I encountered a real-world problem where I want to model the effectiveness of various advertising media of a brand (measured in terms of sales). Basically, the Y in this case is weekly sales, and the X's are media investments in newspaper, magazine, display boards, tv, radio and online, as well as incentive, which is a percentage (like 10% off the original).
There are a few problems with the modelling work:
all variables should have positive coefficients. Typically, more advertising or incentive is at least as good as less advertising/incentive (maybe this is not true if you buy all of the advertising spots in the world, as then your consumer will start to hate your brand, but this is not going to happen here). However, when I fit a typical regression (e.g. lm, glm, gls etc), some coefficients turn out to be negative (as data may be a bit noiser than expected, hence causing this problem?). I wonder if this can be controlled (I know in nonlinear regressions you can set constraints for parameters)
there should be some sort of diminishing marginal return of advertising spendings, but I am not exactly sure how to model that. Some ideas include using a log or square root transformation, another idea may be to use a nonlinear regression and estimator something like a*newspaper^b, where a is some coefficient, and b is an exponent between 0 and 1.
this is serial correlation, but this may not be exactly important here as the goal is only to estimate the parameters (if I use a regression I think I still get the unbiased estimators right? Autocorrelation only screws up the p-values, which is ignored here). Also, how to deal with seasonalities? I don't have much data (2 years) so maybe there is nothing we can do about it, but I have seen adding cos(0.0172*time) + sin(0.0172*time) to the regression equation to adjust for seasonal changes.