Measuring Business Performance
4.1 Central Tendency Methods .1 Average ratio analysis
4.1.3 Random coefficient models
where unexplained variability is represented by εi,which is assumed to be uncorrelated and normally distributed with mean 0, varianceσ2.
Here it is worth noting that,for log-normal variables,the geometric mean is the median. In the case of financial ratios of accounting variables,which are sums of similar transactions,this estimate should be preferred to its competi- tors like the arithmetic average of the ratios or the ratio of arithmetic averages, which implicitly rely on the normal distribution of the ratios or their compo- nents. Elsewhere,it has been shown that,in cross-section,the log-normal model can provide a useful approximation for the ratio of accounting sums (McLeay, 1986).
particularly in order to achieve parsimony in model building,that variance component analysis has been used.
Variance components
Marketing researchers frequently encounter cross-sectional data when devel- oping sales response models. One appropriate approach to analysing such data is to estimate a separate OLS equation for each cross-section. Alternatively, one could pool the data from all cross-sections to estimate a single set of response coefficients for all cross-sections (Leone et al.,1993). However, when data are pooled,the responsiveness of individual cross-sections cannot be evaluated.
Variance component models can be treated as generalizations of ordinary regression models (Harville,1977). Among a number of applications,they are appropriate for unbalanced data having a hierarchical structure where total variability may be separated into components attributable to each ‘cluster’
of observations (Goldstein,1986). In this case,Fieldsend et al. (1987) are suggesting that companies cluster into industrial sectors and,although there might be great variability in a particular financial indicator over the broad cross-section of companies,it is reasonable to expect some similarity in finan- cial characteristics of companies within sectors (Lee,1985). A variance com- ponent model will take into account this ‘within-sector’ homogeneity.
In this respect,variance component analysis treats the variation of sector effects as a component of the unexplained variability,the overall variance being partitioned into components for ‘sector’ as well as the ‘company resid- ual’. Introducing a term associated with the sector in Equation 4.7 gives
lnYij=α+βlnXij+γj+εi (4.8)
which is called the mixed effects model,where the indicesiandjrepresent companyiin sectorj. The regression coefficientαnow represents the intercept for the ‘average’ sector,as the unexplained variability is now represented by two random terms – the sector effectsγjand the company effectsεi,whereγand εare mutually independent and normally distributed with the means 0 and variancesσ2Iandσ2. Thus,for theith company in thejth sector,the model for lnYis composed of the following terms.
1. An industry-specific effectαj=α+γj. 2. βlnXij, whereβis constant over all sectors.
3. A residual company effect,εi.
In other words,while the slope in this mixed effects model remains constant withβas a fixed effect,the interceptαjvaries from sector to sector as a random effect (Fig. 4.3).
Of course, the slope of lnXmay also be allowed to vary, giving the model lnYij=α+βlnXij+γj+δjlnXij+εi (4.9)
where δj is assumed to be a random normal variate with zero mean and varianceσ2s. In the literature,Equation 4.9 is referred to as the random effects model. An alternative presentation is given by
lnYij=αj+βjlnXij+εi (4.10)
where the random slopeβj=β+δjand the random interceptαj=α+γjeach vary from sector to sector. The interpretation of this model is,firstly,that a highαjwill indicate a relatively high value of lnYproportionate to lnXin sectorjand,secondly,that a large difference betweenβjandβwill indicate a relatively greater size effect in that sector.
The fixed effects model (Equation 4.7),the mixed effects model (Equation 4.8) and the random effects model (Equation 4.10) are illustrated in Figs 4.2, 4.3 and 4.4,respectively. In fitting the fixed effects model,no account is taken of the differences in the two sectors at this stage,giving therefore the single regression line (α,β) that is superimposed on the plot of data points.
In Figs 4.3 and 4.4,the fitted lines that are obtained with Equations 4.8 and 4.10 can be compared. Parameter estimates from the ‘between-sector’
mixed effects model provide the regression lines in Fig. 4.3,where the slope remains constant but the intercept varies between the two sectors. Here,there are two fitted linesαA,βandαB,βfor sectors A and B,respectively,showing clearly the differences between the two groups in the intercept. Situated between these fitted lines is the regression line for the ‘average’ sector (α,β) where,as noted before,the estimates ofαandβdiffer from the OLS estimates
Fig. 4.3. The mixed effects regression model.
Fig. 4.4. The random effects regression model.
obtained with the fixed effects model (Equation 4.7). But, looking at the data suggests that there is likely to be some difference in the slopes, reflecting the within-sector variation in the size effect. Fitting the ‘within-sector’ random effects model (Equation 4.10), gives the plot in Fig. 4.4. Here the convergence of the fitted lines for sectors A and B (αA,βAandαB,βB) is evident.
Leoneet al. (1993) introduce a version of the random coefficient model that can be used to estimate separate sets of response coefficients for each cross-section, thereby circumventing the assumption that coefficients are homogeneous in all cross-sections. They demonstrate this approach with an empirical model that relates brand level sales to price and advertising.