Aggregating explanatory variables in regression

I have a number of independent variables $x_1,x_2,...,x_m$ and a dependent variable $y$. My dataset contains some million of rows.

Bear with me if my wording is not precise here, and you are welcome to correct me! I assume that there is no multicollinearity in my data, i.e. $x_i$ cannot be explained by $x_j$ for $i\neq j$.

However, the dependent variable $y$ seems to react similarly on changes of, say, $x_1$ and $x_2$. How is it called, if the change in $x_1$ changes $y$ similarly like a change in $x_2$?

The question here is: Being on the greater search for a way to aggregate my $x_i$ to fewer variables, what would be the right method/procedure/research area/search engine term to find out whether (and how) $x_1$ and $x_2$ explain $y$ similarly?

• You might want to read around formative measurement models. They're a set of variables which, while not necessarily correlating with each other, are thought to be conceptually associated. The variables in a formative measurement model are typically summed (occasionally with weightings) to produce a single variable, similar to what @MaartenBuis suggests in his answer. – Ian_Fin Nov 2 '16 at 9:08
• What I called a sheaf coefficient is a formative measurement model. – Maarten Buis Nov 2 '16 at 9:25

If you mean with "the dependent variable $y$ seems to react similarly on changes of, say, $x_1$ and $x_2$", that the coefficients are very similar, then you can reduce the complexity of your model by constraining those coefficients to be the same. A simple trick is to create a new variable that is the sum of $x_1$ and $x_2$ and add that instead of $x_1$ and $x_2$ (a small note explaining this trick)