Teoría Econométrica II - Práctica 1
Primera Parte: Realiza una gráfica con los primeros 10 valores de las soluciones de las siguientes ecuaciones de diferencias.
Realiza otra gráfica, ahora con los primeros 100 valores.
Resuelve las ecuaciones en diferencias y verifica que coincida la estabilidad dinámica de las soluciones con las gráficas realizadas. Verifica que coincida también el punto de equilibrio si es el caso.
Ejercicio 1 - Xt = −1/2Xt−1 + 1 con X1 = 1
10 valores:
x <- numeric(10)
x[1] <- 1
for(t in 2:10)
x[t] <- (-1/2)*x[t-1]+1
x## [1] 1.0000000 0.5000000 0.7500000 0.6250000 0.6875000 0.6562500 0.6718750
## [8] 0.6640625 0.6679688 0.6660156plot(x, type="l")
100 valores:
x <- numeric(100)
x[1] <- 1
for(t in 2:100)
x[t] <- (-1/2)*x[t-1]+1
x## [1] 1.0000000 0.5000000 0.7500000 0.6250000 0.6875000 0.6562500 0.6718750
## [8] 0.6640625 0.6679688 0.6660156 0.6669922 0.6665039 0.6667480 0.6666260
## [15] 0.6666870 0.6666565 0.6666718 0.6666641 0.6666679 0.6666660 0.6666670
## [22] 0.6666665 0.6666667 0.6666666 0.6666667 0.6666667 0.6666667 0.6666667
## [29] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [36] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [43] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [50] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [57] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [64] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [71] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [78] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [85] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [92] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
## [99] 0.6666667 0.6666667plot(x, type="l")
Interpretación: La gráfica nos muestra que la serie de tiempo tiene un comportamiento convergente con su punto de equilibrio en 0.66666
Ejercicio 2 - Xt =1/4Xt−1 con X1 = 1
10 valores:
x <- numeric(10)
x[1] <- 1
for(t in 2:10)
x[t] <- (1/4)*x[t-1]
x## [1] 1.000000e+00 2.500000e-01 6.250000e-02 1.562500e-02 3.906250e-03
## [6] 9.765625e-04 2.441406e-04 6.103516e-05 1.525879e-05 3.814697e-06plot(x, type="l")
100 valores:
x <- numeric(100)
x[1] <- 1
for(t in 2:100)
x[t] <- (1/4)*x[t-1]
x## [1] 1.000000e+00 2.500000e-01 6.250000e-02 1.562500e-02 3.906250e-03
## [6] 9.765625e-04 2.441406e-04 6.103516e-05 1.525879e-05 3.814697e-06
## [11] 9.536743e-07 2.384186e-07 5.960464e-08 1.490116e-08 3.725290e-09
## [16] 9.313226e-10 2.328306e-10 5.820766e-11 1.455192e-11 3.637979e-12
## [21] 9.094947e-13 2.273737e-13 5.684342e-14 1.421085e-14 3.552714e-15
## [26] 8.881784e-16 2.220446e-16 5.551115e-17 1.387779e-17 3.469447e-18
## [31] 8.673617e-19 2.168404e-19 5.421011e-20 1.355253e-20 3.388132e-21
## [36] 8.470329e-22 2.117582e-22 5.293956e-23 1.323489e-23 3.308722e-24
## [41] 8.271806e-25 2.067952e-25 5.169879e-26 1.292470e-26 3.231174e-27
## [46] 8.077936e-28 2.019484e-28 5.048710e-29 1.262177e-29 3.155444e-30
## [51] 7.888609e-31 1.972152e-31 4.930381e-32 1.232595e-32 3.081488e-33
## [56] 7.703720e-34 1.925930e-34 4.814825e-35 1.203706e-35 3.009266e-36
## [61] 7.523164e-37 1.880791e-37 4.701977e-38 1.175494e-38 2.938736e-39
## [66] 7.346840e-40 1.836710e-40 4.591775e-41 1.147944e-41 2.869859e-42
## [71] 7.174648e-43 1.793662e-43 4.484155e-44 1.121039e-44 2.802597e-45
## [76] 7.006492e-46 1.751623e-46 4.379058e-47 1.094764e-47 2.736911e-48
## [81] 6.842278e-49 1.710569e-49 4.276424e-50 1.069106e-50 2.672765e-51
## [86] 6.681912e-52 1.670478e-52 4.176195e-53 1.044049e-53 2.610122e-54
## [91] 6.525304e-55 1.631326e-55 4.078315e-56 1.019579e-56 2.548947e-57
## [96] 6.372368e-58 1.593092e-58 3.982730e-59 9.956824e-60 2.489206e-60plot(x, type="l")
Interpretación: La gráfica nos muestra que la serie de tiempo tiene un comportamiento convergente con su punto de equilibrio en 0.
