Oracle Inequality for Instrumental Variable Regression

where φ is the parameter of interest which models the relationship while U is an error term. Contrary to usual statistical regression models, the error term is correlated with the explanatory variables X, hence E(U |X) 6= 0, preventing direct estimation of φ. To overcome the endogeneity of X, we assume that there exists an observed random variable W , called the instrument, which decorrelates the effects of the two variables X and Y in the sense that E(U |W ) = 0. It is often the case in economics, where the practical construction of instrumental variables play an important part. For instance [CIN07] present practical situations where prices of goods and quantity in goods can be explained using an instrument. This situation is also encountered when dealing with simultaneous equations, error-in-variable models, treatment model with endogenous effects. It defines the so-called instrumental variable regression model which has received a growing interest among the last decade and turned to be a challenging issue in statistics. In particular, we refer to [HN91], [NP03] [Flo03] for general references on the use of instrumental variables in economics while [HH05], [DFR03] and [FJvB07] deal with the statistical estimation problem.