A Monte Carlo analysis of the coverage accuracy and average length of alternative bootstrap confidence intervals for impulse-response estimators shows that the accuracy of equal-tailed and symmetric percentile-t intervals can be poor and erratic in small samples (both in models with large roots and in models without roots near the unit circle). In contrast, some percentile bootstrap intervals may be both shorter and more accurate. The accuracy of percentile-t intervals improves with sample size, but the sample size required for reliable inference can be very large. Moreover, for such large sample sizes, virtually all bootstrap intervals tend to have excellent coverage accuracy.

Bias-corrected bootstrap confidence intervals explicitly account for the bias and skewness of the small-sample distribution of the impulse response estimator, while retaining asymptotic validity in stationary autoregressions. Monte Carlo simulations for a wide range of bivariate models show that in small samples bias-corrected bootstrap intervals tend to be more accurate than delta method intervals, standard bootstrap intervals, and Monte Carlo integration intervals. This conclusion holds for VAR models estimated in levels, as deviations from a linear time trend, and in first differences. It also holds for random walk processes and cointegrated processes estimated in levels. An empirical example shows that bias-corrected bootstrap intervals may imply economic interpretations of the data that are substantively different from standard methods.

We compare the finite-sample performance of impulse response confidence intervals based on local projections (LPs) and vector autoregressive (VAR) models in linear stationary settings. We find that in small samples, the asymptotic LP interval often is less accurate than the bias-adjusted bootstrap VAR interval, notwithstanding its excessive average length. Although the asymptotic LP interval has adequate coverage in sufficiently large samples, its average length still far exceeds that of bias-adjusted bootstrap VAR intervals with comparable accuracy. Bootstrap LP intervals (with or without bias correction) and asymptotic VAR intervals are shorter on average, but they often lack coverage accuracy in finite samples.

A direct consequence of restricting the price of a good for which secondary markets do not exist is that, in the presence of excess demand, the good will not be allocated to the buyers who value it the most. We demonstrate the empirical importance of this allocative cost for the U.S. residential market for natural gas, which was subject to price ceilings during 1954–89. Using a household-level, discrete-continuous model of natural gas demand, we estimate that the allocative cost in this market averaged $3.6 billion annually, nearly tripling previous estimates of the net welfare loss to U.S. consumers.