Econometric Analysis

Marcelo Fernandes
Queen Mary, University of London

Topic 1: Matrix Algebra

reference:

Greene (1993) Econometric Analysis, chapter 2

Road map

1.1

1. terminology 2. algebraic manipulation 3. geometry of matrices 4. matrix partitioning 5. matrix calculus

Terminology

1.2

matrix

rectangular array of numbers a11  a A = [ank ] =  21  . .  . aN 1


a12 a22 . . . aN 2

··· ··· ... ···

a1K  a2K   .  .  . aN K



dimension number of rows and columns in a matrix, e.g., A is N ×K vector ordered set of numbers arranged either in a row or in a column, e.g., the first row of A is the row vector given by an· = (an1, an2, . . . , anK )

Types of matrices

1.3

square

if A has the same number of columns and rows (N = K) if ank = akn for all n and k (A = A′) square matrix whose only nonzero elements appear on the main diagonal moving from upper left to lower right diagonal matrix with ones on the diagonal, I N square matrix with only zeros either above or below the main diagonal (lower or upper triangular, resp.)

symmetric diagonal

identity triangular

Algebraic manipulation I

1.4

equality

A = B iff ank = bnk for all n and k B = A′ iff bnk = akn for all n and k −→ transpose of a transpose −→ symmetric matrices −→ column/row vectors

transposition

operations

addition and multiplication by inner product −→ conformability + linear combination −→ associative, commutative, and distributive laws −→ transposition: (A + B )′ and (A B )′

Algebraic manipulation II

1.5

sum of values

N x n = ι′ x n=1

example: sample average sum of squares

x′ x example: sample variance x′ y example: sample covariance
1 ¯ Pι = N ι ι′ −→ Pιx = x ι Mι = I N − Pι −→ Mιx = x − x ι ¯ N n=1 xnyn =

N x2 = n=1 n

cross product

projection matrices

Geometry of matrices I

1.6

vector space

any set of vector that is closed under addition and scalar multiplication −→ spanning