Ejercicio 3 - Xt = −1/3Xt−1 con X1 = 2
10 valores:
x <- numeric(10)
x[1] <- 2
for(t in 2:10)
x[t] <- (-1/3)*x[t-1]
x## [1] 2.0000000000 -0.6666666667 0.2222222222 -0.0740740741 0.0246913580
## [6] -0.0082304527 0.0027434842 -0.0009144947 0.0003048316 -0.0001016105plot(x, type="l")
100 valores:
x <- numeric(100)
x[1] <- 2
for(t in 2:100)
x[t] <- (-1/3)*x[t-1]
x## [1] 2.000000e+00 -6.666667e-01 2.222222e-01 -7.407407e-02 2.469136e-02
## [6] -8.230453e-03 2.743484e-03 -9.144947e-04 3.048316e-04 -1.016105e-04
## [11] 3.387018e-05 -1.129006e-05 3.763353e-06 -1.254451e-06 4.181503e-07
## [16] -1.393834e-07 4.646115e-08 -1.548705e-08 5.162350e-09 -1.720783e-09
## [21] 5.735944e-10 -1.911981e-10 6.373271e-11 -2.124424e-11 7.081412e-12
## [26] -2.360471e-12 7.868236e-13 -2.622745e-13 8.742484e-14 -2.914161e-14
## [31] 9.713871e-15 -3.237957e-15 1.079319e-15 -3.597730e-16 1.199243e-16
## [36] -3.997478e-17 1.332493e-17 -4.441642e-18 1.480547e-18 -4.935158e-19
## [41] 1.645053e-19 -5.483509e-20 1.827836e-20 -6.092788e-21 2.030929e-21
## [46] -6.769764e-22 2.256588e-22 -7.521960e-23 2.507320e-23 -8.357733e-24
## [51] 2.785911e-24 -9.286370e-25 3.095457e-25 -1.031819e-25 3.439396e-26
## [56] -1.146465e-26 3.821552e-27 -1.273851e-27 4.246168e-28 -1.415389e-28
## [61] 4.717965e-29 -1.572655e-29 5.242183e-30 -1.747394e-30 5.824648e-31
## [66] -1.941549e-31 6.471831e-32 -2.157277e-32 7.190924e-33 -2.396975e-33
## [71] 7.989915e-34 -2.663305e-34 8.877683e-35 -2.959228e-35 9.864093e-36
## [76] -3.288031e-36 1.096010e-36 -3.653368e-37 1.217789e-37 -4.059297e-38
## [81] 1.353099e-38 -4.510330e-39 1.503443e-39 -5.011478e-40 1.670493e-40
## [86] -5.568309e-41 1.856103e-41 -6.187010e-42 2.062337e-42 -6.874456e-43
## [91] 2.291485e-43 -7.638284e-44 2.546095e-44 -8.486983e-45 2.828994e-45
## [96] -9.429981e-46 3.143327e-46 -1.047776e-46 3.492585e-47 -1.164195e-47plot(x, type="l")
Interpretación: La gráfica nos muestra que la serie de tiempo tiene un comportamiento convergente con su punto de equilibrio en 0
Ejercicio 4 - Xt = 2Xt−1 con X1 = 2
10 valores:
x <- numeric(10)
x[1] <- 2
for(t in 2:10)
x[t] <- 2*x[t-1]
x## [1] 2 4 8 16 32 64 128 256 512 1024plot(x, type="l")
100 valores:
x <- numeric(100)
x[1] <- 2
for(t in 2:100)
x[t] <- 2*x[t-1]
x## [1] 2.000000e+00 4.000000e+00 8.000000e+00 1.600000e+01 3.200000e+01
