Transcript Chapter 20

Time Series Analysis
and Forecasting
Chapter 20
Introduction
• Any variable that is measured over time in
sequential order is called a time series.
• We analyze time series to detect patterns.
• The patterns help in forecasting future
values of the time series.
t
Year
Export for Sweden in
the years
1993 -2004
Source:(www.scb.se)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Export (million skr)
499 507
589 383
700 874
694 152
789 378
842 471
890 690
1 018 544
1 048 444
1 042 952
1 070 177
1 182 568
?
?
Swedish Export
1400000
1200000
1000000
800000
600000
400000
1995
1993
1994
Year
1997
1996
1999
1998
2001
2000
2003
2002
2004
Earlier:
Yt
Now:
X1t
X2t
….
Xkt
Yt
Earlier
values
of Y
Components of a Time Series
• A time series can consist of four
components.
– Long - term trend (T).
– Cyclical effect (C).
– Seasonal effect (S).
– Random variation (R).
A trend is a long term
relatively smooth
pattern or direction,
that persists usually
for more than one
year.
A cycle is a wavelike
pattern describing a
long term behavior
(for more than one year).
6/90 6/93 6/96 6/99 6/02
Cycles are seldom regular, and
often appear in combination with
other components.
Components of a Time Series
The seasonal
component of the
time series exhibits
a short term (less
than one year)
calendar repetitive
behavior.
6/97 12/97 6/98 12/98 6/99
Random variation
comprises the irregular
unpredictable changes
in the time series.
It tends to hide the
other (more predictable)
components.
We try to remove random variation
thereby, identify the other components.
Smoothing Techniques
• To produce a better forecast we need to
determine which components are present
in a time series.
• To identify the components present in the
time series, we need first to remove the
random variation.
• This can be done by smoothing
techniques.
Moving Averages
– A k-period moving average for time
period t is the arithmetic average of the
time series values around period t.
– For example: A 3-period moving average at period t
is calculated by (yt-1 + yt + yt+1)/3
Dow Jones Index
Dow Jones Index
January
February
March
April
May
June
July
August
September
October
November
December
2004
2005
2006
2984.28
3038.65
2996.78
2945.94
2964.20
3042.96
3006.14
3028.03
3097.13
3160.25
3322.93
3395.82
3326.83
3427.49
3353.00
3270.30
3378.14
3348.42
3484.17
3468.54
3538.87
3460.16
3653.10
3638.06
3728.56
Dow Jones Index
January
February
March
April
May
June
July
August
September
October
November
December
2984.28
3038.65
2996.78
2945.94
2964.20
3042.96
3006.14
3028.03
3097.13
3160.25
3322.93
3395.82
3800
3600
3400
3200
3000
Dow Jones Index
MA(DOWJON,3,3)
em
ov
N te
p
Se
ly
Ju
ay
M rc h
a
M ua
n
J a em
ov
N te
p
Se
ly
Ju
ay
M rc h
a
m ua
n
Ja
2800
MONTH
Yt
Three-term MA Five-term MA
y1
--------
-----------
y2
(y1 + y2 + y3)/3
-----------
y3
(y2 + y3 + y4)/3
(y1 + y2 + y3 + y4 + y5)/5
y4
(y3 + y4 + y5)/3
(y2 + y3 + y4 + y5 + y6)/5
y5
(y4 + y5 + y6)/3
(y3 + y4 + y5 + y6 + y7)/5
….
yn
..
--------
…
Centered Moving Average
Sell
Period ($ million)
1
170
2
148
3
141
4
150
5
161
6
137
7
132
8
158
9
157
4-terms MA
4-terms
centered MA
Drawbacks
– The moving average method does not
provide smoothed values (moving
average values) for the first and last set
of periods.
– The moving average method
considers only the observations
included in the calculation of the
average value, and “forgets” the rest.
Exponentially Smoothed Time
Series
St = wyt + (1-w)St-1
St = exponentially smoothed time series at
time t.
yt = time series at time t.
St-1 = exponentially smoothed time series at
time t-1.
w = smoothing constant, where 0  w 1.
• The exponential smoothing method provides
smoothed values for all the time periods
observed.
• When smoothing the time series at time t, the
exponential smoothing method considers all the
data available at t
(yt, yt-1,…).
Exponential Smoothing, w=.2
Small ‘w’ provides
a lot of smoothing
Value
100
50
15
13
11
9
7
5
3
1
0
Exponential Smoothing , w=.7
Big ‘w’ provides
a little smoothing
Value
100
50
0
1
3
5
7
9
11
13
15
Trend and Seasonal Effects
Trend Analysis
• The trend component of a time series
can be linear or non-linear.
• It is easy to isolate the trend component
using linear regression.
– For linear trend use the model
y = b0 + b1t + e.
– For non-linear trend with one (major) change
in slope use a polynomial model, for
example
y = b0 + b1 t + b2 t 2 + e
Trend analysis
The purpose is to
• Describe the trend component in order to
make forecasts
• Detrend the time series in order to make a
season analysis.
Seasonal Analysis
• Seasonal variation may occur within a
year or within a shorter period (month,
week)
• To measure the seasonal effects we
construct seasonal indexes.
• Seasonal indexes express the degree to
which the seasons differ from the average
time series value across all seasons.
