CURSO DE PRONSTICO MACROECONMICO BT13.12 TP- 7: COINTEGRACIN - EL ENFOQUE DE JOHANSEN Respuestas Del 2 al 13 de diciembre de 2013 Braslia, Brasil I In ns st ti it tu ut to o F FM MI I d de el l- 2 - TP-7 Trabajo prctico sobre relaciones de largo plazo: el procedimiento de Johansen I.Demanda de dinero en Inglaterra 1. Estime el VAR de 4 variables con dinero real (mp), inflacin (D(p)), gasto total final (i) y tasa de inters neta (Rnet) para el periodo 1963:1 1989:2. Incluya 5 rezagos y seleccione el nmero ptimo de rezagos como sugerido por el criterio de AIC o SIC. Los comandos de EViews requeridos son: smpl @all varest 1 5 mp d(p) i rnet - 3 - El criterio AIC de este sistema es -23,87; SIC es -21,68.Para seleccionar el numero de rezagos, usted necesita tambin estimar el modelo con 1, 2, 3, 4 rezagos y comparar el AIC/SIC entre todos estos modelos. Afortunadamente, EViews provee una manera simple de hacer estas pruebas. Click en View | Lag Structure | Lag Length Criteria e ingrese 5 en la caja de dilogo que aparece. Note que el criterio AIC se minimice cuando se usa un VAR (2).El AIC es -24.07202.Vamos entonces a usar un modelo VAR (2) para responder las preguntas restantes de este trabajo prctico. - 4 - 2. Haga una prueba de cointegracin con el mtodo de Johansen, comience con su modelo VAR preferido. El primer paso consiste en re-estimar el VAR usando 2 rezagos, como se muestra aqu: Esta estimacin genera el siguiente resultado: - 5 - Para hacer un test de cointegracin, presione sobre View seguido de Cointegration Test. Note que el intervalo de rezagos para las variables diferenciadas endgenas es ahora solo 1 (comparado con 2 para el VAR en niveles), como se requiere.Presione OK para obtener los resultados. Aparecer que hay 1 (y solo 1) vector de co integracin. Nosotros podemos fcilmente rechazar la hiptesis nula de que el primer vector de cointegracin es cero, pero no podemos rechazar la hiptesis nula de que los siguientes son significativamente diferentes - 6 - de cero.La traza y el test del mximo eigenvalor son coherentes en su evaluacin. 3. Si usted encuentra cointegracin, re-estime el VAR imponiendo la restriccin de cointegracin de largo plazo (i.e., hay un vector de cointegracin y el coeficiente de la demanda de dinero con relacin al ingreso es 1). Presione OK para estimar el VECM con un vector de cointegracin. El resultado generado debe ser: - 7 - Note que EViews suministra de regresin de cointegracin de largo plazo, junto con los estadsticos t asociados para analizar la hiptesis nula de que las variables D(p), I, and RNET no deben estar en la regresin de cointegracin de largo plazo. Los errores estndar para los coeficientes tambin se incluyen en los resultados generados por Eviews. Estos sern usados para responder la pregunta 4.Aparte: la regresin de cointegracin de largo plazo se expresa con todas las variables I(1) a la izquierda del signo igual. As, el coeficiente estimado del ingreso es 1.05 (no -1.05!). 