximate 95% confidence bands are computed using the fact that VTrk ~ N(0,1). These can be computed manually using the fact that the corrgm function actually generates a matrix return. The script to generate the intervals is

1 matrix ac = corrgm(g, 12)

2 matrix lb = ac[,1]-1.96/sqrt($nobs)

3 matrix ub = ac[,1]+1.96/sqrt($nobs)

4 matrix all = lb~ac[,1]~ub

5 colnames(all, "Lower AC Upper ")

6 printf "nAutocorrelations and 95%% confidence intervalsn %9.4fn", all

7

The intervals so generated are:

Autocorrelations and 95% confidence intervals Lower AC Upper

image303

Figure 9.9: The 12 period correlogram for U. S. GDP growth.

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296

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494

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692

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213

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411

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609

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.154

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352

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398

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.108

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090

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285

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087

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111

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402

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006

The matrix ac holds the autocorrelations in the first column and the partial autocorrelations in the second. The matrices lb, ub, and all use indexing to use all rows of the first column of ac,

i. e., ac[,1]. This was be dressed up a bit by adding colnames function to add the column names to the matrix.

You can see that zero is not included in the 1st, 2nd, 4th, and last interval. Those are signifi­cantly different from zero at 5% level.

The correlogram can be useful for detecting the order of autocorrelation. A long series of declining autocorrelations with a single significant pacf is often an indication of a short order autocorrelation process. See POE4 for more guidance.

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