endogenous and exogenous variable interpretation

Question

I am running a regression of the form

$$\log\left(Y\right)=x_0 + x_1\beta_1+\log (x_2)\beta_2 +x_3\beta_3+ \epsilon$$

where all the covariates $x_1$,$x_2$,$x_3$ are endogenous. I have an instrument for only one of them ($x_1$) and have no instrument for $x_2,x_3$.

I am estimating the above regression with 2 stage least squares (2SLS) and I get estimates for $\beta_1,\beta_2,\beta_3$.

I was wondering regarding the interpretation of these coefficients. I believe that my estimation is fine, but I can only attach causal interpretation to coefficient $\beta_1$ since I have an instrument for it. The remaining coefficients i.e. $\beta_2,\beta_2$ will only have a correlational interpretation. Is my interpretation correct?

Answer 1

I believe that my estimation is fine

It is not. Since you accept that all regressors are "endogenous" (i.e. correlated with the error term), and you don't have valid instruments for all of them, then your estimator is inconsistent (for all coefficients, since endogeneity contaminates everything): this means that you don't really know what you are actually estimating. At least by the current prevailing consensus on the matter, any interpretation of the coefficients will (should?) not even be heard to begin with.

But I think I understand what your question is and the answer is "no, you cannot have in the same relation some effects correlational only, and some effects causal".

Now, make the assumption that the conditional expectation function of the log-dependent variable with respect to the three regressors is linear:

$$E(\ln y\mid \mathbf x) =\gamma_0 + x_1\gamma_1+ \log (x_2)\gamma_2 +x_3\gamma_3$$

Note that I used different symbols for the unknown coefficients. If you estimate by Ordinary Least Squares this relation, you will always get consistent estimates of the gamma coefficients (if indeed the conditional expectation function is linear). These are all "correlational" effects, as you write, because they only postulate a pure statistical relation between the variables. But the gamma coefficients by design have different values than the beta coefficients in your initial regression.

What this have to do with your question? Well, by saying that your regressors are "endogenous", and more importantly, by saying that this is a problem, you reveal that you are interested in estimating something like the following relationship:

$$\log\left(Y\right)=\beta_0 + x_1\beta_1+\log (x_2)\beta_2 +x_3\beta_3 +(x^*\beta_4+u)$$

where

$$E\big[\log\left(Y\right)\mid \{\mathbf x, x^*\}\big]=\beta_0 + x_1\beta_1+\log (x_2)\beta_2 +x_3\beta_3 +x^*\beta_4$$

but $x^*$ is unavailable, and correlated with the other regressors. So if you have valid instruments for all the regressors, your IV/2SLS estimator would indeed estimate consistently the first four betas. But the betas also are seen to represent a statistical relationship: causality cannot enter the picture through this route.

Answer 2

I will differ here with @AlecosPapadopoulos.

If your instrument for $x_1$, say $v$, is a "good instrument", meaning you can convincingly argue that $cov(x_1,v)\neq0$ and $cov(v,\epsilon)=0$, then 2SLS estimation should allow you to make causal interpretations of the coefficient $\beta_1$. The remaining coefficients will be biased and as such no interpretation is appropriate - so do not interpret these in any way.

Levels of collinearity between the $x$ variables might contaminate the instrument $v$. Think of a scenario where endogeneity is due to $cov(x_2,\epsilon)\neq0$ and/or $cov(x_3,\epsilon)\neq0$. If non-trivial collinearity exists between the $x$ variables, then the 2SLS procedure might yield imprecise estimates of the coefficient of $x_1$. The problem could be so severe so as to result in problems similar to those of weak instruments.