# The minimal possible value of total sum of squares for linear regression

I have a regression model

$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 +u$$

It is known that sample means of both $$x_1$$ and $$x_2$$ are zero, moreover the error term is said to be homoskedastic, the standard error of regression and standard errors of OLS estimates of the slope coefficients are given. Can we say something about total sum of squares (i.e. $$\sum_{i=1}^n (y_i - \bar{y})^2$$), namely, can we somehow estimate its smallest possible value?

What I have tried:

$$\sum_{i=1}^n (y_i - \bar{y})^2 = \sum_{i=1}^n(\hat{\beta_0} + \hat{\beta_1}x_{1i} + \hat{\beta_2}x_{2i} + \hat{u_i} - \bar{y})^2 = \sum_{i=1}^n (\hat{\beta_1}x_{1i} + \hat{\beta_2}x_{2i})^2 + \sum_{i=1}^n \hat{u_i}^2$$.

We know the second term, since we know the standard error. So it remains to estimate the first one. Standard erorrs of OLS estimates of slope coefficients are

$$se(\hat{\beta_1}) = \sqrt{\dfrac{\sum_{i=1}^nx_{2i}^2}{\sum_{i=1}^nx_{1i}^2\sum_{i=1}^nx_{2i}^2- (\sum_{i=1}^nx_{1i}x_{2i})^2}} \hat{\sigma}$$

$$se(\hat{\beta_2}) = \sqrt{\dfrac{\sum_{i=1}^nx_{1i}^2}{\sum_{i=1}^nx_{1i}^2\sum_{i=1}^nx_{2i}^2- (\sum_{i=1}^nx_{1i}x_{2i})^2}} \hat{\sigma}$$

Since we know $$se(\hat{\beta_1}), se(\hat{\beta_2}), \hat{\sigma}$$ we can compute the ratio $$\dfrac{\sum_{i=1}^nx_{1i}^2}{\sum_{i=1}^nx_{2i}^2}$$. Moreover we can express $$\sum_{i=1}^nx_{1i}^2$$ and $$\sum_{i=1}^nx_{2i}^2$$ through $$(\sum_{i=1}^nx_{1i}x_{2i})^2$$. Thus, $$\sum_{i=1}^n (\hat{\beta_1}x_{1i} + \hat{\beta_{2}}x_{2i})^2 = \hat{\beta_1}^2\sum_{i=1}^nx_{1i}^2 + \hat{\beta_2}^2\sum_{i=1}^nx_{2i}^2 + 2 \hat{\beta_1}\hat{\beta_2}\sum_{i=1}^nx_{1i}x_{2i}$$

The idea was to plug the expressions of $$\sum_{i=1}^nx_{1i}^2$$ and $$\sum_{i=1}^nx_{2i}^2$$ into the aforementioned expression and try to minimize it by $$\sum_{i=1}^nx_{1i}x_{2i}$$ treating $$\hat{\beta_1}, \hat{\beta_2}$$ like constants. But I do not think that it is a right way, since once we change $$\sum_{i=1}^nx_{1i}x_{2i}$$ we also change $$\hat{\beta_1}, \hat{\beta_2}$$ because the OLS formulas include $$\sum_{i=1}^nx_{1i}x_{2i}$$. Plugging formulas for OLS estimates also does not seem to be a good idea since these formulas include sample covariances between $$y$$ and $$x_1$$, $$x_2$$ which we do not know. Here I got stuck.

Could you please give me any hints, how to proceed?

Thanks a lot in advance for any help!