Bias correction for regression with t-distributed error I have a GAM /regression model which is originally defined as:
log10(Y)~s(log10(X1))+s(log10(X2))+s(log10(X3)) #using R mgcv 

The response needs to be back transformed. As simply doing exp(Y) would give a biased result, I do the bias correction via: Y_unbiased~exp(Y)*exp(((log(10)*sd(model))^2)/2)
see: https://derekogle.com/FSA/reference/logbtcf.html
Now it seems that for the data on log10-log10 the assumption of a gaussian error is not correct and I still have very heavy tails. The scat() family from mgcv (https://rdrr.io/cran/mgcv/man/scat.html) provides optimal qqplots for the model.
It is based on the t-distribution.
How would now a bias correction look like for a t-distribution ? Is there one at all?
P.S. I also tried to model:
Y~s(log10(X1))+s(log10(X2))+s(log10(X3)) #using R gamlss and a lognormal error distribution
Y~s(log10(X1))+s(log10(X2))+s(log10(X3)) #using R mgcv and a Gamma error distribution with log link

none of those gave as good models i.e. qqplots as the scat family on log10(Y) so how would a bias correction for scat() i.e. the t-distribution look like?
 A: The inherent issue with your model is that your variables are not fully differentiated.  Therefore, they all have unit roots, are not stationary, and lead to overall mispecified models with residuals that probably breach all the assumptions of typical models including residuals not being stationary, being heteroskedastic, autocorrelated, not normally distributed, etc.
The solution is to transform your variables.  First, instead of Log, take the first difference in such logs from one period to another.  This should solve most of the mentioned problems.
If you want to further stabilize the data, you could make additional variable transformations such as:

*

*Standardize the data, so each variable is on a comparable scale or # of standard deviations away from the average of that same variable;


*Normalize the data, so each observed value ranges between 0 (minimum value) and 1 (maximum value) or use another normalization where - 1 (minimum value);


*Remove the seasonality effect from the data.


*Remove the heteroskedasticity from the data if the standard deviation (or technically the variance) changes a lot over time.
For an excellent short video that shows how to do a lot of those transformations in Python, check the URL below
https://www.youtube.com/watch?v=7_Js8h709Dw
Here is an equivalent video for variable transformation using R.
https://www.youtube.com/watch?v=sWDYqnh376o
It is very likely that once you improve the structure of your variables and explore the optimal variable transformation, all your problems related to model biases will go away.
