How can I test "Pollution causes people move to less polluted cities"?

Question

I want to find an answer for the following question: "Air pollution causes people to move to less polluted cities from 1990 to 1991"?

Assume there are four cities: San Diego, Los Angeles, San Francisco, and New York. And their respective air pollution levels are 0, 1, 2, 100. I have data for 1000 people and the data look like this: if a person lived in Sand Diego in 1990 and in Los Angeles in 1991, Pollution1990 will be 0 and Pollution1991 will be 1.

PersonID    City1990   City1991   Pollution1990   Pollution1991
    1        SD         LA             0               1

I thought about checking the following model:

reg Pollution1991   Pollution1990  controls, nocons

and if coefficient is smaller than 1, then I thought I could conclude that people are moving to less-polluted cities.

But then I found that

reg Pollution1990   Pollution1991 controls, nocons

also gives me coefficient smaller than 1. This was strange.

So how can I answer the question "Air pollution causes people to move to less polluted cities from 1990 to 1991"?

Is this even possible to test? Is there a regression towards the mean trap?

Answer 1

The statement of your model is not clear. It should be written such that is understood regardless of the analysis language you use, i.e. y ~ x,z. It is also not clear what is your dependent variable in the regression model. An alternative model which may provide information is to subtract Pollution1991 from Pollution1990 and flag those where the difference is greater or lesser than zero. Doing so will provide a set of zeros if the person did not move, +1 if they moved to a more polluted city and -1 if they moved to a less polluted city. You can use categorical variables with +1 -> "more" and -1 -> "less" and 0 -> "none" (assuming you collected data from people who did not move). You can now calculate the differences in the proportions, no regression needed.