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# Mathematical Economics and Statistics

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 1, 2 модуль

#### Преподаватель

Петросян Ованес Леонович

### Course Syllabus

#### Abstract

Course Mathematical Economics and Statistics is aimed for master students who are willing to obtain a basic knowledge of applied mathematics that is used in Economics. The course consists of Probability Theory, Statistics, Optimization and Dynamical Systems. Topics: 1. Basis of Probability Theory. 2. Statistics: estimation, confidence intervals, hypotheses testing, stochastic processes, time series. 3. Mathematical programming: problem statement, classification mathematical programming problems, linear programming, convex analysis, Kuhn-Tucker theorem. 4. Dynamical systems: difference equations, systems of difference equations, stochastic linear difference equations, basic methods for solving differential equations, dynamic optimization.

#### Learning Objectives

• Being able to perform probabilistic and statistical calculations in standard formulations, give a meaningful interpretation of the results of calculations, process empirical and experimental data.
• Being able to investigate the local behavior and stability of nonlinear dynamical systems in the vicinity of a hyperbolic stationary point.
• Have the skills of probabilistic statistical thinking, have an idea about basic concepts of nonlinear dynamics.

#### Expected Learning Outcomes

• can compute conditional and total probabilities, knows basic laws of probabilities
• can use method of moments and method of of maximum likelihood
• can compute large sample and small sample confidence interval
• can test hypothesis on defined significance level
• able to define MA, ARMA, ARIMA processes
• knows properties of Markov chains, can solve problems
• know key concepts of mathematical programming
• can use simplex algorithm to solve linear programming problem
• can solve nonlinear programming using Lagrange theorem and Kuhn-Tucker conditions
• can solve first order linear difference equations and LDE of order p
• can solve system of linear difference equations
• can solve problems of dynamic programming using Bellman equation

#### Course Contents

• Probability
Set of events. Probability function. Tools for computing sample points. Conditional probability. Total probability. Bayes rule. Discrete random variables. Discrete random distribution: binomial, Poisson, geometric. Continuous random variables: normal, uniform, gamma, exponential, chi-squared. Basic laws of probabilities. Convergences. Law of large numbers. Central limit theorem
• Estimation
Method of moments. Method of maximum likelihood. Relative efficiency. Common Unbiased Point Estimators. Goodness of a Point Estimator
• Confidence intervals
Large Sample Confidence Intervals. Small Sample Confidence Intervals (normal). Sample size. Consistency. Sufficiency. The Rao–Blackwell Theorem.
• Hypothesis testing
Elements of Statistical Test. Common Large Sample Test. Type II Error Probabilities. Attained Significance Level. Neyman-Pearson Lemma. Likelihood Ratio Tests. Student's t-test. Chi-square Test.
• Time series models
Introduction to time series models. Stationarity. MA, ARMA, ARIMA processes. Dickey-Fuller test.
• Markov chains
Introduction to Markov chains. Markov property. Examples of Markov chains. Transition probability matrix. Steady-state analysis and limiting distributions
• Introduction to mathematical programming
Linear programming: introduction. Mathematical programming. Geometric Approach.
• Linear programming
Linear programming: Simplex algorithm. Dual problem.
• Nonlinear programming
Lagrange theorem. Implicit Function Theorem. The Lagrangean. Kuhn-Tucker conditions. Sufficient Conditions. Concave programming.
• Linear difference equations
Introduction to linear difference equations. First order linear difference equations. Solution Algorithm. Steady State and Stability. Linear Difference Equations of Order p. Stability.
• System of linear difference equations
Introduction to systems of linear difference equations. First order system of linear difference equations. Stability. Two dimensional systems of linear difference equations
• Dynamic programming
Problem Statement in dynamic programming. Bellman equation. Examples of problems of dynamic programming

• Test 1
• Test 2
• Homework 1
• Homework 2
• Activity
• Exam

#### Interim Assessment

• Interim assessment (2 module)
0.12 * Activity + 0.4 * Exam + 0.12 * Homework 1 + 0.12 * Homework 2 + 0.12 * Test 1 + 0.12 * Test 2

#### Recommended Core Bibliography

• Ljungqvist, L., & Sargent, T. J. (2012). Recursive Macroeconomic Theory (Vol. 3rd ed). Cambridge, Mass: The MIT Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=550665