To detect the presence of heteroscedasticity using the Park Glejser test, consider the following data.
| Year | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 |
|---|---|---|---|---|---|---|---|
| Yt | 37 | 48 | 45 | 36 | 25 | 55 | 63 |
| Xt | 4.5 | 6.5 | 3.5 | 3 | 2.5 | 8.5 | 7.5 |
The step-by-step procedure for conducting the Park Glejser test:
Step 1: Obtain an estimate of the regression equation
$$\hat{Y}_i = 19.8822 + 4.7173X_i$$
Obtain the residuals from this estimated regression equation:
| Residuals | -4.1103 | -2.5450 | 8.6071 | 1.9657 | -6.6756 | -4.9797 | 7.7377 |
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.
| Sr. No. | Functional Form | Results |
|---|---|---|
| 1) | $|\hat{u}_t| = \beta_1 + \beta_2 X_i +v_i$ | $|\hat{u}_i| = 5.2666-0.00681X_i,\quad R^2=0.00004$ $t_{cal} = -0.014$ |
| 2) | $|\hat{u}_t| = \beta_1 + \beta_2 \sqrt{X_i} +v_i$ | $|\hat{u}_i| = 5.445-0.0962X_i,\quad R^2=0.000389$ $t_{cal} = -0.04414$ |
| 3) | $|\hat{u}_t| = \beta_1 + \beta_2 \frac{1}{X_i} +v_i$ | $||\hat{u}_i| = 4.9124+1.3571X_i,\quad R^2=0.00332$ $t_{cal} = -0.12914$ |
| 4) | $|\hat{u}_t| = \beta_1 + \beta_2 \frac{1}{\sqrt{X_i}} +v_i$ | $\hat{u}_i| = 4.7375+1.0428X_i,\quad R^2=0.00209$ $t_{cal} = 0.10252$ |
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 $X$.
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