# Why are least-squares parameters normally distributed?

I am trying to figure out why the parameter $$\begin{equation*} \hat\beta = (X^TX)^{-1}X^TY \end{equation*}$$ is normally distributed in least-squares prediction. (Where Y is a linear function plus normal noise.) All the examples I've found have said that since \begin{align*} \hat\beta &= (X^TX)^{-1}X^TY \\ &= (X^TX)^{-1}X^T(X\beta + \varepsilon) \\ &= \beta + (X^TX)^{-1}X^T\varepsilon \end{align*} we know that $$\hat\beta-\beta \sim \mathcal{N}(0,\sigma^2 (X^TX)^{-1})$$

I can see how the mean and variance are calculated, but why is this a normal distribution?

• property: a linear transform of a normal is a normal – Xi'an Jan 5 '17 at 5:00
• – Xi'an Jan 5 '17 at 5:02
• Also be aware that under certain regularity conditions, the distribution of $\hat{\beta}$ will be asymptotically normal as the number of observations $n \rightarrow \infty$. For the asymptotic argument, you don't need $\epsilon$ to be normal (but you do need conditions such that a central limit theorem and other asymptotic arguments apply). – Matthew Gunn Jan 5 '17 at 8:39

## 1 Answer

In classical statistics the parameter value $\beta$ in a linear regression model is an unknown constant. The value $\hat{\beta}$ is not a parameter - it is an estimator of the parameter, which is a function of the data. The reason this estimator is normally distributed is that it is a linear function of the underlying error vector (as written in the equation you have shown), which is normally distributed under the model assumptions.