## 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:

$\hat{u}_i$ | $\hat{u}_i^2$ | $p_i$ |

-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.