Box-Cox transformation

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

[\tau(Y;\lambda)=\begin(Y^\lambda-1)/\lambda & \mathrm\ \lambda\neq 0, \\\ln(Y) & \mathrm\ \lambda=0.\end]

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:

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

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

[Q=\big(\alpha K^\lambda + (1-\alpha) N^\lambda\big)^,\,]

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

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

When [\lambda = 1] this produces the linear production function:

[Q=\alpha K + (1-\alpha)N.\,]

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

[Q=K^\alpha N^.\,]

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.

The story of the writing of the paper is told in

DeGroot, M. H. (1987) A Conversation with George Box, Statistical Science, vol. 2, pp. 239-258.

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