Measuring Business Performance
4.1 Central Tendency Methods .1 Average ratio analysis
4.1.2 Simple regression models
In the context of cross-sectional analysis,various authors in the business performance research field consider the case where the ratio of two accounting variablesYand X is compared to some characteristic value,b. If Y is pro- portional toX,then for theith company the difference betweenYiXi−1andb can be interpreted as an effect attributable to the individual company,i.e. as an indication of that particular company’s departure from the norm. Thus,under the assumption of proportionality,inferences may be drawn directly from financial ratios (McDonald and Morris,1984,1985). But Barnes (1982) and Cubbin and Tzanidakis (1998) have suggested that regression analysis may be preferable to simple ratio analysis.
Consider a sample of companies with varying cost/output combinations, denoted as {X,Y}. For a given level of outputY,the observed operational costs Xof each company can be thought of as the costs of an average-efficiency company,plus an efficiency factorureflecting the efficiency difference (‘costs inefficiency’) between the particular company and the average-efficiency company, simply formulated as
{ }
X=n1
∑
X Y, +u (4.1)To measure relative efficiency of a particular company,an estimate or bench- mark of the costs of the average-efficiency company with the same level of output as the company in question is needed. This can be obtained with the help of regression analysis.
Figure 4.1 illustrates observations and a regression through them. Oper- ating costsXare plotted on the horizontal axis and outputYis plotted on the vertical axis. The dots indicate results for companies with certain cost–output combinations. The fitted regression line provides an estimateX*of the average efficiency costs for a given level of outputY. For example,in the case of com- pany P with observed costsXPand outputYPthe estimated average efficiency cost (benchmark) is given byXP*which corresponds to the point where the dotted line intersects the estimated regression line. The difference between the observed and the estimated benchmark costs (XP−XP*) is the estimateuP*of efficiencyuP. Similarly,for each company,an estimateu*of its true efficiency u is given by the difference between the observed and estimated average efficiency costs u*=X−X*. Companies below the regression line are of below-average efficiency whereas companies above the regression line are of above-average efficiency. A scale independent efficiency score for each company can be calculated by expressing the difference between observed and predicted costs as a percentage of the predicted costs,i.e. the efficiency score is given by
e X X
= −X *
* (4.2)
The regression analysis assumes that the average efficiency company increases linearly with outputY:
Y=α+βX+u (4.3)
According to the above,a unit increase in output increases costs byβ,and for zero output there are fixed costsα.1
Fig. 4.1. Regression line and points fitted in an analysis for performance measurement.
1 In an output-oriented model,αwill be negative.
The linear regression model based on the ordinary least squares (OLS) algorithm to make estimates involves some statistical assumptions. This set of statistical assumptions enables,in addition to measurement,the proper statistical estimation and hypothesis testing of the parameters used in the model. The OLS algorithm fits the line by minimizing the sum of squared deviations of the observed costs from the line. Because regression analysis provides conditionally,upon each level of output,an estimate of the average efficiency costs,it is often said to fit an ‘average line’ to the data,or more accurately, a ‘conditionally average line’.
Log transformation
Many accounting variables are sums of similar transactions with constant sign,for example: sales,stocks,creditors and current assets. Unlike other variables,such as earnings,cash flow and net working capital,which can have both positive and negative values,such accounting variables are bounded at zero. Therefore,for large samples of companies,the evidence is that the distribution of the first group of variables may be Pareto-like,or log-normal (Deakin,1976a; Ijiri and Simon,1977). Furthermore,a ratio of two such variables has the same properties (Lev and Sunder,1979). That is,ifXandY are distributed log-normally then,due to the additive properties of normal variables,logY−logX is distributed normally,hence the ratio YX−1 is also log-normally distributed. Therefore,regression studies in the business perfor- mance field,especially when dealing with accounting ratios,adopt the use of log-normal regression analysis.
For example,in the case of a log-normal transformation,Equation 4.3 can be redefined as
lnY=α+βlnX (4.4)
Forβ= 1, this is
lnY−lnX=αand, therefore,Y
X= exp(α) (4.5)
where α is the mean of the logarithms of ratio YX−1. Thus,exp(α) can be referred to as a ‘benchmark’,or characteristic value against which the ratio of theith companyYiXi−1can be compared. The OLS estimate ofαis the average of the logarithms of the ratios
α=
∑
1 n
Y X
i i
ln (4.6)
and so an estimate of the characteristic value exp(α) is obtained as the geometric mean of the ratios YiXi−1. For a bivariate log-normal regression, the model can be summarized as
lnYi=α+βlnXi+εi (4.7)
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).