I'm just starting to learn about linear regression models and time series analysis and came upon the following doubt.
Suppose we have a variable $Y$ that we're trying to model using $p$ explanatory variables $X_i$, maybe using a simple linear model such as:
$$Y = a_0 + a_1X_1 +...+a_pX_p$$
My question is the following: what happens if the explanatory variables are correlated with each other. In the assumptions for this model, I see no mention of this fact, but clearly there has to be some kind of qualitative difference depending on the degree to which the variables are correlated with each other. For example if $X_1 = X_3 ^2$, then what is the point of including both $X_1$ and $X_3$ in the model, intuitively it seems like the other one carries no additional 'information'?
Perhaps depending on how correlated the variables are one would use a different model or approach?