econometría, corrigiendo la heteroscedasticidad

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=

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01020304050607080

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X1

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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


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

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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


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


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


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


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


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


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)

Dependent Variable: LOG(Y/(X1^0.5)) Method: Least Squares Sample: 10 410 Included observations: 401 White Heteroskedasticity-Consistent Standard Errors & Covariance


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

econometría, corrigiendo la heteroscedasticidad

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