Time Series Analysis

Time Series Analysis

TIME – SERIES ANALYSIS :Introduction : Time – series has immense importance in the economic and business fields . Most of the economic and business data like prices , national income , population , imports , exports ,

production , consumption ,sales , profits etc. are collected on the basis of time(such as days , months , years).

Meaning : The set of data collected on the basis of time ( such as days , months , years ) is called as time – series . Under time – series , there are two types of variables (1) independent variables – which represent time and (2) dependent variables – which represent changes taking place in

the value of data (sales etc) with the passage of time .

YEAR PRODUCTION

1975 50 TONNES

1976 52 TONNES

1977 53 TONNES

1978 55 TONNES

1979 60 TONNES

TABLE – 1

YEAR PRODUCTION

1931 27.9 TONNES

1941 31.9 TONNES

1951 36.1 TONNES

1961 43.9 TONNES

1971 54.5 TONNES

TABLE – 2

Utility of time – series :• To study the past behaviour of the data : With the help of time – series

changes occurred in the past are studied .• To forecast future behaviour : With the help of time – series anticipation

of changes going to occur in the future become possible because studies about past prove to be very useful for forecasting future .

• estimation of trade-cycles : Cyclical fluctuations in a time-series give idea about the changes taking place in the business like boom , recession , depression and recovery .

• Comparison with other time-series : It is also important for the comparison of various time-series .

• Study of present variations : It is also helpful in studying the present variations in different economic variables like imports , exports etc .

• Universal utility : It also benefits all classes like businessmen , farmers , consumers , economists etc .

•

Time Series Components

Trend

Seasonal

Cyclical

Irregular

COMPONENTS OF TIME-SERIES :• (1). Secular trend T : It refers to the general tendency of the data to grow or decline over a long period of

time . It is of two types :• (A) Linear trend – when long term rise or fall in a time-series takes place by a constant amount it is called

linear trend . Y = a + bX (B) Parabolic trend – the trend is said to be parabolic when long term rise or fall in a time-series is not

taking place at a definite rate . Y = a + bX + cX

(2). Seasonal variations S: it refers to periodic variations in time-series which occur regularly within a period of 12 months .

(3). Cyclical variations C : it refers to oscillatory variations in a time-series which have a duration anywhere between 2 to 10 years

(4). Irregular variations I – it refers to htose short term variations which occur irregularly due

to certain accidental causes .

Prosperity

Recession Recovery

Cyclical variations

Depression

Trend Line

PHASES OF BUSINESS CYCLE

Trend Component

• Overall Upward or Downward Movement• Data Taken Over a Period of Years

Sales

Time

Upward trend

Cyclical Component

• Upward or Downward Swings• May Vary in Length• Usually Lasts 2 - 10 Years

Sales

Time

Cycle

Seasonal Component

• Upward or Downward Swings• Regular Patterns• Observed Within One Year

Sales

Time (Monthly or Quarterly)

Winter

Irregular Component• Erratic, unsystematic, ‘residual’

fluctuations• Due to random variation or unforeseen

events– Union strike– War

• Short duration & nonrepeating

Random or Irregular Component

• Erratic, Nonsystematic, Random, ‘Residual’ Fluctuations

• Due to Random Variations of – Nature

– Accidents

• Short Duration and Non-repeating

DECOMPOSITION OF TIME-SERIES :• Time-series is composed of four components viz. trend (T) , seasonal variaions (S) ,

cyclical variations (C) and irregular variations (I) .• Models of analysing time-series :• (1) . Additive model – this model is based on the assumption that time-series is the sum

of the four components .• O = T + S + C + I• On the basis of additive model , the anaysis of various components is give below • O – T = S + C +I• O – T – S = C + I• O – T – S – C = I (2) . Multiplicative model – This model is based on the assumption that a time-series is

the product of four components . O = T x S x C x I = TSCI It can be expressed in the following forms O/T = SCI O/TxS = CI O/TxSxC = I

SEMI-AVERAGE METHOD :• In this method , first of all time-series is divided into two equal parts and thereafter, separate arithmetic

mean is calculated for each part . It can be applied in cases of two situations :• (1) . When the number of years in a series is even – when the given number of years in a series is even like

4,6,8 etc. then the series can be easily divided into two equal parts .

2000 2001 2002 2003 2004 2005 2006 2007

412 438 444 454 470 482 490 500

YEAR :SALES (LAKHS)

YEAR SALES

2000200120022003

412438444454

2004200520062007

470482490500

SEMI-TOTAL SEMI-AVERAGE MIDDLE YEAR

1748

1942

X=1748/4=437

X=1942/4=485.5

2001.5

2005.5

(2) . When the number of years in a series is odd :• When the number of years in a series is odd like 5,7,9 etc . Then there will be a problem in

dividing the series into equal parts . In such a case , the mid-year figure is to be dropped . For eg. If 1981 to 1989 (i.e. 9 years ) figures are given , then we will delete 1985 , i.e. 5th year and its corresponding figure and we will make 4-4 years’ parts i.e. 1981-1984 and 1986-1989 . The remaining process will be the same as before .

