Chapter 2
Time series
Dr. A. PHILIP AROKIADOSS
Assistant Professor
Department of Statistics
St. Joseph’s College (Autonomous)
Tiruchirappalli-620 002.
Time series and Trend analysis
A time series consists of a set of observations
measured at specified, usually equal, time interval.
Time series analysis attempts to identify those factors
that exert influence on the values in the series.
.
Time series analysis is a basic tool for
forecasting. Industry and government must
forecast future activity to make decisions and
plans to meet projected changes.
An analysis of the trend of the observations is
needed to acquire an understanding of the
progress of events leading to prevailing
conditions.
.
The trend is defined as the long term underlying
growth movement in a time series.
Accurate trend spotting can only be determined if the
data are available for a sufficient length of time.
Forecasting does not produce
definitive results. Forecasters can
and do get things wrong from
election results and football
scores to the weather.
Time series examples
•
•
•
•
•
•
•
•
Sales data
Gross national product
Share prices
$A Exchange rate
Unemployment rates
Population
Foreign debt
Interest rates
Time series components
Time series data can be broken into these four
components:
1.
2.
3.
4.
Secular trend
Seasonal variation
Cyclical variation
Irregular variation
Components of Time-Series Data
Irregular
fluctuations
Cyclical
Trend
1
2
3
4
5
6
7
8
Seasonal
9
10
11
12
Year
Predicting long term trends without smoothing?
What could go wrong?
Where do you commence your prediction from the bottom of a
variation going up or the peak of a variation going down………..
13
1. Secular Trend
This is the long term growth or decline of the series.
• In economic terms, long term may mean >10 years
• Describes the history of the time series
• Uses past trends to make prediction about the future
• Where the analyst can isolate the effect of a secular
trend, changes due to other causes become clearer
3.08.2008
3.08.2007
3.08.2006
3.08.2005
3.08.2004
3.08.2003
3.08.2002
3.08.2001
3.08.2000
3.08.1999
3.08.1998
3.08.1997
3.08.1996
3.08.1995
3.08.1994
3.08.1993
3.08.1992
3.08.1991
3.08.1990
3.08.1989
3.08.1988
3.08.1987
3.08.1986
3.08.1985
3.08.1984
All Ords
8000
7000
6000
5000
4000
3000
All Ords
2000
1000
0
.
Look out
While trend estimates are often reliable, in
some instances the usefulness of estimates is
reduced by:
• by a high degree of irregularity in original or
seasonally adjusted series or
• by abrupt change in the time series
characteristics of the original data
$A vs $US
during day 1 vote count 2000 US Presidential election
• .
This graph shows the amazing trend of the $A vs $UA during an 18 hour period on
November 8, 2000
2. Seasonal Variation
The seasonal variation of a time series is a pattern of
change that recurs regularly over time.
Seasonal variations are usually due to the differences
between seasons and to festive occasions such as
Easter and Christmas.
Examples include:
• Air conditioner sales in Summer
• Heater sales in Winter
• Flu cases in Winter
• Airline tickets for flights during school vacations
Monthly Retail Sales in NSW Retail Department
Stores
3. Cyclical variation
Cyclical variations also have recurring patterns but with
a longer and more erratic time scale compared to
Seasonal variations.
The name is quite misleading because these cycles can
be far from regular and it is usually impossible to
predict just how long periods of expansion or
contraction will be.
There is no guarantee of a regularly returning pattern.
Cyclical variation
Example include:
•
•
•
•
•
Floods
Wars
Changes in interest rates
Economic depressions or recessions
Changes in consumer spending
Cyclical variation
This chart represents an economic cycle, but we know
it doesn’t always go like this. The timing and length of
each phase is not predictable.
4. Irregular variation
An irregular (or random) variation in a time series
occurs over varying (usually short) periods.
It follows no pattern and is by nature unpredictable.
It usually occurs randomly and may be linked to events
that also occur randomly.
Irregular variation cannot be explained mathematically.
Irregular variation
If the variation cannot be accounted for by secular
trend, season or cyclical variation, then it is usually
attributed to irregular variation. Example include:
–
–
–
–
–
–
Sudden changes in interest rates
Collapse of companies
Natural disasters
Sudden shift s in government policy
Dramatic changes to the stock market
Effect of Middle East unrest on petrol prices
Monthly Value of Building Approvals ACT)
.
Now we have considered the 4 time series
components:
1.
2.
3.
4.
