# Q-Exam Syllabi

## Probability Theory and Statistics for Economists

Required textbook: Statistical Inference, 2nd Edition, George Casella and Roger Berger, 2002.

### 2 Foundation of Probability Theory

• Random Experiments
• Basic Concepts of Probability
• Review of Set Theory
• Fundamental Probability Laws
• Methods of Counting
• Conditional Probability and Independence

### 3 Random Variables and Univariate Probability Distributions

• Random Variables and Distribution Functions
• Discrete Random Variable
• Continuous Random Variables
• Functions of a Random Variable
• Mathematical Expectations
• Moment Generating Function
• Characteristic Function

### 4 Important Parametric Distributions

• Introduction
• Discrete Distributions
• Continuous Probability Distributions

### 5 Random Vectors and Multivariate Probability Distribution

• Random vectors and Joint Probability Distributions
• Marginal Distributions
• Conditional Distributions
• Independence
• Empirical Applications
• Bivariate Transformation
• Expectations Under Multivariate Distributions
• Implications of Independence
• Conditional Expectations

### 6 Introduction to Sampling Theory and Statistics

• Population and Random Sample
• The Sampling Distribution of the Sample Mean
• The Sampling Distribution of the Sample Variance
• Student s t Distribution
• Snedecor s F Distribution
• Sufficient Statistics

### 7 Convergence Concepts and Limit Theories

• Limits and Orders of Magnitude: A Review
• Motivation for Convergence Concepts
• Convergence in Quadratic Mean and Lp-convergence
• Convergence in Probability
• Almost Sure Convergence
• Convergence in Distribution

### 8 Parameter Estimation and Evaluation

• Population and Distribution Model
• Maximum Likelihood Estimation
• Method of Moments and Generalized Method of Moments
• Mean Squared Error Criterion
• Best Unbiased Estimators

### 9 Hypothesis Testing

• Introduction to Hypothesis Testing
• The Wald Test
• The Lagrangian Multiplier Test
• The Likelihood Ratio Test
• A Simple Example

## Principles of Estimation and Inference

Required textbook: Econometrics. Hayashi, F., Princeton University Press (2000)

### 1 Ordinary Least Squares

• Definition and finite sample properties
• Large sample properties in rather general setting.
• Robust test statistics

### 2 Generalized Method of Moments

• Overview, linear single-equation GMM
• OLS and IV as special cases
• Linear single equation
• Hypothesis and specification tests
• 2SLS as special case
• Multiple-equation GMM.
• SUR, FIVE, and RE as special cases
• Panel data: a GMM perspective
• FE, RE, first-difference as special cases

### 3 Extremum Estimators

• Identification: A more formal approach
• Extremum estimators. Overview, consistency, asymptotic normality
• Hypothesis testing: the trinity
• Maximum likelihood: some applications

### 4 The Bootstrap

• What is the bootstrap?
• When does it work?
• When does it improve on asymptotic approximation?
• Bootstrap confidence regions and bootstrap bias correction

## Macroeconomics Q-Exam Syllabus

The qualifying exam in macroeconomics is offered twice each year, in the spring and again in late summer. The qualifying exam is not a final exam for the macroeconomics sequence, nor is it the explicit goal of the courses to teach the exam. The exam covers the topics on the following outline.

While students might expect that much of this material will be covered in Economics 6130 and 6140, not all of it will be, and the instructors may choose to devote time to topics not on the Q syllabus. Students are advised to clarify early on with their instructors what on this list will and will not be covered, so they can plan their studying accordingly.

### Endowment Economy with Complete Markets and under uncertainty

• Social planner’s problem and Pareto efficiency
• Arrow-Debreu equilibrium
• Sequential markets equilibrium
• Asset pricing

References

1. Instructors notes​
2. LS, Chapter 8.
3. SLP, Chapter 15

### Math, Dynamic Programing and Numerical Methods

• Recursive formulation and Bellman equations
• Guess and verify, value function iterations
• Transversality conditions
• Contraction Mapping Theorem
• Blackwell’s Theorem
• Theorem of the Maximum
• Principle of Optimality
• Dynamic Programming with bounded returns
• Benveniste Scheinkman

References

1. Instructors notes
2. LS, Chapters 2-5.
3. SLP, Chapter 3-5.

### Production, Investment and the Neoclassical Growth model

• Neoclassical Growth model in discrete time and under uncertainty
• Social planner’s allocation
• Arrow-Debreu and sequential markets equilibrium
• Balanced growth path

