It is well known that, in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudo-true value and has an asymptotically normal sampling distribution with “sandwich” covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal, and of asymptotic variance that is, in general, different than the sandwich matrix. It is shown that due to this discrepancy, Bayesian inference about the pseudo-true parameter value is, in general, of lower asymptotic frequentist risk when the original posterior is substituted by an artificial normal posterior centered at the MLE with sandwich covariance matrix. An algorithm is suggested that allows the implementation of this artificial posterior also in models with high dimensional nuisance parameters which cannot reasonably be estimated by maximizing the likelihood.

The asymptotic validity of tests is usually established by making appropriate primitive assumptions, which imply the weak convergence of a specific function of the data, and an appeal to the continuous mapping theorem.This paper, instead, takes the weak convergence of some function of the data to a limiting random element as the starting point and studies efficiency in the class of tests that remain asymptotically valid for all models that induce the same weak limit.It is found that efficient tests in this class are simply given by efficient tests in the limiting problem-that is, with the limiting random element assumed observed-evaluated at sample analogues.Efficient tests in the limiting problem are usually straightforward to derive, even in nonstandard testing problems.What is more, their evaluation at sample analogues typically yields tests that coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model.This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests, and tests about regression coefficients under weak instruments.

This paper combines extreme value theory for the smallest and largest k observations for some given k>1 with a normal approximation for the average of the remaining observations to construct a more robust alternative to the usual t-test. The new test is found to control size much more successfully in small samples compared to existing methods. This holds for the canonical inference for the mean problem based on an i.i.d. sample, but also when comparing two population means and when conducting inference about linear regression coefficients with clustered standard errors.

We propose a framework for making Bayesian parametric models robust to local misspecification. Suppose in a baseline parametric model, a parameter of interest has an interpretation in an encompassing semiparametric model. Bayesian and maximum likelihood estimators are generally biased under local misspecification. We propose to augment the baseline likelihood by a multiplicative factor that involves scores for the baseline model, the efficient scores for the encompassing semiparametric model, and an auxiliary parameter that has the same dimension as the parameter of interest. We show that the marginal posterior for the parameter of interest in the augmented model is asymptotically normal with mean equal to the semiparametrically efficient estimator and variance equal to the semiparametric efficiency bound. The suggested augmentation robustifies the baseline parametric model to local misspecification, while preserving the appeal of Bayesian inference. We develop an MCMC algorithm for the augmented model and illustrate the approach in applications.

There are a large number of tests for instability or breaks in coefficients in regression models designed for different possible departures from the stable model. We make two contributions to this literature. First, we consider a large class of persistent breaking processes that lead to asymptotically equivalent efficient tests. Our class allows for many or relatively few breaks, clustered breaks, regularly occurring breaks, or smooth transitions to changes in the regression coefficients. Thus, asymptotically nothing is gained by knowing the exact breaking process of the class. Second, we provide a test statistic that is simple to compute, avoids any need for searching over high dimensions when there are many breaks, is valid for a wide range of data-generating processes and has good power and size properties even in heteroscedastic models.

Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework to formalize the “reasonableness ” of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the degree of unreasonableness of previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are reasonable by construction. We apply our framework to inference about a parameter near a boundary and a local-to-unity autoregressive root. We find that previously suggested confidence intervals are not reasonable, and numerically determine alternative confidence sets that satisfy our criteria.

The paper analyzes the impact of the initial condition on the problem of testing for unit roots. To this end, we derive a family of optimal tests that maximize a weighted average power criterion with respect to the initial condition. We then investigate the relationship of this optimal family to popular tests. We find that many unit root tests are closely related to specific members of the optimal family, but the corresponding members employ very different weightings for the initial condition. The popular Dickey-Fuller tests, for instance, put a large weight on extreme deviations of the initial observation from the deterministic component, whereas other popular tests put more weight on moderate deviations. Since the power of unit root tests varies dramatically with the initial condition, this paper explains the results of comparative power studies of unit root tests. The results allow a much deeper understanding of the merits of particular tests in specific circumstances, and a guide to choosing which statistics to use in practice.

The paper investigates inference in non-linear and non-Gaussian models with moderately time-varying parameters. We show that for many decision problems, the sample information about the parameter path can be summarized by an artificial linear and Gaussian model, at least asymptotically. The approximation allows for computationally convenient path estimators and parameter stability tests. Also, in contrast to standard Bayesian techniques, the artificial model can be robustified so that in misspecified models, decisions about the path of the (pseudo-true) parameter remain as good as in a corresponding correctly specified model.

We develop a framework to assess how successfully standard time series models explain low-frequency variability of a data series. The low-frequency information is extracted by computing a finite number of weighted averages of the original data, where the weights are low-frequency trigonometric series. The properties of these weighted averages are then compared to the asymptotic implications of a number of common time series models. We apply the framework to twenty U.S. macroeconomic and financial time series using frequencies lower than the business cycle.

This paper considers time series Generalized Method of Moments (GMM) models where a subset of the parameters are time varying. We focus on an empirically relevant case with moderately large instabilities, which are well approximated by a local asymptotic embedding that does not allow the instability to be detected with certainty, even in the limit. We show that for many forms of the instability and a large class of GMM models, usual GMM inference on the subset of stable parameters is asymptotically unaffected by the partial instability. In the empirical analysis of presumably stable parameters—such as structural parameters in Euler conditions—one can thus ignore moderate instabilities in other parts of the model and still obtain approximately correct inference.