Is it appropriate to use 3 linear regressions to assess the impact of one common independent variable? I need to assess the impact of cultures on urban economies. The hypothesis is that more entrepreneurship oriented culture will better facilitate the local economy, controlling for other factors (that may contribute to local economic growth). The study area is 3 cities next to/not far from each other but with different cultures (e.g., values, dialects, traditions). My plan is to build 3 regression models using the same set of DV and IVs, and then compare the 3 sets of coefficients (especially the coefficients of the culture indices). However, this method looks tedious to me. Is there a more efficient and elegant way to do this?
Thank you very much!
A.
 A: There is simpler and more elegant method. If the each of view the models you describe would take form
$$
y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k + \varepsilon
$$
then just use dummy variables to encode the cities and define single regression model similar to the above, but with extra interaction terms with the dummies $d_i$
$$
y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k +  \beta_{k+1} d_1 + \beta_{k+2} d_1 X_1 + \beta_{k+3} d_1 X_2 + \dots + \beta_{2k+1} d_1 X_k + \dots +\beta_{(m+1)k+1} d_m X_k + \varepsilon
$$
In such case, beyond the global terms, as in individual models, you would have intercepts $\beta_i d_j$ and coefficients $\beta_l d_j X_p$ for each of the cities.  To test for the differences, just look at the significantly of each of the terms. Of course, depending on your problem, you may choose to define interaction terms only for some variables, if it makes more sense. Interpretation would be the same, as for any other regression model with dummy variables and interaction terms.
A: What you have suggested can be considered a split-sample analysis, whereby you run a separate regression for each of the cities. The above approach proposed by Tim (that is, interacting city dummies with the culture variables) works very well as long as you are okay with constraining the coefficients on the non-interacted variables to being the same across the three cities. This is usually a fine assumption but depends on your research question/context.  
Running a split sample analysis can be considered equivalent to a "fully-interacted" model where every independent variable is interacted with city. This allows coefficients on all variables to vary across cities. If you are running a linear regression model, the coefficients on variables across split samples can be compared using statistical test such as Wald's test. If you are running a non-linear model (e.g. Logit, Tobit), the coefficients do not represent marginal effects, and cannot be directly compared. 
