Beyond Standard Assumptions - Semiparametric Models, A Dyadic Item Response Theory Model, and Cluster-Endogenous Random Intercept Models

In most statistical analyses, quantitative education researchers often make simplifying assumptions regarding the manner in which their data was generated in order to answer some of these questions. These assumptions can help to reduce the complexity of the problem, and allow the researcher to describe their data using a simpler, and often times more interpretable, statistical model. However, making some of these assumptions when they are not true can lead to biased estimates and misleading answers. While the standard sets of assumptions associated with commonly-used statistical models are usually sufficient in a wide range of contexts, it will always be beneficial for education researchers to understand what they are, when they are reasonable, and how to modify them if necessary. This dissertation focuses on three of the most common models used in quantitative education research (viz. parametric models like Linear Models (LMs), Item Response Theory (IRT) models, and Random-Intercept Models (RIMs)), discusses the standard sets of assumptions that accompany these models, and then describes related models with less stringent sets of assumptions. In each of the following three chapters, we either explicitly unpack existing models that are useful but are currently still uncommon in the field of education research, or propose novel models and/or estimation strategies for these models. We begin in Chapter 1 with a common parametric model known as the Gaussian LM, and use it as a scaffold to better understand semiparametric models and their estimation. We begin by reviewing how the coefficients of the Gaussian LM are usually estimated using Maximum Likelihood (ML) or Least-Squares (LS). We then introduce the notion of an $m$-estimator as well as that of a Regular Asymptotically Linear estimator, and show how they relate to the ML estimator. In particular, we introduce the notion of influence functions/curves and discuss their geometry together with concepts such as Hilbert spaces and tangent spaces. We then demonstrate, concretely, how to derive the so-called efficient influence function under the Gaussian LM, and show that it is precisely the influence function of the ML and (Ordinary) LS estimators. This shows that the ML estimator (at least under the Gaussian LM) is efficient. Using the foundation built, we move on from the Gaussian LM by relaxing both the assumption that the residuals are normally distributed, as well as the assumption that they have a constant variance, and define this as the Heteroskedastic Linear Model. Unlike the Gaussian LM, this is a semiparametric model. Where possible, we make use of intuition and analogous results from the parametric setting to help describe the workflow for obtaining an efficient estimator for the coefficients of the Heteroskedastic Linear Model. In particular, we derive the nuisance tangent space for this semiparametric model, and use it to obtain the efficient influence function for our model. We then show how to use the efficient influence function to obtain an efficient estimator (which happens to be the Weighted LS estimator) from the (Ordinary) LS estimator via a one-step approach as well as an estimating equations approach. We then conclude by directing readers to more advanced material, including references on more modern approaches to estimating more general semiparametric models such as Targeted Maximum Likelihood Estimation. In Chapter 2, we focus on a class of measurement models known as Item Response Theory models which are useful for measuring latent traits of a subject based on the subject's response to items. We relax the condition that the responses are only a result of the individual's latent trait (and possibly an external rater), and propose a dyadic Item Response Theory (dIRT) model for measuring interactions of pairs of individuals when the responses to items represent the actions (or behaviors, perceptions, etc.) of each individual (actor) made within the context of a dyad formed with another individual (partner). Examples of its use in education include the assessment of collaborative problem solving among students, or the evaluation of intra-departmental dynamics among teachers. The dIRT model generalizes both Item Response Theory models for measurement and the Social Relations Model for dyadic data. Here, the responses of an actor when paired with a partner are modeled as a function of not only the actor's inclination to act and the partner's tendency to elicit that action, but also the unique relationship of the pair, represented by two directional, possibly correlated, interaction latent variables. We discuss generalizations such as accommodating triads or larger groups, but focus on demonstrating the key idea in the dyadic case. We show that estimation may be performed using Markov-chain Monte Carlo implemented in \texttt{Stan}, making it straightforward to extend the dIRT model in various ways. Specifically, we show how the basic dIRT model can be extended to accommodate latent regressions, random effects, distal outcomes. We perform a simulation study that demonstrates that our estimation approach performs well. In the absence of educational data of this form, we demonstrate the usefulness of our proposed approach using speed-dating data instead, and find new evidence of pairwise interactions between participants, describing a mutual attraction that is inadequately characterized by individual properties alone.Finally, in Chapter 3, we consider the often implicit assumption made when estimating the coefficients of structural Random Intercept Models (RIMs) that covariates at all levels do not co-vary with the random intercepts. A violation of this assumption (called cluster-level endogeneity) leads to inconsistent estimates when using standard estimation procedures. For two-level RIMs with such endogeneity, Hausman and Taylor (HT) devised a consistent multi-step instrumental variable estimator using only internal instruments. We, instead, approach this problem by explicitly modeling the endogeneity using a Structural Equation Model (SEM). In this chapter, we compare, through simulation, the HT and SEM estimators, and evaluate their asymptotic and finite sample properties. We show that the SEM approach is also flexible enough to deal with different exchangeability assumptions for the covariates (e.g., whether the correlations between pairs of all units in a cluster are the same) and investigate how these exchangeability assumptions affect finite sample properties of the HT estimator. For the simulations, we propose a new procedure for generating cluster- and unit-level covariates and random intercepts with a fully flexible covariance structure. We also compare our approach to another common approach known as Multilevel Matching using data from the High School and Beyond survey

