Some Definitions
Some Matrix Operations
Multiplication by a scalar
Matrix Addition (& subtraction)
Transpose of a matrix
Note:
(A')' = A
(A*B)' = B' * A'
Matrix Multiplication
Note:
In general A*B does not equal B*A (commutative law)
(A*B)*C = A*(B*C) (associative law)
A*(B+C) = A*B + A*C
(B+C)*A = B*A + C*A (distributive law)
Identity Matrix
Note:
A * I = A
I * A = A
Inverse of a Matrix
Note:
A^{-1} * A = A * A^{-1} = I
In scalar terms:
a^{-1} = 1/a
a^{-1} * a = 1
The Seven Basic Matrices
Raw Score Matrix (Data Matrix) | [vertical] |
Raw Score SSCP | [square] |
Deviation Score Matrix | [vertical] |
Deviation SSCP | [square] |
Covariance Matrix | [square] |
Standard Score Matrix | [vertical] |
Correlation Matrix | [square] |
Raw Score Matrix (Data Matrix)
Vertical Matrix
Raw Score SSCP
Square Symmetric Matrix
Deviation Score Matrix
Vertical Matrix
Deviation Score SSCP
Square Symmetric Matrix
Covariance Matrix
Square Symmetric Matrix
Standard Score Matrix
Vertical Matrix
Correlation Matrix
Square Symmetric Matrix
Matrix Formulae for Regression Coefficients
b = (X´X)^{-1}X´Y
b = (Z_{x}´Z_{x})^{-1}Z_{x}´Z_{y}
The b Vector
Regression Equations in Matrix Form
Y = Xb + e
Z_{y} = Z_{x}b + Z_{e}
program define matreg version 8.0 syntax varlist(min=2 numeric) [if] [in] [, Level(integer $S_level)] marksample touse /* mark cases in the sample */ tokenize "`varlist'" quietly matrix accum sscp = `varlist' if `touse' local nobs = r(N) local df = `nobs' - (rowsof(sscp) - 1) /* df residual */ matrix XX = sscp[2...,2...] /* X'X */ matrix Xy = sscp[1,2...] /* X'y */ matrix b = Xy * syminv(XX) /* (X'X)-1X'y */ local k = colsof(b) /* number of coefs */ matrix hat = Xy * b' matrix V = syminv(XX) * (sscp[1,1] - hat[1,1])/`df' matrix C = corr(V) matrix seb = vecdiag(V) matrix seb = seb[1, 1...] matrix t = J(1,`k',0) matrix p = t local i = 1 while `i' <= `k' { matrix seb[1,`i'] = sqrt(seb[1,`i']) matrix t[1,`i'] = b[1,`i']/seb[1,`i'] matrix p[1,`i'] = tprob(`df',t[1,`i']) local i = `i' + 1 } display display "Dependent variable: `1'" display display "Regression coefficients" matrix list b display display "Standard error of coefficients" matrix list seb display display "Values of t" matrix list t display display "P values for t" matrix list p matrix drop sscp XX Xy b hat V seb t p C end
use http://www.philender/courses/data/hsbdemo
matreg write read female
regress write read female
Linear Statistical Models Course