Secular trend
Seasonal variation
Cyclical variation
Irregular variation
We now take a closer look at secular trends and 5
techniques available to measure the underlying trend.
Why examine the trend?
When a past trend can be reasonably expected
to continue on, it can be used as the basis of
future planning:
• Capacity planning for increased population
• Utility loads
• Market progress
Measure underlying trend
freehand graph (plot)
Measure underlying trend
Semi-averages
This technique attempts to fit a straight line to describe
the secular trend:
i. Divide data into 2 equal time ranges
ii. Calculate the average of the observations in each of
the 2 time ranges
iii. Draw a straight line between the 2 points
iv. Extend line slightly past the end of the original
observation to make predictions for future years
.
Measure underlying trend
Moving averages
This method is based on the premise that if values in a
time series are averaged over a sufficient period, the
effect of short term variations will be reduced.
That is, short term, cyclical, seasonal and irregular
variations will be smoothed out resulting in an
apparently smooth graph depicting the overall trend.
The degree of smoothing can be controlled by selecting
the number of cases to be included in an average.
The technique for finding a moving average for a particular
observation is to find the average of the m observations before
and after the observation itself.
That is, a total of (2m + 1 ) observations must be averaged each
time a moving average is calculated.
Find:
both 3 and 5 year moving
averages (or mean smoothing)
for this time series:
Year
2001
2002
2003
2004
2005
2006
2007
2008
2009
Sales
13
15
17
18
19
20
20
21
22
13 + 15 +17 = 45
15 + 17 + 18 = 50…. Or 45 +18 - 13 = 50 (pick up the new drop off the old)
13 + 15 + 17 + 18 = 63, 63 / 4 = 15.75
15.75 + 17.25 = 33 ……
33 / 2 = 16.5
Measure underlying trend
Least squares linear regression
A more sophisticated way to of fitting a straight line to
a time series is to use the method of least squares
regression.
Sound familiar?
The observations are the
dependant y variables and
time is the independent x
variable.
.
Least squares regression example
Soft Drink Sales $’000 for Carbonated Pty Ltd
Year
2003
2004
2005
2006
2007
2008
2009
2010
2011
Y
13
15
17
18
19
20
20
21
22
165
x
x^2
xy
Calculation for least squares regression example
.
Least squares regression line
.
Yt
= 18.3 + 1.03 x
What is the predicted Sales figure for 2013?
Hint: what value is x in 2013?
.
Yt
= 18.3 + 1.03 x
What is the predicted Sales figure for 2013?
Hint: what value is x in 2013?
If 2011 = 4, then 2013 = 6, so 6 2nd f ( = 24.53
A quick recap…
.
Line of Best Fit – plot points then draw a line that can be
adjusted by clicking either end of the line
Sales
25
20
15
Sales
10
5
0
2000
2002
2004
2006
2008
2010
2012
Semi Averaging Method
at 2002-3 = 15.75 and 2007-8 = 20.75
Sales
25
20
15
Sales
10
5
0
2000
2002
2004
2006
2008
2010
2012
Moving Averages (or mean smoothing)
The following 2 slides should be reviewed at
home (with strong coffee) …
Measure underlying trend – exponential smoothing
.
The trend line is calculated using this formula
where:
• Sx = the smoothed value for observation x
• Y = the actual value of observation x
• Sx-1 = the smoothed value previously
calculated for observation (x-1)
•  =the smoothing constant where 0    1
.
The exponential smoothing approach has to have a
starting point, so we choose the first smoothed value
(S1) to be the first observation (Y1) in the time series.
The smoothed value of each observation is a function
of the smoothed value of the observation immediately
before it.
Errors made in calculating the smoothed values will
carry on for each successive time period.
Exponential smoothing
 = 0.4
Year
Sales (Y)
Sx-1
2004
12,000
2005
12,500
2006
12,200
4,880
2007
13,000
5,200
2008
13,500
5,400
2009
13,400
5,360
2010
14,000
5,600
(1-) Sx-1
Y
Sx
12,000
12,000
7,200
5,000
Complete the table below
2005 - 12500 x .4 = 5000 , 12000 x .6 = 7200, then Sx = 7200 + 5000 = 12200
Bring Sx figure down to Sx-1 for start of 2006 (figure is 12200)
2006 - 12200 x .4 = 4880, 12200 x .6 = 7320 , Sx = 4880 + 7320 = 12200
2007 - Sx-1 = 12200, 13000 x .4 = 5200, 12200 x .6=7320 , Sx = 5200+7320 = 12520
2008 - Sx-1 = 12520, 13500 x .4=5400, 12520 x .6=7512, etc.
