We discuss three contending transfer pricing models. Let P = “Net” profits, S = Sales, and A = Assets: For simplicity, we don’t consider profit markup on total costs, and exclude random errors:

(1) P(*i*) = α_{0} + α_{1} S(*i*), which is a model of *profit margin*.

(2) P(*i*) = β_{0} + β_{1} A(*i*), which is a model of *profit rate* (in OECD lingo: “return on assets”).

(3) P(*i*) = γ_{0} + γ_{1} S(*i*) + γ_{2} A(*i*), which is a model of *adjusted **profit margin* (profit margin *adjusted* for asset turnover or profit margin *adjusted* for asset instensity).

These rival profit models can be run using OLS (ordinary least squares) and data from *i* = 1, 2, …, N comparable uncontrolled enterprises.

One way to select the best NPI is to run three regression models and select the model with the smallest standard error. However, OLS is not abiding and if Revenue and Assets are correlated, we must use model (3) because otherwise α_{1} or β_{1} is *biased*; and biased coefficients are impermissible in science.

To measure this bias, we test the correlation between Revenue and Assets:

(4) A(*i*) = δ_{0} + δ_{1} S(*i*), which is asset turnover (“asset intensity”).

Empirical tests using Capital IQ Compustat enterprise-level data in RoyaltyStat shows that estimates of δ_{1} ≈ 0.8, which is high positive correlation between revenue and assets.

We substitute (4) into (3) and obtain:

##### (5) P(*i*) = γ_{0} + γ_{1} S(*i*) + γ_{2 }{δ_{0} + δ_{1} S(*i*)}

##### = γ_{0} + γ_{2} δ_{0} + γ_{1 }S(*i*) + γ_{2 }δ_{1} S(*i*)

##### = λ_{0} + λ_{1} S(*i*)

where λ0 = γ0 + γ2 δ0 and λ_{1} = γ_{1} + γ_{2 }δ_{1}.

Above, the factor γ_{2 }δ_{1} is a measure of bias if we use profit model (1) and δ_{1} is different from zero. We can find the bias using model (2) in similar fashion.

Therefore, we must use model (3) when δ_{1} is significant, not zero. Else, we can use (1) or (2) when δ1 = 0, or when δ_{1} is insignificant or negligible (immaterial).

Selecting the most appropriate NPI is not simple like the role of Paladin in Charlemagne’s court. To be defensible against audit scrutiny, we must use statistical analysis to select an unbiased NPI that has the smallest standard error among competing unbiased profit indicators.