Box-Cox Variable Transformation

Readings and References:

• W. H. Greene, Econometric Analysis, 7th Ed., Chapter 7: Nonlinear, Semiparametric, and Nonparametric Regression Models, Prentice-Hall, 2011.
• J. J. Spitzer, "A Primer on Box-Cox Estimation," Review of Economics and Statistics, 1982, 307-313. (Paper)

X^(λ) = (X^λ-1)/λ

Although the range of λ can cover the whole real line, -2 ≤ λ ≤ 2 is the area of interest in many econometric applications. If λ = 2, it is the quadratic transformation. If λ = 0.5, it is a square-root. A linear model corresponds to λ = 1, and the logarithmic transformation is the limiting case that λ -> 0 (by L'Hôspital's rule, lim_λ->0(X^λ-1)/λ = ln(X)).

The value of power λ may not be the same for each of the variables in the model. In particular, the dependent variable and independent variables as a group may take different Box-Cox transformations. Let α = (β,θ,λ)' be the vector of unknown parameters for a regression model:

ε = F(Y,X,α) = Y^(θ) - X^(λ)β

Or, Equivalently,

Y^(θ) = X^(λ)β + ε

where ε ~ normal(0,σ²I). The log-likelihood function is

ll(α,σ²|Y,X) = -½N [ln(2π)+ln(σ²)] -½ (ε'ε/σ²) + (θ-1)∑_i=1,2,...,Nln(|Y_i|)

Note that for each data observation i, the Jacobian term is derived as J_i(θ) = |∂ε_i/∂Y_i| = |Y_i^θ-1|. By substituting out σ² = ε'ε/N, the concentrated log-likelihood function is

ll*(α|Y,X)	= -½N [1+ln(2π)-ln(N)] -½N ln(ε'ε) + (θ-1) ∑i=1,2,...,Nln(|Yi|)
	= -½N [1+ln(2π)-ln(N)] -½N ln(ε*'ε*) where ε* = ε / [(|Y1||Y2|...|YN|)(θ-1)/N]

Given the values of Box-Cox transformation parameters θ and λ, a wide range of model specifications are possible. Of course, θ and λ should be estimated simultaneously with β. The efficient estimator of α = (β,θ,λ)' is obtained by maximizing the above concentrated log-likelihood function. It is equivalent to minimizing the sum of squared weighted errors:

S^*(β|Y,X) = ε^*'ε^*,

where ε^* = wε, and w = 1/[(|Y₁||Y₂|...|Y_N|)^(θ-1)/N].

Based on the estimated parameter vector α = (β,θ,λ)', a Box-Cox model is typically interpreted in terms of the elasticity. That is,

∂ln(Y)/∂ln(X) = (X/Y)(∂y/∂X) = (X^λ/Y^θ)β

Example

M^(θ) = β₀ + β₁ R^(λ) + β₂ Y^(λ) + ε

As described in Greene's Example 10.5 and Table 10.3, M is the real stock of M2, R is the discount interest rate, and Y is the real GNP. Several variations of the Box-Cox transformation parameters may be estimated and tested for the most appropriate functional form of money demand equation:

1. θ = λ, i.e.
   M^(λ) = β₀ + β₁ R^(λ) + β₂ Y^(λ) + ε
2. θ -> 0, i.e.
   ln(M) = β₀ + β₁ R^(λ) + β₂ Y^(λ) + ε
3. λ -> 0, i.e.
   M^(θ) = β₀ + β₁ ln(R) + β₂ ln(Y) + ε
4. θ -> 0 and λ -> 0, i.e.
   ln(M) = β₀ + β₁ ln(R) + β₂ ln(Y) + ε