Transcript LINEAR STRUCTURAL RELATIONS

Regression as Moment Structure
1
Regression Equation
Y = b X + v
Observable Variables
Y
z =
X
Moment matrix
sYY
S =
sYX
sYX
sXX
Moment structure S = S(q)
b2sXX +svv
bsXX
S =
bsXX
sXX
Parameter vector
q = (b, sXX, svv )’
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Sample:
z1, z2, ..., zn
n iid
Sample Moments
S =
n-1 S zi zi’
S =
syy
syx
syx
sxx
Fitting S to S = S(q)
Estimator
q^
S close to S =^S(q) ^
3 moment equations
•syy= b2sXX +svv
•syx= bsXX
•sxx= sXX
with 3 (unknown) parameters
Parameter estimates
^
^
q = (syx/sxx, sXX, syy - (b )2sXX )’
^
b
is
the same as the
usual OLS
estimate of b !
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Regression Equation
Y = b x
+ v
X = x + u
Observable Variables
Y
z =
X
Moment structure S = S(q)
b2sXX +svv
S =
bsXX
Parameter vector
q = (b, sXX, svv , suu )’
bsXX
sXX + suu
new parameter
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Sample:
z1, z2, ..., zn
n iid
Sample Moments
S :=
n-1 S zi zi’
S =
syy
syx
syx
sxx
Fitting S to S = S(q)
Estimator
q^ =
S close to
3 moment equations
•syy= b2sxx +svv
•syx= bsxx
•sxx= sxx + suu
with 4 (unknown) parameters
Parameter estimates
^
S^ = S(q )^
^
b
is
the same as the
usual OLS
estimate of b !
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The effect of measurement error
in regression
v
b
x
X
Y = b (X -u)+ v = bX
Y
u
+ (v - bu)
Note that w is correlated with X,
= cX
+ w,
where
w
= v - bu
unless u or b equals zero
So, the classical LS estimate b of b is neither ubiased, neither
consistent.
In fact,
b ---> sYX/sXX
= b (sxx/sXX )= kb
k is the so called Fiability coefficient (reliability of X).
Since 0 k  1
b suffers from downward bias
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In multiple regression
Regression Equation
Y = b1x1 + b2x2...+ b pxp+ v
Xk
= xk + u k
Observable Variables
b = SXX-1SXY
does not converge to b
b* :=
(SXX - Quu)-1 SXY
Examples with EQS of regression with error in variables
Using suplementary information to assessing the magnitude
of variances of errors in variables.
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Path analysis & covariance
structure
Example with ROS data
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Sample covariance matrix
ROS92
ROS93
ROS94
ROS95
ROS92
72.07
29.56
30.21
27.63
ROS93 ROS94 ROS95
36.21
31.09
24.04
Mean:
6.27
7.35
46.51
35.19 46.62
10.02
8.80
n = 70
bj = ?
F
b1
ROS92
SEM:
b2
b3
ROS93
ROS94
It is
a valid model ?
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Calculations
b1b2= 29.56
b1b3= 30.21
b2b3= 31.09
b1b2/b1b3
= b2/b3 = 29.56/30.21-->
31.09 = b2b3= b3 (.978b3) --> b32=
b3 = 5.64
b2 = .978b3
31.09/.978
In the same way, we obtain
b1=5.34
b2=5.52
Model test in this case is CHI2 = 0, df = 0
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Fitted Model
F
5.34
5.52
R92
43.34
R93
5.80
1
5.64
R94
14.74
CHI2 = 0,
df = 0
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/TITLE
FACTOR ANALYSIS MODEL (EXAMPLE ROS)
/SPECIFICATIONS
CAS=70; VAR=4;
/LABEL
V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95;
/EQUATIONS
V1 = *F1 + E1;
V2 = *F1 + E2;
V3 = *F1 + E3;
/VARIANCES
F1 = 1.0;
E1 TO E3 = *;
/COVARIANCES
/MATRIX
72.07
29.56 36.21
30.21 31.09 46.51
27.63 24.04 35.19 46.62
/END
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ROS92
=V1
=
5.359*F1
.974
5.504
+ 1.000 E1
ROS93
=V2
=
5.516*F1
.650
8.482
+ 1.000 E2
ROS94
=V3
=
5.637*F1
+ 1.000 E3
.753
7.482
VARIANCES OF INDEPENDENT VARIABLES
---------------------------------E
--E1 -ROS92
43.347*I
8.205 I
5.283 I
I
E2 -ROS93
5.789*I
3.924 I
1.475 I
I
E3 -ROS94
14.736*I
4.693 I
3.140 I
I
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… with the help of EQS
RESIDUAL COVARIANCE MATRIX
ROS92
ROS93
ROS94
V
V
V
(S-SIGMA) :
ROS92
V 1
0.000
0.000
0.000
1
2
3
CHI-SQUARE =
ROS93
V 2
0.000
0.000
0.000 BASED ON
ROS94
V 3
0.000
0 DEGREES OF FREEDOM
STANDARDIZED SOLUTION:
ROS92
ROS93
ROS94
=V1
=V2
=V3
=
=
=
.631*F1
.917*F1
.827*F1
+
+
+
.776 E1
.400 E2
.563 E3
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one - factor four- indicators model
F
*
R92
*
*
R93
*
*
*
R94
*
R95
*
CHI2 = ?,
df = ?
p-value = ?
