Minister of Finance
Instructor: Le Thi Ngoc Tu
Group members: Tran Tien Manh
Pham Thi Huyen
Ly Thi Thuy Linh
Nguyen Van Hiep
• Overview
Introduction
• Components of time series
• Smoothing techniques
• Trend analysis
Analysis
• Measuring seasonal effect
• Time-series forecasting with regression
Forecasting
• Application
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Recall Regression Model
X: independent variable
Y: dependent variable
 Time-series:
- Definition: Variable measured over
time in sequential order
- Independent variable: Time

4
Example:
5
Purpose of timeseries analysis
Detect patterns
to forecast the
future value of
the time-series
Applications in
management and
economics
Forecast
interest rates,
U/E rate
Predict the
demand for
products
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Long-term
trend
Cyclical
effect
(T)
(C)
Seasonal
effect
Random
variation
(S)
(R)
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+ Long-term trend: Smooth pattern with
duration > 1 year
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+ Cyclical effect: wavelike pattern about a
long-term trend, duration > 1 year, usually
irregular
Cycles are sequences of points
Volume
above & below the trend line
Time
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+ Seasonal effect: like cycles but short
repetitive periods, duration < 1 year (days,
weeks, months…)
Sales peak in Dec.
10
+ Random variation: irregular changes that we
want to remove to detect other components
Volume
Random
variation that
does not repeat
Time
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
Purpose: Remove random fluctuation
to detect seasonal pattern

2 types:
-
Moving average (MA)
-
Exponential smoothing
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Example of
Moving
average:
Period
t
yt
3-period
MA
4-period
MA
4-period centred
MA
1
12
-
-
-
2
18
15.33
-
-
3
16
19.33
17.5
18.13
4
24
19.00
18.75
18.50
5
17
19.00
18.25
19.38
6
16
19.33
20.5
20.13
7
25
20.67
19.75
-
8
21
-
-
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Linear model:
Techniques
yt = β0 + β1t + Ɛ
Polinomial
model
Trend analysis
Purpose
Isolate the
long-term trend
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Measuring seasonal effect
Values of St x Rt
Quarter
Year
Calculate
=
2005
1
2
MAt : Mulplicative
model:
-
-
1.0254
0.9572
1.0281
1.0318
0.9548
1.0316
1.0212
0.9592
1.0134
1.0481
2006
2007
MAt = T0.9869
t x Ct
2008
2009
1.0012
T x0.9304
Ct x St -x Rt
Yt
0.9900t
Average (Si)
0.9925
Seasonal Index
t (Si)
0.9928
MA
4 Total
1.0239
yt = Tt x Ct x St x Rt
0.9918
3
0.9504
1.0242
T
0.9507
t x Ct1.0246
1.0316
3.9987
1.0319
4.0000
Calculate average of St x Rt
 St
St is adjusted  SIt , so that
average SIt= 1

Forecast of trend & seasonality:
Ft = [ β0 + β1t ] SIt
where:
Ft = forecast for period t
SIt = seasonal index for period t
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Using the following data
about CPI of Viet Nam
from 2005 to 2008 for
forecasting CPI in 2010:
Year
2005
2006
2008
Month
1
2
3
4
5
6
7
8
9
10
11
12
…
…
48
CPI
101.1
102.5
100.1
100.6
100.5
100.4
100.4
100.4
100.8
100.4
100.4
100.8
…
…
99.3
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18

Reasons:
-
CPI is measured over time (monthly)
-
3 components exist
 Technique: Time-series forecasting with
regression
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CPI peaks in
Feb
Random
variation in 2008
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Trend analysis
Using Excel, the trend line is:

yt = 100.551 + 0.016 t
y = 100.551 + 0.016 t
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Measuring seasonal effect
Calculate MAt : Mulplicative model:
yt = Tt x Ct x St x Rt
MAt = Tt x Ct
Yt
MAt
=
Tt x Ct x St x Rt
Tt x Ct
Calculate average of St x Rt
 St
St is adjusted  SIt , so that
average SIt= 1
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 Seasonal
index
Month
1
2
3
SI t
1.0055
1.017
0.997
Month
7
8
9
SI t
4
5
6
0.9998 1.0055 1.0005
10
11
12
0.9975 0.9975 0.9945 0.9928 0.9935 0.9988
 Apply
the formula: Ft = [ β0 + β1t ] SIt
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
Forecast CPI in 2008
Forecast CPI of 2008 did not match actual CPI due to
unexpected events (recession)
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
Forecast CPI in 2010
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y = 100.551 + 0.016 t
Forecasted
CPI
- Long term trend: slight increase in CPI
- Seasonal effect: peak in Feb.
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





‘Time Series Analysis’, Citing or referencing electronic sources of
information, viewed 15 May
2010, http://www.statsoft.com/textbook/time-seriesanalysis/?button=3
Australian Bureau of Statistics, ‘Time Series Analysis: The Basics’, viewed
15 May 2010,
http://www.abs.gov.au/websitedbs/d3310114.nsf/4a256353001af3ed4
b2562bb00121564/b81ecff00cd36415ca256ce10017de2f!OpenDocume
nt#WHAT%20IS%20A%20TIME%20SERIES%3F
‘Introduction to Time Series Analysis’, Citing or referencing electronic
sources of information, viewed 15 May 2010,
http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm
Berenson, M. & Levine, D. 1998, Business Statistics - A first course,
Prentice Hall Press.
Anderson, D., Sweeney, D. & Williams, T. 1999, Statistics for business
and economics, South-Western College Publishing, Ohio.
Selvanathan, A., Selvanathan, S., Keller, G. & Warrack, B. 2004, Australian
business statistics, Nelson Australia Pty Limited.
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