# Time Series

September 27, 2017 | Author: HazemSamir | Category: Time Series, Seasonality, Chart, Index (Economics), Scientific Modeling

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Lesson 1: What is a time series Umberto Triacca Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit` a dell’Aquila, [email protected]

Umberto Triacca

Lesson 1: What is a time series

Time series

What is a time series?

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

If we denote with T the set of time points at which observations are made,

Umberto Triacca

Lesson 1: What is a time series

Time series

If we denote with T the set of time points at which observations are made, with xt the observation made at time t,

Umberto Triacca

Lesson 1: What is a time series

Time series

If we denote with T the set of time points at which observations are made, with xt the observation made at time t, then the time series is represented by the set

{xt ; t ∈ T }

Umberto Triacca

Lesson 1: What is a time series

Time series

No restriction is placed on the nature of T .

Umberto Triacca

Lesson 1: What is a time series

Time series

No restriction is placed on the nature of T . However, two important cases are when

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

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.

Umberto Triacca

Lesson 1: What is a time series

Time series

Discrete time series

Umberto Triacca

Lesson 1: What is a time series

Time series

Discrete time series Continuous time series

Umberto Triacca

Lesson 1: What is a time series

Time series

Discrete time series. A discrete time series is one in which the set of times at which observations are made is a finite set.

Umberto Triacca

Lesson 1: What is a time series

Time series

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.

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series.

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT .

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT . If the time points are equally spaced

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT . If the time points are equally spaced (i.e. ti+1 − ti = ∆ for all i = 1, ..., T − 1, where ∆ > 0 is a constant),

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT . If the time points are equally spaced (i.e. ti+1 − ti = ∆ for all i = 1, ..., T − 1, where ∆ > 0 is a constant), we call the time series regularly sampled.

Umberto Triacca

Lesson 1: What is a time series

Time series

Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT . If the time points are equally spaced (i.e. ti+1 − ti = ∆ 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.

Umberto Triacca

Lesson 1: What is a time series

Time series

When we consider a discrete regularly sampled time series, the observation dates (periods) are denoted by integers t = 1, 2, ..., T

Umberto Triacca

Lesson 1: What is a time series

Time series

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 {x1 , x2 , ..., xT }

Umberto Triacca

Lesson 1: What is a time series

Time series

We will consider only discrete regularly sampled time series

Umberto Triacca

Lesson 1: What is a time series

Time series

An important feature of time series data is the frequency at which the data are collected:

Umberto Triacca

Lesson 1: What is a time series

Time series

An important feature of time series data is the frequency at which the data are collected: the sampling frequency

Umberto Triacca

Lesson 1: What is a time series

Time series

An important feature of time series data is the frequency at which the data are collected: the sampling frequency

Series can be recorded

Umberto Triacca

Lesson 1: What is a time series

Time series

An important feature of time series data is the frequency at which the data are collected: the sampling frequency

Series can be recorded annually,

Umberto Triacca

Lesson 1: What is a time series

Time series

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,

Umberto Triacca

Lesson 1: What is a time series

Time series

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,

Umberto Triacca

Lesson 1: What is a time series

Time series

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,

Umberto Triacca

Lesson 1: What is a time series

Time series

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,

Umberto Triacca

Lesson 1: What is a time series

Time series

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, ...,

Umberto Triacca

Lesson 1: What is a time series

Time series

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.

Umberto Triacca

Lesson 1: What is a time series

Time series

In economics, the most common frequencies are:

Umberto Triacca

Lesson 1: What is a time series

Time series

In economics, the most common frequencies are: annual,

Umberto Triacca

Lesson 1: What is a time series

Time series

In economics, the most common frequencies are: annual, quarterly,

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

Thus we can have annual, quarterly, monthly, weekly, daily time series.

Umberto Triacca

Lesson 1: What is a time series

Time series

Thus we can have annual, quarterly, monthly, weekly, daily time series. the GDP in a country in successive years

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

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.

Umberto Triacca

Lesson 1: What is a time series

The time series graph

A important tool in time series analysis is the time series graph.

Umberto Triacca

Lesson 1: What is a time series

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.

Umberto Triacca

Lesson 1: What is a time series

The time series graph Consider, for example, the monthly time series of the international airline passengers Jan 49 -ˆa Dec 60.

Figure : International airline passengers: monthly totals in thousands. Jan 49 - Dec 60

Umberto Triacca

Lesson 1: What is a time series

The time series graph However, it is often convenient to join up consecutive points in time as in the following figure.

Umberto Triacca

Lesson 1: What is a time series

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

Umberto Triacca

Lesson 1: What is a time series

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. Umberto Triacca

Lesson 1: What is a time series

Time series

Figure : 1

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Lesson 1: What is a time series

Time series

Figure : 2

Umberto Triacca

Lesson 1: What is a time series

Time series

Figure : 3

Umberto Triacca

Lesson 1: What is a time series

Time series

Figure : 4

Umberto Triacca

Lesson 1: What is a time series

The time series graph By plotting data as a function of time, we can gain insight into several data features. These features include:

Umberto Triacca

Lesson 1: What is a time series

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;

Umberto Triacca

Lesson 1: What is a time series

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;

Umberto Triacca

Lesson 1: What is a time series

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;

Umberto Triacca

Lesson 1: What is a time series

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;

Umberto Triacca

Lesson 1: What is a time series

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;

Umberto Triacca

Lesson 1: What is a time series

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.

Umberto Triacca

Lesson 1: What is a time series

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

Umberto Triacca

Lesson 1: What is a time series

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.

Umberto Triacca

Lesson 1: What is a time series

A cyclic component is present in the following time series.

Figure : Sunspots Number. July 1969 to September 2007.

Umberto Triacca

Lesson 1: What is a time series

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.

Umberto Triacca

Lesson 1: What is a time series

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.

Umberto Triacca

Lesson 1: What is a time series

The time series graph

A turning point is present in the unemployment rate in Italy.

Figure : Quarterly unemployment rate in Italy 2000.1-2009.4

Umberto Triacca

Lesson 1: What is a time series

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

Umberto Triacca

Lesson 1: What is a time series

The time series graph

In summary, a time series plot is a valuable tool used in order to identify patterns in the data.

Umberto Triacca

Lesson 1: What is a time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

A question:

Umberto Triacca

Lesson 1: What is a time series

Time series

A question:

Why we want to study a time series?

Umberto Triacca

Lesson 1: What is a time series

Time series

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

Umberto Triacca

Lesson 1: What is a time series

Time series

In order to reach this goal we need of a model of the ”mechanism” that has generated the series.

Umberto Triacca

Lesson 1: What is a time series

Time series

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).

Umberto Triacca

Lesson 1: What is a time series