Box-Cox transformation

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

$\tau(Y;\lambda)=\begin{cases}(Y^\lambda-1)/\lambda & \mathrm{if}\ \lambda\neq 0, \\ \ln(Y) & \mathrm{if}\ \lambda=0.\end{cases}$

The value at Y=1 for any λ is 0, and the derivative with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y=1 at some sort of average value.

The transformation is basically a power transformation, but done in such a way as to make it continuous with the parameter λ at λ=0. It has proved popular in regression analysis, including econometrics.

In regression analysis, one sometimes carries out a series of Box-Cox transformations of the data with a range of λ values, and one then compares the residual sum of squares at these values in order to choose the transformation which gives the best results.

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:

$\tau(Q)=\alpha \tau(K)+ (1-\alpha)\tau(N).\,$