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I want to estimate the following model

$y_{it} = \gamma_{0} + \gamma_{1} x_{it} + \gamma_{2} \theta_{i} x_{it} + u_{it}$

where the $\theta_i$ is unknown but I can estimate it in another regression. Plugging in that estimate and rewriting the equation, I get

$y_{it} = \gamma_{0} + \gamma_{1} x_{it} + \gamma_{2} \hat \theta_{i} x_{it} + u_{it} + \gamma_{2} x_{it} (\theta_i - \hat \theta_{i})$

The last two terms are obviously unobserved, and the last one correlates with $x_{it}$, so my estimates are inconsistent. I am hoping that this is a common problem that has some established fix. Does anyone know literature on that?

Thank you in advance.

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  • $\begingroup$ I think what you are looking for is a "two Stage Least Squares". In R, there is package called twosls for this. You can have a look at subsection 1.0.6 in above link. $\endgroup$ – Stat Dec 17 '13 at 18:27
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I think I was too quick in seeing a problem here; if the first stage is consistent, so is the second stage since $\hat \theta \to_p \theta$, even though it will be biased in finite samples. So I guess this problem is solved.

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The easiest way to correct for this problem is to bootstrap. Assuming you are using the same dataset for estimating $\hat{\theta}$ as for the regression of interest, you want to estimate $\hat{\theta}$ with the bootstrapped sample, this introduces variation between samples which converges to the true variation as the number of replications approaches infinity.

While it is true that your estimate is consistent, your intuition is correct that the standard errors are not correct without correction and this will cause problems with the asymptotics of any hypothesis testing.

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