I am trying to model the effect of advertisement on sales in Stata. The data is weekly and there are around 150 observations. I started by applying an ARMAX(1,0,1) model with the following exogenous variables: investment in advertisement, quantities bought by visit and some seasonal dummies (Q1, Q2, Q3).

I would like to have some ideas regarding the model:

1. Is this the best model to estimate the coefficients accurately?
2. Should I be worried about endogeneity?
3. Is there any way to impose (or test) diminishing returns for the investment in advertisement?

• Generally, if $x_t$ affects $y_t$ and vice versa, having a model of the form $y_t=\beta_0+\beta_1 x_t+\varepsilon_t$ (possibly including other regressors as well) will not work because $x_t$ will be correlated with $\varepsilon_t$ in reality while zero correlation will be mechanically imposed in the model (causing quite a discrepancy). – Richard Hardy Feb 4 '16 at 10:10
3. There are many ways of doing it. Any power function may work like $a^{\alpha}$, where $\alpha\in(0,1)$.