Log and Box-Cox Transformations

Many financial variables — market capitalization, trading volume, firm revenue — are strongly right-skewed: a handful of enormous values sit far out on the right, with everything else bunched near zero. Fitting a linear regression directly on such a variable often violates the assumption that residuals have constant variance, because the model fits the huge values poorly and the errors balloon exactly where the values are largest. The simplest fix is the log transformation: replacing $x$ with $\ln(x)$ compresses large values much more than small ones, pulling in the right tail and often making the relationship with other variables closer to linear.

The Box-Cox transformation generalizes this idea with one parameter, $\lambda$, that is chosen (typically by maximum likelihood) rather than assumed: $x^{(\lambda)} = (x^\lambda - 1)/\lambda$ for $\lambda \neq 0$, and $\ln(x)$ exactly in the limit as $\lambda \to 0$. Trying $\lambda = 1$ leaves the data unchanged, $\lambda = 0.5$ gives a square-root-like compression, and $\lambda = 0$ recovers the ordinary log — so instead of guessing whether a log, square root, or no transform is right, Box-Cox searches over the whole family and picks the value of $\lambda$ that makes the transformed data closest to satisfying the regression assumptions.