Transcript ASSESSMENT OF FIT AND DETECTION OF LACK OF FIT

ESTIMATION, TESTING,
ASSESSMENT OF FIT
1
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
2
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
3
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)
4
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
5
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
6
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
7
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
9
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
10
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
11