We can determine an arm's length profit margin (expressed as operating profit divided by net sales) of a controlled taxpayer (“tested party”) by using a first-order autoregressive model, which we can show (like any stable first-order difference equation) to be equivalent to a range of comparable “routine” profit margins plus a weighted random error time series.

Since classical (Smith, Ricardo) economics, we can let the profit margin in period *t* > 0 be composed of a common (or routine) element plus a surplus, variously called excess, “quasi-rent” or other more nebulous expressions:

(1) Y(*t*) = α + S(*t*),

where the unknown parameter to be estimated (α) represents a “routine” (Ricardo’s return to “risk and trouble”) or normal profit margin, which is expected to be common to the rival enterprises in a similar industry, and S(*t*) represents a surplus profit margin. In competition, measured by a large number of enterprises in the selected industry, S(*t*) is expected to dissipate from the price-reducing effect of innovating entrants. In oligopoly, S(*t*) persists because of barriers to entry, such as high market share resulting from product differentiation or from increasing returns to scale.

We shall model the surplus profit margin as an autoregressive (AR(1)) process, which has well-known stable and cyclical dynamics.

(2) S(*t*) = ρ S(*t* – 1) + V(*t*),

where we expect ABS(ρ) < 1 and V(*t*) ≈ Normal(0, σ), *i*.*e*., the random error is assumed to have a normal distribution with zero mean and constant variance.

We know that equation (1) implies that Y(*t* – 1) = α + S(*t* – 1), which implies further that Y(*t* – 1) − α = S(*t* – 1). We substitute equation (2) into (1), and utilize this transposed expression Y(*t* – 1) − α = S(*t* – 1) to obtain:

(3) Y(*t*) = α + ρ S(*t* – 1) + V(*t*),

= α + ρ (Y(*t* – 1) – α) + V(*t*),

= α + ρ Y(*t* – 1) – ρ α + V(*t*),

= (1 – ρ) α + ρ Y(*t* – 1) + V(*t*),

(4) Y(*t*) = C + ρ Y(*t* – 1) + V(*t*), where C = (1 – ρ) α.

Model (4) is called AR(1) with a constant drift, and is more parsimonious than (3) because we reduced from two (S(*t* – 1), Y(*t*)) to one variable per comparable observation, the time-series Y(*t*). Equation (4) can be estimated by using OLS such that we obtain C, ρ, and σ. See *e*.*g*. Maurice Kendall & Keith Ord, *Time Series* (3^{rd} edition), Edward Arnold, 1990, section 7.2 (Fitting autoregressions).

**Using the Lag Operator to Show Invertibility**

The lag operator (L) is useful in time series analysis to convert an AR(p) process into a moving average MA(q) process. We multiply L to a variable producing one-period lag value of the same variable:

(5) L Y(*t*) = Y(*t* – 1).

Applying L to Y(*t* – 1) = L (L Y(*t*)) = L^{2} Y(*t*) = Y(*t* – 2). In general, we obtain the formula:

(6) L* ^{k}* = Y(

*t*–

*k*).

The L-operator does not change a constant; thus, L C = C, where C is a constant such as the intercept of (4).

Here is an interesting invertible development. We can use the L-operator such that (4) becomes:

(7) (1 – ρ L) Y(*t*) = C + V(*t*), and obtain:

Y(*t*) = (1 – ρ L)^{−1} (C + V(*t*)), which becomes:

Y(*t*) = (1 – ρ L)^{−1} C + (1 + ρ L + ρ^{2} L^{2} + ρ^{3} L^{3} + …) V(*t*),

Y(*t*) = λ + V(*t*) + ρ V(*t* − 1) + ρ^{2} V(*t* – 2) + ρ^{3} V(*t* – 3) + …, where λ = (1 – ρ L)^{−1} C.

(8) Y(*t*) = λ + φ_{0} V(*t*) + φ_{1} V(*t* − 1) + φ_{2} V(*t* – 2) + φ_{3} V(*t* – 3) + …, where φ_{0} = 1.

Since the L-operator does not change the constant, we can simplify the intercept of (8) and obtain:

(9) λ = (1 – ρ L)^{−}^{1} C = (1 – ρ)^{−}^{1} C = C / (1 – ρ).

Equation (8) shows that (4) is equivalent to a constant (aka “routine”) profit margin (λ), common to the rival comparable enterprises in the selected industry, *plus* a decreasing weighted indefinite order of random shocks, V(*t*) for *t* > 0. Since the one-period lag coefficient is expected to be less than one, and so this dynamic system is stable, the effect of past random shocks dissipate as we consider back years from the audit year. The difference between our dynamic or autoregressive model (4) or (8) and the naïve constant mean profit margin model, postulated in the OECD *Transfer Pricing Guidelines*, ¶ 2.90 (“net profit divided by sales”), is that the estimated constant (“routine”) operating profit margin (λ) and the random error structure are radically different. To see this difference, take the expected value and consider the variance of both models.

Proof is in the pudding! In a given transfer pricing case, we can test the explanatory power of the naïve versus the AR(1) model by subjecting them to a simple Friedman (1953, p. 7) type demarcation criterion that “the only relevant test of the validity of a hypothesis is comparison of its predictions with experience.” See Milton Friedman, “The Methodology of Positive Economics” in his *Essays in Positive Economics*, University of Chicago Press, 1953.

**Ednaldo Silva**(Ph.D.) is founder and managing director at 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: esilva@royaltystat.com

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