Valid Confidence Intervals and Inference in the Presence of Weak Instruments

We investigate confidence intervals and inference for the instrumental variables model with weak instruments. Wald-based confidence intervals perform poorly in that the probability they reject the null is far greater than their nominal size. In the worst case, Wald-based confidence intervals always exclude the true paremeter value. Confidence intervals based on the LM, LR, and Anderson-Rubin statistics perform far better than the Wald. The Anderson-Rubin statistic always has the correct size, but LM and LR statistics have somewhat greater power. Performance of the LM and LR statistics is improved by a degrees-of- freedom correction in the overidentified ccase. We show that the practice of "pre-testing" by looking at the significance of the first - stage regression leads to extremely poor results when the instruments are very weak.confidence intervals, instrumental variables, pre-testing, weak instruments

Mean Reversion in Stock Prices? A Reappraisal of the Empirical Evidence

Recent research based on variance ratios and multiperiod-return autocorrelations concludes that the stock market exhibits mean reversion in the sense that a return in excess of the average tends to be followed by partially offsetting returns in the opposite direction. Dividing history into pre-1926, 1926-46, and post-1946 subperiods, we find that the mean-reversion phenomenon is a feature of the 1926-46 period, but not of the post-1946 period which instead exhibits persistence of returns. Evidence for pre-1926 data is mixed. The statistical significance of test statistics is assessed by estimating their distribution using stratified randomization. Autocorrelations of multiperiod returns imply a forecast of future returns, which is presented for post-war three-year returns using 1926-46, full sample, and sequentially updated coefficient estimates. The correlation between actual and forecasted returns is negative in each case. We conclude that evidence of mean reversion in U.S. stock returns is substantially weaker than reported in the recent literature. If mean-reversion continues to be a feature of the stock market, then the experience of the past forty years has been an aberration.

Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator

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.

The Distribution of the Instrumental Variables Estimator and Its t-RatioWhen the Instrument is a Poor One

When the instrumental variable is a poor one, in the sense of being weakly correlated with the variable it proxies, the small sample distribution of the IV estimator is concentrated around a value that is inversely related to the feedback in the system and which is often further from the true value than is the plim of OLS. The sample variance of residuals similarly becomes concentrated around a value which reflects feedback and not the variance of the disturbance. The distribution of the t-ratio reflects both of these effects, stronger feedback producing larger t-ratios. Thus, in situations where OLS is badly biased, a poor instrument will lead to spurious inferences under IV estimation with high probability, and generally perform worse than OLS.

Fiscal Policy Under Imperfect Competition: A Survey

This paper surveys the link between imperfect competition and the effects of fiscal policy on output, employment and welfare. We examine static and dynamic models, with and without entry under a variety of assumptions using a common analytical framework. We find that in general there is a robust relationship between the fiscal multiplier and welfare, the tantalizing possibility of Pareto improving fiscal policy is much more elusive. In general, the mechanisms are supply side, and so welfare improving policy, whilst possible, is not a general result