Freyberger, J. (2015), "Asymptotic theory for differentiated products demand models with many markets",
Journal of Econometrics, 185 (1), 162-181.
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This paper develops asymptotic theory for differentiated product demand models with a large number of markets T. It takes into account that the predicted market shares are approximated
by Monte Carlo integration with R draws and that the observed market shares are approximated from a sample of N consumers. The estimated parameters are root-T consistent
and asymptotically normal as long as R and N grow fast enough relative to T. Both approximations yield additional bias and variance terms in the asymptotic expansion. I propose
a bias corrected estimator and a variance adjustment that takes the leading terms into account. Monte Carlo simulations show that these adjustments should be used in applications
to avoid severe undercoverage caused by the approximation errors.
Freyberger, J. and Horowitz, J. L. (2015), "Identification and shape restrictions in nonparametric instrumental variables estimation",
Journal of Econometrics, 189 (1), 41-53.
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This paper is concerned with inference about an unidentified linear functional, L(g), where the function g satisfies the relation Y = g(X) + U; E(U | W) = 0. In much applied research, X and W are discrete, and W has fewer points of support than X. Consequently, L(g) is not identified nonparametrically and can have any value in (-∞,∞).
This paper uses shape restrictions, such as monotonicity or convexity, to achieve interval identification of L(g). The paper shows that under shape restrictions, L(g) is contained in an interval whose bounds can be obtained by solving linear programming problems. Inference about L(g) can be carried out by using the bootstrap.
An empirical application illustrates the usefulness of the method.
"On completeness and consistency in nonparametric instrumental variable models"
Accepted for publication at Econometrica
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This paper provides positive testability results for the identification condition in a nonparametric instrumental variable model, known as completeness, and it links the outcome of the test to properties of an estimator of the structural function. In particular, I show that the data can provide empirical evidence in favor of both an arbitrarily small identified set as well as an arbitrarily small asymptotic bias of the estimator. This is the case for a large class of complete distributions as well as certain incomplete distributions. As a byproduct, the results can be used to estimate an upper bound of the diameter of the identified set and to obtain an easy to report estimator of the identified set itself.
"Nonparametric panel data models with interactive fixed effects"
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This paper studies nonparametric panel data models with multidimensional, unobserved individual effects when the number of time periods is fixed.
I focus on models where the unobservables have a factor structure and enter an unknown structural function nonadditively.
A key distinguishing feature of the setup is to allow for the various unobserved individual effects to impact outcomes differently in different time periods.
When individual effects represent unobserved ability, this means that the returns to ability may change over time. Moreover, the models allow for heterogeneous marginal effects of the covariates on the outcome.
The first set of results in the paper provides sufficient conditions for point identification when the outcomes are continuously distributed. These results lead to identification of marginal and average effects.
I provide further point identification conditions for discrete outcomes and a dynamic model with lagged dependent variables as regressors.
Using the identification conditions, I present a nonparametric sieve maximum likelihood estimator and study its large sample properties.
In addition, I analyze flexible semiparametric and parametric versions of the model and characterize the asymptotic distribution of these estimators. Monte Carlo experiments demonstrate that the estimators perform well in finite samples.
Finally, in an empirical application, I use these estimators to investigate the relationship between teaching practice and student achievement. The results differ considerably from those obtained with commonly used panel data methods.
"Compactness of infinite dimensional parameter spaces" (with Matt Masten)
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We provide general compactness results for many commonly used parameter spaces in nonparametric estimation. We consider three kinds of functions: (1) functions with bounded domains
which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded domains. In all three cases we
provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a ||·||s-norm bound are compact under a norm ||·||c-norm .
We apply these results to nonparametric mean regression and nonparametric instrumental variables estimation.
"Uniform confidence bands: characterization and optimality" (with Yoshiyasu Rai)
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This paper studies optimal uniform confidence bands for functions g(x,b0), where b0 is an unknown parameter vector.
While there are many different 1-a confidence bands for the same function, not all 1-a confidence bands are taut in the sense that
it might be possible to weakly decrease the width of the band for all x and to strictly decrease it for some x. We provide a simple characterization
of a general class of taut 1-a uniform confidence bands, allowing for both nonlinear and nonparametric functions. Specifically, we show that all taut bands can
be obtained from projections on confidence sets for b0 and we characterize the class of confidence sets which yield taut bands.
Using our simple and constructive characterization of these sets, we then present a computational method for selecting an approximately optimal confidence band
for a given objective function, such as minimizing the weighted area. We illustrate the wide applicability of these results in two numerical applications.
"Dissecting characteristics nonparametrically" (with Andreas Neuhierl and Michael Weber)
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We propose a nonparametric method to test which characteristics provide independent information for the cross section of expected returns. We use the adaptive group LASSO to select characteristics and to estimate how they affect expected returns nonparametrically. Our method can handle a large number of characteristics, allows for a flexible functional form, and is insensitive to outliers. Many of the previously identified return predictors do not provide incremental information for expected returns, and nonlinearities are important. Our proposed method has higher out-of-sample explanatory power compared to linear panel regressions, and increases Sharpe ratios by 50%.