Number of books published in a year (noBook) is my dependent variable and I have independent variables including the age of the author (age).
The coefficient of age is positive and it is significant. When I add age^2 to the model, age and age^2 are significant but the sign of age becomes negative.
Could you please advise why it is like that? And, which model should I consider?
Best Answer
The fact that the coefficient of age becomes negative doesn't tell you much on its own. In the new model, the predictive effect of the author's age isn't just represented by an expression of the form $\beta\textit{age}$, but by a sum $\beta_{1}\mathit{age}^2+\beta_{2}\mathit{age}$. Assuming that the coefficient of age was positive in your first model, I would expect that over the range that you are considering, this quadratic expression is still increasing (check the derivative $\frac{\partial}{\partial\textit{age}}$ to make sure), indicating that nbooks increases with age, even if $\beta_{2}$ is negative.
(This reasoning still holds even if you have interaction terms, but the partial derivative above will take a little more time to compute, and you may have to consider a few more cases).