Abstract

A companion post established that the sample mean of ratio-based profit level indicators (PLI), m(i) = Y(i)/X(i), is biased by a/H when the underlying relationship Y = a + bX has a nonzero intercept. This post derives the variance of m(i) and shows that division by X induces heteroscedasticity: Var[m(i)] = σ²/X(i)². Comparables with small bases inflate both the bias and the variance. The proper remedy is regression inference on the slope coefficient b, whose 68% confidence interval is b̂ ± SE(b̂)—unbiased, efficient, and with known coverage probability. Appendix A provides a step-by-step derivation of the variance formula for readers less familiar with statistical algebra.

1. Setup

From the structural economic model relating profit Y (OIADP) to base X (REVT, XOPR, PPENT) across N comparable enterprises:

Y(i) = a + b \cdot X(i) + \varepsilon(i), \quad \varepsilon(i) \sim (0, \sigma^2)

The ratio-based profit level indicator is:

m(i) = \frac{Y(i)}{X(i)} = b + \frac{a}{X(i)} + \frac{\varepsilon(i)}{X(i)}

A prior post derived E[m̄] = b + a/H, where H is the harmonic mean of the denominator. This post derives the variance of m(i) and contrasts it with more defensible regression inference on the slope b. (See Appendix A for the detailed algebra.)

2. Variance of a Single Ratio Observation

Conditional on the independent variable X(i), the only stochastic part in m(i) is ε(i)/X(i). Under the standard homoscedasticity assumption Var[ε(i)] = σ²:

\mathrm{Var}[m(i)\mid X(i)] = \mathrm{Var}\!\left[\frac{\varepsilon(i)}{X(i)}\right] = \frac{\sigma^2}{X(i)^2}

\mathrm{Var}[m(i)] = \frac{\sigma^2}{X(i)^2}

The variance of each ratio observation is inversely proportional to the square of the base. A comparable with X(i) = 10 has 100 times the variance of a comparable with X(i) = 100. This is heteroscedasticity by construction—not a violation of model assumptions, but an algebraic consequence of dividing by X.

3. Variance of the Sample Mean of Ratios

The sample mean is:

\bar{m} = \frac{1}{N} \sum m(i) = b + \frac{a}{H} + \frac{1}{N} \sum \frac{\varepsilon(i)}{X(i)}

Taking variances with ε(i) independent across i:

\mathrm{Var}[\bar{m}] = \frac{1}{N^2} \sum \mathrm{Var}\!\left[\frac{\varepsilon(i)}{X(i)}\right] = \frac{\sigma^2}{N^2} \sum \frac{1}{X(i)^2}

Define the mean of reciprocal squares:

Q \equiv \frac{1}{N} \sum \frac{1}{X(i)^2}

Then:

\mathrm{Var}[\bar{m}] = \frac{\sigma^2 \cdot Q}{N}

Compare this to the variance of the OLS slope estimator b̂:

\mathrm{Var}(\hat{b}) = \frac{\sigma^2}{\sum (X(i) - \bar{X})^2} = \frac{\sigma^2}{S_{XX}}

The ratio mean variance depends on 1/X²; the regression slope variance depends on S_XX, the sum of squared deviations. A sample with small X values inflates Var[m̄], while large spread in X reduces Var[b̂].

4. The Induced Heteroscedasticity Problem

Even if the original regression model is homoscedastic (Var[ε(i)] = σ² for all i), the ratio m(i) = Y(i)/X(i) is heteroscedastic by construction:

\mathrm{Var}[m(i)] = \frac{\sigma^2}{X(i)^2}

The heterogeneous variances have two consequences for transfer pricing practice:

(1) Unweighted statistics on m(i) are inefficient (unreliable). The sample mean, median, and quartiles give equal weight to all observations, but observations with small X carry disproportionate noise. A single small comparable can dominate the interquartile range.

