# OLS - Why coefficient Beta has Normal Distribution but not t-Distribution?

I have a little trouble understanding the solution to the following problem. To my understanding, coefficients in OLS should follow t-distribution. However, the solution says it follows Normal.

Please see the answer in the picture

• Where did this problem come from? Commented Jan 27, 2023 at 4:27

The result shown is correct. Indeed, in general, for any true error variance $$\sigma^2$$ and under the usual linear model assumptions,

$$\hat\beta \sim N_p(\beta, \sigma^2(X^\top X)^{-1}),$$ where $$\beta = (\beta_1,\ldots,\beta_{p})$$. On the other hand, for a single component of $$\hat\beta$$, say $$\hat\beta_r$$, we have that

$$\hat\beta_r \sim N(\beta_r, \sigma^2(X^\top X)^{-1}_{rr}),$$

where $$(X^\top X)^{-1}_{rr}$$ denotes the $$r$$th element of the diagonal of the matrix $$(X^\top X)^{-1}$$, $$r=1,\ldots,p$$. Now the $$t$$-Student distribution comes in when you wish to perform inference on $$\beta_r$$. Indeed, given the pivot

$$\frac{(n-p) S^2}{\sigma^2}\sim \chi_{n-p}^2,$$ where $$S^2 = \frac{\sum_i e_i^2}{n-p},$$ with $$e_i$$ the i-th residual, under the assumption of the linear model, we have the following pivot for $$\beta_r$$

$$\frac{\frac{\hat\beta_r-\beta_r}{\sqrt{\sigma^2(X^\top X)_{rr}^{-1}}}}{\sqrt{S^2/\sigma^2}} = \frac{\hat\beta_r-\beta_r}{\sqrt{S^2(X^\top X)_{rr}^{-1}}}\sim t_{n-p}\,\tag{*}$$

It is in (*) where you encounter the $$t$$-Student distribution concerning the OLS estimator.

$$\hat\beta_3 \sim N(\beta_3, 0.022 \sigma^2)$$ is the distribution of the estimate $$\hat\beta_3$$ conditional on the values of $$\beta_3$$ and $$\sigma$$.

In inference, you often do not know $$\beta_3$$ and $$\sigma$$ and compute a statistic $$\frac{\hat\beta_3}{\hat{\sigma}}$$. That is the statistic which is t-distributed (if the null hypothesis, $$\beta_3 = 0$$, is true, otherwise it is non-central t-distributed).

For your 'problem 1' the idea is to ignore this for a while and just experience/understand/observe how the estimate $$\hat\beta_3$$ is distributed, when we know the actual values of $$\beta_3$$ and $$\sigma$$. It is a sort of thought experiment.

• (+1) for the speed. Commented Jan 26, 2023 at 8:59

In OLS, we typically assume the following for the error terms $$u$$:

• A(1): $$E(u) = 0$$
• A(2): $$u \sim N(0, \sigma^2I)$$

this leads to the distribution $$\widehat{\beta} \sim N(\beta, \operatorname{Var}(\widehat{\beta}))$$ in small samples, if assumption A(2) holds true and even approximate for large samples, if A(2) is violated