References

1. Instructors notes
2. Romer, Chapter 8
3. LS, Chapter 11.1-11.3, 11.9. 15.1-15.3, 15.5.

### Overlapping Generations Models

• Arrow-Debreu and sequential markets equilibrium
• Samuelson and classical economies. Balasko-Shell
• Ricardian Equivalence

References

1. Instructors notes
2. LS Chapter 9

### Textbooks

LS: Lars Ljunqvist and Thomas J. Sargent. Recursive Macroeconomic Theory, 3rd edition, MIT Press, 2012.
SLP: Nancy Stokey and Robert Lucas, with Edward Prescott, Recursive Methods in Economic Dynamics, Harvard University Press, 1989.
David Romer, Advanced Macroeconomics, 3rd edition, McGraw Hill, 2006.

### The classical monetary model

• The representative household
• Firms and production

References

1. Gali Ch. 2

### The basic New Keynesian model

• Monopolistic competition
• CES demand systems
• Calvo pricing and the New Keynesian Phillips Curve
• Monetary non-neutrality

References

1. Instructor’s notes
2. Gali Ch. 3

### Monetary policy

• Fluctuations, welfare, efficiency
– Gali Ch. 4

• Cost-push shocks
• Discretion vs commitment
– Gali Ch. 5

### Time Series

• Linear Difference Equations
• Lag operators

References

1. Hamilton Ch.1-2.
2. Brockwell and Davis Ch.1

### Hilbert space, projections

• Time series as objects of a Hilbert space
• The projection theorem

References

1. Brockwell and Davis Ch.2

### State space models and the Kalman filter

• State Space Models
• The Kalman Filter

References

1. Instructor’s notes

### Textbooks

Brockwell, P.J. and Davis, R.A., Time series: theory and methods, Springer Science & Business Media, 2005.
Gali, Jordi, Monetary Policy, Inflation, and the Business Cycle, Princeton University Press, 2015.

### Dynamic Optimization

• Convex analysis
• Necessary and Sufficient Conditions
• Applications Kuhn-Tucker Theorem
• Welfare Theorems
– References: M(1), HM2(1)

### One Sector Growth Model (Ramsey - Cass - Koopmans)

• Intro to the One sector Growth Model
• Extension to Infinite Horizon Economies
• Saddle path, existence and uniqueness of a stable manifold.
• Recursive representation, existence of equilibrium.
• Shooting algorithms and dynamic programming
– References: M(2, 3), HM2(1),HM1(26), BSiM (3, Appendix, Math Methods).

### Competitive Equilibrium and Heterogeneity

• Competitive equilibrium, Arrow-Debreu economies
• Recursive equilibrium
• Heterogeneity and Gorman Aggregation
– References: M(4), LS(5), HM1(11,26).

### Exogenous Long Run Growth

• Pontryagrin’s maximum principle.
• AK model.
• Saddle path, uniqueness and stability.
• Recursive representation in continuous time.
• Application: Human capital accumulation (Ben-Porath).
– References: BSiM(3), A(7,8), AH(2)

### Endogenous Long Run Growth

• Increasing Returns (Romer, 1986)
• Externalities in Human Capital (Lucas, 1988)
– References: A(10, 11), AH (3), HM2 (1).

### Innovation and Long Run Growth

• Variety innovation (Romer, 1990)
– References: A(11), AH (4, 5), K(9.4.3).

### Textbooks

Handbook of Macroeconomics Vol. 1 (1999) (HM1)
Handbook of Macroeconomics Vol. 2 (2016) (HM2)
Rody Manuelliâ˘AZ´ s notes, Notes on Discrete Time Economic Models (M)
Stokey, Lucas and Prescott (1989), Recursive Methods in Economic Dynamics (SLP)
Ljungqvist and Sargent (2004), Recursive Macroeconomics Theory (LS)
Barro and Sala-i-Martin (2004), Economic Growth (BSiM)
Aghion and Howitt (2008), The Economics of Growth (AH)
Acemoglu (2009), Introduction to Modern Macroeconomic Growth (A)
Krueger (2012), Macroeconomic Theory (K)

### Aggregation of technology and preferences

• Elementary results on aggregation of technology and preferences
• Gorman aggregation
• (In)determinacy of the wealth distribution
• The Negishi method