(2) Confidence intervals based on m require a weighted approach. The standard error of m̄ is not σ/√N but rather σ√Q / √N, which can be much larger when the sample includes small-base comparables.

5. The 68% Confidence Interval for the Slope b

Under classical linear regression assumptions, the OLS slope estimator is normally distributed:

\hat{b} \sim \mathcal{N}\!\left(b, \frac{\sigma^2}{S_{XX}}\right)

where S_XX = Σ(X(i) − X̄)². The standard error is:

\mathrm{SE}(\hat{b}) = \frac{s}{\sqrt{S_{XX}}}

with s the residual standard error (s² = RSS/(N−2)).

The 68% confidence interval corresponds to the central 68% of the t-distribution. For large N, t_{0.16}(N−2) ≈ 1.0, so:

b \in \left[ \hat{b} - \mathrm{SE}(\hat{b}), \; \hat{b} + \mathrm{SE}(\hat{b}) \right] \quad (68\% \text{ CI})

For the exact two-sided 68% interval with finite samples:

\hat{b} \pm t_{0.16}(N-2)\cdot \mathrm{SE}(\hat{b})

For N = 20, t_{0.16}(18) ≈ 1.015; for N = 50, t_{0.16}(48) ≈ 1.005. The ±1 SE rule is a reasonable approximation.

6. Contrast: Ratio IQR vs. Regression 68% Interval

7. Numerical Illustration

Consider a five-company sample with a = 5, b = 0.10, and H = 43.5 (the harmonic mean of the bases). Assume σ = 2 for illustration.

The ratio interval is not only displaced from b by the intercept bias—it is also wider due to the heteroscedasticity induced by small-base comparables.

8. Conclusion

The variance of the ratio PLI reveals a second defect beyond bias: induced heteroscedasticity. Small comparables inflate both the bias (via 1/X) and the variance (via 1/X²). The 68% confidence interval for the regression slope b is:

The ratio interquartile range has none of these properties when a ≠ 0.

Appendix A: Step-by-Step Derivation of Var[m(i)] = σ²/X(i)²

This appendix provides the complete algebraic derivation of the variance formula for the ratio PLI. Each step is explained in plain language. No prior knowledge of variance algebra is assumed beyond basic arithmetic.

A.1 Starting Point: The Ratio Formula

We begin with the structural relationship between profit Y and the base X:

Y(i) = a + b \cdot X(i) + \varepsilon(i)

where a is the intercept (a fixed number), b is the slope (the true profit ratio we want to measure), and ε(i) is a random error term with mean zero and variance σ². We can accommodate non-constant variances with GLS.

The ratio-based profit level indicator is obtained by dividing both sides by X(i):

m(i) = \frac{Y(i)}{X(i)}

A.2 Substituting the Model into the Ratio

Step 1: Replace Y(i) with its model expression:

m(i) = \frac{a + b \cdot X(i) + \varepsilon(i)}{X(i)}

Step 2: Distribute the division across each term in the numerator:

m(i) = \frac{a}{X(i)} + \frac{b \cdot X(i)}{X(i)} + \frac{\varepsilon(i)}{X(i)}

Step 3: Simplify the middle term (X(i) cancels):

m(i) = \frac{a}{X(i)} + b + \frac{\varepsilon(i)}{X(i)}

Rearranging to put the constant term first:

m(i) = b + \frac{a}{X(i)} + \frac{\varepsilon(i)}{X(i)}

This equation shows that the ratio m(i) consists of three parts:

A.3 Two Essential Variance Rules

Before continuing, we need two basic facts about variance. These are standard results that apply to any random variable.

Rule 1: The variance of a constant is zero.

If c is a fixed number (not random), then Var[c] = 0. This makes sense: a constant doesn’t vary, so its variance is zero.

Rule 2: When you multiply a random variable by a constant, the variance is multiplied by the square of that constant.

If Z is a random variable and c is a constant, then Var[c · Z] = c² · Var[Z]. If we double all values, the spread doubles, but the variance (which measures squared deviations) quadruples.

