Term selection in a model

Question

In selecting terms to include in a model, say a linear one, should we always test the significance of the main effects first, keep only the significant ones, then consider the possible interactions between them?

For example, if I have five covariates $x_1$ - $x_5$, and I choose the significant main effects in some way. Assume the first three are significant, $\hat{Y} = x_1 + x_2 + x_3$. Now I consider the interaction between $x_1$ - $x_3$. Finally I may end up with something like $\hat{Y} = x_1 + x_2 + x_3 + x_1 \cdot x_3$.

Is it justifiable to do this? A problem may be the significant interaction between $x_4$ and $x_2$ ?

Answer 1

The common practice for linear model is, if you want to have higher order term, like interaction or quadratic term then even if the lower term, for example x4 is not significant according to certain criteria.

so if you suspect there might be higher order interactions, include them in the model first then do model selection.

Of course, you have to explore the relationship of those variables first, like in R using pairs function to plot the data points etc.