ESTIMATION, TESTING, ASSESSMENT OF FIT

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ESTIMATION, TESTING, ASSESSMENT OF FIT
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Estimation

• How do we fit S(q)?
• Choose q so that the reproduced S, S(q), is as
  close as possible to S,
• i.e. Choose a fit function F (S,S) to be
  minimized with respect to q

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Covariance structure analysis
The model imply a specific covariance structure
S S (q), q in Q, for the covariance matrix S of
the observed variables z
The minimun chi-square method estimates q
F(q) (s - s(q)) V ((s
- s(q)) min!
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Asymptotic distribution free (ADF) analysis
The ADF analysis has the inconvenience of having
to manipulate a matrix of high dimension and of
using fourth order moments which may lead to
lack robustness against small sample size
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Normal theory statistics
Under normality the asymptotic covariance matrix
of s is given by G 2 D(S?S)D ( G
avar (s z N) ) where D is the
Moore-Penrose inverse of the duplication matrix
D
Normal theory fit function FML (q) log S(q)
trace S S(q)-1 p This is equivalent to
using MD with V 2-1 D(S -1? S -1)D,
The normal theory statistics are all
asymptotically equivalent
When z is normally distributed, minimization of
FML yield maximum likelihood estimators, and
nFML (q) is a likelihood ratio test statistic
for the test of H0SS(q), q in Q, against S is
unrestricted
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Asymptotic theory
For an asymptotically optimal weight matrix V
(i.e. VGV V) avar (qV ) n-1 (DVD)-1
moerover, df ajs equal to 1, and the rest are
equal to zero nFV ?2 df
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Kinds of Estimates

• Non-Iterative
• Stepwise ad-hoc methods which use reference
  variables and instrumental variables techniques
  to estimate the parameters
• Iterative
• minimize a fit (discrepancy) function F(S,S) of S
  and S where
• S Observed moment matrix
• S Theoretical moment matrix implied by the
• model, a function of the parameters of the
  model

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Fit functions

• Three specific functions
• Unweighted Least Squares (ULS)
• Generalized Least Squares (GLS)
• Maximun Likelihood (ML)

p number of observable variables
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Kinds of Estimates

• Non-Iterative
• 1. IV Instrumental Variables method
• 2. TSLS Two-Stage Least Squares Method
• Iterative
• 3. ULS Unweighted Least Squares Method
• 4. GLS Generalized Least Squares Method
• 5. ML Maximum Likelihood Method
• 6. WLS Weighted Least Squares Method
• 7. DWLS Diagonally Weighted Least
  Squares Method

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Assessement of fit

• 1. Examine
• a) Parameter Estimates
• b) Standard Errors
• If anything is unreasonable, either the model
  is fundamentally wrong or the data is not
  informative

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Assessement of fit

• 2. Measures of Overall Fit
• a) ?2, DF, and P-value
• b) Goodness-of-fit Index Adjusted
• Goodness-of-fit-index
• c) Root Mean Square Residual
• 3. Detailed Assessment of Fit
• a) Residuals
• b) Standarized Residuals
• c) Modification Indices
• e) Parameter change

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Assessement of fit
METHOD ME LS, GLS, ML are always computed in
this order even if the methods are permuted ME
EGLS, LS gives LS, GLS, EGLS in sequence ME
AGLS gives LS, AGLS ME ERLS, EGLS, ELS gives
LS, ELS, GLS, EGLS, ML, ERLS.