Building and Testing a Multiple Linear Regression Model 99
values also depend on the sample size and the number of independent vari-
ables in the multiple regression.^8
For example, suppose that there are 12 independent variables in a regres-
sion, there are 200 observations, and that the significance level selected is
5%. Then according to the Durbin-Watson critical value table, the critical
values are
dL = 1.643 and dU = 1.896
Then the tests in the previous table can be written as:
Null Hypothesis Range for Computed d Decision Rule
No positive autocorrelation 0 < d < 1.643 Reject the null hypothesis
No positive autocorrelation 1.643 ≤ d ≤ 1.896 No decision
No negative autocorrelation 2.357 < d < 4 Reject the null hypothesis
No negative autocorrelation 2.104 ≤ d ≤ 2.357 No decision
No autocorrelation 1.896 < d < 2.104 Accept the null hypothesis
Modeling in the Presence of Autocorrelation If residuals are autocorrelated, the
regression coefficient can still be estimated without bias using the formula
given by equation (3.10) in Chapter 3. However, this estimate will not be
optimal in the sense that there are other estimators with lower variance of
the sampling distribution. Fortunately, there is a way to deal with this prob-
lem. An optimal linear unbiased estimator called the Aitken’s generalized
least squares (GLS) estimator can be used. The discussion about this estima-
tor is beyond the scope of this chapter.
The principle underlying the use of such estimators is that in the pres-
ence of correlation of residuals, it is common practice to replace the stan-
dard regression models with models that explicitly capture autocorrelations
and produce uncorrelated residuals. The key idea here is that autocorrelated
residuals signal that the modeling exercise has not been completed. That is,
if residuals are autocorrelated, this signifies that the residuals at a generic
time t can be predicted from residuals at an earlier time.
Autoregressive Moving Average Models There are models for dealing with
the problem of autocorrelation in time series data. These models are called
autoregressive moving average (ARMA) models. Although financial time
(^8) See N. Eugene Savin and Kenneth J. White, “The Durbin-Watson Test for Serial
Correlation with Extreme Sample Sizes or Many Regressors,” Econometrica 45
(1977): 1989–1996.