There is no prerequisite, however, to ensure that students can achieve the educational objectives of the course, it is advisable to participate in the "crash courses" at the beginning of the academic year.
The course aims to strengthen the knowledge of probability and statistics acquired in a three-year degree course and to develop topics useful to applications in the economic and financial fields. At the end of the course, the student will gain the ability to:
- understand basic probability axioms and rules and the moments of discrete and continuous random variables as well as be familiar with common named discrete and continuous random variables;
-derive the probability density function of transformations of random variables, random vector, and stochastic processes and use these techniques to generate data from various distributions;
- understand discrete-time Markov chains and methods of finding the equilibrium probability distributions, calculate probabilities of absorption and expected hitting times for discrete-time Markov chains with absorbing states;
- understand the stochastic Poisson processes knowing some application of these processes in the economics and financial fields;
- know the maximum likelihood method and how to apply it to multivariate random variables and to stochastic process to estimate the parameters involved and to validate the proposed model.
The program will include the following themes:
1) probability and conditional probability;
2) random variables and random vector, conditional distributions, correlation, and conditional expectations, the transformation of random variables, the distribution of the maximum and the minimum of random variables;
3) convergence in quadratic mean, convergence in probability, almost sure convergence and convergence in distribution, the law of large numbers, and the central limit theorem.
4) stochastic processes, Poisson processes, Markov chains;
5) the maximum likelihood method, likelihood and log-likelihood function, maximum likelihood estimates (MLE), Score function and Fisher information, statistical properties of MLE estimators, numerical calculation of MLE;
6) classical linear regression theory, classical model assumptions, statistical properties of the Ordinary Least Squares (OLS) estimator, maximum likelihood method, t-test and the F-test, confidence intervals, categorical predictors, analysis of variance, the goodness of fit, transformations, model selection methods, residuals analysis and diagnostics, multicollinearity, applications in economics and finance are used to illustrate the applications of the statistical procedures; 7) logistic regression, model specification, parameter interpretation, and parameter estimation via the MLE;
8) Inference for the Poisson processes and Markov chains, the MLE, and its properties.
The course combines traditional lectures with practical sessions during which the students will have the possibility to familiarize themselves with the techniques presented in the lectures (using R software). Both traditional lectures and practical sessions are aimed at fostering participation and class discussion.
The procedure and content of the exam will be the same for both attending and non-attending students.
The final exam consists of a written test that can be taken in any exam session. It consists of:
- 5/6 multiple-choice questions. For each question 4 possible answers are provided, of which only one is correct
- 4/5 numerical questions
- one or more theoretical questions with open answers (let's say, 150 words)
- one or more practical numerical exercises
The questions and exercises cover the entire course program.
If the course will be done remotely or blended, changes will be possibly made compared to what is stated in the syllabus to make the course and exams accessible also in these forms.