4. Recuerde las teoras estndar de la demanda de dinero, segn las cuales el dinero es demandado al menos por dos razones: como un inventario para suavizar las diferencias entre el flujo de ingreso y el de gasto, y como uno entre varios activos en un portfolio de inversin. Ambas demandas llevan a una especificacin de largo plazo de la siguiente forma: - 8 - ( , )netMfI RP La ecuacin de arriba aparece comnmente en forma log lineal, adems con la tasa de inters incluida ya sea en forma de logaritmos o de niveles: 0 1 2 3netm p i R p . Anticipatedsignsandmagnitudesofcoefficientsare 11 (quantitytheory)or 10.5 (Baumol-Tobinframework),20 ,and 30 .Thesignsof 2 and 3 shouldbenegative because other assets and goods are both alternative to money.(i)Test if unit income elasticity holds (i.e, test if 11 holds). Thet-statisticissimply:(1.055372-1.0000)/0.08508,whichislessthan1.Henceunit income elasticity holds. (ii)Test if 10.5 holds. The t-statistic is simply:(1.05372-0.5000)/0.08508, which is greater than 2.Hence we can reject the null hypothesis that income elasticity is 0.50. II. Demanda de dinero en Chile 1.Vector autorregresivo Vector Autoregression Estimates Date: 11/17/11 Time: 11:44 Sample (adjusted): 1987Q2 2001Q3 Included observations: 58 after adjustments Standard errors in ( ) & t-statistics in [ ] MRYI MR(-1) 0.736129 0.128767 0.075759 (0.21563) (0.07427) (0.13720) [ 3.41390][ 1.73374][ 0.55216] MR(-2)-0.162745-0.000530 0.139404 (0.27041) (0.09314) (0.17206) [-0.60185][-0.00569][ 0.81019] MR(-3) 0.097831 0.122463-0.044315 (0.27310) (0.09407) (0.17377) [ 0.35823][ 1.30189][-0.25502] - 9 - MR(-4)-0.574110-0.327410 0.331565 (0.28350) (0.09765) (0.18039) [-2.02509][-3.35295][ 1.83803] MR(-5) 0.256111 0.073856-0.202057 (0.21201) (0.07303) (0.13490) [ 1.20800][ 1.01137][-1.49778] Y(-1) 0.616935 0.824288-0.091889 (0.44579) (0.15355) (0.28366) [ 1.38393][ 5.36831][-0.32394] Y(-2) 0.345816-0.017962-0.106351 (0.51234) (0.17647) (0.32601) [ 0.67497][-0.10179][-0.32622] Y(-3)-0.269813 0.193976 0.087440 (0.49400) (0.17015) (0.31434) [-0.54618][ 1.14001][ 0.27818] Y(-4) 0.794865 0.119430-0.407605 (0.51801) (0.17842) (0.32961) [ 1.53446][ 0.66936][-1.23662] Y(-5)-0.780320-0.122142 0.175218 (0.38310) (0.13196) (0.24377) [-2.03684][-0.92562][ 0.71878] I(-1)-0.788893-0.132124 0.129088 (0.33682) (0.11601) (0.21432) [-2.34220][-1.13887][ 0.60232] I(-2) 0.028673 0.122208 0.372110 (0.31755) (0.10938) (0.20206) [ 0.09030][ 1.11732][ 1.84161] I(-3) 0.219448 0.297205 0.101869 (0.36311) (0.12507) (0.23105) [ 0.60436][ 2.37633][ 0.44090] I(-4)-0.276165-0.163024 0.665597 (0.37284) (0.12842) (0.23724) [-0.74071][-1.26945][ 2.80559] I(-5) 0.123916-0.001558 0.018856 (0.32746) (0.11279) (0.20836) [ 0.37842][-0.01382][ 0.09050] C-7.203673 0.073866 3.559893 (2.38811) (0.82256) (1.51957) [-3.01648][ 0.08980][ 2.34270] R-squared 0.992228 0.998703 0.708346 Adj. R-squared 0.989452 0.998240 0.604184 Sum sq. resids 0.047884 0.005681 0.019388 - 10 - S.E. equation 0.033765 0.011630 0.021485 F-statistic 357.4728 2156.583 6.800416 Log likelihood 123.5845 185.4028 149.8048 Akaike AIC-3.709810-5.841474-4.613960 Schwarz SC-3.141412-5.273076-4.045562 Mean dependent 6.104436 15.68210-0.070660 S.D. dependent 0.328773 0.277243 0.034150 Determinant resid covariance (dof adj.) 3.32E-11 Determinant resid covariance 1.26E-11 Log likelihood 480.9193 Akaike information criterion-14.92825 Schwarz criterion-13.22306 2. Nmero de rezagos VAR Lag Order Selection