Ejemplo VEC
WILSON SANDOVAL
r Sys.Date()
Cargar las librerias
## Loading required package: MASS
## Loading required package: strucchange
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
## Loading required package: sandwich
## Loading required package: urca
## Loading required package: lmtest
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library(forecast)
library(urca)
library(highcharter)
## Highcharts (www.highcharts.com) is a Highsoft software product which is
## not free for commercial and Governmental use
##
## Attaching package: 'highcharter'
## The following object is masked from 'package:lmtest':
##
## unemployment
Cargar la base de datos
Utilizaron las siguientes series: productividad laboral definida como
la diferencia logarítmica entre el PIB y el empleo, el registro de
empleo, tasa de desempleo y salarios reales, definidos como el logaritmo
del salario real índice. Estas series están representadas por “prod”,
“e”, “U” y “rw”, respectivamente. Se toman los datos de la base de datos
de la OCDE y abarca desde el primer trimestre de 1980 hasta el cuarto
trimestre de 2004
data(Canada)
Canada=as.data.frame(Canada)
layout(matrix(1:4, nrow = 2, ncol = 2))
plot.ts(Canada$e, main = "Employment", ylab = "", xlab = "")
plot.ts(Canada$prod, main = "Productivity", ylab = "", xlab = "")
plot.ts(Canada$rw, main = "Real Wage", ylab = "", xlab = "")
plot.ts(Canada$U, main = "Unemployment Rate", ylab = "", xlab = "")
.
var_canada<-ts(Canada[,c(1,2,3,4)],frequency = 12)
plot.ts(var_canada)
Pruebas de estacionariedad
- pruebas de estacionariedad empleo
adf.empleo <- ur.df(var_canada[,1], type = "trend", selectlags = "BIC")
summary(adf.empleo)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.84769 -0.24745 -0.02081 0.24187 0.82344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 47.681128 17.445770 2.733 0.00776 **
## z.lag.1 -0.051256 0.018785 -2.729 0.00786 **
## tt 0.019217 0.007005 2.743 0.00754 **
## z.diff.lag 0.753011 0.075724 9.944 1.61e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3937 on 78 degrees of freedom
## Multiple R-squared: 0.5674, Adjusted R-squared: 0.5508
## F-statistic: 34.11 on 3 and 78 DF, p-value: 3.462e-14
##
##
## Value of test-statistic is: -2.7286 4.0771 3.8115
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
##
## Augmented Dickey-Fuller Test
##
## data: var_canada[, 1]
## Dickey-Fuller = -2.148, Lag order = 4, p-value = 0.5152
## alternative hypothesis: stationary
- pruebas de estacionariedad Productividad
adf.product <- ur.df(var_canada[,2], type = "trend", selectlags = "BIC")
summary(adf.product)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.22554 -0.40892 0.02578 0.41669 1.70908
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.281115 14.506222 2.019 0.04697 *
## z.lag.1 -0.073036 0.036127 -2.022 0.04664 *
## tt 0.014219 0.006151 2.312 0.02344 *
## z.diff.lag 0.310251 0.109678 2.829 0.00594 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6797 on 78 degrees of freedom
## Multiple R-squared: 0.1467, Adjusted R-squared: 0.1139
## F-statistic: 4.471 on 3 and 78 DF, p-value: 0.005977
##
##
## Value of test-statistic is: -2.0216 2.4483 2.6786
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
pruebas de estacionariedad
salario
adf.salario <- ur.df(var_canada[,3], type = "trend", selectlags = "BIC")
summary(adf.salario)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.14517 -0.55886 -0.00263 0.50749 2.96709
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.38961 7.77366 3.009 0.00353 **
