Determining statistical significance of linear regression coefficient in the presence of multicollinearity Suppose I have a bunch of cities with different population sizes, and I wanted to see if there was a positive linear relationship between the number of liquor stores in a city and the number of DUIs.  Where I'm determining whether this relationship is significant or not based on a t-test of the estimated regression coefficient.
Now clearly the pop. size of a city is going to be positively correlated with both the number of DUIs as well as the number of liquor stores.  Thus if I run a simple linear regression on just liquor stores and see if its regression coefficient is statistically significant, I will likely run into a problem of multicollinearity, and over-estimate the effect of liquor stores on DUIs.
Which of the two methods should I use to correct for this?


*

*I should divide the number of liquor stores in the city by its population in order to get a liquor store per capita value and then regress on that.

*I should regress on both liquor stores and size, and then look to see if the liquor store coefficient is significant when controlling for size.

*Some other method?
I honestly can't decide which seems more sensible.  I vacillate between them, depending on which one I think about I'm able to convince myself that that's the right way.
On the one hand liquor stores per capita seems like the right variable to use, since DUIs are committed by individuals, but that doesn't seem very statistically rigorous.  On the other hand, controlling for size seems statistically rigorous, but rather indirect.  Furthermore, if I rescale after computing the liquor stores per capita variable, I get very similar regression coefficients between the two methods, but method 1 produces a smaller p-value.
 A: I would regress the "DUI per capita" (Y) on "liquer stores per capita" (X) and "population size" (Z). This way your Y reflects the propensity to drunk driving of urban people, while X is the population characteristic of a given city. Z is a control variable just in case if there's size effect on Y. I don't think you are going to see multicollinearity issue in this setup.
This setup is more interesting than your model 1. Here, your base is to assume that the number of DUIs is proportional to population, while $\beta_Z$ would capture nonlinearity, e.g. people in larger cities are more prone to drunk driving. Also X reflects cultural and legal environment directly, already adjusted to size. You may end up with roughly the same X for cities of different sizes in Sough. This also allows you introduce other control variables such as Red/Blue state, Coastal/Continental etc.
A: If estimating your model with ordinary least squares, your second regression is rather problematic.
And you may want to think about how the variance of your error term varies with city size. 
Regression (2) is equivalent to your regression (1) where observations are weighted by the square of the city's population:
For each city $i$, let $y_i$ be drunk driving incidents per capita, let $x_i$ be liquor stores per capita, and let $n_i$ be the city's population.
Regression (1) is: 
$$y_i = a + b x_i + \epsilon_i $$
If you run regression (2) without a constant, you've essentially scaled each observation of regression (1) by the population, that is, you're running:
$$ n_i y_i = a n_i + b n_i x_i + u_i $$
This is weighted least squares, and the weights you're applying are the square of the city's population. That's a lot of weight you're giving the largest cities?! 
Note that if you had an observation for each individual in a city and assigned each individual the average value for the city, that would be equivalent to running a regression where you are weighting each city by population (not population squared).
