Las variables son
cualesquiera:
Y=
X1=
X2=
X3=
Se esperaría que:
crece X1 implicará decrece Y
crece X2 implicará decrece Y
crece X3 implicará decrece Y
Hay que justificar teóricamente cada una de estas relaciones
Y
720
720
720
700
700
700
680
680
680
660
660
660
640
640
640
620
620
620
600
600
600
X3
X2
X1
720
720
720
720
700
700
700
700
680
680
680
680
660
660
660
660
640
640
640
640
620
620
620
620
600
80
70
600
80
70
600
80
70
600
60
50
60
50
60
50
60
50
40
40
40
40
30
20
30
20
30
20
30
20
10
10
10
10
0
0
90
80
70
60
50
40
30
20
10
0
26
0
90
80
70
60
50
40
30
20
10
0
26
0
90
80
70
60
50
40
30
20
10
0
26
24
24
24
22
22
22
20
20
20
18
18
18
16
16
16
14
14
14
12
12
12
X2
X3
Y
1)
90
80
70
60
50
40
30
20
10
0
26
24
X1
22
20
18
16
14
12
Y
X3
80
70
X2
X1
26
24
22
X1
20
18
16
14
12
600
620
640
660
Y
680
700
720
2)
Dependent Variable: Y
Method: Least Squares
Date: 11/15/08 Time: 18:09
Sample: 1 420
Included observations: 420
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
X1
698.9330
-2.279808
9.467491
0.479826
73.82451
-4.751327
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.051240
0.048970
18.58097
144315.5
-1822.250
0.129062
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
654.1565
19.05335
8.686903
8.706143
22.57511
0.000003
Si se conoce la varianza
• Divídase el modelo entre la desviación
típica conocida.
En el caso de que se desconozca la
varianza
• Aplicar Mínimos Cuadrados ponderados
• Veamos el caso más conocido,
cuando la varianza no se conoce,
entonces hay que indentificar el
patrón.
• Patrones de la varianza:
CASO 1)
CASO 2)
Caso 3)
Caso 4)
Sin corrección de heterocedasticidad
Dependent Variable: Y
Method: Least Squares
Date: 11/15/08 Time: 18:09
Sample: 1 420
Included observations: 420
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
X1
698.9330
-2.279808
9.467491
0.479826
73.82451
-4.751327
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.051240
0.048970
18.58097
144315.5
-1822.250
0.129062
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
654.1565
19.05335
8.686903
8.706143
22.57511
0.000003
Intentando corregir la heterocedasticidad
Dependent Variable: Y/(X1^0.5)
Included observations: 420
Variable
Coefficient
Std. Error
t-Statistic
Prob.
1/(X1^0.5)
X1^0.5
699.4881
-2.308074
9.371808
0.479454
74.63747
-4.813960
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
0.794067
0.793574
4.246278
7536.906
-1202.290
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
148.1814
9.346010
5.734713
5.753952
0.135110
Regresando a
nuestro caso
teníamos esto
Hay un problema atípico con los primero y ultimos datos
Pretendemos corregir quitando 10 datos de cada extremo
Aún con la corrección existe heterocedasticidad grafica y según White
Dependent Variable: Y/(X1^0.5)
Method: Least Squares
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
1/(X1^0.5)
X1^0.5
686.3607
-1.657306
10.12647
0.508579
67.77890
-3.258697
0.0000
0.0012
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
0.800102
0.799601
3.982701
6328.901
-1122.158
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
147.9025
8.896714
5.606773
5.626693
0.061076
200
180
160
10
140
5
120
0
-5
-10
-15
50
100
150
Residual
200
250
Actual
300
350
Fitted
400
3)
Dependent Variable: Y
Method: Least Squares
Sample: 10 410
Included observations: 401
El modelo con las tres variables
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
X1
X2
X3
688.7375
-0.884708
-0.465897
-0.767716
5.729231
0.290357
0.031491
0.050488
120.2146
-3.046967
-14.79469
-15.20588
0.0000
0.0025
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.637617
0.634878
10.73487
45749.26
-1518.757
1.043932
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
653.7394
17.76552
7.594797
7.634637
232.8415
0.000000
Dependent Variable: Y/(X1^0.5)
Method: Least Squares
Sample: 10 410
Included observations: 401
Corrigiendo como en el anterior
por la variable heterocedástica X1,
asi que dividimos entre la raíz de
x1.
