Time Series Analysis

A time series is a collection of data recorded over a period of time – weekly, monthly, quarterly or yearly.

Examples:

1. The hourly temperature recorded at a locality for a period of years.2. The weekly prices of wheat in Dhaka.3. The monthly consumption of electricity in a certain town.4. The monthly total passengers carried bay a train.5. The quarterly sales of a certain fertilizer.6. The annual rainfall at a place for a number of years.7. The students enrolment of a college over a number of years.

Application

Time Series Analysis is used for many applications such as:

Economic Forecasting Sales ForecastingBudgetary Analysis Stock Market AnalysisYield Projections Process and Quality ControlInventory Studies Workload ProjectionsUtility Studies Census Analysis

and many, many more...

Components of a time series

There are four components of a time series:

1. The trend2. The cyclical variation3. The seasonal variation and 4. The irregular variation.

Secular trend

The smooth long-term direction of a time series. The long-term trends of sales, employment, stock prices and other business and economic series follow various patterns. Some move steadily upward, others decline, and still others stay the same over time.

Example: Wheat production of Chittagong district from 1991 to 1995.

Year 1991 1992 1993 1994 1995

Production (M. ton)

20 22 25 26 32

1990 1991 1992 1993 1994 1995 1996

20

22

24

26

28

30

32

Year

Pro

duct

ion

(M. t

on)

Example: Number of officers in Sonali bank from 1985 to 1990.

Year 1985 1986 1987 1988 1989 1990

Officer 1000 700 500 400 300 300

1984 1986 1988 1990

300

400

500

600

700

800

900

1000

Year

No.

of o

ffice

r

Cyclical variation

A typical business cycle consists of a period of prosperity followed by periods of recession, depression, and then recovery.

In a recession, for example, employment, production, the Dow Jones Industrial Average, and many other business and economic series are below the long term trend lines. Conversely, in periods of prosperity they are above their long-term trend lines.

Example: Following figure shows the number of batteries sold by National Battery Sales, Inc. from 1988 to 2005. The cyclical nature of business is highlighted.

Seasonal variation

Patterns of change in a time series within a year. These patterns tend to repeat themselves each year. The unit of time reported is either quarterly or monthly.

Example: Men’s and boy’s clothing have extremely high sales just prior to Eid-ul-fitr and relatively low sales after it.

Example: Following figure shows the quarterly sales, in millions dollars, of a sporting goods company that specializes in selling baseball and softball equipment for high schools, colleges and youth leagues. There is a distinct seasonal pattern to their business. Most of the sales are in the first and second quarters of the of the year, when schools and organizations are purchasing equipment for the upcoming season.

Irregular variation

The irregular variations occur in a completely unpredictable manner as they are caused by some unusual events such as floods, droughts, strikes, fires, earthquakes, wars and political events and so forth.

Example: Monthly Value of Building Approvals, Australian Capital Territory (ACT).

Measures of trend

To measure a trend which can be represented by a straight line or some type of smooth curve, the following methods are used:

1. Freehand curve2. Semi average method3. Moving average method and4. Method of least squares

Freehand curve

Plot the given data on a graph paper and join the plotted points by segments of straight line. Draw a straight line freehand passing through the plotted points in a way such that the general direction of change in values is indicated.

Example: Export quantity in ton of a food from 1971 to 1978:

Year 1971 1972 1973 1974 1975 1976 1977 1978

Export(ton) 70 90 103 75 85 115 120 130

1970 1972 1974 1976 19780

20

40

60

80

100

120

Year

Ton

Semi average method

Divide the values in the series into two equal parts. Find the average values of each part and place the average values against the respective mid points of the two parts. Plot these two average values on the graph of the original values and draw a straight line connecting the two points and extend the line to cover the whole series.

Example: Export (Lac ton) information of Bangladesh from 1990 to 1998:

Year Export Total Average

1990 100

1991 110

1991.5 485 121.25

1992 125

1993 150

1994 130

1995 135

1996 160

1996.5 600 150

1997 145

1998 160

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 199960

80

100

120

140

160

Year

Exp

ort

Moving average method

The moving average method is not only useful in smoothing a time series to see its trend, it is the basic method used in measuring the seasonal fluctuation. The moving average only smooths the fluctuations in the data. This is accomplished by ‘moving’ the average values through the time series.

The k period moving averages are defined as the averages calculated using the k consecutive values of the observed series.

Each k period moving average is placed against the middle of its time period.

These average values are plotted on the graph of the original values and the line connecting these points is the moving average trend.

Example: From the information below draw a 3-year moving average trend.

Year Export 3-year total

3-year average

1973 9

1974 10 31 10.3

1975 12 33 11

1976 11 40 13.3

1977 17 44 14.7

1978 16 53 17.7

1978 20 57 19

1980 21

1972 1973 1974 1975 1976 1977 1978 1979 1980 19818

10

12

14

16

18

20

22

Year

Exp

ort

original

3-year

Exercise 1: From the information below draw 4-year moving average trend.

Year Export 4-year mid average 4-year average

1982 52

1983 63

67

1984 75 70.75

74.5

1985 78 77.87

81.25

1986 82 83.75

86.25

1987 90 89

91.75

1988 95

1989 100

Method of least squares

A trend can be represented by a mathematical equation of the form of a straight line. The straight line equation can be represented by

bxay '

y’ is the projected value of y and y is the time series variable.a is the y intercept. It is the estimated value when x=0.b is the slope of the line, or the average change in y.x is the value of time.

If we can convert x is such a way that the sum of converted x is zero, then the value of a and b can be estimated as:

2,

x

xybya

Example: From the information below using least square method draw the trend.

Year (X)

Production (y)

x xy y’

1985 27 -4 -108 16 25.58

1986 29 -3 -87 9 26.63

1987 30 -2 -60 4 27.68

1988 24 -1 -24 1 28.73

1989 28 0 0 0 29.78

1990 25 1 25 1 30.83

1991 34 2 68 4 31.88

1992 35 3 105 9 32.93

1993 36 4 144 16 33.98

Total 268 0 63 60 268

2x

xy

a

b

05.178.29'

78.299/268

05.160/63

1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

24

26

28

30

32

34

36

Year

Pro

du

ctio

n

original

least square

Exercise 2: From the information below draw a trend using least square method.

Year (X)

Production (y)

x xy y’

1988 800 -7 -5600 49 743.74

1989 750 -5 -3750 25 730.35

1990 800 -3 -2400 9 716.96

1991 550 -1 -550 1 703.57

1992 600 1 600 1 690.18

1993 650 3 1950 9 676.78

1994 675 5 3375 25 663.40

1995 750 7 5250 49 650.00

Total 5575 0 -1125 168 5574.98

2x

xy

a

b

696.6875.696'

875.6968/5575

696.6168/1125