# Hausman specification test

The Hausman specification test , also known as the Durbin-Wu-Hausman test , is a test procedure from mathematical statistics . It is a test for endogeneity , i.e. a test for the connection between the explanatory (independent) variables and the disturbance variable. It was developed by Jerry Hausman in 1978 in order to decide for panel data models whether a panel data model with fixed effects ( English fixed effects model , short FE model) or a panel data model with random effects ( English random effects model , short RE model) is present (see linear panel data models ). The former assumes an individual ( to be determined by means of a regression ) deviation from the panel mean value for each individual observed , while this deviation in the RE model represents a normally distributed random variable .

## Test statistics

The null hypothesis that there is an RE model is rejected when the test statistic ${\ displaystyle \ lambda}$

${\ displaystyle \ lambda = \ left ({\ hat {\ varvec {\ beta}}} _ {FE} - {\ hat {\ varvec {\ beta}}} _ {RE} \ right) '\ left (\ operatorname {Cov} ({\ hat {\ varvec {\ beta}}} _ {FE}) - \ operatorname {Cov} ({\ hat {\ varvec {\ beta}}} _ {RE}) \ right) ^ {-1} \ left ({\ hat {\ varvec {\ beta}}} _ {FE} - {\ hat {\ varvec {\ beta}}} _ {RE} \ right) \ sim \ chi ^ {2 } (K)}$

is greater than the corresponding quantile of the chi-square distribution with degrees of freedom . ${\ displaystyle K}$

The variables used are defined as follows:

## Endogeneity test

If the estimators are not biased (i.e. the following applies and there is therefore no endogeneity), the estimator in the model with fixed effects is always consistent (i.e., the more the number of observations increases, it approaches the true value of the parameter), while the estimator in the model with random effects is only consistent, but also efficient , if and are uncorrelated. The Hausman specification test compares the regressors of the two methods. If they differ significantly , the null hypothesis is rejected. An estimate using fixed effects is therefore advisable. ${\ displaystyle \ operatorname {E} [\ varepsilon x_ {it}] = 0}$${\ displaystyle \ alpha _ {i}}$${\ displaystyle x_ {i, t}}$${\ displaystyle \ operatorname {E} [\ alpha | x_ {it}] = 0}$

When testing for endogeneity , a simple variant of the Hausman specification test is the examination of individual variables with the help of a residual test . The following two theses are tested against each other:

The test consists of two stages: First, the variable to be examined is regressed on all exogenous variables of 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 model expanded in this way is estimated using the least squares method . If the coefficient of the residual variable is significant, there is a correlation between the disturbance variable and the investigated regressor, i.e. the null hypothesis must be rejected and the existence of endogeneity regarded as confirmed.

## literature

• Marno Verbeek (2004): A Guide to Modern Econometrics. 2nd edition, Chichester: John Wiley & Sons.
• Katja Wolf (2005): Comparison of estimation and test procedures under alternative specifications of linear panel models. Lohmar / Cologne: Eul.
• Jerry A. Hausman (1978): Specification Tests in Econometrics. In: Econometrica 46/6, pp. 1251-1271.

## Individual evidence

1. Ludwig Fahrmeir , Thomas Kneib , Stefan Lang: Regression: Models, Methods and Applications. , Springer Verlag 2009, p. 253
2. Ludwig Fahrmeir , Thomas Kneib , Stefan Lang: Regression: Models, Methods and Applications. , Springer Verlag 2009, p. 253
3. ^ Wooldridge, Jeffrey M. (2003): Introductory Econometrics: a Modern Approach. 2nd edition, Australia / Cincinnati (Ohio): South-Western College Pub.