[sigue habiendo
heterocedasticidad según WHITE]
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
1/(X1^0.5)
X2/(X1^0.5)
X3/(X1^0.5)
-7.153723
703.2368
-0.467614
-0.781395
2.557214
11.25696
0.031908
0.050741
-2.797468
62.47130
-14.65500
-15.39953
0.0054
0.0000
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.924314
0.923742
2.456818
2396.274
-927.4300
1.039133
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
147.9025
8.896714
4.645536
4.685376
1616.112
0.000000
White Heteroskedasticity Test:
F-statistic
7.284180
Probability
0.000000
Obs*R-squared
Probability
0.000000
57.58004
Dependent Variable: Y/(X1^0.5)
Method: Least Squares
Sample: 10 410
Included observations: 349
Excluded observations: 52
Decidimos en aplicar logaritmos a
las explicativas [persiste el
problema de heterocedasticida]
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(1/(X1^0.5))
LOG(X2/(X1^0.5))
LOG(X3/(X1^0.5))
384.6893
157.8823
-0.923572
-2.381607
4.035467
2.706767
0.091112
0.125357
95.32709
58.32876
-10.13667
-18.99855
0.0000
0.0000
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.927127
0.926493
2.307772
1837.405
-785.0606
1.253277
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
146.7935
8.511957
4.521837
4.566022
1463.087
0.000000
White Heteroskedasticity Test:
F-statistic
3.093285
Probability
0.001383
Obs*R-squared
Probability
0.001701
26.48571
Dependent Variable: LOG(Y/(X1^0.5))
Method: Least Squares
Sample: 10 410
Included observations: 349
Excluded observations: 52
Aplicamos también logaritmos a
la explicada, [parece mejorar el
problema, gráfica residuos].
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(1/(X1^0.5))
LOG(X2/(X1^0.5))
LOG(X3/(X1^0.5))
6.594585
1.066644
-0.006472
-0.015861
0.027035
0.018133
0.000610
0.000840
243.9321
58.82254
-10.60386
-18.88709
0.0000
0.0000
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.928304
0.927680
0.015460
0.082462
961.9507
1.205134
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
4.987370
0.057490
-5.489689
-5.445505
1488.985
0.000000
Dependent Variable: LOG(Y/(X1^0.5))
Añadimos la corrección
Method: Least Squares
Sample: 10 410
automática de e-views de
Included observations: 349
estándar consistentes de
Excluded observations: 52
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(1/(X1^0.5))
LOG(X2/(X1^0.5))
LOG(X3/(X1^0.5))
6.594585
1.066644
-0.006472
-0.015861
0.030013
0.020039
0.000629
0.000926
219.7246
53.22880
-10.28736
-17.11997
0.0000
0.0000
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.928304
0.927680
0.015460
0.082462
961.9507
1.205134
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
4.987370
0.057490
-5.489689
-5.445505
1488.985
0.000000
“errores
White”
El problema parece solucionarse. Ya no
hay heterocedasticidad, [NO podemos
rechazar la hipótesis nula de
homocedasticidad en los residuales].
White Heteroskedasticity Test:
F-statistic
Obs*R-squared
1.526877
13.59612
Probability
Probability
0.136933
0.137435
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 11/16/08 Time: 18:39
Sample: 10 410
Included observations: 349
Excluded observations: 52
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(1/(X1^0.5))
(LOG(1/(X1^0.5)))^2
(LOG(1/(X1^0.5)))*(L
OG(X2/(X1^0.5)))
(LOG(1/(X1^0.5)))*(L
OG(X3/(X1^0.5)))
LOG(X2/(X1^0.5))
(LOG(X2/(X1^0.5)))^2
(LOG(X2/(X1^0.5)))*(
LOG(X3/(X1^0.5)))
LOG(X3/(X1^0.5))
(LOG(X3/(X1^0.5)))^2
0.018755
0.024256
0.007918
0.000132
0.014430
0.019164
0.006362
0.000428
1.299691
1.265712
1.244596
0.309322
0.1946
0.2065
0.2141
0.7573
0.000199
0.000380
0.524360
0.6004
0.000224
1.19E-05
-1.13E-05
0.000644
9.73E-06
1.44E-05
0.347518
1.222272
-0.783328
0.7284
0.2225
0.4340
0.000267
8.97E-06
0.000567
9.74E-06
0.471107
0.921757
0.6379
0.3573
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.038957
0.013443
0.000338
3.88E-05
2298.816
1.777731
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.000236
0.000341
-13.11642
-13.00596
1.526877
0.136933
4 PUNTO)
VERIFICAR SI X2 y X3 son NO significativas. Lo
haremos mediante la prueba de Wald que esta
en E-views. Se rechaza la hipótesis nula de
que los estimadores de X2 Y X3 SEAN
AMBOS CERO.
Estimation Equation:
=====================
LOG(Y/(X1^0.5)) = C(1) + C(2)*LOG(1/(X1^0.5)) +
C(3)*LOG(X2/(X1^0.5)) + C(4)*LOG(X3/(X1^0.5))
Wald Test:
Equation: Untitled
Test Statistic
F-statistic
Chi-square
Value
df Probability
305.3015
610.6031
(2, 345)
2
0.0000
0.0000
Value
Std. Err.
-0.006472
-0.015861
0.000629
0.000926
Null Hypothesis Summary:
Normalized Restriction (= 0)
C(3)
C(4)
Restrictions are linear in coefficients.
FIN
Dependent Variable: LOG(Y/(X1^0.5))
Method: Least Squares
Sample: 10 410
Included observations: 401
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
LOG(1/(X1^0.5))
6.628711
1.098396
0.043969
0.029473
150.7579
37.26773
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.798377
0.797872
0.026800
0.286583
883.3649
0.054502
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
4.994771
0.059611
-4.395835
-4.375915
1579.945
0.000000
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Econometría, corrigiendo la heteroscedasticidad