Instrumental Variables: An Econometrician’s Perspective
Econometricians had begun to develop the concept of instrumental variables, long before statisticians, for drawing causal inferences without the assumption of random assignment. They were interested in studying the effect of economic policies on human behavior. From the very beginning, they acknowledged that the level of treatment is often actively influenced by economic agents (e.g.. households). Therefore, the units in question are not only different in terms of treatment received, but the active decision to seek out that treatment. This sort of a setting refers to an endogenous treatment. This of course gives rise to selection bias, which is completely Unobservable. Instrumental variables were not given attention by the statistics community because of how economic-theory-centric they were.
Statistics literature traditionally looked at assignment as randomised: possibly because they looked at inanimate or passive objects like plots of land. The Fisher and Neyman concept of potential outcomes of randomised experiments was extended to general observational studies by scholars like Rubin, Rosennbaum, by using covariates, propensity scores etc. Unconfoundedness conditional on covariates, is analogous to selection-on-observables or exogeneity in econometric literature. In contrast, economic literature studied economic agents that are active agents that make their own decisions subject to constraints of resources and also personal judgement. That these agents will behave ‘rationally’ to maximise utility is a traditional starting point (often not very practical). As an example, if we as statisticians were interested in observing the difference in productivity of fishing and hunting, we do a simple ATE. Maybe we condition on characteristics like income. However, economists start by assuming that each individual will choose the occupation the maximises their productivity, and then get bounds for ATE. This has some faults: how can individual always know which occupation will serve them better? Also, they might find other factors (aside from productivity) that they might take into consideration like family history or something even irrational. The link between the two approaches is that individuals with similar covariates are still comparable. Instrumental variables are thus forces that influence agents to choose different treatments without changing the possible potential outcomes. This assumption is a bit controversial because it is fundamentally untestable.
Imbens explains the concept of simultaneous equations of supply and demand in the context of endogeneity with an example of a fish market with small supply and a large demand. If we wanted to estimate the effect of a new tax on fish sold, the problem that an investigator would have is that we only observe one potential outcome - nobody yet pays the tax. So we never observe Y(with tax) for any of the units. But by using observed data and general regression (or perhaps some other method), we can come up with an estimate of how much demand will fall. But this doesn’t take into account that prices are not independent of quantities traded, that is, violation of unconfoundedness. Maybe on a particular day there was less fish yield, or conversely maybe buyers bought in bulk. Therefore we would need to adjust for how both the buyers and sellers react to the change in price in order to estimate the effect of tax increase, through the help of demand and supply equations. Instrumental variables come into the picture by identifying factors that affect supply but not demand (like ocean conditions) and vice versa. A more modern example that Imbens provides is one of a randomised experiment with non compliance, interrelated with encouragement design. If we consider encouragement (yes/no) and acceptance of encouragement binary, we have four potential situations with respect to the different arrangements. Encouragement assignment obviously need to be correlated with treatment for any substantial inferences. (weak IVs arise when this correlation is low). We assume that encouragement assignment is independent of potential outcomes. (we can always assume that encouragement assignment is unconfounded conditional on some covariates X). Under both conditions, we can talk about the intention to treat, under the assumption of exclusion restriction, that encouragement assignment does not affect potential outcome of treatments (which is again, controversial), of monotonicity which essentially means that we rule out the case of defiers or those who end up doing the opposite of encouragement: which means that the probability that a patient receives a vaccine, say for example, will not lower just because their doctor received a letter of encouragement.
So we can finally understand what complier effect means: out of all the different cases, how did the average outcome change when response to encouragement was compliant with encouragement? Although we can’t identify the complier population directly, we can identify those who don’t fall into it. Hence, we can consider this a ‘second-best’ analysis setting.