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