## z.lag.1 -0.05421 0.01925 -2.816 0.00615 **
## tt 0.03134 0.01797 1.744 0.08512 .
## z.diff.lag 0.17639 0.10670 1.653 0.10233
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8466 on 78 degrees of freedom
## Multiple R-squared: 0.3674, Adjusted R-squared: 0.3431
## F-statistic: 15.1 on 3 and 78 DF, p-value: 7.717e-08
##
##
## Value of test-statistic is: -2.8163 13.4198 11.3011
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
- pruebas de estacionariedad Desempleo
adf.desempleo <- ur.df(var_canada[,1], type = "trend", selectlags = "BIC")
summary(adf.desempleo)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.84769 -0.24745 -0.02081 0.24187 0.82344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 47.681128 17.445770 2.733 0.00776 **
## z.lag.1 -0.051256 0.018785 -2.729 0.00786 **
## tt 0.019217 0.007005 2.743 0.00754 **
## z.diff.lag 0.753011 0.075724 9.944 1.61e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3937 on 78 degrees of freedom
## Multiple R-squared: 0.5674, Adjusted R-squared: 0.5508
## F-statistic: 34.11 on 3 and 78 DF, p-value: 3.462e-14
##
##
## Value of test-statistic is: -2.7286 4.0771 3.8115
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
Con una diferencia
diff.adf.empleo <- ur.df(diff(var_canada[,1]), type = "trend", selectlags = "BIC")
summary(diff.adf.empleo)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9077 -0.2442 -0.0408 0.3000 0.7114
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.086812 0.092323 0.940 0.3500
## z.lag.1 -0.364701 0.080222 -4.546 2e-05 ***
## tt 0.001341 0.001870 0.717 0.4756
## z.diff.lag 0.322536 0.108388 2.976 0.0039 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3914 on 77 degrees of freedom
## Multiple R-squared: 0.2273, Adjusted R-squared: 0.1972
## F-statistic: 7.549 on 3 and 77 DF, p-value: 0.0001715
##
##
## Value of test-statistic is: -4.5462 6.9125 10.3669
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
diff.adf.product <- ur.df(diff(var_canada[,2]), type = "trend", selectlags = "BIC")
summary(diff.adf.product)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.08255 -0.41492 0.03547 0.42292 1.77919
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.007671 0.160585 -0.048 0.962
## z.lag.1 -0.723368 0.139237 -5.195 1.63e-06 ***
## tt 0.003058 0.003456 0.885 0.379
## z.diff.lag -0.024356 0.114799 -0.212 0.833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6981 on 77 degrees of freedom
## Multiple R-squared: 0.3679, Adjusted R-squared: 0.3433
## F-statistic: 14.94 on 3 and 77 DF, p-value: 9.378e-08
##
##
## Value of test-statistic is: -5.1952 9.1153 13.6696
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
diff.adf.salario <- ur.df(diff(var_canada[,3]), type = "trend", selectlags = "BIC")
summary(diff.adf.salario)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.2737 -0.5911 -0.0613 0.4972 3.4851
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.502206 0.344946 4.355 4.05e-05 ***
## z.lag.1 -0.791156 0.141679 -5.584 3.38e-07 ***
## tt -0.017532 0.005305 -3.305 0.00145 **
## z.diff.lag 0.020803 0.114310 0.182 0.85607
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8924 on 77 degrees of freedom
## Multiple R-squared: 0.388, Adjusted R-squared: 0.3642
## F-statistic: 16.27 on 3 and 77 DF, p-value: 2.764e-08
##
##
## Value of test-statistic is: -5.5841 10.4507 15.5933
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
diff.adf.desempleo <- ur.df(diff(var_canada[,1]), type = "trend", selectlags = "BIC")
summary(diff.adf.desempleo)
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9077 -0.2442 -0.0408 0.3000 0.7114
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.086812 0.092323 0.940 0.3500
## z.lag.1 -0.364701 0.080222 -4.546 2e-05 ***
## tt 0.001341 0.001870 0.717 0.4756
## z.diff.lag 0.322536 0.108388 2.976 0.0039 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3914 on 77 degrees of freedom
## Multiple R-squared: 0.2273, Adjusted R-squared: 0.1972
## F-statistic: 7.549 on 3 and 77 DF, p-value: 0.0001715
##
##
## Value of test-statistic is: -4.5462 6.9125 10.3669
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2 6.50 4.88 4.16
## phi3 8.73 6.49 5.47
Se puede concluir que todas las series temporales están integradas de
orden uno
Una vez las series son estacionarias, se procede a realiza la
estimacion
VARselect(Canada,lag.max=10, type = "both")
## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 2 1 3
##
## $criteria
## 1 2 3 4 5
## AIC(n) -6.457999837 -6.862930587 -6.9695915550 -6.826230019 -6.696865092
## HQ(n) -6.159906898 -6.366109023 -6.2740413655 -5.931951204 -5.603857651
## SC(n) -5.710735482 -5.617489996 -5.2259747277 -4.584436956 -3.956895792
## FPE(n) 0.001570168 0.001052827 0.0009575467 0.001128994 0.001329669
## 6 7 8 9 10
## AIC(n) -6.687175990 -6.55982329 -6.584746129 -6.493397603 -6.523612352
## HQ(n) -5.395439924 -5.06935860 -4.895552811 -4.605475660 -4.436961783
## SC(n) -3.449030454 -2.82350152 -2.350248120 -1.760723357 -1.292761870
## FPE(n) 0.001412901 0.00172546 0.001859561 0.002330702 0.002704777
Según el AIC y el FPE, el número de retraso óptimo es \(p = 3\), mientras que el criterio HQ indica
\(p = 2\) y el criterio SC indica una
longitud de retraso óptima de \(p =
1\).