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… with the help of EQS
/TITLE
FACTOR ANALYSIS MODEL (EXAMPLE ROS)
! This line is not read
/SPECIFICATIONS
CAS=70; VAR=4;
/LABEL
V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95;
/EQUATIONS
V1 = *F1 + E1;
V2 = *F1 + E2;
V3 = *F1 + E3;
V4 = *F1 + E4;
/VARIANCES
F1 = 1.0;
E1 TO E4 = *;
/COVARIANCES
/MATRIX
72.07
29.56 36.21
30.21 31.09 46.51
27.63 24.04 35.19 46.62
/END
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… with the help of EQS
ROS92
=V1
=
ROS93
=V2
=
ROS94
=V3
=
ROS95
=V4
=
4.998*F1
.966
5.175
4.837*F1
.622
7.779
+ 1.000 E1
6.417*F1
.653
9.833
+ 1.000 E3
+ 1.000 E2
5.393*F1
+ 1.000 E4
.710
7.590
VARIANCES OF INDEPENDENT VARIABLES
---------------------------------E
--E1 -ROS92
47.090*I
8.437 I
5.581 I
I
E2 -ROS93
12.810*I
2.775 I
4.616 I
I
E3 -ROS94
5.332*I
3.017 I
1.767 I
I
E4 -ROS95
17.536*I
3.682 I
4.763 I
D
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Fitted Model
F
4.84
4.99
R92
47.10
6.42
R93
12.81
R94
5.33
5.40
R95
17.54
CHI2 = 6.27,
df = 2
p-value = .043
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/TITLE
FACTOR ANALYSIS MODEL (EXAMPLE ROS)
/SPECIFICATIONS
CAS=70; VAR=4;
/LABEL
V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95;
/EQUATIONS
V1 = *F1 + E1;
V2 = *F1 + E2;
V3 = *F1 + E3;
V4 = *F1 + E4;
/VARIANCES
F1 = 1.0;
E1 TO E4 = *;
/COVARIANCES
/CONSTRAINTS
(V1,F1)=(V2,F1)=(V3,F1)=(V4,F1);
/MATRIX
72.07
29.56 36.21
30.21 31.09 46.51
27.63 24.04 35.19 46.62
/END
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… estimation results
ROS92
=V1
=
5.521*F1
.528
10.450
+ 1.000 E1
ROS93
=V2
=
5.521*F1
.528
10.450
+ 1.000 E2
ROS94
=V3
=
5.521*F1
.528
10.450
+ 1.000 E3
ROS95
=V4
=
5.521*F1
.528
10.450
+ 1.000 E4
CHI-SQUARE = 12.425 BASED ON 5 DEGREES OF FREEDOM
PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.02941
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... EQS use an iterative optimization method
ITERATIVE SUMMARY
ITERATION
1
2
3
4
5
6
7
8
9
10
PARAMETER
ABS CHANGE
21.878996
5.741889
2.309283
0.477505
0.147232
0.056361
0.014530
0.005784
0.001423
0.000598
ALPHA
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
FUNCTION
1.39447
0.43985
0.19638
0.18079
0.18014
0.18008
0.18007
0.18007
0.18007
0.18007
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Exercise:
a)
Write the covariance structure for the one - factor four- indicators
model
b)
From the ML estimates of this model, shown in previous slides,
compute the fitted covariance matrix.
c)
In relation with b), compute the residual covariance matrix
Note:
For c), use the following sample moments:
ROS92
ROS93
ROS94
ROS95
ROS92
72.07
29.56
30.21
27.63
36.21
31.09
24.04
Mean:
6.27
7.35
n = 70
ROS93 ROS94 ROS95
46.51
35.19 46.62
10.02
8.80
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one - factor four- indicators model with means
F
1
*
*
R92
*
*
R93
*
*
*
R94
*
*
*
*
R95
*
CHI2 = ?,
df = ?
p-value = ?
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/TITLE
FACTOR ANALYSIS MODEL (EXAMPLE ROS data)
/SPECIFICATIONS
CAS=70; VAR=4; ANALYSIS = MOMENT;
/LABEL
V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95;
/EQUATIONS
V1 = *V999+ *F1 + E1;
V2 = *V999+ *F1 + E2;
V3 = *V999+ *F1 + E3;
V4 = *V999+ *F1 + E4;
/VARIANCES
F1 = 1.0;
E1 TO E4 = *;
/COVARIANCES
/CONSTRAINTS
!
(V1,F1)=(V2,F1)=(V3,F1)=(V4,F1);
/MATRIX
72.07
29.56 36.21
30.21 31.09 46.51
27.63 24.04 35.19 46.62
/MEANS
6.27 7.35 10.02 8.80
/END
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ROS92
=V1
=
6.270*V999 + 4.998*F1
1.022
.966
6.135
5.175
ROS93
=V2 =
7.350*V999 + 4.837*F1
.724
.622
10.146
7.779
ROS94
=V3 = 10.020*V999 + 6.417*F1
.821
.653
12.204
9.833
ROS95
=V4 =
8.800*V999 + 5.393*F1
.822
.710
10.706
7.591
VARIANCES OF INDEPENDENT VARIABLES
---------------------------------E
--E1 -ROS92
47.092*I
8.437 I
5.582 I
I
E2 -ROS93
12.810*I
2.775 I
4.616 I
I
E3 -ROS94
5.332*I
3.017 I
1.767 I
I
E4 -ROS95
17.535*I
3.682 I
4.763 I
+ 1.000 E1
+ 1.000 E2
+ 1.000 E3
+ 1.000 E4
D
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