SEM, una eina classica i moderna per l`anàlisi multivariant aplicada

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
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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
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o Conclusions
Examples with Coupon data
(Bagozzi, 1994)
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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
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1.498
Variables
/LABELS
V1 = Intentions1;
V2 = Intentions2;
V3 = Behavior;
V4 = Attitudes1;
V5 = Attitudes2;
V6 = Attitudes3;
F1 = Attitudes
F2 = Intentions
V3 = Behavior
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SEM indicadores múltiples
E4
D2
V4
V1
E5
E6
V5
F1
F2
V2
E3
V6
V3
F1 = Attitudes
F2 = Intentions
V3 = Behavior
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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
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... 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
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STRUCTURAL
EQUATION
MODELING
LINEAR
STRUCTURAL
RELATIONS
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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)
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Computer programs
•
•
•
•
•
•
•
•
•
LISREL
EQS
LISCOMP / Mplus
COSAN
MOMENTS
CALIS
AMOS
RAMONA
Mx
•
•
•
•
•
•
•
•
•
Jöreskog and Sörbom
Bentler
Muthén
McDonalds
Schoenberg
SAS
Arbunckle
Browne
Neale
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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
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... books
•
•
•
•
•
•
•
Bollen (1989)
Dwyer (1983)
Hayduk (1987)
Mueller (1996)
Saris and Stronkhorst (1984)
Dunn, Everitt and Pickles (1993)
...
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... 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.
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Type of variables
Manifest Variables:
Yi , Xi
Measurement Model:
X3
X4
λ32
ξ2
λ32
Measurement error, disturbances:
εi , δi
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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
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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)
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... 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
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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
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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
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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
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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
...
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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.
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A multiple group SEM: model
and statistical analysis
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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
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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,
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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)
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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)
'
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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.
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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)
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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),
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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 )
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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)
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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
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SEM BASIC MODELS
Factor analysis
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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".
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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;
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Single-Factor Model
*
V1
*
V2
*
*
*
V3
*
V4
*
*
*
V5
*
*
V6
*
F1
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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
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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
.
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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
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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
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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
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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
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P-value = 0.43804
Asymptotic Robustness
Monte Carlo Evaluation
> Single sample, one-factor model
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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
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.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.
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Non-normal data (factors and errors chi2 1df):
histogram of y1
Skewness = 1.1243
Kurtosis = 4.9154
(norm = 3)
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Normal probability plot of Y1
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Non-normal data
(factors and errors are chi2 with 1df)
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Scatterplot matrix of
obs. Variables
gplotmatrix(Y,[],[])
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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
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Factors and Errors non-normal
AR holds (df = 9) H0 holds
n = 2000
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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
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non-AR ( df = 14), H0 Holds
n = 120
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Factors and Errors non-normal
AR holds (df = 9), no H0
n = 2000
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Factors and Errors non-normal
no-AR holds (df = 14) no H0,
n = 800
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Simultaneous equations
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True value a = - .2
Path Diagram
U2
* a
V1
V3
V2
V4
V5
V6
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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
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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
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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
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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
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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)
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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.
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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).
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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
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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
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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
*
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Modelo de indicadores y causas múltiples (MIMIC).
Renta
Ocup
Re1
PS
Afi
e1
e2
u
Educ
Ami
e3
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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.
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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
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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
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Growth curve model
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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
*
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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
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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
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ISI 54TH SESSION, BERLIN, 13-20 AUGUST, 2003
Albert Satorra
Universitat Pompeu Fabra. Barcelona
&
Juan Carlos Bou
Universitat Jaume I. Castelló
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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
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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
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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)
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ROA across time
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Scatterplots and correlations
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Intraclass-correlation
(within industry)
Variable
Y1
Y2
Y3
Y4
Y5
Y6
Correlation
0.070
0.082
0.085
0.107
0.121
0.088
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Firm level
Industry level
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