We have this model:

$$D=\alpha_1+\gamma_1 Z + \epsilon_1$$

$$Y=\alpha_2+\gamma_2 D + \epsilon_2$$

Notations are as usual: Y is the outcome, D is the treatment, Z is the instrument

Two conditions for instrument variable to work in the literature: relevance (there is a correlation between Z and D) and exogeneity (no correlation between Z and error term $\epsilon_2$)

However, it seems no one cares about endogeneity in the first stage regression, which is the correlation between Z and $\epsilon_1$. Why is that the case? I have never seen any applied micro paper addressing this.


Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and $cov(Z,\epsilon_1)\neq 0$ would violate them (see A Guide to Modern Econometrics by Verbeek).

Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have $F$-statistics above $10$ (as a rule of thumb). Also the instrument should have an effect on $Y$ only through $D$ - the so called exclusion restriction (see Mostly Harmless Econometrics by Angrist and Pischke).

  • $\begingroup$ relevance is equal to weak instrument. relevance mean the the correlation between Z and D must be significantly larger than 0. exogeneity condition is similar to exclusion restriction. they are just different names for the same thing $\endgroup$ – FARRAF Sep 27 '20 at 22:35
  • $\begingroup$ @FARRAF no it’s not equal. Even with non-zero correlation, and individual betas being significant using t test, the F-statistics can be <10 $\endgroup$ – 1muflon1 Sep 27 '20 at 22:36

I'm going to assume that by "conditions for instrumental variables to work" you mean "instrumental variables is consistent." However, there are other properties to consider, like performance in small samples, etc. $ \newcommand{\Cov}{\text{Cov}} $

In this simple case, the IV estimator in sample of size $n$ is $$ \hat \gamma_2 = \frac{\sum_{i=1}^n (Z_i - \bar Z)(Y_i - \bar Y)}{\sum_{i=1}^n (Z_i - \bar Z)(D_i - \bar D)}. $$ For the instrumental variables estimator to be consistent, we only need $$ \text{Cov}(Z, \epsilon_2) = 0 \quad \text{ (instrument exogeneity)} \tag{1} $$ and $$ \text{Cov}(Z, D) \neq 0. \quad \text{ (instrument relevance)} \tag{2} $$ As long as these conditions are satisfied, it doesn't matter if $Z$ is correlated with $\epsilon_1$. The IV estimator will be consistent. To see this, let's analyze the model in terms of populations, \begin{align} \Cov(Z,Y) &= \gamma_2 \Cov(Z, D) + \Cov(Z, \epsilon_2), \end{align} and solve $$ \gamma_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)} - \frac{\Cov(Z, \epsilon_2)}{\Cov(Z, D)}. $$ If conditions (1) and (2) hold, then $\frac{\Cov(Z, \epsilon_2)}{\Cov(Z, D)} = 0$ and the IV estimate
$$ \gamma_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)} $$ is well-defined. The estimate is consistent since the probability limit is the population analog, $$ \hat \gamma_2 \overset{p}{\rightarrow} \frac{\Cov(Z, Y)}{\Cov(Z, D)} = \gamma_2. $$

Note that weak instruments become a problem as soon as we relax the assumption that $\text{Cov}(Z, \epsilon_2) = 0$ exactly, as you might want to do when analyzing the properties of the estimator in small samples.

So, are there any consequences if $Z$ is correlated with $\epsilon_1$? Well, it means that a regression of $D$ onto $Z$ will give a biased and inconsistent estimate of $\gamma_1$. However, the usual assumption underlying IV is that you don't really care about estimating $\gamma_1$. You want to consistently estimate $\gamma_2$, which you can do as long as assumptions (1) and (2) hold, as shown above.


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.