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.

A collection of time series are "related" if they follow similar stochastic processes and/or they are statistically dependent. This paper proposes a Related Time Series (RTS) forecasting model that exploits these relationships. The model's foundation is a set of univariate Gaussian autoregressions, one for each series, which are then augmented to incorporate stochastic volatility, heavy-tailed innovations, additive outliers, time-varying parameters and common factors. The model is estimated and forecasts are computed using Bayesian methods with hierarchical priors that pool information across series. Computationally efficient MCMC methods are proposed. The RTS model is applied to three datasets and yields encouraging pseudo-out-of-sample forecasting results.

Standard extreme value theory implies that the distribution of the largest observations of a large cross section is well approximated by a parametric model, governed by a location, scale and shape parameter. The extremes of a panel of independent cross sections are all governed by the same parameters as long as the underlying distribution as well as the size of the cross sections are time invariant. We derive inference about these parameters, and tests of the null hypothesis of time invariance, under asymptotics that do not require the number of extremes or the number of time periods to increase. We further apply Hamiltonian Monte Carlo techniques to estimate the path of time-varying parameters. We illustrate the approach in four examples of U.S. data: damages from weather-related disasters, financial returns, city sizes and firm sizes.

This paper proposes a model for, and investigates the consequences of, strong spatial dependence in economic variables. Our findings echo those of the corresponding "unit root" time series literature: Spatial unit root processes induce spuriously significant regression results, even with clustered standard errors or spatial HAC corrections. We develop large-sample valid unit root and stationarity tests that can detect such strong spatial dependence. Finally, we use simulations to study strategies for valid inference in regressions with persistent spatial data, such as spatial analogues of first-differencing transformations. Regressions from Chetty, Hendren, Kline, and Saez (2014) are used to illustrate the issues and methods.

This paper combines extreme value theory for the smallest and largest ν observations for some given ν ≥ 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 in the presence of moderately heavy tails. 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.

The social cost of carbon dioxide (SC-CO₂) measures the monetized value of the damages to society caused by an incremental metric tonne of CO₂ emissions and is a key metric informing climate policy. Used by governments and other decision-makers in benefit–cost analysis for over a decade, SC-CO₂ estimates draw on climate science, economics, demography and other disciplines. However, a 2017 report by the US National Academies of Sciences, Engineering, and Medicine (NASEM) highlighted that current SC-CO₂ estimates no longer reflect the latest research. The report provided a series of recommendations for improving the scientific basis, transparency and uncertainty characterization of SC-CO₂ estimates. Here we show that improved probabilistic socioeconomic projections, climate models, damage functions, and discounting methods that collectively reflect theoretically consistent valuation of risk, substantially increase estimates of the SC-CO₂. Our preferred mean SC-CO₂ estimate is $185 per tonne of CO₂ ($44–$413 per tCO₂: 5%–95% range, 2020 US dollars) at a near-term risk-free discount rate of 2%, a value 3.6 times higher than the US government's current value of $51 per tCO₂. Our estimates incorporate updated scientific understanding throughout all components of SC-CO₂ estimation in the new open-source Greenhouse Gas Impact Value Estimator (GIVE) model, in a manner fully responsive to the near-term NASEM recommendations. Our higher SC-CO₂ values, compared with estimates currently used in policy evaluation, substantially increase the estimated benefits of greenhouse gas mitigation and thereby increase the expected net benefits of more stringent climate policies.

We consider inference about a scalar coefficient in a linear regression with spatially correlated errors. Recent suggestions for more robust inference require stationarity of both regressors and dependent variables for their large sample validity. This rules out many empirically relevant applications, such as difference-in-difference designs. We develop a robustified version of the recently suggested SCPC method that addresses this challenge. We find that the method has good size properties in a wide range of Monte Carlo designs that are calibrated to real world applications, both in a pure cross sectional setting, but also for spatially correlated panel data. We provide numerically efficient methods for computing the associated spatial-correlation robust test statistics, critical values, and confidence intervals.

