We can consider a first-order autoregressive (AR(1)) model to determine an arm’s length profit margin of a “tested party” subject to transfer pricing audit compliance:

##### (1) Y(*t*) = α + ρ Y(*t* – 1) + V(*t*)

for a selected audit period *t* = 1 to T, such that |ρ| < 1 and the random error V(*t*) is Normal(0, σ). This means that the *expected value *of the random error, E(V(*t*)) = 0, and *standard deviation*,* *STDV(V(*t*)) = σ. See Michael Clements & David Hendry, *Forecasting Economic Time Series*, Cambridge University Press, 1998, p. 81 (normal assumption of the random errors is not necessary).

Assuming a stationary condition when Y(*t*) = Y(*t* – 1), we obtain:

(2) E(Y(*t*)) = α / (1 − ρ), and

(3) STDV(Y(*t*)) = SQRT(σ^{2} / (1 – ρ^{2})) for all *t* considered during the audit analysis.

Given data for Y(*t* – 1), *i*.*e*., information about a comparable profit margin in a year prior to the audit year, the conditional expectation of the dependent variable (such as an arm’s length profit margin) is:

(4) E(Y(*t*) given Y(*t* – 1)) = α + ρ Y(*t* – 1), and we can forecast one period-ahead using the formula:

(5) Ŷ(T + 1) = α + ρ Y(T),

where the intercept (α), slope coefficient (ρ), and STDV of the random errors (σ) are estimated using comparable company Y(*t*) data. We must have one time series equation per company selected as comparable to the tested party. The variable Ŷ(T + 1) is a one-period ahead forecast of a comparable profit margin. Equation (5) shows the folly of forecasting two, three or more periods ahead, because they would be forecasts based on prior forecasts!

If we forecast the profit margin associated with certain identifiable intangibles transferred among affiliates, which in the U.S. would be subject to the “commensurate with income” (CWI or periodic adjustments) rule, the forecasting error must be bound. *E*.*g*., using one-year ahead (T + 1) periodic adjustments test:

##### (6) Y(T + 1) ≤ 0.2 Ŷ(T + 1),

where the error tolerance of the difference between the actual and forecasted profit margin is 20%; otherwise, the value of the intangible calculated with the forecasted profit margin is subject to the periodic adjustments rule, valid for each of five years ahead of the transferred intangibles. See U.S. Treas. Reg. § 1.482-4(f)(2) (Periodic adjustments), located at*: http://www.ecfr.gov/cgi-bin/retrieveECFR?gp=1&SID=30b7720aa9068e3cc13d303855959759&ty=HTML&h=L&mc=true&r=SECTION&n=se26.8.1_1482_64*

The OECD *Transfer Pricing Guidelines* (2010) do not include a CWI rule, but we believe that it’s a matter of time before this slip is corrected because allowing value forecast without considering error bounds is inconsistent with basic principles of economic and statistical forecasting. We know that every forecast is subject to errors; therefore, allowing forecasts to assess corporate income tax connected to an affiliate transfer of intangibles but not subject such forecasts to error limits is absurd. Such profligacy is likely to induce self-serving prophecies to avoid income tax.