**GoldFeld-Quandt Test of Heteroscedasticity**

**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 *μ _{i}*.

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

- Order the observations according to the magnitude of
*X*(the independent variable which is the proportionality factor). - 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*. - 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)/2*–*K*df, where*K*is 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)/2*–*K*df, where*K*is total number of parameters. - 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 *μ _{i}*‘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.

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