Reading 11, 3.4 Coefficient of Determination - Method 2

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

The references refer to the CFA text book.

3.4 Coefficient of determination (“CD”)

Why CD?

The standard error of estimate (SEE) gives some indication of how certain one can be about a particular prediction of Y using the regression equation.

It does not tell us how well the independent variable explains variation in the dependent variable. The coefficient of determination does this, measuring the fraction of the total variation in the dependent variable that is explained by the independent variable.

There are 2 methods to calculate CD. Method 1 was discussed in the previous post

Method 2:  For use in a linear regression with more than ONE independent variable

Total variation = Unexplained variation Plus Explained variation

Therefore

Explained variation = Total variation Minus Unexplained variation

The CD is the fraction of the total variation that is explained by the regression.

Thus CD =                            Explained variation(Divided by)

           Total variation

Thus CD also =            Total variation Minus Unexplained variation (Divided by)

Total variation

Thus CD, (after dividing by Total variation) = 1 minus  Unexplained variation(Divided by)
                                                                                  Total variation

Formulas used

Total variation is calculated as:

(We use the mean to calculate total variation)

Unexplained variation is:

(Using the difference between actual data and the expected data)

Example Method 2
(Link to the excel file on box.net)

Data
Assume that Revenue is the independent variable (X)
					2010	2009	2008	2007	2006
	Revenue US $m				52798	50211	59473	47473	39099
	Earnings per ordinary share (diluted) (US cents)				227.8	105.4	274.8	228.9	172.4

Calculation

Year	Revenue $ (X)	Earnings - Expected based on regression analysis in 3.1	Earnings - Actual		Total variation (based on the mean)	Unexplained variation
2010	52798	215.22	227.80		672.88	158.18
2009	50211	203.65	105.40		9304.53	9653.11
2008	59473	245.08	274.80		5320.24	883.11
2007	47473	191.40	228.90		731.16	1406.09
2006	39099	153.94	172.40		867.89	340.70
Average	49810.8	201.86	201.86
Sum					16896.71	12441.20
			12441.20
Formula	Explained variation =	1 Minus	16896.71
	=	1 Minus	0.7363
	=	0.26

NB, the answer is very similar to that of the first method. This is because only one variable is used. If more than one variable is used, Method 1 can not be used.