•

1981 1982 1983 1984 1985 1986 1987

20 22 27 26 30 29 40

YEARPROFIT (000)

1981 201982 221983 271984 261985 301986 291987 40

YEAR PROFIT(000) SEMI-TOTAL SEMI-AVG MIDDLE YR

= 69

X = 69 /3 = 23

= 1982

= 99 X = 99/3 = 33 = 1986

• Merits and demerits of semi-average method :• Merits :• (1) . It is an easy method• (2) . It is free from bias • (3) . Trend values thus obtained are definite .• (4). Less time and effort is involved in drawing the trend line .• Demerits :• (1) . It is based on straight line trend assumption which does not always

hold true .• (2) . It is affected by extreme values .• (3) . It ignores the effect of cyclical fluctuations .

Moving Average Method

MOVING AVERAGE METHOD :• This method is very widely used in practice . Under this method , moving averages

are calculated . In moving averages , one has to decide what moving year average – 3 year , 4 year , 5 year , 7 year should be taken up . It is studied in two different situations :

• (1) . Odd period average method - when period of moving average is odd , then the following steps need to be taken ;

• (a) add up the values of corresponding to first 3 years in the time-series and put the sum before the middle year (i.e. 2nd year)

• (b) leaving the 1st year value , add up 2nd , 3rd and 4th year value and put the sum in front of the middle year (i.e. 3rd year) . Carry this process further till we reach the last value of the series .

• (c) moving totals thus obtained are to be divided by the period of the moving average and show the trend values of different years .

Moving Average Graph

02468

93 94 95 96 97 98Year

SalesActual

1981 1982 1983 1984 1985 1986 1987

412 438 446 454 470 483 490

YEARPRODUCTION

YEAR PRODUCTION 3 YEARLY MOVING TOTALS

3 YEARLY MOVING AVERAGE

191981 412

1982 438

1983 446

1984 454

1985 470

1986 483

1987 490

----- -----

412+438+446=1296 1296/3 = 432

438+446+454 =1338 1338/3 = 446

446+454+470 =1370 1370/3 = 457

454+470+483 =1407 1407/3 = 469

470+483+490 =1443 1443/3 = 481

----- ------

EVEN PERIOD MOVING AVERAGE :• When moving average period is even , say 4 years , then moving averages have to

be centered . It can be computed by two methods :• (a) . First method – the computation procedure of 4 yearly moving averages is as

follows :• (1) . Add up first 4 values corresponding to the first 4 years and put the sum in

between 2nd and 3rd year . The next total (i.e. from 2nd to the 5th year total) is to be put in between 34d and 4th year . Carry on this process till the last value of the series .

• (2) . Now add up the 1st and 2nd 4 year totals and put them in front of 3rd year . Similarly , add up 2nd and 3rd 4 year total and put them in front of the 4th year . Carry on this process till the last value

• (3) . 8 years’ totals thus obtained are to be divided by 8 . These values are 4 yearly moving averages and show the trend values for different years .

YEARSALES (CRORES)

1970 1971 1972 1973 1974 1975 1976 1977 1978 1979

7 8 9 11 10 12 8 6 5 10

YEAR SALES (CRORES) 4 YRLY MVNG TOTALS

2 PERIOD MVNG TTLS OF 4 YRLY

4 YRLY MVNG AVGS CENTERED

1970 7 ----- ------ ------1971 8 ----- ------ ------

7+8+9+11=351972 9 35+38=73 73/8 = 9.125

8+9+11+10=381973 11 38+42=80 80/8 = 10

9+11+10+12=421974 10 42+41=83 83/8 = 10.375

11+10+12+8=41

1975 12 41+36=77 77/8 = 9.625

10+12+8+6=36

1976 8 36+31=67 67/8 = 8.375

12+8+6+5=31

1977 6 31+29=60 60/8 = 7.5

8+6+5+10=29

1978 5 ----- -------- --------

1979 101979 10 ----- -------- --------

Freehand Curve Method

Freehand Curve [Graph]

02468

93 96 97 98 99

Year

ProductionActual

METHODS OF MEASURING TREND : (1) . Free-hand curve method – this is the simplest method of trend fitting . This method is also called trend

fitting by inspection .

Merits and demerits of freehand curve method are : Merits : (a). It is simple. (b). It is flexible. (c) . No mathematical formula is used in this method . (d) . It is also used for forecasting about future. Demerits : (a) . It is based on subjective judgements . So bias may affect its findings . (b) . There is lack of accuracy. (c) . Long-run movement obtained from this method is not definite.

