Heteroskedasticity代写

Lecture 3: Test for Heteroskedasticity

Heteroskedasticity代写 In Lecture one, we have learned that one of the assumptions of OLS regression is variance of the unobserved…

  • What is heterskedasticity? Heteroskedasticity代写

In Lecture one, we have learned that one of the assumptions of OLS regression is variance of the unobserved error terms, conditional on the explanatory variables, being constant, i.e.,  (Note  is constant).

 

Homoskedasticity fails whenever the variance of the error terms changes across different segments of the population, where the segments are determined by the different values of the explanatory variables, i.e.,

heteroskedasticity代写
heteroskedasticity代写
  • Consequences of heteroskedasticity for OLS

Heteroskedasticity does not cause bias or inconsistency in the OLS estimates.

The interpretation of our goodness-of-fit, R_square, is also unaffected by the presence of heteroskedasticity.

However, the estimators of the variance are biased without the homoskedasticity assumption. The usual OLS t-statistics do not have t-distributions in the presence of heteroskedasticity and the problem is not resolved by using large sample size.

 

  • Testing for heteroskedasticity Heteroskedasticity代写

Over the years, many possible approaches have been suggested, such as Goldfeld-Quandt test, Breusch-Pagan test, and White test etc..

We will restrict ourselves to more modern test, i.e., regessions using squared residuals.

 

 The idea of testing for heteroskedasticy is based on the use of squared OLS residuals as a proxy for unobservable squared disturbances using the following specification

Heteroskedasticity代写
Heteroskedasticity代写

Z can be any variable that is included in the explanatory variables X, or a transformation of such variables, or a new variable that does not appear in X.

In principle, the parameters ….can be estimated from a regression of  on Z. If the squared OLS regressed on Z, we obtain an estimated model, which can be written as

heteroskedasticity代写
heteroskedasticity代写

If the parameters ….are all zero, variance is a constant, equal to and there is no heteroskedasticity in the error terms to the original model.

Heteroskedasticity代写
Heteroskedasticity代写

This suggests that one can test the null hypothesis of no heterodskedasticity by testing

We can use a large sample of test:

heteroskedasticity代写
heteroskedasticity代写

Under the null hypothesis, is approximately with (p-1) degrees of freedom. So if the value of nR2 is greater than 5% critical value for a distribution with p-1 degrees of freedom, the null hypothesis of no heteroskedasticity is rejected at 5%significant level.

 

  • How to conduct the estimation in the presence of heteroskedasticty?

 

Feasible GLS (FGLS)