1. The 1989 argument and the rule
In 1989, in my second year at the IRS in Manhattan, I wrote a research paper on Kitchin’s inventory cycles. Its thesis was that year-by-year transfer pricing audits are indefensible. Operating profits (then gross profits) oscillate with the inventory cycle of three to four years that Kitchin (1923) identified and that Metzler (1941) and Abramovitz (1950) later formalized. A single fiscal year is one point on an oscillation, one phase of the short-run cycle. It carries no information about the central tendency of the tested party or of its comparables, and the positions of the two in the cycle need not coincide.
The 1992 proposed section 482 regulations adopted this view, and the 1994 final regulations codified it in Treas. Reg. § 1.482-1(f)(2)(iii). Multiple-year data are required whenever business cycles or product life cycles affect results, and the comparable profits method uses the taxable year under review plus two or more preceding years. The OECD Guidelines followed in Chapter III (paras. 3.75–3.79). Three years are not an arbitrary convention. They approximate the Kitchin period, so that a three-year window contains one full oscillation of inventories, prices, and operating margins.
Of my three points about comparables, this one matters most in practice. The other two (comparables are going-concern entities, not transactions; the arm’s-length range is a statistical interval, not an interquartile range of ratios) concern the specification of the test. The multiple-year point concerns the data admitted to the test. This note states the point in the algebra of the comparables regression, where it becomes a testable restriction rather than an appeal to authority.
2. Notation
Let i = 1, \ldots, n index comparable companies and t = 1, \ldots, T index fiscal years, so the pooled sample size is N = nT. Observables are Compustat annual items written as Latin majuscules; parameters are Greek minuscules.
\begin{aligned}
R(i,t) &= \text{REVT, net sales}
\\[8pt]
C(i,t) &= \text{COGS, cost of goods sold}
\\[8pt]
G(i,t) &= \text{XSGA, selling, general and administrative expense}
\\[8pt]
X(i,t) &= C(i,t) + G(i,t) = \text{XOPR, total operating cost (excludes depreciation)}
\\[8pt]
D(i,t) &= \text{DP, depreciation and amortization}
\\[8pt]
K(i,t) &= \text{PPENT, net property, plant and equipment}
\\[8pt]
I(i,t) &= \text{CAPX, capital expenditures}
\\[8pt]
V(i,t) &= \text{INVT, inventories at fiscal year end}
\\[8pt]
\Delta V(i,t) &= V(i,t) - V(i,t-1), \ \text{inventory investment}
\\[8pt]
\mathrm{PURC}(i,t) &= \text{cost of goods produced or purchased during the year (not reported; see (2))}
\end{aligned}The baseline comparables regression tests capital intensity through the coefficient on net fixed assets:
\quad (1)\qquad R(i,t) = \alpha + \beta\, X(i,t) + \gamma\, K(i,t) + \varepsilon(i,t).
Here \alpha is the fixed-cost intercept, \beta = 1 + k the gross-of-markup coefficient on operating cost, and \gamma the return on net fixed assets whose 68% confidence interval (estimate \pm standard error) decides whether capital intensity is a comparability factor at all. Equation (1) contains no cyclical term.
3. The inventory identity
Cost of goods sold is cost of goods produced (or purchased for resale) adjusted for the change in inventories:
\quad
(2)\qquad
C(i,t) = V(i,t-1) + \mathrm{PURC}(i,t) - V(i,t) = \mathrm{PURC}(i,t) - \Delta V(i,t).Adding G to both sides gives the operating-cost identity
\quad
(3)\qquad
X(i,t) = \mathrm{PURC}(i,t) + G(i,t) - \Delta V(i,t).Inventory investment is already inside the regressor X of equation (1), with an implicit coefficient of -\beta. Adding the inventory level V to (1) would not test the cycle: it would estimate a return on the inventory stock, the unconstrained form of the working capital adjustment, and it would be collinear with X within each company because inventory turnover is stable over three years. The cycle lives in the flow \Delta V, Metzler’s state variable, and identity (3) dictates how it is moved to the right-hand side.
4. Structural equation and reduced form
Price is a markup on the cost of what is produced, and the inventory phase of the company shifts revenue given cost. The structural equation is written on cost of production, \mathrm{PURC} + G, with a separate coefficient \theta on inventory investment:
\quad
(4)\qquad
R(i,t) = \alpha + \beta\,[\mathrm{PURC}(i,t) + G(i,t)] + \theta\, \Delta V(i,t) + \gamma\, K(i,t) + \varepsilon(i,t).If \theta = 0, inventory accumulation is priced like any other cost. If \theta = -\beta, the markup is earned on goods sold only and inventory accumulation is revenue-neutral; this is the maintained hypothesis of (1). Any other value of \theta is the Kitchin signature: the position of the company in its inventory cycle moves revenue independently of cost.
