Is It Acceptable to Use Different Box-Cox Transformations for Each Variable?

[chalaf]

Hello everyone,

I am currently conducting gold price forecasting using a multi-input transfer function model with several financial variables as inputs.

Based on the Box–Cox transformation analysis, I obtained different optimal lambda (λ) values for each variable:

* Y (Gold Price): λ = -1 → transformation (1/Y)

* X1 : λ = 1 → no transformation (original scale)

* X2 : λ = 0.5 → square root transformation

My question is:

In transfer function modeling, is it statistically acceptable to use different Box–Cox lambda values for different variables according to each variable’s optimal characteristic?

Or should the Box–Cox transformation be standardized using the same lambda value across all variables in the model?

Thank you very much for your guidance and suggestions.

[sbxkoenk]

On the proposal of @Ksharp , you have posted your original question here again.

The original question is here:
https://communities.sas.com/t5/Statistical-Procedures/Is-It-Acceptable-to-Use-Different-Box-Cox-Tran...

It would have been better if the entire thread (entire topic) had been moved from one board (Statistical Procedures) to the other (Forecasting & Econometrics), but @Ksharp  probably doesn’t have the necessary permissions to do so.

Perhaps it would be best to close your other thread (in that other board).

Thanks,
Koen

[sbxkoenk]

No action required !

It’s fine now.

I’ve merged the identical (original) post/question from that other board (Statistical Procedures) with this one.
That does mean your question now appears twice in the thread, but people will see through that.

Let's hope you get good answers here!

BR,

Koen

[chalaf]

Hello everyone,

I am currently conducting gold price forecasting using a multi-input transfer function model with several financial variables as inputs.

Based on the Box–Cox transformation analysis, I obtained different optimal lambda (λ) values for each variable:

* Y (Gold Price): λ = -1 → transformation (1/Y)
* X1 : λ = 1 → no transformation (original scale)
* X2 : λ = 0.5 → square root transformation

My question is:

In transfer function modeling, is it statistically acceptable to use different Box–Cox lambda values for different variables according to each variable’s optimal characteristic?

Or should the Box–Cox transformation be standardized using the same lambda value across all variables in the model?

Thank you very much for your guidance and suggestions.

[Ksharp]

My two cents:
I think it is OK.
Box-Cox Transformations is just to transform non-normal distribution into normal distribution. It has nothing to do with other variables. But the explanation of parameter estimator of that variable is another story !

[chalaf]

Thank you for your explanation.

I would like to ask further about the parameter estimation aspect.

If different Box-Cox transformations are applied to each variable in a multi-input transfer function model, how should the parameter estimates be interpreted statistically?

Does applying different transformations affect:

* the consistency or comparability of parameter estimates,

* the interpretation of transfer function coefficients,

* the forecasting performance after back-transformation,

* or potentially introduce bias into the model estimation or forecasting results?

Thank you very much for your insight.

[Ksharp]

Sorry. Your question is too tough for me.
If your requirement is regarding FORECASTING , you could post your question at Forecasting Forum:

https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/bd-p/forecasting_econometrics

[sbxkoenk]

Which procedure do you use for your multi-input transfer function model (dynamic regression model)?

I ask this because in PROC ARIMA for example ... you define transfer function (numerator and denominator) orders individually for each input.

Often, you specify the same functional transformation to the inputs as is used for the variable to be forecast. But that's not a requirement!

BR,

Koen

[chalaf]

Thank you for your explanation.

I am using PROC ARIMA for the multi-input transfer function modeling.

For each input series, I identify the appropriate transfer function orders individually through the prewhitening and cross-correlation analysis steps.

My concern was mainly about whether using different Box-Cox transformations for each variable could statistically affect the validity of the parameter estimation or potentially introduce bias in the forecasting results.

[sbxkoenk]

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@chalaf wrote:

My concern was mainly about whether using different Box-Cox transformations for each variable could statistically affect the validity of the parameter estimation or potentially introduce bias in the forecasting results.

---

On your 1st point (... could statistically affect the validity of the parameter estimation):
I do not think so. I wouldn't know why.
But interpretation and explain-ability will be a challenge.

On your 2nd point (... potentially introduce bias in the forecasting results):
There's a bias when back-transforming your point forecasts (response variable) back to the original scale. This happens due to your transformation of the original target variable, regardless of what you do to the inputs.
See here: 
5.6 Forecasting using transformations
https://otexts.com/fpp3/ftransformations.html

(from "Forecasting: Principles and Practice (3rd ed)"

by Rob J Hyndman and George Athanasopoulos

Monash University, Australia)