Time Series

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Time-Series Analysis and
Forecasting – Part III
To read at home
Time-Series Data
 Numerical data ordered over time
 The time intervals can be annually, quarterly,
daily, hourly, etc.
 The sequence of the observations is important
Example 13
Year:
2005 2006 2007 2008 2009
Sales:
75.3 74.2 78.5 79.7 80.2
Time-Series Plot
A time-series plot is a two-dimensional
plot of time series data
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1975
 the horizontal axis
corresponds to the
time periods
U.S. Inflation Rate
Inflation Rate (%)
 the vertical axis
measures the variable
of interest
Year
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-3
The problem of comparability of
levels of time series
Jointing (смыкание) of time
series
Since time series is formed during
the long period of time, its levels are
frequently incomparable
Reasons of the incomparability
1. Change of prices.
2. Different methods of calculation of
the same indicator.
3. Change of «borders» (organizational,
administrative)
The method of jointing time series is often used
to ensure the comparability of data. It is
necessary to have a transitional link
(переходное звено) for jointing time series.
Transitional link – is the period of time, for
which the investigated indicator was calculated
using the old method (in old borders) and the
new method (in new borders). A transitional
coefficient for this transitional is calculated, the
transitional coefficient spreads over all the
previous period of time
Example 14
Production of oil, mln t
2005
2006
2007
2008
2009
Before merger
6600
6700
6900
-
-
After merger
-
-
7500
7800
7900
Transitional coefficient
7500
К
 1,087
6900
y06  6700 1,087  7283
y05  6600 1,087  7174
Production of oil, mln t
2005
2006
2007
2008
2009
Before merger
6600
6700
6900
-
-
After merger
-
-
7500
7800
7900
Comparable series
7174
7283
7500
7800
7900
Analysis of the main tendency
of time series
The levels of time series are formed
under the influence of lots of
factors. They can be divided into 5
groups
Time-Series Components
Time Series
Trend
Component
Secular
Component
Seasonality
Component
Cyclical
Component
Irregular
Component
1. Determining (Определяющие)
factors have a constant and strong
influence on the examined indicator.
They determine the main tendency (the
trend) of time series
The trend component
Trend Component
 Long-run increase or decrease over time
(overall upward or downward movement)
 Data taken over a long period of time
Sales
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Time
Chap 19-17
Trend Component
(continued)
 Trend can be upward or downward
 Trend can be linear or non-linear
Sales
Sales
Time
Downward linear trend
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Time
Upward nonlinear trend
Chap 19-18
Real fluctuations around the trend
There are 4 kinds of charge or variations involved in time
series analysis. They are:
2.Secular trend Ut. In the secular trends the value of variable
tends to increase or decrease over a long period of time. The
steady increase of the cost of living recorded by the
Сonsumer price index is an example of secular trend. From
year to year, the cost of living varies a great deal, but if we
examine long-period, we see that the trend is toward a steady
increase.
3. Cyclical fluctuation Vt. The most common example of
cyclical fluctuation is the business cycle. Over time, there
are years when the business cycle hits a peak above the
trend line. At other times, business activity is likely to slump,
hitting a low point below the trend line. The time between
hitting peaks and falling to low points is at least one year
and it can be as many as 15 or 20 years.
Cyclical Component
 Long-term wave-like patterns
 Regularly occur but may vary in length
 Often measured peak to peak or trough to
trough
1 Cycle
Sales
Year
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-22
4. Seasonal variation St involves pattern of change
within year that tend to be repeated from year to
year. For example the consumption of drinks,
juices, ice cream and other. Seasonal factors give
rise to oscillations relative to the main tendency
Seasonal Component
 Short-term regular wave-like patterns
 Observed within 1 year
 Often monthly or quarterly
Sales
Summer
Winter
Summer
Spring
Winter
Spring
Fall
Fall
Time (Quarterly)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-24
variation εt. The value of variable may be
completely unpredictable changing in random
manner. For example, the Iraqi situation in 1990, the
ruble devaluation in 1998 and the others. Random
factors cause the random fluctuations of levels of
series (for example, weather factor)
5. Irregular
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-25
5. Irregular variation εt. The value of variable may be completely
unpredictable changing in random manner. For example, the Iraqi
situation in 1990, the ruble devaluation in 1998 and the others.
Thus each value of time series could be presented as follows:
yt  Ut  Vt  St   t.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-26
Irregular Component
 Unpredictable, random, “residual” fluctuations
 Due to random variations of
 Nature
 Accidents or unusual events
 “Noise” in the time series
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-27
Time-Series Component Analysis
 Used primarily for forecasting
 Observed value in time series is the sum or product of
components
 Additive Model
Xt  Tt  St  Ct It
 Multiplicative model (linear in log form)
Xt  TtStCtIt
where
Tt = Trend value at period t
St = Seasonality value for period t
Ct = Cyclical value at time t
It = Irregular (random) value for period t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-28
Smoothing the Time Series
 Calculate moving averages to get an overall
impression of the pattern of movement over
time
 This smooths out the irregular component
Moving Average: averages of a designated
number of consecutive
time series values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-29
Method of interval
enlargement
Method of interval enlargement
consists in replacement of initial
levels of series by the average
values, which are calculated for
the enlarged intervals
Example 15
Month
yt
1
5.1
2
5.4
3
5.2
4
5.3
5
5.6
6
5.8
7
5.6
8
5.9
9
6.1
10
6.0
11
5.9
12
6.2
Quarterly sums
Average
monthly value
(per quarter)
15.7
5.23
16.7
5.57
17.6
5.87
18.1
6.03
The End of Part III
 To be continued
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-33
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