Met 2212- Multivartate Statistics Ho-Testing OLS-regression LISREL IS GREEK TO ME The SEM model LISREL SOFTWARE Ulf H. Olsson Professor of Statistics THE LISREL MODEL y y x x Ulf H. Olsson THE LISREL MODEL Branch Loan Savings Satisfaction Loyalty Satisfaction 11Branch 12 Loan 13Savings 1 Loyalty 21Satisfaction 2 Ulf H. Olsson THE LISREL MODEL 1 11 2 21 12 1 11 12 22 2 21 22 1 13 1 2 23 2 3 Ulf H. Olsson THE LISREL MODEL (Factor Model) y1 11 0 1 y2 21 0 2 1 y 0 3 3 31 2 y4 0 41 4 y 0 51 5 5 Ulf H. Olsson THE LISREL MODEL (Factor Model) x1 11 x2 21 x 3 31 x4 0 x 0 5 x6 0 x7 0 x8 0 x9 0 0 0 0 42 52 62 0 0 0 1 2 3 1 4 2 5 3 6 73 7 8 83 93 9 0 0 0 0 0 0 Ulf H. Olsson THE LISREL MODEL 11 12 13 21 22 23 Corr ( i , j ) i, j 31 32 33 Ulf H. Olsson THE LISREL MODEL cov( i , j )i, j ; Assumed to be diagonal cov( i , j )i, j ; Assumed to be diagonal Ulf H. Olsson THE LISREL MODEL Cov( i , j ) i, j Assumed to be diagonal Ulf H. Olsson Greek Letters CAP / low er Name & Description • C A P / ALPHA (AL-fuh) First letter of the Greek alphabet. l o w BETA (BAY-tuh) e r • N GAMMA (GAM-uh) a m e DELTA (DEL-tuh) & D e EPSILON (EP-sil-on) The second form of the lower case epsilon is used as the “set membership” symbol. s c r i Ulf H. Olssonp t Greek Letters ZETA (ZAY-tuh) ETA (AY-tuh) THETA (THAY-tuh) IOTA (eye-OH-tuh) KAPPA (KAP-uh) Ulf H. Olsson Greek Letters LAMBDA (LAM-duh) MU (MYOO) NU (NOO) XI (KS-EYE) OMICRON (OM-i-KRON) Rarely used because it looks like an ‘o.’ Ulf H. Olsson Greek Letters PI (PIE) RHO (ROW) SIGMA (SIG-muh) TAU (TAU) Ulf H. Olsson Greek Letters UPSILON (OOP-si-LON) PHI (FEE) The two versions of lower-case Phi are used interchangeably. CHI (K-EYE) PSI (SIGH) OMEGA (oh-MAY-guh) Last letter of the Greek alphabet. Ulf H. Olsson Parameter Function y ( I B) 1 (' )( I B' ) 1 y ' ( ) 1 ' ( I B ' ) y' x y ( I B) 1 x ' x x ' Ulf H. Olsson Multivariate Normal Distribution f ( z ) (2 ) p/2 1 / 2 1 1 exp z ' z 2 Ulf H. Olsson The Maximum Likelihood Estimator TML log ( ) tr S ( ) log S t 1 Ulf H. Olsson Measurement Error in Linear Multiple Regression Models •Ulf H Olsson •Professor Dep. Of Economics The stadard linear multiple regression Model y Z ; is fixed but unknown; Z contains the regressors ( N g ) E ( | Z ) 0; Assume Z is mean centered Ulf H. Olsson Measurement Error/Errors-in-variables Z is not observable; Instead we observe X : X Z V ; V is a matrix( N g ) of measurement error The rows of V are assumed i.i.d with zero exp ectation and cov ariance matrix E (V | Z ) 0; and E ( | V ) 0 Ulf H. Olsson The consequences of neglecting the measurent error The OLS estimators of and : 2 b ( X ' X ) 1 X ' y 1 s y ' ( I N X ( X ' X ) 1 X ' ) y N 2 Asymptotically N N g Ulf H. Olsson The consequences of neglecting the measurent error • The probability limits of the two estimators when there is measurement error present: y ( X V ) X ( V ) X u u V •The disturbance term shares a stochastic term (V) with the regressor matrix •=> u is correlated with X and hence E(u|X)0 Ulf H. Olsson The consequences of neglecting the measurent error • Lack of orthogonality – crucial assumption underlying the use of OLS is violated ! Let 1 1 S Z Z ' Z and S X X ' X N N p lim N S Z Z X p lim N 1 X ' X p lim N S X Z N Ulf H. Olsson The consequences of neglecting the measurent error • The inconsistency of b Let p lim N b 1 X'X X'y p lim N X N 1 1 X X ( X ) 1 X Z Ulf H. Olsson The consequences of neglecting the measurent error • The inconsistency of b Let 1 X 1 1 1 X ( X Z )( Z ); 1 ( X Z ) is the Re liability of X ; 1 ( Z ) is the Noise to Signal Ratio Ulf H. Olsson The consequences of neglecting the measurent error / X 1 Z is number between 0 and 1 for g 1 • The inconsistency of b • Bias towards zero (attenuation) for g=1 • In multiple regression context things are less clear cut. Not all estimates are necessarilly biased towards zero, but there is an overall attenuation effect. Ulf H. Olsson The consequences of neglecting the measurent error The inconsiste ncy of s 2 1 s y ' ( I N X ( X ' X ) 1 X ' ) y N 1 ( y ' y y ' X ( X ' X ) 1 Xy) N 2 1 ' S Z ' Z ' b ' S X b N N 2 •In the limit we find: Ulf H. Olsson The consequences of neglecting the measurent error The inconsiste ncy of s Let p lim N s 2 2 ' Z ' X 2 1 ' X Z 2 ' 2 2 •The estimator is biased upward Ulf H. Olsson The consequences of neglecting the measurent error The effect on R 2 1 1 Z Z X Z 0 ' Z ' Z X Z 0 1 Since X Z We obtain ' Z ' Z ' Z ' Z 2 2 ' Z ' Z Ulf H. Olsson The consequences of neglecting the measurent error An overall attenuation in b s is biased upward 2 R underestimates the exp lanatory power Ulf H. Olsson