Autocorrelation in Time Series Data

The REG Procedure

Autocorrelation in Time Series Data

When regression is performed on time series data, the errors may not be independent. Often errors are autocorrelated; that is, each error is correlated with the error immediately before it. Autocorrelation is also a symptom of systematic lack of fit. The DW option provides the Durbin-Watson d statistic to test that the autocorrelation is zero:

$d = \frac{ \sum_{i=2}^n (e_i - e_{i-1})^2}{\sum_{i=1}^n e_i^2}$

The value of d is close to 2 if the errors are uncorrelated. The distribution of d is reported by Durbin and Watson (1951). Tables of the distribution are found in most econometrics textbooks, such as Johnston (1972) and Pindyck and Rubinfeld (1981).

The sample autocorrelation estimate is displayed after the Durbin-Watson statistic. The sample is computed as

$r = \frac{\sum_{i=2}^n e_i e_{i-1}}{\sum_{i=1}^n e_i^2}$

This autocorrelation of the residuals may not be a very good estimate of the autocorrelation of the true errors, especially if there are few observations and the independent variables have certain patterns. If there are missing observations in the regression, these measures are computed as though the missing observations did not exist.

Positive autocorrelation of the errors generally tends to make the estimate of the error variance too small, so confidence intervals are too narrow and true null hypotheses are rejected with a higher probability than the stated significance level. Negative autocorrelation of the errors generally tends to make the estimate of the error variance too large, so confidence intervals are too wide and the power of significance tests is reduced. With either positive or negative autocorrelation, least-squares parameter estimates are usually not as efficient as generalized least-squares parameter estimates. For more details, refer to Judge et al. (1985, Chapter 8) and the SAS/ETS User's Guide.

The following SAS statements request the DW option for the US population data (see Figure 55.55):

   proc reg data=USPopulation;
      model Population=Year YearSq / dw;
   run;

The REG Procedure

Model: MODEL1

Dependent Variable: Population

Durbin-Watson D	1.264
Number of Observations	19
1st Order Autocorrelation	0.299

Figure 55.56: Regression Using DW Option

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