1971 1972 1973 1974 1975 1976 1977 1978 1979 1980

150 155 165 152 174 150 174 175 160 180

YEAR :PRODUCTION

Least Square Method :

LEAST SQUARE METHOD• This is the best method of trend fitting in a time series and is

most used in practice. This is a mathematical method and a trend line in this method is fitted or obtained in such a way that the following two conditions are fulfilled:

1. (y- )=0, i.e., the sum of the deviations of the actual values of Y and computed trend values ( ) is zero.

2. (y- is least, i.e. the sum of square of deviations of the actual and computed trend values from this line is least.

The trend line thus fitted under this method is called as the line of best fit.

(A)FITTING OF STRAIGHT LINE TREND

• A straight line trend can be expressed by the following equation

Y=a+bXWhere Y= trend values X=unit of time a is the Y-intercept and b is the slope of line Y= Na+ b x XY= a X + b

• There are two methods of computing Straight line trend by using Least square method:1. Direct method: The procedure to compute straight line trend in this method is as follows- (i) Any year is taken as the year of origin. Usually first year or before that is taken as zero, deviations of

other years are marked on 1,2,3…etc. Time deviations are denoted by X.

(ii) Then X, Y, XY and XY and are computed. (iii) the values computed are put in the following normal equations: Y= Na+b X XY- a X+b

The values of A and B are determined by solving the above set two normal equations

(iv) Finally, the calculated values of a and b are put in Y = a + bX and trend values are computed.

Example 1

Fit a straight line trend by the method of least square to following data:

Also obtain the trend values.

1979 1980 1982 1983 1984 1985

5 7 9 10 12 17

YEARPRODUCTION LAKH TONS

Solution :YEAR PRODUCTION

LAKH TONS (Y)DEVIATIONS FROM 1978 (X)

XY

1979 5 1 5 1

1980 7 2 14 4

1981 9 3 27 9

1982 10 4 40 16

1983 12 5 60 25

1984 17 6 102 36

N = 6 Y = 60 X = 21 XY = 248 = 91

The straight line trend is ddefined by the equationY= a+bXTwo normal equations are Y= Na+b X XY- a X+b Substituting the vales we get60=6a+21b ..(i)248=21a+91b ..(ii)Solving the two equations (i) and (ii)Multiplying (i) by 7 and (ii) by 2 and subtracting420= 42a+147b496= 42a+182b- - - .-76= -35b

b=-76/-35=2.17

By stubstituting the values of b60=6a+21(2.17)6a=14.43a=2.40Hence, the trend equation isY= 2.40+2.17X; Origin= 1978,X unit= 1 year.Computing he trend valuesFor 1979, X= 1,Y= 2.40+2.17(1)= 4.57For 1980, X= 2,Y= 2.40+2.17(2)= 6.74For 1981, X= 3,Y= 2.40+2.17(3)= 8.91For 1982, X= 4,Y= 2.40+2.17(4)= 11.08For 1983, X= 5,Y= 2.40+2.17(5)= 13.25For 1984, X= 6,Y= 2.40+2.17(6)= 15.42

(B) Fitting Of Second Degree Parabolic Trend Or Quadratic Trend

There may be many such situations in economics and business fields in which a straight line trend may not represent the long-term tendency of the time series data. In such cases, a second degree parabolic trend or quadratic trend is fitted. The equation is:

Y = a + b X + cX2

Where, a is the Y- intercept, b is the slope of the curve at the origin and c is the rate of change in slope.Under the method of least square the values of the constants a, b and c are obtained by solving the following three normal equations:

Y = N a + b X + c X2

XY = a X + b X2 + c X3

X2 Y = a X2 + b X3 + c X4

SHORTCUT METHOD

Short-cut Method:If the time deviations are taken from the middle year (or

arithmetic mean of two middle years), the values ofX and X3 would be zero.

TYPE 1: ODD Number of YearsEXAMPLE:- Fit a second degree parabolic trend to the

following data.Year Production

(Y)Deviations from 1985

(X)

XY X2 Y X2 X3 X4

1983 5 -2 -10 20 4 -8 16

1984 7 -1 -7 7 1 -1 1

1985 4 0 0 0 0 0 0

1986 9 1 9 9 1 1 1

1987 12 2 24 48 4 8 16

N = 5 27 0 16 84 10 0 34

The second degree parabolic trend is given by the equationY = a + b X + c X2

The three normal equations are:Y = N a + b X + c X2

XY = a X + b X2 + c X3

X2 Y = a X2 + b X3 + c X4

Substituting the values, we get37 = 5a + b(0) + c(10) 37 = 5a + 10c …(i)16 = a(0) + b(10) + c(0) 16 = 10b …(ii)84 = a(10)+ b(0) + c(34) 84 = 10a + 34c ...(iii)