A.4 What Is Constant and What Is Random?

In the expression m(i) = b + a/X(i) + ε(i)/X(i), we treat X(i) as a known, fixed value for each entity. This is the standard assumption in regression analysis: we condition on the observed values of X.

Constants (not random): a, b, X(i), and therefore a/X(i) and 1/X(i).

Random: ε(i) only. The error term is the sole source of randomness.

A.5 Computing Var[m(i)] Step by Step

Step 1: Write down what we want to find:

\mathrm{Var}[m(i)] = \mathrm{Var}\!\left[b + \frac{a}{X(i)} + \frac{\varepsilon(i)}{X(i)}\right]

Step 2: Apply Rule 1—the first two terms are constants, so they contribute zero variance:

\mathrm{Var}[b] = 0 \quad (\text{b is a constant})

\mathrm{Var}\!\left[\frac{a}{X(i)}\right] = 0 \quad (\text{a and } X(i) \text{ are constants})

Therefore:

\mathrm{Var}[m(i)] = \mathrm{Var}\!\left[\frac{\varepsilon(i)}{X(i)}\right]

Step 3: Rewrite the remaining term to see the constant multiplier:

\frac{\varepsilon(i)}{X(i)} = \left(\frac{1}{X(i)}\right)\varepsilon(i)

Here, 1/X(i) is a constant (since X(i) is fixed), and ε(i) is the random variable.

Step 4: Apply Rule 2—multiply the variance by the square of the constant:

\mathrm{Var}\!\left[\left(\frac{1}{X(i)}\right)\varepsilon(i)\right] = \left(\frac{1}{X(i)}\right)^2 \mathrm{Var}[\varepsilon(i)]

Step 5: Substitute the known variance of ε(i):

By assumption, Var[ε(i)] = σ² (the same for all comparable entities). Therefore:

\mathrm{Var}[m(i)] = \left(\frac{1}{X(i)}\right)^2 \sigma^2

Step 6: Simplify the notation:

\left(\frac{1}{X(i)}\right)^2 = \frac{1}{X(i)^2}

Final result:

\mathrm{Var}[m(i)] = \frac{\sigma^2}{X(i)^2}

A.6 What This Result Means

The formula Var[m(i)] = σ²/X(i)² tells us that the variance of the ratio depends on the square of the base X(i) in the denominator. This has damaging consequence:

An entity with X = 10 (e.g., $10 million in sales) contributes 100 times more variance to the ratio distribution than a firm with X = 100 (e.g., $100 million in sales). When transfer pricing practitioners compute the interquartile range of ratios, small comparables exert outsized influence on the spread—not because they are economically different, but because of this purely algebraic effect.

A.7 Why Regression Avoids This Problem

The OLS slope estimator b̂ does not divide by X(i). Its variance is:

\mathrm{Var}(\hat{b}) = \frac{\sigma^2}{\sum (X(i) - \bar{X})^2}

This formula depends on the spread of X values, not on their reciprocals. A widespread in X reduces the variance of b̂, while including small X values inflates the variance of m̄. This is why regression-based inference is both more efficient and more robust than ratio-based methods.

Summary

The variance of the ratio m(i) = Y(i)/X(i) is σ²/X(i)² because: (1) only the error term ε(i) is random; (2) dividing ε(i) by the constant X(i) multiplies its variance by (1/X(i))². Small-base comparables, therefore, carry disproportionate noise, making ratio-based statistics unreliable even when the underlying model is well-behaved.

References

Aitken, A. C. (1935). On least squares and linear combination of observations. Proceedings of the Royal Society of Edinburgh, 55, 42–48.

Greene, W. H. (2018). Econometric Analysis (8th ed.). Pearson.

Silva, E. A. (2026). Why ratio quartiles are a biased arm’s length benchmark—and what to do instead. EdgarStat Blog. Why Ratio Quartiles Are a Biased Arm’s Length Benchmark — and What to Do Instead