Modelos VAR
Juan F
29 de agosto de 2017
Modelo de Vectores Autorregresivos (VAR: Vector autoregressive models)
Definición
Dado un conjunto de K variables endogenas \((y_{1t},...,y_{kt},...,y_{Kt})\) for k = 1,…K. The VAR(p)-process is then defined as:
\(y_t = (A_1y_{t-1}+...+A_py_{t-p},...,yKt)\) for k = 1,…K. The VAR(p)-process is then defined as:
library(vars)## 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: lmtestdata(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 = "")
Para seleccionar el Rezago del modelo se utiliza la siguiente función
VARselect(Canada, lag.max = 5, type = "const")## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 2 2 3
##
## $criteria
## 1 2 3 4 5
## AIC(n) -5.817851996 -6.35093701 -6.397756084 -6.145942174 -5.926500201
## HQ(n) -5.577529641 -5.91835677 -5.772917961 -5.328846166 -4.917146309
## SC(n) -5.217991781 -5.27118862 -4.838119523 -4.106417440 -3.407087295
## FPE(n) 0.002976003 0.00175206 0.001685528 0.002201523 0.002811116var.2c <- VAR(Canada, p = 2, type = "const")
summary(var.2c)##
## VAR Estimation Results:
## =========================
## Endogenous variables: e, prod, rw, U
## Deterministic variables: const
## Sample size: 82
## Log Likelihood: -175.819
## Roots of the characteristic polynomial:
## 0.995 0.9081 0.9081 0.7381 0.7381 0.1856 0.1429 0.1429
## Call:
## VAR(y = Canada, p = 2, type = "const")
##
##
## Estimation results for equation e:
## ==================================
## e = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## e.l1 1.638e+00 1.500e-01 10.918 < 2e-16 ***
## prod.l1 1.673e-01 6.114e-02 2.736 0.00780 **
## rw.l1 -6.312e-02 5.524e-02 -1.143 0.25692
## U.l1 2.656e-01 2.028e-01 1.310 0.19444
## e.l2 -4.971e-01 1.595e-01 -3.116 0.00262 **
## prod.l2 -1.017e-01 6.607e-02 -1.539 0.12824
## rw.l2 3.844e-03 5.552e-02 0.069 0.94499
## U.l2 1.327e-01 2.073e-01 0.640 0.52418
## const -1.370e+02 5.585e+01 -2.453 0.01655 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.3628 on 73 degrees of freedom
## Multiple R-Squared: 0.9985, Adjusted R-squared: 0.9984
## F-statistic: 6189 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation prod:
## =====================================
## prod = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## e.l1 -0.17277 0.26977 -0.640 0.52390
## prod.l1 1.15043 0.10995 10.464 3.57e-16 ***
## rw.l1 0.05130 0.09934 0.516 0.60710
## U.l1 -0.47850 0.36470 -1.312 0.19362
## e.l2 0.38526 0.28688 1.343 0.18346
## prod.l2 -0.17241 0.11881 -1.451 0.15104
## rw.l2 -0.11885 0.09985 -1.190 0.23778
## U.l2 1.01592 0.37285 2.725 0.00805 **
## const -166.77552 100.43388 -1.661 0.10109
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6525 on 73 degrees of freedom
## Multiple R-Squared: 0.9787, Adjusted R-squared: 0.9764
## F-statistic: 419.3 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation rw:
## ===================================
## rw = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## e.l1 -0.268833 0.322619 -0.833 0.407
## prod.l1 -0.081065 0.131487 -0.617 0.539
## rw.l1 0.895478 0.118800 7.538 1.04e-10 ***
## U.l1 0.012130 0.436149 0.028 0.978
## e.l2 0.367849 0.343087 1.072 0.287
## prod.l2 -0.005181 0.142093 -0.036 0.971
## rw.l2 0.052677 0.119410 0.441 0.660
## U.l2 -0.127708 0.445892 -0.286 0.775
## const -33.188339 120.110525 -0.276 0.783
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7803 on 73 degrees of freedom
## Multiple R-Squared: 0.9989, Adjusted R-squared: 0.9987
## F-statistic: 8009 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation U:
## ==================================
## U = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## e.l1 -0.58076 0.11563 -5.023 3.49e-06 ***
## prod.l1 -0.07812 0.04713 -1.658 0.101682
## rw.l1 0.01866 0.04258 0.438 0.662463
## U.l1 0.61893 0.15632 3.959 0.000173 ***
## e.l2 0.40982 0.12296 3.333 0.001352 **
## prod.l2 0.05212 0.05093 1.023 0.309513
## rw.l2 0.04180 0.04280 0.977 0.331928
## U.l2 -0.07117 0.15981 -0.445 0.657395
## const 149.78056 43.04810 3.479 0.000851 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2797 on 73 degrees of freedom
## Multiple R-Squared: 0.9726, Adjusted R-squared: 0.9696
## F-statistic: 324 on 8 and 73 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## e prod rw U
## e 0.131635 -0.007469 -0.04210 -0.06909
## prod -0.007469 0.425711 0.06461 0.01392
## rw -0.042099 0.064613 0.60886 0.03422
## U -0.069087 0.013923 0.03422 0.07821
##
## Correlation matrix of residuals:
## e prod rw U
## e 1.00000 -0.03155 -0.1487 -0.6809
## prod -0.03155 1.00000 0.1269 0.0763
## rw -0.14870 0.12691 1.0000 0.1568
## U -0.68090 0.07630 0.1568 1.0000plot(var.2c)
Raices del modelo
roots(var.2c)## [1] 0.9950338 0.9081062 0.9081062 0.7380565 0.7380565 0.1856381 0.1428889
