Obtaining 2nd Derivatives for LogLikelihood Function

Question

L(β;σ^2) = -T/2*(log(2π))-T/2*(log(σ^2))-(1/2σ^2)*(y-Xβ)'(y-Xβ)

Can someone clarify that ∂^2 L/∂β∂σ^2 = (1/σ^4)*(X'Xβ-X'y) ??

My tutor has it as -(My answer) but I can't work out why. I'll assume this is a mistake unless this can be proved otherwise? I have sent an e-mail to him but he normally takes a couple of days!

Thanks

I was going to typeset your formulas so it's easier to read, but you are missing parentheses in the first line. please look at it closely. — symplectomorphic, Apr 30 '14 at 21:22

score 0 · Accepted Answer · answered Apr 30 '14 at 22:08

0

Differentiate L wrt $\beta$ gives $X'(y-X\beta)/\sigma^2$. (There are two minus signs which give the +). This is easy if $\beta$ is just a number, but also works if it is a vector. Differentiate wrt to $\sigma^2$ gives minus the quoted result as $\partial (1/x) /\partial x = -1/x^2$

answered Apr 30 '14 at 22:08

user121049

1,599

Which = the result I got right? – hseager Apr 30 '14 at 22:18
I failed to notice that you had flipped the terms around in the brackets of your answer, so what you wrote above is correct. – user121049 May 01 '14 at 07:16
Brilliant. I've also found another lecturers notes from a different university which clarifies my answer. Thanks for your help. – hseager May 01 '14 at 12:10

Obtaining 2nd Derivatives for LogLikelihood Function

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