Should the control variables in an econometric regression be correlated with both the dependent and the primary independent variables? If, for instance, my dependent variable is some happiness index, and my independent variable is a dummy for whether they experienced some randomly occurred natural disaster. I am trying to analyze the effect of having experienced that disaster on the happiness index.

I am using a fixed effects model.

I have basically two interrelated questions:


What I read so far is that in regressions, control variables need to be correlated with both (primary) x and y variables. If that is the case I do not see a lot of variables that are correlated with both the disaster and happiness, besides:


Wealth (the poor are affected more by the disaster; wealth is also correlated to happiness)
Some disaster-induced variable like dummies for whether the disaster affected their water source, whether they had to live in a temporary shelter (tents), whether their livestock was killed, etc., which are all correlated with happiness.
However, I also think that if I control for both of these factors, then I am controlling for the mechanism in which an individual's happiness could be affected by the disaster, i.e. taking away the effect of the earthquake-induced factors in the second point above. Am I thinking correctly?

Is it plausible to control for factors like age, education, etc.? These factors influence happiness (may even influence post-disaster happiness), but these variables like age are not directly correlated with whether they experienced disaster or not. Am I thinking correctly?
First of all, you mentioned that the disasters you are dealing with are "randomly" occurring.  So I want to provide the following as food for thought:

There is an important question here about what your "disaster" variable actually measures and at what level it was "randomized" (if at all).  If it is measured in spatial terms (was an individual at this particular place when disaster occurred), it is hard to imagine that this would be a binary variable (you experienced a disaster or not) and not something measured on a scale (how much damage you area experienced), unless treatment and control groups are far away from each other.  But in that case, it is hard to imagine that this variable would be random (both groups having equal probability of experiencing the same or similar type of disaster).  If it is measured in temporal terms (was there a disaster on day x) and there is no spatial variation in the data, then it is also not clear whether this variable is random because a lot of places experience the same disaster year after year and changes can be made over time in the same location based on previous experiences to better plan for future events.

But if the variable is, in fact, random, then by the beauty of randomized treatment you don't need to worry about control variables because the treatment variable is exogenous.

Question 1:

In answering your questions, I am going to assume that your "disaster" variable is not random, which seems like a more reasonable assumption.  I am also thinking of "experiencing a disaster" not just as whether you were in that place or time when the disaster occurred, but as something that had a physical/physiological/mental affect on you.  If it meant just being in that place or time then it is hard to think of anything affecting someone being in that place or time without specific context.

So we have to think about possible confounding variables.  Do any variables exist that affect whether someone "experiences" a disaster and that also independently affect someone's happiness level?  Yes, wealth fits that criteria, as a wealthier individual may have better safeguards in place (like earthquake or flood insurance) to allow them to deal with the impact of a disaster.  Other possible variables are age and health level.  Health affects happiness and during times of disasters those living in nursing homes or having chronic conditions like kidney failure or COPD are more affected by loss of power, for example.  Similarly, whether someone lives alone can be a confounder.  Occupation can be possibly, as emergency workers may be more impacted by disasters or some other professionals may be better trained to deal with the consequences of disasters.

Your thinking about not controlling for disaster-induced variables is correct.  You should not control for variables that are themselves not fixed at the time of "treatment".  They should be allowed to change when your "treatment" variable changes.

Question 2:

Like I said above, I would think of age as a confounding variable in the relationship between experiencing a disaster and happiness.  But if you have variables that you believe do not directly affect "experience" of a disaster but that do affect happiness, then it is still recommended to include them in the model because this reduces the error variance in the model (i.e., standard errors of coefficient estimates will be reduced).