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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?

Model 1 enter image description here

Model 2 enter image description here

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