On Identification of the Linear Projection Model from Two-Sample Data

Résumé

We investigate what can learned about the coefficients (β,δ) in the linear projection model Y=Xβ+Z′δ+ε from data consisting of two independent random samples with common variables; the first sample gives information on variables (Y,Z) but not X, while the second sample gives information on (X,Z) and not Y. Here Y is a scalar response variable, X is a scalar covariate, Z is a vector of other covariates, and ε is an error term uncorrelated with the covariates (X,Z). Complications arise because joint realizations of the variables (Y,X) are unobserved. The existing literature suggests to overcome these complications by assuming either that there exist an instrumental variable observed in both samples, or that Y and X are independent conditional on Z. Our contribution is to sharply characterize the identified set of the coefficients (β,δ) when the latter assumptions are not invoked. This set represents the limit of what can learned about the coefficients (β,δ) given the model and the available data. Its characterization is useful to evaluate the sensitivity of inferences to failure of the assumptions currently adopted. We show that the identified set is not a singleton, so the coefficients (β,γ) are set not point identified by the linear projection model. This result contrasts with the existing literature, where the assumptions have the power to point identify the coefficients of interest. We also discuss inference procedures for the identified set of the coefficients (β,γ).