Box-Cox transformation

In statistics, the Box-Cox transformation of the variable Y given the "Box-Cox parameter" λ ≥ 0 is defined as

<math>\tau(Y;\lambda)=\begin{cases}(Y^\lambda-1)/\lambda & \mathrm{if}\ \lambda\neq 0, \\

\ln(Y) & \mathrm{if}\ \lambda=0.\end{cases}<math>

This transformation has proved popular in regression analysis, including econometrics.

Economists often characterize production relationships by some variant of the Box-Cox transformation.

Consider a common representation of production Q as dependent on services provided by a capital stock K and by labor hours N:

<math>\tau(Q)=\alpha \tau(K)+ (1-\alpha)\tau(N).\,<math>

Solving for Q by inverting the Box-Cox transformation we find

<math>Q=\big(\alpha K^\lambda + (1-\alpha) N^b\big)^{1/\lambda},\,<math>

which is known as the constant elasticity of substitution (CES) production function.

The CES production function is a homogeneous function of degree one.

When b = 1 this produces the linear production function:

<math>Q=\alpha K + (1-\alpha)N.\,<math>

When λ → 0 this produces the famous Cobb-Douglas production function:

<math>Q=K^\alpha N^{1-\alpha}.\,<math>

References

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. Journal of Royal Statistical Society, Series B, vol. 26, pp. 211-–246.