Self testing exercise 1
 = 0.3
Year
Sales (Y)
2004
2005
2006
2007
2008
2009
2010
12,000
13,000
14,000
15,000
16,000
17,000
18,000
Sx-1
(1-a) Sx-1
Ya
Sx
12,000
12,000
8,400
3,900
4,200
4,500
4,800
5,100
5,400
The ABS has some useful information on Seasonal Data
.
A seasonally adjusted series involves estimating and removing
the cyclical and seasonal effects from the original data.
Seasonally adjusting a time series is useful if you wish to
understand the underlying patterns of change or movement in a
population, without the impact of the seasonal or cyclical
effects.
For example, employment and unemployment are often
seasonally adjusted so that the actual change in employment
and unemployment levels can be seen, without the impact of
periods of peak employment such as Christmas/New Year when
a large number of casual workers are temporarily employed.
.
Suppose there were some adverse publicity in
December about ice-cream.
It would be incorrect simply to compare the sales of
ice-cream in June with those in December to determine
the effect of the bad publicity. December would have
higher sales in any case, because it is warmer.
Useful comparisons of sales could only be made after
seasonal variation had been allowed for. The true
impact of the publicity could then be determined.
Show how “D = A / Si” can be re-arranged to be Si = A / D – memory trigger
the word “SAID” except the I is the division symbol or the word “DAISY”
with “I” as the division symbol
You tube clip calculating Seasonal indices
http://www.youtube.com/watch?v=7Ug2zA0M-1Y&feature=related
SI - simple average method
The simple average method takes the average
for each period (period mean) over a period of
at least three years, and expresses that as an
index by comparing it to the average of all
periods over the same period of time.
Note: indices can be based on periods such as
months or weeks.
Yearly average 1994 = 43 + 64 + 63 + 41 = 211/4 = 52.75
Yearly average 1995 = 46 + 64 + 67 + 43 = 220 /4 = 55.0
Yearly average 1996= 51 + 69 + 75 + 39 = 234/4 = 58.5
Yearly average 1997= 55 + 73 + 79 + 48 = 255 / 4 =63.75
.
To calculate quarterly SI using same technique
as video:
1994 Q1 Seasonal Index is
1994 Q1 sales/ average for 1994
= 43 / 52.75
= 0.82
Q1
Q2
Q3
Q4
1994
0.82
1.21
1.19
0.78
1995
0.84
1.16
1.22
0.78
1996
0.87
1.18
1.28
0.67
1997
0.86
1.15
1.24
0.75
total
3.39
4.70
4.93
2.98
.
total
3.39
4.70
4.93
2.98
Divide by 4 0.85
1.18
1.23
0.75
to get SI
Alternate approach to calculate Seasonal Index number for each quarter
note: same result!
D = A / SI x 100
60 = 51 / 84.8 x 100
D = A / SI x 100
393 = 362 / 92.1 x 100
So even though Actual sales in the 4th quarter were $357 – once we take
the seasonal factor out of the equation we can see it was the best
performing quarter.
Projected figures will
always be performed on
deseasonalised figures
D = A / SI x 100
Therefore A = D x SI / 100
D = A / SI x 100
D = 18530 / 94.3 x 100 = 19650
D = A / SI x 100
Therefore A = D x SI / 100
A = 2070 x 87.2 / 100 = 1805
Quarter 1 = 1.68 + 1.75 + 1.80 + 1.76 + 1.85 = 8.84 / 5 = 1.768
Quarter 2 = 1.89 + 2.84 + 2.96 +3.03 + 3.08 = 13.8 / 5 = 2.76
Quarter 3 = 2.06 + 2.1 + 2.08 + 2.14 +2.19 = 10.57 / 5 = 2.114
Quarter 4 = 3.25 + 3.3 + 3.32 + 3.39 + 3.38 = 16.64 / 5 = 3.328
Average for the quarters = 1.768 + 2.76 + 2.114 + 3.328 = 9.97 / 4 = 2.4925
Index for Quarter 1 = 1.768 / 2.4925 x 100 = 70.93
D = A / SI x 100
1997 Quarter 1 = D = 1.85 / 70.93 x 100 = 2.61
11.24 / 4 = Quarterly deseasonalised data = 2.81
D = A / SI x 100 Therefore A = D x SI / 100
Quarter 1 = A = 2.81 x 70.93 / 100 = 1.99