First Year Courses
Crash courses - Elective
Mathematics (50 hours)
Paolo Gibilisco
Linear Algebra: Systems of linear equations. Matrix Algebra. Algebra of square ma- trices. Transpose and its properties. Determinant. Groups, fields, vector spaces. Linear independence and basis. Dimension of vector spaces. Linear transformations. Kernels. Scalar products. Cauchy-Schwartz inequality. Eigenvalues, eigenvectors and the characteristic polynomial of a square matrix. Basic properties of eigenspaces. Symmetric, and orthogonal matrices. Positive definite matrices. Projection operators. Cholesky decomposition. Diagonalizable matrices. The spectral theorem. Calculus: Series. The complex numbers. Complex series and the complex exponential. The Euler formula. Differentiability for functions of several variables: examples and counterexamples. The gradient. The Jacobian matrix. The chain rule for differentials. Mixed partial derivative. The Schwartz (Young) theorem. Integration in n dimension. The Fubini theorem. The change of variable formula. Integration using polar coordinates. Differentiation un- der the integral sign. Introduction to differential equations. The Cauchy problem. The L^2 scalar product on R^2, on C[0, 1] and for random variables. Quasiconcave functions. Implicit functions. The contraction mapping principle. Optimization: The Taylor polynomial in n-dimensions. The Hessian matrix. Uncon- strained optimization: necessary and sufficient conditions for maxima and minima. Constrained optimization. Lagrangian function and Lagrange multiplier. Introduction to Kuhn-Tucker. The Envelope Theorem.
Statistics (36 hours)
Maura Mezzetti
Brief review of probability: Random samples and asymptotic methods. Sampling and sums of random variables. Laws of large numbers and central limit theorem. Principles of Data Reduction: Sufficiency. The Likelihood Principle: The Likelihood Function Point Estimation. Methods of Finding Estimators: Methods of Moments. Maximum Likelihood Estimators. Finite Sample Properties: Unbiasedness and Efficiency. Asymptotic Properties: Asymptotic Unbiasedness, Consistency and Efficiency. Comparison between estimators: Bayesian estimators. Fisher Information and the Cramer-Rao theorem. Confidence Intervals Hypothesis Testing: Methods of Finding Tests: Neyman Pearson lemma. Large sample tests: Likelihood Ratio Tests, Score Test, Wald Test. Methods of Evaluating Tests: the Power Function, Most Powerful Tests. The p-value. Nonparametric Inference
Mathematical Methods
Mathematics for Economics (20 hours)
Annalisa Fabretti
Topology and Metric Spaces: Definitions and examples. Open and closed set. Convergence of sequences, Cauchy sequences, completeness and compactness. Linear Spaces: Convex sets. Normed linear spaces. Separation Theorem. Functions and mappings: Continuity and hemi upper and lower continuity. Fixed point theorems.
Probability theory (20 hours)
Paolo Gibilisco
Elements of a probability space. Conditional probability, total probability formula, Bayes formula. Independent events. Random variables and their properties. Distribution function and densities function of a random variable. Geometric approach to the conditional expectation. Sequences of random variables. Convergence in probability and in law. The (weak) law of large numbers. The characteristic function. Central limit theorem. Multivariate Gaussian distribution. Conditional expectation for the bivariate gaussian.
Differential and difference equations (20 hours)
Elisabetta Tessitore
First-order differential equations, Cauchy's Theorem, definitions and examples. Methods of solving: separable variables and linear differential equations. Higher order differential equations and systems of differential equations, stability and Routh's Theorem. Difference equations and systems of difference equations, stability and Schur's Theorem.
Microeconomics
Microeconomics I (30 hours)
Eloisa Campioni
Consumer theory: Preference and choice. Budget constraint and optimal consumption. Demand and comparative statics. The utility maximization program. Walrasian demand and the weak axion of revealed preferences. The expenditure minimization program. Demand, indirect utility and expenditure. Welfare evaluation of economic changes. Production theory: Production sets. Profit maximization and cost minimization. The geometry of costs. Efficient production. Competitive equilibrium and its basic welfare properties: Markets and prices. Welfare Theorems. Pareto optima and social welfare optima. First-order conditions for Pareto optimality. An elementary proof of existence. Applications.
