Time Series Analysis_Economics

Time Series AnalysisIf the data are arranged according to time order (chronological order) then it is known as the Time Series.

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Year

Pro

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n

Components of time series

• Long term trend / Secular trend (T)

• Cyclic Variation (C )

• Seasonal Variation (S)

• Random Variation (R)

Long term trend

• The upward or downward pattern of

movement of the data is known as trend.

• It’s duration is more than one year may be several years.

• Trend presents in the data due to the

changes in technology, population, wealth

value, effect of competition etc.

• Systematic in nature.

Seasonal variation

• Short term regular periodic variation,

• Occur within short period of time (yearly, quarterly, monthly, weekly or daily data).

• More or less systematic.

• Occur due to seasons, weather, festivals, social customs, religions, culture, etc

Cyclic Variation• Regular patterns that repeat over a long period

of time.• Movement are cyclical.• Occur more than one year.• Four phases exist such as; prosperity (growth),

recession (contraction) , depression (trough) and recovery (expansion)

• Due to the combination of numerous factors, however, economic booms and depression are the major causes.

Random (Irregular) Fluctuation

• Variations due to accidents, random or simply due to the chance factors.

• Unsystematic in nature

• Occur due to famines, strikes, war, political situations etc.

• Almost impossible to measure or isolate the value.

• Short duration and non-repetitive in nature.

Analysis of Time Series

Case I: If the data are given in the form of more than one year then

• Additive model

• Multiplicative model

Where Yi = Value at time i.

Ti =Value of the trend component

Ci = Value of cyclic component

Ii = Value of the irregular component

iiii ICTy

iiii ICTy **

Case II: If the data are given in the form of less than one year then the

• Additive model

• Multiplicative model

Where Yi = Value at time i.

Ti =Value of the trend component

Ci = Value of cyclic component

Si = Value of seasonal component

Ii = Value of the irregular component

iiiii ISCTy

iiiii ISCTy ***

Smoothing in Time Series

Smoothing is a procedure to remove the

effect of several components (such as

seasonal and irregular) associated with time

series data. In smoothing process, therefore,

we attempt to remove the effect of irregular

Components of the time series.

Smoothing the annual time series

• First plot the given annual data and examine the tendency over the time. If the time series are fairly stable with no significant trend, cyclic and seasonal effect.

• There may be the two cases:– Data series may move long term upward or

downward movement

– Data may oscillate about the horizontal line over the time period.

In such situation, the best way of removing the

random fluctuation is to smooth the time

series are MOVING AVERAGES and

EXPONENTIAL SMOOTHING

Moving Average MethodConcept: Any large irregular component of

time series at any point of time will have a less significant effect on trend if the observation at that point of time is averaged with such values immediately after and before the observation under the consideration.

Procedure:

• Determine the moving average period (such as 3-yearly, 5-yearly, 7-yearly, etc)

• Obtain the total of moving values and calculate the average of the total of moving values.

• Place the moving average at the middle value of the time series.

• Continue the process until all the moving average is not computed.

Note:

• Remember that one value at the beginning and the last of the data will not computed in case of 3-yearly moving averages.

