Slides_Quan201_Topic6_Heteroskedasticity slides

QUAN201 - Introduction to Econometrics¶

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Reference: Wooldridge, Chapter 8

Recall Assumption MLR.5: Homoskedasticity. The error $u$ has the same variance given any value of the explanatory variables:
$Var(u|x_1,x_2,...,x_k)=\sigma^2$
If this is not the case, there is heteroskedasticity.
What does this mean?

Consider the following model: $wage=\beta_0+\delta_0 female +u$
Homoskedasticity means that the variance of the error term $u$ (and in this case the variance of wages) is the same for females and males.
Is this realistic? Discuss.
Let's look at the summary statistics,

Heteroskedasticity leads to incorrect inference-
- t-statistics, F-statistics, confidence intervals and p-values are no longer valid.
OLS is no longer the most efficient estimator (OLS is no longer BLUE), i.e., there might be other estimators like WLS that are more efficient.
Heteroskedasticity does not
- cause bias or inconsistency
- affect the interpretation of $R^2$

Imagine that you estimate the following linear model, $y= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...+ \beta_k x_k + u$ where all the other CLM assumptions hold. Most importantly the assumption that $E[u|\textbf{x}]=0$ holds implying that the OLS is unbiased and consistent ( $\textbf{x}=<x_1,x_2,...,x_k>$ ).
The null hypothesis, $H_0: \, Var(u|\textbf{x})=E[u^2|\textbf{x}]=\sigma^2$ says that the error variance is constant, i.e., homoskedastic.
[Note: $Var(u|\textbf{x})=E[u^2|\textbf{x}]$ is not generally true, but here follows from $E[u|\textbf{x}]=0$ .]
The idea of the Breusch-Pagan test is to examine whether $u^2$ (the variance of the error) is related to one or more variables: $u^2=\delta_0 + \delta_1 x_1 + \delta_2 x_2 +...+\delta_k x_k + \upsilon$

Then the null hypothesis $H_0: \, Var(u|\textbf{x})=E[u^2|\textbf{x}]=\sigma^2$ suggests using, $H_0: \delta_1 = \delta_2 = ...=\delta_k=0$
This means that the independent variables are not (linearly) related to the variance of error term $u$ .
The main issue is that it is impossible to know the error terms $u$ .
How can we obtain estimates for the error term to test this hypothesis? We use the OLS residuals $\hat{u}^2$ as estimates of the error term. The equation then becomes $\hat{u}^2 = \delta_0 + \delta_1 x_1 + \delta_2 x_2 +...+\delta_k x_k + \upsilon$
We can estimate this equation and test $H_0$ with an F-test. If the F-test rejects $H_0$ we have evidence for heteroskedasticity.

$\widehat{Var(\beta_j)} = \frac{\sum_{i=1}^n \hat{r}_{ij}^2 \hat{u}_{ij}^2 }{SSR_j^2}$ where,

$\hat{r}_{ij}$ is the residual for person $i$ from a regression of $x_j$ on all the other independent variables.
$SSR_j$ is the sum of squared residuals from same regression of $x_j$ on all the other independent variables.