# QUANTITATIVE METHODS FOR ECONOMIC APPLICATIONS

# QUANTITATIVE METHODS FOR ECONOMIC APPLICATIONS

Mathematics

Corso di Matematica Generale

Statistics

Basic knowledge of classical statistical principles and methods (estimation and hypothesis testing) and knowledge of basic matrix algebra and calculus.

Mathematics

Basic knowledge in mathematical calculus for advanced methods used in economical and statistical theory.

Statistics

The aim of the course is to address statistical modelling and, in particular, introduce students to generalized linear models, which provide a unifying framework for many statistical techniques commonly adopted in economics and finance. At the end of the course, students will be able to model the relationship between a univariate response variable and a set of explanatory variables measured on various scales, to estimate the parameters of the model understanding their meaning and to check the adequacy of the model to data. Moreover, students will acquire the basic notion of the R software (R Core Team, 2019) to apply generalized linear models related techniques.

Mathematics

I Section (3 credits):

Complex numbers: trigonometric form, powers, roots and complex exponential. Linear algebra: vectors, matrices, vector spaces, bases, basis changes. Linear applications and linear systems: Kernel and Image. Theorems of Sylvester, Cramer, Rouchè-Capelli. Eigenvalues and eigenvectors, characteristic polynomial. Diagonalizable matrices. Symmetric matrices.

II Section (3 credits):

Functions of a vector variable: limits, continuity, differential calculus: differentiable functions, partial and directional derivatives, Taylor polynomial. Implicit functions. Free and constrained maximization, under equality and inequality constraints. Conditions of the I and II order, quadratic forms. Kuhn-Tucker theorem.

Statistics

Random variables, expected value, variance. Random vectors, mean vector and variance-covariance matrix.

Parametric statistical model. Point estimators, finite sample properties (unbiasedness, mean squared error, efficiency) and large sample properties (asymptotic unbiasedness, consistency, asymptotic efficiency).

Likelihood function, maximum likelihood estimators and their properties. Score and Fisher information. Fisher scoring algorithm.

Confidence interval estimators, pivotal quantity, interval estimators based on the asymptotic properties of maximum likelihood estimators.

Testing hypotheses: test statistic, power function, type I and II errors, uniformly most powerful tests, consistency. P.value. Generalized likelihood ratio test, score test, Wald test.

Exponential family of distribution, properties of expectation and variance.

Generalized linear models, maximum likelihood estimation of model parameters, hypothesis testing for model parameters. Deviance, testing model goodness of fit.

Normal linear model (multiple linear regression, analysis of variance, general linear model).

Logistic regression, Poisson regression.

Generalized linear models will be fitted to dataset using the R-environment (R Core Team, 2019).

Mathematics

Reading texts from the teacher available in the WEB page:

http://docenti.unisi.it/marcolonzi/pubblications/libri/

Statistics

Textbook

Dobson, J. and Barnett, A.G. (2018) An Introduction to Generalized Linear Models, fourth ed., CRC Press.

Additional reading

Faraway, J.J. (2016) Extending the Linear Model with R, second ed., CRC Press.

Mathematics

Lessons and exercises in the classroom. Talks with the students.

Statistics

Lectures and computer lab hours.

Mathematics

The verification of learning is divided into two parts.

Intermediate test: Written test consisting of five exercises on the first part of the program: from complex numbers to the whole part of Linear Algebra.

The test has five possible outcomes: Insufficient, Sufficient, Good, Distinct, Excellent.

Final exam: Written test consisting of eight exercises concerning the whole program. After passing the written test, the oral exam must be taken, having as objective the verification of the knowledge and understanding of both the theoretical foundations and the practical applications of the topics covered by the program.

Statistics

The exam consists in a written and oral tests. The written test is composed by exercises on the inferential techniques introduced in the first part of the course, aiming to evaluate the theoretical knowledge of the methodologies, and by exercises on interpreting output obtained using R, aiming to assess the capability of interpreting the results of the analysis.

When the written test is satisfactory, the student will face an oral test, based on questions on the main course topics, aimed to evaluate the ability of applying the inferential techniques, and particularly the introduced models, in economic, business and financial contexts.

Two written midterms formed by exercises are scheduled. The topics of the first are the inferential techniques introduced in the first part of the course, the second midterm deals with interpreting R outputs.

If both midterms are satisfactory, the written test is passed.

The preparatory courses of Statistics and Mathematics are highly recommended.

( timetable at https://economics.unisi.it/en/study/calendar-and-timetable)