1 ESTIMATION, TESTING, ASSESSMENT OF FIT

2 Estimation How do we fit  (  )? –Choose  so that the reproduced   (  ), is as close as possible to S, i.e.: Choose a fit function F = (S,  ) to be minimized with respect to 

3 Covariance structure analysis The model imply a specific covariance structure:  =  (  ),  in  for the covariance matrix  of the observed variables z The minimun chi-square method estimates  F(  ) = (s -  (  ))’ V ((s -  (  )) = min! - if plimV -1 =  = avar (s), V (or F) is said to be asymptotically optimal. In which case - estimators are asymptotically efficient - nF= nF(  ) is asimptotically chi-squared distributed ^ ^

4 Asymptotic distribution free (ADF) analysis A consistent estimator of the asymptotic covariance matrix if s is given by the following sample “fourth-order” matrix  = (n-2) -1 (n-1) -1  (b j -b)(b j -b), b j = vech(z j -z)(z j -z)’, b and z are the mean of the b j ’s and z j ’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) /2 ·55 ·(51)

5 Normal theory statistics Under normality the asymptotic covariance matrix of s is given by:  = 2 D + (  )D + ’ (  * = avar (s | z ~ N) ) where D + is the Moore-Penrose inverse of the “duplication” matrix D Normal theory fit function : F ML (  ) = log |  (  ) | + trace {S  (  ) -1 } – p This is equivalent to using MD with V = 2 -1 D’(  -1  -1 )D  The normal theory statistics are all asymptotically equivalent When z is normally distributed, minimization of F ML yield maximum likelihood estimators, and nF ML (  ) is a likelihood ratio test statistic for the test of H 0 :  =  (  ),  in , against  is unrestricted

6 Asymptotic theory Assume: s  , in probability; n 1/2 (s -   ) N(0,  ), in distrib. If   = s(  0 ), then  V is a consistent estimator of  0 and asymptotically normal with asymptotic covariance matrix given by avar (  V ) = n -1 (  V  ’) -1  ’V  V  (  V  ’) -1 nF V   j T j, in distrib. For an asymptotically optimal weight matrix V (i.e. V  V = V): avar (  V ) = n -1 (  V  ’) -1 ; moerover, df  j ’s equal to 1, and the rest are equal to zero: nF V ~  2 df

7 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,  ) of S and  where: S = Observed moment matrix  = Theoretical moment matrix implied by the model, a function of the parameters of the model Kinds of Estimates

8 Fit functions Three specific functions: –Unweighted Least Squares (ULS): –Generalized Least Squares (GLS): –Maximun Likelihood (ML): p = number of observable variables

9 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

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

11 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 Assessement of fit

12 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. Assessement of fit