Microeconomics II (30 hours)
Juha Kasperi Tolvanen
Static games of complete information: main solution concepts. Existence theorem for a Nash equilibrium. Applications: models of imperfect competition. Dynamic games of complete information. Representation through extensive form and backward induction. Subgame perfection. Introduction to repeated games. Applications: bargaining models, repeated oligopoly. Games of incomplete information and Bayesian equilibria. Applications: first and second-price auctions, revenue equivalence. Dynamic games with incomplete information and perfect Bayesian equilibria. Applications: A model of screening in the labor market, reputational bargaining, political reputation and diplomatic brinkmanship.
Macroeconomics
Macroeconomics I (30 hours)
Gaetano Bloise
1. Basic principles of dynamic programming: The Principle of Optimality. The Contraction Mapping Theorem and the Bellman Equation. Concavity and differentiability of the value function. First-order conditions. The Euler Equation and the Transversality Condition. An introduction to unbounded returns and recursive preferences. Extension to uncertainty. 2. Foundations of neoclassical growth: The planning program and the Euler equation. Competitive equilibrium and Welfare Theorems. Steady-state equilibrium and transitional dynamics. Technological change, policy and comparative dynamics. 3. Asset prices and consumption: The stochastic Euler equation. Optimal saving, intertemporal substitution and wealth effect. Hall’s random walk theory of consumption. Precautionary saving and prudence. Lucas’ model of asset prices. The Modigliani-Miller Theorem. Government debt and Ricardian Equivalence.
Macroeconomics II (30 hours)
Andrey Alexandrov
1. Standard RBC model. Local solutions: linearization, linear rational expectations, introduction to Dynare, model solution, simulation, impulse responses. Empirical: obtaining macro data, extracting cyclical and trend components. Global solutions: recursive form, value function iteration, policy function iteration, off-grid methods. 2. Heterogeneous agent models. Bewley-Huggett-Aiyagari model: stationary equilibrium, transition dynamics, MIT shocks. 3. Nominal rigidities. New Keynesian model: welfare cost of inflation, optimal rate of inflation, optimal monetary stabilization policy, divine coincidence.
Finance
Quantitative finance (60 hours)
Katia Colaneri, Stefano Herzel, Paolo Pigato, Alessandro Ramponi
The purpose of this course is to provide a formal yet accessible introduction to continuous-time financial mathematics. This course begins with an overview of discrete-time models which are used to introduce the concept of absence of arbitrage, dynamic hedging and pricing. The binomial model will be used as an illustration of the main results. A second part covers basic facts about stochastic processes and introduces Brownian motion, discussing its sample path properties and quadratic variation. The third part analyses the Black-Scholes model market completeness and incompleteness and the fundamental theorems of asset pricing. Finally interest rate models and their applications to interest rate and currency derivatives are discussed. Theoretical content will be complemented by exercise classes and coding sessions.
Econometrics
The linear regression model (32 hours)
Alessandro Casini
1. Geometric Interpretation of Least-squares: vector spaces and projections. 2. Properties of the Least-squares Projection. 3. The Basic Linear Model; Finite-Sample Results; the Gauss-Markov Theorem. 4. Restricted Least-Squares. 5. Normal Distribution Theory. 6. A Brief Review of Asymptotic Results. 7. The Basic Linear Model; Asymptotic Results. 8. Instrumental Variables. 9. Review of Time Series. 10. Non-Spherical Errors. 11. Generalized Methods of Moments (GMM).
Principles of Bayesian Inference (10 hours)
Maura Mezzetti
Bayesian Inference: Exchangeability and De Finetti’s theorem; Frequentist vs Bayesian. Bayesian Inference: Prior to Posterior; Posterior Predictive Distributions; Choosing a prior distribution. Estimation, hypothesis testing and model choice: Estimation as a decision problem; Credible intervals; Hypothesis testing; Bayes factors. Exchangeability and hierarchical models: Hierarchical models; Regression and linear models; Model selection.
Panel Data (18 hours)
Federico Belotti
1. Panel Data Models. 2. Estimation of Short T Linear Static Panel Data Models. 3. Estimation of Short T Linear Static Panel Data Models: further extensions. 4. Test of Hypotheses with Panel Data . 5. Estimation of Short T Linear Dynamic Panel Data Models. 6. Estimation of Short T Non-Linear Static and Dynamic Panel Data Models.