Università degli Studi di Roma "Tor Vergata"

The course has four parts: Calculus, Linear Algebra, Optimization and Probability. The main goals of the course are the study of:
- integration in several variables (Fubini, change of variable formula, polar coordinates, …);
- linear transformations, eigenvalues, eigenvectors, projections and the spectral theorem;
- unconstrained and constrained optimization (Taylor formula in several variables, Kuhn-Tucker);
- limit theorems in probability and conditional expectation (weak law of large numbers, central limit theorem, multivariate gaussian).

The detailed program is available in the website of the course.
5. Learning outcomes
Upon completion of the course the student will have the mathematic background to understand the notions required in Statistics, Econometrics and in the other parts of Economics and Finance where a quantitative approach is needed.

Linear Algebra. 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, skew-symmetric and orthogonal matrices. Positive definite matrices. Diagonalization

Probability. Elements of a probability space. Algebras of events and information about random experiments. Introduction to combinatorial calculus. Finite probability spaces, probability measures, introduction to Kolmogorov theory. Conditional probability, total probability formula, Bayes formula. Independent events. Random variables and their properties. Probability distribution, distribution function and densities function of a random variable. Expectation and variance of a random variable and their properties. Expectation and variance for the main kinds of random variables. Covariance and scale-invariance of the correlation coefficient. Random vectors and their properties. Probability distribution, distribution functions and densities functions of a random vector. Independent random variables, covariance and correlation. Conditional expectation of a random variable and its properties. Conditional expectation as best estimator. 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.

Calculus. Series. Power series. The complex numbers. Complex power series and 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 under the integral sign. Introduction to differential equations. The Cauchy problem. Metric spaces. Normed spaces. Inner product spaces. The Sup norm on continuous functions. Pointwise convergence versus uniform convergence. The L^2 scalar product on R^2, on C[0,1] and for random variables. Density of polynomials in the space of continuous functions: the Bernstein proof of Weierstrass theorem by means of the weak law of large numbers.

Optimization. Taylor polynomial in n-dimensions. The Hessian matrix. Unconstrained optimization: necessary and sufficient conditions for maxima and minima. Constrained optimization. Lagrangian function and Lagrange multiplier. Introduction to Kuhn-Tucker

The course is an introduction to the fundamental principles and tools of statistical inference, i.e. how to draw conclusions from
data subject to random variation. Topics include: random sampling; principles of data reduction; point and interval estimation (likelihood theory); hypothesis testing; condense intervals and nonparametric inference.

Growth Theory
Solow Model Review. The Ramsey Cass Koopmans Model I and II. The Diamond Model. The Romer 86 model. The Romer 90 model I and II. Human Capital and Growth.
Business Cycle
Choice under uncertainty. Game theory (introduction). Adverse selection, screening, signalling. Insurance. Moral hazard, principal agent.

The first part of the course will familiarize students with the workhorses of empirical work in economics, namely the linear regression model and the ordinary least squares (OLS) estimator, and with the problems that arise when the model assumptions are violated, in particular when the regressors cannot be regarded as exogenous, that is, uncorrelated with the regression errors. The second part of the course will pay special attention to the method of instrumental variables as a way of solving the endogeneity problem and the specific issues that arise with the use of this method.
Programme
i) Static Regression. Introduction and review. The classical linear model and the OLS estimator. Sampling properties of OLS. GLS and feasible GLS. Diagnostic procedures. Hypothesis testing and model selection. ii) Instrumental Variables. The instrumental variables (IV) method. Estimation of causal effects. Properties of conventional IV estimators under weak instruments. Robust inference under weak instruments. The generalized method of moments (GMM). Weak identification and robust inference in GMM.

MatLab environment. Variables and operators. Matrices and matrix operations. Graphical functions. File management, input and output commands.

Programming elements: algorithms, loops and conditional statements (if, else, switch, for, while), preallocation and vectorization. Scripts and functions, MatLab build-in functions.
Elements of numerical calculus and simulation.

Part 2) Introduction to Stata
Instructor: Stefano Gagliarducci

Course content: dataset management, descriptive statistics, graphics, loops, linear regression

Required reading (textbook): none

Prerequisites: Statistics

Part 3) Applied Economics with Stata
Instructor: Stefano Gagliarducci

Course content: instrumental variable models (IV), panel data models, difference-in-difference models (DiD), dynamic panel models, models for discrete and limited dependent variables, regression discontinuity design (RDD)

Growth Theory
Solow Model Review. The Ramsey Cass Koopmans Model I and II. The Diamond Model. The Romer 86 model. The Romer 90 model I and II. Human Capital and Growth.
Business Cycle
Choice under uncertainty. Game theory (introduction). Adverse selection, screening, signalling. Insurance. Moral hazard, principal agent.

