Demo of pre-selection self assessment test for Data Science prospective students

This course has the target of providing the students with the modern techniques of measuring
quantitatively advanced topics in economic statistics. In particular, our focus will be on
three main interrelated directions: 1) the analysis of production and efficiency, specifically
in the private but also in the public sectors, 2) economic dynamics of sectorial systems
founded on micro data, 3) growth, ICT and technology in the modern economy.
This course uses statistical methods, both stochastic and deterministic, to analyze
topics such as productivity, efficiency and growth at micro, sectorial, and for coherence at
macro level. We first take into exam data from firms that will be useful for the mentioned
three-level study, then, as regards the efficiency analysis of productive units, such data will
be employed in order to evaluate mergers and acquisitions of plants and firms and management
of productive factors. Efficiency will be evaluated from the sides of costs, profits
and revenues. As for the sectorial analysis, static and dynamic models will be considered
to allow for forecasts and simulations in each sector for variables like production, labour,
capital, raw materials, prices and capital gains. As a consequence, an aggregate analysis
on the production, growth and prices will follow. We also deal with ICT and technical
progress in the production process considering how and if the associated externalities are
effective. We will use the following techniques for data analysis: accounting rules for the
database, panel data econometrics, (possibly) time series analysis for systems of equations,
methods for differential equation systems. Topics on private and also public sectors will
contribute to explain the relationship between economic structure and the actual crisis.
Specifically, lectures also include the examinations of cases study concerning the efficiency
and productivity analysis on the recent patterns of the banking sector in the international
context.
Two main parts characterize the course. The first part considers the sectorial productivity
analysis, the second part the efficiency and productivity analysis of firms (micro
level).

Learning goals.

General learning targets: 

The course is organised as a series of classes where the students will have the possibility to solve and discuss the solutions of a series of advanced exercises on the theory of stochastic processes.  

Knowledge and understanding.

At the end of the course the students will acquire the ability to solve autonomously simple and more advanced exercises on the theory of stochastic processes.  

Applying knowledge and understanding.

During the course the students will exercise the ability to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts. 

Making judgements At the end of the course the students will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space.  

Communication skills.

The students will exercise the intuition and the communication skills necessary to describe phenomena in the mathematical language of stochastic processes. 

In particular the student will acquire familiarity with the main ideas that are behind the stochastic model, e.g., the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour. Learning skills.

The students will acquire autonomy in studying more advanced theoretical aspects of stochastic processes and in applying the main ideas of stochastic processes to the subsequent studies in the area of statistics and finance.

Prerequisites:

- elementary probability theory,

- analysis (measure theory and linear functional analysis). 

- ordinary and partial differential equations,

- linear algebra.  

Random walks 

Exercises on: Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions.    

Brownian motion

Exercises on: path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Markov chains

Exercises on: transition matrix, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states.  

Branching processes

Exercises on: expectation and variance of the population size, probability of extinction of the population.   

   

Poisson processes 

Exercises on the main properties of Poisson processes.  

The course is organised in a series of taught classes where the students will have the possibility to solve and discuss the solutions of a series of simple and more advanced exercises on the theory of stochastic processes.   

The students are required to solve some exercises on the topics studied during the course. 

In the solution of the exercises the students must show that they have acquired the techniques to solve the problems, the intuition, the technical language, and that they are able to present the topics with all necessary details.  

These courses are meant to provide students with the elements of Probability theory and Fundamentals of R whose knowledge is required for the international MS programmes Statistical Methods and Applications and Data Science.

The courses are taught in early September (17-22) before the formal start of the academic year.

Attendance is strongly recommended since the concepts covered in the course will be assumed during the MS programmes.


This course uses statistical methods, both stochastic and deterministic, to analyze topics such as productivity, efficiency and growth at micro, sectorial, and macro level. To begin with, first it will be examined data from firms that will be useful for the mentioned three-levels study. As regards the efficiency analysis of productive units, such data will be employed in order to evaluate mergers and acquisitions of plants and firms and management of productive factors. To this aim, efficiency will be evaluated from the sides of costs, profits and revenues. As for the sectorial analysis, static and dynamic models will be considered to allow for forecasts and simulations in each sector for variables like production, labour, capital, raw materials, prices and capital gains. Coherently, an aggregate analysis on the production, growth and prices will follow. We will also deal with ICT and technical progress in the production process by considering how and if the associated externalities are effective. Specific focus on private, in particular banking, and public sectors will contribute to explain the relationship between economic structure and the actual crisis. We will use the following techniques for data analysis: panel data econometrics, time series analysis, methods for differential equation systems, non parametric methods. Lessons will refer to Italian and European data in the international context.

The course provides a broad introduction to stochastic processes. 

In particular the aim is  

- to give a rigorous introduction to the theory of stochastic processes, 

- to discuss the most important stochastic processes in some depth with examples and applications, 

- to give the flavour of more advanced work and applications,    

- to apply these ideas to answer basic questions in several applied situations including biology, finance and search engine algorithms.

At the end of the course the students 

- will be familiar with the basic concepts of the theory of stochastic processes in discrete and continuous time and will be able to apply various techniques to study stochastic models that appear in applications,

- will have the tools to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts,

- will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space, 

- will acquire the necessary language skills to read academic books on the topic and research papers,

- will acquire the rationale behind the stochastic model studied (e.g. the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour...) that is necessary to communicate to specialist and non-specialist audiences.     

- will acquire the methodology and the language to study in a manner that may be largely autonomous and to apply the methodology to the subsequent studies in the area of statistics and finance.

Program:  

Random walks

Definition, Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions, short introduction on Black–Scholes model.    

Brownian motion  

Definition and existence, Brownian motion as a limit of a simple random walk, path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Markov chains

Definition, homogeneous Markov chains, transition matrix, examples of Markov chains, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states, short introduction on Monte Carlo method, MCMC and search engine algorithms. 

Branching processes

Definition, expectation and variance of the population size, geometric branching, probability of extinction of the population.   

 

Poisson processes (Point processes)

Definition and main properties. 

Percolation Theory 

Recommended books:

- G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. 3rd edn, OUP, 2001

- P. Mörters and Y. Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, 2010 

Helpful books:

- D. Williams. Probability with Martingales. CUP, 1991  

The examination consists of three questions on three different topics studied during the course. 

The students are required to answer the questions showing that they have acquired the intuition and the technical language, and that they are able to present the topics and the proofs with all necessary details.