RoyaltyStat Blog

Profit Margin with Heterogeneous Variance

Posted by Ednaldo Silva

In transfer pricing, we may encounter a situation in which the statistical residuals among the selected comparables do not have a common variance. This phenomenon is called heteroskedascity. To correct this problem, we can transform or deflate the relevant variables and measure them as ratios. E.g., suppose that we have comparable company coordinated pairs of data on sales (S) and “net” (operating) profits (P), and their bivariate scatter diagram suggests a linear relationship:

     (1)     Pi = α + β Si + Ui

considering i = 1, 2, …, N comparables. Profits and sales are measured using the same currency units, such as million USD or million EUR.

We suggested elsewhere that assuming that the intercept is null (α = 0) and proceeding to compute quartiles of the profit margin defined by the ratios (Mi = P/ Si) is likely to be controverted under audit scrutiny. Here we explore another problem in which the variance of the residual errors (Ui) of equation (1) may not be constant. E.g., they may show an open megaphone with increasing company size.

As a solution, we can estimate a reciprocal regression function under a Glesjer assumption that the variance of the residual errors is proportional to the square of the explanatory variable, such as company sales:

     (2)     Var(Ui) = σi2 = σ2 Si2

In practice, this means that we can deflate equation (1) and test a hypothesis that the profit margin is a reciprocal function of net sales instead of being constant among the selected comparables:

     (3)     Mi = (α / Si) + β + Vi

In equation (3) above, the profit margin is Mi = P/ Si and the regression residuals Vi = (U/ Si) are assumed to be well-behaved {≈ NID(0, σ2)} because Var(Vi) = Var(U/ Si) = σ2. See J. Johnston, Econometric Methods (3rd edition), McGraw-Hill, 1984 [1963], pp. 302-304. Else see Jan Kmenta, Elements of Econometrics (2nd edition), Macmillan, 1986 [1971], p. 283, or G. Maddala, Econometrics, McGraw-Hill, 1977, section 12-3.

The ratio regression equation (3) can be estimated in the same way as the level model (1), except that the roles of the intercept and slope coefficients are reversed.  Since the reciprocal model (3) is not usual in empirical economics research (but see Dennis Mueller, “The Persistence of Profits above the Norm,” Economica, New Series, Vol. 44, No. 176 (Nov., 1977)), another way of stabilizing the variance among the N residuals is to use a power function instead of the ordinary linear equation (1) (See Maddala, supra, p. 265: “There are two remedies that are often suggested and used for heteroskedascity. One is to transform the variables into logs, and the other is to deflate all variables by some measures of ‘size’”):

     (4)     Pi = α Si β Ui

which we can estimate by using the double logarithmic transformation of the dependent and independent (or explanatory) variables:

      (5)     LN (Pi) = γ + β LN (Si) + Zi

where γ = LN α, Zi = LN (Ui) and Var(Zi) = σ2.

We tend to get reliable results by estimating the comparable profit margin using equation (5), especially when the selection of comparables includes small and large companies measured by sales, total assets, or number of employees. For more learning on this matter, read (among many other published works): Sidney Alexander, “The Effect of Size of Manufacturing Corporation on the Distribution of the Rate of Return,” Review of Economics and Statistics, Vol. 31, No. 3 (Aug., 1949) and John Eatwell, “Growth, Profitability and Size (The Empirical Evidence)” in Robin Marris & Adrian Wood (eds.), The Corporate Economy, Harvard University Press, 1971.

In summary, we can better survive audit scrutiny by reducing empirical controversy in applied transfer pricing analysis. It follows that our recommended practice is to select the most reliable model of the profit margin by comparing the alternative regression results of equations (1), (3), and (4).  In RoyaltyStat®, we can select company comparables and test these models online. As posited on several other blogs, we are suspicious of the prevailing naive approach to transfer pricing consisting of computing quartiles of the selected profit indicator without testing the empirical functional form of the adopted model or adjusting for the existence of bivariate or multivariate influencing factors.

Published on Jun 23, 2016 8:24:25 AM

Ednaldo Silva (Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at:

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Topics: Net Profit Indicator