When verifying whether a potential instrumental variable is valid, how should I check if it meets the exclusion restriction? To reject that it meets the exclusion restriction, do I simply have to come up with a excluded variable that affects the dependent variable? Or does the excluded variable have to be an omitted variable, namely that it is correlated with the independent and dependent variables of the model?
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There are two criteria for good instruments:
- The instrument $z$ is correlated with the endogenous variable $x$ (relevance).
- The instrument $z$ affects dependent variable $y$ only through $x$. In other words, $z$ itself does not cause $y$. This is the exclusion restriction.
You can check 1 (relevance) statistically, but must make good arguments to support 2 (exclusion).
For example, suppose we want to estimate the effect of police ($x$) on crime ($y$) in a cross-section of cities. One issue is that places with lots of crime will hire more police. We therefore seek an instrument $z$ that is correlated with the size of the police force, but unrelated to crime.
One possible $z$ is number of firefighters. The assumptions are that cities with lots of firefighters also have large police forces (relevance) and that firefighters do not affect crime (exclusion). Relevance can be checked with the reduced form regressions, but whether firefighters also affect crime is something to be argued for. Theoretically, they do not and are therefore a valid instrument.
If you are curious about this specific example, see
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1$\begingroup$ Great explanation. But I am curious, can't we argue that the number of firefighters is correlated with our Y: crime, because some crimes such as arson and looting often lead to fires and thus cities with more crime will also need to hire more firefighters? $\endgroup$– nesta13Dec 13, 2021 at 17:49
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2$\begingroup$ @nesta13, I'm not sure your example would invalidate the instrument because you're suggesting a relationship about arson causing cities to hire firefighters rather than a relationship about firefighters reducing crime. But, generally, the type of concern you're raising is important. In fact, you might argue that firefighters directly reduce arson and thus affect crime. In this case, a supplementary analysis that limits the data to certain crimes (such as violent crime, which is more plausibly unrelated to firefighters) would be a good "robustness" check. $\endgroup$– ChrisPDec 13, 2021 at 21:20