VAR(1)
Canada <- Canada[, c("prod", "e", "U", "rw")]
p1ct <- VAR(Canada, p = 1, type = "both")
p1ct
##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation prod:
## =========================================
## Call:
## prod = prod.l1 + e.l1 + U.l1 + rw.l1 + const + trend
##
## prod.l1 e.l1 U.l1 rw.l1 const trend
## 0.96313671 0.01291155 0.21108918 -0.03909399 16.24340747 0.04613085
##
##
## Estimated coefficients for equation e:
## ======================================
## Call:
## e = prod.l1 + e.l1 + U.l1 + rw.l1 + const + trend
##
## prod.l1 e.l1 U.l1 rw.l1 const
## 0.19465028 1.23892283 0.62301475 -0.06776277 -278.76121138
## trend
## -0.04066045
##
##
## Estimated coefficients for equation U:
## ======================================
## Call:
## U = prod.l1 + e.l1 + U.l1 + rw.l1 + const + trend
##
## prod.l1 e.l1 U.l1 rw.l1 const trend
## -0.12319201 -0.24844234 0.39158002 0.06580819 259.98200967 0.03451663
##
##
## Estimated coefficients for equation rw:
## =======================================
## Call:
## rw = prod.l1 + e.l1 + U.l1 + rw.l1 + const + trend
##
## prod.l1 e.l1 U.l1 rw.l1 const trend
## -0.22308744 -0.05104397 -0.36863956 0.94890946 163.02453066 0.07142229
summary(p1ct, equation = "e")
##
## VAR Estimation Results:
## =========================
## Endogenous variables: prod, e, U, rw
## Deterministic variables: both
## Sample size: 83
## Log Likelihood: -207.525
## Roots of the characteristic polynomial:
## 0.9504 0.9504 0.9045 0.7513
## Call:
## VAR(y = Canada, p = 1, type = "both")
##
##
## Estimation results for equation e:
## ==================================
## e = prod.l1 + e.l1 + U.l1 + rw.l1 + const + trend
##
## Estimate Std. Error t value Pr(>|t|)
## prod.l1 0.19465 0.03612 5.389 7.49e-07 ***
## e.l1 1.23892 0.08632 14.353 < 2e-16 ***
## U.l1 0.62301 0.16927 3.681 0.000430 ***
## rw.l1 -0.06776 0.02828 -2.396 0.018991 *
## const -278.76121 75.18295 -3.708 0.000392 ***
## trend -0.04066 0.01970 -2.064 0.042378 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.4701 on 77 degrees of freedom
## Multiple R-Squared: 0.9975, Adjusted R-squared: 0.9973
## F-statistic: 6088 on 5 and 77 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## prod e U rw
## prod 0.469517 0.06767 -0.04128 0.002141
## e 0.067667 0.22096 -0.13200 -0.082793
## U -0.041280 -0.13200 0.12161 0.063738
## rw 0.002141 -0.08279 0.06374 0.593174
##
## Correlation matrix of residuals:
## prod e U rw
## prod 1.000000 0.2101 -0.1728 0.004057
## e 0.210085 1.0000 -0.8052 -0.228688
## U -0.172753 -0.8052 1.0000 0.237307
## rw 0.004057 -0.2287 0.2373 1.000000
Diagnostico VAR(1)
ser11 <- serial.test(p1ct, lags.pt = 16, type = "PT.asymptotic")
ser11$serial
##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object p1ct
## Chi-squared = 233.5, df = 240, p-value = 0.606
norm1 <-normality.test(p1ct)
norm1$jb.mul
## $JB
##
## JB-Test (multivariate)
##
## data: Residuals of VAR object p1ct
## Chi-squared = 9.9189, df = 8, p-value = 0.2708
##
##
## $Skewness
##
## Skewness only (multivariate)
##
## data: Residuals of VAR object p1ct
## Chi-squared = 6.356, df = 4, p-value = 0.1741
##
##
## $Kurtosis
##
## Kurtosis only (multivariate)
##
## data: Residuals of VAR object p1ct
## Chi-squared = 3.5629, df = 4, p-value = 0.4684
arch1 <- arch.test(p1ct, lags.multi = 5)
