What is the difference between the following two t-statistics? When we compute t-stat for a specific estimate in the multiple regression, we usually use:
$$ t = \frac{
\hat\beta_k - \beta_k}{  se(\hat\beta_k)}.$$ where $se(\hat\beta_k)$ is from variance-covariance matrix: $\sqrt{s^2(X'X)^{-1}_{kk}}$ and $s^2$ can be obtained by $\frac{SSE}{n-k-1}$.
But can we instead use $$t = \frac{\hat\beta_k - \beta_k}{ \sqrt{s^2/n}}?$$
What's the difference between  $\sqrt{s^2(X'X)^{-1}_{kk}}$ and  $\sqrt{s^2/n}$ and why don't we use both standard errors for t-statistic in case of regression analysis? (I computed both but the numbers turned out to be different.)
 A: Given the $n\times p$ design matrix $X$, the $n\times 1$ vector of response values $y$, the ordinary least squares (OLS) estimator of $\beta = (\beta_1,\ldots,\beta_p)$ is
$$\hat\beta = (X^\top X)^{-1} Xy,$$
which is equivalent to the maximum likelihood estimator. The OLS estimator can be shown to be, unbiased and normally distributed with a variance equal to
$$\text{var}(\hat\beta) = \sigma^2 (X^\top X)^{-1}.$$
As $\sigma^2$ is typically unknown, we replace it with its unbiased estimator
$$s^2 = e^\top e /(n-p),$$
where $e = y\top y−(X\hat\beta)^\top y$ is the vector of residuals.
Thus, in practice we almost always use
$$
\widehat{\text{var}}(\hat\beta) = s^2(X^\top X)^{-1},
$$
instead of in of $\text{var}(\hat\beta)$. This plug-in destroys the normality of $\hat\beta$ but not all is lost. Indeed the quantity
$$
(*)\quad\quad\quad\frac{\hat\beta_k - \beta_k}{\sqrt{s^2 (X^\top X)^{-1}_{kk}}} = \frac{\frac{\hat\beta_k - \beta_k}{\sqrt{\sigma^2 (X^\top X)^{-1}_{kk}}}}{\sqrt{s^2/\sigma^2}}\sim \frac{N(0,1)}{\sqrt{\frac{\chi_{n-p}^2}{n-p}}} \sim t_{n-p}
$$
is an exact pivot. This pivot is used to conduct inference on $\beta$.
Thus, the standard error of $\beta_{k}$ is $\sqrt{s^2 (X^\top X)^{-1}_{kk}}$, which equals $\sqrt{s^2/n}$ only when $\beta_k$ is the intercept, i.e.
$$
\widehat{\text{se}}(\hat\beta_1) = \sqrt{s^2/n}.
$$