Equation (4) cannot be estimated as written because \mathrm{PURC} is unobserved. Substituting (3), \mathrm{PURC} + G = X + \Delta V, and collecting terms:
\quad (5)\qquad R(i,t) = \alpha + \beta\, X(i,t) + (\beta + \theta)\, \Delta V(i,t) + \gamma\, K(i,t) + \varepsilon(i,t),
estimated by OLS as
\quad (6)\qquad R(i,t) = \lambda_0 + \lambda_1 X(i,t) + \lambda_2 \Delta V(i,t) + \lambda_3 K(i,t) + \varepsilon(i,t),
with
\quad (7)\qquad \lambda_0 = \alpha, \qquad \lambda_1 = \beta, \qquad \lambda_2 = \beta + \theta, \qquad \lambda_3 = \gamma.
The system is exactly identified. Writing b_j for the OLS estimate of \lambda_j and c for the implied estimate of \theta,
\quad
(8)\qquad
c = b_2 - b_1, \qquad \mathrm{SE}(c) = \big[\operatorname{Var}(b_2) + \operatorname{Var}(b_1) - 2\,\operatorname{Cov}(b_2, b_1)\big]^{1/2},where the variances and covariance are read from the firm-clustered sandwich covariance matrix of (6). The three hypotheses are linear restrictions on the reduced form:
- \lambda_2 = 0 (\theta = -\beta): markup on goods sold; inventory accumulation neutral. This is equation (1).
- \lambda_2 = \lambda_1 (\theta = 0): markup on goods produced.
- \lambda_2 < 0 (\theta < -\beta): Kitchin phase. Abramovitz found that inventory investment lags sales, so inventories are drawn down at the onset of the upswing and accumulate at the downturn; revenue given cost is then high when \Delta V < 0 and low when \Delta V > 0.
Each restriction is assessed by the 68% interval b_2 \pm \mathrm{SE}(b_2) and, for the second, c \pm \mathrm{SE}(c). No p-values are reported.
5. What a single-year indicator confounds
Equation (1) is equation (6) under the restriction \lambda_2 = 0. When the restriction fails, the baseline estimates of \beta and \gamma absorb the omitted inventory term. Writing \tilde X and \Delta\tilde V for the regressors after partialling out the intercept and K, the probability limit of the baseline slope is (Weisberg 2014, ch. 3)
\quad
(9)\qquad
\operatorname{plim} b_1 = \beta + \lambda_2\, \frac{\operatorname{Cov}(\Delta\tilde V, \tilde X)}{\operatorname{Var}(\tilde X)},with the analogous expression for the estimate of \gamma. The bias is the product of the structural inventory effect and the cyclical covariance between inventory investment and operating cost. This is the algebraic content of the 1989 argument: a single-year profit indicator confounds the pricing of the controlled transaction with the position of the company in the cycle, and the confounding term cannot be removed after the fact because it is a function of a variable the single-year ratio never observes. Pooling company-year observations over the Kitchin period is the only remedy, and it also supplies the degrees of freedom that a reliable interval estimate requires.
6. Flows versus stocks: the perpetual inventory method
A probable objection is that with the flow \Delta V on the right-hand side, the stock K should be replaced by a flow as well, so that both sides of the regression are measured in dollars per year. Two corrections to the objection, then the nesting that disposes of it.
First, the flow counterpart of K is not \Delta K but capital expenditure. By the perpetual inventory method,
\quad
(10)\qquad
K(i,t) = K(i,t-1) + I(i,t) - D(i,t), \qquad \text{so} \qquad I(i,t) = \Delta K(i,t) + D(i,t).The parallel to identity (3) is exact: (3) moves accumulation out of cost of goods sold; (10) moves accumulation out of investment spending. Since XOPR excludes depreciation (\mathrm{OIBDP} = \mathrm{REVT} - \mathrm{XOPR}), D is not already in X, and I and \Delta K differ on the right-hand side by a capital-consumption flow with its own coefficient.
Second, dimensional homogeneity is satisfied by the stock specification. R, X and \Delta V are dollars per year; K is dollars; \gamma is therefore a rate per year and \gamma K is dollars per year. Parameters carry dimensions: \beta and \lambda_2 are pure numbers, \gamma is a rate of return. The objection has no dimensional footing. What it has is substantive content: it proposes replacing a profit-rate term (a return on capital advanced) by an accelerator term (revenue responding to investment spending). That changes the hypothesis. The capital-intensity claim under test is a claim about the stock; a coefficient on \Delta K would neither confirm nor refute it.