We propose a method for constructing confidence intervals that account for many forms of spatial correlation. The interval has the familiar "estimator plus and minus a standard error times a critical value" form, but we propose new methods for constructing the standard error and the critical value. The standard error is constructed using population principal components from a given "worst-case" spatial correlation model. The critical value is chosen to ensure coverage in a benchmark parametric model for the spatial correlations. The method is shown to control coverage in finite sample Gaussian settings in a restricted but nonparametric class of models and in large samples whenever the spatial correlation is weak, that is, with average pairwise correlations that vanish as the sample size gets large. We also provide results on the efficiency of the method.

We develop a Bayesian latent factor model of the joint long-run evolution of GDP per capita for 113 countries over the 118 years from 1900 to 2017. We find considerable heterogeneity in rates of convergence, including rates for some countries that are so slow that they might not converge (or diverge) in century-long samples, and a sparse correlation pattern ("convergence clubs") between countries. The joint Bayesian structure allows us to compute a joint predictive distribution for the output paths of these countries over the next 100 years. This predictive distribution can be used for simulations requiring projections into the deep future, such as estimating the costs of climate change. The model's pooling of information across countries results in tighter prediction intervals than are achieved using univariate information sets. Still, even using more than a century of data on many countries, the 100-year growth paths exhibit very wide uncertainty.

We introduce a generalization of the popular local-to-unity model of time series persistence by allowing for p autoregressive (AR) roots and p−1 moving average (MA) roots close to unity. This generalized local-to-unity model, GLTU(p), induces convergence of the suitably scaled time series to a continuous time Gaussian ARMA(p, p−1) process on the unit interval. Our main theoretical result establishes the richness of this model class, in the sense that it can well approximate a large class of processes with stationary Gaussian limits that are not entirely distinct from the unit root benchmark. We show that Campbell and Yogo's (2006) popular inference method for predictive regressions fails to control size in the GLTU(2) model with empirically plausible parameter values, and we propose a limited-information Bayesian framework for inference in the GLTU(p) model and apply it to quantify the uncertainty about the half-life of deviations from purchasing power parity.

We consider inference about a scalar coefficient in a linear regression model. One previously considered approach to dealing with many controls imposes sparsity, that is, it is assumed known that nearly all control coefficients are (very nearly) zero. We instead impose a bound on the quadratic mean of the controls' effect on the dependent variable, which also has an interpretation as an R²-type bound on the explanatory power of the controls. We develop a simple inference procedure that exploits this additional information in general heteroskedastic models. We study its asymptotic efficiency properties and compare it to a sparsity-based approach in a Monte Carlo study. The method is illustrated in three empirical applications.

We suggest approximating the distribution of the sum of independent and identically distributed random variables with a Pareto-like tail by combining extreme value approximations for the largest summands with a normal approximation for the sum of the smaller summands. If the tail is well approximated by a Pareto density, then this new approximation has substantially smaller error rates compared to the usual normal approximation for underlying distributions with finite variance and less than three moments. It can also provide an accurate approximation for some infinite variance distributions.

Consider a small-sample parametric estimation problem, such as the estimation of the coefficient in a Gaussian AR(1). We develop a numerical algorithm that determines an estimator that is nearly (mean or median) unbiased, and among all such estimators, comes close to minimizing a weighted average risk criterion. We also apply our generic approach to the median unbiased estimation of the degree of time variation in a Gaussian local-level model, and to a quantile unbiased point forecast for a Gaussian AR(1) process.

We develop inference methods about long-run comovement of two time series. The parameters of interest are defined in terms of population second moments of low-frequency transformations ("low-pass" filtered versions) of the data. We numerically determine confidence sets that control coverage over a wide range of potential bivariate persistence patterns, which include arbitrary linear combinations of I(0), I(1), near unit roots, and fractionally integrated processes. In an application to U.S. economic data, we quantify the long-run covariability of a variety of series, such as those giving rise to balanced growth, nominal exchange rates and relative nominal prices, the unemployment rate and inflation, money growth and inflation, earnings and stock prices, etc.

