FORECASTING
INTRODUC TION
Forecasting is prediction of events which are to occur in the future and there are many variables to be predicted in an organization e.g.
- Demand /sales
- Costs
- Personnel requirements
- Future technology etc.
The need for forecasting arises from the fact that decisions are about the future which in many cases in uncertain. Thus forecasting can be viewed as an effort to reduce uncertainty so as to make more accurate decisions. This gives the firm the competitive advantage in the market.
QUALITATIVE AND QUANTITATIVE TECHNIQUES
A convenient classification of forecasting techniques is between those that are broadly qualitative and those that are broadly quantitative. These classifications are by no means rigid or exclusive but serve
as a means of identification.
Quantitative Techniques
These are techniques of varying levels of statistical complexity which are based on analyzing past data
of the item to be forecast e.g. sales figures, stores issues, costs incurred. However sophisticated the technique used, there is the underlying assumption that past patterns will provide some guidance to the future. Clearly for many operational items (material usage, sales of existing products, costs) the past does serve as a guide to the future, but there are circumstances for which no data are available, e.g.
the launching of a completely new product, where other, more qualitative techniques are required.
These techniques are dealt with briefly first and then the detailed quantitative material follows.
Qualitative Techniques
These are techniques which are used when data are scarce, e.g. the first introduction of a new product. The techniques use human judgment and experience to turn qualitative information into quantitative estimates. Although qualitative techniques are used for both short and long term purposes, their use becomes of increasing importance as the time scale of the forecast lengthens. Even when past data are available (so that standard quantitative techniques can be used), longer, term forecasts require judgment, intuition, experience, flair etc, that is, qualitative factors, to make them more useful. As the time scale lengthens, past patterns become less and less useful meaningful. The qualitative methods briefly dealt with next are the Delphi Method, Market Research and Historical Analogy.
Delphi Method
This is a technique mainly used for longer term forecasting, designed to obtain expert consensus for a particular forecast without the problem of submitting to pressure to conform to a majority view. The procedure is that a panel of experts independently answer a sequence of questionnaires in which the responses to one questionnaire are use to produce the next questionnaire. Thus any information available to some experts and not others is passed on to all, so that their subsequent judgments are refined as more information and experience become available.
Market Research/Market Survey
Widely used procedures involving opinion surveys, analyses of market data, questionnaires designed to gauge the reaction of the market to a particular products’, design, price, colour etc. Market research is often very accurate for the relatively short term, but longer term forecasts based purely on surveys are likely to be suspect because peoples’ attitudes and intentions change.
Historical Analogy
Where past data on a particular item are not available, e.g. for a new product, data on similar products are analysed to establish the life cycle and expected sales of the new product.
Clearly, considerable care is needed in using analogies which relate to different products in different time periods, but such techniques may be useful in forming a broad impression in the medium to long term.
Note on Qualitative Methods
a) When past quantitative data are unavailable for the item to be forecast, then inevitably much more judgment is involved in making forecasts.
b) Some of the qualitative techniques mentioned above use advanced statistical techniques, e.g. some of the sampling methods use in Market Research. Nevertheless, any such method may prove to be a relatively poor forecaster, purely due to the lack of appropriate quantitative data relating to the factor being forecast.
CORRELATION ANALYSIS
Before we study the quantitative forecasting techniques, it is necessary to study correlation, which among others issues, is a way of selecting predictor variables in some priority fashion.
Correlation analysis is concerned with finding if there is a relationship between any 2 variables, say X and Y e.g.
- Level of advertising and sales (+ve)
- Interest rates and savings (+ve)
- Prices and demand for a good etc (-ve)
It seeks to answer 2 fundamental questions in forecasting:
1. What nature, if any, of the relationship between 2 variables i.e in what direction is it?
–ve? or +ve?
2. What is the strength or the degree of the correlation between 2 variables? This enables ranking of predictor or independent variables to be carried out.
