#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!!!

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