Performs a forward or an inverse Box‑Cox (power) transformation.

Routine BCTR performs a forward or inverse Box‑Cox transformation of the n = NOBS observations {Zt} for t = 1, 2, …, n.

The forward transformation is useful in the analysis of linear models or models with nonnormal errors or nonconstant variance (Draper and Smith 1981, page 222). In the time series setting, application of the appropriate transformation and subsequent differencing of a series may enable model identification and parameter estimation in the class of homogeneous stationary autoregressive‑moving average models. The inverse transformation may later be applied to certain results of the analysis, such as forecasts and probability limits of forecasts, in order to express the results in the scale of the original data. A brief note concerning the choice of transformations in ARIMA models is given in Box and Jenkins (1976, page 328). The class of power transformations discussed by Box and Cox (1964) is defined by

where Zt+ ξ > 0 for all t. Since

the family of power transformations is continuous.

Let λ = POWER and ξ = SHIFT; then, the computational formula utilized by routine BCTR is given by

where Zt+ ξ > 0 for all t. The computational and Box‑Cox formulas differ only in the scale and the origin of the transformed data. Consequently, the general analysis of the data is unaffected (Draper and Smith 1981, page 225).

The inverse transformation is computed by

where {Zt} now represents the result computed by BCTR for a forward transformation of the original data using parameters λ and ξ.

Consider the Airline Data (Box and Jenkins 1976, page 531) consisting of the monthly total number of international airline passengers from January 1949 through December 1960. Routine BCTR is used to compute a forward Box‑Cox transformation of the first 12 observations. In the transformation SHIFT and POWER are each set to zero, which corresponds to taking natural logarithms of the data.

The estimated standard errors of forecasts (lead times 1 through 12 at origin July 1957) using the transformed Airline Data (Box and Jenkins 1976, page 311) may be converted back to their original scale using routine BCTR. The backward Box‑Cox transformation with SHIFT and POWER each set to zero corresponds to using the exponential function.