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The selected profit indicator under the CPM (TNMM) can be adjusted for asset intensity by augmenting the structural profit equation (2):

However, estimating structural equations (like the underlying linear equation producing the univariate quartiles of the selected profit indicator) produces omitted variable biased results. Therefore, we suggest using the reduced-form equation (3):

(1) R(*t*) = C(*t*) + P(*t*)

(2) P(*t*) = α + β R(*t*) + γ A(*t*−1) + U(*t*)

R(*t*) = α + C(*t*) + β R(*t*) + γ A(*t*−1) + U(*t*)

R(*t*) − β R(*t*) = α + C(*t*) + γ A(*t*−1) + U(*t*)

(3) R(*t*) = λ_{0} + λ_{1} C(*t*) + λ_{2} A(*t*−1) + V(*t*)

where the intercept λ_{0} = α / (1 – β), the slope coefficient λ_{1} = 1 / (1 – β) > 1 is the operating profit *markup*, and λ_{2} = γ / (1 – β) is the asset intensity coefficient.

If the estimated partial slope coefficient λ_{2} is significant (the regression coefficient is statistically different from zero), asset intensity is relevant to explain the behavior of the operating profit markup or profit margin; otherwise, asset adjustment has no economic merit and should be abandoned.

The asset adjusted operating profit *margin* can be obtained by solving for β from λ_{1}^{:}

(4) λ_{1} = 1 / (1 – β) ⇔ β = (λ_{1} – 1) / λ_{1}

The variables necessary to estimate the reduced-form equation (3) can be found per company in EdgarStat®:

(a) R(*t*) is Revenue in year *t* = 1 to T audit years (plus at least two prior or post audit years).

(b) C(*t*) is total cost, which is the sum of direct cost and indirect expenses (including depreciation and amortization). Thus, the indirect least squares slope coefficient β is the operating profit margin after depreciation and amortization (EBIT margin).

(c) P(*t*) is Operating Profit after depreciation and amortization (EBIT).

(d) A(*t*−1) is Property, plant, and equipment (Net PPE) at the beginning of the year.

(e) The unknown variables U(*t*) and V(*t*) are the random uncertainty of the structural and reduced-form equations.

The OLS (ordinary least squares) regression (3) can be computed in EdgarStat online (interactive applications) using the Multiple Regression function.

We apply the principle of parsimony (Occam’s razor) adopting the simplest model if the augmented model does not add significant information to explain the behavior of the selected profit indicator (EBIT). For this purpose, simplicity is measured by the number of estimated coefficients, and significant information is measured by the partial Newey-West *t*-statistics and the adjusted R^{2 }comparing the augmented trivariate versus the simpler bivariate model.

The Newey-West *t*-Statistics are corrected for autocorrelation among the regression residuals. See https://www.econometrics-with-r.org/15-4-hac-standard-errors.html

**Example 1**: Manufacturer, SIC Code 3841 (Surgical and medical instruments)

Manufacturers employ Net PPE (they are asset intensive), and we test the relevance of this asset factor in the reduced-form profit equation. We take Baxter International, Inc. (BAX) as an example:

(3.1) R(*t*) ≈ 1.1705 C(*t*) + 0.1035 A(*t*−1) – 398.3

The Newey-West *t*-statistics are 20.2986, 1.1556, −3.1511, Count = 38 years of data (1982-2020), and the adjusted R^{2} = 0.978.

We conclude that asset adjustment is not significant because the Newey-West *t*-Statistics of 1.1556 is less than the critical value, and report the simpler bivariate model:

(3.2) R(*t*) ≈ 1.2111 C(*t*) – 332.7

The Newey-West *t*-statistics are 40.0, −2.5697, Count = 39 years of data, and R^{2} = 0.977.

**Example 2**: Distributor, SIC Code 5045 (Computers, peripherals & software)

Distributors do not employ much Net PPE (they are not asset intensive) but we test the relevance of this asset factor in the reduced-form profit equation because some analysts are obsessed with return on assets. We take CDW Corp. (CDW) as an example:

(3.3) R(*t*) ≈ 1.0943 C(*t*) – 0.2615 A(*t*−1) – 379.8

The Newey-West *t*-statistics are 248.6, −5.5561, −7.8022, Count = 8 years of data (2010-2019), and the adjusted R^{2} = 0.999.

Despite the statistical significance of Net PPE and the high adjusted R^{2}, we are concerned with the perverse (negative coefficient of asset intensity) result. Thus, we try the more parsimonious bivariate regression alternative for comparison:

(3.4) R(*t*) ≈ 1.0882 C(*t*) – 358

The Newey-West *t*-statistics are 390, −9.822, Count = 10 years of data, and R^{2} = 0.999.

We select the bivariate model (3.4) instead of the augmented model (3.3) because we obtain significant results with a frugal model containing fewer estimated coefficients.

Overfitting (applying the principle of parsimony) the workhorse bivariate model with a trivariate model and settling with the simpler model if significant information is not obtained with the augmented model is explored *ad nauseum* in S. Pandit and S. Wu, *Time Series and System Analysis with Applications*, John Wiley & Sons, 1983.

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