aegisknight
New member
- Joined
- Mar 14, 2013
- Messages
- 2
I am half way through my first graduate econometrics course, but I'm having some significant gaps in my stats background to get everything.
In particular, I am still struggling a bit on how projections work. Generally in applications I have no problem using them (I've become quite familiar with OLS, and am moving on to GLS and FGLS), but here I am trying to re-write some of my notes, and I am having a very hard time putting into words exactly what a projection is, and how it works.
I've been looking around the internet, and I can't get the distinction between any arbitrary projection, an idempotent projection, and an orthogonal projection, specifically in the context of basic linear regressions.
Since the professor spent about 5 minutes on the first day going over it, I guess I should have learned this at a more basic course level, but I've never worked with it before.
What I have so far is that a projection is idempotent if p^2 = p, and it is a matrix which projects a vector onto another matrix (or something like that).
I also have this proof showing that p is symmetric and idempotent:
p' = (x(x'x)^(-1) x')' = (x')'((x'x)^-1)'x'...
x(x'x)^-1 x' = p
I get the matrix algebra easily enough, and if my understanding of idempotency from above is accurate (which I believe it is, given proofs I have done relating to the OLS) then I understand that as well. But I don't know how I am proving symmetry, and its implications.
In particular, I am still struggling a bit on how projections work. Generally in applications I have no problem using them (I've become quite familiar with OLS, and am moving on to GLS and FGLS), but here I am trying to re-write some of my notes, and I am having a very hard time putting into words exactly what a projection is, and how it works.
I've been looking around the internet, and I can't get the distinction between any arbitrary projection, an idempotent projection, and an orthogonal projection, specifically in the context of basic linear regressions.
Since the professor spent about 5 minutes on the first day going over it, I guess I should have learned this at a more basic course level, but I've never worked with it before.
What I have so far is that a projection is idempotent if p^2 = p, and it is a matrix which projects a vector onto another matrix (or something like that).
I also have this proof showing that p is symmetric and idempotent:
p' = (x(x'x)^(-1) x')' = (x')'((x'x)^-1)'x'...
x(x'x)^-1 x' = p
I get the matrix algebra easily enough, and if my understanding of idempotency from above is accurate (which I believe it is, given proofs I have done relating to the OLS) then I understand that as well. But I don't know how I am proving symmetry, and its implications.