How to perform linear regression on clusters of data Suppose I have 2 clusters of data: $\{(Y_{1i}, X_{1i})\}_{i=1}^{n_1}$ and $\{(Y_{2i}, X_{2i})\}_{i=1}^{n_2}$, and I'm interested in running a simple linear regression on each cluster.
I assume that
$$Y_{1i} = \beta_{10} + \beta_{11}X_{1i}+\epsilon_{1i}$$
$$Y_{2i} = \beta_{20} + \beta_{21}X_{2i}+\epsilon_{2i},$$
where $\epsilon_{1i}, \epsilon_{2i}$ have mean 0 given $X$. To estimate the intercept and slope coefficients, I can minimize the empirical squared error in the two clusters separately:
$$argmin_{\beta_{10}, \beta_{11}} \frac{1}{n_1}\sum_{i=1}^{n_1}(Y_{1i} - \beta_{10}-\beta_{11}X_{1i})^2$$
$$argmin_{\beta_{20}, \beta_{21}} \frac{1}{n_2}\sum_{i=1}^{n_2}(Y_{2i} - \beta_{20}-\beta_{21}X_{2i})^2$$
Now suppose I assume that the intercept and slope coefficients are identical between the two clusters, i.e.,  $\beta_{10} = \beta_{20} = \beta_0$ and $\beta_{11} = \beta_{21} = \beta_1$. Is this equivalent to running a single linear regression model on the pooled data? i.e., I would minimize:
$$argmin_{\beta_{0}, \beta_{1}} \frac{1}{n_1 + n_2}\sum_{i=1}^{n_1 + n_2}(Y_{i} - \beta_{0}-\beta_{1}X_{i})^2$$

This depends on exactly what you mean by "equivalent" in: Is this equivalent to running a single linear regression model on the pooled data
if you mean whether you would get the same results from running regression on pooled data. Then the answer is yes.
