A power function has the form:
(1) Y = X β
where the slope parameter β is determined from data comparable to the tested party. Y is the dependent variable and X is the independent (explanatory or predictor) variable.
The contrived linear function: Operating Profits = β Revenue, with the forced zero intercept specified in the U.S. and OECD guidelines, and the quadratic and cubic functions are power functions. Other power functions include the reciprocal and squared-root functions.
Typical power functions in economics include:
Here, we examine a power function that is prevalent in economics:
(2) Y = α X β
and take the first derivative of (2) to obtain the slope coefficient:
(3) d Y / d X = β α X β – 1
Slope = β α (X β / X) = β (Y / X).
Except for the special case in which α = β = 1, we can use the double-logarithms regression to estimate equation (2):
(4) LN(Y) = LN(α) + β LN(X)
Now, we consider the best regression fit among two rival operating profits functions using actual company-level annual data:
(5) Y = α + β X,
versus the power equation (2) or its regression version (4).
Chart 1 contains the linear regression: Y = 0.071 X, with the Newey-West t-statistics of the slope = 7.8158 and the R2 = 0.8032. The alpha-intercept is not significant, so we don’t report it.
Chart 1: Linear Regression
Chart 2 contains in this case a superior (more reliable) double-logarithms regression (4): LN(Y) = 1.0219 LN(X) − 2.938, with the Newey-West corrected t-statistics of the intercept = − 8.6162, the t-statistics of the slope coefficient = 26.035, and the R2 = 0.9043. Each regression contains 167 annual paired X and Y observations.
Chart 2: Double-Logrithms Regression
As takeaway, we must examine the comparable company data using bivariate scatterplots instead of assuming the special (contrived) linear equation without an intercept:
(6) Y = β X,
where the intercept is forced to be α = 0.
The regression results above were computed online in EdgarStat using the historical pairs of X = Revenue and Y = Operating Profits (after depreciation and amortization) of five U.S. retailers :