Double Sqrt Transformations and Related Slope Estimates – An Alternative to Double Log Regressions

Introduction

In certain cases, the data generation process for operating profit (Y) given Net Sales (X) for a company is given by the following equation:

Y^{0.5} = A + B X^{0.5} + u

This could be the case, for example, when the data is rightly skewed. In this case, for large values this compresses the effect on the right side of the distribution and stretches out small values on the left side of the distribution.

Approximation

Consider again the data generation process for company j:

Y_j^{0.5} = A + BX^{0.5}_j + u_j

Running the regression across N comparable companies in the set, one obtains R squared, slope coefficients A and B together with relevant standard errors and other statistics.

We now pursue an approximation to the data generation process above. Let’s consider first order Taylor approximation around Y* where this is the mean operating profit across the N comparable companies in the set.

Y_j^{0.5} \approx {Y^*}^{0.5} + \frac{1}{2\sqrt{Y^*}}(Y_j-Y^*)

and similarly:

X_j^{0.5} \approx {X^*}^{0.5} + \frac{1}{2\sqrt{X^*}}(X_j-X^*)

where X* is the mean Net Sales across the N comparable companies in the set.

Plug into above:

{Y^*}^{0.5} + \frac{1}{2\sqrt{Y^*}}(Y_j-Y^*) \approx A + B({X^*}^{0.5} + \frac{1}{2\sqrt{X^*}}(X_j-X^*)) + u_j

Rearranging, one obtains:

{Y_j} \approx E + F X_j + v_j

where E is a linear function of A, B, X* and Y* but more importantly:

F = B \sqrt{\frac{Y^*}{X^*}}

and the error term:

v_j = 2 \sqrt{Y^*} u_j

Methodology

Consider the following steps:

-Step 1: Run a simple regression and collect C, D, X* and Y*:

Y_j = C + D X_j + w_j

-Step 2: Run the double sqrt regression and collect B:

Y_j^{0.5} = A + BX^{0.5}_j + u_j

-Step 3: Get the adjusted slope coefficient F by the formula above:

F = B \sqrt{\frac{Y^*}{X^*}} = B \sqrt{(\frac{C}{X^*}+D)}

Conclusion

Here, we propose a linear approximation to the case where the data generation process is double square root. The linear approximation provides the convenient interpretation of the marginal effect of increase in net sales to operating profits which are not present in the original formulation. We show that the adjusted slope coefficient F for the linear regression is given by the formula:

F = B \sqrt{\frac{Y^*}{X^*}} = B \sqrt{(\frac{C}{X^*}+D)}

In a future companion blog we shall address the issue of the standard error of F.