Computing Seasonal Indexes
•
Remove the effects of the seasonal and
random variations by regression analysis
yˆ t = b0 + b1t
>
>
• For each time period compute the ratio
yt/yt
This is based on the Multiplicative Model.
which removes most of the trend variation
• For each season calculate the average of yt/yt
which provides the measure of seasonality.
• Adjust the average above so that the sum of averages
of all seasons is 1 (if necessary)
Computing Seasonal Indexes
• Example 20.3 (Xm20-03)
– Calculate the quarterly seasonal indexes
for hotel occupancy rate in order to
measure seasonal variation.
– Data:
Year
Quarter
1996
1
2
3
4
1997
1
2
3
4
Rate
0.561
0.702
0.8
0.568
0.575
0.738
0.868
0.605
Year
Quarter
1998
1
2
3
4
1999
1
2
3
4
Rate
0.594
0.738
0.729
0.6
0.622
0.708
0.806
0.632
Year
Quarter
2000
1
2
3
4
Rate
0.665
0.835
0.873
0.67
Computing Seasonal Indexes
• Perform regression analysis for the model
y = b0 + b1t + e where t represents the time,
and y represents the occupancy rate.
Time (t) Rate
1
0.561
2
0.702
3
0.800
4
0.568
5
0.575
6
0.738
7
0.868
8
0.605
.
.
.
.
Rate
yˆ  .639368  .005246 t
0
5
10
15
20
The regression linet represents trend.
25
>
The Ratios yt / yt
yt
Ratio
yˆ t
.561
.645
.561/.645=.870
.702
.650
.702/.650=1.08
………………………………………………….
=.639368+.005245(1)
No trend is observed, but
seasonality and randomness
still exist.
yt
Rate/Predicted rate
ˆy t
1.5
1
0.5
19
17
15
13
11
9
7
5
3
0
1
t
1
2
3
The Average Ratios by Seasons
Rate/Predicted rate
0.870
1.080
1.221
0.860
0.864
1.100
1.284
0.888
0.865
1.067
1.046
0.854
0.879
0.993
1.122
0.874
0.913
1.138
1.181
0.900
• To remove most of the random variation
but leave the seasonal effects,average
the terms y t / yˆ t for each season.
Rate/Predicted rate
1.5
1
0.5
0
1
3
5
7
9
11
13
15
17
19
Average ratio for quarter 1: (.870 + .864 + .865 + .879 + .913)/5 = .878
Average ratio for quarter 2:
(1.080+1.100+1.067+.993+1.138)/5 =
1.076
Average ratio for quarter 3:
(1.221+1.284+1.046+1.122+1.181)/5 =
1.171
Average ratio for quarter 4: (.860 +.888 +
.854 + .874 + .900)/ 5 = .875
Adjusting the Average Ratios
• In this example the sum of all the averaged ratios
must be 4, such that the average ratio per season is
equal to 1.
• If the sum of all the ratios is not 4, we need to adjust
them proportionately.
Suppose the sum of ratios is equal to 4.1. Then each
ratio will be multiplied by 4/4.1.
In our problem the sum of all the averaged ratios is equal to 4:
.878 + 1.076 + 1.171 + .875 = 4.0.
No normalization is needed. These ratios become the
seasonal indexes.
Interpreting the Seasonal Indexes
• The seasonal indexes tell us what is the ratio
between the time series value at a certain
season, and the overall seasonal average.
17.1% above the
• In our problem:
annual average
7.6% above the
annual average
Annual average
occupancy (100%)
117.1%
12.5% below the
annual average
107.6%
12.2% below the
annual average
87.8%
87.5%
Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarter 1 Quarter 2 Quarter 3 Quarter 4
The Smoothed Time Series
• The trend component and the seasonality
component are recomposed using the
multiplicative model. This is used for
forecasting.
yˆ t  Tˆt  Sˆ t  (.639  .0052 t )Sˆ t
In period #1 ( quarter 1):
In period #2 ( quarter 2):
yˆ 1  Tˆ1  Sˆ 1  (.639  .0052 (1))(.878 )  .566
yˆ 2  Tˆ2  Sˆ 2  (.639  .0052 (2))(1.076 )  .699
• We can also use indicator variables in order to
analyze the seasonal effects.
Quarter I1
I2
I3
1
1
0
0
2
0
1
0
3
0
0
1
4
0
0
0
Coefficientsa
Model
1
(Constant)
I1
I2
I3
T
Unstandardized
Coefficients
B
Std. Error
,555
,025
3,512E-03
,024
,139
,024
,205
,024
5,037E-03
,002
a. Dependent Variable: RATE
Standardized
Coefficients
Beta
,015
,609
,897
,293
t
22,350
,143
5,741
8,510
3,348
Sig .
,000
,888
,000
,000
,004
Seasonal analysis
The purpose is to
• Describe the seasonal component in order
to make forecasts
• Deseasonalize the time series (makes it
for example easier to compare timeseries
over seasons)
Deseasonalized Time Series
Seasonally adjusted time series = Actual time series
Seasonal index
By removing the seasonality, we can
identify changes in the other components of
the time series, that might have occurred
over time.
Deseasonalized Time Series
y1 / SI1  .561/ .870  .639
y 2 / SI2  .708 1.076  .652
y 5 / SI1  .575 .870  .661
In period #5 ( quarter 1):
There was a gradual increase in occupancy rate
In period #1 ( quarter 1):
In period #2 ( quarter 2):
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25