White Test - rileywheadon/ffa-framework GitHub Wiki

The White Test is used to detect changes in the variance of a time series.

• Null hypothesis: The variance of the time series is constant (homoskedasticity).
• Alternative hypothesis: The variance of the time series is time-dependent (heteroskedasticity).

Consider a simple linear regression model:

$$y_{i} = \beta_{0} + \beta_{1} x_{i} + \epsilon_{i}$$

Use ordinary least squares to fit the model. Then compute the squared residuals:

$${\hat{\epsilon}}{i}^{2} = (y{i} - \hat{y}_{i})^{2}$$

Next, fit an auxillary regression model to the squared residuals. This model should include each regressor, the square of each regressor, and the cross products between all regressors. Since $x$ is our only regressor,

$${\hat{\epsilon}}{i}^{2} = \alpha{0} + \alpha_{1}x_{i} + \alpha_{2}x_{i}^{2} + u_{i}$$

Next, we compute the coefficient of determination $R^2$ for the auxillary model. The test statistic is $nR^2 \sim \chi_{d}^2$, where $n$ is the number of observations and $d = 2$ is the number of regressors, excluding the intercept. If $nR^2 > \chi_{1-\alpha }$, we reject the null hypothesis and conclude that the time series exhibits heteroskedasticity.

For more information, the following sources may be useful:

• White Test Deep Dive
• Marno Verbeek, A Guide to Modern Econometrics (2004)
• William H. Greene, Econometric Analysis, 5th Edition (2002)