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1 ESTIMATION, TESTING, ASSESSMENT OF FIT

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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

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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 ^ ^

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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) 10 55 1/2 ·55 ·(51)

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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

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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

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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

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8 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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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

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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

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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

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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

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