Seminari del Servei d’Estadística, UAB 8 d’octubre, 2003 SEM, una eina classica i moderna per l’anàlisi multivariant aplicada Albert Satorra Universitat Pompeu Fabra Departament d’Economia i Empresa 1 de 110 Temes que tractarem: o o o o o Un exemple: regressió amb errors a les variables Elements bàsics de SEM Estimació i Contrast Robustesa asimptòtica Tipus de models SEM Anàlisi factorial “Path Analysis” Equacions simultànies Models de corbes de creixement o Una aplicació de SEM a l’anàlisi de rendiments empresarials, dades de dos nivells 2 de 110 o Conclusions Examples with Coupon data (Bagozzi, 1994) 3 de 110 Ejemplo: En un estudio de Bagozzi, Baumgartner, and Yi (1992), sobre “coupon usage” se dispone de la matriz de varianzas y covarianzas siguiente de dos muestras de mujeres: Sample A: Action oriented women (n = 85) Intentions #1 4.389 Intentions #2 3.792 4.410 Behavior 1.935 1.855 2.385 Attitudes #1 1.454 1.453 0.989 1.914 Attitudes #2 1.087 1.309 0.841 0.961 Attitudes #3 1.623 1.701 1.175 1.279 Sample B: State oriented women (n = 64) Intentions #1 3.730 Intentions #2 3.208 3.436 Behavior 1.687 1.675 2.171 Attitudes #1 0.621 0.616 0.605 Attitudes #2 1.063 0.864 0.428 Attitudes #3 0.895 0.818 0.595 1.373 0.671 0.912 1.480 1.220 1.397 0.663 1.971 4 de 110 1.498 Variables /LABELS V1 = Intentions1; V2 = Intentions2; V3 = Behavior; V4 = Attitudes1; V5 = Attitudes2; V6 = Attitudes3; F1 = Attitudes F2 = Intentions V3 = Behavior 5 de 110 SEM indicadores múltiples E4 D2 V4 V1 E5 E6 V5 F1 F2 V2 E3 V6 V3 F1 = Attitudes F2 = Intentions V3 = Behavior 6 de 110 E1 E2 INTENTIO=V1 = 1.000 F2 + 1.000 E1 INTENTIO=V2 = 1.014*F2 + 1.000 E2 .088 CHI-SQUARE = 5.426, 7 DEGREES OF FREEDOM PROBABILITY VALUE IS 0.60809 11.585 BEHAVIOR=V3 ATTITUDE=V4 = = .330*F2 + .492*F1 .103 .204 3.203 2.411 1.020*F1 + 1.000 E3 VARIANCES OF INDEPENDENT VARIABLES ---------------------------------E --- + 1.000 E4 E1 -INTENTIO E2 -INTENTIO E3 -BEHAVIOR E4 -ATTITUDE E5 -ATTITUDE E6 -ATTITUDE .136 7.501 ATTITUDE=V5 = .951*F1 + 1.000 E5 .117 8.124 ATTITUDE=V6 = 1.269*F1 + 1.000 E6 .127 10.005 INTENTIO=F2 = 1.311*F1 .214 6.116 D --.649*I D2 .255 I 2.542 I I .565*I .257 I 2.204 I I 1.311*I .213 I 6.166 I I .875*I .161 I 5.424 I I .576*I .115 I 5.023 I I .360*I .132 I 2.729 I -INTENTIO 2.020*I .437 I 4.619 I I I I I I I I I I I I I I I I I I I I I + 1.000 D2 7 de 110 ... adding parameters ? LAGRANGE MULTIPLIER TEST (FOR ADDING PARAMETERS) ORDERED UNIVARIATE TEST STATISTICS: NO -1 2 3 4 5 6 7 8 9 CODE ---2 2 2 2 2 2 2 2 2 12 12 20 20 20 20 0 0 0 PARAMETER --------V2,F1 V1,F1 V4,F2 V5,F2 V6,F2 V3,F2 F1,F1 F2,D2 V1,F2 CHI-SQUARE ---------1.427 1.427 0.720 0.289 0.059 0.000 0.000 0.000 0.000 PROBABILITY ----------0.232 0.232 0.396 0.591 0.808 1.000 1.000 1.000 1.000 PARAMETER CHANGE ---------------0.410 -0.404 0.080 -0.045 -0.025 0.000 0.000 0.000 0.000 8 de 110 STRUCTURAL EQUATION MODELING LINEAR STRUCTURAL RELATIONS 9 de 110 Terminología • LINEAR LATENT VARIABLE MODELS • T.W. Anderson (1989), Journal of Econometrics • MULTIVARIATE LINEAR RELATIONS • T.W. Anderson (1987), 2nd International Temp. Conference in Statistics • LINEAR STATISTICAL RELATIONSHIPS • T.W. Anderson (1984), Annals of Statistics, 12 • COVARIANCE STRUCTURES • • • • • Browne, Shapiro, Satorra, ... Jöreskog (1973, 1977) Wiley (1979) Keesling (1972) Koopmans and Hovel (1953) 10 de 110 Computer programs • • • • • • • • • LISREL EQS LISCOMP / Mplus COSAN MOMENTS CALIS AMOS RAMONA Mx • • • • • • • • • Jöreskog and Sörbom Bentler Muthén McDonalds Schoenberg SAS Arbunckle Browne Neale 11 de 110 Computer programs and Web adresses • SEM software: – – – – – EQS http://www.mvsoft.com LISREL http://www.ssicentral.com MPLUS http://www.statmodel.com/index2.html AMOS http://smallwaters.com/amos/ Mx http://www.vipbg.vcu.edu/~vipbg/dr/MNEALE.shtml Web addresses of interest: SEMNET; http://www.gsu.edu/~mkteer/semfaq.html Jason Newsom's SEM Reference List http://www.ioa.pdx.edu/newsom/semrefs.htm 12 de 110 ... books • • • • • • • Bollen (1989) Dwyer (1983) Hayduk (1987) Mueller (1996) Saris and Stronkhorst (1984) Dunn, Everitt and Pickles (1993) ... 13 de 110 ... many research papers • Austin and Wolfle (1991): Annotated bibliography of structural equation modeling: Technical Works. BJMSP, 99, pp. 85-152. • Austin, J.T. and Calteron, R.F. (1996). Theoretical and technical contributions to structural equation modeling: An updated annotated bibliography. SEM, pp. 105-175. 14 de 110 Type of variables Manifest Variables: Yi , Xi Measurement Model: X3 X4 λ32 ξ2 λ32 Measurement error, disturbances: εi , δi 15 de 110 The form of structural equation models: Latent constructs: - Endogenous - Exogenous ηi ξi Structural Model: - Regression of η1 on ξ2: γ12 - Regression of η2 on η1: β12 Structural Error: ζi 16 de 110 LISREL model: η(m x 1) = Β(m x m) η(m x 1) + Γ(m x n) ξ(n x 1) + ζ(m x 1) y(p x 1) = Λy(p x m) η(m x 1) + ε(p x 1) x(q x 1) = Λx(q x n) ξ(n x 1) + δ(q x 1) 17 de 110 ... path diagram δ1 X1 δ2 X2 δ3 ξ1 ε1 ε2 ε3 Y1 Y2 Y3 γ11 ζ1 η1 β31 θ21 η3 X3 δ4 X4 δ5 X5 ξ2 ζ2 γ22 β32 η2 Y6 ε6 Y7 ε7 ζ3 Y4 Y5 ε4 ε5 18 de 110 Aspects of SEM 1. Substantive theory – Concepts – Constructs – Formalization 2. Basic Issues – Causality – Model building • Theory Driven vs Data Driven – Exploratory vs Confirmatory Analysis – Units of measurement and Standardization – Scale Types 19 de 110 Aspects of SEM 3. Statistical – Statistical Specification of Model – Identification of Models and Parameters – Fitting and Testing of Model • Assessment of Fit • Detection of Specification Errors – Sequential Model Fitting – Testing Structural Hypothesis 4. Computational 20 de 110 Main virtues of SEM • Unifies several multivariate methods into one analytic framework • Allows for latent variables in a statistical model and measurement error • Expresses the effects of latent variables on each other and the effect of latent variables on observed variables • Allows testing substantive hypothesis involving causal relationships among construct or latent variables 21 de 110 Type of data • Flexibility on the type of data: – – – – – – – – Continuous and ordinal variables multiple sample Informative missingness (MCA, MAR) Finite mixture distributions Multilevel models Samples with complex design General longitudinal type of data ... 22 de 110 Ordinal Variables Is is assumed that there is a continuous unobserved variable x* underlying the observed ordinal variable x. A threshold model is specified, as in ordinal probit regression, but here we contemplate multivariate regression. It is the underlying variable x* that is acting in the SEM model. 23 de 110 A multiple group SEM: model and statistical analysis 24 de 110 Introducción a SEM: • Datos: • Matriz de datos (“raw data”) • Estadísticos suficientes (medias muestrales, varianzas y covarianzas) vars Momentos Muestrales:: Indiv. Matriz datos (n x p) • Vector de medias • Matriz S de var. y cov. (p x p) • Matriz de momentos de cuarto orden: Γ (p* x p*) p* = p(p+1)/2, p=20--> p* =210 25 de 110 SEM: ηi = B η +Γ ξ ( g) zi = U ν ( g) ( g) ( g) ( g ) ( g) i i ( g) ( g) i i=1,2, ...., ng, g=1,2, ..., G donde: zi(g): vector de variables observables, ηi(g) : vector de variables endógenas ξi(g) : vector de variables exógenas vi(g) = (ηi(g)’, ξi(g)’)’: vector de variables observables y latentes, U(g): matriz de selección completamente especificada, B(g), Γ(g) y Φ(g) = E(ξi(g) ξi(g)’): matrices de parámetros del modelo ng : tamaño muestral en la muestra g ésima, 26 de 110 G: no. de grupos El modelo general: zi (g) =G = Λ ξi (g) ( g ) −1 (g) (I − B ) Γ I (g) (g) ξ i (g) donde: ( g) −1 ( g) ( g) ( g) (I − B ) Γ Λ = G I Φ (g) = var ξ (g) 27 de 110 Estructura de momentos: Las matrices de momentos Σ pueden ser expresadas como: ( g) Σ (g) = Ezi zi ,' ( g) ( g) g = 1, ..., G, = Λ Φ Λ ' = Σ (θ ) (g) (g) (g) (g) donde (θ) es el vector de parámetros. El análisis prosigue ajustando Σ(g)(θ) ≈ S (g) 1 = ng ∑ ng z i =1 i (g) zi (g) ' 28 de 110 Función de ajuste: GLS: { ng 1 (g) (g) ( g ) −1 FGLS (θ ) = ∑ tr ( S − Σ ) S 2 n } 2 donde: Σ(g) = Σ(g)(θ), n = n1 + ... + nG El minimizador θ^de FGLS (θ) es el estimador de mínima distancia. Es asintóticamente óptimo cuando zi(g) tiene distribución normal. 29 de 110 Matriz de pesos: V (g) 1 = D ' (Σ ( g ) −1 ⊗ Σ ( g ) −1 ) D. 2 La matriz de “duplicadora” D es el operador de vectorización que suprime elementos redundantes debido a simetría, (ver Magnus y Neudecker; 1981) 30 de 110 Varianza asintótica de los estimadores: La expresión general de la matriz de varianza de los parámetros estimados es: 1 ˆ avar (ϑ ) = J n −1 R ' V Γ VRJ −1 R = Jacobiano de vech Σ(θ), 31 de 110 Varianzas asintóticas de los momentos: La expresión general del estimador de Γ es n n Γˆ = diag (1) Γˆ (1) , … , ( G ) Γˆ ( G ) n n donde Γˆ ( g ) = 1 n( g ) (g) (g) h ∑ i hi n ( g ) − 1 i =1 con hi (g) = vech ( zi (g) − s ( g ) )( zi (g) − s ( g ) )' y s ( g ) = vechS ( g ) 32 de 110 Contraste chi-cuadrado de bondad de ajuste T = nFGLS ( s, σˆ ) Cuando el modelo es adecuado y los supuestos sobre la distribución se cumplen, T es un estadístico chi-cuadrado con r grados de libertad, donde r es el número de restricciones independientes del modelo en la matriz de momentos (r = nº momentos- nº de parámetros) 33 de 110 Estadísticos de contraste: Bajo supuestos de distribución generales, una versión escalada de este estadístico que se distribuye aproximadamente como una chi-cuadrado a pesar de la no normalidad ha sido desarrollado por Satorra y Bentler (1994). Este estadístico escalado se define como −1 T =c T donde { } c = r −1tr (Vˆ − VˆRˆ Jˆ −1 Rˆ 'Vˆ )Γˆ 34 de 110 SEM BASIC MODELS Factor analysis 35 de 110 Factor Analysis Charles Spearman, 1904 Acording to the two-factor theory of intelligence, the performance of any intellectual act requires some combination of "g", which is available to the same individual to the same degree for all intellectual acts, and of "specific factors" or "s" which are specific to that act and which varies in strength from one act to another. If one knows how a person performs on one task that is highly saturated with "g", one can safely predict a similar level of performance for a another highly "g" saturated task. Prediction of performance on tasks with high "s" factors are less accurate. Nevertheless, since "g" pervades all tasks, prediction will be significantly better than chance. Thus, the most important information to have about a person's intellectual ability is an estimate of their "g". 36 de 110 Spearman, 1904 Variables CLASSIC FRENCH ENGLISH MATH DISCRIM MUSIC Correlation matrix = = = = = = V1 V2 V3 V4 V5 V6 1 .83 1 .78 .67 1 .70 .64 .64 1 .66 .65 .54 .45 1 .63 .57 .51 .51 .40 1 cases = 23; 37 de 110 Single-Factor Model * V1 * V2 * * * V3 * V4 * * * V5 * * V6 * F1 38 de 110 EQS code for a factor model /Title confirmatory factor analysis: 1 factor ! (Spearman, 1904 ) eqs/exer3.eqs /Specifications var = 6; cases = 23; /Label v1 = classic; v2 = french; v3 =english; v4 = math; V5 = discrim; V6=music; /equations V1 = *f1 + e1; V2 = *f1+ e2; V3 = *f1 + e3; V4 = *f1 + e4; V5 = *f1 + e5; V6 = *f1 + e6; /variances f1 = 1; e1 to e6 = *; /matrix 1 .83 1 .78 .67 1 .70 .64 .64 1 .66 .65 .54 .45 1 .63 .57 .51 .51 .40 1 /LMTEST /end 39 de 110 NT analysis RESIDUAL COVARIANCE MATRIX CLASSIC FRENCH ENGLISH MATH DISCRIM MUSIC V V V V V V 1 2 3 4 5 6 CLASSIC V 1 0.000 -0.001 0.005 -0.006 -0.001 0.003 6 MUSIC V 6 0.000 (S-SIGMA) : FRENCH V 2 0.000 -0.029 0.003 0.054 0.005 ENGLISH V 3 0.000 0.046 -0.015 -0.017 MATH V 4 DISCRIM V 5 0.000 -0.056 0.030 0.000 -0.049 CHI-SQUARE = 1.663 BASED ON 9 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.99575 THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS 1.648 MUSIC V . 40 de 110 Loadings’ estimates, s.e. and z-test statistics CLASSIC =V1 = .960*F1 .160 6.019 +1.000 E1 FRENCH =V2 = .866*F1 .171 5.049 +1.000 E2 ENGLISH =V3 = .807*F1 .178 4.529 +1.000 E3 =V4 = .736*F1 .186 3.964 +1.000 E4 DISCRIM =V5 = .688*F1 .190 3.621 +1.000 E5 MUSIC = .653*F1 .193 3.382 +1.000 E6 MATH =V6 41 de 110 Estimates of unique-factors E1 -CLASSIC E2 -FRENCH E3 -ENGLISH E4 - E5 -DISCRIM E6 -MUSIC MATH .078*I .064 I 1.224 I I .251*I .093 I 2.695 I I .349*I .118 I 2.958 I I .459*I .148 I 3.100 I I .527*I .167 I 3.155 I I .574*I .180 I 3.184 I I I I I I I I I I I I I I I I I I I I I I I I I I 42 de 110 STANDARDIZED SOLUTION: CLASSIC FRENCH ENGLISH MATH DISCRIM MUSIC =V1 =V2 =V3 =V4 =V5 =V6 = = = = = = .960*F1 .866*F1 .807*F1 .736*F1 .688*F1 .653*F1 + + + + + + .279 .501 .591 .677 .726 .758 E1 E2 E3 E4 E5 E6 43 de 110 Data of Lawley and Maxwell M0: GAELIC /TITLE Lawley and Maxwell data /SPECIFICATIONS CAS=220; VAR=6; ME=ML; /LABEL v1 =Gaelic; v2 = English; v3 = Histo; /EQUATIONS v4 =aritm; V1= *F1 + E1; v5 =Algebra; V2= *F1 + E2; v6 =Geometry; V3= *F1 + E3; /EQUATIONS V4= *F2 + E4; V1= *F1 + E1; V5= *F2 + E5; V2= *F1 + E2; V6= *F2 + E6; V3= *F1 + E3; /VARIANCES V4= *F1 + E4; F1 = 1; F2=1; E1 TO E6 = *; V5= *F1 + E5; /COVARIANCES V6= *F1 + E6; F1, F2 = *; /VARIANCES F1 = 1; E1 TO E6 = *; /COVARIANCES /MATRIX 1 .439 .410 .288 .329 .248 .439 1 .351 .354 .320 .329 .410 .351 1 .164 .190 .181 .288 .354 .164 1 .595 .470 .329 .320 .190 .595 1 .464 .248 .329 .181 .470 .464 1 /END M1: M0, Single factor model CHI-SQUARE = 52.841, 9 df P-value LESS THAN 0.001 =V1 = ENGLISH =V2 = HISTO =V3 = ARITM =V4 = ALGEBRA =V5 = GEOMETRY=V6 = .687*F1 .076 9.079 .672*F1 .076 8.896 .533*F1 .076 7.047 .766*F2 .067 11.379 .768*F2 .067 11.411 .616*F2 .069 8.942 + 1.000 E1 + 1.000 E2 + 1.000 E3 + 1.000 E4 + 1.000 E5 + 1.000 E6 COVARIANCES AMONG INDEPENDENT VARIABLES --------------------------------------I F2 F2 .597*I I F1 F1 .072 I 8.308 M1, Two factor model with correlated factors: CHI-SQUARE = 7.953, 8 df 44 de 110 P-value = 0.43804 Asymptotic Robustness Monte Carlo Evaluation > Single sample, one-factor model 45 de 110 True model: a two-factor model phi 1 1 F1 1 F2 .8 1 Y3 Y4 .8 .8 .8 Y1 .3 Y2 .3 .3 .3 Y5 .3 Y6 46 de 110 .3 Distribution of factors and errors • F1, F2 non-normal, chi2 of 1 df, transformed to be of variance 1 and with the desired correlation phi. • Errors are independent chi2 of 1 df scaled to have the desired variance, .3. 47 de 110 Non-normal data (factors and errors chi2 1df): histogram of y1 Skewness = 1.1243 Kurtosis = 4.9154 (norm = 3) 48 de 110 Normal probability plot of Y1 49 de 110 Non-normal data (factors and errors are chi2 with 1df) 50 de 110 Scatterplot matrix of obs. Variables gplotmatrix(Y,[],[]) 51 de 110 MA: the * are unconstrained parameters (df =9) MB:var. of unique factors restr. by equal. (df = 14) Model analyzed: * F 1 * Y1 * * * Y4 Y3 Y2 * * * * * Y5 * Y6 52 de 110 * A case where H0 holds but two different models: df = 9 or 14 53 de 110 Factors and Errors non-normal AR holds (df = 9) H0 holds n = 2000 54 de 110 Factors and Errors non-normal AR holds (df = 9) , H0 holds n = 120 55 de 110 Factors and Errors non-normal non-AR ( df = 14), H0 holds n = 2000 56 de 110 non-AR ( df = 14), H0 Holds n = 120 57 de 110 Factors and Errors non-normal AR holds (df = 9), no H0 n = 2000 58 de 110 Factors and Errors non-normal no-AR holds (df = 14) no H0, n = 800 59 de 110 Simultaneous equations 60 de 110 True value a = - .2 Path Diagram U2 * a V1 V3 V2 V4 V5 V6 61 de 110 Limitations of OLS regression When ignoring simultaneous equations, i.e. OLS: MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS V2 =V2 = .335*V1 .028 11.899 + -.011*V4 .030 -.365 + .312*V6 .018 17.050 + 1.000 D2 62 de 110 EQS analysis: GOODNESS OF FIT SUMMARY /TITLE example of SEM /SPECIFICATIONS CASES = 500 ; VAR = 6; /EQUATIONS V1 = .5*V2 + *V3 + *V5 +D1 ; V2 = -.5*V1 + *V4 + *V6 + D2 ; /VARIANCES D1 = *; D2 = *; V3 = *; V4 = *; V5 = *; V6=*; /COVARIANCES V3 to V6 = *; /MATRIX 2.5123 0.9345 0.5414 1.4768 0.4544 1.5910 2.1110 0.6925 1.4842 2.1037 1.5067 0.5093 0.5278 1.4727 0.3683 0.4155 -0.0683 0.0235 0.9376 /END 1.5566 0.0847 63 de 110 Simultaneous Equations M E A SU R E M E NT E Q U A TI O N S WI T H S T AN D A R D E R R O R S A N D T ES T S T A TI S T I C S V1 =V 1 = . 8 6 3* V 2 .046 18.757 + . 5 12 * V 3 . 0 19 2 6 . 7 65 + . 51 2 * V 5 . 02 1 2 4 . 44 5 + 1 .0 0 0 D 1 V2 =V 2 = - . 1 4 6* V 1 .058 -2.541 + . 4 71 * V 4 . 0 59 7 . 9 83 + . 48 9 * V 6 . 02 8 1 7 . 23 0 + 1 .0 0 0 D 2 CH I - S Q UA R E = 0 . 21 3 B AS E D O N 3 D E GR E E S OF F R EE D O M PR O B A B IL I T Y V A L U E F O R TH E C HI - S Q U AR E S TA T I S TI C I S 0. 9 7 5 4 4 64 de 110 Modelo de regresión con errores en las variables. En relación a los datos de una muestra de “managers” de granjas agrícolas en el estado de Iowa (USA), Fuller (1987, Ejemplo 3.1.2) considera el siguiente modelo: y = α + β1 x 1 + β 2 x 2 + u Y = y + e1 X1 = x 1 + e 2 X 2 = x 2 + e3 en donde: Y ≡ ln (tamaño de la granja), X1 ≡ ln (número de años de experiencia), X 2 ≡ ln (número de años de educación). La primera ecuación explica la variación de Y en términos de los 65 de 110 valores verdaderos de las variables y . A efectos de proteger la confidencial de la información, se añadió error aleatorio a los datos. Los coeficientes de fiabilidad de Y, y son .89, .80 y .83 respectivamente. e2 X1 u Exper Tamaño e3 X2 Educ Y e1 Magnitud del error de medida: Θ ε = diagonal ( .0997, .2013, .1808) 66 de 110 El resultado del análisis para una muestra de 176 individuos es el siguiente: Ecuación de regresión y 0 0.439 (.112) 0.313 (.107) x = 0 0 0 1 x 2 0 0 0 0 . 