### INFORMATION AND COURSE DESCRIPTIONS

The certificate in economic theory, econometrics and mathematical economics is divided into 3 parts. In Part I, economics students take mathematics courses that help them gain exposure to rigorous college-level mathematics while mathematics students take economics courses that expose them to modern economics. These are generally elementary (undergraduate) level courses in economics and mathematics.

In Part II, more advanced (graduate level) courses are presented. All students take Part II courses, and students with a grade of **B or better** in Part I mathematics courses (or equivalent courses elsewhere) are particularly encouraged to take stochastic calculus.

In Part III, students produce a supervised dissertation in any topics within economics or finance or related areas in statistics/mathematics under the supervision of our researchers. Our researchers work in the areas of macroeconomics (theory and applied), financial economics, development, microeconomic (theory and applied), econometrics (theory and applied, including financial), asset pricing (empirical and theoretical), corporate finance, market microstructure, statistics (theory and applied), and applied mathematics related to business problems.

## PART I

### Elementary Pure and Applied Mathematics

**Real Analysis**

Methods of proof writing. Set theory. Real numbers and basic properties. Sequences: convergence, subsequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. Riemann integral. Fundamental Theorem of Calculus. Power series and Taylor series. Convergence of sequences and series of functions**.** Differentiation and Mean Value theorems. Infimum and supremum. Applications. Introduction to measure theory, measurability, Lebesgue integral. Elementary treatment of metric spaces.

**Multivariable Calculus**

** **Partial and total derivatives. Functions of several variables, multiple integration. Limits and continuity in several variables. Differentiability. Triple integrals and several differentials. Change of variables

**Linear Algebra**

** **Systems of equations, matrix algebra, and formalization of the properties of matrices; vector spaces, linear transformations; subspaces, linear independence, bases for vector spaces, dimension, rank; eigenvectors, eigenvalues, matrix diagonalization.

**Differential Equations**

** **First order differential equations, higher order linear differential equations, boundary value and initial value problems, qualitative analysis of solutions, and applications of differential equations. Differential equations, along with the methods of solutions and applications. Systems of linear first-order differential equations. First and higher order difference equations

**Functional Analysis & Optimization **

** **Normed spaces, completeness, functionals, fixed point, operators; compactness, properties of functionals. Convexity and concavity. Contraction mapping theorem. Blackwell’s sufficient conditions for a contraction. Functions and operators. Optimization: constrained and unconstrained optimization, Karush–Kuhn–Tucker conditions, applications. Optimization in several variables. Lagrange multipliers.

### Elementary Economic Theory and Econometrics

**Microeconomic Theory**

Theoretical treatment of individuals and firms’ decisions. Principles of microeconomics, interactions between individual households and business firms. Supply and demand, determination of prices. Market structure, market failure and income distribution. Resource allocation in a market economy, supply and demand interactions, light utility maximization, profit maximization, elasticity, perfect competition, monopoly power, imperfect competition, competitive markets, market failures, and government interventions to address failures and how optimal. Market structure and income distribution will also be considered.

**Macroeconomic Theory**

Principles of economics, general concept of opportunity cost, analyzing markets using the law of supply and demand, supply and demand models and policy applications, unemployment, inflation, and output. National income concepts, factors influencing national income, employment, price, and applications to current problems. Economic growth. Solow growth model, consumption, two period consumption-saving problem, fiscal policy, Ricardian equivalence, money, Neoclassical business cycle model, money, inflation, and interest rates, IS-LM, the Phillips Curve. History of financial crises, recessions, consequences, and solutions

**Econometric Theory**

** **An introduction to probability and statistical methods for empirical work in economics. Economic data sources, economic applications, regression analysis, testing, and forecasting. The two-variable regression model, multiple regression. Techniques for dealing with violations of the regression model’s assumptions, including autocorrelation, heteroscedasticity, specification error, and measurement error. Dummy variables, discrete-choice models, time series models. Introduction to simultaneous equations.

**Applied Econometrics**

** **Hands-on implementation of econometric models learned in econometric theory. Descriptive statistics, estimation and hypothesis testing, implementation of regression models. Use of statistical software packages. Basic empirical work in economics, data analysis, regression analysis, testing, and forecasting. Statistical analysis of economic data. Estimation and testing of models. Econometric analysis for empirical research, appropriate use of data, and specification. At least 1 statistical/econometric package chosen from R, Stata, EViews

## PART II

### Graduate Theory Courses

**Dynamic Programming**

Dynamic optimization methods. Formulation and solutions to Discrete time: deterministic models, Discrete time: stochastic models, Continuous time models. Comparison of discrete and continuous time models.

**Macroeconomic Theory**

** **Discrete-time models, sequential and recursive formulation, Lagrange, and Bellman techniques. Solving and analyzing Neoclassical Model, Real Business Cycle Model, Overlapping Generations Model, New Keynesian Model, log-linearization of deterministic and stochastic models. Continuous-time models. Solutions to continuous time models. Hamiltonian and Hamilton Jacobi Bellman. Introduction to models with wedges; incorporation of wedges into macroeconomics models. Simulation and estimation of DSGE models with DYNARE-MATLAB.

**Microeconomic Theory**

** **Consumer and producer behavior, decision making under uncertainty, welfare analysis, and general equilibrium theory. Game theory, the economics of information, and applications.

**Econometric Theory**

** **Asymptotic theory, estimation, and testing. Extensions of linear models, predetermined, exogenous regressors; classical GLS; instrumental variables and GMM estimators. Models for analysis of cross-sectional and panel data. Empirical applications.

**Stochastic Calculus – with Asset Pricing Applications**

** **Martingales in discrete time. Continuous time Martingales. Brownian motion. Stochastic integration, Ito calculus and functional limit theorems**.** Girsanov Theorem. Stochastic differential equations. Diffusions processes. Fractional Brownian motion. Harmonic functions.

**Mathematical Statistics & Measure-theoretic Probability**

Abstract probability spaces: sample spaces, sigma-algebras, probability measures. Random variables: distribution functions, discrete & continuous distributions. Expectations and Lebesgue integral: convergence theorems and properties. Modes of convergence. Functions of random variables, order statistics, sampling distributions, central limit theorem, quality of estimators, interval estimation, sufficient statistics, minimum-variance unbiased estimator, maximum likelihood, large-sample theory

## PART III

**Mini Dissertation in Economics and Finance**

Each student chooses a preferred area from material learned so far and completes a dissertation under the guidance of our participating faculty.