Compute the 3-yearly and 7-yearly moving average from the following information

Year Sales Year Sales

1970 5.3 1982 6.2

1971 7.8 1983 7.8

1972 7.8 1984 8.3

1973 8.7 1985 9.3

1974 6.7 1986 8.6

1975 6.6 1987 7.8

1976 8.6 1988 8.1

1977 9.1 1989 7.9

1978 9.5 1990 7.5

1979 9 1991 7

1980 7.1 1992 7.2

1981 6.8    

Year Sales3yearly Total

3- yearly MA

7early Total

7-yearly MA

1970 5.3 - - - -

1971 7.8 20.9 7 - -

1972 7.8 24.3 8.1 - -

1973 8.7 23.2 7.7 51.5 7.4

1974 6.7 22 7.3 55.3 7.9

1975 6.6 21.9 7.3 57 8.1

1976 8.6 24.3 8.1 58.2 8.3

1977 9.1 27.2 9.1 56.6 8.1

1978 9.5 27.6 9.2 56.7 8.1

1979 9 25.6 8.5 50.1 7.2

1980 7.1 22.9 7.6 41.5 5.9

1981 6.8 13.9 4.6 32.4 4.6

1982 6.2 14 4.7 31.6 4.5

1983 7.8 22.3 7.4 40.2 5.7

1984 8.3 25.4 8.5 48 6.9

1985 9.3 26.2 8.7 56.1 8.0

1986 8.6 25.7 8.6 57.8 8.3

1987 7.8 24.5 8.2 57.5 8.2

1988 8.1 23.8 7.9 56.2 8.0

1989 7.9 23.5 7.8 54.1 7.7

1990 7.5 22.4 7.5  - - 

1991 7 21.7 7.2  -  -

1992 7.2  -  -  -  -

0

12

34

5

67

89

10

Sales

average

7y-av

Decision: It is obvious that 7-yearly moving average fits the data well

than 3-years moving average.

Limitations of Moving Averages

• It avoids the values for the first and last years of the data.

• It always provides the same weights for all the observations irrespective of the number of the time periods taken into consideration

Exponential smoothing

Exponential smoothing is just the modified form of moving average in which it assigns more weights for the time series data, which are more important.

For example, it is logical to assign more weight for the most recent data as compared with too old data for the future prediction.

Therefore, exponential smoothing uses the moving average with appropriate weights assigned to the values taken into consideration in order to arrive at a more accurate forecast.

Mathematically, the form of exponential smoothing is

S1 =Y1

S2 = y1 +(1- )S1

In general, we haveSt = y1 +(1- )St-1

Where, St = exponential smoothed time series.

Yi = time series at time t. St-1 = exponentially smoothed time series at time t-1. Alpha ( )= smoothing constant

The value of this constant is decided by the

decision maker on the basis of degree of

smoothing required. Smaller value of alpha

means a greater degree of smoothing and

large value of alpha means very little

smoothing.

If =1, there is no smoothing at all.

Example: Following information gives the sales of

petrol in 16 months. Apply the exponential smoothing

technique when alpha is 0.2 and 0.7.

Month Sales Month Sales

1 39 9 41

2 37 10 69

3 61 11 49

4 58 12 66

5 18 13 54

6 56 14 42

7 82 15 90

8 27 16 66

month sales

Smoothing when alpha = 0.2

Smoothing when alpha = 0.7

1 39 39 39

2 37 38.6 37.6

3 61 43.1 54.0

4 58 46.1 56.8

5 18 40.5 29.6

6 56 43.6 48.1

7 82 51.2 71.8

8 27 46.4 40.4

9 41 45.3 40.8

10 69 50.1 60.6

11 49 49.8 52.5

Solution:

Contd…

12 66 53.1 61.9

13 54 53.3 56.4

14 42 51.0 46.3

15 90 58.8 76.9

16 66 60.2 69.3

0

20

40

60

80

100 sales

Smoothing w henalpha = 0.2Smoothing w henalpha = 0.7

Decision: From the graph, It is obvious that that smoothing is less for alpha=0.7 , while smoothing is too much when alpha = 0.2. Therefore, alternative value of Alpha may have better explanation.

Analysis of TrendTrend is one of the important factors of

analyzing time series analysis.

• This is particularly important because it is

used as a forecasting model and it has

various advantages over the other

components.

• In practice, we use two types of trend

equations: linear and curvilinear model to

study the trend.

Least Square fitting of Linear Trend

As in regression analysis, we can estimate the

linear trend equation by using principle of least

square. Let the linear equation be

And its’ estimated or fitted equation is

The two constants and are estimated

Coefficients.

ii xy 10

ixbby 10ˆ 0b 1b

Where

Y = Actual value of the time series in period time t.

n= number of the periods.

= Average value of the time series.

= Average value of the time (xi)

y

x

As mentioned previously,

Then the intercept can be obtained as

221 )( xxn

yxxynb

xbyb 10

Example: A car fleet owner have been in the

fleet for several different years. The manager

wants to establish if there is linear relationship

between the age of car and repair in hundred of

dollar for a given year. The following is the

information provided.

Obtain the repair cost for the 6 years old car.