Introduction to Dynamic Macroeconomics
The dynamics of aggregate supply and demand. Rational expectations and the Lucas Critique. Solving rational expectations models. The central bank and monetary policy rules. Microfoundations of incomplete nominal adjustment.
Consumption and Investment
Stochastic implications of the Permanent Income Hypothesis. The overlapping generations model with money. Fixed Capital Investment. Inventory investment. Credit Rationing.

Growth Theory
Solow Model Review. The Ramsey Cass Koopmans Model I and II. The Diamond Model. The Romer 86 model. The Romer 90 model I and II. Human Capital and Growth.
Business Cycle
Introduction to dynamic programming. Business Cycle facts. Real Business Cycles theory. Money and business cycles. New Keynesian modules. Business cycles and the labor market.

Univariate Time Series
Univariate time series analysis: Basic concepts. Stationarity, autocorrelation, Linear indeterministic processes. Nonstationary time series analysis: ARIMA models. Seasonal models. Unit root tests. The Beveridge-Nelson decomposition. Forecasting and the evaluation of forecasts. Univariate analysis of financial time series: Volatility and conditional heteroscedasticity. GARCH and IGARCH models.
Multivariate Time Series
Stationary and Ergodic Multivariate Time Series. Multivariate Wold Representation. Vector Autoregression (VAR) Models. Identification and Estimation of VAR models. Forecasting. Structural VAR Models. Impulse Response Functions. Forecast Error Variance Decompositions. Shocks Identification Using the Choleski Factorization. The Cointegrated VAR. Maximum Likelihood Inference on the Cointegrated VAR. The Common Trends Representation

MatLab environment. Variables and operators. Matrices and matrix operations. Graphical functions. File management, input and output commands.

Programming elements: algorithms, loops and conditional statements (if, else, switch, for, while), preallocation and vectorization. Scripts and functions, MatLab build-in functions.
Elements of numerical calculus and simulation.

Part 1) Introduction to Matlab
Instructor: Marianna Brunetti

Course content : Introduction to Matlab. Matrices and operations with matrices. Relations and Booleans. Scripts and Functions. Controlling the flow: cycles and conditions. Generation of random numbers. Working with real data: exporting (save) and importing (load) external data. Graphics. LS estimation of univariate and multivariate linear models. Applications and examples: generation of random numbers, European options implied volatility, portfolio allocations, correlation analysis, OLS estimation.

Required reading (textbook):
Handouts will be provided during the class
Additional reading include:
- Brian R. Hunt, Ronald L. Lipsman, Ronald J. Lipsman, Jonathan Rosenberg, Kevin R Coombes (2006), A guide to MATLAB: for beginners and experienced users, Cambridge University Press
- Brandimarte Paolo (2006), Numerical Methods in Finance and Economics: A MATLAB-Based Introduction

Conoscenza del quadro istituzionale e regolamentare dei mercati finanziari
nell’ambito dell’Unione Europea e in rapporto all’ordinamento nazionale
Capacità di reperire, comprendere e interpretare le fonti della disciplina
dell’Unione Europea dei mercati finanziari e la loro influenza sulla disciplina
nazionale Capacità di identificare e distinguere problemi, obiettivi e strumenti di
regolamentazione e vigilanza europee dei mercati finanziari, con riguardo sia
all’appello al pubblico risparmio sia alla prestazione dei servizi di investimento
Capacità di identificare, distinguere e descrivere caratteristiche e funzioni
dell’offerta al pubblico di sottoscrizione, di vendita e di acquisto di prodotti
finanziari
Individuazione e comprensione degli interessi tutelati dal sistema di trasparenza e
informazione obbligatorie nell’appello al pubblico risparmio e nella quotazione di
strumenti finanziari
Individuazione e comprensione degli interessi tutelati dal sistema di riserva di
attività e di regole di condotta nella prestazione di servizi di investimento
Individuazione e comprensione delle alternative di tutela nel sistema di regole
dell’offerta pubblica di acquisto in rapporto alle diverse forme di acquisizione del
controllo di società quotate
Individuazione e comprensione del ruolo e dei limiti della regolamentazione delle
agenzie di rating

The focus of this course is to give the students a practical analysis in the management of investment
portfolios.
The course aims to provide students with an understanding of:
- asset allocation strategy;
- market timing strategy;
- performance valuation of mutual funds;
- risk management techniques in asset management.

In order to be admitted to the graduation session, students must earn 96 credit points -at least- provided for the exams listed in the program structure and must not have any pending financial obligations to the University of Rome "Tor Vergata". The dissertation is evaluated on contents, presentation and discussion. The evaluation is expressed out of 110. The minimum grade to obtain the final degree is 66/110. The Board can unanimously award honours ("cum laude") as a special distinction.

In order to reduce plagiarism we will check each dissertation with the Turitin software that reports an originality index. Students will have the opportunity to run the check informally, as in a draft form, before officially presenting the Turnitin results to their supervisor. Students are in charge of presenting directly their supervisor the results