# standard errors for computing means via OLS regression

Suppose I have a variable $$y_{i,t}$$ which is say a well-being index for individual of type $$i$$ in month $$t$$. There are 2 types of individual, $$A$$,$$B$$, which are mutually exclusive, and three non-overlapping periods $$t1$$, $$t2$$, $$t3$$, so that the entire sample periods consists of those three periods.

If I want to estimate the mean of $$y$$ for each type of individual and each period, along with standard errors for the hypothesis that the mean is 0 for that individual type and period, I can either do it normally (selecting the relevant part of the dataset and computing the mean), or I could run a regression:

$$y_{it} = \beta_1 D_A D_{t1} + \beta_2 D_A D_{t2} + \beta_3 D_A D_{t3} + \beta_4 D_B D_{t1} + \beta_5 D_B D_{t2} + \beta_6 D_B D_{t4}+\epsilon_{it}$$

where $$D_A$$ equals one if individual is of type $$A$$ and 0 otherwise, and $$D_{t1}$$ is 1 if we are in time $$t1$$ and 0 otherwise. The remaining variables are defined analogously.

My interpretation is that $$\beta_1$$ is the mean of the dependent variable for individuals of type $$A$$ in period $$1$$. Is this the correct interpretation?

Also, is it necessary to adjust the standard errors in any way? Since periods and individual types are mutually exclusive, I assume not.

Your interpretation of $$\beta_1$$ is correct. It should be pretty easy to verify by comparing the 6 coefficients to the 6 type x period means.