# 2SLS with endogenous interaction terms

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?

## 1 Answer

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.

• 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? – ChinG Jul 5 '18 at 14:37
• @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. – Alecos Papadopoulos Jul 5 '18 at 16:46
• 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 – ChinG Jul 5 '18 at 20:01
• @ChinG We do not apply 2SLS here, especially since you have only one instrument. My answer deals with a one-stage estimator. – Alecos Papadopoulos Jul 5 '18 at 23:59