## A Bouquet of Essays

- Author(s): Guggisberg, Michael Ryan
- Advisor(s): Poirier, Dale
- et al.

## Abstract

This bouquet of essays contains four chapters.

In the first chapter I present a brief summary of the literature of misspecified models. I discuss what various estimators are actually estimating when the model is misspecified. Further I discuss corrections to standard errors and when they are useful. I briefly cover hypothesis testing in the presence of misspecified models. I cover both frequentist and Bayesian approached. I show that a misspecified model can indeed be useful and discuss some misconceptions with misspecified models.

The second chapter investigates the impact of misspecification in discrete choice models. I derive necessary and sufficient conditions for consistency of the maximum likelihood estimator from the misspecified model. A corollary is that the misspecified estimator is consistent for the correct sign, under certain conditions. It also follows that Huber-White standard errors can be used to obtain asymptotically conservative type one errors when testing the nullity of the coefficient.

The third chapter builds a Bayesian model for multiple-output quantiles using a commonly accepted definition for the quantile. The prior can be elicited as the ex-ante knowledge of Tukey depth, the first prior of its kind. I apply the model to the Tennessee Project STAR experiment and find there is a joint increase in \emph{all quantile subpopulations} for reading and mathematics scores given a decrease in the number of students per teacher. This result is consistent with, and much stronger than, the results from previous studies.

The four chapter I investigate if United States Supreme Court Justices recuse themselves strategically. I create a new structural model of recusals. Using this model I find causal evidence that justices recuse themselves strategically. I then calibrate and simulate the model to find the frequency of cases where at least one justice has a conflict of interest but does not recuse. I find at most 47\% of cases have at least one justice with a conflict of interest that did not recuse.

It was difficult to come up with an overarching theme for this bouquet of essays -- hence the title. The closest theme would be `model misspecification.' The first chapter provides an overview of misspecified models, the second chapter investigates a misspecified discrete choice models and the third chapter purposefully uses a misspecified model to get an interesting estimator. However, the closest the fourth chapter gets to `model misspecification' is the use of the Kullback-Leibler distance in a simulation. The Kullback-Leibler distance is often used in the misspecified literature but there is nothing about the distance that makes it inherently related to misspecified models.