The goal of this paper is to develop techniques to simplify semiparametric inference. We do this by deriving a number of numerical equivalence results. These illustrate that in many cases, one can obtain estimates of semiparametric variances using standard formulas derived in the well-known parametric literature. This means that for computational purposes, an empirical researcher can ignore the semiparametric nature of the problem and do all calculations as if it were a parametric situation. We hope that this simplicity will promote the use of semiparametric procedures.

It is shown that in a nonparametric nonseparable triangular system, the conditional moment restriction (CMR) does not identify the average structural function (ASF). The CMR identifies the ASF only if the model is structurally separable in observable covariates and unobservable random errors. This excludes, for instance, random coefficient models in which the CMR in general does not identify the average response. An implication of our results is that empirical researchers should use methods other than CMR if they want to estimate the average response in models that are not additively separable.

The problem of when to control for continuous or high-dimensional discrete covariate vectors arises in both experimental and observational studies. Large-cell asymptotic arguments suggest that full control for covariates or stratification variables is always efficient, even if treatment is assigned independently of covariates or strata. Here, we approximate the behavior of different estimators using a panel-data-type asymptotic sequence with fixed cell sizes and the number of cells increasing to infinity. Exact calculations in simple examples and Monte Carlo evidence suggest this generates a substantially improved approximation to actual finite-sample distributions. Under this sequence, full control for covariates is dominated by propensity-score matching when cell sizes are small, the explanatory power of the covariates conditional on the propensity score is low, and/or the probability of treatment is close to 0 or 1. Our panel-asymptotic framework also provides an explanation for why propensity-score matching can dominate covariate matching even when there are no empty cells. Finally, we introduce a random-effects estimator that provides finite-sample efficiency gains over both covariate matching and propensity-score matching.

We provide a framework for evaluating and improving multivariate density forecasts. Among other things, the multivariate framework lets us evaluate the adequacy of density forecasts involving cross-variable interactions, such as time-varying conditional correlations. We also provide conditions under which a technique of density forecast “calibration” can be used to improve deficient density forecasts, and we show how the calibration method can be used to generate good density forecasts from econometric models, even when the conditional density is unknown. Finally, motivated by recent advances in financial risk management, we provide a detailed application to multivariate high-frequency exchange rate density forecasts.