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Essays in semiparametric and high dimensional methods

Qiu, Chen (2020) Essays in semiparametric and high dimensional methods. PhD thesis, The London School of Economics and Political Science (LSE).

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Abstract

Chapter 1 is concerned with estimation of functionals of a latent weight function that satisfies possibly high dimensional multiplicative moment conditions. A leading example is functionals of the stochastic discount factor in asset pricing. This chapter proposes to estimate the latent weight function by an information theoretic approach combined with the `1-penalization technique to deal with high dimensional moment conditions under sparsity. This chapter derives asymptotic properties of the proposed estimator and illustrates the proposed method by a theoretical example on treatment effect analysis and empirical example on the stochastic discount factor. In Chapter 2, I introduce a semiparametric framework called the average regression functional, defined as a continuous linear function of a conditional expectation function. This framework is relevant to many empirical problems, including estimating average treatment effects, regression discontinuity design away from cut-off and measurement error with auxiliary data. I develop a new minimax methodology to estimate average regression functionals. Embedded in a penalized series space, this new strategy exploits a minimax property of a key nonparametric component of the average regression functional and aims to directly control main remainder bias. I then construct a new class of estimators, called minimax learners and show they are straightforward to implement due to their minimum distance representation. In Chapter 3, I separately study in detail asymptotic properties of minimax learners as the ratio of controls to sample size goes to zero, constant and infinity. Root-n normality is established under weak conditions for all three cases. In simulations where selection bias is mild, minimax learners behave stably, maintain small mean square error and do not over control; if selection bias is substantial, minimax learners are able to correctly reduce mean square error as more relevant controls are added. As an empirical illustration, Chapter 4 revisits the work of Ferraz and Finan (2011) that studies the effect of electoral accountability on corruption. With plausibly exogenous treatment, one of their main empirical strategies is OLS with many controls. I find estimates from OLS change considerably as more covariates are sequentially added to the regression. Minimax learners, on the other hand, perform stably and lead to economically coherent conclusions, even when the number of controls is much larger. Other popular off-the-shelf shrinkage methods do not work as well as minimax learners.

Item Type: Thesis (PhD)
Additional Information: © 2020 Chen Qiu
Library of Congress subject classification: H Social Sciences > HB Economic Theory
Q Science > QA Mathematics
Sets: Departments > Economics
Supervisor: Otsu, Taisuke
URI: http://etheses.lse.ac.uk/id/eprint/4129

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