Power transformations and ARIMA

[ac_1]

Hi Tom,

For ARIMA modelling,

If I am to model log-returns i.e 1st differences of a logged series, based on:
(i) A transformation 'typically' applied to the series in the academic literature, and
(ii) @BJTRANS, choice of transformation which produces approximate uniform variability i.e. the log transformation.
Then on BOXJENK I would input the depvar as 'log of the series' and use DIFFS=1.
After generating forecasts on the transformed data I would exp the forecasts, i.e. back to the original scale.

However, on any series:
(a) If @BJTRANS says e.g. the square-root of the series stabilises the variance the best, then
(b) use @BJDIFF for an automated choice for the DIFFS, SDIFFS and CONSTANT options, then
(c) on BOXJENK input the depvar as 'sqrt of the series' and the appropriate choices via @BJDIFF.
After generating forecasts on the transformed series I would square the forecasts, i.e. back to the original scale.

Questions:
(1) I want IID-Normal residuals, am I aiming for a transformation e.g. via @BJTRANS which produces a Normal distribution in the LEVELS series, or a Normal distribution for the DIFFS series (DIFFS>=1), or both?
(2) Also, is it more appropriate to stick with a transformation 'standard' in the academic literature for the series or that suggested from the empirical data?
(3) Both a log transformation and a square-root transformation are part of the family of Box-Cox/Yeo-Johnson power transformations, depending on lambda, used to stabilize variance and make the data more normal distribution-like https://en.wikipedia.org/wiki/Power_transform. How would I apply a Box-Cox/Yeo-Johnson transformation to choose an optimal value for lambda, and then back-transform the forecasts in RATS?

thanks,
Amarjit

[TomDoan]

Questions:
(1) I want IID-Normal residuals, am I aiming for a transformation e.g. via @BJTRANS which produces a Normal distribution in the LEVELS series, or a Normal distribution for the DIFFS series (DIFFS>=1), or both?

Neither. Normal residuals are the residuals, not either the levels or differences. Note, however, that the distribution of the residuals is what it is. If they aren't normal, no transformation of y is going to change that.

(2) Also, is it more appropriate to stick with a transformation 'standard' in the academic literature for the series or that suggested from the empirical data?

If the standard is fairly obvious (such as log's for series like GDP), then you should stick with it. For series like the unemployment rate or interest rates, there is no obvious transformation.

(3) Both a log transformation and a square-root transformation are part of the family of Box-Cox/Yeo-Johnson power transformations, depending on lambda, used to stabilize variance and make the data more normal distribution-like https://en.wikipedia.org/wiki/Power_transform. How would I apply a Box-Cox/Yeo-Johnson transformation to choose an optimal value for lambda, and then back-transform the forecasts in RATS?

The backtransformation of the forecasts is done by solving backwards through the Box-Cox transformation. @BJTRANS does an AR(1) on the transformed data which should take care of the gross serial correlation in most series. If you want, you can do a more complete search over lambda, which would probably mean fitting a more complete AR or ARMA model to each set of transformed data. If you do that, you need to make sure to correct the log likelihood for the Jacobian term of the Y transformation.