Rather than argue, nest both. Write K(i,t) = K(i,t-1) + \Delta K(i,t) and free the two coefficients:
\quad (11)\qquad R(i,t) = \alpha + \beta\, X(i,t) + \lambda_2\, \Delta V(i,t) + \gamma_1\, K(i,t-1) + \gamma_2\, \Delta K(i,t) + \varepsilon(i,t).
- \gamma_1 = \gamma_2: the stock specification (6), with end-of-year K(t).
- \gamma_2 = 0: return on beginning-of-year capital, K(t-1), the correct timing for a profit rate on capital advanced.
- \gamma_1 = 0: the objector’s flow-only specification.
Substituting \Delta K = I - D gives the CAPX form, \gamma_2 I - \gamma_2 D, and freeing the coefficient on D tests whether the markup applies to capital consumption as it does to operating cost. The same nesting applies on the inventory side, V(t) = V(t-1) + \Delta V(t), so the general equation is
\quad (12)\qquad R = \alpha + \beta X + \phi_1 V(t-1) + \phi_2 \Delta V + \gamma_1 K(t-1) + \gamma_2 \Delta K + \varepsilon,
and (6) is the restriction \phi_1 = 0, \gamma_1 = \gamma_2.
Two cautions. \Delta K is the state variable of the Juglar cycle of seven to eleven years (Juglar 1862; Schumpeter 1939), not of the Kitchin cycle; inside a three-year window it is a trend term, not a phase term, so \gamma_2 does not carry the cyclical interpretation that \lambda_2 does. And seven slopes on N = 3n observations is thin at n = 8; the general form (12) is for the larger samples that an entity database can supply, the restricted forms for a typical comparables set.
7. Estimation notes
Data. The lag in \Delta V consumes one fiscal year, so T = 3 company-year observations per comparable require four years of data.
Covariance. \Delta V is measured in dollars and inherits the heteroskedasticity of X; the three years of each company are serially dependent by construction of the cycle. The covariance estimator is the firm-clustered sandwich (X'X)^{-1} S (X'X)^{-1}, with S the middle matrix formed from company-level score sums.
Aggregate phase. Because the Kitchin cycle is economy-wide, (6) may be augmented with year effects \alpha_t. Year effects absorb the common phase; \Delta V then measures the idiosyncratic phase of each comparable. A nonzero \lambda_2 after year effects is the strong result.
8. Two corollaries
First, an isolated loss year of a comparable is cyclical evidence, not a selection defect; persistent losses are a different matter and raise the going-concern question. Second, comparability adjustments that ignore the cycle (working capital adjustments computed for a single year are the common case) adjust noise: they regress on the level V in a year in which \Delta V, the variable that carries the information, is unobserved.
References
Abramovitz, M. (1950). Inventories and Business Cycles, with Special Reference to Manufacturers’ Inventories. New York: NBER. The empirical foundation of the inventory cycle in U.S. manufacturing; source of the lag of inventory investment behind sales.
Juglar, C. (1862). Des crises commerciales et de leur retour périodique en France, en Angleterre et aux États-Unis. Paris: Guillaumin. The fixed-investment cycle of seven to eleven years. First edition on Gallica, https://gallica.bnf.fr/ark:/12148/bpt6k1060720; the revised 1889 edition (ed. L. Frobert, ENS Éditions, 2014) is open access at https://books.openedition.org/enseditions/1382.
Kitchin, J. (1923). Cycles and trends in economic factors. Review of Economic Statistics, 5(1), 10–16. Identification of the 40-month cycle in prices and interest rates.
Metzler, L. A. (1941). The nature and stability of inventory cycles. Review of Economic Statistics, 23(3), 113–129. The accelerator model in which inventory investment is the state variable of the oscillation.
OECD (2022). Transfer Pricing Guidelines for Multinational Enterprises and Tax Administrations, Chapter III, paras. 3.75–3.79. Multiple-year data.
Schumpeter, J. A. (1939). Business Cycles. New York: McGraw-Hill, vol. 1, ch. IV. The three-cycle schema (Kitchin, Juglar, Kondratieff) and the accessible account of Juglar’s finding.
Treas. Reg. § 1.482-1(f)(2)(iii). Multiple year data.
Weisberg, S. (2014). Applied Linear Regression, 4th ed. Hoboken: Wiley. Omitted-variable algebra and added-variable (partial) regression.