*Models should have mathematical beauty *(*they must be parsimonious*).

Paraphrasing Paul Dirac (1955), *Physical laws should have mathematical beauty*, quoted in Abraham País, Maurice Jacob, David Olive, Michael Atiyah, *Paul Dirac* (The Man and his Work), Cambridge University Press, 1998, p. 46.

Economic model-building involves three issues:

- Mathematical completeness of the model, which requires that the model have the same number of equations as endogenous variables.
- Specification of the variables and parameters of the model, which means that the parameters are uniquely determined.
- Statistical estimation of the parameters of the model, which requires that the parameter estimates must be unbiased and have minimum variance.

See Wonnacott & Wonnacott, 1970, pp. 101-102, 188-189. These namesake authors state (p. 313): “sound *prior specification* of the model is very important—in many ways more important than the details of the statistical techniques”. See also Kmenta, 1986, pp. 652-653 and Lohmöller, 1989, pp. 1-25 and Hill, Griffiths & Lim, 2018, section 1.7 (Empirical research), pp. 11-13.

We can avoid *biased* estimation of the parameters if the model is well-specified, such as avoiding the postulate that the dependent variable is a function of one explanatory variable when it is really dependent on multiple explanatory variables. See Wonnacott & Wonnacott, 1970, pp. 312-313.

In general, we appreciate well-specified parsimonious models with few parameters to be estimated.

Here, we argue that the standard estimation of return on assets is biased because of the failure to recognize that assets are endogenous.

#### Misspecification of return on assets

The standard return on assets model is postulated with a *structural* *equation*:

##### (1) P(*t*) = ρ K(*t* – 1)

in which P(*t*) denotes operating profits in period *t* = 1 to T and K(*t* – 1) denotes operating assets in the beginning of the period. We exclude the error term from equation (1) to facilitate exposition. See: https://bookdown.org/ccolonescu/RPoE4/simplelm.html#the-general-model

See Harrington, 1993, pp. 17-18 and Bernstein & Wild, 2000, Chapter 6 (Return on invested capital), pp. 234-256 for the accounting treatment of equation (1) and (4) below.

First, equation (1) is misspecified because the idea that assets generate profits without employee compensation is preposterous. Equation (1) excludes compensation of employees as an important explanatory variable of profits.

Second, even if we grant the absurd postulate that profits are generated without labor input (compensation), equation (1) is misspecified because we suspect that K(*t*) is an endogenous variable.

Thus, to complete the equation system in which the endogenous variables are functions of the exogenous variables, we need to add an *identity equation* for K(*t*):

##### (2) K(*t*) = β K(*t* – 1) + G(*t*)

in which β = (1 – δ), the parameter δ denotes the depreciation rate, and G(*t*) denotes gross investment (CAPX).

In our scenario, the variables P(*t*) and K(*t*) are endogenous, and G(*t*) is exogenous.

In company GAAP or IFRS accounts, K(*t*) is property, plant and equipment, which is reported as an aggregate composite variable on the balance sheet, and G(*t*) is CAPX reported on the cash flow statement.

We substitute equation (2) into (1) and obtain the reduced-form equation of operating profits depending on two factors, K(*t*) and G(*t*), and not on K(*t*) alone:

##### P(*t*) = ρ [β K(*t* – 2) + G(*t *− 1)]

##### (3) P(*t*) = φ K(*t* – 2) + ρ G(*t *− 1)

in which φ = ρ β and the rate of return ρ is the partial derivative of P(*t*) with respect to G(*t *− 1). Thus, based on equation (3), we can interpret ρ as the rate of return on investment and not on fixed assets.

If we ignore equation (2), and estimate regression equation (1) alone, we obtain bias estimates of the rate of return on operating assets. See Hill, Griffiths & Lim, 2018, section 6.3.2 (Omitted variable bias), including mathematical proof of bias on pp. 276-277.

We can escape this omitted variable bias if ρ in equation (3) is not significant (meaning that ρ is zero except for random fluctuation); but even in this special case the assets measure is lagged two-periods, K(*t* – 2), and the rate of return on assets includes the depreciation rate, φ = ρ β = ρ (1 – δ), such that ρ = φ / β and we would need to assume δ to find ρ.

We can get consistent estimates of (1) by using *instrumental variables* (IV) or by using two-stage least squares (TSLS), but implementation is cumbersome and not free of controversy regarding the choice of proxy variables. Thus, using TSLS we may not get an unique estimate of the return on assets for the same set of comparable data. See Wonnacott & Wonnacott, 1970, pp. 190-192, 357-364 (TSLS) and Kmenta, 1986, pp. 357-361 (IV) and 681-687 (TSLS). See also: https://bookdown.org/ccolonescu/RPoE4/random-regressors.html#the-instrumental-variables-iv-method

In sum, equation (1) is misspecified because G(*t*) is excluded from model (1). Also, we consider equation (1) misconceived because labor input is omitted. The gravity of omitting labor cost from the return on assets function is equivalent to the fustian positing of a production function without labor input. See Kmenta, 1986, pp. 443-446 regarding the *statistical bias* resulting from the *omission of a relevant explanatory variable* from the model. See also: https://bookdown.org/ccolonescu/RPoE4/further-inference-in-multiple-regression.html#omitted-variable-bias

#### Using two-period average asset base

Often the assets base K(*t*) of the rate of return on assets is defined in the same period as P(*t*) or the assets base is defined (it is a messy expression) as an average of two periods:

2 P(*t*) = ρ [K(*t*) + K(*t* – 1)] or

##### (4) P(*t*) = ρ X(*t*)

where X(*t*) = 0.5 [K(*t*) + K(*t* – 1)] from averaging two values.

