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Las variables son cualesquiera:
Se esperaría que:crece X1 implicará decrece Y crece X2 implicará decrece Ycrece X3 implicará decrece Y
Hay que justificar teóricamente cada una de estas relaciones
Y=
X1=
X2=
X3=
600
620
640
660
680
700
720
X3
Y
600
620
640
660
680
700
720
X2
600
620
640
660
680
700
720
X1
600
620
640
660
680
700
720
Y
600
620
640
660
680
700
720
600
620
640
660
680
700
720
600
620
640
660
680
700
720
0
10
20
30
40
50
60
70
80
X3
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0102030405060708090
X2
0102030405060708090
0102030405060708090
0102030405060708090
12
14
16
18
20
22
24
26
Y
X1
12
14
16
18
20
22
24
26
X3
12
14
16
18
20
22
24
26
X2
12
14
16
18
20
22
24
26
X1
1)
12
14
16
18
20
22
24
26
600 620 640 660 680 700 720
Y
X1
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 698.9330 9.467491 73.82451 0.0000 X1 -2.279808 0.479826 -4.751327 0.0000
R-squared 0.051240 Mean dependent var 654.1565 Adjusted R-squared 0.048970 S.D. dependent var 19.05335 S.E. of regression 18.58097 Akaike info criterion 8.686903 Sum squared resid 144315.5 Schwarz criterion 8.706143 Log likelihood -1822.250 F-statistic 22.57511 Durbin-Watson stat 0.129062 Prob(F-statistic) 0.000003
2)
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)
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 698.9330 9.467491 73.82451 0.0000 X1 -2.279808 0.479826 -4.751327 0.0000
R-squared 0.051240 Mean dependent var 654.1565 Adjusted R-squared 0.048970 S.D. dependent var 19.05335 S.E. of regression 18.58097 Akaike info criterion 8.686903 Sum squared resid 144315.5 Schwarz criterion 8.706143 Log likelihood -1822.250 F-statistic 22.57511 Durbin-Watson stat 0.129062 Prob(F-statistic) 0.000003
Dependent Variable: Y/(X1^0.5) Included observations: 420
Variable Coefficient Std. Error t-Statistic Prob.
1/(X1^0.5) 699.4881 9.371808 74.63747 0.0000 X1^0.5 -2.308074 0.479454 -4.813960 0.0000
R-squared 0.794067 Mean dependent var 148.1814 Adjusted R-squared 0.793574 S.D. dependent var 9.346010 S.E. of regression 4.246278 Akaike info criterion 5.734713 Sum squared resid 7536.906 Schwarz criterion 5.753952 Log likelihood -1202.290 Durbin-Watson stat 0.135110
Intentando corregir la heterocedasticidad
Sin corrección de heterocedasticidad
Regresando a nuestro caso
teníamos esto
Hay un problema atípico con los primero y ultimos datos
Pretendemos corregir quitando 10 datos de cada extremo
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) 686.3607 10.12647 67.77890 0.0000 X1^0.5 -1.657306 0.508579 -3.258697 0.0012
R-squared 0.800102 Mean dependent var 147.9025 Adjusted R-squared 0.799601 S.D. dependent var 8.896714 S.E. of regression 3.982701 Akaike info criterion 5.606773 Sum squared resid 6328.901 Schwarz criterion 5.626693 Log likelihood -1122.158 Durbin-Watson stat 0.061076
-15
-10
-5
0
5
10
120
140
160
180
200
50 100 150 200 250 300 350 400
Residual Actual Fitted
Aún con la corrección existe heterocedasticidad grafica y según White
Dependent Variable: Y Method: Least Squares Sample: 10 410 Included observations: 401
Variable Coefficient Std. Error t-Statistic Prob.
C 688.7375 5.729231 120.2146 0.0000 X1 -0.884708 0.290357 -3.046967 0.0025 X2 -0.465897 0.031491 -14.79469 0.0000 X3 -0.767716 0.050488 -15.20588 0.0000
R-squared 0.637617 Mean dependent var 653.7394 Adjusted R-squared 0.634878 S.D. dependent var 17.76552 S.E. of regression 10.73487 Akaike info criterion 7.594797 Sum squared resid 45749.26 Schwarz criterion 7.634637 Log likelihood -1518.757 F-statistic 232.8415 Durbin-Watson stat 1.043932 Prob(F-statistic) 0.000000
El modelo con las tres variables
3)
Dependent Variable: Y/(X1^0.5) Method: Least Squares Sample: 10 410 Included observations: 401
Variable Coefficient Std. Error t-Statistic Prob.
C -7.153723 2.557214 -2.797468 0.0054 1/(X1^0.5) 703.2368 11.25696 62.47130 0.0000
X2/(X1^0.5) -0.467614 0.031908 -14.65500 0.0000 X3/(X1^0.5) -0.781395 0.050741 -15.39953 0.0000
R-squared 0.924314 Mean dependent var 147.9025 Adjusted R-squared 0.923742 S.D. dependent var 8.896714 S.E. of regression 2.456818 Akaike info criterion 4.645536 Sum squared resid 2396.274 Schwarz criterion 4.685376 Log likelihood -927.4300 F-statistic 1616.112 Durbin-Watson stat 1.039133 Prob(F-statistic) 0.000000
Corrigiendo como en el anterior por la variable heterocedástica X1, asi que dividimos entre la raíz de x1.
[sigue habiendo heterocedasticidad según WHITE]
White Heteroskedasticity Test:F-statistic 7.284180 Probability 0.000000
Obs*R-squared 57.58004 Probability 0.000000
Dependent Variable: Y/(X1^0.5) Method: Least Squares Sample: 10 410 Included observations: 349 Excluded observations: 52
Variable Coefficient Std. Error t-Statistic Prob.
