# Working Papers

• Abstract: This paper develops a point-identified structural model to study the outcome of a structural change (e.g., merger, entry or exit), which is a critical task in economics. The model incorporates micro-level Nash bargaining, endogenous network formation, suppliers’ capacity constraints, and multiple continuous purchase behaviors for a market with business-to-business transactions. I provide a detailed discussion on model identification, implementation, and algorithms for estimation and counterfactual. To operationalize the model, I construct a comprehensive data set of power plants and coal companies from 1998 to 2008. Next, using the data, I estimate the model and evaluate the Arch-Triton merger in 2005, which the FTC initially challenged but failed to block. In contrast with a static network model without cost synergies, my model suggests that the equilibrium prices under a new stable network may increase in the absence of the merger due to capacity constraints. Moreover, divestiture remains a valid merger remedy since equilibrium prices would always increase otherwise.

• Abstract: We propose a new unconstrained maximization approach to approximate the value and policy functions in a single-agent dynamic programming problem. The method follows the scheme of Howard’s iteration and alternates between maximizing an unconstrained objective function over the (possibly approximate) value function given a policy function and the (possibly approximate) policy function given a value function. Our method is flexible since researchers can choose which part to approximate while solving the other part exactly, which helps utilize the characteristics of a specific problem. When polynomial approximations are adopted, the objective function is concave across various models, including the discrete choice in industrial organization and the continuous choice in Macroeconomics. We show that our algorithm is hill-climbing with a convergence guarantee under concavity. An upper bound then follows from an analogous minimization problem over the approximate value function. Simulations on both IO-type and Macro-type model demonstrate that our computational method provides significant computational gains, especially with colossal state space and without high-speed computing devices.

• Abstract: We present a Frisch-Waugh-Lovell-type (FWL) theorem for empirical likelihood estimation with instrumental variables. Our theorem is similar to the standard FWL theorem for partitioning-out exogenous regressors (including dummy variables), but the partitioning-out procedure uses the empirical likelihood weights at the solution instead of the original empirical distribution. We show how to leverage this result to simplify computation via an iterative estimation algorithm, where exogenous variables are partitioned out using weighted least squares, and these weights are updated between iterations. Furthermore, we show that iterations converge locally to the full empirical likelihood estimate at a stochastically super-linear rate. We also provide a feasible iterative constrained optimization algorithm for calculating empirical-likelihood-based confidence intervals and discuss its properties. Monte Carlo simulations on both point estimation and confidence intervals demonstrate that our iterative algorithms are robust and generate results within numerical tolerance of the full empirical likelihood estimator in the finite sample, while offering substantial savings on computation time as the number of exogenous regressors grows.

• [4] Addressing Endogeneity Issues in a SAR Model using Copulas, with Yanli Lin.

Abstract: We provide a new copula method to tackle possible endogeneity issues in a spatial autoregressive (SAR) model, which might originate from an endogenous spatial weight matrix, endogenous heterogeneity specification, or endogenous regressors. Model specifications and estimations for the three variants of a SAR model are presented. Using copula endogeneity correction technique, we propose 3-stage estimation methods and establish their consistency and asymptotic normality. Monte Carlo experiments are performed to investigate finite sample performance of the estimators. We then apply our method to an empirical study of spatial spillovers in regional productivity with endogenous spatial weights constructed by the proximity of a meaningful socioeconomic characteristic - years of education.

Online Supplement Materials

• Abstract: Based on the fast development in machine learning techniques, we propose a semi-parametric two-stage estimator to solve the endogeneity in models when implementing the usual linear-IV is impossible. Those models include binary discrete-choice with endogenous regressors and multinomial Logit model with zero market share. Our first stage is a flexible estimation of the expected outcomes, conditional on the regressors and instruments, whereas the second stage is a simple linear-IV regression with first-stage results inserted. After adding an analytical-calculated correction term, the first-stage error up to $o(N^{-1/4})$ does not affect the asymptotic distribution of the second-stage structural parameter estimates, allowing for a variety of non-parametric methods in the first stage. We demonstrate the behavior of our estimator through a pile of simulations. Moreover, we demonstrate the performance by a simple application using the publicly-available 2011 to 2015 transaction-level data between coal mining companies and coal-fired power plants in the US.

• [6] QML Estimation of a Spatial Autoregressive Model with Endogenous Heterogeneity, with Yanli Lin and Yang Yang.

Abstract: This paper develops a spatial autoregressive (SAR) model with endogenous heterogeneity. The scalar spatial coefficient is allowed to be a bounded function of a spatial unit’s own endogenous characteristics. Of particular interest is a structural interaction model with nonlinear and endogenous spillover effects. We propose a quasi-maximum likelihood estimation (QMLE) with a control function approach for this model, and investigate the asymptotic properties of the estimator which is verified to have good performance in finite sample Monte Carlo simulations. We apply our model to an empirical study of regional economic performance in about 1,400 subnational regions from 110 different countries and detect that the spillovers of regional productivity are heterogeneous and depend on an endogenous variable - years of education.

# Computational Notes

• [7] Counterfactual Calculation in Multiple Discrete Continuous Choice with Multiple Budget Constraints.

Abstract: I extended the efficient greedy algorithm provided by \textcite{PinjariBhat2011} for single constraint to an iterative one, which can accommodate more than one budget constraints in the multiple discrete continuous choice model. This can be used to accommodate the case where there is an extra binding purchase obligation from the currently-hold long-run contract. Monte Carlo simulation has demonstrated that, for a problem with extra binding minimum purchase, which seems prohibitively large to solve by traversing all the possible choice sets, the elapsed time for this iterative method is less than 0.2s.

# Work in Progress

• [8] Non-semiparametric Demand Approximation without the Exact or Lp Sparsity. (Second-year paper, draft under construction.)
• Short Description: Nonparametric demand estimation for market-level data with price endogeneity has become an important task following Berry and Haile (2014). Based on the work of Compiani (2019) and Bakhitov (2022), I propose a new estimator for high dimensional data that requires less sparsity of coefficients and performs better in the finite sample. A possible application regarding the first-order approximation of the after-merger price effect is provided. Besides, the importance of having shape restrictions in the nonparametric estimation is discussed, where the penalty tends to be a substitute for the shape restrictions if the approximation is already flexible enough.

• [9] Spatial Autoregressive Model with Copulas in Dynamic Panels, with Yanli Lin and Yang Yang.

Short Description: Panel data constitutes the main stream of research in economics. In this study, we expand the semi-parametric copulas method into a dynamic panel data setting.