Modeling Quantile Dependence: A New Look at the Money-Output Relationship

Do money supply shocks influence output growth asymmetrically? At different levels of output growth, would the influence of the same monetary policy stance vary? To address these questions, we propose a series-estimation method that models the quantile of output growth on the quantile of money supply shock, where restrictive (expansive) policies are represented by the left (right) tail of the shockÂ’s distribution. Generally, we find that each quantile of output growth responds more to restrictive than expansive money supply shocks. For M2 money supply, both restrictive and expansive shocks become even more effective when applied to output growth in its tails.monetary policy, output growth, quantile regression, quantile dependence, series estimation.

Autarkic Indeterminacy and Trade Determinacy

Most existing evidences for indeterminacy are obtained from analyzing models that do not consider trade. This paper considers an extension of Nishimura and Shimomura (Journal of Economic Theory, 2002) Heckscher-Ohlin framework by removing sector-specific externalities in one country while maintaining all other assumptions previously made by the authors. We show that even though indeterminacy arises under autarky, it can be eliminated when trade takes place with another country exhibiting saddle-path stability. Consequently, support for indeterminacy from calibrating an autarkic framework should be treated with some degree of caution.Indeterminacy, Trade, Two-Country, Heckscher-Ohlin

Indeterminacy and market instability

This note shows that indeterminacy arising from an economy exhibiting production with social constant returns to scale may be related to the instability of the consumption goods market equilibrium. Furthermore, trade does not contribute to indeterminacy indeterminacy arises becasue each country's equilibrium path is already indeterminate before trade.

Some decomposition numbers and v-decomposition numbers for q-Schur algebras

Master'sMASTER OF SCIENC

Indeterminacy and Market Instability

This note shows that indeterminacy arising from an economy exhibiting production with social constant returns to scale may be related to the instability of the consumption goods market equilibrium. Furthermore, trade does not contribute to indeterminacy; indeterminacy arises becasue each country’s equilibrium path is already indeterminate before trade.Indeterminacy, Market Instability

Autarkic Indeterminacy and Trade Determinacy

We extend the model of Nishimura and Shimomura (2002) to consider a two-country framework where under autarky indeterminacy arises in one country but determinacy in the other, and show that indeterminacy could be eliminated when trade takes place between the two.Indeterminacy, Trade, Two-Country Framework.

