1

I'm trying to understand the calculation of the t-statistic in a linear regression when you are testing against a specific value, i.e. using the formula $$t=\frac{\hat{\beta}-\beta}{\text{S.E.}(\hat{\beta})}.$$ t-statistic states that "$\text{S.E.}(\hat{\beta})$ is the standard error of the estimator ${\hat {\beta }}$ for $\beta$."

I understand how this works for $\beta=0$ (using $\sigma^2(X^TX)^{-1})$ but I'm not sure how to generalise the calculation of $\text{S.E.}(\hat{\beta})$.

mrepic1123
  • 489
  • 6
  • 1
    The equation you wrote is the generalization for $\beta != 0$. If $E(\hat{\beta}) = \beta$ then $E(\hat{beta} - \beta) = 0$. If I'm not right, can you be more specific about what you mean by "generalization"? Can you tell us what kind of equation you are looking for? – R Carnell Jan 20 '24 at 21:30
  • @RCarnell apologies that was not clear, I meant the calculation of the standard error. Does this remain unchanged when $\beta!=0$? – mrepic1123 Jan 20 '24 at 21:35
  • 1
    The calculation of the standard error of $\hat{\beta}$ is only dependent on the sample. So, yes, if the true $\beta$ is non-zero, the equation still works. – R Carnell Jan 20 '24 at 21:46

0 Answers0