time series ( ts )
DESCRIPTION
Time series ( TS ). What is the time serie ??. data about the socio - economic phenomenon - in chronological order in time properly assembled time series data must meet the comparability of data: in time (the same length of periods) in space (the same territorial units, regions) - PowerPoint PPT PresentationTRANSCRIPT
doc.Ing. Zlata Sojková, CSc. 1
Time series(TS)
doc.Ing. Zlata Sojková, CSc. 2
Vývoj miery nezamestnanosti SR - mesačné údaje za rr. 93-február 2000 v %
10
12
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22
doc.Ing. Zlata Sojková, CSc. 3
What is the time serie??
data about the socio - economic phenomenon - in chronological order in time
properly assembled time series data mustmeet the comparability of data:
in time (the same length of periods)in space (the same territorial units, regions)and substantive comparability (same
methodology)
doc.Ing. Zlata Sojková, CSc. 4
Denote the value of the investigated variable:y1, y2 , y3 , ... yt …… yT, where t = 1, 2, …. T, T is the number of periods, t is formal time variable, which specifies the order of
the value , e.g.
Rok Yt t1995 3110 11996 3570 21997 3860 31998 3870 41999 3770 5
GDP SR per capita. Years 95-99 v US$
doc.Ing. Zlata Sojková, CSc. 5
Interval T S (yearly data about G DP o f S R )
instantaneous T S(developm ent of popu la tion )
Absolute variables
relative indicators(ra te of g row th of G D P )
average values(developm ent o f average wages )
deried variables
Tim e Series
Basic types of time series depending on the nature of data
doc.Ing. Zlata Sojková, CSc. 6
By the length of the period we distinguish:
Long-term time series - yearly data, etc.Short-term time series - quarterly, monthly
data....etc...
doc.Ing. Zlata Sojková, CSc. 7
Basic characteristics of time series analysis
Absolute rate of growth (decline): absolute increase (decline) – first
differences y t = y t - y t -1
second first differences (acceleration) y t
2 = y t - y t -1
doc.Ing. Zlata Sojková, CSc. 8
Relative rate of growth coefficients of growth: k t = y t / y t - 1
coefficient of increase: k t = k t - 1
growth rate (growth coef .in % ): Tt = k t . 100
Increase rate: T t
=Tt - 100, resp. T t = (k t - 1 ) . 100
doc.Ing. Zlata Sojková, CSc. 9
GDP SR for years rr.95-99 in US$ per capita and year.
Rok GNPSR (US$) coefficientcoefficient growth increase1995 3110 of growth of increse in % rate rate in %1996 3570 1,148 114,79 0,15 14,791997 3860 1,081 108,12 0,08 8,121998 3870 1,003 100,26 0,00 0,261999 3770 0,974 97,42 -0,03 -2,58
In 1997 compared to 96 increased GDP per capita on 108,12%
In 1997 compared to 96 increased GDP per capita by 8,12%
doc.Ing. Zlata Sojková, CSc. 10
From individual growth rates can be calculated:
average growth rate
1-T 2 1-T
1 ...k . k .k k _ 4
k = (1,148.1,081. 1,003 . 0,974) = 1.0493
For the period 95-99 was growth of GDP in SR per year approximately 4,9%
doc.Ing. Zlata Sojková, CSc. 11
Components of time seriesTime series are created as the effect of important and not
important factors on investigated phenomenon. These factors can be divided:
Trend – determine the main direction of development t.j. trend in TS (Tt )
Periodic – cause regular fluctuations around the trend values of TS, can be divided: Cyclic (C t )- in long-term TS (economic cycles) Seasonal (S t )- in short-term TS (seasonal fluctuations
of prices, seasonal demand…..),
doc.Ing. Zlata Sojková, CSc. 12
Random effects (E t ) – random, iregular. These factors affect the development of each investigated variable in statistics.
=> We can decompose TS into three components: Trend (Tt ) Periodic (C t ), resp. (S t ) Random (E t )
Between components may be:Adittive relationship: Yt = T t + St + E t
Multiplicative relationship: Yt = T t . St . Et
doc.Ing. Zlata Sojková, CSc. 13
Analysis of trend and seasonal component (if occurs in TS)Using standard decomposition approach
Analysis of trend in TS Analysis of trend using decomposition approach is based on: analytical smoothing of investigated values by appropriate
trend function.analogy to the simple regression analysis, the estimated
values are a function of time variable t, yt
, = f (t) „trend function“ is then used not only to evaluating the
quality of forecasts "ex post", but also to forecast ex-ante
doc.Ing. Zlata Sojková, CSc. 14
doc.Ing. Zlata Sojková,CSc. 21
Some types of simple trend functions
t21o
't
1bo
't
t1o
't
221o
't
1o't
1o't
b . bb y
t .b y
b .b y
t.bt . bb y
tlog . bb y
t/bb y
Historical data „Ex ante“ prognosis
doc.Ing. Zlata Sojková, CSc. 15
Statistical evaluation of appropriatness of function
y var.celk. TF l.var.vysvet
)yy(
)y'y( i
t2T
1tt
2T
1tt
yt
• using index of correlation i yt , resp. • index of determination iyt
2
Which reflects the quality of “ex-post” forecast
•Priority is assesment of the fittnes of trend function. It is necessary to consider future development of examined variable y.