Modeling Quantile Dependence

Thesis advisor: Zhijie XiaoIn recent years, quantile regression has achieved increasing prominence as a quantitative method of choice in applied econometric research. The methodology focuses on how the quantile of the dependent variable is influenced by the regressors, thus providing the researcher with much information about variations in the relationship between the covariates. In this dissertation, I consider two quantile regression models where the information set may contain quantiles of the regressors. Such frameworks thus capture the dependence between quantiles - the quantile of the dependent variable and the quantile of the regressors - which I call models of quantile dependence. These models are very useful from the applied researcher's perspective as they are able to further uncover complex dependence behavior and can be easily implemented using statistical packages meant for standard quantile regressions. The first chapter considers an application of the quantile dependence model in empirical finance. One of the most important parameter of interest in risk management is the correlation coefficient between stock returns. Knowing how correlation behaves is especially important in bear markets as correlations become unstable and increase quickly so that the benefits of diversification are diminished especially when they are needed most. In this chapter, I argue that it remains a challenge to estimate variations in correlations. In the literature, either a regime-switching model is used, which can only estimate correlation in a finite number of states, or a model based on extreme-value theory is used, which can only estimate correlation between the tails of the returns series. Interpreting the quantile of the stock return as having information about the state of the financial market, this chapter proposes to model the correlation between quantiles of stock returns. For instance, the correlation between the 10th percentiles of stock returns, say the U.S. and the U.K. returns, reflects correlation when the U.S. and U.K. are in the bearish state. One can also model the correlation between the 60th percentile of one series and the 40th percentile of another, which is not possible using existing tools in the literature. For this purpose, I propose a nonlinear quantile regression where the regressor is a conditional quantile itself, so that the left-hand-side variable is a quantile of one stock return and the regressor is a quantile of the other return. The conditional quantile regressor is an unknown object, hence feasible estimation entails replacing it with the fitted counterpart, which then gives rise to problems in inference. In particular, inference in the presence of generated quantile regressors will be invalid when conventional standard errors are used. However, validity is restored when a correction term is introduced into the regression model. In the empirical section, I investigate the dependence between the quantile of U.S. MSCI returns and the quantile of MSCI returns to eight other countries including Canada and major equity markets in Europe and Asia. Using regression models based on the Gaussian and Student-t copula, I construct correlation surfaces that reflect how the correlations between quantiles of these market returns behave. Generally, the correlations tend to rise gradually when the markets are increasingly bearish, as reflected by the fact that the returns are jointly declining. In addition, correlations tend to rise when markets are increasingly bullish, although the magnitude is smaller than the increase associated with bear markets. The second chapter considers an application of the quantile dependence model in empirical macroeconomics examining the money-output relationship. One area in this line of research focuses on the asymmetric effects of monetary policy on output growth. In particular, letting the negative residuals estimated from a money equation represent contractionary monetary policy shocks and the positive residuals represent expansionary shocks, it has been widely established that output growth declines more following a contractionary shock than it increases following an expansionary shock of the same magnitude. However, correctly identifying episodes of contraction and expansion in this manner presupposes that the true monetary innovation has a zero population mean, which is not verifiable. Therefore, I propose interpreting the quantiles of the monetary shocks as having information about the monetary policy stance. For instance, the 10th percentile shock reflects a restrictive stance relative to the 90th percentile shock, and the ranking of shocks is preserved regardless of shifts in the shock's distribution. This idea motivates modeling output growth as a function of the quantiles of monetary shocks. In addition, I consider modeling the quantile of output growth, which will enable policymakers to ascertain whether certain monetary policy objectives, as indexed by quantiles of monetary shocks, will be more effective in particular economic states, as indexed by quantiles of output growth. Therefore, this calls for a unified framework that models the relationship between the quantile of output growth and the quantile of monetary shocks. This framework employs a power series method to estimate quantile dependence. Monte Carlo experiments demonstrate that regressions based on cubic or quartic expansions are able to estimate the quantile dependence relationships well with reasonable bias properties and root-mean-squared errors. Hence, using the cubic and quartic regression models with M1 or M2 money supply growth as monetary instruments, I show that the right tail of the output growth distribution is generally more sensitive to M1 money supply shocks, while both tails of output growth distribution are more sensitive than the center is to M2 money supply shocks, implying that monetary policy is more effective in periods of very low and very high growth rates. In addition, when non-neutral, the influence of monetary policy on output growth is stronger when it is restrictive than expansive, which is consistent with previous findings on the asymmetric effects of monetary policy on output.Thesis (PhD) — Boston College, 2009.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics