Proxy and Instrumental Variable Methods in a Regression Model with One of the Regressors Missing

Suppose Y has a linear regression on X1, X2, but observations are only available on (Y, X1). If large scale data on (X1, X2) are available, which do not include Y, and if the regression of X2, given X1, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X1) ≔ E(X2|X1) for X2, or an instrument φ(X1) which is uncorrelated with X2. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared.