1 NIVELES VS INCREMENTOS EN LA ESTIMACIÓN DE MODELOS
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NIVELES VS INCREMENTOS EN LA ESTIMACIÓN DE MODELOS DE REGRESIÓN: ALGUNAS REGLAS BÁSICAS a) Senderos aleatorios con deriva Procesos no cointegrados PGD: DX1=nrnd+0.1 DY1=2*DX1+nrnd 1) La estimación en niveles produce inconsistencia Dependent Variable: Y1 Method: Least Squares Date: 10/26/03 Time: 10:46 Sample: 1 1000 Included observations: 1000 Variable C X1 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient Std. Error t-Statistic Prob. 4.763699 1.683017 0.626139 0.006858 7.608055 245.4027 0.0000 0.0000 0.983698 0.983682 8.244128 67829.72 -3527.439 0.016348 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 144.4675 64.53723 7.058877 7.068693 60222.47 0.000000 2) La estimación en niveles con AR(1) permite obtener un estimador consistente de “beta” Dependent Variable: Y1 Method: Least Squares Date: 10/26/03 Time: 10:46 Sample(adjusted): 2 1000 Included observations: 999 after adjusting endpoints Convergence achieved after 6 iterations Variable Coefficient Std. Error t-Statistic Prob. C X1 AR(1) -29.09405 2.005744 0.996594 11.19131 0.031999 0.002155 -2.599701 62.68160 462.4503 0.0095 0.0000 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Inverted AR Roots 0.999758 0.999757 1.003531 1003.047 -1419.539 1.929940 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 144.6121 64.40725 2.847926 2.862661 2054957. 0.000000 1.00 1 3) La estimación en incrementos permite obtener estimadores consistentes de “alfa” y de “beta” Dependent Variable: DY1 Method: Least Squares Date: 10/26/03 Time: 11:08 Sample: 1 1000 Included observations: 1000 Variable Coefficient Std. Error t-Statistic Prob. C DX1 -0.026062 2.005375 0.032100 0.031950 -0.811919 62.76605 0.4170 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.797877 0.797674 1.005847 1009.704 -1423.767 1.928309 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.245172 2.236176 2.851534 2.861350 3939.577 0.000000 Procesos cointegrados PDD: X2=X1 Y2=2+2*X2+nrnd 4) La estimación en niveles permite obtener estimadores consistentes de “alfa” y de “beta” Dependent Variable: Y2 Method: Least Squares Date: 10/26/03 Time: 10:48 Sample(adjusted): 2 1000 Included observations: 999 after adjusting endpoints Variable C X2 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient Std. Error t-Statistic Prob. 1.945017 2.000413 0.075729 0.000829 25.68375 2412.867 0.0000 0.0000 0.999829 0.999829 0.994220 985.5089 -1410.728 1.984434 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 168.1614 75.94304 2.828284 2.838108 5821927. 0.000000 2 5) La estimación en incrementos permite obtener estimadores consistentes de “beta” Dependent Variable: D(Y2) Method: Least Squares Date: 10/26/03 Time: 11:14 Sample(adjusted): 3 1000 Included observations: 998 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. 0.001460 1.995341 0.044759 0.044521 0.032625 44.81830 0.9740 0.0000 C D(X2) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.668517 0.668184 1.401251 1955.651 -1751.791 3.029499 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.270045 2.432583 3.514610 3.524442 2008.680 0.000000 Procesos no cointegrados y regresión espuria PDD: X3=X1 DY3=0.2+nrnd Representación gráfica 250 200 150 100 50 0 -50 250 500 X3 750 1000 Y3 3 6) La regresión en niveles produce una “regresión espuria” Dependent Variable: Y3 Method: Least Squares Date: 10/26/03 Time: 10:49 Sample: 1 1000 Included observations: 1000 Variable Coefficient Std. Error t-Statistic Prob. C X3 -33.06879 1.914118 0.955650 0.010467 -34.60345 182.8652 0.0000 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.971020 0.970991 12.58267 158007.1 -3950.258 0.029608 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 125.8182 73.87665 7.904517 7.914332 33439.68 0.000000 7) La estimación en incrementos permite detectar que se trata de un regresión espuria Dependent Variable: D(Y3) Method: Least Squares Date: 10/26/03 Time: 10:50 Sample(adjusted): 2 1000 Included observations: 999 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C D(X3) 0.242879 -0.000255 0.032701 0.032532 7.427279 -0.007825 0.0000 0.9938 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.000000 -0.001003 1.024172 1045.782 -1440.380 1.933089 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.242845 1.023659 2.887647 2.897470 6.12E-05 0.993758 4 8) La estimación en niveles con AR(1) “también” permite detectar que se trata de una regresión espuria Dependent Variable: Y3 Method: Least Squares Date: 10/26/03 Time: 10:51 Sample(adjusted): 2 1000 Included observations: 999 after adjusting endpoints Convergence achieved after 10 iterations Variable Coefficient Std. Error t-Statistic Prob. C X3 AR(1) 939.9565 -0.001320 0.999701 1202.666 0.032592 0.000439 0.781561 -0.040514 2275.315 0.4347 0.9677 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Inverted AR Roots 0.999808 0.999807 1.024449 1045.298 -1440.148 1.933432 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 125.9441 73.80617 2.889186 2.903921 2589533. 0.000000 1.00 5 b) Senderos aleatorios sin deriva Procesos no cointegrados PGD: DX1=nrnd, DX2=nrnd, DX3=nrnd DYX=2·DX1+2·DX2+2·DX3+nrnd 1) La estimación en niveles produce inconsistencia Dependent Variable: Y1 Method: Least Squares Date: 10/30/03 Time: 21:17 Sample: 2 10000 Included observations: 9999 Variable Coefficient Std. Error t-Statistic Prob. C X1 X2 X3 -21.77510 1.979500 1.742952 1.954095 0.474370 0.005623 0.003815 0.008529 -45.90317 352.0070 456.8162 229.1080 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 0.974153 0.974146 16.03864 2571093. -41933.20 0.004048 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 56.15612 99.74733 8.388278 8.391163 125570.3 0.000000 2) La estimación en niveles con un proceso AR(1) permite obtener un estimador consistente de “beta” Dependent Variable: Y1 Method: Least Squares Date: 10/30/03 Time: 21:18 Sample: 2 10000 Included observations: 9999 Convergence achieved after 5 iterations Variable Coefficient Std. Error t-Statistic Prob. C X1 X2 X3 AR(1) -53.83252 1.998631 1.998373 1.997127 0.998509 6.893216 0.009829 0.009891 0.009783 0.000488 -7.809492 203.3319 202.0385 204.1507 2044.063 0.0000 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 Inverted AR Roots 0.999902 0.999902 0.986375 9723.523 -14048.30 1.995995 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 56.15612 99.74733 2.810940 2.814546 25558122 0.000000 1.00 6 3) La estimación en incrementos permite obtener estimadores consistentes de “alfa” y de “beta” Dependent Variable: DY1 Method: Least Squares Date: 10/30/03 Time: 21:19 Sample: 2 10000 Included observations: 9999 Variable Coefficient Std. Error t-Statistic Prob. C DX1 DX2 DX3 -0.005638 1.998815 1.998862 1.997274 0.009870 0.009832 0.009890 0.009784 -0.571224 203.2963 202.1018 204.1329 0.5679 0.0000 0.0000 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.925257 0.925235 0.986787 9732.615 -14052.97 1.997093 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.009089 3.608888 2.811675 2.814559 41243.38 0.000000 Procesos cointegrados 4) La estimación en niveles permite obtener estimadores consistentes de “alfa” y de “beta” Dependent Variable: Y2 Method: Least Squares Date: 10/30/03 Time: 21:21 Sample: 2 10000 Included observations: 9999 Variable C Z1 Z2 Z3 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient Std. Error t-Statistic Prob. 2.010660 1.999780 1.999926 1.999436 0.029734 0.000352 0.000239 0.000535 67.62257 5673.465 8362.585 3740.010 0.0000 0.0000 0.0000 0.0000 0.999914 0.999914 1.005303 10101.29 -14238.85 1.967612 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 108.3463 108.2299 2.848855 2.851739 38623823 0.000000 7 5) La estimación en incrementos permite obtener estimadores consistentes de “beta” Dependent Variable: D(Y2) Method: Least Squares Date: 10/30/03 Time: 21:22 Sample(adjusted): 3 10000 Included observations: 9998 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C D(Z1) D(Z2) D(Z3) -0.000324 1.972066 2.005368 1.998500 0.014103 0.014049 0.014132 0.013980 -0.022999 140.3743 141.9003 142.9558 0.9817 0.0000 0.0000 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.857663 0.857621 1.409940 19867.38 -17619.33 2.994308 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.015185 3.736604 3.525371 3.528256 20073.27 0.000000 Procesos no cointegrados y regresión espuria PGD: W1=X1, W2=X2, W3=X3 DY3=nrnd Representación gráfica 200 100 0 -100 -200 -300 2500 5000 Y3 W1 7500 10000 W2 W3 8 6) La regresión en niveles produce una “regresión espuria” Dependent Variable: Y3 Method: Least Squares Date: 10/30/03 Time: 21:25 Sample: 2 10000 Included observations: 9999 Variable Coefficient Std. Error t-Statistic Prob. C W1 W2 W3 44.98617 -0.267909 -1.019153 -1.011407 1.340604 0.015892 0.010783 0.024104 33.55664 -16.85775 -94.51754 -41.96022 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 0.600160 0.600040 45.32633 20534493 -52321.03 0.001530 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) -62.47053 71.67082 10.46605 10.46894 5000.836 0.000000 7) La estimación en incrementos permite detectar que se trata de un regresión espuria Dependent Variable: D(Y3) Method: Least Squares Date: 10/30/03 Time: 21:26 Sample(adjusted): 3 10000 Included observations: 9998 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C D(W1) D(W2) D(W3) -0.023560 -0.026440 0.005314 -0.003316 0.010001 0.009963 0.010022 0.009914 -2.355704 -2.653905 0.530202 -0.334509 0.0185 0.0080 0.5960 0.7380 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.000743 0.000443 0.999855 9991.096 -14183.09 2.010209 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) -0.023201 1.000076 2.837986 2.840871 2.475989 0.059502 9 8) La estimación en niveles con AR(1) “también” permite detectar que se trata de una regresión espuria Dependent Variable: Y3 Method: Least Squares Date: 10/30/03 Time: 21:28 Sample(adjusted): 3 10000 Included observations: 9998 after adjusting endpoints Convergence achieved after 13 iterations Variable Coefficient Std. Error t-Statistic Prob. C W1 W2 W3 AR(1) -980.9110 -0.026421 0.005302 -0.003332 0.999975 5170.456 0.009963 0.010023 0.009915 0.000139 -0.189715 -2.651854 0.528985 -0.336111 7194.753 0.8495 0.0080 0.5968 0.7368 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Inverted AR Roots 0.999805 0.999805 0.999925 9991.495 -14183.29 2.010080 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) -62.47687 71.67161 2.838226 2.841832 12837631 0.000000 1.00 10
