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Timothy B. Armstrong
Job Market Candidate
Stanford University
Department of Economics
579 Serra Mall
Stanford, CA 94305
541-351-8412
[email protected]
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Research papers
Asymptotically Exact Inference in Conditional Moment Inequality Models
(Job Market Paper)
This paper derives the rate of convergence and asymptotic distribution for a class of Kolmogorov-Smirnov style test statistics for conditional moment inequality models for parameters on the boundary of the identified set under general conditions. In contrast to other moment inequality settings, the rate of convergence is faster than root-n, and the asymptotic distribution depends entirely on nonbinding moments. The results require the development of new techniques that draw a connection between moment selection, irregular identification, bandwidth selection and nonstandard M-estimation. Using these results, I propose tests that are more powerful than existing approaches for choosing critical values for this test statistic. I quantify the power improvement by showing that the new tests can detect alternatives that converge to points on the identified set at a faster rate than those detected by existing approaches. A monte carlo study confirms that the tests and the asymptotic approximations they use perform well in finite samples. In an application to a regression of prescription drug expenditures on income with interval data from the Health and Retirement Study, confidence regions based on the new tests are substantially tighter than those based on existing methods.
Weighted
KS Statistics for Inference on Conditional Moment
Inequalities
This paper proposes confidence regions for the identified set in conditional moment inequality models using Kolmogorov-Smirnov statistics with a truncated inverse variance weighting with increasing truncation points. The new weighting differs from those proposed in the literature in two important ways. First, confidence regions based on KS tests with the weighting function I propose converge to the identified set at a faster rate than existing procedures based on bounded weight functions in a broad class of models. This provides a theoretical justification for inverse variance weighting in this context, and contrasts with analogous results for conditional moment equalities in which optimal weighting only affects the asymptotic variance. Second, the new weighting changes the asymptotic behavior, including the rate of convergence, of the KS statistic itself, requiring a new asymptotic theory in choosing the critical value, which I provide. To make these comparisons, I derive rates of convergence for the confidence regions I propose along with new results for rates of convergence of existing estimators under a general set of conditions. A series of examples illustrates the broad applicability of the conditions. A monte carlo study examines the finite sample behavior of the confidence regions.
Large
Market Asymptotics for Differentiated Product Demand
Estimators with Economic Models of Supply
IO economists often estimate demand for differentiated products using data sets with a small number of large markets. By modeling demand as depending on a small number of product characteristics, one might hope to obtain increasingly precise estimates of demand parameters as the number of products in a single market grows large. In this paper, I address the question of consistency and asymptotic distributions of IV estimates of demand in a small number of markets as the number of products increases in some commonly used demand models under conditions on economic primitives. I show that, under the common assumption of a Bertrand-Nash equilibrium in prices, product characteristics lose their identifying power as price instruments in the limit in many of these models, giving inconsistent estimates in these cases. I find that consistent estimates can still be obtained for many of the cases I consider, but care must be taken in modeling demand and choosing instruments. For cases where consistent estimates can be obtained, I provide sufficient conditions for consistency and asymptotic normality of estimates of parameters and counterfactual outcomes. A monte carlo study confirms that the asymptotic results provide an accurate description of the behavior of estimators in market sizes of practical importance.
Bounds
in Auctions with Unobserved Heterogeneity
(revise and resubmit at Quantitative Economics)
Many empirical studies of auctions rely on the assumption that the researcher observes all variables that make auctions differ ex ante. When there is unobserved heterogeneity, the direction of the bias this causes is known only in a few restrictive examples. In this paper, I show that ignoring unobserved heterogeneity in a first price sealed bid auction with symmetric independent private values gives bounds on several quantities of economic interest under surprisingly general conditions. These include bidder profits (which can be used to recover bid preparation costs in entry models) and the efficiency loss of assigning the object randomly. I then turn to estimation of these bounds, and show that, when only the winning bid is available, the rate of convergence can be slower than the square root of the number of auctions observed and depends on the number of bidders. These results apply more generally to estimation of functionals of a distribution from repeated observations of an order statistic and may be of independent interest. I apply these methods to bound the efficiency loss from replacing a set of procurement auctions for highway construction in Michigan with random assignment.
Work in progress
Inference for M-Estimators under Set Identification
Semiparametric Approaches to Auction Data (joint with
Dominic Coey and Paulo Somaini)
A Fast Bootstrap Method for Parametric and
Semiparametric Models (joint with Marinho
Bertanha and Han Hong)
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