arch1$arch.mul
##
## ARCH (multivariate)
##
## data: Residuals of VAR object p1ct
## Chi-squared = 570.14, df = 500, p-value = 0.01606
plot(stability(p1ct), nc = 2)
VEC r = 1
vec <- ca.jo(Canada[, c("rw", "prod", "e", "U")], type = "trace", ecdet = "trend", K = 3, spec = "transitory")
summary(vec)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , with linear trend in cointegration
##
## Eigenvalues (lambda):
## [1] 4.505013e-01 1.962777e-01 1.676668e-01 4.647108e-02 -2.869035e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 3 | 3.85 10.49 12.25 16.26
## r <= 2 | 18.72 22.76 25.32 30.45
## r <= 1 | 36.42 39.06 42.44 48.45
## r = 0 | 84.92 59.14 62.99 70.05
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## rw.l1 prod.l1 e.l1 U.l1 trend.l1
## rw.l1 1.00000000 1.0000000 1.0000000 1.000000 1.0000000
## prod.l1 0.54487553 -3.0021508 0.7153696 -7.173608 0.4087221
## e.l1 -0.01299605 -3.8867890 -2.0625220 -30.429074 -3.3884676
## U.l1 1.72657189 -10.2183403 -5.3124427 -49.077208 -5.1326687
## trend.l1 -0.70918872 0.6913363 -0.3643533 11.424630 0.1157125
##
## Weights W:
## (This is the loading matrix)
##
## rw.l1 prod.l1 e.l1 U.l1 trend.l1
## rw.d -0.084814510 0.048563998 -0.02368721 -0.0016583070 -5.214289e-12
## prod.d -0.011994081 0.009204887 -0.09921487 0.0020567547 -8.557489e-12
## e.d -0.015606038 -0.038019448 -0.01140202 -0.0005559337 -1.893956e-12
## U.d -0.008659911 0.020499658 0.02896325 0.0009140795 3.357504e-12
vec.r1 <- cajorls(vec, r = 1)
vec.r1
## $rlm
##
## Call:
## lm(formula = substitute(form1), data = data.mat)
##
## Coefficients:
## rw.d prod.d e.d U.d
## ect1 -0.084815 -0.011994 -0.015606 -0.008660
## constant 55.469125 8.274808 10.331308 5.687832
## rw.dl1 -0.012082 0.004707 -0.078491 0.017263
## prod.dl1 -0.074493 0.234441 0.200953 -0.138916
## e.dl1 -0.634084 -0.246544 0.821558 -0.646846
## U.dl1 0.063137 -0.979868 0.003379 -0.191125
## rw.dl2 -0.157388 -0.190264 -0.095835 0.080354
## prod.dl2 -0.251940 -0.029520 0.048273 -0.002909
## e.dl2 0.081197 -0.580473 -0.459693 -0.019741
## U.dl2 -0.230009 -0.128101 -0.103415 -0.262685
##
##
## $beta
## ect1
## rw.l1 1.00000000
## prod.l1 0.54487553
## e.l1 -0.01299605
## U.l1 1.72657189
## trend.l1 -0.70918872
vec <- ca.jo(Canada[, c("rw", "prod", "e", "U")], type = "eigen", ecdet = "trend", K = 3, spec = "transitory")
summary(vec)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , with linear trend in cointegration
##
## Eigenvalues (lambda):
## [1] 4.505013e-01 1.962777e-01 1.676668e-01 4.647108e-02 -2.869035e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 3 | 3.85 10.49 12.25 16.26
## r <= 2 | 14.87 16.85 18.96 23.65
## r <= 1 | 17.70 23.11 25.54 30.34
## r = 0 | 48.50 29.12 31.46 36.65
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## rw.l1 prod.l1 e.l1 U.l1 trend.l1
