Chapter 5 (time series analysis) student

CHAPTER 5

TIME SERIES
ANALYSIS
1


Time series data
 Numerical

data obtained at regular
time intervals
 The time intervals can be annually,
quarterly, daily, hourly, etc.

2


Example
Annual sales of a firm for 5 successive
years
Year

2000

2001

2002

2003

2004

Sales

75.3

74.2

78.5

79.7

80.2

3


Example


Number of registered journeys for a Home
Removals firm:
Qtr 1

Qtr 2

Qtr 3

Qtr 4

Year 1

73

90

121

98

Year 2

69

92

145

107

Year 3

86

111

157

122

Year 4

88

109

159

131
4


Time series cycle




General pattern which broadly repeats
Regularly occur but may vary in length
Often measured peak to peak or trough to
trough
1 Cycle
Sales

Year

5


Standard time series models




There are various types of model that can
be used to describe time series data.
Two main models:
Additive model:
y=t+s+r

Multiplicative
model
y=txSxR

y: is a given time series value
wher
e

t: is the trend component
s: is the seasonal component
r: is the residual component

6


Time series components
Time-Series
Trend
Component

Seasonal

Component

Residual
Component

7


Trend component


The underlying, long-term tendency of the
data series (overall upward or downward
movement)



Some techniques are used for extracting a
trend from a given time series: semiaverage, regression (least square method),
moving average…

8


Trend Component


Example

Sales

trend
d
r
a
Upw

Time

9


Trend Component



Trend can be upward or downward
Trend can be linear or non-linear

Sales

Sales

Time
Downward linear trend

Time
Upward nonlinear trend
10

Seasonal Component




Short-term cyclic fluctuation
Observed within 1 year (Often monthly or quarterly)
Sometimes observed within 1 week (daily)

Sales
Summer
Winter
Spring

Fall

Time (Quarterly)
11


Examples





Daily ‘seasons’ over a weekly cycle for
sales in a supermarket
Monthly ‘season’ over a yearly cycle for

the number of registered guests of a
resort.
Quarterly ‘season’ over a yearly cycle for
sales of confectionary.

12


Example


Number of registered journeys for a Home
Removals firm:
Qtr 1

Qtr 2

Qtr 3

Qtr 4

Year 1

73

90

121

98

Year 2

69

92

145

107

Year 3

86

111

157

122

Year 4

88

109

159

131

13


Year 1

Year 2

Year
3

Year 4

14


Residual component



Unpredictable, random, “residual”
fluctuations
Due to random factors such as

15


Time series trend






The underlying, long-term tendency of the
data series (overall upward or downward
movement)
The object of finding time series trend: to
enable the underlying tendency of the
data to be highlighted.
Techniques for extracting the trend:
- least square regression
- moving average
16


The method of least squares
regression


-

-

Consider the time series data as bivariate
The procedure for conducting the method
Step 1: Take the physical time points as
values of the independent variable x
(coded as 1, 2, 3, etc).
Step 2: Take the data values themselves
as values of the dependent variable y

17


The method of least squares
regression
-

Step 3: Calculate the regression line of y
on x, y= a+bx
Step 4: Translate the regression line as
t=a+bx, where any given value of time
point x will yield a corresponding value of
the trend, t.

18


Example


Estimate a trend line using regression
analysis

Year

1999
2000
2001
2002

2003
2004

Time
Period
(x)
1
2
3
4
5
6

Sales
(y)
20
40
30
50
70
65
19


Example
UK outward passenger movements by sea
(millions). Estimate a trend line using
regression analysis.
Qtr 1

Qtr 2

Qtr 3

Qtr 4

Year 1

2.2

5.0

7.9

3.2

Year 2

2.9

5.2

8.2

3.8

Year 3

3.2

5.8

9.1

4.1
20


The method of moving
average





The method of obtaining a time series
trend involves calculating a set of average
(known as moving average)
Each average is calculated by moving from
one overlapping set of values to the next
Used for smoothing seasonal variation
(length of period for computing means (L)
should be equal to the number of seasons)
- For quarterly data, L = 4
- For monthly data, L = 12
21


Notes


Each moving average value calculated must
correspond with an appropriate time point
(median of the time point)
- for moving average with an odd-numbered
period (3, 5, 7, etc), the relevant time point
is that corresponding to the 2nd, 3rd, 4th, etc
- for moving average with an evennumbered period, there is no obvious
corresponding time point. The technique
“centering” will be used
22


Moving average (odd-numbered
period)
Time
point

Original
data

1
2

12
10

3
4

11
11

5
6

9
11

7
8

10
10

9
10

11
10

Moving
totals

Moving
average

23

Moving average (even-numbered
period)
Time
point

Data
value

1
2

9
14

3
4

17
12

5
6

10
14

7
8

19

15

9
10

10
16

…

Moving
totals

Moving
averages

Centering
averages

24


Example
UK outward passenger movements by sea
(millions). Calculate trend value using
moving average centering.
Qtr 1

Qtr 2

Qtr 3

Qtr 4

Year 1

2.2

5.0

7.9

3.2

Year 2

2.9

5.2

8.2

3.8

Year 3

3.2

5.8

9.1

4.1
25