New results on the exact small sample distribution of the instrumental variable estimator are presented by studying an important special case. The exact closed forms for the probability density and cumulative distribution functions are given. There are a number of surprising findings. The small sample distribution is bimodal. with a point of zero probability mass. As the asymptotic variance grows large, the true distribution becomes concentrated around this point of zero mass. The central tendency of the estimator may be closer to the biased least squares estimator than it is to the true parameter value. The first and second moments of the IV estimator are both infinite. In the case in which least squares is biased upwards, and most of the mass of the IV estimator lies to the right of the true parameter, the mean of the IV estimator is infinitely negative. The difference between the true distribution and the normal asymptotic approximation depends on the ratio of the asymptotic variance to a parameter related to the correlation between the regressor and the regression, error. In particular, when the instrument is poorly correlated with the regressor, the asymptotic approximation to the distribution of the instrumental variable estimator will not be very accurate.

This paper reexamines the empirical evidence for mean-reverting behavior in stock prices. Comparison of data before and after World War II shows that mean reversion is entirely a prewar phenomenon. Using randomization methods to calculate significance levels, the authors find that the full sample evidence for mean reversion is weaker than previously indicated by Monte Carlo methods under a normal assumption. Further, the switch to mean-averting behavior after the war is about to be too strong to be compatible with sampling variation. The authors interpret these findings as evidence of a fundamental change in the stock returns process.

We examine a variety of models in which the variance of a portfolio's excess return depends on a state variable generated by a first-order Markov process. A model in which the state is known to economic agents is estimated. It suggests that the mean excess return moves inversely with the level of risk. We then estimate a model in which agents are uncertain of the state. The estimates indicate that agents are consistently surprised by high-variance periods, so there is a negative correlation between movements in volatility and in excess returns.