# Category: Test of Heteroscedasticity

Different available test of heteroscedasticty, Detection of Heteroscedasticity using Graphical techniques will be presented in this category.

## The Breusch-Pagan Test (Numerical Example)

To perform the Breusch-Pagan test for the detection of heteroscedasticity, use the data from the following file Table_11.3.

Step 1:

The estimated regression is $\hat{Y}_i = 9.2903 + 0.6378X_i$

Step 2:

The residuals obtained from this regression are:

The estimated $\tilde{\sigma}^2$ is $\frac{\sum u_i^2}{n} = \frac{2361.15325}{30} = 78.7051$.

Compute a new variable $p_i = \frac{\hat{u}_i^2}{\hat{\sigma^2}}$

Step 3:

Assuming $p_i$ is linearly related to $X_i(=Z_i)$ and run the regression of $p_i=\alpha_1+\alpha_2Z_{2i}+v_i$.

The regression Results are: $\hat{p}_i=-0.74261 + 0.010063X_i$

Step 4:

Obtain the Explained Sum of Squares (ESS) = 10.42802.

Step 5:

Compute: $\Theta = \frac{1}{2} ESS = \frac{10.42802}{2}= 5.2140$.

The Breusch-Pagan test follows Chi-Square Distribution. The $\chi^2_{tab}$ value at a 5% level of significance and with ($k-1$) one degrees of freedom is 3.8414. The $\chi_{cal}^2$ is greater than $\chi_{tab}^2$, therefore, results are statistically significant. There is evidence of heteroscedasticity at a 5% level of significance.

## White General Heteroscedasticity Test (Numerical Example)

We will consider the following data, to test the presence of heteroscedasticity using the White General Heteroscedasticity test.

To perform the white test, the general procedure is

Step 1: Run a regression and obtain $\hat{u}_i$ of this regression equation.

The regression model is: $income = \beta_1+\beta_2\, educ + \beta_3\, jobexp + u_i$

The Regression results are: $Income_i=-7.09686 + 1.93339 educ_{i} + 0.649365 jobexp_{i}$

Step 2: Run the following auxiliary regression

$$\hat{u}_i^2=\alpha_1+\alpha_2X_{2i}+\alpha_3 X_{3i}+\alpha_4 X_{2i}^2+\alpha_5X_{3i}^2+\alpha_6X_{2i}X_{3i}+vi$$

that is, regress the squared residuals on a constant, all the explanatory variables, the squared explanatory variables, and their respective cross-product.

Here in auxiliary regression education, $Y$ is income, $X_2$ is educ, and $X_3$ is jobexp.

The results from auxiliary regression are:

$$Y=42.6145 -0.10872\,X_{2i} – 5.8402\, X_{3i} -0.15273\, X_{2i}^2 + 0.200715\, X_{3i}^2 + 0.226517\,X_{2i}X_{3i}$$

Step 3: Formulate the null and alternative hypotheses

$H_0: \alpha_1=\alpha_2=\cdots=\alpha_p=0$

$H_1$: at least one of the $\alpha$s is different from zero

Step 4: Reject the null and conclude that there is significant evidence of heteroscedasticity when the statistic is bigger than the critical value.

The statistic with computed value is:

$$n \cdot R^2 \, \Rightarrow = 20\times 0.4488 = 8.977$$

The statistics follow asymptotically $\chi^2_{df}$, where $df=k-1$. The Critical value is $\chi^2_5$ at 5% level of significance is  11.07.

Since the calculated value is smaller than the tabulated value, therefore, the null hypothesis is accepted. Therefore, on the basis of the White general heteroscedasticity test, there is no heteroscedasticity.

## Park Glejser Test: Numerical Example

To detect the presence of heteroscedasticity using the Park Glejser test, consider the following data.

The step by step procedure of conducting Park Glejser test:

Step 1: Obtain estimate the regression equation

$$\hat{Y}_i = 19.8822 + 4.7173X_i$$

Obtain the residuals from this estimated regression equation:

Take the absolute values of these residuals and consider it as your dependent variables to perform the different functional forms suggested by Glejser.

Step 2: Regress the absolute values of $\hat{u}_i$ on the $X$ variable that is thought to be closely associated with $\sigma_i^2$. We will use the following function forms.

Since none of the residual regression is significant, therefore, the hypothesis of heteroscedasticity is rejected. Therefore, we can say that there is no relationship between the absolute value of the residuals ($u_i$) and the explanatory variable $X$.

How to perform White General Heteroscedasticity?

## Numerical Example: Goldfeld-Quandt Test

Data is taken from the Economic Survey of Pakistan 1991-1992. The data file link is at the end of this numerical example of the Goldfeld-Quandt Test.

For an illustration of the Goldfeld-Quandt test, data given in the file should be divided into two sub-samples after dropping (removing/deleting) the middle five observations.

Sub-sample 1 consists of data from 1959-60 to 1970-71.

Sub-sample 2 consists of data from 1976-77 to 1987-1988.

The sub-sample 1 is highlighted in green colour, and sub-sample 2 is highlighted in blue color, while the middle observation that has to be deleted is highlighted in red.

The Step by Step procedure to conduct the Goldfeld-Quandt test is:

Step 1: Order or Rank the observations according to the value of $X_i$. (Note that observations are already ranked.)

Step 2: Omit $c$ central observations. We selected 1/6 observations to be removed from the middle of the observations.

Step 3: Fit OLS regression on both samples separately and obtain the Residual Sum of Squares (RSS) for each sub-sample.

The Estimated regression for the two sub-samples are:

Sub-sample 1: $\hat{C}_1 = 1010.096 + 0.849 \text{Income}$

Sub-sample 2: $\hat{C}_2 = -244.003 + 0.88067 \text{Income}$

Now compute the Residual Sum of Squares for both sub-samples.

Residual Sum of Squares for Sub-Sample 1 is $RSS_1=2532224$

Residual Sum of Squares for Sub-Sample 2 is $RSS_2=10339356$

The F-Statistic is $\lambda=\frac{RSS_2/n_2}{RSS_1/n_1}=\frac{10339356}{2532224}=4.083$

The critical value of $F(n_1=10, n_2=10$ at 5% level of significance is 2.98.

Since the computed F value is greater than the critical value, heteroscedasticity exists in this case, that is, the variance of the error term is not consistent, rather it depends on the independent variable, GNP.

Your assignment is to perform this Numerical Example of the Goldfeld-Quandt test using any statistical software and confirm the results.