doc.Ing. Zlata Sojková, CSc. 16
Analysis of seasonal component in TSDecomposition approach It is assumed:Multiplicative model of TS: Yt = Tt . St . Et
Analysis of trend in TS (if present) by suitable trend function: Tt = yt
, = f(t)Analysis of seasonal component using seasonal
indices:
where y t , are values obtained from fitted trend
function for t = 1,2…T
, yy S ,
t
tt
doc.Ing. Zlata Sojková, CSc. 17
Year Qartal Revenue1 184
1987 2 1733 1604 1891 191
1988 2 1853 1794 2001 205
1989 2 1923 2004 2291
1990 234
In table are data about Revenue of chosen company for 3 years. Analyse development of revenues in recent years and make forecast for year 1990.
How to makeForecastfor 1999
Yt = Tt . St . Et
Tt = yt, = f(t)
We create variable t = 1,2,…,12
?
doc.Ing. Zlata Sojková, CSc. 18
Development of revenues (obvious seasonal fluctuations)
150
200
250
1 3 5 7 9 11
Tržby v tis.Sk
doc.Ing. Zlata Sojková, CSc. 19
Procedure for analysis and prediction:First, we analyze the trend using suitable trend function. From
the chart can be concluded that linear function is sufficient. We will use Excel (Tools- data Analysis -Regression)We calculate „predicted“ values according to trend function
(also for year 1990)Seasonal Indices S t are calculated by dividing real value of
revenues y t by value y t ‘ predicted according to trend
functionWe make average of seasonal indices (to objectify seasonal
component) and correct them for the sum of 4 (correction for accurancy)
doc.Ing. Zlata Sojková, CSc. 20
Regression StatisticsMultiple R 0.772R Square 0.595Adjusted R Square 0.555Standard Error 11.591Observations 12
ANOVAdf SS MS F Significance F
Regression 1 1975.470 1975.47 14.7045 0.00329Residual 10 1343.446 134.34Total 11 3318.917
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 166.424 7.134 23.3297 0.0000 150.530 182.319X Variable 1 3.717 0.969 3.8346 0.0033 1.557 5.876
Aproximately 60% of revenue variability is explained by trend, rest 40% is the variability caused by seasonal and random fluctuations
We use trend function coefficients for „ex-post“ and „ex-ante“ forecast of trend
Result of trend analysis
doc.Ing. Zlata Sojková, CSc. 21
Rok Kvartál Tržby v tis.Sk t yt´ St = yt / yt' St priemerne St pr. korig y*1 184 1 170.141 1.0814558 1.046494 1.046400 178.036
1987 2 173 2 173.858 0.995066 0.972797 0.972709 169.1133 160 3 177.575 0.9010298 0.931975 0.931891 165.484 189 4 181.291 1.0425206 1.049093 1.048999 190.1741 191 5 185.008 1.0323869 Suma St priem. 193.593
1988 2 185 6 188.725 0.9802626 4.000359724 4 183.5753 179 7 192.442 0.9301517 Korekcny faktor: 179.3354 200 8 196.159 1.0195836 0.999910077 205.771 205 9 199.875 1.0256395 209.15
1989 2 192 10 203.592 0.9430623 198.0363 200 11 207.309 0.9647441 193.1894 229 12 211.026 1.0851762 221.3661 13 214.742 224.706
1990 2 14 218.459 212.4973 15 222.176 207.0444 16 225.893 236.961
ForecastOf trend
ForecastY t ‘ . St priem.
Predicted values of trend
Analysis of seasonality and forecastSeasonalindices
Forecast for trend and seasonality
doc.Ing. Zlata Sojková, CSc. 22
Vývoj tržieb v rr. 87 89 a prognóza na r.90
150
170
190
210
230
250
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4kvartály
Tržb
y v
tis. S
k
Real values
Forecast of trend
„ex-ante“forecast fortrend and seasonality
doc.Ing. Zlata Sojková, CSc. 23
…
It was justbriefly
introductionin TS
analysis….. …good luck...