The General Structural Equations model with latent variables

1
The General Structural Equations model with
latent variables

• by
• Willem E.Saris

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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
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The measurement of the endogenous variables
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The Measurement model for the exogenous variables
x1

• d1

Perception environmental damage
x2

• d2

• d3

x3
Understanding politics
x4

• d4

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A General Approach illustrated
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The LISREL model
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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 Ly CLy Qe
• C (I-B)-1(GFG Y)(I-B)-1
• Sxx Lx FLx Qd
• Syx Ly(I-B)-1GFLx
• 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

• 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 dfgt 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 dfgt0
• Eventual misspecifications can be detected
• Corrections in the models can be introduced