The chi square test for regression coefficients

[Matt Rice]

I am sorry for bothering you with details of this question. I am studying turnout disparity between rural and urban areas in a developing country. My theory suggests that rural and urban areas have fundamentally different modes of participation. (I developed a hypothesis based on this theory) Currently, I have a pooled sample of turnout and several independent variables in about 700 districts. In order to test hypothesis, I think I need to split the sample between urban and rural areas. I generated a dichotomous variable of urban/rural districts. I run two regressions (one for urban and the other for rural areas), and then compared the coefficients. I run the two regressions with “suest test”. Among several independent variables, only for one regression coefficient the prob>chi2 is less than .05, and this variable is the main independent variable according to the theory. For the rest of variables (which are the controls) the the prob>chi2 is larger than .05. (the implied homoscedasticity is unacceptable for this variable, I run robust regression, and the result is almost the same).
I have a difficult time to interpret the result. If “only one” coefficient (which happens to be main independent variable) varies between rural and urban districts. Can I conclude that my hypothesis about the fundamentally different modes turnout (between urban and rural districts) is correct? OR all other control variables should interact with urban/rural variables and have the chi2 less than .05 to support my idea about the different modes of turnout?

Many thanks

[Clyde Schechter]

Well, it depends on what your idea about different models of turnout was. If your idea was that urban and rural would differ in all of these ways, or in many of them, then your results are not very supportive of that. (Bear in mind, however that lack of statistical significance does not mean no difference; it just means that the data do not estimate the difference with sufficient precision for you to determine the direction of any difference that might exist.) But if your idea was that they would differ in this one particular independent variable, then your data support that and the other results are beside the point.

[Matt Rice]

Many Thanks Clyde,
Just one more question. Is it better to study the theory via "suest test"? or alternatively add the interaction term between the dichotomous variable of urban/rural and all other independent variables? (because if urban/rural variable interact with other variables, it means that the impact of independent variables differs in rural and urban districts).

[Clyde Schechter]

The interaction method has the advantage of being applicable to any kind of regression analysis, where as -suest- only supports some models. (In particular, -suest- does not support -xtreg, fe-, which is often the model you want to use.) The other difference in the aproaches is that -suest- takes the two separate models at face value, whereas the interaction approach may impose some tacit constraints. For example, if the underlying regressions or OLS linear regressions, the interaction approach constrains the residual variance to be the same in urban and rural. If that assumption is not acceptable, then the interaction approach must be avoided.

If none of the above considerations apply, ll else equal, I find the interaction approach more convenient to code and work with, but that is just my personal taste.

[Matt Rice]

Thanks a lot Clyde,
I was working on the scatter plots and I examined the relation between the main the independent variable (with the unacceptable homoscedasticity) and the dependent variable. Is there any connection between the difference between the coefficients on the one hand and heteroscedasticity on the other hand? And how can I interpret the heteroscedasticity and the differences in the coefficients together?

[Clyde Schechter]

Is there any connection between the difference between the coefficients on the one hand and heteroscedasticity on the other hand?

No, I don't think so.

And how can I interpret the heteroscedasticity and the differences in the coefficients together?

Well, the first thing to understand is that the heteroscedasticity does not lead to bias in the estimation of the coefficients. The effect of heteroscedasticity is to undermine the basis for calculation of standard errors, confidence intervals, and p-values. If you are using the -suest- approach to combining the two models and you also need good standard errors, you can specify the -vce(robust)- option in -suest- and get results that are robust to heteroscedasticity. (N.B. You cannot use -suest- with models that have themselves used -vce(robust)-, but you can specify -vce(robust)- with -suest- to get heteroscedasticity-robust standard errors.)

[Matt Rice]

Many Thanks Clyde, Your response helped me a lot to refine my thinking about the problem.