How Much Does Industry matte presents summation over the subscript normally Developing the corresponding expression for in that position. For example, nik. is the total Ey? and substituting into(9) gives number of observations of business-unit ik, and the total number of observations w= n Define the average return yi of industry i as the arithmetic average of all observations in Es2=02+ industry i. Thus, yi= r./n;.., and, using(7) That is, the observed average industry return is the sum of u, the industry effect a; and weighted averages of the business-unit effects and errors variance components and describes the precise ithin th to compute yi for each of the 242 industries an to then calculate the sample variance sy among A and equating the expected and computed these industry returns. The result is sa=61.90. values of sy gives How good an estimate is this of o2? Examining (8), it should be clear upon refection that overestimates o2. The variance among industry E=2+0.1950+0.050302=6190.(1) returns will be oa plus terms in o, and o2. That is, industry returns vary from one another because That is, the observed variance in industry returns of industry effects and because of the random is expected to be the true variance in industry impact of business-unit effects and errors on effects plus (approximately) one-fifth of the ed industry returns. To develop an variance in business-unit effects, plus Esx in terms of variance com-(approximately) one-twentieth of the error vari ponents, first note that ance Next consider fik=rik/nik, the average return 52=2iyi-y.a (9) of business-unit ik. The varia s of busine unit returns(sample A)is 1 7.49. Using steps parallel to those taken above, this sample variance ow conside The independence assump- can be equated to its expected value tions assure he expectations of cross- products are zero(e.g, E(a, dik)=0), and that within families of effects, expectations of products E=0.95802+o+0.25902=18749.(12) E(a, a )=0 if i=j, and 0 otherwise. Squaring That is, the variance among business-unit returns (8)and taking the expectation, yields is expected to be the true variance among business-unit effects, plus(approximately) the variance in industry effects, plus(approximately) one-fourth the error variance Finally the same method can be used to obtain for Es2. the expected total observed ariance in returns I5 E∑y=la2+l2 E2=0.99502+0.999%+2=279.35.(13) Is Note that Es2*o?+oi+o? because s is calculated u the sample average rather than the true p