DETECTION OF
HETROSCEDASTICITY
Several test procedures have been proposed to detect heteroscedasticity
in a regression model. The following tests are commonly used to detect heteroscedasticity
in the regression model.
The Spearman’s Rank
Correlation Test
The Spearman’s rank correlation test
is applied when there is a simple linear regression model and the sample size is
small. The following test is suggested to detect heteroscedasticity.
Test procedure:
Arrange the absolute
values of ei and their corresponding Xi in ascending order
of magnitude and assign ranks. If the value of re.x is high, suggest the
existence of heteroscedasticity.
Test
criterion:
Practice Question
Given the data below
ii. Test the existence of heteroscedasticity using the spearman’s rank correlation test.
Solution:
The estimated regression line is given by:
vi. Remarks:
As the t-calculated value does not fall in the rejection region, we have not
sufficient evidence to reject H0. There is no evidence of heteroscedasticity.
Goldfield-Quantt Test
The Goldfield-Quantt test is applicable when the sample
size is large and disturbance is a function of a single explanatory variable.
That is
To perform the test, the following necessary steps are
taken:
1. Write
the explanatory variables in ascending order of magnitude and also write their
corresponding response variable.
2. Omit the central number of observations (say). The remaining (n-C) is divided into
two parts.
(n-c)/2 for the lower part and (n-c)/2 for the upper part.
3. Fit the regression line to the data and compute the residual sum of squares for each part
separately and test the hypothesis:
i. H0: The disturbance term
is Homoscedastic vs. H1: The disturbance term
is Hetroscedastic
ii. The significance level,
iii. The
test statistic;
Practice Question
The following table presents consumption (Y) and pocket (X)
for 20 male students. Test the existence of heteroscedasticity.
Determine the presence of heteroscedasticity at 5% using the data above.
Solution: State the null and alternative hypotheses as:
V. Computation: Arrange independent variables in
ascending order of magnitude and write their corresponding dependent variable
values.
Breusch-Pagan Test
The
Goldfield-Quantt test
is used to detect heteroscedasticity. When the variance
of the disturbance term is a function of a single explanatory variable.
That is
Now if the variance of the disturbance term is a linear function of more
than one explanatory variable.
Then we use the Breusch-Pagan test, which is designed to detect any linear form of
heteroskedasticity. Here we assume the heteroscedasticity process is a function of
one or more independent variables included in a regression model.
Breusch-Pagan can use auxiliary regression analysis by regressing the
squared of residuals from the original model on a set of original regressors.
This assumption can be expressed as:
BP test
procedure:
A weakness of the BP test is that it assumes the heteroskedasticity is a linear function of the independent
variables.
Consider the multiple linear regression model:
Run the OLS and find the
estimates of the model parameters and calculate the estimate of the disturbance term.
To test
the assumption of heteroscedasticity, White can use auxiliary regression
analysis by regressing the squared of residuals from the original model on the set of original regressors, the squared of regressors, and the cross products of
the regressors.
Determine the coefficient of determination of the auxilary regression and test the hypothesis as:
- Read More: Autocorrelation
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