Time Series

Analysis: 

Importance of time series:

1.     Analysis of causes and conditions prevailing during occurrence of past changes, one can easily determine the future policies and programs

2.     Estimation of future trends on the basis of analysis or past trends.

3.     Trends of trade cycles are studied and their effect can be reduced to a considerable extent.

4.     Comparative study with the other time series.

 

B.) Cyclical Fluctuations: The fluctuations or changes occurring in economic activities are known as cyclical fluctuations. Although they are also regular in nature but period of reoccurrence is more than a year. There are four stages in cyclical fluctuations:- Boom, Recession, Depression, Recovery.

Irregular or random fluctuations: When changes in time series occur due to some unforeseen causes then it is called irregular or random fluctuations. Prediction of irregular fluctuations is very difficult as they occur accidentally. They are of two types:

A)  Episodic Movement: It occurs due o some unforeseen situation for ex war, flood drought, earthquake etc.

B)   Accidental Movements: Accidental movements are of random nature.

Time Series decomposition model• Analysis include two steps:– Identify factors which influence the variations in

the series.– Isolating, analyzing & measuring the effect of

these factors independently.

• Purpose of decomposition is to break a series into components:– trend value (T), seasonal variations (S), cyclical

fluctuations (C) and irregular fluctuations(I).

• Two models are:– Multiplicative Model.– Additive Model.

Multiplicative Model• Y= T*C*S*I• This model is effective where the effect of

C,S and I is measured in relative sense instead of absolute sense.

• All variables are interdependent.• The geometric mean is less than 1.• EX: Sales = 423.36, Mean is 400, Current

cycle .90 and seasonality is 1.20, Random fluctuation absent.

• Expected value of sales = 400*.90*1.20=432.

• Id the random factor decreases sale by 2% then actual sales will be 432*.98= 423.36

Additive ModelY= T+C+S+IT, C, S, I are absolute quantities and can

have positive or negative values.It is assumed that all four components are

independent

ANALYSIS OF TIME SERIES:

 Study of components of time series is known as analysis of time series. The original data (O) is a combination of, trend value (T), seasonal variations (S), cyclical fluctuations (C) and irregular fluctuations(I). Analysis of all these components of time series separately is known as analysis of time series.

Measures Of Secular Trend:

  1.     Freehand Curve Method

2. Smoothing Methods:

Semi Average Method

Moving Average Method

4.     Trend Projection Methods

Least Square Method

FREEHAND CURVE METHOD:

It is the most simplest method according to this method firstly time series is plotted on the graph paper, keeping in view the direction of fluctuation of the time series straight line or curve is drawn passing through the midpoints. The line or curve represents the secular trend.

Ex 1. Find out the secular trend by freehand curve method:

 Yr 1985 86 87 88 89 90

Prod (lacs) 15 23 19 27 24 30

YEARS

P

R

OD

U

C

T

I

O

N

SEMI-AVERAGE METHOD: Acc to this method the original data is divided into two parts. Then the mean of both the parts is calculated separately.

 Ex2. Determine the trend by applying the semi-average method. Yr. 1990 91 92 93 94 95Sales 50 58 54 71 66 74

Moving Average Method:

This is the simple and most widely used method for the calculation of trend values. In this firstly we have to decide what will be the period of moving average? The period can be odd i.e 3, 5, 7, 9, 11 or even i.e 2, 4, 6, 8, 10 years

Years Production/Sales. 3 yearly moving total

3 yearly moving average

1

2

3

4

5

6

7

8

X1

X2

X3

X4

X5

X6

X7

X8

X1+X2+X3

X2+X3+X4

X3+X4+X5

.

.

.

.

X1+X2+X3/3

X2+X3+X4/3

X3+X4+X5/3

.

.

.

.

Table Format

Ex 4 Find the secular trend by 3 yearly moving average. Yr Sales (in lacs)2001 452002 402003 482004 622005 542006 752007 502008 82

4 Yearly Moving Average Table Format

Year Production/sales

4 Yearly Moving Total Un centered

4 Yearly Moving Average Un centered

4 Yearly Centered Moving Average

(Trend)

1

2

3

4

5

6

X1

X2

X3

X4

X5

X6

X1+X2+X3+X4

X2+X3+X4+X5

X3+X4+X5+X6

A=X1+X2+X3+X4/4

B=X2+X3+X4+X5/4

C=X3+X4+X5+X6/4

A+B/2

B+C/2

Ex5: Find the secular trend by 4yearly moving average

  Yr Value

  2001 506

            2002 620

            2003 1036

2004 673

            2005 588

            2006 696

2007   1116

           2008 738

           2009 663

Trend Projection MethodsIt is best represented by straight line is termed as long run directions(upward, downward, constant), of any business activity over a period of several years.

Reasons to study trend:Helps in describing long term general

directions of any business activity over a long period of time.

It helps in making intermediate and long term forecasting projections.

Least Square Method:

It is calculated to be the best method for calculating the trend values. The trend line obtained by this methods is called line of best fit. This line can be a straight line or parabolic curve.

Least Square Method: This method can be used in both cases when no. of years are odd as well as even.

 Y = N a + b X

XY = a X + b X2

 Function Equation Y = a + b x

Year Prod /Sales

X X2 XY Trend Values

Yc = a+bX

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Y X X2 XY Yc

Table format of least square method

Ex 6 Fit the straight line trend by least square method:

Yr Production(lacs)

2002 5

2003 7

2004 9

2005 10

2006 12

2007 17

Ex7 Fit the straight-line trend by least square method. Also predict the sales for year 2010

  Yr Sales (000)

2003 80

2004 90

2005 92

2006 83

2007 94

2008 99

2009 92

Fit the straight-line trend by least square method by assuming 2005 as the base year . Also predict the sales for year 2010

Yr Value

  2001 506

            2002 620

            2003 1036

2004 673

            2005 588

            2006 696

2007   1116

           2008 738

           2009 663