We consider inference about tail properties of a distribution from an iid sample, based on extreme value theory. All of the numerous previous suggestions rely on asymptotics where eventually, an infinite number of observations from the tail behave as predicted by extreme value theory, enabling the consistent estimation of the key tail index, and the construction of confidence intervals using the delta method or other classic approaches. In small samples, however, extreme value theory might well provide good approximations for only a relatively small number of tail observations. To accommodate this concern, we develop asymptotically valid confidence intervals for high quantile and tail conditional expectations that only require extreme value theory to hold for the largest k observations, for a given and fixed k. Small-sample simulations show that these "fixed-k" intervals have excellent small-sample coverage properties, and we illustrate their use with mainland U.S. hurricane data. In addition, we provide an analytical result about the additional asymptotic robustness of the fixed-k approach compared to k_n → ∞ inference.

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 and a notion of bet-proofness to formalize the "reasonableness" of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the violations of bet-proofness for previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are bet-proof by construction. We apply our framework to several nonstandard problems involving weak instruments, near unit roots, and moment inequalities. We find that previously suggested confidence intervals are not bet-proof, and numerically determine alternative bet-proof confidence sets.

Long-run forecasts of economic variables play an important role in policy, planning, and portfolio decisions. We consider forecasts of the long-horizon average of a scalar variable, typically the growth rate of an economic variable. The main contribution is the construction of prediction sets with asymptotic coverage over a wide range of data generating processes, allowing for stochastically trending mean growth, slow mean reversion, and other types of long-run dependencies. We illustrate the method by computing prediction sets for 10- to 75-year average growth rates of U.S. real per capita GDP and consumption, productivity, price level, stock prices, and population.

We consider the construction of set estimators that possess both Bayesian credibility and frequentist coverage properties. We show that under mild regularity conditions there exists a prior distribution that induces (1−α) frequentist coverage of a (1−α) credible set. In contrast to the previous literature, this result does not rely on asymptotic normality or invariance, so it can be applied in nonstandard inference problems.

Suppose estimating a model on each of a small number of potentially heterogeneous clusters yields approximately independent, unbiased, and Gaussian parameter estimators. We make two contributions in this setup. First, we show how to compare a scalar parameter of interest between treatment and control units using a two-sample t-statistic, extending previous results for the one-sample t-statistic. Second, we develop a test for the appropriate level of clustering; it tests the null hypothesis that clustered standard errors from a much finer partition are correct. We illustrate the approach by revisiting empirical studies involving clustered, time series, and spatially correlated data.

This paper considers nonstandard hypothesis testing problems that involve a nuisance parameter. We establish an upper bound on the weighted average power of all valid tests, and develop a numerical algorithm that determines a feasible test with power close to the bound. The approach is illustrated in six applications: inference about a linear regression coefficient when the sign of a control coefficient is known; small sample inference about the difference in means from two independent Gaussian samples from populations with potentially different variances; inference about the break date in structural break models with moderate break magnitude; predictability tests when the regressor is highly persistent; inference about an interval identified parameter; and inference about a linear regression coefficient when the necessity of a control is in doubt.

Applied work routinely relies on heteroscedasticity and autocorrelation consistent (HAC) standard errors when conducting inference in a time series setting. As is well known, however, these corrections perform poorly in small samples under pronounced autocorrelations. In this article, I first provide a review of popular methods to clarify the reasons for this failure. I then derive inference that remains valid under a specific form of strong dependence. In particular, I assume that the long-run properties can be approximated by a stationary Gaussian AR(1) model, with coefficient arbitrarily close to one. In this setting, I derive tests that come close to maximizing a weighted average power criterion. Small sample simulations show these tests to perform well, also in a regression context.

Consider inference about the pre and post break value of a scalar parameter in a time series model with a single break at an unknown date. Unless the break is large, treating the break date estimated by least squares as the true break date leads to substantially oversized tests and confidence intervals. To develop a suitable alternative, we first establish convergence to a Gaussian process limit experiment. We then determine a nearly weighted average power maximizing test in this limit experiment, and show how to implement a small sample analogue in GMM time series models.

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.

