When does the standard errors of OLS estimates decreases when we have more explanatory variables? So far my understanding is that standard errors of $\hat\beta_j$ increase when we have more independent variables in the regression model because variation among those independent variables will increase ($\rightarrow R^2_j \uparrow$). But I saw the opposite thing happened when I add more explanatory variables in the RHS of the simple difference-in-differences model:
$$y_t=\beta_0+\delta_0x_1+\beta_1 x_2+\delta_1x_1x_2+u $$
I can't understand how this could be possible?

This happens mostly with misspecified models that have large residuals because of a misspecified model and that can be reduced by adding more variables.
As simple illustration can be made with simple linear regression.


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*The linear model fit $x=a+bx$ in the upper left corner is more precise in the estimate at x=0. The estimate for the standard error is small because the residuals are small.

*The model that uses only an intercept $x=a$ indicates a larger error. The reason is that the residuals are much larger. With the linear model fit these residuals are a lot reduced.

*The lower right image: to our eye it seems weird that the model $y=a$ is doing bad but that is because our brain sees the linear relationship and we imagine a line going through 0. The model doesn't see this and includes no information about $x$. For the model it looks like the $x$ data are scrambled.

Another illustration that shows it dramatically is from Why is the intercept in multiple regression changing when including/excluding regressors?
A smaller model can have bias which increases the error.