References: Instructor notes, GLV, LS (8)

### Consumption insurance and the permanent income hypothesis

• Full risk-sharing versus autarky
• Complete markets versus partial risk-sharing via a single risk-free bond
• Marginal utility of consumption as a random walk
• Consumption dynamics under parametric income processes
• Identification of income shocks from panel data

References: Instructor notes, GLV, LS (18)

### Precautionary savings and the income fluctuation problem

• Prudence
• Borrowing constraints
• The natural borrowing limit
• Consumption dynamics under deterministic income fluctuations
• Consumption dynamics under stochastic income fluctuations

References: Instructor notes, GLV, LS (17)

### Stationary equilibrium in an economy with idiosyncratic risk and incomplete markets

• Definition of recursive competitive equilibrium
• Existence and (lack of) uniqueness
• Computational strategies for computing stationary equilibrium
• Model calibration

References: Instructor notes, GLV, LS (2,18)

### Applications of Bewley models

• Contribution of precautionary savings to aggregate capital stock
• Contribution of precautionary savings to wealth inequality
• Entrepreneurship
• Optimal taxation
• Optimal quantity of government debt

References: Instructor notes, GLV

### Transitional dynamics in a Bewley model

• Backwards induction and computation of transition path
• Applications (i.e., welfare change from tax reform)
• Decomposition of welfare changes

References: Instructor notes, GLV, LS (10)

### Bewley models with aggregate uncertainty

• Formulation of economic environment, state space, household problem
• Approximate equilibrium and computational algorithm
• Near-aggregation in Krusell-Smith

References: Instructor notes, GLV, LS (18)

### More extensions of Bewley models

• Indivisible labor and insurance within households
• Lifecycle economies
• Endogenous borrowing constraints
• Constrained efficiency in the Bewley model

References: Instructor notes, GLV

### Textbooks

LS: Lars Ljunqvist and Thomas J. Sargent. Recursive Macroeconomic Theory, 4th edition, MIT Press, 2018.
GLV: Gianluca Violante. Lecture notes on heterogeneous agent models, mimeo, 2014.

## Microeconomics Q-Exam Syllabus

The qualifying exam in microeconomic theory is offered twice each year, in the spring and again in late summer. The 'Q' exam is not a final exam for the microeconomic theory sequence; nor is it the explicit goal of the courses to teach the exam. The exam covers the topics on the following outline.

While students might expect that much of this material will be covered in Economics 6090 and 6100, not all of it will be, and the instructors may choose to devote time to topics not on the Q syllabus. Students are advised to clarify early on with their instructors what on this list will and will not be covered, so they can plan their studying accordingly.

### 1. Consumer Theory

• Preferences and representation with utility functions
• Budget sets
• Revealed preference
• Choice based on preferences and on utility
• Demand functions–derivation (maximization) and properties of demand
• Comparative statics
• Duality–indirect utility, expenditure functions and Hicksian demand
• Integrability
• Consumer surplus
• Aggregation and properties of aggregate demand

### 2. Theory of the Firm

• Objective of the firm
• Technology
• Cost functions
• Profit maximization
• Input demand and output supply–derivation and properties
• Duality–profit functions and cost functions
• Aggregation

### 3. Decision Making under Uncertainty

• Objective uncertainty–probabilities as objects of choice
• Objective expected utility representation
• Risk aversion and measures of risk aversion
• Insurance and gambling
• Subjective uncertainty—Anscombe and Aumann structure
• Subjective expected utility representation

### 4. General Equilibrium — Analysis

• The existence problem
• Welfare analysis — Pareto optimality
• Comparative statics
• Equilibrium of plans, prices and price expectations (Radner equilibrium)

### 5. General Equilibrium — Examples

• Single agent economies
• Bilateral pure exchange
• The 2 × 2 production model
• Linear economies
• State-preference equilibrium models of uncertainty
• Intertemporal equilibrium
• Overlapping generations models

### 6. Externalities and Market Failure

• Public goods
• Information
• Non-convexities
• Market power

### 7. Non-cooperative Games

• The normal form
• Domination and iterative dominance
• Nash equilibrium
• Games of incomplete information, and normal and extensive form equilibrium concepts
• Repeated games and the Folk Theorem
• Dynamic games of complete information and Subgame Perfect Nash Equilibria