The Generalised Method of Moments and the transformation of data

It is not unusual to find empirical work for which conventional goodness of fit measures show that the conditional distribution of one set of variables on another is incompatible with the Gaussian (that is, normal) probability density. This has the important implication that conditional expectations will not, in general, be linear functions of the variables held fixed. In this paper the inverse hyperbolic sine transformation is used in conjunction with the Generalised Method of Moments (GMM) to implement asymptotically efficient parameter estimation based on the Gaussian probability density. Two examples are provided of the effectiveness of these procedures in conforming data to Gaussian distributional assumptions. The first involves the book to market ratios of equity of a large sample of publicly listed North American firms covering the period from 2005 until 2019; the second is based on an analysis of the U.S. money supply, stock prices and inflation covering the period from 1871 to 2018.