Econometrics/Stats: Proving a Matrix is Symmetric and Idempotent

[Elesirdur]

So, I've got a question for the math/stats/econ folks on here. How does one prove a matrix is symmetric and/or idempotent when one cannot see the actual values of the matrix? I know the properties of a symmetric matrix (A' = A) and an idempotent matrix (BB = B) but proving that outside of specific knowledge of the contents of the matrix seems to be eluding me.

The specific problem in question is asking to prove the symmetry and idempotentce of the P and M matrix from the OLS regression where P = X[(X'X)^-1]X' and M = I - P, and where X is a n by k matrix.

The ending size of the P "projection matrix" and the M matrix are both n by n, so they meet one of the necessary conditions for symmetry and idempotence by being square matrices. However, I don't really know how to prove symmetry or idempotence beyond that.

Any help?

[BadKarma]

I nominate this title as the most CONFUSING FUCKING TITLE of 2012 so far.

[Elesirdur]

Welcome to Econometrics lol

[Omnipotent]

I use this shit at least 3x a day in the business world.

[BadKarma]

Not a biz major, and I am fucking atrocious with math. :c I should take the time to learn one of these days though.

[Joyo]

http://en.wikipedia.org/wiki/Transpose
You need to use the rules X'' = X , (AB)' =B'A' and (A+B)' = A'+B'

Symmetry:
P' = (X[(X'X)^-1]X')' = X''[(X'X)^-1]'X' = X[(X'X)'^-1]X' = X[(X'X'')^-1]X' = X[(X'X)^-1]X' = P
M' = (I - P)' = I' - P' = I - P = M

Idempotence:
PP = X[(X'X)^-1]X'X[(X'X)^-1]X' = X I [(X'X)^-1]X' = X[(X'X)^-1]X' = P
MM = (I-P)(I-P) = II -2IP + PP = I - 2P + P = I - P = M

That should be it.

[Callisto]

I had to google at least three different terms to make sure this wasn't a joke topic with made-up words that needed to get moved to Spam.

[Ashmada]

Not a biz major, and I am fucking atrocious with math. :c I should take the time to learn one of these days though.

The first step is not to let yourself be intimidated by words.
Take "idempotence": idem=same; potence=multiplication (in this context)-> matrix that, when multiplied by itself results in itself.
Symmetric just means it's symmetric along the diagonal (high left, low right), that's all.

Most of the people that "aren't good at maths" are just giving up too easily, at least for stuff this level. It does get more complicated eventually .

[Elesirdur]

http://en.wikipedia.org/wiki/Transpose
You need to use the rules X'' = X , (AB)' =B'A' and (A+B)' = A'+B'

Symmetry:
P' = (X[(X'X)^-1]X')' = X''[(X'X)^-1]'X' = X[(X'X)'^-1]X' = X[(X'X'')^-1]X' = X[(X'X)^-1]X' = P
M' = (I - P)' = I' - P' = I - P = M

Idempotence:
PP = X[(X'X)^-1]X'X[(X'X)^-1]X' = X I [(X'X)^-1]X' = X[(X'X)^-1]X' = P
MM = (I-P)(I-P) = II -2IP + PP = I - 2P + P = I - P = M

That should be it.

Wow, thank you so much. My econometrics text simply states that "the symmetry and idempotence of the M matrix can be checked simply" and then never says how. My university really needs to re-jig the way linear algebra is taught for econ students. So much of what we do doesn't facilitate the proofs and properties we need for econometrics.

http://i10.photobucket.com/albums/a1...-hand-here.jpg

[Cayos]

Check out my school's Linear Algebra page. It's an online class, so everything you need will be on this site, and it even has some example problems.

Click "Online Text" on the left, and then find the section you're on.

http://www.emporium.vt.edu/math1114/

[madpersian]

Out of curiosity, is this grad or undergrad econometrics?

[Elesirdur]

Undergraduate, I'm taking a BSc program in Economics that requires theory courses in econometrics, not just applied econometrics. You do get this kind of stuff in grad econometrics, but it's a real crash course if you haven't seen the theory already.

[madpersian]

Ah okay. Thanks. I'm in undergrad but I was going to take a grad econometrics course because our undergrad course isn't nearly as rigorous as what you are doing.

[Elesirdur]

You'll get this theory stuff in graduate econometrics no doubt. Being good at your linear algebra insofar as dealing with matrices as variables (stochastic or not) will make things a lot easier. Honestly, when it comes to doing actual research this linear algebra is pretty "useless" but what comes out of it when dealing with omitted variable bias, etc. or even developing your own estimation methods is important. Sucks to study though.

[BerenTebogo]

Yeah I came late but was gonna say this is just basic undergrad linalg. One of the reasons I actually respect econ majors, real math.

[Chrius]

I had to google at least three different terms to make sure this wasn't a joke topic with made-up words that needed to get moved to Spam.

Econometrics/Stats: Proving a Matrix is Symmetric and Idempotent

What was the 3rd and potentially more term?
inb4 the matrix not being real

[Ferion]

Yeah I came late but was gonna say this is just basic undergrad linalg. One of the reasons I actually respect econ majors, real math.

Most Econ majors at my school were just students who got rejected from the business school. In one of my econ classes the professor put the formula for a density function of a continuous random variable on one of the slides on the first day, by the second class probably a third of the students had dropped the class.

[Pirian]

Econ is part of the business school at my school, lol. What does your school classify it as, a social science? <.<

[Ferion]

Econ is part of the business school at my school, lol. What does your school classify it as, a social science? <.<

Yeah pretty much haha. I mean don't get me wrong, you can definitely structure the major more towards the business world and make it pretty difficult if you wanted. I combined an econ major with a math major so I was able to take the classes where you start using non-linear models and actually do problems like the one asked about in this thread. If you really wanted to though you could have gotten away with only taking Calc I and the hardest thing related to math you would have to do is take intro to econometrics, which is pretty much just intro to stats.

[Elesirdur]

Econ programs seem to vary quite a lot depending on the school. My understanding is that in Canada, unless you're doing a Math minor or focus, are in an Honours program, or are doing a BSc program like I am (which are not widely seen programs, though it is more than just marketing at my university) you can get a BA in Economics by just taking calculus for social scientists and an applied econometrics course as the hardest math/stats you need to do. I find it funny when I take 300 level courses that don't require the more advanced econ math (linear algebra, multivariate calculus, differential calculus, etc) where the prof puts up a derivative on the board and most of the class start to freak out. Since I'm on a second degree and a compressed schedule, I'm just doing all the technical and theory classes but I have no idea what the BA students do in the upper-year courses without the math or econometrics requirements. Probably similar to the history and polisci classes I did for my first degree lol

Since there are some Econ folks here, I might as well share this. I'm taking a really interesting course in computational economics which focuses on using computer programs to develop static and dynamic models and using them for problem solving and research. This is something that's rarely taught or introduced at the undergrad level, but if you're thinking of going on to grad school, or even in a business application, it's really applicable. The text for the course "Computational Economics" by Kendrick, Mercado, and Amman walks you through using Excel, Mathematica, MATLAB, Maple, and GAMS with examples to demonstrate how to use the programs for economics applications. It's a bit pricey, but if your uni library has it it's worth a good read. It's supposedly very good stuff to have on your resume that you can use these analytical packages (and stat software like EViews or SPSS, though EViews is better for econ/finance applications) as they are used in both research and industry.

GoogleBooks like for Computational Economics, preview only: http://books.google.ca/books?id=tKoI...page&q&f=false