Learning objectives

The module aims to introduce students to time series analysis, in order to understand how economic phenomena evolve over time. It will present the main econometric tools used to make forecasts and assess their accuracy on economic and financial time series. The use of statistical and econometric professional packages will complement the study of theoretical concepts. At the end of the module, students will prove to be able to critically interpret dynamic models for the analysis and forecast of economic and financial variables, in response to real problems.

Prerequisites and basic notions

Knowledge of the following elementary topics is required:
- Calculus: derivatives, integrals, series
- Linear algebra: matrices, rank, systems of equations
- Descriptive and inferential statistics

Program

1. Introductory topics
Review of univariate and multivariate statistics
Joint, marginal and conditional density
Correlation versus Dependence
The multivariate Normal
Distributional properties of time-series
Non-normality tests
Serial correlation, Ljung-Box and Box-Pierce test statistics
Markov property
2. Stationary linear time-series models I
Weak and strong stationarity
White noise, random walk, random walk with trend
The autocovariance of a weakly stationary process
AR(1) model: conditions for stationarity, autocovariance and autocorrelation.
AR(2) model: vector representation, conditions for stationarity, autocovariance and autocorrelation.
3. Stationary linear time-series models II
The AR(p) model: vector representation, conditions for stationarity, autocovariance and autocorrelation
The Yule-Walker equations
MA(q) model: stationarity, autocovariance and autocorrelation
Invertibility of MA(1) and identification issues
ARMA(p,q) model: stationarity, autocovariance and autocorrelation
The Wold decomposition theorem
Short versus long memory processes
4. Estimation, Identification and Diagnostic
LLN and CLT for dependent process
Consistency and asymptotic normality of the sample mean and sample autocovariance
Yule-Walker estimation of AR(p) processes
OLS estimation of AR(p) process
Violation of strict exogeneity in time-series models
Maximum-likelihood estimation
MLE of sample mean and sample variance under normality
Asymptotic properties of MLE
Conditional Maximum-likelihood estimation
Exact and conditional likelihood estimation of the AR(1) model
Conditional likelihood estimation of the MA(1) model
Quasi-maximum likelihood
Partial autocorrelation and information criteria
Diagnostic
5. Forecasting
Loss functions and mean square error
Forecasting based on conditional expectations
Forecasting with AR, MA and ARMA models
Multistep ahead forecasts
Direct versus iterated forecasts
Density forecasts
Some remarks on non-linear time-series models and realized volatility

Bibliography

Didactic methods

The course aims to provide an overview of the main tools of time-series analysis, with a special emphasis on applications involving the forecast of economic, financial, and business data. The main topics of the course are introduced following a bottom-up approach, starting from motivational examples and discussing in a second step the methodology in rigorous form. Applications are illustrated using publicly available datasets and the MATLAB software.

Learning assessment procedures

The exam consists of a written exam and a group homework that will be assigned to students at the end of the course.

Evaluation criteria

It is assessed the understanding of the main topics of the course, the capacity to communicate them in a rigourous form, as well as the ability to apply them in practice through the use of data.

Criteria for the composition of the final grade

70% written exam + 30% project

Exam language

Inglese/English