Download ASSESSMENT OF FIT AND DETECTION OF LACK OF FIT

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!
- if plimV-1 = G = avar (s),
V (or F) is said to be asymptotically optimal.
In which case
- estimators are asymptotically efficient
^
^ distributed
- nF= nF(q) is asimptotically chi-squared
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Asymptotic distribution free (ADF) analysis
A consistent estimator of the asymptotic covariance matrix if s is
given by the following sample “fourth-order” matrix
–
–
^
-1
-1
G = (n-2) (n-1) S (bj-b)(bj-b),
– -z)’,
–
bj = vech(zj-z)(z
j
–
b and z– are the mean of the bj’s and zj’s
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
z
s
G
p
p*=1/2p(p+1)
1/2p*x(p*+1)
10
55
1/2 ·55 ·(51)
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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 H0:S=S(q), q in Q, against S is unrestricted
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Asymptotic theory
Assume:
s
s0 , in probability; n1/2(s - s0)
N(0,G), in distrib.
If s0 = s(q0), then qV is a consistent estimator of q0 and asymptotically
normal with asymptotic covariance matrix given by
avar (qV) = n-1 (DVD’)-1D’VGVD (DVD’)-1
nFV
S ajTj, in distrib.
For an asymptotically optimal weight matrix V (i.e. VGV = V):
avar (qV ) = n-1 (DVD’)-1;
moerover, df aj’s 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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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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