Spelled out, equation (4) is an ugly expression; and it is not sensible to define X(*t*) = 0.5 [K(*t*) + K(*t* – 1)] as the single determinant of P(*t*). See Bernstein & Wild, 2000, p.244.

Besides showing the dependent variable multiplied by 2 (obtained from taking the average of two adjacent values of K(*t*)), equation (4) has the current and one-period lagged measure of assets multiplied by the same regression coefficient. Although this assumption may be reasonable if we assume that the level of assets is in equilibrium between two adjacent periods, it is better to let the data provide information instead of punishing the data with preconceived ideas.

Since bias results have no place in science, we should estimate return on assets using equation (3) and not estimate equation (1). The regression estimates of equation (3) may be disappointing, and the naïve infatuation with return on assets as the appropriate profit indicator to benchmark the tested party may not find empirical support from the selected comparables (also, equation (3) may not survive Occam’s razor).

#### Empirical estimates

We estimated equation (3) using a group of major U.S. retailers that we have been studying, and the results are not impressive compared to our markup model discussed on prior blogs. See https://blog.royaltystat.com/transfer-pricing-methods-based-on-operating-profits

The dependent variable P(*t*) = OIBAM = OIADP + AM, where OIADP is operating income [profit] after depreciation and AM is the amortization of acquired intangibles. We use all available data per company from 1978 to 2019. We use the OLS (ordianry least squares) estimators and HAC (heteroskedasticity and autocorrelation consistent) standard errors. See Hill, Griffiths & Lim, 2018, section 9.5.2 (HAC standard errors), pp. 448-450. The Newey-West *t*-Statistics (based on HAC standard errors) are writen within parentheses.

Company | GVKEY | Count | φ | ρ | Adjusted R^{2} |

Best Buy | 2184 | 35 | 0.2982 | 1.2711 | 0.8712 |

Newey-West t-statistics | (2.8796) | (3.8056) | |||

Conn’s Inc. | 156614 | 16 | 0.4558 | 0.6223 | 0.2039 |

(1.3015) | (0.9811) | ||||

Home Depot | 5680 | 38 | 0.3585 | 0.3389 | 0.6848 |

(3.2387) | (0.7765) | ||||

Kohl’s Corp. | 25283 | 24 | 0.0874 | 0.9413 | 0.7596 |

(3.0405) | (3.8748) | ||||

Lowe’s | 6829 | 40 | 0.1983 | 0.2678 | 0.8206 |

(4.5099) | (1.1950) | ||||

Macy’s | 4611 | 40 | 0.2086 | −0.3348 | 0.6879 |

(5.6155) | (−0.8754) | ||||

Target Corp. | 3813 | 40 | 0.1332 | 0.4157 | 0.9275 |

(11.3543) | (3.3598) | ||||

Walmart Inc. | 11259 | 40 | 0.1091 | 0.7772 | 0.9673 |

(4.0703) | (3.8423) |

The annual company data series used in the computations above were retrieved from RoyaltyStat/Compustat, and the parameters of equation (3) were estimated using RoyaltyStat’s interactive regression function. The estimates of the return on assets do not appear within reasonable ranges, and several coefficients are not significant. These results suggest that assets are not profit drivers in major U.S. retailing.

We distrust equation (3), and find that it is too vulnerable to impeachment. The regression results on the table above show that model (3) does not perform well with the sample data examined; we get no additional knowledge about company operating profit determination by employing this more complex multivariate model (3) than using the simpler to estimate and interpret profit markup equation discussed on several prior blogs.

The profit markup model alternative that we suggest has several important advantages over the return on assets equation (3). First, the regression parameters of the profit markup model have easier interpretation; second, the markup model has parsimony of expression. Moreover, the parameters of the profit markup equation are pure numbers; they are not dependent on the demensional units of the heterogeneous assets variable K(*t*).

#### References

Leopold Bernstein & John Wild, *Analysis of Financial Statements* (5^{th} edition), McGraw-Hill, 2000.

Diana Harrington, *Corporate Financial Analysis* (4^{th} edition), Richard Irwin, 1993.

Carter Hill, William Griffiths & Guay Lim, *Principles of Econometrics* (5^{th} edition), Wiley, 2018.

Jan Kmenta, *Elements of Econometrics* (2^{nd} edition), Macmillan, 1986.

Jean-Bernd Lohmöller, “Basic Principles of Model Building: Specification, Estimation, Evaluation,” in Herman Wold (editor), *Theoretical Empiricism*, Paragon House, 1989.

Ronald Wonnacott & Thomas Wonnacott, *Econometrics*, John Wiley & Sons, 1970.