# Decomposition of the Time Series Presentation

Preparing of the project, which aims to forecast the economic variable chosen by the team ofstudents. Project us research tasks include:decomposing a time series for a projected variableover a historical period, proposing two or more alternative forecasting methods, comparing theireffectiveness in forecasting the variable by analysing selected ex post forecasterrors, setting exante(point)forecasts using a bettermethod,graphical presentation of ex postand ex ante forecasts in relation to actual data. summary and critical evaluation of the prospective analysis carried out.

DECOMPOSITION OF THE TIME SERIES
TREND STRAIGHT LINE – LEAST
SQUARES METHOD
LEAST SQUARES METHOD
𝑁
2
2

ฎ 𝑏0 + 𝑏1 *X
σ
SSE= σ𝑁
𝑒
=
(𝑦

𝑌
)
, AND 𝑌=
𝑖=1 𝑖
𝑖=1 𝑖
LEAST SQUARES METHOD

ത _2 = σ 𝑋 2 – (σ 𝑋)
SS_x = σ(𝑋 − 𝑋)
𝑁

(σ 𝑌)
SS_y = σ(𝑌 − 𝑌)2 = σ 𝑌 2 𝑁
2
2
ത – 𝑌)=
ത σ 𝑋𝑌 −
• SS_xy = σ(𝑋 − 𝑋)(Y
• 𝑏1 =
SS_xy
SS_x
• 𝑏0 = 𝑌ത – 𝑏1 * 𝑋ത
σ𝑋 σ𝑌
𝑁
EXAMPLE 1
Incomes
25
y = 1.75x – 3509
20
R² = 0.9754
Year
2011
2012
2013
2014
2015
2016
2017
2018
15
10
5
0
2010
Incomes
2011
2012
2013
2014
2015
2016
2017
2018
10
13
13
15
18
19
20
23
2019
FORECASTING BASED ON THE DESIGNATED
TREND LINE
• y=1,75x-3509
• 𝑍𝑡 = 𝑏0 + 𝑏1 t + 𝑎𝑡
• 𝑍𝑡 =1.75t – 3509
• 𝑍2019 = 1.75 * 2019 – 3509 = 24,25
EXAMPLE 2
Year
Incomes
2011
2012
2013
2014
2015
2016
2017
10
13
17
30
46
68
90
Incomes
100
y = 2.2738×2 – 9145.4x + 9E+06
R² = 0.9985
90
80
70
60
50
40
30
20
10
0
2010
2011
2012
2013
2014
2015
2016
2017
2018
QUESTION
• Can the trend analysis itself be the only method used for forecasting monthly ice cream
sales in Poland?
MOVING AVERAGE
DEFINITION OF THE MOVING AVERAGE
• The moving average of a time series is the average of k items of observations, which
changes as we move through time.
EXAMPLE OF CALCULATING THE MOVING AVERAGE
WHEN K=5
• 1
2
3
4
5
6
7
8
9
10
• 12
15
21
14
12
25
31
23
18
36
• (12+15+21+14+12)/5 =14,8
(15+21+14+12+25)/5 =17,4
(21+12+12+25+31)/5=20,6
(14+12+25+31+23)/5 =21
(12+25+31+23+18)/5=21,8
(25+31+23+18+36)/5=26,6
• moving average : 14.8
17.4
20.6
21
21.8
26.6
K=4
Quart Incom
Year er
es
2016Q1
13
Q2
36
Q3
60
Q4
20
2017Q1
16
Q2
30
Q3
72
Q4
15
2018Q1
20
Q2
37
Q3
66
Q4
18
Moving
average
32.25
33
31.5
34.5
33.25
34.25
36
34.5
35.25
Incomes
80
70
60
50
40
30
20
10
0
Q1
Q2
Q3
2016
Q4
Q1
Q2
Q3
2017
Q4
Q1
Q2
Q3
2018
Q4
MULTIPLICATIVE MODEL
• Z=T * S * C * I
• Z/MA = S * I
• Calculating a moving average smoothes out the seasonality.
• In the case of smoothing the series with a moving average for monthly or quarterly data, we will get a
series containing cyclical fluctuations and a trend.
• MA = T * C
• Hence the series will be without seasonal component and irregular component.
• The seasonal component and the irregular component are smoothed out with a moving average.

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