2
$\begingroup$

Suppose I have a regression model: $$y_{i}=x_{1i}\beta_{1}+x_{1i}D_{i}\beta_{2}+\epsilon $$ where $\mathbb{E}\left[\epsilon_{i}|x_{i}\right]\neq0$ , and there is a problem of endogeneity. In the above, think of the dummy variable being equal to $1$ when $i$ is male. As such, this model allows for a differential effect of $x_{1i}$ on $y_{i}$, if the individual is a male or female. In order to obtain consistent estimates, suppose I have an instrument $z_{1i}$, which is both a relevant instrument for $x_{1i}$, and satisfies the exclusion restriction (such that $cov(z_{i},\epsilon_{i})=0).$ If there were no interaction term, it would be trivial to estimate the above using Two stage least squares, or simply just computing: $$ \hat{\beta}=\left(\boldsymbol{z'}x\right)^{-1}\left(\boldsymbol{z'y}\right) $$

However, the complication arises due to the interaction term. How would I go about obtaining estimates for both $\beta_{1}$ and $\beta_{2}$? One option is to construct another instrument $D_{i}Z_{1i}$. Would such a procedure be valid?

$\endgroup$
1
$\begingroup$

Let $X$ be the original regressor matrix and $Z = [z \;\; zd]$ be the intstruments matrix, and consider the IV estimator

$$\hat \beta_{IV} = (Z'X)^{-1}Z'y = (Z'X)^{-1}Z'(X\beta + u) = \beta + (Z'X)^{-1}Z'u$$

As long as the things that are included in $Z$ are orthogonal to the error term, we get consistency... provided that $Z'X$ remains invertible, something that you can easily check in your case.

$\endgroup$
  • $\begingroup$ Thank you. With regards to the projection of X and XD in the first stage, would they each be projected onto Z and ZD in order to obtain fitted values? $\endgroup$ – ChinG Jul 5 '18 at 14:37
  • $\begingroup$ @ChinG If I understand your question correctly, fitted values for the dependent variable should be obtained by using $x_i$ and $d_i$, alongside the IV estimates. $\endgroup$ – Alecos Papadopoulos Jul 5 '18 at 16:46
  • $\begingroup$ I think I was not clear in my question: X is endogenous, so I assume so is XD. I only have one instrument Z. It seems according to your response that I can use ZD as an instrument as well. However, in the first stage, will I regress X on Z and ZD in one regression, and XD on Z and ZD in the second? This way I will obtain fitted values for X and XD as separate regressions on both instruments in the first stage. Thereafter I use these fitted values on the RHS in the second. I feel as if I am missing something basic. Thanks a lot @Alecos Papadopoulos $\endgroup$ – ChinG Jul 5 '18 at 20:01
  • $\begingroup$ @ChinG We do not apply 2SLS here, especially since you have only one instrument. My answer deals with a one-stage estimator. $\endgroup$ – Alecos Papadopoulos Jul 5 '18 at 23:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.