Solving (i), (ii) & (iii), we geta = 5.98 = 6 (approx.)b = 1.6c = 0.71Thus, the second degree parabolic trend is Y = 6 + 1.6X + 0.71 X2, Origin = 1985, X unit = 1 yearComputation of Trend ValuesFor 1983, X = -2, Y = 6 + 1.6(-2) + 0.7(-2)2 = 5.64For 1984, X = -1, Y = 6 + 1.6(-1) + 0.7(-1)2 = 5.11For 1985, X = 0, Y = 6 + 1.6(0) + 0.7(0)2 = 6For 1986, X = +1, Y = 6 + 1.6(+1) + 0.7(+1)2 = 8.31For 1987, X = +2, Y = 6 + 1.6(+2) + 0.7(+2)2 = 12.04

TYPE 2 : EVEN Number of YearsEXAMPLE:- Fit a second degree parabola to the following

data.

Year Price (Y)

Deviations from

1966.5

Deviations multiplied by 2 (X)

XY X2Y X2 X3 X4

1994 100 -2.5 -5 -500 2500 25 -125 625

1995 170 -1.5 -3 -321 963 9 -27 81

1996 128 -0.5 -1 -128 128 1 -1 1

1997 140 +0.5 +1 +140 140 1 +1 1

1998 181 +1.5 +3 +543 1629 0 +29 81

1999 192 +2.5 +5 +960 4800 25 +125 625

N = 6 848 0 694 10160 70 0 1414

Substituting the values in the above mentioned three normal equations, we get

848 = 6a + b(0) + 70c 848 = 6a + 70c …(i)694 = a(0) + b(70) + c(0) 694 = 70b …(ii)10160 = 70a + b(0) + 1414c 10160 = 70a + 1414c …(iii)Solving (i), (ii) & (iii), we geta = 136.125b = 9.914c = 0.4464Thus, second degree parabola is

Y= 136.125 + 9.914X + 0.4464X2,Origin:1996.5, X unit= ½ year

(B) Fitting Of Exponential Trend

If the time series is increasing or decreasing by a constant percentage rather than constant absolute amount, the fitting of exponential trend is considered appropriate. Such tendency is found in many economic and business data. The equation is

Y = abX

Where a is Y- intercept and b is the slope of the curve at the origin of X.In the logarithmic form, the above equation is written as under:log Y = log a + X log bWhen plotted on a semi logarithmic graph, the curve gives a straight line (or called logarithmic straight line). However, on an arithmetic scale chart the curve gives a non-linear trend.Under the method of least square, the values of the constants a & b are obtained by solving the following two normal equations:

log Y = N log a + log b X(X log Y) = log a X + log b X2

The equation of the exponential trend isY = abX ….(i)

Taking log both sides, we havelog Y = log a + X log b .…(ii)

Since, X = 0. Therefore,log a = 1.2341 & log b = 0.075

Thus, the exponential trend equation in log form is:log Y = 1.2341 + 0.075X

Here, Origin = 1993, X unit = 1 year

TYPE 2: EVEN number of yearsEXAMPLE: Fit an exponential trend to the following data.

Year Population (X)

X log Y X log Y X2

1941 31.9 -5 1.5038 -7.5190 25

1951 36.1 -3 1.5575 -4.6725 9

1961 43.9 -1 1.6425 -1.6425 1

1971 54.8 +1 1.7388 1.7388 1

1981 68.3 +3 1.8344 5.5032 9

1991 84.4 +5 1.9263 9.6315 25

N = 6 0 10.2033 3.0395 70

The equation of the exponential trend isY = abX ….(i)

In log form the equation is written aslog Y = log a + X log b ….(ii)

Since, X = 0. Thereforelog a = 1.70 & log b = 0.043

Thus, the exponential trend equation in log form is:log Y = 1.70 + 0.043X

Here, Origin = 1996, X unit = 5 years

Merits and Demerits of Least Square Method:

MERITS:

(i) This method is far better than moving average method because the trend values for all the years are obtained. Not even a single initial or terminal trend values is left over in this method.

(ii) It results in a mathematical equation which may be used for forecasting.

(iii) It is widely used method of fitting a curve to the given data. The results obtained are reliable and appropriate.

DEMERITS:

(i) The computation process in this method is complex which is not easily understandable.

(ii) This method does not have the attribute of flexibility. If some figures are added to or subtracted from the original data, all computations have to be redone.

(iii) It is difficult to select an appropriate type of equation in this method. Results based on inappropriate selection of equation are likely to be misleading.