# linearmodels.iv.results.IVResults.wu_hausman¶

- IVResults.wu_hausman(
*variables=None*)[source]¶ Wu-Hausman test of exogeneity

- Parameters:
- Returns:
`WaldTestStatistic`

Object containing test statistic, p-value, distribution and null

Notes

Test statistic is difference between sum of squared OLS and sum of squared IV residuals where each set of residuals has been projected onto the set of instruments in the IV model.

Start by defining

\[\delta & = \hat{\epsilon}'_e P_{[z,w]} \hat{\epsilon}_e - \hat{\epsilon}'_c P_{z} \hat{\epsilon}_c\]where \(\hat{\epsilon}_e\) are the regression residuals from a model where

`vars`

are treated as exogenous, \(\hat{\epsilon}_c\) are the regression residuals from the model leaving`vars`

as endogenous, \(P_{[z,w]}\) is a projection matrix onto the exogenous variables and instruments (z) as well as`vars`

, and \(P_{z}\) is a projection matrix only onto z.The test statistic is then

\[\frac{\delta / q}{(\hat{\epsilon}'_e\hat{\epsilon}_e - \delta) / v}\]where \(q\) is the number of variables iv, \(v = n - n_{endog} - n_{exog} - q\). The test statistic has a \(F_{q, v}\) distribution.