Wednesday, March 13, 2019

Forecasting After an Inverse Hyperbolic Sine Transformation

There are all sorts of good reasons why we sometimes transform the dependent variable (y) in a regression model before we start estimating. One example would be where we want to be able to reasonably assume that the model's error term is normally distributed. (This may be helpful for subsequent finite-sample inference.)

If the model has non-random regressors, and the error term is additive, then a normal error term implies that the dependent variable is also normally distributed. But it may be quite plain to us (even from simple visual observation) that the sample of data for the y variable really can't have been drawn from a normally distributed population. In that case, a functional transformation of y may be in order.

So, suppose that we estimate a model of the form

              f(yi) = β1 + β2 xi2 + β3 xi3 + .... + βk xik + εi ;    εi ~ iid N[0 , σ2] .                         (1)

where f(.) is usually a 1-1 function, so that f-1(.) is uniquely defined. Examples include f(y) = log(y), (where, throughout this post, log(a) will mean the natural logarithm of 'a'.); and f(y) = √(y) (if we restrict ourselves to the positive square root).

Having estimated the model, we may then want to generate forecasts of y itself, not of f(y). This is where the inverse transformation, f-1(y), comes into play.