Inverse Cube Root Transformation: Theory and Application to Time Series Data

Abstract

Constant variance assumption is one of the necessary assumptions of several time series models. The violation of this assumption often necessitates data transformation. This paper investigated the properties of the inverse cube root transformation of a time series based on the multiplicative model. The probability density function of the inverse cube root of the left truncated normally distributed error components of the model was derived; the relationship between the variances of the errors in both the transformed and untransformed multiplicative models was examined through simulations. The error variance for the untransformed model was empirically found to be nine (9) times the error variance for the transformed model. Other results obtained in this paper revealed that an inverse cube roots transformation of a time series that results in normally distributed errors is possible when 0 <  < 0.38. In other words, the variance of the inverse cube root transformed error component was greater than that of the untransformed component for all .