The course provides basic tools to interpret and analyze time series data: preliminary analyses; stochastic processes; ARIMA and seasonal ARIMA models; GARCH-like models for financial time series.
Di Fonzo, T. e Lisi F. (2013). Serie storiche economiche. Analisi statistiche e applicazioni, Carocci Editore, Roma.
Additional readings can be provided during the course.
Learning Objectives
KNOWLEDGE:
Analysis of time series data: fundamental concepts; ARIMA and seasonal ARIMA models; GARCH models; forecasting.
EXPERTISE:
To face the analysis of time series data: to set the analysis in its background; preliminary checks; modeling and diagnostic checks; practical use of the model (forecasts; simulations).
To transfer the results of the analyses to other people by using an appropriate language.
To interpret analyses of time series data made by other people (report, scientific articles).
To read the literature on the topics of the course.
Prerequisites
1st course in Statistics; 2nd course in Statistics; R environment
Teaching Methods
Traditional lessons plus practical lessons in the computer room guided by the teacher.
Traditional lessons: 47 hours
Practical lessons: 25 hours
Lessons, written using a pen tablet, made available on the Moodle page.
Further information
Further details and material available on the Moodle page of the course (http://e-l.unifi.it/). Students need to ask the teacher for the permission to login.
Type of Assessment
The exam consists of an oral conversation. The student must prepare a written report (including a statistical analysis of an economic and a financial time series ) to be delivered to the teacher at least a week before the oral exam.
The instructions on how to prepare the report are available on Moodle. In the oral exam, we pay particular attention on the level of knowledge and the logical abilities, in genereal and on pracical examples.
Course program
Time series data.
- What a time series (ts) is: examples of ts plots; definition; decomposition (additive or multiplicative) of a ts in trend, cycle, seasonal and irregular components; ts transformations (index numbers, change rates, logarithms, lag and diff operators, moving averages).
Stochastic processes.
- What is behind a ts: definition of stochastic process (sp) or data generating process (dgp); strongly and weakly stationary processes; estimation of expectations of a stationary sp and ergodicity conditions
- Examples of stationary and non-stationary sp: White Noise (WN); WN + constant; Random Walk (RW); RW + drift; Seasonal RW; linear-trend; AR(1); MA(1); study of their stationarity properties.
Linear sp (ARIMA).
- Identification and diagnostic tools: ACF; Portmanteau (Ljung-Box) test; Normality checks (qq-plot, Jarque-Bera test).
- MA(inf) processes: definition; properties; Wold decomposition.
- AR(p) processes: definition; properties; Yule-Walker equations; stationarity condition; MA(inf) and AR(inf) representation.
- MA(q) processes: definition; properties; AR(inf) processes and invertibility; MA(inf) and AR(inf) representation.
- PACF.
- ARMA(p,q) processes: definition; common roots and plot check.
- Non-stationarity and ARIMA(p,d,q) processes: definition; properties.
- Pure seasonal processes and pure seasonal ARIMA(P,D,Q)S: definition; properties.
- Seasonal ARIMA(p,d,q)x(P,D,Q)S: definition; properties; long-ARMA representation.
ARIMA complements.
- From sp to ts analysis: conditional vs unconditional expectations; simulations; residuals.
- Variable transformations: motivations; variance stability transformation; sqrt, log as relevant cases.
- External regressors: calendar effects; theoretical aspects.
- Unit root tests: DF, ADF and KPSS tests.
- Parameter estimation.
- Outliers.
Financial time series.
- Generalities: prices, returns and characteristics of the corresponding ts; volatility.
- GARCH sp: definition; stationarity and non-negativity properties; persistence; variance targeting.
- Diagnostics: ARCH test; Leverage test; Nyblom test; BDS test.
- Additional sp: GJR-GARCH; T-GARCH; Family-GARCH.
- Alternative error distributions.
- Other measures of volatility: the hi-lo measures of Parkinson (1980) and Garman-Klass (1980).
Forecasting.
- Generalities: definition; ex-ante vs ex-post forecasts; best predictor as the MSE minimizer; prediction error; variance of prediction.
- Forecasting with ARIMA: forecasts; different behavior of MA/AR components and final prediction equation; different behavior of stationary/non-stationary sp's; variance around forecasts; different behavior of stationary/non-stationary sp's; confidence intervals.
- Forecasting with GARCH models.
- Ex-post checks and error measures: Mean Error (ME), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Percentage Error (RMSPE); naive forecasts and scaled measures.
- Additional ex-post checks: Mincer-Zarnowitz diagnostic, Diebold-Mariano test.
R statistical environment for time series analysis.