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Applying Regression Analysis to Transfer Pricing's Cup Method

Posted by Ednaldo Silva

The comparable uncontrolled price (CUP) method is described in US Treas. Reg. §1.482-3(b).

We hold that the empirical verification of CUP transactions is best measured by regression analysis.

CUP Rules Aren’t Will-o’-the-Wisp (Ignis fatuss)

The CUP method is used to evaluate the amount charged in a controlled transaction involving the transfer of tangible property by reference to the amount charged in one or more comparable uncontrolled transactions.

This can be tested using the bivariate regression:

     (1)     Y(i, t) = α + β X(i, t) + V(i, t)

     (2)     V(i, t) ≈ Normal(0, σ2)

We assume that the intercept is zero (α = 0), and test the CUP “law of one-price” hypothesis that the slope coefficient is unity (β = 1).

The residual errors V(i, t) must be examined to determine if they are well-behaved. The usual assumptions are that the residual errors V(i, t) are independent (no serial correlation), they have zero mean and constant variance (σ2), and follow a normal distribution. See Weisberg (2014), pp. 23, 25, 35-38, 204-218. 

The variables Y(i, t) and X(i, t) denote the controlled and uncontrolled prices per unit of the product (tangible property). The subscript i = 1 to N denotes the product number measured by SKU or other specific product code; and t = 1 to T denotes the time (e.g., day) of the paired controlled and uncontrolled unit price transactions.

The two assumptions that α = 0 and β = 1, which we can submit to statistical significance (reliability) testing, can reduce equation (1) to:

     (3)     Y(i, t) = X(i, t) + V(i, t),

which means that the controlled price per unit of the product purchased or sold is identical to the uncontrolled unit price of the same product plus a residual error whose mean is zero and thus has an expected negligible effect on measuring the controlled price being tested for arm's length behavior.

Applicable CUP Rules

The application of the CUP method is discussed in Treas. Reg. §1.482-3(b)(2)(ii) and (iii).

Treas. Reg. §1.482-3(b)(2)(ii) (Comparability) has two sub-sections (which we paraphrase in extenso):

     (A) The comparability between controlled and uncontrolled transactions is determined by applying the provisions of Treas. Reg. §1.482-1(d). Although all the factors described in Treas. Reg. §1.482-1(d)(3) must be considered, similarity of products will have the greatest effect on comparability under the CUP method.

     In addition, comparability under the CUP method depends on close similarity respecting contractual terms or economic conditions, or adjustments to account for any differences.

     The results derived from applying the CUP method can be the most direct and reliable measure of an arm’s length price for the controlled transaction if an uncontrolled transaction has no differences with the controlled transaction that would affect the price, or if there are only minor differences that have a definite and reasonably ascertainable effect on price and for which appropriate adjustments are made.

     If such adjustments cannot be made, or if there are more than minor differences between the controlled and uncontrolled transactions, the CUP method may be used, but the reliability of the results as a measure of the arm’s length price will be reduced. Further, if there are material product differences for which reliable adjustments cannot be made, the CUP method cannot provide a reliable measure of an arm’s length result.

     (B) Adjustments for differences between controlled and uncontrolled transactions. If there are differences between the controlled and uncontrolled transactions that would affect price, adjustments must be made to the price of the uncontrolled transaction according to the comparability provisions of Treas. Reg. §1.482-1(d)(2). Specific examples of the adjustment factors that may be relevant to the CUP method include the (1) Quality of the product; (2) Contractual terms (e.g., scope and terms of warranties provided, sales or purchase volume, credit terms, transport terms); (3) Level of the market (i.e., wholesale, retail, etc.); (4) Geographic market in which the transaction takes place; (5) Date of the transaction; (6) Intangible property associated with the sale; (7) Foreign currency risks; and (8) Alternatives realistically available to the buyer and seller.

In addition, Treas. Reg. §1.482-3(b)(2)(iii) (Data and assumptions) imposes conditions about data quality. The reliability of the results derived from the CUP method is affected by the completeness and accuracy of the data used and the reliability of the assumptions made to apply the method. See §1.482-1(c) (Best method rule).

