RoyaltyStat Blog

Alternative Functional Forms of Comparable Profits

Posted by Ednaldo Silva

We can test several bivariate (X, Y) regression functions to obtain the most reliable estimate of comparable operating profits. The explanatory variable X can be sales, costs or assets of the selected comparable companies. The dependent variable Y can be their operating profits before or after depreciation, and the slope coefficient is an estimate of the comparable operating profit indicator:

     (1)     Linear: Y = α + β X, slope = β

     (2)     Semilog: Y = α + β LN(X), slope = β / X

     (3)     Hyperbola: Y = α – β / X, slope β / X2

     (4)     Doublelog: LN(Y) = α + β LN(X), slope = β (Y / X)

     (5)     Loghyperbola: LN(Y) = α – β / X, slope = β (Y / X2)

See Goldberger (1964), § 5.2 (Functional Forms), pp. 213-218. We show the slope of (4) on a prior blog:

We can test these different operating profit functions in RoyaltyStat/Compustat because we have integrated an online regression function with current and historical listed company financials. Thus, we can produce the most reliable estimate of a comparable (arm's length) profit indicator by using the maximum adjusted R2 rule. See Maddala (1977), p. 462. According to the maximum adjusted R2 model selection rule, we can compare models with the same dependent variable, such as we can compare models (1), (2) and (3) or we can compare models (4) and (5), and determine the most reliable profit indicator. The linear model (1) prescribed by the OECD may not produce the most reliable profit indicator under specific facts and circumstances.

Also, instead of selecting the bivariate regression function à priori such as from prior guidance from the OECD, by convention or from lazy habit mislabeled “best practices,” we can consider the lessons from Anscombe (1973) quartet. Anscombe created four datasets that produce identical regression results yet appear different when graphed. Each dataset consists of eleven (X, Y) points that demonstrate the importance of graphing data before analyzing them and the effect of outliers on statistical estimates. Only one of the four datasets produce a good statistical fit using linear model (1).

In sum, even a simple two variables (bivariate compared with multivariate) regression analysis can offer many advantages over a trope calculation of quartiles using the profit ratio Y / X postulated by the OECD. The purpose of testing these different models of the comparable operating profit function is to obtain more reliable estimates of the slope parameter, and of its accompanying standard error, together with a better statistical fit measured by the maximum adjusted R2.


Francis Anscombe, “Graphs in Statistical Analysis,” American Statistician, Vol. 27, No. 1 (February 1973). Stable URL:

Arthur Goldberger, Econometric Theory, John Wiley & Sons, 1964.

G. Maddala, Econometrics, McGraw-Hill, 1977.

Published on Sep 4, 2019 1:54:06 PM

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