# Is Hausman-Taylor Estimator with different within- and between-group effects biased?

Background

Assume the simple two-level linear model $$y_{ij}=\beta_0 + \beta_1X_{ij}+\beta_2X_{ij}^{end}+\beta_3Z_j+\beta_4Z_j^{end}$$ where $$X_{ij}$$ and $$Z_j$$ are exogenous and $$X^{end}_{ij}$$ and $$Z^{end}_j$$ endogenous.

In essence the Hausman-Taylor estimator first computes the within-group estimator for $$\beta_1$$ and $$\beta_2$$ using FE regression, and then deducts the effect of $$X_{ij}$$ and $$X_{ij}^{end}$$ from $$y_{ij}$$ to get the residual term $$\hat\eta_{ij}=y_{ij}-\hat\beta_1X_{ij}+\hat\beta_2X_{ij}^{end} .$$ Calculating $$\bar\eta_{j}$$ as the cluster mean of $$\hat\eta_{ij}$$, one gets the 2SLS model $$\bar\eta_{j}=\beta'_0+\beta_3Z_{j}+\beta_4Z_{j}^{end} .$$ $$Z_{j}^{end}=\gamma_0+\gamma_3Z_{j}+\gamma_1 X_{ij} .$$ to obtain the unbiased estimator for $$\beta_4$$.

However, if the true within-group estimator $$\beta_1^W$$ is different from the true between-group estimator $$\beta_1^B$$, then we failed to remove part of the direct effect of $$\bar X_{j}$$ on $$\bar\eta_{j}$$. In other words, $$\bar X_j$$ has still a direct effect on $$\bar\eta_{j}$$ and the exclusion restriction of the IV approach is violated.

Question

1. Is my reasoning right and the Hausman-Taylor estimator makes the assumption that the within- and the between-group effect of the IV are identical; i.e. $$\beta_1^W=\beta_1^B$$?

2. Can one test whether $$\beta_1^W=\beta_1^B$$ by fitting the within-between RE model $$y_{ij}=\beta_0 + \beta_1^W(X_{ij}-\bar X_j) + \beta^B_1\bar X_{j}+\beta_2X_{ij}^{end}+\beta_3Z_j+\beta_4Z_j^{end}$$ and comparing $$\beta_1^W$$ and $$\beta_1^B$$ statistically?

STATA example

Using a classic dataset for the impact of union membership and education on logged wage (lwage), I tried this out in STATA using the xthtaylor command. In the first model I used exper as time-variant IV and in the second model I sepearte experience into a de-meaned within component diff_exper and a cluster mean component mean_exper as it is done in within-between RE modelling described in my question 2. The results show that the estimator for the endogenous cluster-level variable education (educt) indeed changes! Does this imply that my argument in question 1 appears to be correct?