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FORECASTINGBusiness Decision Techniques

BUSS 0204

The components of time series

A time series is a series of figures or values recorded over time.

Any pattern found in the data is then assumed to continue into the future and an extrapolative forecast is produced.

There are four components of a time series: ◦ trend,

◦ seasonal variations,

◦ cyclical variations and

◦ random variations.

Standard time series models

1. The time series additive model: y = t + c + s + I

2. The time series multiplicative model: y = t × c × s × I

Where y is a given time series valuet is the trend componentc is the cyclic components is the seasonal componenti is the irregular component

Description of time series components

1. TrendThis is a long term movement

2. Cyclical variationRepeating up and down movements due to interactions of factors influencing economy.

3. Seasonal variationThe effect of seasons – spring, summer. Autumn and winter – on the series.

4. Irregular or random variationThese are disturbances due to ‘everyday’ unpredictable influences, such as weather conditions, illness and so on.

Based on the data, plot the time series.

Numbers of car sell in XYZ Company

Qtr 1 Qtr 2 Qtr 3 Qtr 4

Year 1 73 90 121 98

Year 2 69 92 145 107

Year 3 86 111 157 122

Year 4 88 109 159 131

0

20

40

60

80

100

120

140

160

180

Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4

Year 1 Year 2 Year 3 Year 4

Freq

uen

cy

Year

Numbers of car sell in XYZ Company

A

B

C

Finding the Trend (T) Moving Average (MA) eliminates S and I,

leaving T and C. In practice C rarely exists (Keynesian intervention), leaving T

12 months/ 4 quarter centred MA

8

Year Sales value 4 quarter

moving total

4 quarter moving

average

2009

2010

2011

2012

2013

2014

52

23

84

55

36

67

Finding the Trend (T) – odd

Year Quarter Sales

value

4 quarter

moving

total

4 quarter

moving

average

Centered

4 quarter

moving

average

2013

2014

1

2

3

4

1

2

3

4

75

32

48

85

93

60

92

47

10

Finding the Trend (T) – even

Moving Average (Centred)

12 month centred moving average or 4 quarter moving average eliminates S

Averaging process eliminates I

C is often absent from data

So left with T (Trend)

11

Finding Trend

Centred moving average gives us the trend (T)

These points can be plotted on a scatter diagram

12

Finding the seasonal component using the additive model

Step 1 : The additive model for time series analysis is Y = T + S + R

Step 2 : If we deduct the trend from the additive model, we get Y - T = S + R .

Step 3 : If we assume that R, the random, component of the time series is relatively small and therefore negligible, then S = Y - T

Therefore, the seasonal component, S = Y - T (the de-trended series).

Example: The trend and seasonal variationsAbsence and sickness records of HJA Corp.

Required Find the seasonal variation for each of the

10 days, and the average seasonal variation for each day of the week using the moving averages method.

Week Monday Tuesday Wednesday Thursday Friday

1 4 7 8 11 18

2 3 8 10 13 21

Finding the seasonal component using the multiplicative model

Step 1 Calculate the moving total for an appropriate period.

Step 2 Calculate the moving average (the trend) for the period. (Calculate the mid-point of two moving averages if there are an even number of periods.)

Step 3 Calculate the seasonal variation. For a multiplicative model, this is Y/T.

Step 4 Calculate an average of the seasonal variations.

Step 5 Adjust the average seasonal variations so that they add up to an average of 1.

Therefore, the seasonal component, S = Y/T (the de-trended series)

ForecastingStep 1 Plot a trend line: use the line of best

fit method, linear regression analysis or the moving averages method.

Step 2 Extrapolate the trend line. This means extending the trend line outside the range of known data and forecasting future results from historical data.

Step 3 Adjust forecast trends by the applicable average seasonal variation to obtain the actual forecast.

(a) Additive model – add positive variations to and subtract negative variations from the forecast trends.

(b) Multiplicative model – multiply the forecast trends by the seasonal variation.