699 Ψ = 0 0 0 0 . 805 − 0 . 449 y e x + x 1 1 x 2 x 2 − 0 . 449 0 . 858 0 en donde los números entre paréntesis indican errores tipo. Si ignorásemos el error de medida, es decir, si estimásemos por MCO la regresión de Y sobre X1 y X2, las estimaciones de los correspondientes coeficientes de regresión resultan ser .300 (.076) y .198 (.075), que son de magnitud sensiblemente menor que los obtenidos teniendo en cuenta el error de medida. 67 de 110 Ecuaciones simultáneas En un estudio sobre Desarrollo educacional, Sewell, Haller & Ohlendorf (1970) analizan una muestra de 3500 individuos en base al modelo siguiente: Y1 = γ 11 X 1 + u1 Y2 = β 21 Y1 + γ 21 X 1 + γ 22 X 2 + u 2 Y13212 ≡≡≡ X Y2 = β 31 Y1 + β 31 Y2 + u3 en donde: Y1 = éxito académico (AP), Y2 = influencias significativas de otras personas (SO), Y3 = aspiraciones educacionales (EA), X1 = habilidad mental (MA), X2 = estatus socioeconómico (SES). 68 de 110 Y1 u3 X1 Y3 e2 X2 y2 Y2 u2 La ecuación de medida será: Y1 1 Y 0 2 Y3 = 0 X 1 0 X 2 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Y1 0 y e 2 2 Y3 + 0 X1 0 X 2 0 Θ ε = diagonal(0, *, 0, 0, 0 ) El modelo propuesto impone 2 restricciones de sobreidentificación obteniéndose chi2=7.14. Sin introducir error de medida en Y2, el valor del estadístico chi-cuadrado es 186.39 con 3 df, valor extraordinariamente alto que conduce por tanto al rechazo 69 de 110 del modelo que ignora error de medida. Mimic model Joreskog & Goldberger, JASA 1979 y =social participation X1 = Income X2 = Occupation X3 = Education Y1= Church attendance Y2 = Membership Y3 = Frieds Seen X1 X2 X3 β1 β2 λ1 λ2 λ3 y β3 e Y1 u1 Y2 u2 Y3 u 70 de3110 Mimic model Y1 λ 11 Y λ 2 2 Y3 λ 3 = X1 0 X 2 0 X 3 0 y 0 X 0 1 = X 2 0 X 3 0 0 0 0 0 1 0 0 0 0 0 1 0 ∗ ∗ ∗ 0 0 0 0 0 u1 0 y u 2 0 X 1 u 3 + ⋅ 0 X 2 h1 0 X 3 h2 1 h 3 ∗ ∗ ∗ y e 0 0 0 X 1 X 1 + ⋅ 0 0 0 X 2 X 2 0 0 0 X 3 X 3 ~ ~ Φ Ψ 0 * 0 0 * 71 de 110 Modelo de indicadores y causas múltiples (MIMIC). Renta Ocup Re1 PS Afi e1 e2 u Educ Ami e3 72 de 110 Utilizando el método de estimación de la máxima verosimilitud, obtenemos las siguientes estimaciones: y = .269 X 1 + .114 X 2 + .386 X 3 + u (.066) (.065) (.070) Y1 = .402 y + e1 (.046) Y2 = .634 y + e 2 (.060) Y3 = .346 y + (.046) e3 Un aspecto a destacar aquí es que el modelo en cuestión impone 6 restricciones de sobreidentificación sobre la matriz de varianzas y covarianzas de las variables observables. El correspondiente estadístico es igual a 12.36 con un “P-VALUE” igual a 0.052. 