122 Time Series Analysis
INTRODUCTION
Businesses and governments use statistical analysis of information collected at regular
intervals over extensive periods of time to plan future policies. For example, sales values or
unemployment levels recorded at yearly, quarterly or monthly intervals are examined in an
attempt to predict their future behaviour. Such sets of values observed at regular intervals
over a period of time are called time series.
The analysis of this data is a complex problem as many variable factors may influence the
changes. The first step is to plot the observations on a scattergraph, which differs from those
scattergraphs we have considered previously as the points are evenly spaced on the time
axis in the order in which they are observed, and the time variable is always the independent
variable. This scattergraph gives us a good visual guide to the actual changes, but is of very
little help in showing the component factors causing these changes or in predicting future
movements of the dependent variable.
Statisticians have constructed a number of mathematical models to describe the behaviour
of time series, and several of these will be discussed in this study unit.
A. STRUCTURE OF A TIME SERIES
These mathematical models assume that the changes are caused by the variation of four
main factors; they differ in the relationship between these factors. It will be easier to
understand the theory in detail if we relate it to a simple time series so that we can see the
calculations necessary at each stage.
Consider a factory employing a number of people in producing a particular commodity, say
thermometers. Naturally, at such a factory during the course of a year some employees will
be absent for various reasons. The following table shows the number of days lost through
sickness over a five-year period. Each year has been broken down into four quarters of three
months. We have assumed that the number of employees at the factory remained constant
over the five years.
Table 9.1: Days lost through sickness at a thermometer factory
Year Quarter Days Lost Year Quarter Days Lost
2003 1 30 2004 1 40
2 20 2 25
3 15 3 18
4 35 4 45
2005 1 45 2006 1 50
2 30 2 32
3 22 3 28
4 55 4 60
2007 1 60
2 35
3 30
4 70
We will begin by plotting a time-series graph for the data, as shown in Figure 9.1.
, Time Series Analysis 123
Note the following characteristics of a time-series graph:
It is usual to join the points by straight lines. The only function of these lines is to help
your eyes to see the pattern formed by the points.
Intermediate values of the variables cannot be read from the graph.
Every time-series graph will look similar to this, but a careful study of the change of
pattern over time will suggest which model should be used for analysis.
Figure 9.1: Time series of days lost through sickness
80
70
60
50
Lost days
40
30
20
10
0
Year 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
2003 2004 2005 2006 2007
There are four factors that influence the changes in a time series – trend, seasonal
variations, cyclical fluctuations, and irregular or random fluctuations. Now we will consider
each in turn.
Trend
This is the change in general level over the whole time period and is often referred to as the
secular trend. You can see in Figure 9.1 that the trend is definitely upwards, in spite of the
obvious fluctuations from one quarter to the next.
A trend can thus be defined as a clear tendency for the time series data to travel in a
particular direction in spite of other large and small fluctuations. An example of a linear trend
is shown in Figure 9.2. There are numerous instances of a trend, for example the amount of
money collected from UK taxpayers is always increasing; therefore any time series
describing income from tax would show an upward trend.
, 124 Time Series Analysis
Figure 9.2: Example of trend
Value of variable (y)
Time (x)
Seasonal Variations
These are variations which are repeated over relatively short periods of time. Those most
frequently observed are associated with the seasons of the year, e.g. ice cream sales tend to
rise during the summer months and fall during the winter months. You can see in our
example of employees' sickness that more people are sick during the winter than in the
summer.
If you can establish the variation throughout the year then this seasonal variation is likely to
be similar from one year to the next, so that it would be possible to allow for it when
estimating values of the variable in other parts of the time series. The usefulness of being
able to calculate seasonal variation is obvious as, for example, it allows ice cream
manufacturers to alter their production schedules to meet these seasonal changes. Figure
9.3 shows a typical seasonal variation that could apply to the examples above.
