RoyaltyStat Blog

Properties of the AR(1) Model of the Profit Margin

Posted by Ednaldo Silva on Oct 31, 2016 10:13:44 AM

It’s useful to study the mean and variance of the first-order autoregressive model (AR(1)), which is postulated as univariate:

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

where the slope coefficient ABS(ρ) < 1 and the random error V(t) ≈ Normal(0, σ) with zero mean and constant variance.

The mean and variance properties are useful in order to know the central value and data spread of the variable examined, and to compute a confidence interval representing an arm's length range of acceptable results of such a variable. In transfer pricing, the variable Y(t) can be the operating profit margin of a selected (or composite) comparable examined during the taxpayer's audit year t = 1, 2, 3, …, T, including prior or post audit years to account for short to medium term business fluctuations. In fact, we developed a profit margin theory leading to this AR(1) model on a prior blog, dated also on Oct. 31, 2016.

Taking the expected value of (1), we obtain:

     (2)     E(Y(t)) = α + ρ E(Y(t – 1)),

because E(V(t)) = 0, meaning that overall the random error is expected to have a negligible effect explaining the behavior of Y(t).

Assuming stability (aka stationary condition), E(Y(t)) = E(Y(t – 1)) = μ is a constant. Thus,

     (3)     μ = α + ρ μ or E(Y(t)) = μ = (1 – ρ)−1 α = α / (1 – ρ).

The variance of (1) is obtained by using certain elementary rules of the Var operator:

     (4)     Var(Y(t)) = ρ2 Var(Y(t – 1)) + Var(V(t))

                              = ρ2 Var(Y(t – 1)) + σ2, where σ2 = Var(V(t)).

Assuming stability again, Var(Y(t)) = Var(Y(t – 1)), such that we obtain:

     (5)     Var(Y(t)) = σ2 / (1 – ρ2), if ρ2 < 1.

The variance of the AR(1) model is equal to the variance of the random error (σ) weighted by the reciprocal of one minus the slope coefficient squared. We can use (5) to obtain an arm’s length range of operating profit margins derived from comparable taxpayers and determine the “true taxable income” of the “tested party”.  E.g., we can calculate a 50% confidence interval using an estimated dependent variable:

     (6)     Ŷ(t) ± 0.675 SQRT(σ2 / (1 – ρ2)).

The multiplier 0.675 is an approximation. A more exact multiplier for a given confidence level (probability) and count can be obtained by using the Excel function, TINV(probability, degrees of freedom). E.g., TINV(0.5, 12) = 0.695, because we know that the AR(1) regression has (count – 2) degrees of freedom. Maurice Kendall & Keith Ord, Time Series (3rd edition), Edward Arnold, 1990, call the AR(p) model a Markov process when p = 1 (section 5.9), and a Yule process when p = 2 (section 5.14). For an academic reference, Kendall & Ord show the mean of AR(1) on equation 5.19 and the variance on equation 5.22, p. 56.


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: [email protected]

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Topics: Net profit