## [8] 0.1428889Las dos primeras están cercanas a 1.
Respuesta al impulso
Canada2 <- Canada[, c(3, 1, 4, 2)]
var.2c.alt <- VAR(Canada2, p = 2, type = "const")
summary(var.2c.alt)##
## VAR Estimation Results:
## =========================
## Endogenous variables: rw, e, U, prod
## Deterministic variables: const
## Sample size: 82
## Log Likelihood: -175.819
## Roots of the characteristic polynomial:
## 0.995 0.9081 0.9081 0.7381 0.7381 0.1856 0.1429 0.1429
## Call:
## VAR(y = Canada2, p = 2, type = "const")
##
##
## Estimation results for equation rw:
## ===================================
## rw = rw.l1 + e.l1 + U.l1 + prod.l1 + rw.l2 + e.l2 + U.l2 + prod.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## rw.l1 0.895478 0.118800 7.538 1.04e-10 ***
## e.l1 -0.268833 0.322619 -0.833 0.407
## U.l1 0.012130 0.436149 0.028 0.978
## prod.l1 -0.081065 0.131487 -0.617 0.539
## rw.l2 0.052677 0.119410 0.441 0.660
## e.l2 0.367849 0.343087 1.072 0.287
## U.l2 -0.127708 0.445892 -0.286 0.775
## prod.l2 -0.005181 0.142093 -0.036 0.971
## const -33.188339 120.110525 -0.276 0.783
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7803 on 73 degrees of freedom
## Multiple R-Squared: 0.9989, Adjusted R-squared: 0.9987
## F-statistic: 8009 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation e:
## ==================================
## e = rw.l1 + e.l1 + U.l1 + prod.l1 + rw.l2 + e.l2 + U.l2 + prod.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## rw.l1 -6.312e-02 5.524e-02 -1.143 0.25692
## e.l1 1.638e+00 1.500e-01 10.918 < 2e-16 ***
## U.l1 2.656e-01 2.028e-01 1.310 0.19444
## prod.l1 1.673e-01 6.114e-02 2.736 0.00780 **
## rw.l2 3.844e-03 5.552e-02 0.069 0.94499
## e.l2 -4.971e-01 1.595e-01 -3.116 0.00262 **
## U.l2 1.327e-01 2.073e-01 0.640 0.52418
## prod.l2 -1.017e-01 6.607e-02 -1.539 0.12824
## const -1.370e+02 5.585e+01 -2.453 0.01655 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.3628 on 73 degrees of freedom
## Multiple R-Squared: 0.9985, Adjusted R-squared: 0.9984
## F-statistic: 6189 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation U:
## ==================================
## U = rw.l1 + e.l1 + U.l1 + prod.l1 + rw.l2 + e.l2 + U.l2 + prod.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## rw.l1 0.01866 0.04258 0.438 0.662463
## e.l1 -0.58076 0.11563 -5.023 3.49e-06 ***
## U.l1 0.61893 0.15632 3.959 0.000173 ***
## prod.l1 -0.07812 0.04713 -1.658 0.101682
## rw.l2 0.04180 0.04280 0.977 0.331928
## e.l2 0.40982 0.12296 3.333 0.001352 **
## U.l2 -0.07117 0.15981 -0.445 0.657395
## prod.l2 0.05212 0.05093 1.023 0.309513
## const 149.78056 43.04810 3.479 0.000851 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.2797 on 73 degrees of freedom
## Multiple R-Squared: 0.9726, Adjusted R-squared: 0.9696
## F-statistic: 324 on 8 and 73 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation prod:
## =====================================
## prod = rw.l1 + e.l1 + U.l1 + prod.l1 + rw.l2 + e.l2 + U.l2 + prod.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## rw.l1 0.05130 0.09934 0.516 0.60710
## e.l1 -0.17277 0.26977 -0.640 0.52390
## U.l1 -0.47850 0.36470 -1.312 0.19362
## prod.l1 1.15043 0.10995 10.464 3.57e-16 ***
## rw.l2 -0.11885 0.09985 -1.190 0.23778
## e.l2 0.38526 0.28688 1.343 0.18346
## U.l2 1.01592 0.37285 2.725 0.00805 **
## prod.l2 -0.17241 0.11881 -1.451 0.15104
## const -166.77552 100.43388 -1.661 0.10109
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6525 on 73 degrees of freedom
## Multiple R-Squared: 0.9787, Adjusted R-squared: 0.9764
## F-statistic: 419.3 on 8 and 73 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## rw e U prod
## rw 0.60886 -0.042099 0.03422 0.064613
## e -0.04210 0.131635 -0.06909 -0.007469
## U 0.03422 -0.069087 0.07821 0.013923
## prod 0.06461 -0.007469 0.01392 0.425711
##
## Correlation matrix of residuals:
## rw e U prod
## rw 1.0000 -0.14870 0.1568 0.12691
## e -0.1487 1.00000 -0.6809 -0.03155
## U 0.1568 -0.68090 1.0000 0.07630
## prod 0.1269 -0.03155 0.0763 1.00000irf.rw.eU <- irf(var.2c.alt, impulse = "rw", response = c("e","U"), boot = TRUE)
names(irf.rw.eU)## [1] "irf" "Lower" "Upper" "response" "impulse"
## [6] "ortho" "cumulative" "runs" "ci" "boot"
## [11] "model"plot(irf.rw.eU)