Figure 9.3: Seasonal variations
Value of variable (y)
Time (x)
INTRODUCTION
Businesses and governments use statistical analysis of information collected at regular
intervals over extensive periods of time to plan future policies. For example, sales values or
unemployment levels recorded at yearly, quarterly or monthly intervals are examined in an
attempt to predict their future behaviour. Such sets of values observed at regular intervals
over a period of time are called time series.
The analysis of this data is a complex problem as many variable factors may influence the
changes. The first step is to plot the observations on a scattergraph, which differs from those
scattergraphs we have considered previously as the points are evenly spaced on the time
axis in the order in which they are observed, and the time variable is always the independent
variable. This scattergraph gives us a good visual guide to the actual changes, but is of very
little help in showing the component factors causing these changes or in predicting future
movements of the dependent variable.
Statisticians have constructed a number of mathematical models to describe the behaviour
of time series, and several of these will be discussed in this study unit.
A. STRUCTURE OF A TIME SERIES
These mathematical models assume that the changes are caused by the variation of four
main factors; they differ in the relationship between these factors. It will be easier to
understand the theory in detail if we relate it to a simple time series so that we can see the
calculations necessary at each stage.
Consider a factory employing a number of people in producing a particular commodity, say
thermometers. Naturally, at such a factory during the course of a year some employees will
be absent for various reasons. The following table shows the number of days lost through
sickness over a five-year period. Each year has been broken down into four quarters of three
months. We have assumed that the number of employees at the factory remained constant
over the five years.
Table 9.1: Days lost through sickness at a thermometer factory
Year Quarter Days Lost Year Quarter Days Lost
2003 1 30 2004 1 40
2 20 2 25
3 15 3 18
4 35 4 45
2005 1 45 2006 1 50
2 30 2 32
3 22 3 28
4 55 4 60
2007 1 60
2 35
3 30
4 70
We will begin by plotting a time-series graph for the data, as shown in Figure 9.1.
, Time Series Analysis 123
Note the following characteristics of a time-series graph:
It is usual to join the points by straight lines. The only function of these lines is to help
your eyes to see the pattern formed by the points.
Intermediate values of the variables cannot be read from the graph.
Every time-series graph will look similar to this, but a careful study of the change of
pattern over time will suggest which model should be used for analysis.
Figure 9.1: Time series of days lost through sickness
80
70
60
50
Lost days
40
30
20
10
0
Year 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
2003 2004 2005 2006 2007
There are four factors that influence the changes in a time series – trend, seasonal
variations, cyclical fluctuations, and irregular or random fluctuations. Now we will consider
each in turn.
Trend
This is the change in general level over the whole time period and is often referred to as the
secular trend. You can see in Figure 9.1 that the trend is definitely upwards, in spite of the
obvious fluctuations from one quarter to the next.
A trend can thus be defined as a clear tendency for the time series data to travel in a
particular direction in spite of other large and small fluctuations. An example of a linear trend
is shown in Figure 9.2. There are numerous instances of a trend, for example the amount of
money collected from UK taxpayers is always increasing; therefore any time series
describing income from tax would show an upward trend.
, 124 Time Series Analysis
Figure 9.2: Example of trend
Value of variable (y)
Time (x)
Seasonal Variations
These are variations which are repeated over relatively short periods of time. Those most
frequently observed are associated with the seasons of the year, e.g. ice cream sales tend to
rise during the summer months and fall during the winter months. You can see in our
example of employees' sickness that more people are sick during the winter than in the
summer.
If you can establish the variation throughout the year then this seasonal variation is likely to
be similar from one year to the next, so that it would be possible to allow for it when
estimating values of the variable in other parts of the time series. The usefulness of being
able to calculate seasonal variation is obvious as, for example, it allows ice cream
manufacturers to alter their production schedules to meet these seasonal changes. Figure
9.3 shows a typical seasonal variation that could apply to the examples above.
Figure 9.3: Seasonal variations
Value of variable (y)
Time (x)
