Introductory Econometrics ECON 4720
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This 3 page Class Notes was uploaded by Nola Williamson on Monday September 21, 2015. The Class Notes belongs to ECON 4720 at University of Virginia taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/209780/econ-4720-university-of-virginia in Economcs at University of Virginia.
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Date Created: 09/21/15
Matrix Algebra August 2006 1 Motivational Example Consider the model7 n ytZzn6a 112T 1 11 Consider some examples We want to nd the OLS estimates of 1 2 As in the univariate case7 we can de ne T ar min 61 m g gnarqmg n 2 111 i 2 9012752 i1 with FOCs T n 29m 1 Zia5 0 11 j1 for each m i 1271 In general7 solving for 152 can be done without matrix algebra but is very tedious and messy Matrix algebra provides a cleaner7 more elegant way to handle this problem 2 Basic Structures and Notation Rewrite equation 1 as 11 3011 9012 3017 5 1 61 12 9021 22 9021 g 62 11 T1 T2 39 39 39 JETn n 6T Tgtlt1 x 1 Tgtlt1 This can be written more concisely as yX86 Where 3 Properties am is the tith element in the ith row and ith column of X Transp ose 11 21 90111 X 9012 9022 W2 3011quot 21quot 39 MT ngtltT Note that I X X Diagonal matrices and identity matrices Addition C A B 3 cij lij bij Vij nXm nXm nXm Note that two matrices can be added only if their dimensions are the same Multiplication Note that two matrices can be multiplied ie are multiplication con formable only if the number of columns in the rst matrix equal the num ber of rows in the second matrix Also note that in general BC 7E CB Note that I AB B A and that A AI A can can calm can can Can c A 1gtlt1ngtltm 39 cam Gang canm Linear lndependence Consider n vectors 1 2 n De ne cixi to be a linear combination of the 35 s Where each Ci is a scalar If the only set 0102 cn 0 is 00 0 then the 35 s ar linearly independent Otherwise they are linearly dependent Example Let 1 l 2 3 1 2 2 3 3 4 x4 8 2 3 2 6 7 Then an and 12 are linearly independent hand 13 are linearly dependent 2x1 7 3 0 and 1 2 and x4 are linearly dependent 901 212 7 x4 0 Rank is the number of independent column vectors or row vectors in a matrix For example i 111213x4 from equation 2 then Rank X 2 If all of the columns of a matrix are linearly independent then the matrix is of full rank lnvertability Given a matrix A if 3B AB I then A B l and B A 1 If A has full rank then A 1 exists Note that a AB 1 B lA l if A and B are both full rank b only square matrices are invertible and c A lVl A A 1 144