# #StackBounty: #regression #least-squares #sufficient-statistics Sufficient Statistic for \$beta\$ in OLS

### Bounty: 100

I have the classical regression model

$$y = beta X + epsilon$$
$$epsilon sim N(0, sigma^2)$$

where $$X$$ is taken to be fixed (not random), and $$hatbeta$$ is the OLS estimate for $$beta$$.

It is known that $$(y^T y, X^T y)$$ pair is a complete sufficient statistic for $$x_0^T beta$$, for some input $$x_0$$.

Can we conclude that $$(y^T y, X^T y)$$ is also a sufficient statistic for $$beta$$, and why? I think for this to work $$X^T X$$ should be full rank. I mean a 1 to 1 transformation of a sufficient statistic is still a sufficient statistic, but it is still a sufficient statistic for $$x_0^T beta$$. Based on what are we going to conclude the sufficiency of $$hatbeta$$ for $$beta$$ itself?

Get this bounty!!!

This site uses Akismet to reduce spam. Learn how your comment data is processed.