Working papers

Researchers frequently estimate regressions when key variables are missing or observed only through intervals. Existing approaches typically resolve this incompleteness through imputation or auxiliary assumptions on the incomplete-data process. We instead ask what the maintained regression model and incomplete data alone imply. For a broad class of conditional moment models, we derive an operational characterization of the sharp identified set, develop a Hausdorff-consistent estimator with uniformly valid confidence regions, and provide tractable computational methods for common regression models. An application to a multiple-price-list design shows that incomplete-data-robust inference can remain informative, while also demonstrating that sharpness matters: non-sharp outer sets can substantially overstate the uncertainty implied by the observed data.

Set identification leaves multiple structural parameter values observationally equivalent, often generating wide identified sets and correspondingly ambiguous economic conclusions. Bayesian analysis offers a way to reduce this ambiguity by using prior information to weight these values differently, but economic reasoning may only partially identify such prior beliefs. This paper develops a robust Bayesian framework that imposes only economically meaningful restrictions on selected prior marginals while remaining agnostic about otherwise unspecified prior features. In set-identified models, the resulting posterior bounds reduce to linear optimization problems and can be approximated by reweighting simulated structural draws. Applied to oil-market SVARs, we find that demand shocks robustly dominate supply shocks, while the stronger conclusion that supply shocks are nearly irrelevant requires much tighter prior discipline on short-run elasticities.

We introduce the power-prior framework to vector autoregressions (VARs) as a flexible way to incorporate external data into a focal-country analysis. External likelihoods enter with estimated discount factors, allowing the degree of borrowing to adapt to the similarity between external and current data. We extend existing theoretical results on power priors to VARs, show their connection to simple hierarchical structures, and establish that the resulting priors preserve conjugacy and compatibility with modern structural identification methods. The power prior allows borrowing foreign information without estimating a full multi-country system. In an application to long-run Phillips curves, we show that cross-country borrowing substantially tightens posterior credible sets, while pseudo-out-of-sample exercises across two datasets and distinct evaluation periods demonstrate forecast performance competitive with established multi-country models.

Quantile nowcasts of GDP growth are central to monitoring macroeconomic tail risk, but producing them often requires computationally demanding models or bootstrap procedures. We propose conformal quantile nowcasting, a computationally inexpensive, model-agnostic approach that adds empirical quantiles of past real-time forecast errors to current point nowcasts. Obtaining multiple quantiles requires no additional model estimation or simulation, making the computational cost essentially that of point nowcasting. The procedure is a one-sided version of split conformal prediction, with finite-sample calibration under exchangeability, approximate calibration under weak dependence, and non-crossing quantiles by construction. In simulations, it is robust to many regressors and outliers, both of which distort residual bootstraps. In weekly nowcasts of the 5% quantile of US GDP growth, conformalized MIDAS performs comparably to a Bayesian mixed-frequency model with stochastic volatility, while a MIDAS bootstrap is miscalibrated. For daily stock returns, the method performs on par with a dynamic multiple quantile model.

This article introduces mcs, a Stata command for implementing the model confidence set procedure of Hansen, Lunde, and Nason (2011). The procedure starts from a user-specified set of competing models and returns a subset that contains the best model, or models, with a chosen level of confidence, where best is defined by a user-supplied loss function. Unlike procedures that select a single winning model, the model confidence set explicitly recognizes that the data may not be informative enough to separate closely performing alternatives. The command works directly with loss variables stored in a Stata dataset, supports the commonly used T_max and T_R statistics, implements bootstrap inference, reports the sequential elimination path, and stores the resulting superior set in returned results. The command is useful for forecast comparison, model selection, and any setting in which several competing procedures can be evaluated by a common loss criterion. We illustrate the syntax and output using reproducible examples and discuss practical choices concerning the confidence level, loss functions, and bootstrap settings.