Doc.Ing. Zlata Sojková, CSc.1 Time series (TS). doc.Ing. Zlata Sojková, CSc.2

doc.Ing. Zlata Sojková, CSc. 1

Time series(TS)


Vývoj miery nezamestnanosti SR - mesačné údaje za rr. 93-február 2000 v %

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12

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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)


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$


Interval T S (yearly data about G D P o f S R )

instantaneous T S(developm ent of population )

Absolute variables

relative indicators(rate of grow th of G D P )

average values(developm ent of average w ages )

deried variables

Tim e Series

Basic types of time series depending on the nature of data


By the length of the period we distinguish:

Long-term time series - yearly data, etc.Short-term time series - quarterly, monthly

data....etc...


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


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


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%


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%


Components of time series

Time 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…..),


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


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. 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


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.


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

, y

y S

,t

tt


Year Qartal Revenue

1 1841987 2 173

3 1604 189

1 1911988 2 185

3 1794 200

1 2051989 2 192

3 2004 229

11990 2

34

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

?


Development of revenues (obvious seasonal fluctuations)

150

200

250

1 3 5 7 9 11

Tržby v tis.Sk


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)


Regression StatisticsMultiple R 0.772R Square 0.595Adjusted R Square 0.555Standard Error 11.591Observations 12

ANOVA

df SS MS F Significance FRegression 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


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


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

Trž

by v

tis

. S

k

Real values

Forecast of trend

„ex-ante“forecast fortrend and seasonality


…

It was justbriefly

introductionin TS

analysis….. …good luck...

Documents

Doc.Ing. Zlata Sojková, CSc.1 Time series (TS). doc.Ing. Zlata Sojková, CSc.2