a high correlation coefficient between the dependent variable and a control variable I'm running a cross-sectional regression, and one of my control variables is highly correlated with the dependent variable ($\rho \approx 0.70$).
What things should I be concerned about, given this high correlation coefficient?
It's a very important control variable so I'm reluctant to omit it. 
 A: If it's highly correlated with the outcome and not correlated with the other predictors, you should almost certainly include it, as it increases the power to detect the effect of other other predictors. In a randomized trial, the baseline measure of the outcome variable is the most important control variable.
If it is correlated with the other predictors, it is a theoretical question whether to include it in the model. If you include it in the model it changes the meaning of your other parameters, and whether you want that or not depends on what you want to know. 
Illustrative case:
Your outcome is height, your predictor is gender (dummy coded; female is reference category.
You predict height with gender. The regression parameter you get is the difference between the heights of males and females in your sample.
You also know hair length. Hair length is correlated with both gender and height. You put that into the model. The parameter estimate associated with gender now means .... I have no idea. 
It turns out this is a sample of kids. You also know their age, and age is uncorrelated with gender, but correlated with height. So you put age into the model.  The parameter estimate associated with gender does not change, but the standard error drops (hence you have more power)
The size of the correlation between hair length and age, or height and age, is not relevant to the decision of whether to include them in the model. 
A: You can add it, check if it improves somehow your model. Anyway if it isn't higher than 0.800 or lower than -0.800 you might want to keep it in your model.
Check if you can use some statistics like p-value to support your decision.