## rw.l1 1.00000000 1.0000000 1.0000000 1.000000 1.0000000
## prod.l1 0.54487553 -3.0021508 0.7153696 -7.173608 0.4087221
## e.l1 -0.01299605 -3.8867890 -2.0625220 -30.429074 -3.3884676
## U.l1 1.72657189 -10.2183403 -5.3124427 -49.077208 -5.1326687
## trend.l1 -0.70918872 0.6913363 -0.3643533 11.424630 0.1157125
##
## Weights W:
## (This is the loading matrix)
##
## rw.l1 prod.l1 e.l1 U.l1 trend.l1
## rw.d -0.084814510 0.048563998 -0.02368721 -0.0016583070 -5.214289e-12
## prod.d -0.011994081 0.009204887 -0.09921487 0.0020567547 -8.557489e-12
## e.d -0.015606038 -0.038019448 -0.01140202 -0.0005559337 -1.893956e-12
## U.d -0.008659911 0.020499658 0.02896325 0.0009140795 3.357504e-12
vec.r1 <- cajorls(vec, r = 1)
vec.r1
## $rlm
##
## Call:
## lm(formula = substitute(form1), data = data.mat)
##
## Coefficients:
## rw.d prod.d e.d U.d
## ect1 -0.084815 -0.011994 -0.015606 -0.008660
## constant 55.469125 8.274808 10.331308 5.687832
## rw.dl1 -0.012082 0.004707 -0.078491 0.017263
## prod.dl1 -0.074493 0.234441 0.200953 -0.138916
## e.dl1 -0.634084 -0.246544 0.821558 -0.646846
## U.dl1 0.063137 -0.979868 0.003379 -0.191125
## rw.dl2 -0.157388 -0.190264 -0.095835 0.080354
## prod.dl2 -0.251940 -0.029520 0.048273 -0.002909
## e.dl2 0.081197 -0.580473 -0.459693 -0.019741
## U.dl2 -0.230009 -0.128101 -0.103415 -0.262685
##
##
## $beta
## ect1
## rw.l1 1.00000000
## prod.l1 0.54487553
## e.l1 -0.01299605
## U.l1 1.72657189
## trend.l1 -0.70918872
vecm <- ca.jo(Canada[, c("prod", "e", "U", "rw")], type = "trace",
ecdet = "trend", K = 3, spec = "transitory")
SR <- matrix(NA, nrow = 4, ncol = 4)
SR[4, 2] <- 0
LR <- matrix(NA, nrow = 4, ncol = 4)
LR[1, 2:4] <- 0
LR[2:4, 4] <- 0
svec <- SVEC(vecm, LR = LR, SR = SR, r = 1, lrtest = FALSE,
boot = TRUE, runs = 100)
summary(svec)
##
## SVEC Estimation Results:
## ========================
##
## Call:
## SVEC(x = vecm, LR = LR, SR = SR, r = 1, lrtest = FALSE, boot = TRUE,
## runs = 100)
##
## Type: B-model
## Sample size: 81
## Log Likelihood: -161.838
## Number of iterations: 7
##
## Estimated contemporaneous impact matrix:
## prod e U rw
## prod 0.58402 0.07434 -0.152578 0.06900
## e -0.12029 0.26144 -0.155096 0.08978
## U 0.02526 -0.26720 0.005488 0.04982
## rw 0.11170 0.00000 0.483771 0.48791
##
## Estimated standard errors for impact matrix:
## prod e U rw
## prod 0.09126 0.10702 0.22242 0.07370
## e 0.06836 0.06127 0.17681 0.03814
## U 0.05484 0.04236 0.04912 0.02651
## rw 0.14754 0.00000 0.63775 0.08119
##
## Estimated long run impact matrix:
## prod e U rw
## prod 0.7910 0.0000 0.0000 0
## e 0.2024 0.5769 -0.4923 0
## U -0.1592 -0.3409 0.1408 0
## rw -0.1535 0.5961 -0.2495 0
##
## Estimated standard errors for long-run matrix:
## prod e U rw
## prod 0.1539 0.00000 0.0000 0
## e 0.2265 0.18255 0.5664 0
## U 0.1109 0.09259 0.1504 0
## rw 0.1908 0.16210 0.2666 0
##
## Covariance matrix of reduced form residuals (*100):
## prod e U rw
## prod 37.4642 -2.096 -0.2512 2.509
## e -2.0960 11.494 -6.9273 -4.467
## U -0.2512 -6.927 7.4544 2.978
## rw 2.5087 -4.467 2.9783 48.457
Pronostico
predictions <- predict(p1ct, n.ahead = 8, ci = 0.95)
plot(predictions)