A Real CUP Illustration

We tested the “law of one-price” model (1) using a random sample of 64 ordered pairs of controlled and uncontrolled prices in the taxpayer's audit year, and we obtained the regression results:

     (5)     PriceCust1 = 0.01801 + 1.0118 PriceCust0, R2 = 0.988

                   SE Coef.    0.0574      0.0207

                   t-Stats.      0.3135   48.8368

The regression slope coefficient (estimated β = 1.0118) is significant because its t-statistics = 48.8 far exceeds the t-critical value ≈ 2 for 62 degrees of freedom obtained from a t-table.  Using this t-critical value ≈ 2, we computed a CUP interval based on the standard error (Newey-West SE Coef.) of the regression slope coefficient: ± (2 × 0.0207) = ± 0.0414, as follows:

     (6a)     Estimated β ± 2 × SE (Estimated β), or

     (6b)     95% Confidence Interval for the regression slope coefficient: 0.9704 ≤ β ≤ 1.0532

The lower limit of this confidence interval = (1.0118 − 0.0414) = 0.9704, and the upper limit = (1.0118 + 0.0414) = 1.0532. See Wonnacott and Wannacott (1969), p. 244, James et. al. (2013), p. 66, and Weisberg (2014), p. 31.

The CUP hypothesis that β = 1 is inside of this factual-based 95% confidence interval. Therefore, we can conclude (with strong statistical confidence) that the taxpayer has established (as a factual matter) the existence of CUP transactions during the audit year under examination and that no transfer pricing adjustment was warranted. The chart below provides a visual image of this real CUP examination.

Special CUP Rule for Quoted Prices

The CUP method includes a special rule under Treas. Reg. §1.482-2(b)(5) (Indirect evidence of comparable uncontrolled transactions), which contains two conditional sub-sections:

     (i)  A CUP may be derived from data from public exchanges or quotation media, but only if the following requirements are met:

          (A) The data is widely and routinely used in the ordinary course of business in the industry to negotiate prices for uncontrolled sales;

          (B) The data derived from public exchanges or quotation media is used to set prices in the controlled transaction in the same way it is used by uncontrolled taxpayers in the industry; and

          (C) The amount charged in the controlled transaction is adjusted to reflect differences in product quality and quantity, contractual terms, transportation costs, market conditions, risks borne, and other factors that affect the price that would be agreed to by uncontrolled taxpayers.

     (ii) Limitation. The use of data from public exchanges or quotation media may not be appropriate under extraordinary market conditions.

Statistical Significance

The existence of actual CUP transactions can be demonstrated in a reliable way by testing the statistical significance of the “law of one-price” between controlled and uncontrolled individual (disaggregated) product sales or purchases. Here, we show that regression analysis is a reliable tool to assay claimed CUP transactions and establish their verifiable existence.

Without the rigor offered by regression analysis, we may be lost in Ulysses Gaze and not reach CUP island.


The expression Ulysses’ Gaze (1995) refers to the masterful quest film directed by Theo Angelopoulos

Ednaldo Silva, “Applying Regression Analysis to the Cup Method,” Tax Management Transfer Pricing Report, Vol. 10, No. 15, November 28, 2001. This article uses unidentifiable data from an actual transfer pricing case docketed in the US Tax Court (but settled in pretrial negotiations) in which this author served as the taxpayer's economic expert witness and utilized the CUP method to demonstrate that the IRS transfer pricing audit deficiency was unwarranted. See (Bloomberg BNA, November 26, 1997):

IRS Concedes Adjustment in Case of Houston Electronics Distributor: The Internal Revenue Service Oct. 22 [1997] conceded the $21.7 million in deficiencies it had assessed a Houston electronics distributor, most of which stemmed from sales the Service attributed to the distributor, Eletex Inc., from a Mexican manufacturer, according to stipulated decisions filed in Tax Court. [Eletex Inc. v. Comr., No. 7245-96, T.C., and Conductores Monterrey S.A. de C.V. v. Comr., No. 27498-96, T.C., stipulated decisions filed 10/22/97]. The stipulated decision in Eletex’s case said the parties agreed no deficiencies or penalties were due from the company for 1988-93.” Reproduced with permission. Published November 26, 1997. Copyright 1997 by The Bureau of National Affairs, Inc. (800-372-1033) <>

Download the article.

Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani, An Introduction to Statistical Learning, Springer, 2013 (corrected at 4th printing 2014). Chapter 3 (Linear regression) contains a lucid and well-illustrated introduction to simple and multiple regression, including usual measures of data fit quality.

Treas. Reg. §1.482-3 (Methods to determine taxable income in connection with a transfer of tangible property):

Sanford Weisberg, Applied Regression Analysis (4th edition), Wiley, 2014. This book was written by a statistician (non-economist) and his approach is less cluttered than introductory econometrics textbooks. 

Thomas Wonnacott and Ronald Wonnacott, Instroductory Statistics, Wiley, 1969. The authors were gifted writers. Chapter 7 (Estimation), Chapter 12 (Regression theory) and Chapter 13 (Multiple regression) contain lucid explanations of these topics. We were fortunate to be assigned this wonderful book as the first undergraduate text in statistics and regression analysis.


Fitted Line PrCust1 versus PrCust0


Published on Oct 7, 2019, 11:42:15 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: CUP Method