Methods of correlation
1. Scatter diagram.
2. Pearsonian co-efficient of correlation ()
3. Spearmans rank correlation coefficient , ()
Scatter diagram
Also called a scatter graph, this is obtained by plotting values of X and Y on a graph. We have X (independent variable) on the horizontal axis and Y (dependant variable) on the vertical axis.
Example
The output X (in units) and the associated TC, Y for a product over 10 weeks are as follows, Y is in millions of shillings.
N0. / X / Y1
2
3
4
5
6
7
8
9
10 / 15
12
20
17
12
25
22
9
18
30 / 180
140
230
190
160
300
270
110
240
320
Issues of concern
1. Is there a relationship between X and Y?
2. If there is a relationship between X and Y can we obtain it in equation form so that if it persists into the future, we can use X to predict Y?
3. Can we assess the reliability and validity of the prediction model and the predictions it gives?
4.
Scatter diagram
Y
350 *
300 *
250 * *
200 * *
150 * *
100 *
X
5 10 15 20 25 30 35
Importance of scatter diagram
1. It gives a measure of approximate correlation. In the example, generally, as X increases, Y also
increases (+ve correlation.) It is also a strong correlation.
2. It suggests the best mathematical or functional form for X and Y. In this example, a linear fit seems the best.
Other possibilities (scatter diagrams)
Y 1) Strong quadratic correlation
x = Outlier
x - outlier
0 X
2) Weak linear –ve correlation
Y
X
Y 3) No correlation
X
3. A further usefulness of a scatter diagram is that it helps in identifying and hence weeding out outlier cases i.e. cases which are unlikely to recur e.g.
- Production when there was a strike
- Sales when there was a riot or a celebration.
Weaknesses of a scatter diagram
- It cannot be used to represent two or more independent variables simultaneous due to the
limitation of the cartesian plane.
Pearsonian Coefficient of Correlation (Product Moment - Coefficient of Correlation)
measures the nature and strength of the relationship between 2 variables on a scale from -1 to +1 such that:
If = -1 perfect indirect (inverse) correlation
= +1 perfect direct correlation
= 0 no correlation; these are variables which have no relationship with one
another.
Formula
r = Cov (x, y)
=
Cov = Covariance
Calculations for
No / X / Y / x - / /1
2
3
4
5
6
7
8
9
10 / 15
12
20
17
12
25
22
9
18
30 / 180
140
230
190
160
300
270
110
240
320 / -3
-6
2
-1
-6
7
4
-9
0
12 / -34
-74
16
-24
-54
86
56
-104
26
106 / 102
444
32 24
324
602
224 936
0
1272
/ 180 / 2140 / 0 / 0 / 3960
= = = 18 = 214
Cov (x, y) = = 396
Interpretation – There is a +ve association between output, X and TC ,Y
Limitation – Co-variance is not rigidly defined i.e. it can take any value making interpretation and comparison difficult e.g. ranking of predictors is impossible.
Remedy
By dividing the Cov by the product of the standard deviations of x and y, we obtain a statistic which is strictly in the range -1 to +1. This is the Pearsonian coefficient of correlation (due to the German mathematician, Karl Pearson).
= 6.13
= 66.06 r = = 0 .98
Interpretation - There is a very strong +ve correlation between X (output) and Y (TC). Hence X is a very good predictor of Y.
Simplified Formula for rp
Given that the n in the numerator and the ones in the denominator will cancel out, we can have a simplified formula thus:
Comments
1) r is unitless, thus it is possible to compare correlation for any types of units.
2) Since r is strictly from -1 to +1, the scale of the data is immaterial e.g. Suppose X is in Kgs and Y in Shs.
= = Unitless
3) Maxim/Adage – High correlation does not mean causation but causation certainly implies high correlation. The fact that X and Y have high correlation does not mean changes in X cause changes in Y because it is possible to get a high correlation purely by chance (also known as a spurious or a non-sensical correlation)
- However, sometimes, a seemingly spurious correlation may be explainable through an
intervening variable e.g.
