Box-Cox transformation of firm size data in statistical analysis

TL;DR

This paper applies the Box-Cox transformation to firm size data, such as number of employees and sales, to improve normality and linearity, revealing their approximate log-normal distribution.

Contribution

It demonstrates that Box-Cox transformation effectively normalizes firm size data and shows their linear relationship, with parameters close to zero indicating log-normality.

Findings

Transformed firm size data exhibit strong linearity.

Box-Cox parameters are close to zero, indicating log-normal distribution.

Firm size indicators like employees and sales have similar properties.

Abstract

Firm size data usually do not show the normality that is often assumed in statistical analysis such as regression analysis. In this study we focus on two firm size data: the number of employees and sale. Those data deviate considerably from a normal distribution. To improve the normality of those data we transform them by the Box-Cox transformation with appropriate parameters. The Box-Cox transformation parameters are determined so that the transformed data best show the kurtosis of a normal distribution. It is found that the two firm size data transformed by the Box-Cox transformation show strong linearity. This indicates that the number of employees and sale have the similar property as a firm size indicator. The Box-Cox parameters obtained for the firm size data are found to be very close to zero. In this case the Box-Cox transformations are approximately a log-transformation. This suggests that the firm size data we used are approximately log-normal distributions.