ACCA P3考試：Linear regression

Least squares linear regression is a method of fitting a straight line to a set of points on a graph. Typical pairs of graph axes could include:

• total cost v volume produced

• quantity sold v selling price

• quantity sold v advertising spend.

The general formula for a straight line is y = ax +b. So, ‘y’ could be total cost and ‘x’ could be volume. ‘a’ gives the slope or gradient of the line (eg how much the cost increases for each additional unit), and ‘b’ is the intersection of the line on the y axis (the cost that would be incurred even if production were zero).

You must be aware of the following when using linear regression:

• The technique guarantees to give the best straight line possible for any set of points. You could supply a set of people’s ages and their telephone numbers and it would purport to a straight-line relationship between these. It is, therefore, essential to investigate how good the relationship is before relying on it. See later when the coefficients of correlation and determination are discussed.

• The more points used, the more reliable the results. It is easy to draw a straight line through two points, but if you can draw a straight line through 10 points you might be on to something.

• A good association between two variables does not prove cause and effect. The association could be accidental or could depend on a third variable. For example, if we saw a share price rise as a company’s profits increase we cannot, on that evidence alone, conclude that an increase in profits causes an increase in share price. For example, both might increase together in periods of economic optimism.

• Extrapolation is much less reliable than interpolation. Interpolation is filling the gaps within the area we have investigated. So, if we know the cost when we make 10,000 units and the cost when we make 12,000 units, we can probably make a reasonable estimate of the costs when we make 11,000 units. Extrapolation, on the other hand, is where you use data to predict what will occur in areas outside the region you have investigated.

We have no experimental data for those areas and therefore run the risk that things might change there. For example, if we have never had production of more than 12,000 units, how reliable will estimates of costs be when output is 15,000 units? Overtime might have to be paid, machines might break down, more production errors might be made.

• Remove other known effects, such as inflation, before performing the analysis, or the results are likely to be distorted. Total cost Volume b, fixed costs A, slope is the variable cost per unit per unit