X Y Z = “causes changes in”
Hence X Z so that we expect a high correlation between X and Z, although it is not direct.
Y is the intervening variable e.g. alcohol driver accident.
4) Ranking of predictor variables.
Example
Predictor / / RankX1
X2
X3
X4 / -0.99
0.93
-0.40
0.52 / (1)
(2)
(4)
(3)
NB: When ranking predictor or independent variables, the direction of the correlation is immaterial. What only matters is the size or the magnitude of the correlation.
- The pearsonian coefficient of correlation is also called Product moment coefficient of correlation.
SPEARMANS COEFFICENT OF RANK CORRELATION,
This coefficient of correlation was developed by the British psychologist, Charles Edward Spearman in 1904. It is most suitable when data is of ordinal nature i.e. where only ranking is possible.
Types of data
1) Nominal – These are labels e.g. car no’s in a safari rally.
2) Ordinal – e.g. ranking of school plays or beauty contest candidates.
3) Interval – e.g. temperature scale – has no natural zero (it is a arbitrary)
4) Ratio – e.g. marks scored in an exam or the age of a person – has natural zero.
Formula
= where = difference between the pairs of ranked values
n = no. of pairs of rankings
Like the product – moment coefficient of correlation, can take values from –1 to +1.
Example
Two managers are asked to rank a group of employees in order of potential for promotion.
The rankings are as follows:
Employee / Manager1 (R1) / Manager
2 (R2) / (R1-R2)2 =
A
B
C
D
E
F
G
H
I
J / 10
2
1
4
3
6
5
8
7
9 / 9
4
2
3
1
5
6
8
7
10 / 1
4
1
1
4
1
1
0
0
_1_
= / 14
Required:
Compute the coefficient of rank correlation and comment on the value.
Solution
= = =
= 0.92
Comment
There is a very high +ve correlation between the ranking of the 2 managers, meaning they largely agree.
for tied ranking
The formula is modified to be:
More than one tie
where t = number of tied rankings
Example on tied rankings - More than one tie
Two judges in a schools drama festival awarded marks to eight schools in order of preference. From these, determine the Spearmans coefficient of rank correlation rs. Thus, comment on whether the two judges generally agree on the schools performance or not.
School / Judge 1 / Judge 2A
B
C
D
E
F
G
H / 72
79
77
77
73
71
77
70 / 69
83
81
69
68
66
65
81
Solution
Ranking the schools / Average ts and calculation of d2School / R1 / R2 / School / R1 / R2 / d
A
B
C
D
E
F
G
H / 6
1
2
2
5
7
2
8 / 4
1
2
4
6
7
8
2 / A
B
C
D
E
F
G
H / 6
1
3
3
5
7
3
8 / 4.5
1
2.5
4.5
6
7
8
25 / 2.25
0
0.25
2.25
1
0
25
30.25
1 - = = 0.24
Comment
There is a very low +ve correlation between the marks awarded to the schools by the two judges, meaning that generally, they do not agree on the schools’ performances.
Example
The scores of 8 Sec. 4 students in QA and Tax tests where as follows:
Score %Student / QA / Tax
I
II
III
IV
V
VI
VII
VIII / (2) 70
(7) 41
(6) 45
(1) 83
(3) 65
(3) 65
(5) 59
(8) 39 / (3) 72
(6) 54
(4) 66
(2) 80
(5) 62
(1) 89
(8) 45
(7) 50
( ) Ranking
Required:
Do those students who score highly in QA also do so in Tax or is it vice versa?
Use spearman’s rank correlation coefficient ,
Solution
The tied rankings are added and divided by their no.
RanksStudent / QA / Tax /
I
II
III
IV
V
VI
VII
VIII / 2
7
6
1
3.5
3.5
5
8 / 3
6
4
2
5
1
8
7 / 1
1
4
1
2.25
6.25
9
_1___
= t = 2 (no. of tied rankings)