4.7Powerregression

Table4.3.ANOVAtable

Sourceofvariation Sumofsquares(SS) Degreesoffreedom(d.f.) Meansumofsquares (MSS) F

Regression(explained) ESS =1792 1 1792 10.77 Residual(unexplained) RSS = 832 5 166.4 Total TSS =2624 6

It can also be shown that: TD ¼ ED þ UD (4:12)

TD2 ¼ ED2 þ UD2 (4:13)

Expression (4.13) can also be written:

Total sum of squares ¼ explained sum of squares þ unexplained sum of

squares or:

TSS ¼ ESS þ RSS (4:14) where RSS is the residual sum of squares and is unexplained by the regression line. These relationships are frequently illustrated in an analysis of variance, or ANOVA, table. For the data in Table 4.1 this is shown in Table 4.3. Variance, or mean sum of squares, is given by variation (SS) divided by degrees of freedom. The values in the second column are calculated in Table 4.13 in Appendix A.3. The last three columns will be explained in more detail in section 4.9.

The definition of the coefficient of determination at the beginning of this subsection indicates that it is given by: R2 ¼ ESS=TSS (4:15) For the data set above, R2 ¼ 1792/2624 ¼ 0.6829

This means that 68.29 per cent of the variation in sales is explained by the linear relationship with price. The corollary of this is that 31.71 per cent of the variation in sales is unexplained by the relationship, and therefore it must be explained by other omitted variables. Thus when R2 is low it means that other variables play an important part in affecting the dependent variable and should preferably be taken explicitly into account (if they can be identified and measured) in a multiple regression analysis, as described in section 4.9.

4.7 Power regression

It was assumed in the above analysis that the relationship between the variables was linear. Demand relationships are usually considered to be in linear or powerformasseeninthelastchapter.Rarelydowehaveastrong apriori belief regardingwhichmathematicalformiscorrectfromthepointofviewof economictheory;therefore,wetendtoseewhichformfitsthedatabestin