is it possible to use an existing non-linear relationship (GAM) to estimate back one of the independent variable? Hi I have used a set of data to estimate the relationship between v0 (response variable) and v1-v6 (independent variables) through a non-linear model using GAM. The model is fitted with data between 1980-2010 with a good deviance explanation rate. Interestingly, I have more historical data of v0-v5 from 1700-1980, but not for v6. My question is can we use v0-v5 and the fitted relationship to estimate v6 data from 1700-1980? if yes, how to do so? any help or thoughts will be appreciated!
v0 = s(v1)+s(v2)+s(v3)+s(v4)+s(v5)+s(v6)
where s()represents a smooth function of a variable


Turning my comment into an answer:
Multiple imputation is often used to estimate missing values of predictors. But it's much more liable to fail when you are extrapolating beyond the range of your data i.e. it's easier to fill in data from say 1990-1995 because you have values on either side to constrain possibilities. Basically, you'd have to make strong assumptions, such as the covariance being the same from 1700-1980 as from 1980-2010. These may or may not be justified - it depends on the nature of your data, and you'd need to explain that in detail for us to provide a good answer. I would say in most cases what you are suggesting would not be advisable.
That said, it does not mean that you need to ignore all data before 1980 - you have data on 6 out of 7 variables for this time period! I would be inclined to fit 2 models: one for 1980 - 2010 with all variables, and another for 1700 - 2010 omitting $v6$. Once you've evaluated these two models, you could decide how best to proceed.