Car Age (x) Repair (y)

1 1 4

2 3 6

3 3 7

4 5 7

5 6 9

Solution:

Now we compute linear trend equation to analyse

the time data for this problem.

car age (x)Repair (y) (Rs. 000) xy x2

1 1 4 4 1

2 3 6 18 9

3 3 7 21 9

4 5 7 35 25

5 6 9 54 36

 Total 18 33 132 80

xyis

elseriestimefittedtheHence

xbyband

b

xxn

yxxynb

87.047.3ˆ

mod

47.3306*87.0606

87.0)18(80*5

33*18132*5

)(

10

21

221

Estimation: Now we have to estimate the

repair cost for 6 years old car. Therefore, the

estimated repair cost is

3.47+0.87*6 =Rs. 8.69 thousands

where x is time and it is 6 years

Fitting of Quadratic Model

Sometimes the straight line equation may be

inappropriate to describe the time series

data and time series are best described by

the curves. To overcome this problem, we

prefer the parabolic curve, which is explained

mathematically by a second-degree equation.

The general form of estimated second degree

equation is 2ˆ ctbtay

Least square estimates of the coefficients

The least square estimate of the coefficients

of second degree trend is given by

4322

32

2

tctbtayt

tctbtaty

tctbany

Where a= estimated y-intercept

b= estimated linear effect on y

c= estimated curvilinear effect on y

0

50

100

150

200

250

1 2 3 4 5 6 7

Example: Following data provides the total sales of

Quartz watch company in different years.

Choose an appropriate fitting line of trend and

estimate the sales for year 2000.

Year Sales

1991 13

1992 24

1993 39

1994 65

1995 106

Solution: Since the nature of the data shows not

uniform increase in the data level therefore,

second degree equation may be the best fit for the

data. Now first convert the given year in

appropriate time period by assuming the year 1993

as 0, we have Year t= year-1993

1991 -2

1992 -1

1993 0

1994 1

1995 2

Sales (y) Year (t) t2 t4 Y*t t2*y

13 -2 4 16 -26 52

24 -1 1 1 -24 24

39 0 0 0 0 0

65 1 1 4 65 65

106 2 4 16 212 424

247 0 10 37 227 565

Now we estimate the value of the constant by substituting these values in the normal equations as mentioned previously, then we get a= 39.3, b= 22.7 and c= 5.07Now the fitted trend equation is

207.57.223.39ˆ tty

Here we have to forecast the volume of sales for

the year 2000, therefore, we have

t = 2000-1993=7 then

y= 39.3+22.7t+5.07t2

Estimated value for the year 2000 =39.3+22.7*

7+5.07*72 =446.6

Year Sales Fitted value

1991 13 14

1992 24 22

1993 39 39

1994 65 67

1995 106 105

Now comparing the observed and fitted

values through graph

0

50

100

150

200

250

1 2 3 4 5

Fitted value

Sales

Facts to be considered during the selection of

fitting model • Linear trend is used when the time series

increases or decreases by equal absolute amount. On plotting the data, it gives the straight line.

• In other words, a linear trend is assumed to be perfect one when the consecutive difference of the observations in the series are if same. For example;

Year 1992 1993 1994 1995 1996 1997 1998 1999 2000

Age 33 36 39 42 45 48 51 54 57

I Diff. - 3 3 3 3 3 3 3 3

• Second degree model is preferred when it shows curvilinear (either concave upward or downward) graph on plotting in logarithmic scale.

• In other words, second degree model is appropriate when second difference between consecutive pairs of observations in the series are same throughout. For example;

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999

weights

30 31 33.5 37.5 43.0 50.0 58.5 68.5 80.0

I Diff

II Dif

-

-

1.0

1.5

2.5

1.5

4.0

1.5

5.5

1.5

7.0

1.5

8.5

1.5

10.0

1.5

11.5

1.5

Example

Following is the information about the sales

obtained during five years.

Compute the second degree trend equation

and determine the trend value for 1971.

Year Sales

1921 11.4

1931 12.1

1941 13.9

1951 17.3

1961 18.0

Solution: First, we need to obtain the normal

equations to get the value of the constants.

Year Sales t= (x-1941)/10 t2 t3 t4 t*y t3*y

1921 11.4 -2 4 -8 16 -22.8 45.6

1931 12.1 -1 1 -1 1 -12.1 12.1

1941 13.9 0 0 0 0 0 0

1951 17.3 1 1 1 1 17.3 17.3

1961 18 2 4 8 16 36 72

 Total 72.7 0 10 0 34 18.4 147

Now the normal equation are

211.084.1312.14ˆ

11.0

84.1

312.14

,

34100.147

104.18

7.72105

tty

isequationtrendfittedtheNow

c

b

a

getweequationsthesesolvingOn

ca

b

ca