Criteria Endogenous variables: MR Y IExogenous variables: C Date: 11/17/11 Time: 11:37 Sample: 1986Q1 2003Q4 Included observations: 58 LagLogLLRFPEAICSCHQ 0 224.1494NA9.79e-08-7.625841-7.519267-7.584328 1 431.4694 386.0442 1.05e-10-14.46446-14.03816-14.29841 2 450.2985 33.11318 7.50e-11-14.80340 -14.05737*-14.51281 3 457.7298 12.30012 7.97e-11-14.74930-13.68356-14.33417 4 475.651227.80910*5.94e-11* -15.05694*-13.67147 -14.51727* 5 480.9193 7.629645 6.89e-11-14.92825-13.22306-14.26405 * indicates lag order selected by the criterion LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion 3. Var preferido Vector Autoregression Estimates Date: 11/17/11 Time: 12:09 Sample (adjusted): 1987Q1 2001Q3 Included observations: 59 after adjustments Standard errors in ( ) & t-statistics in [ ] MRYI - 11 - MR(-1) 0.764290 0.128622 0.084323 (0.21073) (0.07298) (0.12910) [ 3.62682][ 1.76248][ 0.65315] MR(-2)-0.077833 0.031209 0.121894 (0.27024) (0.09359) (0.16556) [-0.28801][ 0.33348][ 0.73627] MR(-3) 0.091718 0.107878-0.038508 (0.27936) (0.09674) (0.17114) [ 0.32832][ 1.11510][-0.22501] MR(-4)-0.363828-0.243609 0.137789 (0.18869) (0.06535) (0.11560) [-1.92814][-3.72801][ 1.19196] Y(-1) 0.386984 0.776398 0.073183 (0.40444) (0.14006) (0.24777) [ 0.95683][ 5.54328][ 0.29536] Y(-2) 0.313521-0.022372-0.175086 (0.49778) (0.17239) (0.30496) [ 0.62983][-0.12978][-0.57413] Y(-3)-0.345700 0.102563 0.122027 (0.48710) (0.16869) (0.29841) [-0.70971][ 0.60801][ 0.40892] Y(-4) 0.299258 0.117384-0.376722 (0.36208) (0.12539) (0.22182) [ 0.82650][ 0.93616][-1.69833] I(-1)-0.644503-0.118019 0.113764 (0.30631) (0.10608) (0.18765) [-2.10408][-1.11258][ 0.60624] I(-2) 0.112467 0.156622 0.415341 (0.29538) (0.10229) (0.18096) [ 0.38075][ 1.53111][ 2.29520] I(-3) 0.339369 0.307985 0.125678 (0.35158) (0.12175) (0.21539) [ 0.96527][ 2.52956][ 0.58349] I(-4)-0.138102-0.109313 0.473938 (0.29705) (0.10287) (0.18198) [-0.46491][-1.06262][ 2.60429] C-6.698234 0.290788 3.723569 (2.18371) (0.75623) (1.33781) [-3.06736][ 0.38452][ 2.78334] R-squared 0.991497 0.998574 0.692133 Adj. R-squared 0.989278 0.998201 0.611820 Sum sq. resids 0.055361 0.006639 0.020778 - 12 - S.E. equation 0.034692 0.012014 0.021253 F-statistic 446.9645 2683.583 8.617924 Log likelihood 121.9394 184.5050 150.8491 Akaike AIC-3.692862-5.813729-4.672851 Schwarz SC-3.235099-5.355967-4.215088 Mean dependent 6.094334 15.67316-0.070116 S.D. dependent 0.335035 0.283287 0.034112 Determinant resid covariance (dof adj.) 3.42E-11 Determinant resid covariance 1.62E-11 Log likelihood 481.7678 Akaike information criterion-15.00908 Schwarz criterion-13.63579 4. Prueba de Cointegracin Date: 11/17/11 Time: 11:55 Sample (adjusted): 1986Q3 2001Q3 Included observations: 61 after adjustments Trend assumption: Linear deterministic trend Series: MR Y ILags interval (in first differences): 1 to 1 Unrestricted Cointegration Rank Test (Trace) HypothesizedTrace0.05 No. of CE(s)EigenvalueStatisticCritical ValueProb.** None * 0.326842 33.11330 29.79707 0.0200 At most 1 0.098475 8.970995 15.49471 0.3681 At most 2 0.042470 2.647278 3.841466 0.1037 Trace test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegration Rank Test (Maximum Eigenvalue) HypothesizedMax-Eigen0.05 No. of CE(s)EigenvalueStatisticCritical ValueProb.