Extending the scope of instrumental variable methods

Since Philip Wright's proposal of the linear instrumental variables (IV) estimator in 1928, the scope of application of IV models has greatly expanded. There are now nonlinear IV models in use with both additive and nonadditive latent variables, employing both semiparametric and nonparametric specifications. A notable feature of Wright's linear IV estimator is that it is applicable to incomplete models that leave the genesis of endogenous explanatory variables unspecified. However, inversion of the linear model enables the value of unobservable heterogeneity to be expressed as a function of observable variables. Only recently has the application of incomplete IV models been extended to cases in which this inversion produces a non-singleton set of values of unobservable variables. This situation arises in many models such as those featuring discrete choice, counts, or censored outcomes, or when there are multiple sources of heterogeneity as in random coefficient models, or when economic considerations result in inequality restrictions on values taken by observed and unobserved variables. This paper provides an introduction to the methods set out for analysing such generalized instrumental variable models in Chesher and Rosen (2017). An illustrative example is provided in which an outcome is determined by a random coefficients linear model with an endogenous explanatory variable and instrumental variable exclusion and independence restrictions.