Three essays on bias reduction for inference in econometrics
Creators: name: Choi, Jungjun
Publisher: name: No Publisher Supplied
Year: 2022

description: The chapters that constitute my dissertation can be briefly summarized as follows. Chapter 1 studies the inferential theory for estimating low-rank matrices when the matrices have missing components. It shows that the least square estimation of eigenvectors following the nuclear norm penalization attains the asymptotic normality without shrinkage bias. Chapter 2 introduces debiased estimators for the integrated eigenvalues and eigenvectors of the correlation matrix-based PCA for high-frequency data and proposes some statistical tests using the debiased estimators. Chapter 3 analyzes several different biases that emerge from the (possibly) low-precision nonparametric ingredient in a semiparametric model and suggests two bias robust inference procedures, based on multi-scale jackknife and analytical bias correction, respectively. Importantly, all three chapters provide bias reduction methods for inference. In social science including economics, researchers typically consider not just the estimated value but the result of the hypothesis test to verify their argument. However, sometimes, biases in estimators complicate statistical inference and invalidate statements based on simple distributional approximations. In this case, bias reduction is essential for performing valid statistical inference. For this reason, my dissertation proposes several methods that can reinstate valid inference unbiasedly in various statistics contexts. What follows is a more detailed summary of each of chapters. Chapter 1, a joint work with Hyukjun Kwon and Yuan Liao, studies the inferential theory for estimating low-rank matrices when the matrices have missing components. It also provides an inference method for the average treatment effect as an application. We show that the least square estimation of eigenvectors following the nuclear norm penalization attains the asymptotic normality without shrinkage bias. The key contribution of our method is that it does not require sample splitting, which has several disadvantages like loss of estimation quality in finite sample cases. In addition, this paper allows dependent observation patterns and heterogeneous observation probabilities. Empirically, we apply the proposed procedure to estimating the impact of the presidential vote on allocating the U.S. federal budget to the states. Chapter 2, a co-authored paper with Xiye Yang, studies the difference between the covariance matrix-based PCA and the correlation matrix-based PCA in the context of high-frequency data and shows that one will get misleading results if one uses the analytical results of the covariance matrix-based PCA while applying PCA on the correlation matrix. This research is important because it is a common practice to conduct PCA using standardized data, which is equivalent to applying PCA to the correlation matrix rather than the covariance matrix. We derive the analytical forms of the asymptotic biases and variances of the estimators for the integrated eigenvalues and eigenvectors and provide debiased estimators for the integrated eigenvalues and eigenvectors using the analytical bias correction and multi-scale jackknife method, respectively. Furthermore, we propose a novel jackknife-type estimator of the asymptotic variance of the integrated volatility functional estimator, which shows much better finite sample performances compared to other existing ones. In Chapter 3, a joint work with Xiye Yang, we analyze several different biases that emerge from the (possibly) low-precision nonparametric ingredient in a semiparametric model. We show that both the variance part and the bias part of the nonparametric ingredient can lead to some biases in the semiparametric estimator under conditions weaker than typically required in the literature. We then propose two bias-robust inference procedures, based on multi-scale jackknife and analytical bias correction, respectively. We also extend our framework to the case where the semiparametric estimator is constructed by some discontinuous functionals of the nonparametric ingredient. The simulation study shows that both bias correction methods have good finite-sample performance.