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On binscatter
Cattaneo, Matias D.; Crump, Richard K.; Farrell, Max H.; Feng , Yingjie (2019-02-01)
Binscatter is a popular method for visualizing bivariate relationships and conducting informal specification testing. We study the properties of this method formally and develop enhanced visualization and econometric binscatter tools. These include estimating conditional means with optimal binning and quantifying uncertainty. We also highlight a methodological problem related to covariate adjustment that can yield incorrect conclusions. We revisit two applications using our methodology and find substantially different results relative to those obtained using prior informal binscatter methods. ...
Staff Reports , Paper 881

Working Paper
Bayesian Estimation and Comparison of Conditional Moment Models
Simoni, Anna; Shin, Minchul; Chib, Siddhartha (2019-12-09)
We provide a Bayesian analysis of models in which the unknown distribution of the outcomes is speci?ed up to a set of conditional moment restrictions. This analysis is based on the nonparametric exponentially tilted empirical likelihood (ETEL) function, which is constructed to satisfy a sequence of unconditional moments, obtained from the conditional moments by an increasing (in sample size) vector of approximating functions (such as tensor splines based on the splines of each conditioning variable). The posterior distribution is shown to satisfy the Bernstein-von Mises theorem, subject to a ...
Working Papers , Paper 19-51

Working Paper
Explaining Machine Learning by Bootstrapping Partial Marginal Effects and Shapley Values
Palmer, Nathan M.; Modig, Zach; Cook, Thomas R. (2024-08-06)
Machine learning and artificial intelligence are often described as “black boxes.” Traditional linear regression is interpreted through its marginal relationships as captured by regression coefficients. We show that the same marginal relationship can be described rigorously for any machine learning model by calculating the slope of the partial dependence functions, which we call the partial marginal effect (PME). We prove that the PME of OLS is analytically equivalent to the OLS regression coefficient. Boot- strapping provides standard errors and confidence intervals around the point ...
Research Working Paper , Paper RWP 21-12

Working Paper
A Uniformly Valid Test for Instrument Exogeneity
Gospodinov, Nikolay; Dovonon, Prosper (2025-09-25)
This paper studies the limiting behavior of the test for instrument exogeneity in linear models when there is uncertainty about the strength of the identification signal. We consider the test for conditional moment restrictions with an expanding set of constructed instruments. We establish the uniform validity of the standard normal asymptotic approximation, under the null, of this specification test over all possible degrees of model identification. As a result, this allows the researcher to use standard inference for testing instrument exogeneity without the need of any prior knowledge if ...
FRB Atlanta Working Paper , Paper 2025-9

Working Paper
Nonparametric Local Projections
Goncalves, Silvia; Herrera, Ana María; Kilian, Lutz; Pesavento, Elena (2024-11-20)
Nonlinearities play an increasingly important role in applied work when studying the responses of macroeconomic aggregates to policy shocks. Seemingly natural adaptations of the popular local linear projection estimator to nonlinear settings may fail to recover the population responses of interest. In this paper we study the properties of an alternative nonparametric local projection estimator of the conditional and unconditional responses of an outcome variable to an observed identified shock. We discuss alternative ways of implementing this estimator and how to allow for data-dependent ...
Working Papers , Paper 2414

Report
Beta-Sorted Portfolios
Cattaneo, Matias D.; Wang, Weining; Crump, Richard K. (2023-07-01)
Beta-sorted portfolios—portfolios comprised of assets with similar covariation to selected risk factors— are a popular tool in empirical finance to analyze models of (conditional) expected returns. Despite their widespread use, little is known of their econometric properties in contrast to comparable procedures such as two-pass regressions. We formally investigate the properties of beta-sorted portfolio returns by casting the procedure as a two-step nonparametric estimator with a nonparametric first step and a beta-adaptive portfolios construction. Our framework rationalizes the ...
Staff Reports , Paper 1068

Working Paper
Macroeconomic Indicator Forecasting with Deep Neural Networks
Cook, Thomas R.; Smalter Hall, Aaron (2017-09-29)
Economic policymaking relies upon accurate forecasts of economic conditions. Current methods for unconditional forecasting are dominated by inherently linear models {{p}} that exhibit model dependence and have high data demands. {{p}} We explore deep neural networks as an {{p}} opportunity to improve upon forecast accuracy with limited data and while remaining agnostic as to {{p}} functional form. We focus on predicting civilian unemployment using models based on four different neural network architectures. Each of these models outperforms bench- mark models at short time horizons. One model, ...
Research Working Paper , Paper RWP 17-11

Working Paper
Nonlinear Budget Set Regressions for the Random Utility Model
Blomquist, Soren; Kumar, Anil; Liang, Che-Yuan; Newey, Whitney K. (2022-09-28)
This paper is about the nonparametric regression of a choice variable on a nonlinear budget set when there is general heterogeneity, i.e., in the random utility model (RUM). We show that utility maximization makes this a three-dimensional regression with piecewise linear, convex budget sets with a more parsimonious specification than previously derived. We show that the regression allows for measurement and/or optimization errors in the outcome variable. We characterize all of the restrictions of utility maximization on the budget set regression and show how to check these restrictions. We ...
Working Papers , Paper 2219

Report
800,000 Years of Climate Risk
Adrian, Tobias; Boyarchenko, Nina; Giannone, Domenico; Prasad, Ananthakrishnan; Seneviratne, Dulani; Xiao, Yanzhe (2022-09-01)
We use a long history of global temperature and atmospheric carbon dioxide (CO2) concentration to estimate the conditional joint evolution of temperature and CO2 at a millennial frequency. We document three basic facts. First, the temperature–CO2 dynamics are non-linear, so that large deviations in either temperature or CO2 concentrations take a long time to correct–on the scale of multiple millennia. Second, the joint dynamics of temperature and CO2 concentrations exhibit multimodality around historical turning points in temperature and concentration cycles, so that prior to the start of ...
Staff Reports , Paper 1031

Working Paper
The Income-Achievement Gap and Adult Outcome Inequality
Nielsen, Eric R. (2015-05-14)
This paper discusses various methods for assessing group differences in academic achievement using only the ordinal content of achievement test scores. Researchers and policymakers frequently draw conclusions about achievement differences between various populations using methods that rely on the cardinal comparability of test scores. This paper shows that such methods can lead to erroneous conclusions in an important application: measuring changes over time in the achievement gap between youth from high- and low-income households. Commonly-employed, cardinal methods suggest that this ...
Finance and Economics Discussion Series , Paper 2015-41

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