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:
| $\hat{u}_i$ | $\hat{u}_i^2$ | $p_i$ |
|---|---|---|
| -5.31307 | 28.22873 | 0.358665 |
| -8.06876 | 65.10494 | 0.827201 |
| 6.49801 | 42.22407 | 0.536485 |
| 0.55339 | 0.30624 | 0.003891 |
| -6.82445 | 46.57318 | 0.591743 |
| 1.36447 | 1.86177 | 0.023655 |
| 5.79770 | 33.61333 | 0.427079 |
| -3.58015 | 12.81744 | 0.162854 |
| 0.98662 | 0.97342 | 0.012368 |
| 8.30908 | 69.04085 | 0.877209 |
| -2.25769 | 5.09715 | 0.064763 |
| -1.33584 | 1.78446 | 0.022673 |
| 8.04201 | 64.67391 | 0.821724 |
| 10.47524 | 109.73066 | 1.3942 |
| 6.23093 | 38.82451 | 0.493291 |
| -9.09153 | 82.65588 | 1.050197 |
| -12.79183 | 163.63099 | 2.079039 |
| -16.84722 | 283.82879 | 3.606231 |
| -17.35860 | 301.32104 | 3.828481 |
| 2.71955 | 7.39595 | 0.09397 |
| 2.39709 | 5.74604 | 0.073007 |
| 0.77494 | 0.60052 | 0.00763 |
| 9.45248 | 89.34930 | 1.135241 |
| 4.88571 | 23.87014 | 0.303286 |
| 4.53063 | 20.52658 | 0.260804 |
| -0.03614 | 0.00131 | 1.66E-05 |
| -0.30322 | 0.09194 | 0.001168 |
| 9.50786 | 90.39944 | 1.148584 |
| -18.98076 | 360.26909 | 4.577455 |
| 20.26355 | 410.61159 | 5.217089 |
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 degree 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.
