## ‘Metrics Monday: Dealing with Imperfect Instruments I

Happy New Year! After running out of easy, off-the-top-of-my-head topics for this series, I have decided to go with a friend’s suggestion of blogging econometrics papers whose results are useful for applied work.

Given that I am working on a paper in which I am dealing with an instrumental variable that is only plausibly exogenous–that is, the exclusion restriction is likely to hold, but there is a small chance that it does not–I thought I should begin the year with two posts on how to deal with imperfect instruments.

This does not mean that these posts will discuss what to do with plain-old bad instrumental variables (IVs), i.e., instruments for which the exclusion restriction clearly does not hold. Again, this post and the next will discuss situations where your IV most likely meets the exclusion restriction, but wherein there is a small chance that it does not.

Let’s start with the results in Conley et al. (REStat, 2012; see here for a non-gated version). The core idea is as follows: You are interested in the effect of treatment $D$ on outcome  $Y$, with or without controls  $X$. You are interested in estimating

(1) $Y = \beta_{0}{D} +\epsilon_{0},$

from which I am omitting the constant and the controls for brevity. Specifically, you are interested in the causal effect of the endogenous treatment $D$ on $Y$, and you have a plausibly exogenous instrument $Z$.

In the equation

(2) $Y = \beta_{1}{D} + \gamma_{1}{Z}+\epsilon_{1},$

parameters $\beta$ and $\gamma$ are not jointly identified because $D$ is endogenous. For a strictly exogenous IV–one whose exclusion restriction is met–we have that $\gamma = 0$.

The problem with an IV that is only plausibly exogenous is that $\gamma$, though it is likely to be small, is unlikely to be zero. So how do you go about the problem? One way to do it is to incorporate prior information about what $\gamma$ looks like, and in their paper, Conley et al. present four different ways to do that by (i) specifying only a range of possible values for $\gamma$, (ii) imposing a distribution on $\gamma$, and (iv) adopting a full Bayesian approach, which requires imposing priors on all the parameters (in my example, you’d need to have a prior for both $\gamma$ and for $\beta$).

Then, it is possible to either obtain a point estimate or confidence interval, depending on the method chosen, for $\beta$, the estimand of interest. If the point estimate is different from zero, or if the confidence interval excludes zero, then this is a sign that the 2SLS estimate is robust to a small departure from the strict exogeneity assumption–one wherein the IV is only plausibly but not strictly exogenous.

These posts are meant to be short, so I cannot possibly go into the details of Conley et al.’s four methods. If you are interested in the method, read the paper. There is also a Stata add-on called –plausexog– that can be used to implement some of the methods delineated above (as far as I have played with it, -plausexog- does not allow using a full Bayesian approach).

Again, a word of caution: This is not a cure for a bad IV, and no amount of using this method will turn a bad IV into a good one. Moreover, for all its benefits, this method can involve a certain amount of arbitrary decisions when it comes to incorporating prior information.

Another useful insight in Conley et al.’s paper is the following:

… The sensitivity of the 2SLS estimator $\beta$ to violations of the exclusion restriction depends on the strength of the instrument … The desire to use instruments that are strong but may violate the exclusion restriction provides a direct motivation for the methods in this paper.

In other words, because a weak IV biases the 2SLS estimate of $\beta$ away from the true value of $\beta$, it is sometimes preferable to use a strong IV that is only plausiby exogenous to a weak IV that is strictly exogenous. See Bound et al. (1995; click here for an ungated version.)

HT: Ag econ wunderkind Nate Hendricks, whose seminar here last fall introduced me to the method.

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