More Results on the Exact Small Sample Properties of the Instrumental Variable Estimator

On the Bimodality of the Exact Distribution of the TSLS Estimator

“…the degree of endogeneity), a parameter which affects the presence of bimodality in the density of the TSLS estimator (e.g. Maddala and Jeong (1992) and Woglom (2001) …”

“…Bimodality is one of these unexpected properties: Wickens (1978) Solution 6.19, pp. 351-355, Nelson andStartz (1990), Maddala and Jeong (1992) and Woglom (2001) have shown that the density of the TSLS estimator may be bimodal in a just-identified structural equation.…”

“…First, we look at the claim that the exact finite sample distribution of the TSLS estimator cannot be easily interpreted (e.g. Woglom (2001) p. 1381). We show that the possible bimodality of the density of the TSLS estimator can be easily understood, and generalized to over-identified models, using the exact results reviewed by Phillips (1983).…”

The exact distribution of the TSLS estimator for a non-Gaussian just-identified linear structural equation

“…Following Nelson and Startz (1990), Maddala and Jeong (1992) and Woglom (2001), we focus on a structural equation that is just-identified and has only one endogenous variable. In this simple case, the TSLS estimator of the coefficient of the endogenous variable has the form of a ratio of two correlated random variables.…”

“…The density of β is two modes, but one of them could be very small. The potential bimodality of β has been studied by Nelson and Startz (1990), Maddala and Jeong (1992) and Woglom (2001) It is clear from equations (17) and (18) This implies that the TSLS estimator has a distribution with thick tails so that the chance of observing extreme values is relatively large. Clearly, caution needs to be exercised when studying the properties of β using Monte Carlo simulations.…”