It's useful to model company profits using a first-order autoregressive AR(1) process. However, “duality” (invertibility) between an AR(1) model and a weighted sum of random errors tempers theoretical or long-term ambitions. Duality is a metamorphosis from one dynamic process to another such that an AR(1) model can be converted into a moving average of random errors model. Moving average models lack X-factors explanation.

**First-Order Dynamics**

Consider a simple AR(1) model of company profit ratios:

(1) Y(*t* + 1) = α + β Y(*t*)

reflecting time *period t* = 0, 1, 2, 3, …, T.

In transfer pricing, T ≤ 5 years; thus, we are concerned with short-term behavior of comparable company profit ratios. The parameters (α and β) can be estimated using regression analysis and comparable company data for the variable Y(*t*). See Saber Elaydi, *Introduction to Difference Equations* (3^{rd} edition), Springer, 2005, p. 4, eq. 1.2.8, which contains solutions of (1) for different ranges of the slope coefficient.

**Fixed-Point Equilibrium**

The expected (or equilibrium) value of Y(*t*) is a fixed-point that is invariant to time. This means that as long as we have no disturbance in system (1), the equilibrium of Y(*t*) remains a fixed-point. Disturbances include perturbations of the parameters (α and β) or changes of the variable Y(*t*) itself. The equilibrium of Y(*t*) depends on the estimated parameters:

(4a) Y(*t*) = Y(0), which means that Y(*t*) is stuck on its initial position, if β = 1 and α = 0.

(4b) Y(*t*) = {α / (1 – β)}, a fixed-point, if β ≠ 1.

If β = 1 and α ≠ 0, a fixed-point equilibrium does not exist, and the variable Y(*t*) increases indefinitely if α > 1; or decreases indefinitely if α < 1. This is a random walk with an increasing or decreasing drift, depending on the intercept. In statistics, we have the Dickey-Fuller test of the “unit root” hypothesis for β = 1. This special condition is unlikely to be a good reflection of company profit margins. However, random walk is vogue modeling of financial markets. See an influential book by Paul Cootner (ed.), *The Random Character of Stock Market Prices*, MIT Press, 1964.

If the slope coefficient is in the closed range (0 < β < 1), the variable Y(*t*) is convergent. Alpha Chiang & Kevin Wainwright, *Fundamental Methods of Mathematical Economics* (4^{th} edition), McGraw-Hill, 2005, pp. 552-553, contain a useful table covering many numerical regions of β and their respective charts. None of the charts is a good representation of Y(*t*) as company profits, so more complex models need to be tested if the time horizon of the relevant variable is long (e.g., if T exceeds three years when the correlation between Y(*t*) and its past values is likely to become tenuous).

**AR and MA Duality**

In prior blogs we postulated that company profit ratios have two components: (*i*) a “routine” component that is common to all the comparables; (*ii*) plus a surplus profit that is earned by innovators who benefit from significant market share:

(5) Y(*t*) = μ + S(*t*), which implies that S(*t*) = Y(*t*) – μ.

Massimo Pivetti, *An Essay on Money and Distribution*, St. Martin's Press, 1991, contains a historical discussion of this two component profits model.

Assume that the surplus (or excess) profit ratio S(*t*) can be represented by an AR(1) model.

(6) S(*t* + 1) = α + β S(*t*) + U(*t*),

which is like equation (1) except that we have added a random component, U(*t*).

We can assume that U(*t*) is normal and has zero expected value and fixed variance. However, this normality assumption is not necessary. Model (6) is the work-horse of Dennis Mueller and his followers. See Dennis Mueller, *Profits in the Long Run*, Cambridge University Press, 1986, p. 13, equation 2.7; and Dennis Mueller (ed.), *The Dynamics of Company Profits*, Cambridge University Press, 1990, p. 35, equation 3.2. This edited book contains several chapters using model (6) to study company profits in the United States, Canada, Germany, France, Japan, and the United Kingdom.

As a historical curiosum, our familiarity with Dennis Mueller's research on company profits (when we taught economics at the Graduate Faculty of the New School in New York) led us to introduce Compustat to the IRS as the principal database to find company comparables in transfer pricing. In 1989, we tested Moody's Industrials (now Mergent) and Compact Diclosure (acquired by Thomson Reuters), and selected Compustat because of its advanced data normalization standards. Before our arrival from academia, IRS economists were using *RMA Annual Statement Studies* to find industry *gross* profits to determine corporate transfer pricing assessments based on the resale price or cost plus method. We introduced an operating profits method (initially called CPI [comparable profits interval], and then CPM [comparable profits method]; later called TNMM by the OECD) based on company comparables. This company-based profit analysis persuaded the IRS Commissioner, Shirley Peterson (1992), to establish the dictum at a briefing regarding the CPM, which survived in the U.S. 26 CFR Section 1.482-1(d)(2) transfer pricing provision: “unadjusted industry average returns themselves cannot establish arm’s length results.”

Let's return to our AR(1) model of company profit ratios scaled by sales or assets:

Define L S(*t*) ≡ S(*t* – 1) and L* ^{k}* ≡ S(

*t*–

*k*), where L is a lag or time-shift operator. For simplicity, we disregard the intercept and (6) becomes:

(7) (1 – β L) S(*t*) = U(*t*) from which we obtain:

(8) S(*t*) = (1 – β L)^{−1} U(*t*).

We know that (1 – β L)^{−1} = (1 + β L + β^{2} L^{2} + β^{3} L^{3} + …) when T is sufficiently long. See Germund Dahlquist & Åke Björk, *Numerical Methods*, Prentice-Hall, 1974, p. 66, Table 3.1.1. The properties of the L shift operator can be found on their p. 255, eq. 7.1.1.

Therefore, model (6) can be subject to a revealing metamorphosis:

(9) S(*t*) = (1 + β L + β^{2} L^{2} + β^{3} L^{3} + …) U(*t*), or

(10) S(*t*) = U(*t*) + β U(*t −* 1) + β^{2} U(*t* − 2) + β^{3} U(*t *− 3) + …

Remember that we set the intercept of (6) at zero for convenience. Thus, the company profit equation (5) is converted into a fixed point element representing a common (routine) return on investments plus a weighted sum of randon events, sans any X-factor explanation. This development shows that an AR(1) model is a metamorphosed version of a MA(*k*) process of indefinite *k*-order.

In essence, the AR(1) model may not be a satisfactory work-horse because it’s more likely to capture *short-term* company profit behavior during which the correlation of Y(*t*) and its immediate past values is strong. We are not sure about long-run movements of Y(*t*). In transfer pricing, AR(1) is defensible because we are concerned with short-term behavior of company profits. Thinking long-term, AR(1) may not be satisfactory because the duality of AR(1) and MA(*k*) robes the autoregressive model of any explanatory power in the sense that the behavior of Y(*t*) can be explained by a weighted sum of random errors.

In summary, a first-order dynamic AR(1) model produces well-known behavior regarding equilibrium, drift or trend, and oscillations. This behavior may reflect short-term reaction to random events, and not long-term gravitation of company profits. In the short-term world of transfer pricing, an AR(1) model is likely to produce more reliable measures of company profits than the static linear model of profits vs. revenue, costs, or assets (aka “capital employed”) prescribed by the OECD and certain acolyte tax authorities.

**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

**RoyaltyStat**provides premier online databases of**royalty rates**extracted from unredacted agreements,**normalized company financials**(income statement, balance sheet, cash flow), and**annual reports**. We provide high-quality data, built-in analytical tools, customer training and attentive technical support.