Did splitting my linear regression sample improve my model? Approach 1: I regress $Y$ on $X$, obtaining estimates $\hat{\beta}$ and its covariance matrix $S$.
Approach 2: I first partition $X$ into $X_1, X_2$ using some method (e.g. k-means).
Then I separately regress $Y$ on $X_1$ and $X_2$, obtaining results $\hat{\beta_1}, S_1, \hat{\beta_2}, S_2$.
Did approach 2 improve over approach 1?
I've been looking at the determinant of the covariance matrix, aka generalized variance, as a scalar measure of model performance (intuitively I want my estimates to have low variances). Since smaller samples naturally lead to higher-variance estimates, is there a way to take that into account?
 A: Turning my comments above into an answer: 
Splitting data into subsets and fitting models to each subset cannot improve your models, parameter estimates or inferences provided your original model is properly specified. As Andrew Gelman puts it here, while considering the case of data being sampled multiple times from the same process: 

From a Bayesian standpoint, the result is the same, whether you
  consider all the data at once, or stir in the data one-third at a
  time. The problem would come if you make intermediate decisions that
  involve throwing away information, for example if you take parts of
  the data and just describe them as statistically significant or not.

As @whuber points out in the comments below, the assumption that the original model (fit to the full dataset) is specified properly is crucial. If splitting the dataset and fitting models to the subsets leads to improvement, it would be evidence that the original model was misspecified. I should note that there are, of course, other ways of identifying model misspecification.
