Essays in semiparametric estimation and inference with monotonicity constraints

Xu, Mengshan (2021) Essays in semiparametric estimation and inference with monotonicity constraints. PhD thesis, London School of Economics and Political Science.

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Abstract

Chapter 1 studies semiparametric estimation of partially linear single index models with a monotone link function. Our estimator is an extension of the scoretype estimator developed by Balabdaoui, Groeneboom, and Hendrickx (2019) for monotone single index models, which profiles out the unknown link function by isotonic regression. We show that our estimator for the finite-dimensional components is tuning-parameter-free, √n-consistent, and asymptotically normal. Furthermore, by introducing a single smoothing parameter, we propose an asymptotically efficient estimator for the finite-dimensional components. Chapter 2 proposes an empirical likelihood inference method for monotone index models. We construct the empirical likelihood function based on the modified score function of a monotone index model, where the monotone link function is estimated by isotonic regression. It is shown that the empirical likelihood ratio statistic converges to a weighted chi-squared distribution. We suggest inference procedures based on an adjusted empirical likelihood statistic that is asymptotically pivotal, and a bootstrap calibration with recentering. A Monte-Carlo simulation study illustrates the usefulness of the proposed inference methods. The models in Chapter 1 and 2 can be regarded as special cases of the framework analyzed in Chapter 3, which studies a general semiparametric estimator, where the associated moment condition contains a nuisance monotone function estimated by isotonic regression. We show that the properties of the isotonic estimator satisfy the framework of Newey (1994). As a result, the proposed estimator is √n-consistent, asymptotically normally distributed, and tuning-parameter-free. Furthermore, in a number of relevant cases, the estimator is efficient. The estimator generalizes the estimation methods of existing semiparametric models with monotone nuisance functions. We also apply the estimator to the case of inverse probability weighting, where the propensity scores are assumed to be monotone increasing. Simulations show that the proposed estimator has desired properties. Furthermore, we establish the asymptotic validity of the bootstrap, which ensures that the estimator is tuning-parameter-free in both estimation and inference.

Item Type:	Thesis (PhD)
Additional Information:	© 2021 Mengshan Xu
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/4289

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