Building and Testing a Multiple Linear Regression Model 95
Tests For Constant Variance of the Error Term
(Homoscedasticity)
The second test regarding the residuals in a linear regression analysis is that
the variance of all squared error terms is the same. As we noted earlier, this
assumption of constant variance is referred to as homoscedasticity. How-
ever, many time series data exhibit heteroscedasticity, where the error terms
may be expected to be larger for some observations or periods of the data
than for others.
There are several tests that have been used to detect the presence of
heteroscedasticity. These include the
■ (^) White’s generalized heteroscedasticity test
■ (^) Park test
■ (^) Glejser Test
■ (^) Goldfeld-Quandt test
■ (^) Breusch-Pagan-Godfrey Test (Lagrangian multiplier test)
These tests will not be described here.
If heteroscedasticity is detected, the issue then is how to construct mod-
els that accommodate this feature of the residual variance so that valid
regression coefficient estimates and models are obtained for the variance of
the error term. There are two methodologies used for dealing with hetero-
scedasticity: weighted least squares estimation technique and autoregressive
conditional heteroscedastic (ARCH) models. We describe the first method
here. We devote an entire chapter, Chapter 11, to the second methodology
because of its importance for not just testing for heteroscedasticity but in
forecasting volatility.
Weighted Least Squares Estimation Technique A potential solution for correcting
the problem of heteroscedasticity is to give less weight to the observations
coming from the population with larger variances and more weight to the
observations coming from observations with higher variance. This is the
basic notion of the weighted least squares (WLS) technique.
To see how the WLS technique can be used, let’s consider the case of the
bivariate regression given by
yt = β 0 + β 1 xt + εt (4.22)
Let’s now make the somewhat bold assumption that the variance of
the error term for each time period is known. Denoting this variance by σt^2