We consider inference about tail properties of a distribution from an i.i.d. 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 any approach that relies on consistent estimation of the tail index parameter.Low-Frequency Econometrics. (Joint with MARK WATSON.)
Many questions in economics involve long-run or "trend" variation and covariation in time series. Yet, time series of typical lengths contain only limited information about this long-run variation. This paper suggests that long-run sample information can be isolated using a small number of low-frequency trigonometric weighted averages, which in turn can be used to conduct inference about long-run variability and covariability. An advantage of this approach is that, because the low-frequency weighted averages have large-sample normal distributions, large sample valid inference can often be conducted using familiar small sample normal inference procedures. Moreover, the general approach is applicable for a wide range of persistent stochastic processes that go beyond the familiar I(0) and I(1) models.Nearly Weighted Risk Minimal Unbiased Estimation. (Joint with YULONG WANG.)
We study non-standard parametric estimation problems, such as the estimation of the AR(1) coefficient close to the unit root. 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 demonstrate the usefulness of our generic approach by also applying it to estimation in a predictive regression, estimation of the degree of time variation, and long-range quantile point forecasts for an AR(1) process with coefficient close to unity.Credibility of Confidence Sets in Nonstandard Econometric Problems. (Joint with ANDRIY NORETS.)
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 property 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 most previously suggested confidence intervals are not bet-proof, and numerically determine alternative bet-proof confidence sets.Forecasts in a Slightly Misspecified Finite Order VAR. (Joint with JAMES STOCK.)
We propose a Bayesian procedure for exploiting small, possibly long-lag linear predictability in the innovations of a finite order autoregression. We model the innovations as having a log-spectral density that is a continuous mean-zero Gaussian process of order 1/sqrt(T). This local embedding makes the problem asymptotically a normal-normal Bayes problem, resulting in closed-form solutions for the best forecast. When applied to data on 132 U.S. monthly macroeconomic time series, the method is found to improve upon autoregressive forecasts by an amount consistent with the theoretical and Monte Carlo calculations.Forthcoming and Published Papers
Measuring Uncertainty about Long-Run Predictions. Accepted for publication at the Review of Economic Studies. (Joint with MARK WATSON.)
Coverage Inducing Priors in Nonstandard Inference Problems. Accepted for publication at the Journal of the American Statistical Association. (Joint with ANDRIY NORETS.)
Inference with Few Heterogenous Clusters. Review of Economics and Statistics 98 (2016), 83 – 96. (Joint with RUSTAM IBRAGIMOV.)
Nearly Optimal Tests when a Nuisance Parameter is Present Under the Null Hypothesis. Econometrica 83 (2015), 771 – 811. (Joint with GRAHAM ELLIOTT and MARK WATSON.)
HAC Corrections for Strongly Autocorrelated Time Series. Journal of Business & Economic Statistics 32 (2014), 311 – 322. Comments and Rejoinder.
Pre and Post Break Parameter Inference. Journal of Econometrics 180 (2014), 141 – 157. (Joint with GRAHAM ELLIOTT.) 2012 working paper version.
Risk of Bayesian Inference in Misspecified Models, and the Sandwich Covariance Matrix, Econometrica 81 (2013), 1805 – 1849.
Low-Frequency Robust Cointegration Testing, Journal of Econometrics 174 (2013), 66 – 81. (Joint with MARK WATSON.)
Measuring Prior Sensitivity and Prior Informativeness in Large Bayesian Models, Journal of Monetary Economics 59 (2012), 581 – 597.
Efficient Tests under a Weak Convergence Assumption, Econometrica 79 (2011), 395 – 435. (Formerly circulated under the title "An Alternative Sense of Asymptotic Efficiency".)
Estimation of the Parameter Path in Unstable Time Series Models, Review
Economic Studies 77 (2010), 1508 – 1539. Supplement. Correction. (Joint with PHILIPPE-EMMANUEL
t-statistic Based Correlation and Heterogeneity Robust Inference, Journal of Business & Economic Statistics 28 (2010), 453 – 468. Supplement. (Joint with RUSTAM IBRAGIMOV.)
Valid Inference in Partially Unstable GMM Models, Review of Economic Studies 76 (2009), 343 – 365. (Joint with HONG LI.)
Comment on "Unit Root Testing in Practice: Dealing with Uncertainty over the Trend and Initial Condition" by D. I. Harvey, S. J. Leybourne and A. M. R. Taylor, Econometric Theory 25 (2009), 643 – 648.
Testing Models of Low-Frequency Variability, Econometrica 76 (2008), 979 – 1016. (Joint with MARK WATSON.)
The Impossibility of Consistent Discrimination between I(0) and I(1) Processes, Econometric Theory 24 (2008), 616 – 630.
A Theory of Robust Long-Run Variance Estimation, Journal of Econometrics 141 (2007), 1331 – 1352. (Substantially different 2004 working paper).
Confidence Sets for the Date of a Single Break in Linear Time Series Regressions, Journal of Econometrics 141 (2007), 1196 – 1218. (Joint with GRAHAM ELLIOTT.)Minimizing the Impact of the Initial Condition on Testing for Unit Roots, Journal of Econometrics 135 (2006), 285 – 310. (Joint with GRAHAM ELLIOTT.)
Efficient Tests for General Persistent Time Variation in Regression Coefficients, Review of Economic Studies 73 (2006), 907 – 940. Formerly circulated under the title “Optimally Testing General Breaking Processes in Linear Time Series Models”. (Joint with GRAHAM ELLIOTT.)
Are Forecasters Reluctant to Revise their Predictions? Some German Evidence, Journal of Forecasting 25 (2006), 401 – 413. (Joint with GEBHARD KIRCHGÄSSNER.)
Size and Power of Tests for Stationarity in Highly Autocorrelated Time Series, Journal of Econometrics 128 (2005), 195 – 213.
Tests for Unit Roots and the Initial Condition, Econometrica 71 (2003), 1269 – 1286. (Joint with GRAHAM ELLIOTT.)Ecological Tax Reform and Involuntary Unemployment: Simulation Results for Switzerland, Schweizerische Zeitschrift für Volkswirtschaft und Statistik 134 (1998), 329 – 359. (Joint with GEBHARD KIRCHGÄSSNER and MARCEL SAVIOZ.)