LR - Tests - Derivation of Student T-Statistic

When 𝜎 is UNKNOWN, the normal test for 𝛽ˆ is modified to become a student-t test.

A student-t random variable, with 𝜈 degrees of freedom, can be written as:

• $t=\frac{z}{\sqrt{{\mathcal{X}}_{𝜈}^2/𝜈}}$

where:

• 𝑧 is a standard normal random variable
• 𝒳_𝜈² is a chi-squared random variable
• 𝑧 and 𝒳_𝜈² are independent

We can use:

• $z=\frac{\widehat{𝛽}}{𝜎\sqrt{(X^{T}X{)}^{-1}}}$
• ${\mathcal{X}}_{n-p}^2=(n-p)\frac{{\widehat{𝜎}}^2}{{𝜎}^2}$

Thus:

• $t=\frac{z}{\sqrt{{\mathcal{X}}_{𝜈}^2/𝜈}}$
• $t=\frac{\frac{\widehat{𝛽}}{𝜎\sqrt{(X^{T}X{)}^{-1}}}}{\sqrt{(n-p)\frac{{\widehat{𝜎}}^2}{{𝜎}^2}/(n-p)}}$
• $t=\frac{\frac{\widehat{𝛽}}{𝜎\sqrt{(X^{T}X{)}^{-1}}}}{\frac{\widehat{𝜎}}{𝜎}}$
• $t=\frac{\widehat{𝛽}}{\widehat{𝜎}\sqrt{(X^{T}X{)}^{-1}}}$

Under the null hypothesis, this is a standard student-t variable with 𝑛-𝑝 degrees of freedom.

Reject the null hypothesis if:

• $|T|>qt(0.975,n-p)$