Hausman-Test

The Hausman specification test (test for endogeneity, that is a test for the relationship between the explanatory (independent) variables and the disturbance ) is a test method of mathematical statistics. It was developed in 1978 by Jerry Hausman to decide on panel data models, if more of a fixed-effects model (FE - model ) or a random-effects model is available (RE ) model (see Linear panel data models). The former assumed for each subject individual an individual ( by regression to be determined ) deviation from the panel mean, and this deviation in the RE model is a normally distributed random variable.

The null hypothesis that an RE model is effective is rejected if the test statistic is greater than the corresponding percentile of the distribution with K degrees of freedom:

The variables used are defined herein as follows:

  • : Constant regression parameters ( intercept)
  • : Number of regressors in the panel data model
  • : Vector of estimated regression coefficients K of the random-effects estimation
  • : Vector of estimated regression coefficients K of the fixed-effects estimation
  • : Estimated variance - covariance matrix of the FE estimator
  • : Error terms, deviation between estimated and observed value

If the estimators are not distorted ( ie true ), the fixed-effects estimator is always consistent (ie leads with increasing number of observations closer to the true value of the parameter approach ), while the random-effects estimator only consistent, but in addition also is efficient even if and are uncorrelated. The Hausman test compares the regressors of the two methods. If they differ significantly, the null hypothesis is rejected. Thus, an estimate using fixed effects is advised.

When testing for endogeneity is a simple variant of the Hausman test, the examination of individual variables using a residuals tests dar. The following two propositions are tested against each other:

The test consists of two stages: First, to be examined variable is regressed on all exogenous variables in the model. The residuals of this regression are then used in the second stage of the test in the output equation as an additional regressor. The thus expanded model is estimated using the method of least squares. If the coefficient of Residuenvariablen significantly, there is correlation between disturbance and the examined regressor, that is, the null hypothesis must be rejected and the existence of endogeneity as confirmed are considered.

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