Estimate model with dummies for whole sample versus model without dummies for each subsample Assume the true (cross-section) regression model is
$$ Y_i = \alpha + \alpha_mD_m + \beta X_i + \beta_m X_iD_m + \epsilon_i $$
where $D_m$ is a dummy for gender (equal to 1 if individual is a male).
Assume I have a random sample of the above population, of size $n$. Now, instead of estimating the above, I estimate separate models for each gender:
$$ Y_i = \rho + \phi X_i + \mu_i $$
for female individuals and
$$ Y_i = \rho_m + \phi_m X_i + u_i $$
for male individuals.
Is it always the case that the probability limit of the estimators converge to the true parameter, as below?
$$ \rho \overset{p}{\rightarrow} \alpha $$
$$ \rho_m \overset{p}{\rightarrow} \alpha_m $$
$$ \phi \overset{p}{\rightarrow} \beta $$
$$ \phi_m \overset{p}{\rightarrow} \beta_m $$
I cannot think of any reason why the above is not true.
I ask this because I wonder if instead of estimating a model for a whole sample full of dummies to account for group effects, I can estimate a simpler model for different groups. Naturally, assuming that $n$ for each group is sufficiently high.
 A: If I understood you correctly, your main question is whether there are disadvantages of fitting separate models for different subgroups (for instance men and women) rather than fitting the data with a single model plus dummy variables.
A problem of splitting the problem in multiple sub-models is that you use your data for the confounding variables (thereby I mean those things you control for, but are not interested in in your study) ineffectively. Because you use less data for each sub-model to find the correct estimators for your confounding variables; their confidence intervals are larger and therewith the individual models are not as good as they could be.
Using the gender example, two separate models are technical the same as a  single model which has gender as dummy variable, your confounding variables PLUS interaction variables between gender and each confounding variable. But in most cases, that is not we want. Thus it is usually better to put it all together in a single model rather than fitting separate models.
