# Transformation for stabilizing variance in time series

In most forecasting packages there are two transformations easily available for stabilizing variance: the square root and the logarithnic. Is there any procedure (e.g. test) that can tell the user which one of the two transformations to choose? Finally, the variance stabilization transofrmation should be apllied to time series data (if they exhibit multiplicative variance) no matter what is the model to be used (e.g. it should be used in ARIMA, exponential smoohting, unobserverd components, classical decomposition etc if the time series exhibit multiplicative variance)?

You might want to read about Box-Cox transformation: $$y \mapsto \left\{ \begin{eqnarray} \frac{y^\lambda-1}{\lambda (\dot y)^{\lambda-1}}, & \lambda \neq 0 \\ \dot y \ln y, & \lambda = 0 \end{eqnarray} \right.$$ where $\dot y$ is the geometric mean of the data. It generalizes both the square root and the log transformation, and admits a likelihood ratio test to select the best fitting parameter. Of course you must have the data that are always positive for the power and the log to be applicable. In general, you can shift your data $y\mapsto y+a$. The shift parameter $a$, however, is difficult to identify, and usually has to be used as a fixed value in practice.
In Stata's Box-Cox model, you can specify the same or different shape parameter $\lambda$ to be applied to the dependent and explanatory variables, although I suspect it will only work with AR models specified explicitly with lags, rather than the general ARIMA models.