## [6] 6.400000e+01 1.280000e+02 2.560000e+02 5.120000e+02 1.024000e+03
## [11] 2.048000e+03 4.096000e+03 8.192000e+03 1.638400e+04 3.276800e+04
## [16] 6.553600e+04 1.310720e+05 2.621440e+05 5.242880e+05 1.048576e+06
## [21] 2.097152e+06 4.194304e+06 8.388608e+06 1.677722e+07 3.355443e+07
## [26] 6.710886e+07 1.342177e+08 2.684355e+08 5.368709e+08 1.073742e+09
## [31] 2.147484e+09 4.294967e+09 8.589935e+09 1.717987e+10 3.435974e+10
## [36] 6.871948e+10 1.374390e+11 2.748779e+11 5.497558e+11 1.099512e+12
## [41] 2.199023e+12 4.398047e+12 8.796093e+12 1.759219e+13 3.518437e+13
## [46] 7.036874e+13 1.407375e+14 2.814750e+14 5.629500e+14 1.125900e+15
## [51] 2.251800e+15 4.503600e+15 9.007199e+15 1.801440e+16 3.602880e+16
## [56] 7.205759e+16 1.441152e+17 2.882304e+17 5.764608e+17 1.152922e+18
## [61] 2.305843e+18 4.611686e+18 9.223372e+18 1.844674e+19 3.689349e+19
## [66] 7.378698e+19 1.475740e+20 2.951479e+20 5.902958e+20 1.180592e+21
## [71] 2.361183e+21 4.722366e+21 9.444733e+21 1.888947e+22 3.777893e+22
## [76] 7.555786e+22 1.511157e+23 3.022315e+23 6.044629e+23 1.208926e+24
## [81] 2.417852e+24 4.835703e+24 9.671407e+24 1.934281e+25 3.868563e+25
## [86] 7.737125e+25 1.547425e+26 3.094850e+26 6.189700e+26 1.237940e+27
## [91] 2.475880e+27 4.951760e+27 9.903520e+27 1.980704e+28 3.961408e+28
## [96] 7.922816e+28 1.584563e+29 3.169127e+29 6.338253e+29 1.267651e+30plot(x, type="l")
Interpretación: La gráfica nos muestra que la serie de tiempo tiene un comportamiento divergente, con una tendencia positiva.
Ejercicio 5 - Xt = -1/4Xt−1+3 con X1 = 1
10 valores:
x <- numeric(10)
x[1] <- 1
for(t in 2:10)
x[t] <- (-1/4)*x[t-1]+3
x## [1] 1.000000 2.750000 2.312500 2.421875 2.394531 2.401367 2.399658 2.400085
## [9] 2.399979 2.400005plot(x, type="l")
100 valores:
x <- numeric(100)
x[1] <- 1
for(t in 2:100)
x[t] <- (-1/4)*x[t-1]+3
x## [1] 1.000000 2.750000 2.312500 2.421875 2.394531 2.401367 2.399658 2.400085
## [9] 2.399979 2.400005 2.399999 2.400000 2.400000 2.400000 2.400000 2.400000
## [17] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [25] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [33] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [41] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [49] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [57] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [65] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [73] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [81] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [89] 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000 2.400000
## [97] 2.400000 2.400000 2.400000 2.400000plot(x, type="l")
Interpretación: La gráfica nos muestra que la serie de tiempo tiene un comportamiento convergente con su punto de equilibrio en 2.4.