# Can you extrapolate values of the dependent variable with a GAM?

I'm trying to find issues where GLMs are better than GAMs and came to the idea that GLMs can make predictions beyond the scope of the data used to feed the model (i.e, extrapolations), while GAMs cannot:

Suppose we have a set of X and Y observations. The X observations are spread inside the domain [x0, x1]. If we fit a GLM to X vs Y we obtain a mathematical relation between X and Y (in the most simple case, Y = b0*X + b1). Therefore, we can obtain for every X_i of our choice a modelled Y_i. We surely should have a good estimate if X_i is inside [x0,x1] but nothing speaks about giving a try also for values outside this range (another story is that the estimate is "good").

Now, GAMs are based on smooth functions obtained from the X-Y scatter, but they give no (simple) mathematical relation between X and Y. You get an Y estimate for each X observation you have and can make a nice plot. Surely you can interpolate any Y value between observations to obtain an estimate of your choice, but considering we have only X data inside the range [x0, x1] you cannot predict (or extrapolate) a Y value with a GAM for an X value lying outside the range [x0, x1]. With no mathematical relation linking X and Y, you cannot extrapolate!

So, if I understand correctly and the answer to my question is "no", I would say the extrapolation or predicting potential of a GLM is surely a very strong advantage in comparison to a GAM!

• I think you're correct, as long as you can assume that the estimates are the same even for higher or lower values of x than observed. For example, if we consider prices on used cars and all our data are from cars that are 5-10 years old, then we might see a nice (negative) linear relationship between car age and price. But this will be quite useless to assess prices on cars that are 1-2 years old, as the reduction in price is more steep initially. So assuming that the estimated coefficient remains stable, I think you're correct. – JonB Oct 14 '15 at 13:14
• Using natural splines you can do some limited extrapolation. – kjetil b halvorsen May 9 at 10:57