Goldfeld–Quandt Test

The Goldfeld-Quandt test is an econometric test for heteroscedasticity in a linear regression. It is most useful when you can order observations by some variable and you suspect the error variance increases (or decreases) as that ordering variable rises.

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When The Test Is A Good Fit

The Goldfeld-Quandt test is designed for cases like:

households ordered by income (variance of consumption errors may rise with income),
firms ordered by size (variance of profit shocks may rise with size),
time ordered by a trend proxy (variance may rise in later periods).

If there is no plausible ordering variable, tests like Breusch-Pagan or White are often more natural.

Mechanics (Step By Step)

Suppose you have the regression y = Xβ + u with k parameters.

Order the data by the variable you think is linked to the error variance (for example, income or firm size).
Split the ordered sample into a low group and a high group, often dropping the middle r observations to make the two groups more distinct.
Run OLS separately on the low group and the high group.
Compare residual variances with an F statistic:

\[ F = \frac{\text{SSR}_2/(n_2-k)}{\text{SSR}_1/(n_1-k)} \]

where SSR1 and SSR2 are sums of squared residuals from the two regressions and n1, n2 are the group sample sizes.

Under the null hypothesis of homoscedasticity, this statistic is compared to an F distribution (with degrees of freedom n2-k and n1-k).

Interpretation And Caveats

A large F (relative to the critical value) suggests the variance differs across the ordered groups.
Results can be sensitive to:
- which ordering variable you choose,
- how many middle observations you drop,
- whether the model is correctly specified.

Also, heteroscedasticity does not bias OLS coefficients in the standard exogeneity setup, but it does make usual standard errors and t-tests unreliable unless corrected.

What To Do If You Detect Heteroscedasticity

Common responses include:

use heteroscedasticity-robust (Huber-White) standard errors,
model the variance (for example, weighted least squares) if you have a good variance model,
transform variables (often logs) when scale effects drive heteroscedasticity.

Knowledge Check

### What is the null hypothesis in the Goldfeld-Quandt test? - [x] The regression errors have constant variance (homoscedasticity) - [ ] The regression has omitted variables - [ ] The errors are autocorrelated - [ ] The regressors are perfectly collinear > **Explanation:** The test asks whether residual variance is the same in the low and high ordered groups. ### Before running the Goldfeld-Quandt test, you typically: - [x] Order observations by a variable suspected to be linked to the error variance - [ ] Randomly permute the data to eliminate patterns - [ ] Difference the dependent variable - [ ] Drop all high-leverage points > **Explanation:** The method is designed for situations where variance changes systematically with an ordering variable (income, size, time). ### The Goldfeld-Quandt test statistic is essentially a ratio of: - [ ] regression coefficients - [ ] sample means - [x] residual variances (SSR adjusted for degrees of freedom) across two subsamples - [ ] R-squared values from two models > **Explanation:** The test compares variance estimates from two parts of an ordered sample to see if variance is systematically different.