# Basic Statistics and Data Analysis

## Breusch-Pagan Test for Heteroscedasticity

Breusch–Pagan test (named after Trevor Breusch and Adrian Pagan) is used to test for heteroscedasticity in a linear regression model.

Assume our regression model is $Y_i = \beta_1 + \beta_2 X_{2i} + \mu_i$ i.e we have simple linear regression model, and $E(\mu_i^2)=\sigma_i^2$, where $\sigma_i^2=f(\alpha_1 + \alpha_2 Z_{2i})$

That is $\sigma_i^2$ is some function of the non-stochastic variable Z‘s. f() allows for both the linear and non-linear forms of the model. The variable Z is the independent variable X or it could represent a group of independent variables other than X.

Step to Perform Breusch-Pagan test

1. Estimate the model by OLS and obtain the residuals $\hat{\mu}_1, \hat{\mu}_2+\cdots$
2. Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
3. Run the regression $\frac{e_i^2}{\hat{\sigma^2}}=\beta_1+\beta_2 Z_i + \mu_i$ and compute explained sum of squares (ESS) from this regression
4. Test the statistical significance of ESS/2 by $\chi^2$-test with 1 df at appropriate level of significance (α).
5. Reject the hypothesis of homoscedasticity in favour of heteroscedasticity if $\frac{ESS}{2} > \chi^2_{(1)}$ at appropriate level of α.

Note that the

• Breusch-Pagan test is valid only if μi‘s are normally distributed.
• For k independent variables, ESS/2 have ($\chi^2$) Chi-square distribution with k degree of freedom.
• If the μi‘s (error term) are not normally distributed, White test is used.

References:

• Breusch, T.S.; Pagan, A.R. (1979). “Simple test for heteroscedasticity and random coefficient variation”. Econometrica (The Econometric Society) 47 (5): 1287–1294.

# GoldFeld-Quandt Test of Heteroscedasticity

The Goldfeld-Quandt test is one of two tests proposed in a 1965 paper by Stephen Goldfeld and Richard Quandt. Both a parametric and nonparametric test are described in the paper, but the term “Goldfeld–Quandt test” is usually associated only with the parametric test.
GoldFeld-Quandt test is frequently used as it is easy to apply when one of the regressors (or another r.v.) is considered the proportionality factor of heteroscedasticity. GoldFeld-Quandt test is applicable for large samples.The observations must be at least twice as many as the parameters to be estimated. The test assumes normality and serially independent error terms μi.

The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. So data is divided in h subgroups. Usually data set is divided into two parts or groups, and hence the test is sometimes called a two-group test.

The procedure of conducting GoldFeld-Quandt Test is

1. Order the observations according to the magnitude of X (the independent variable which is the proportionality factor).
2. Select arbitrarily a certain number (c) of central observations which we omit from the analysis. (for n=30, 8 central observations are omitted i.e. 1/3 of the observations are removed). The remaining n – c observations are divided into two sub-groups of equal size i.e.(n – c)/2, one sub-group includes small values of X and other sub-group includes the large values of X, as data set is arranged according to the magnitude of X.
3. Now Fit the separate regression to each of the sub-group, and obtain the sum of squared residuals form each of them. So$\sum c_1^2$
Show sum of squares of Residuals from sub-sample of low values of X with (n – c)/2K df, where Kis total number of parameters.

$\sum c_2^2$
Show sum of squares of Residuals from sub-sample of large values of X with (n – c)/2K df, where K is total number of parameters.

4. Compute the Relation $F^* = \frac{RSS_2/df}{RSS_2/df}=\frac{\sum c_2^2/ ((n-c)/2-k)}{\sum c_1^2/((n-c)/2-k) }$

If Variances differs, F* will have a large value. The higher the observed value of F* ratio the stronger the hetro of the μi‘s.

References

• Goldfeld, Stephen M.; Quandt, R. E. (June 1965). “Some Tests for Homoscedasticity”. Journal of the American Statistical Association 60 (310): 539–547
• Kennedy, Peter (2008). A Guide to Econometrics (6th ed.). Blackwell. p. 116
• Cook, R. Dennis; Weisberg, S. (April 1983). “Diagnostics for heteroscedasticitiy in regression”. Biometrika 70 (1): 1–10.