73 de 110 Panel data xtα = γt + βx(t-1)α + lα + µtα Xtα = xtα + vtα t = 1,2, ..., T α = 1,2,..., N Anderson (1986) xtα budget of household α at time t lα individual (unobserved) characteristic of household α 74 de 110 Panel data X 1 1 X 0 2 X 3 = 0 X 4 0 l1 0 0 1 0 0 0 x1 γ 1 0 x γ β 2 2 x 3 = γ 3 + 0 x4 γ 5 0 l1 0 0 0 0 1 0 0 0 x1 v1 0 x2 v2 0 ⋅ x3 + v3 0 x4 v4 1 l1 0 0 0 0 1 0 0 0 β 0 0 0 0 0 β 0 ~ 0 0 x1 µ1 0 1 x2 µ 2 0 1 ⋅ x3 + µ 3 0 1 x4 µ 4 0 0 l1 l1 Ψ ~ Φ 75 de 110 Growth curve model 76 de 110 Abilities at four time points: (Dependence between ability scores at 6,7,9 and 11) Growth curve model * * V1 * V3 V4 * 1 * D1 V2 * 1 * 1 D2 * 1 * 1 Slope F2 Intercept F1 * 77 de 110 Estimates ABIL6 =V1 = 1.000 F1 + 1.000 E1 ABIL7 =V2 = 1.000 F1 + 1.000 F2 + 1.000 E2 ABIL9 =V3 = 1.000 F1 + 2.187*F2 .069 31.739 + 1.000 E3 ABIL11 =V4 = 1.000 F1 + 3.656*F2 .119 30.755 + 1.000 E4 INTERCEP=F1 SLOPE =F2 = = 18.042*V999 .470 38.417 7.822*V999 .305 25.612 + 1.000 D1 + 1.000 D2 78 de 110 Estimates of variances E D --- E1 -ABIL6 E2 -ABIL7 E3 -ABIL9 E4 -ABIL11 --8.907*I D1 .625 I 14.248 I I 8.907*I D2 .625 I 14.248 I I 8.907*I .625 I 14.248 I I 8.907*I .625 I 14.248 I -INTERCEP -SLOPE 31.568*I 3.725 I 8.474 I I 2.028*I .361 I 5.621 I I I I I I I I I 79 de 110 80 de 110 ISI 54TH SESSION, BERLIN, 13-20 AUGUST, 2003 Albert Satorra Universitat Pompeu Fabra. Barcelona & Juan Carlos Bou Universitat Jaume I. Castelló 81 de 110 This talk • Introduction: permanent and transitory components of profits (ROA) • Data & model • Substantive hypotheses • SEM: one- and two-level analyses • Variance decomposition of profits: – temporary vs permanent – Industry vs firm levels 82 de 110 Introduction • actual profit rates differ widely across firms, both between and within industries. • Some firms show what can be regarded as ``abnormal returns'', i.e. returns that deviate substantially from the mean return level of all the firms. • According to economic theory, in a ``competitive market'' these differences should disappear as the time passes. • How much evidence exists of the persistence of abnormal returns, or how much variation of the returns can be attributed to permanent and time-vanishing components 83 de 110 Data • Initial sample: 5000 Spanish firms (excluding finance and public companies) • Screened database: 4931 firms • Financial Profit data were collected for each firm (Return On Assets, ROA) • 6 Time Period: 1995 – 2000 • Firms were classified by 4-digit SIC code • Number of Industries: 342 (quasi average number of firms: 14.28) 84 de 110 ROA across time 85 de 110 Scatterplots and correlations 86 de 110 Intraclass-correlation (within industry) Variable Y1 Y2 Y3 Y4 Y5 Y6 Correlation 0.070 0.082 0.085 0.107 0.121 0.088 87 de 110 Firm level Industry level 88 de 110