How to investigate the effect of predictors that correlate using regression? I have 7 predictors and I would like to know how each of them affects Y variable. However, some of them correlate:

How should I proceed?

*

*Fit them individually?
y ~ A
y ~ B
...
Then I can have pseudo effects? E.g. C has an effect in real life only, however, it correlates with D; thus, D shows pseudo effect in regression?


*Fit them together?
y ~ A + B + C + D + E + F + G
This shows no effects of C and D, since they correlate?
 A: There are multiple things you can do. I would recommend looking at the problem from the perspective of the new causal revolution. You are interested in the causal effect of $\{A,B,C,D,E,FF,G\}$ on $Y,$ but you're unsure if there are causal relationships among the explanatory variables. The very first thing I would do is draw a causal diagram. This is just a Directed Acyclic Graph (DAG), where node $A$ causing node $B$ is represented by the simple arrow $A\to B.$ NEVER underestimate the power of a DAG in analyzing cause-and-effect. Once you have your DAG, you can start to think about what would make the most sense to do, to isolate the causal effect in which you're interested. For example: suppose you have the following DAG:

This is called a mediation scenario. You do NOT have a back-door path from $X$ through $Z$ to $Y,$ because the arrow between $X$ and $Z$ points to $Z.$ This is therefore not a confounding situation, even though $X$ and $Z$ would likely be correlated. There is no need to condition on $Z.$ In fact, if you want the true causal effect of $X$ on $Y,$ you should NOT condition on $Z.$ On the other hand, suppose you had this situation:

Now you have a backdoor path: $X\leftarrow Z\to Y,$ and you must condition on $Z.$
Now I've used this term "conditioning" a couple of times. In a linear regression scenario, conditioning looks like simply including the variable in the model. So in the mediation example (the first one above), not conditioning on $Z$ means your model is $Y=mX+b.$ In the confounding example (the second one with the backdoor path), conditioning on $Z$ means your model is $Y=mX+nZ+b.$
This should get you started, I hope. If you draw a DAG for your situation, please do include it in your question.
A: The first way is to possibly construct a good theory first. Without a good theory, two things that should not have any good relationship may have a correlation. For example, my electric bill is high may be correlated to that the rice was overcooked, but not the reason for one over another (maybe all of them are due to the third factor, my rice cooker is broken). Having a valid hypothesis with explanation may give you a good groundwork.
That said if you are given a pool of n variables that may or may not correlate, a good way to handle it may be to select criteria, like maximizing the modified R^2, and the first regress the data individually with respect to each of the variable, $Y$ ~$X_i$ for $i=1,2,..,n$. You then select the variable that has the greatest R^2. Without losing generality, assume the variable you choose is $X_i$l, then you regress for each $i$, with $Y$~$X_1+X_i$ for $i=2,3....,n$. Repeat until you think the model is good enough.
