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# ECONOMETRIC APP ECON 483

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## Popular in Economcs

This 8 page Class Notes was uploaded by Miss Adeline Weimann on Wednesday September 9, 2015. The Class Notes belongs to ECON 483 at University of Washington taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/192492/econ-483-university-of-washington in Economcs at University of Washington.

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Date Created: 09/09/15
Matrix Algebra Review Erie Zivot Department of Economics University of Washington January 3 2000 This version May 11 2006 1 Matrix Algebra Review A matrix is just an array of numbers The dimension of a matrix is determined by the number of its rows and columns For example a matrix A with n rows and m columns is illustrated below 111 112 aim 121 122 azm mm anl anZ anm where 0 denotes the 139 row and jth column element of A A vector is simply a matrix with 1 column For example 1 2 x nxl a is an n X 1 vector with elements 173727 7x Vectors and matrices are often written in bold type or underlined to distinguish them from scalars single elements of vectors or matrices The transpose of an n X m matrix A is a new matrix with the rows and columns of A interchanged and is denoted A or AT For example 1 4 1 2 3 7A 2 5 2x3 4 5 6 3x2 3 6 1 x 2 x 1 2 3 3X1 3 1X3 A symmetric matrix A is such that A A Obviously this can only occur if A is a square matrix ie the number of rows of A is equal to the number of columns For example consider the 2 X 2 matrix H53 A A12 Clearly 2 1 11 Basic Matrix Operations 111 Addition and subtraction Matrix addition and subtraction are element by element operations and only apply to matrices of the same dimension For example let Harem Then 49 2074290769 AB l21ll07l l2017l l28l7 49 20 472 970 2 9 A B l21l7l07ll270177ll276l39 112 Scalar Multiplication Here we refer to the multiplication of a matrix by a scalar number This is also an element by element operation For example let 0 2 and H331 Then C39A 228 2l 113 Matrix Multiplication Matrix multiplication only applies to conformable matrices A and B are conformable matrices of the number of columns in A is equal to the number of rows in B For example if A is m X n and B is m Xp then A and B are conformable The mechanics of matrix multiplication is best explained by example Let 1 2 1 2 1 d B 2x2 3 4 an 2x3 3 4 2 12 121 AB 2x22gtlt3 34 342 7 112312241122 3143 3244 3142 7 10 5 7 15 22 11233 Then The resulting matrix C has 2 rows and 3 columns In general if A is n X m and B iszpthenCAB isngtltp As another example let 12 2 213373 4 and2 1i 12 5 2132I2Ex31 7 3 46 Then 7 1526 7 3546 7 17 7 39 39 As a nal example let 1 4 x 2 y 5 3 6 Then x y1 2 3 5 14253632 6 12 The Identity Matrix The identity matrix plays a similar role as the number 1 Multiplying any number by 1 gives back that number In matrix algebra pre or post multiplying a matrix A by a conformable identity matrix gives back the matrix A To illustrate let 1 0 12 l o 1 l denote the 2 dimensional identity matrix and let A 111 112 121 122 denote an arbitrary 2 X 2 matrix Then 1 0 all 112 011lan 111 112 A 121 122 and 111 112 1 0 AI2 121 12210 13 Representing Summation Using Vector Notation Consider the sum TL Ek1k 191 Let x 175L n be an n X 1 vector and 1 171 be an n X 1 vector of ones Then 1 TL x 15 1nl 11kk 191 and 551 n 1 x11 lx1xnxk 1471 Next consider the sum of squared 5L values TL 2 3 2 2 2 191 This sum can be conveniently represented as w H H Last consider the sum of cross products TL 2 193419 1341 39 39 39 nyn 191 This sum can be compactly represented by 341 n xyx1 1y1xnynkyk yn 191 Note that x y y x 14 Representing Systems of Linear Equations Using Matrix Algebra Consider the system of two linear equations 1 3 VV 1 2x 7 y 1 Equations 1 and 2 represent two straight lines which intersect at the point 5L g and y This point of intersection is determined by solving for the values of SC and 31 such that xy 2x 7 yl The two linear equations can be written in matrix form as 1 illillel Azb 1Soving for ac gives as 23 Substituting this value into the equation as y 1 gives 23 y l and solving for y gives y 13 Solving for at then gives as 01 5 where Al fllzllmblil If there was a 2 X 2 matrix B with elements bij such that B A I2 where I2 is the 2 X 2 identity matrix then we could solve for the elements in Z as follows In the equation A Z b pre multiply both sides by B to give BAZ Bb 7 311 312 17b11391512391 y 7 J21 322 1 7 b211b221 If such a matrix B exists it is called the inverse of A and is denoted A71 ln tuitively the inverse matrix A 1 plays a similar role as the inverse of a number Suppose a is a number eg a 2 Then we know that i a a710 1 Similarly in matrix algebra A lA I2 where I2 is the identity matrix Next consider solving the equation ax 1 By simple division we have that St 2118 071 Similarly in matrix algebra if we want to solve the system of equation Ax b we multiply by A 1 and get x Ailb Using B A71 we may express the solution for Z as z A lb As long as we can determine the elements in A 1 then we can solve for the values of SC and y in the vector Z Since the system of linear equations has a solution as long as the two lines intersect we can determine the elements in A 1 provided the two lines are not parallel There are general numerical algorithms for nding the elements of A 1 and typical spreadsheet programs like Excel have these algorithms available However if A is a 2 X 2 matrix then there is a simple formula for A71 Let A be a 2 X 2 matrix such that A 111 112 121 122 Then 1 a 7a A1 ail detA a11a22ia21a12 where detA denotes the determinant of ABy brute force matrix multiplication we can verify this formula 1 1 Ha a a 7 22 12 11 12 A 1A 111122 121112 quot121 111 121 122 1221111 1121121 122112 112122 111122 121112 quot121111 Jr 1111121 a21a12 111122 1 1221111 1121121 0 111122 121112 0 a21a12 111122 a11a22ia21a12 0 021a12011022 111122 1121 112 7 1 0 0 1 39 Let s apply the above rule to nd the inverse of A in our example 1 71 71 l A 1 ll 2 l A lA WIHWIH Notice that wlmwlp WILWIH 11 1 1 N H l H H 11 H 1 1 G H H O 11 Our solution for Z is then condemn coltle 11 H 1 1 W H 11 sothatccandy In general if we have n linear equations in n unknown variables we may write the system of equations as 111531 112532 39 11115811 b1 121531 122532 39 12115811 2 an11 1112532 39 anan bu which we may then express in matrix form as 111 112 39 39 39 1111 1 51 121 122 39 39 39 1211 2 52 anl anZ 39 39 39 arm Lin bu or A x b nxn nxl n x1 The solution to the system of equations is given by x A lb where A lA I and I is the n X n identity matrix H the number of equations is greater than two then we generally use numerical algorithms to nd the elements in A l 2 Further Reading Excellent treatments of portfolio theory using matrix algebra are given in lngersol 1987 Huang and Litzenberger 1988 and Campbell Lo and MacKinlay 1996 References l H 1 Campbell JY Lo AW and MacKinlay AC 1997 The Econometrics of Financial Markets Priceton New Jersey Princeton University Press 2 Huang C F and Litzenbeger RH 1988 Foundations for Financial Eco nomics New York North Holland E lngersoll JE 1987 Theory of Financial Decision Making Totowa New Jersey Rowman amp Little eld

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