Lesson 1: What is a time series

Transcription

1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it

2 What is a time series?

3 A time series is a set of observations on a given variable x collected over the time.

4 If we denote with T the set of time points at which observations are made, with x t the observation made at time t, then the time series is represented by the set {x t ; t T }

5 No restriction is placed on the nature of T. However, two important cases are when T is a finite set of time points; T is a closed time interval.

6 Discrete time series Continuous time series

7 Discrete time series. A discrete time series is one in which the set of times at which observations are made is a finite set. Continuous time series. A continuous time series is obtained when the observations are made continuously over some closed time interval.

8 Consider a discrete time series. Let the times at which observations are made be satisfying t 1, t 2,..., t T t 1 < t 2 <... < t T. If the time points are equally spaced (i.e. t i+1 t i = for all i = 1,..., T 1, where > 0 is a constant), we call the time series regularly sampled. Otherwise, the sequence of observations is called irregularly sampled time series.

9 When we consider a discrete regularly sampled time series, the observation dates (periods) are denoted by integers t = 1, 2,..., T Thus a discrete regularly sampled time series on one variable x with T observations, will be represented by the sequence {x 1, x 2,..., x T }

10 We will consider only discrete regularly sampled time series

11 An important feature of time series data is the frequency at which the data are collected: the sampling frequency Series can be recorded annually, quarterly, monthly, weekly, daily,..., every minutes.

12 In economics, the most common frequencies are: annual, quarterly, monthly, daily

13 Thus we can have annual, quarterly, monthly, weekly, daily time series. the GDP in a country in successive years the money supply in a country in successive quarters the number of international airline passengers in successive months the returns of IBM stock on successive days.

14 The time series graph A important tool in time series analysis is the time series graph. A time series graph is a scatter plot with the observation values on the y-axis and the corresponding dates on the x-axis.

15 The time series graph Consider, for example, the monthly time series of the international airline passengers Jan 49 -â Dec 60. Figure : International airline passengers: monthly totals in thousands. Jan 49 - Dec 60

16 The time series graph However, it is often convenient to join up consecutive points in time as in the following figure. Figure : International airline passengers: monthly totals in thousands. Jan 49 - Dec 60 This graph is called a time series plot.

17 Figure : 1

18 Figure : 2

19 Figure : 3

20 Figure : 4

21 The time series graph By plotting data as a function of time, we can gain insight into several data features. These features include: 1 Trend: a tendency, a long term change in the mean level; 2 Seasonality: an intra-year fluctuation that is more or less stable year after year with respect to timing, direction and magnitude; 3 Cycle: an inter-year periodic fluctuation around the trend, revealing a succession of phases of expansion and contraction; 4 Outliers: values that do not appear to be consistent with the rest of the data; 5 Turning points: different trends within a series; 6 Discontinuities: an unexpected shift, a sudden change of level.

22 The time series graph An increasing trend is apparent in the number of divorces in Italy Figure : Annual number of divorces in Italy from 1995 to 2007

23 The time series graph A clear seasonal component is contained in the series presented in the following Figure. Figure : Industrial production - total industry (excluding construction) in Italy, Index (2005 = 100) from January 1990 to February 2010.

24 A cyclic component is present in the following time series. Figure : Sunspots Number. July 1969 to September 2007.

25 The time series graph Figure : Daily returns of the Dow Jones Industrials index from 80/01/02 to 89/12/29 Direct inspection of figure suggests the series of daily returns of the Dow Jones Industrials index contains some outliers. In particular, the crash of October 19, 1987.

26 The time series graph A turning point is present in the unemployment rate in Italy. Figure : Quarterly unemployment rate in Italy

27 The time series graph The annual U.S. price index for gasoline is shown in this Figure. It appears from the graph the presence of a structural break in the data. Figure : Annual U.S. price index for gasoline from 1960 to 1995

28 The time series graph In summary, a time series plot is a valuable tool used in order to identify patterns in the data. bigskip A good picture is better than 1000 words

29 A question: Why we want to study a time series?

30 The main objective for analyse a time series is to forecast its future values

31 In order to reach this goal we need of a model of the mechanism that has generated the series. A model of the so-called Data Generator Process (DGP).