732A34 Time series analysis Fall semester ECTS-credits Course tutor and examiner: Anders Nordgaard Course web: Course literature: Cryer J.D., Chan K-S.: Time Series Analysis – With Applications in R. 2nd ed. ISBN Complementary handouts

Organization of this course: Weekly “meetings”: Mixture between lectures, (computer) exercises and seminars A great portion of self-studying Assignments (every second week) Individual written exam Access to a computer is necessary. Optimal: Bring your own laptop to the meetings

Examination The course is examined by 1.Assignments (3 in total) 2.Final written exam Assignments will be marked Passed or Failed. If Failed, corrections must be done for the mark Pass. Written exam marks are given according to ECTS grades. The final grade will be the same grade as for the written exam.

Communication Contact with course tutor is best through Office in Building B, Entrance 27, 2nd floor, corridor E (the small one close to Building E), room 3E:485. Normal working hours: When teaching response almost all weekdays and occassionally in weekends All necessary information will be communicated through the course web. Always use the English version. The first page contains the most recent information (messages) Solutions to assignments should be ed. Note! Course tutor is away from Linköping on 3-4 September September September

Assignments A number of exercises will be given as assignments to be individually carried out. There will not be any supervision for these assignments since they are part of the examination, but they can be carried out in the computer rooms or at home. No other statistical software than R will be needed. The solutions to the assignments should be submitted in forms of written reports. The core text of these reports may contain graphs and tables, but the latter should be constructed from scratch (i.e. no copying and pasting from R or other software). Besides such components the text should be completely your own and easy to read. Direct outputs from the software (except graphs) can only be included in form of attachments. In the marking of these reports, emphasis will be put on the English language. It will not be sufficient to simply give short answers to the detailed questions of the exercises.

Time series What kind of patterns can visually be detected? Is the development stable or non-stable?

What kind of patterns can visually be detected? Is the development stable or non-stable?

Non-independent observations (correlations structure) Systematic variation within a year (seasonal effects) Long-term increasing or decreasing level (trend) Irregular variation of small magnitude (noise) Characteristics:

Economic indicators: Sales figures, employment statistics, stock market indices, … Meteorological data: precipitation, temperature,… Environmental monitoring: concentrations of nutrients and pollutants in air masses, rivers, marine basins,… Sports statistics? Electromagnetic och thermal fields Where can time series be found? Time series analysis Estimate/Investigate different parts of a time series in order to –understand the historical pattern –judge upon the current status –make forecasts of the future development –judge upon the quality of data

MethodThis course? Classical decomposition(Yes) Time series regressionYes Exponential smoothingNo ARIMA modelling (Box-Jenkins)Yes Non-parametric and semi-parametric analysisNo Transfer function and intervention modelsYes State space modellingNo Heteroscedastic models: ARCH, GARCHYes Advanced econometric methods: CointegrationNo Spectral domain analysisNo Data mining techniquesNo Methodologies

Decomposition ytyt ytyt A time series can be thought of as built-up by a number of components

What kind of components can we think of? Long-term? Short-term? Deterministic? Purely random?

Decomposition – Analyse the observed time series in its different components: Trend part(TR) Seasonal part(SN) Cyclical part(CL) Irregular part(IR) Cyclical part: State-of-market in economic time series In environmental series, usually together with TR Multiplicative model: yt=TR t ·SN t ·CL t ·IR t Suitable for economic indicators Level is present in TR t or in TC t =(TR∙CL) t SN t, IR t (and CL t ) works as indices  Seasonal variation increases with level of y t

Additive model: y t =TR t +SN t +CL t +IR t More suitable for environmental data Requires constant seasonal variation SN t, IR t (and CL t ) vary around 0 Additive or multiplicative model?

Example 1: Sales figures, additive decomposition sales figures jan-98-dec observed (blue), deseasonalised (magenta) observed (blue), estimated trend (green) observedTRSNfittedIR

A more theoretical description A time-series is a special case of a stochastic process. A stochastic process is a family of random variables coupled with a deterministic index variable t: t can be continuous or discrete. Y can be continuous- or discrete- valued.

Examples Y t = Number of events (e.g. number of telephone calls arrived) up to time t.  Point process (usually modelled as a Poisson process) Y t =Number of customers in a queue at time-point t (Birth-and- death process) Y n = The number of offspring in generation n of a population starting with an initial population Y 0.  Markov chain (Y n depends only on Y n – 1 ) Assume you score +1 if you toss a coin and get “heads” and –1 if you get “tails”. Let Y n = The sum of scores after n tosses.  Random walk Y t = The temperature outdoors at time point t (infinitesimal resolution)

When t is discrete a stochastic process is called a sequence and constitute a model for an observed time series. (Sometimes the sequence itself is referred to as the time series) Mean (value) function: (Auto)covariance function: (Auto)correlation function:

Example Random walk

Note that  t,s and  t,s are symmetric functions, i.e.

Stationarity A stochastic process is said to be strictly stationary if the joint probability distribution of is the same as the joint probability distribution of for any set of time points (t 1, …, t n ) no matter of the value of k

A stochastic process is said to be weakly stationary (or second-order stationary) if stationary non-stationary Roughly: Constant mean and constant variance

White noise A stochastic process that is a sequence of independent and identically distributed (i.i.d.) random variables e 1, e 2, … is called a white noise process. By definition a white noise process is strictly stationary Independent random variables Of interest in the construction of models for general processes