## 10.3 Heteroskedasticity

### 10.3.1 Breusch-Pagan test

A4 implies

$E(\epsilon_i^2|\mathbf{x_i})=\sigma^2$

$\epsilon_i^2 = \gamma_0 + x_{i1}\gamma_1 + ... + x_{ik -1}\gamma_{k-1} + error$

and determining whether or not $$\mathbf{x}_i$$ has any predictive value

• if $$\mathbf{x}_i$$ has predictive value, then the variance changes over the levels of $$\mathbf{x}_i$$ which is evidence of heteroskedasticity
• if $$\mathbf{x}_i$$ does not have predictive value, the variance is constant for all levels of $$\mathbf{x}_i$$

The Breusch-Pagan test for heteroskedasticity would compute the F-test of total significance for the following model

$e_i^2 = \gamma_0 + x_{i1}\gamma_1 + ... + x_{ik -1}\gamma_{k-1} + error$

A low p-value means we reject the null of homoskedasticity

However, Breusch-Pagan test cannot detect heteroskedasticity in non-linear form

### 10.3.2 White test

test heteroskedasticity would allow for a non-linear relationship by computing the F-test of total significance for the following model (assume there are three independent random variables)

\begin{aligned} e_i^2 &= \gamma_0 + x_i \gamma_1 + x_{i2}\gamma_2 + x_{i3}\gamma_3 \\ &+ x_{i1}^2\gamma_4 + x_{i2}^2\gamma_5 + x_{i3}^2\gamma_6 \\ &+ (x_{i1} \times x_{i2})\gamma_7 + (x_{i1} \times x_{i3})\gamma_8 + (x_{i2} \times x_{i3})\gamma_9 + error \end{aligned}

A low p-value means we reject the null of homoskedasticity

Equivalently, we can compute LM as $$LM = nR^2_{e^2}$$ where the $$R^2_{e^2}$$ come from the regression with the squared residual as the outcome