C 384.6893 4.035467 95.32709 0.0000 LOG(1/(X1^0.5)) 157.8823 2.706767 58.32876 0.0000
LOG(X2/(X1^0.5)) -0.923572 0.091112 -10.13667 0.0000 LOG(X3/(X1^0.5)) -2.381607 0.125357 -18.99855 0.0000
R-squared 0.927127 Mean dependent var 146.7935 Adjusted R-squared 0.926493 S.D. dependent var 8.511957 S.E. of regression 2.307772 Akaike info criterion 4.521837 Sum squared resid 1837.405 Schwarz criterion 4.566022 Log likelihood -785.0606 F-statistic 1463.087 Durbin-Watson stat 1.253277 Prob(F-statistic) 0.000000
White Heteroskedasticity Test:F-statistic 3.093285 Probability 0.001383
Obs*R-squared 26.48571 Probability 0.001701
Decidimos en aplicar logaritmos a las explicativas [persiste el problema de heterocedasticida]
Dependent Variable: LOG(Y/(X1^0.5)) Method: Least Squares Sample: 10 410 Included observations: 349 Excluded observations: 52
Variable Coefficient Std. Error t-Statistic Prob.
C 6.594585 0.027035 243.9321 0.0000 LOG(1/(X1^0.5)) 1.066644 0.018133 58.82254 0.0000
LOG(X2/(X1^0.5)) -0.006472 0.000610 -10.60386 0.0000 LOG(X3/(X1^0.5)) -0.015861 0.000840 -18.88709 0.0000
R-squared 0.928304 Mean dependent var 4.987370 Adjusted R-squared 0.927680 S.D. dependent var 0.057490 S.E. of regression 0.015460 Akaike info criterion -5.489689 Sum squared resid 0.082462 Schwarz criterion -5.445505 Log likelihood 961.9507 F-statistic 1488.985 Durbin-Watson stat 1.205134 Prob(F-statistic) 0.000000
Aplicamos también logaritmos a la explicada, [parece mejorar el problema, gráfica residuos].
Dependent Variable: LOG(Y/(X1^0.5)) Method: Least Squares Sample: 10 410 Included observations: 349 Excluded observations: 52 White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
C 6.594585 0.030013 219.7246 0.0000 LOG(1/(X1^0.5)) 1.066644 0.020039 53.22880 0.0000
LOG(X2/(X1^0.5)) -0.006472 0.000629 -10.28736 0.0000 LOG(X3/(X1^0.5)) -0.015861 0.000926 -17.11997 0.0000
R-squared 0.928304 Mean dependent var 4.987370 Adjusted R-squared 0.927680 S.D. dependent var 0.057490 S.E. of regression 0.015460 Akaike info criterion -5.489689 Sum squared resid 0.082462 Schwarz criterion -5.445505 Log likelihood 961.9507 F-statistic 1488.985 Durbin-Watson stat 1.205134 Prob(F-statistic) 0.000000
Añadimos la corrección automática de e-views de “errores estándar consistentes de White”
White Heteroskedasticity Test:
F-statistic 1.526877 Probability 0.136933 Obs*R-squared 13.59612 Probability 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 0.018755 0.014430 1.299691 0.1946 LOG(1/(X1^0.5)) 0.024256 0.019164 1.265712 0.2065
(LOG(1/(X1^0.5)))^2 0.007918 0.006362 1.244596 0.2141 (LOG(1/(X1^0.5)))*(L
OG(X2/(X1^0.5))) 0.000132 0.000428 0.309322 0.7573
(LOG(1/(X1^0.5)))*(LOG(X3/(X1^0.5)))
0.000199 0.000380 0.524360 0.6004
LOG(X2/(X1^0.5)) 0.000224 0.000644 0.347518 0.7284 (LOG(X2/(X1^0.5)))^2 1.19E-05 9.73E-06 1.222272 0.2225 (LOG(X2/(X1^0.5)))*(LOG(X3/(X1^0.5)))
-1.13E-05 1.44E-05 -0.783328 0.4340
LOG(X3/(X1^0.5)) 0.000267 0.000567 0.471107 0.6379 (LOG(X3/(X1^0.5)))^2 8.97E-06 9.74E-06 0.921757 0.3573
R-squared 0.038957 Mean dependent var 0.000236 Adjusted R-squared 0.013443 S.D. dependent var 0.000341 S.E. of regression 0.000338 Akaike info criterion -13.11642 Sum squared resid 3.88E-05 Schwarz criterion -13.00596 Log likelihood 2298.816 F-statistic 1.526877 Durbin-Watson stat 1.777731 Prob(F-statistic) 0.136933
El problema parece solucionarse. Ya no hay heterocedasticidad, [NO podemos rechazar la hipótesis nula de homocedasticidad en los residuales].
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 Value df Probability
F-statistic 305.3015 (2, 345) 0.0000 Chi-square 610.6031 2 0.0000
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
C(3) -0.006472 0.000629 C(4) -0.015861 0.000926
Restrictions are linear in coefficients.
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.
4 PUNTO)
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 6.628711 0.043969 150.7579 0.0000 LOG(1/(X1^0.5)) 1.098396 0.029473 37.26773 0.0000
R-squared 0.798377 Mean dependent var 4.994771 Adjusted R-squared 0.797872 S.D. dependent var 0.059611 S.E. of regression 0.026800 Akaike info criterion -4.395835 Sum squared resid 0.286583 Schwarz criterion -4.375915 Log likelihood 883.3649 F-statistic 1579.945 Durbin-Watson stat 0.054502 Prob(F-statistic) 0.000000