Transcript No Slide Title

The General Structural Equations model
with latent variables
by
Willem E.Saris
Specification of a full SE model
• A full SE model consists of three parts:
• a structural model
• a measurement model for the endogenous variables
• a measurement model for the exogenous variables
• This approach will be illustrated by an example
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The structural model
1
Environmental
Values
(1)
Perception
Environmental
damage
(1)
Environment friendly
behavior (3)

3
Influence
(2 )
1
2
3
1
0
0
b31
2
0
0
b32
3
0
0
0
1
11
0
0
2
0
22
0
Understanding
politics
(2)
with E(’) = 0, E(’) = E(’) =
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The measurement of the endogenous
variables
e1
e2
e3
y1
y2
y4
e5
y5
e6
2
3
y1
11
0
0
y2
21
0
0
y3
0
32
0
y4
0
42
0
Environment friendly
y5
behavior (3)
0

53
y6
0
0
63
Environmental
Values
(1)
y3
e4
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1
y6
Influence
2
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or
y =  y
’’) =e
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The Measurement model for the
exogenous variables
x1
x2
x3
x4

11
21
0
0

0
0
32
42

Perception
environmental
damage
or
x =  x
with E(’)=0, E(’)=
Understanding
politics
x1
1
x2
2
x3
3
x4
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A General Approach illustrated
1
e1
y1
e2
y2
e3
y3
x1
Environmental
Values
(1)
Perception
Environmental damage
(1)
x2
Environment friendly
behavior (3)
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e4
y4
e5
y5
e6
y6
Influence
2

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1
2
3
x3
3
x4
4
Understanding
politics
(2)
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The LISREL model

() 

 y
y
e

1. The structural model:


x
x


 =  +  + 
2. A measurement model for the endogenous variables:
y =  y + e
3. A measurement model for the exogenous variables:
x =  x + 
The parameters of the model can be found in:  y , x 
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Decomposition of the full system
• The covariance matrix of the observed variables is
denoted by S. This matrix consists of three
submatrices Syy Syx = Sxy ‘ Sxx
Syy = y Cy’ + 
C = (I-B)-1(’ I-B)’-1
Sxx = x x’ + 
Syx = y(I-B)-1x’
This result holds for all SE models !!
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The decomposition of the covariance
matrix in the parameters
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Specific models of the full model
•
•
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•
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•
•
•
•
regression
recursive models
Non-recursive models
factor analysis
second order factor models
mimic models
panel models
Multiple groups models
latent growth models
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Identification
• If the measurement models are identified
and
the structural model are identified
• then the full model is identified
• Then the models can be estimated
• If df> 0 the model can also be tested
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Identification
• Measurement models are identified
- if each latent variables has 3 indicators
- if each latent variables has two indicators but the
latent variables are correlated
• For the structural part of the model the
identification rules of the econometric literature can
be applied
• A practical rule is:
If the standard errors of the parameters can be
estimated then the model is identified
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Estimation and testing of the full
system
• With respect to estimation nothing has been
changed
• Different estimators are available: ULS, ADF, ML
• All three are consistent but they have different
advantages and disadvantages as discussed before.
• Testing also does not change but we will discuss it
in more detail because it is essential and not so
simple as presented so far.
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SEM Approach
• A model is specified with observed and latent
variables
• If the model is identified the parameters can be
estimated including the effects between latent
variables i.e. corrected for measurement error.
• A test of the model can be performed if df>0
• Eventual misspecifications can be detected
• Corrections in the models can be introduced
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