** None * 0.326842 24.14231 21.13162 0.0183 At most 1 0.098475 6.323718 14.26460 0.5720 At most 2 0.042470 2.647278 3.841466 0.1037 Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):- 13 - MRYI 27.87319-31.46176 29.51600 0.090161-5.587283-55.64145 5.303948-4.846485-23.69761 Unrestricted Adjustment Coefficients (alpha):D(MR)-0.012547 0.002949-0.005307 D(Y) 0.004527 0.000832-0.002183 D(I) 0.006582 0.003947 0.003017 1 Cointegrating Equation(s): Log likelihood 469.9297 Normalized cointegrating coefficients (standard error in parentheses) MRYI 1.000000-1.128746 1.058939 (0.03853) (0.43154) Adjustment coefficients (standard error in parentheses) D(MR)-0.349725 (0.12217) D(Y) 0.126191 (0.04732) D(I) 0.183460 (0.08004) 2 Cointegrating Equation(s): Log likelihood 473.0915 Normalized cointegrating coefficients (standard error in parentheses) MRYI 1.000000 0.000000 12.52785 (3.28647) 0.000000 1.000000 10.16075 (2.90134) Adjustment coefficients (standard error in parentheses) D(MR)-0.349459 0.378275 (0.12167) (0.13949) D(Y) 0.126266-0.147087 (0.04722) (0.05414) D(I) 0.183816-0.229132 (0.07868) (0.09019) 5. Estimacin de un vector de correccin de errores Vector Error Correction Estimates Date: 11/17/11 Time: 11:55 - 14 - Sample (adjusted): 1987Q1 2001Q3 Included observations: 59 after adjustments Standard errors in ( ) & t-statistics in [ ] Cointegrating Eq: CointEq1 MR(-1) 1.000000 Y(-1)-1.121098 (0.04248) [-26.3932] I(-1) 0.875591 (0.52427) [ 1.67013] C 11.53762 Error Correction:D(MR)D(Y)D(I) CointEq1-0.610875-0.001385 0.306698 (0.18159) (0.06729) (0.11093) [-3.36396][-0.02058][ 2.76483] D(MR(-1)) 0.372853 0.123303-0.204653 (0.20815) (0.07714) (0.12715) [ 1.79127][ 1.59853][-1.60954] D(MR(-2)) 0.274469 0.134237-0.083755 (0.19963) (0.07398) (0.12195) [ 1.37486][ 1.81451][-0.68680] D(MR(-3)) 0.320607 0.195197-0.116605 (0.17539) (0.06500) (0.10714) [ 1.82796][ 3.00325][-1.08836] D(Y(-1))-0.191823-0.120382 0.421531 (0.35285) (0.13076) (0.21554) [-0.54364][-0.92065][ 1.95570] D(Y(-2)) 0.106373-0.157355 0.243615 (0.34950) (0.12952) (0.21350) [ 0.30435][-1.21493][ 1.14107] D(Y(-3))-0.249645-0.069600 0.383805 (0.35431) (0.13130) (0.21643) [-0.70459][-0.53009][ 1.77331] D(I(-1))-0.109127-0.125155-1.119105 (0.32475) (0.12034) (0.19837) [-0.33604][-1.03997][-5.64137] D(I(-2))-0.005117 0.013137-0.663904 (0.42094) (0.15599) (0.25714) [-0.01216][ 0.08422][-2.58193] - 15 - D(I(-3)) 0.222292 0.197987-0.492513 (0.27735) (0.10278) (0.16942) [ 0.80148][ 1.92632][-2.90700] C 0.005273 0.011987-0.013644 (0.01015) (0.00376) (0.00620) [ 0.51940][ 3.18614][-2.20007] R-squared 0.328156 0.408123 0.591167 Adj. R-squared 0.188188 0.284815 0.505993 Sum sq. resids 0.056489 0.007757 0.021079 S.E. equation 0.034305 0.012713 0.020956 F-statistic 2.344515 3.309791 6.940731 Log likelihood 121.3444 179.9134 150.4249 Akaike AIC-3.740489-5.725877-4.726268 Schwarz SC-3.353151-5.338539-4.338931 Mean dependent 0.018173 0.015139-0.002047 S.D. dependent 0.038074 0.015032 0.029815 Determinant resid covariance (dof adj.) 3.66E-11 Determinant resid covariance 1.97E-11 Log likelihood 475.9829 Akaike information criterion-14.91467 Schwarz criterion-13.64702