On the Exact Small Sample Distribution of the Instrumental Variable Estimator

The purpose of this note is to point out that the bimodality is a consequence of the special case considered by N-S. The exact finite sample distribution of the IV estimator has been derived in the economic literature earlier, but its tabulation or plotting involves some assumptions about the nuisance parameters. In this note we plot the densities of the exact finite sample distribution of the IV estimator for several parameter values and show that the bimodality feature noted by N-S is a consequence of a singular covariance matrix and not of a poor instrument.