Standard inference in cointegrating models is fragile because it relies on an assumption of an I(1) model for the common stochastic trends, which may not accurately describe the data's persistence. This paper considers low-frequency tests about cointegrating vectors under a range of restrictions on the common stochastic trends. We quantify how much power can potentially be gained by exploiting correct restrictions, as well as the magnitude of size distortions if such restrictions are imposed erroneously. A simple test motivated by the analysis in Wright (2000) is developed and shown to be approximately optimal for inference about a single cointegrating vector in the unrestricted stochastic trend model.

In large Bayesian models, such as modern DSGE models, it is difficult to assess how much the prior affects the results. This paper derives measures of prior sensitivity and prior informativeness that account for the high dimensional interaction between prior and likelihood information. The basis for both measures is the derivative matrix of the posterior mean with respect to the prior mean, which is easily obtained from Markov Chain Monte Carlo output. We illustrate the approach by examining posterior results in the small model of Lubik and Schorfheide (2004) and the large model of Smets and Wouters (2007).

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.

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 general approach to robust inference about a scalar parameter of interest when the data is potentially heterogeneous and correlated in a largely unknown way. The key ingredient is the following result of Bakirov and Szekely (2005) concerning the small sample properties of the standard t-test: For a significance level of 5% or lower, the t-test remains conservative for underlying observations that are independent and Gaussian with heterogenous variances. One might thus conduct robust large sample inference as follows: partition the data into q ≥ 2 groups, estimate the model for each group, and conduct a standard t-test with the resulting q parameter estimators of interest. This results in valid and in some sense efficient inference when the groups are chosen in a way that ensures the parameter estimators to be asymptotically independent, unbiased and Gaussian of possibly different variances. We provide examples of how to apply this approach to time series, panel, clustered and spatially correlated data.

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.

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.

An I(0) process is commonly defined as a process that satisfies a functional central limit theorem, i.e., whose scaled partial sums converge weakly to a Wiener process, and an I(1) process as a process whose first differences are I(0). This paper establishes that with this definition, it is impossible to consistently discriminate between I(0) and I(1) processes. At the same time, on a more constructive note, there exist consistent unit root tests and also nontrivial inconsistent stationarity tests with correct asymptotic size.

Long-run variance estimation can typically be viewed as the problem of estimating the scale of a limiting continuous time Gaussian process on the unit interval. A natural benchmark model is given by a sample that consists of equally spaced observations of this limiting process. The paper analyzes the asymptotic robustness of long-run variance estimators to contaminations of this benchmark model. It is shown that any equivariant long-run variance estimator that is consistent in the benchmark model is highly fragile: there always exists a sequence of contaminated models with the same limiting behavior as the benchmark model for which the estimator converges in probability to an arbitrary positive value. A class of robust inconsistent long-run variance estimators is derived that optimally trades off asymptotic variance in the benchmark model against the largest asymptotic bias in a specific set of contaminated models.

This paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression. We establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude. Given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon, we proceed to develop an appropriate alternative. We suggest constructing confidence sets by inverting a sequence of tests. Each of the tests maintains a specific break date under the null hypothesis, and rejects when a break occurs elsewhere. By inverting a certain variant of a locally best invariant test, we ensure that the asymptotic critical value does not depend on the maintained break date. A valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number.

The outcome of popular unit root tests depends heavily on the initial condition, i.e. on the difference between the initial observation and the deterministic component. In some applications it is difficult to rule out small or large values of the initial condition a priori, so this dependence can be quite difficult to deal with in practice. We explore a number of methods for constructing unit root tests whose properties are less affected by the initial condition. We show that no nontrivial test can remain completely unaffected, and instead derive an asymptotically efficient unit root test whose power varies relatively little as a function of the initial condition.

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.

Tests of stationarity are routinely applied to highly autocorrelated time series. Following Kwiatkowski et al. (J. Econom. 54 (1992) 159), standard stationarity tests employ a rescaling by an estimator of the long-run variance of the (potentially) stationary series. This paper analytically investigates the size and power properties of such tests when the series are strongly autocorrelated in a local-to-unity asymptotic framework. It is shown that the behavior of the tests strongly depends on the long-run variance estimator employed, but is in general highly undesirable. Either the tests fail to control size even for strongly mean reverting series, or they are inconsistent against an integrated process and discriminate only poorly between stationary and integrated processes compared to optimal statistics.

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.