I run a fixed and random effects model with the same variables and got similar results. The Hausmann test indicated (=1) that I should use the Random Effects Model.
However, when I added a lag variable to the Random Effects Model, the Hausmann test indicated (<2.2e-16) I should use the Fixed Effects Model. The estimates for the variabels changed slightly.
What should I do?
You should not use a lag of the dependent variable in a mixed or random effects model because it is correlated with the random intercept. Correlations between predictors and the error term is referred to as endogeneity in statistical terms and this is a bad thing because it violates the regression model assumption that errors are independent of the predictors. If you want to include a lagged dependent variable in your model, you need to switch to a different modeling framework. Specifically, you should investigate what are called dynamic models. See this earlier CV thread for more information. See also this blog post from Paul Allison on the topic.
