Reading 11, 3.7 Prediction intervals

Level 2, Volume 1, Quantitative methods, Reading 11, Correlation & Regression

In previous examples a regression model was calculated, where BHP revenue is the independent variable and Earnings per ordinary share is the dependent variable.

Using the linear regression formula:

Yi = bo + b1Xi, it was concluded that the linear regression formula for BHP is:

Yi =     -20.96370784 +  0.004473.Xi

In 2010, BHP revenue was $52,798b. If it is assume that, given the rise in commodities prices, investors are expecting the revenue to increase to $60,000b.

Using the regression model, 

Yi =     -20.96370784 +  0.004473.Xi, it is therefore calculated that

Yi = 247.44 cents earnings per share

It would however be pretty useful if we could use statistics to provide a level of confidence around this number. For example, as an investor it would be great if we could say that we are 95% confident that the earnings number would be for example between 200 cents earnings per share and 270 cents per share.

This range is what is calculated by Prediction intervals.

There are 4 steps to calculating the Prediction Interval.

Step 1 – Use the linear regression model to forecast value

Calculate the prediction, e.g that was done above. It was calculated that the Earnings per share would total 247.44 cents per share, if the Revenue for 2011 totalled $60,000b.

Step 2 – Calculate the estimated variance of the prediction error

Note the following:

• The formula above calculates the variance! We are using Standard Deviation in Step 3. One should therefore obtain the square root of this answer, to get the standard deviation

• Note that the formula makes reference to X.This is important, as it refers to the independent value, in this case Revenue per year. Values for the Prediction, Mean and Standard deviation of the independent variable is required!

Calculating the estimated variance of the prediction error:

Inputs	Values
Standard Error of Estimate	64.4
N	5
X	60000
Mean X	49810
Variance of the Mean X	55 662 400

Estimated variance of the prediction error =

(64.4)² * (1 + (1/5)+ ((60000-49810)²/(4 X 55 662 400)))

=6825

AND REMEMBER, to get to Standard Deviation:

= √6825

= 82.61

Step 3 – Determine the critical value

N = 5, df therefore is 3

We want 95% confidence

This gives a critical value of 3.182

Step 4

247.44 +/- (3.182)(82.61)

=510.30502 AND

-15.42502

Comment

The range appears at face value to be extremely wide, to the extent that it in practise might have little value. Some the following factors should be considered:

1) The initial correlation between Revenue & Earnings per share was calculated at a low 0.51 in previous examples. This is because of an outlier year in 2009.

·         2) The sample is small, impacting on the interval 

3)   3) The level of confidence is high, at 95%.