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.
"Nonparametric panel data models with interactive fixed effects"
Revise and resubmit at the Review of Economic Studies
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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.
"On completeness and consistency in nonparametric instrumental variable models"
Revise and resubmit at Econometrica
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This paper provides a first test for the identification condition in a nonparametric instrumental
variable model, known as completeness, by linking the outcome of the test to consistency
of an estimator. In particular, I show that uniformly over all distributions for which the
test rejects with probability bounded away from 0, an estimator of the structural function
is consistent. This is the case for a large class of complete distributions as well as certain
sequences of incomplete distributions. As a byproduct of this result, the paper makes two
additional contributions. First, I present a definition of weak instruments in the nonparametric
instrumental variable model, which is equivalent to the failure of a restricted version
of completeness. Second, I show that the null hypothesis of weak instruments, and thus
failure of a restricted version of completeness, is testable and I provide a test statistic and a
bootstrap procedure to obtain the critical values. Finally, I demonstrate the finite sample
properties of the tests and the estimator in Monte Carlo simulations.
"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.