Let’s assume I have a polynomial regression but I’m not sure which polynomial degree d to use. In econometrics, one would probably try out the various regressions, each with different degrees and then check the significance of the individual terms, compare R^2 value and so on. Assuming I have enough data points, if I want to explore d = 1,..., 20, then I’ll need to do twenty individual regressions and compare them.

Would be wrong, if I start from the very largest model, say d=20, and then use lasso regression to select only the terms that have non-zero coefficients?

My question is absolutely related to the following old and unanswered question in stackexchange: https://stats.stackexchange.com/questions/224707/selecting-polynomial-terms-in-regression