DEMONSTRATION FINITE-SAMPLE PROPERTIES OF LEAST SQUARES $\frac{(N-k)S^2}{\sigma^2}\sim\chi^2[n-K]$

Question

Im a Student of Economics, and I have a concern. In the solution of

$\frac{(n-K)S^2}{\sigma^2}\sim\chi^2[n-K]$

How can I show that if the matrix is ​​symmetric and idempotent between

$(I-H)=|| (I-H)Y||^2=Y'(I-H)Y$

Answer 1

You should give more context to your questions. (What is $n$? $K$? etc. I was only able to figure out the context from your previous question.)

I think you mean "idempotent," not "independent."

Your last line also does not make much sense. $I-H$ is a matrix, while the other two quantities are real numbers. Also, $Y'(I-H) Y$ is a number, so it does not make sense to take the norm of it.

If $H$ is the hat matrix in linear regression then the following are true:

1. $H$ is symmetric and idempotent. Consequently, $I-H$ is also symmetric and idempotent.
2. $\|(I-H)Y\|^2 = ((I-H)Y)' ((I-H)Y) = Y' (I-H)' (I-H) Y = Y' (I-H)(I-H) Y = Y'(I-H) Y$, where we have used the fact that $I-H$ is symmetric and idempotent.