Laboratorio No. 6
Ricardo Cruz 1048220
2023-03-14
Primero se carga las librerías a utilizar y la serie de tiempo.
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: lmtestlibrary(fpp2)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## ── Attaching packages ────────────────────────────────────────────── fpp2 2.5 ──## ✔ ggplot2 3.4.0 ✔ fma 2.5
## ✔ forecast 8.20 ✔ expsmooth 2.3## library(TSA)## Registered S3 methods overwritten by 'TSA':
## method from
## fitted.Arima forecast
## plot.Arima forecast##
## Attaching package: 'TSA'## The following objects are masked from 'package:stats':
##
## acf, arima## The following object is masked from 'package:utils':
##
## tarseries<-uschange
autoplot(uschange[,c(2,5)])
ts.plot(series[,c(2,5)], xlab="Tiempo",col=c(1,3))
Se escogen las variables con mayor causalidad así como Ingreso y Desempleo y se extraen de la serie de tiempo de uschange.
Búsqueda de parámetros y elección del modelo candidato
a <- VARselect(uschange[,c(2,5)], lag.max=15,type="const")
a$selection## AIC(n) HQ(n) SC(n) FPE(n)
## 8 1 1 8La función Varselect sufiere que el mejor modelo candidato es de 8 lags, pero al utilizar 8 lags el modelo presenta autocorrelación e incluye variables que no son significativas al modelo. por esa razón usaremos 5 lags.
modelo1<-VAR(uschange[,c(2,5)],p=5,type=c("const"))
modelo_s<-summary(modelo1)summary(modelo1,equation="Income")##
## VAR Estimation Results:
## =========================
## Endogenous variables: Income, Unemployment
## Deterministic variables: const
## Sample size: 182
## Log Likelihood: -262.292
## Roots of the characteristic polynomial:
## 0.7776 0.7776 0.7661 0.7661 0.7488 0.7488 0.672 0.672 0.6364 0.5646
## Call:
## VAR(y = uschange[, c(2, 5)], p = 5, type = c("const"))
##
##
## Estimation results for equation Income:
## =======================================
## Income = Income.l1 + Unemployment.l1 + Income.l2 + Unemployment.l2 + Income.l3 + Unemployment.l3 + Income.l4 + Unemployment.l4 + Income.l5 + Unemployment.l5 + const
##
## Estimate Std. Error t value Pr(>|t|)
## Income.l1 -0.13069 0.07628 -1.713 0.08848 .
## Unemployment.l1 -0.47812 0.22736 -2.103 0.03693 *
## Income.l2 0.08289 0.07790 1.064 0.28883
## Unemployment.l2 0.39251 0.23868 1.644 0.10191
## Income.l3 0.03524 0.07602 0.464 0.64351
## Unemployment.l3 -0.65814 0.24414 -2.696 0.00773 **
## Income.l4 -0.09359 0.07556 -1.239 0.21717
## Unemployment.l4 -0.14117 0.24531 -0.575 0.56571
## Income.l5 -0.12761 0.07471 -1.708 0.08944 .
## Unemployment.l5 0.39764 0.22214 1.790 0.07522 .
## const 0.87038 0.14889 5.846 2.49e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.8936 on 171 degrees of freedom
## Multiple R-Squared: 0.1347, Adjusted R-squared: 0.08413
## F-statistic: 2.663 on 10 and 171 DF, p-value: 0.004781
##
##
##
## Covariance matrix of residuals:
## Income Unemployment
## Income 0.7985 -0.05280
## Unemployment -0.0528 0.09033
##
## Correlation matrix of residuals:
## Income Unemployment
## Income 1.0000 -0.1966
## Unemployment -0.1966 1.0000summary(modelo1,equation="Unemployment")##
## VAR Estimation Results:
## =========================
## Endogenous variables: Income, Unemployment
## Deterministic variables: const
## Sample size: 182
## Log Likelihood: -262.292
## Roots of the characteristic polynomial:
## 0.7776 0.7776 0.7661 0.7661 0.7488 0.7488 0.672 0.672 0.6364 0.5646
## Call:
## VAR(y = uschange[, c(2, 5)], p = 5, type = c("const"))
##
##
## Estimation results for equation Unemployment:
## =============================================
## Unemployment = Income.l1 + Unemployment.l1 + Income.l2 + Unemployment.l2 + Income.l3 + Unemployment.l3 + Income.l4 + Unemployment.l4 + Income.l5 + Unemployment.l5 + const
##
## Estimate Std. Error t value Pr(>|t|)
## Income.l1 -0.050174 0.025657 -1.956 0.05215 .
## Unemployment.l1 0.436087 0.076472 5.703 5.08e-08 ***
## Income.l2 -0.004766 0.026202 -0.182 0.85589
## Unemployment.l2 0.156143 0.080280 1.945 0.05342 .
## Income.l3 -0.023081 0.025569 -0.903 0.36796
## Unemployment.l3 0.067925 0.082118 0.827 0.40930
## Income.l4 -0.012685 0.025414 -0.499 0.61833
## Unemployment.l4 -0.183030 0.082510 -2.218 0.02785 *
## Income.l5 0.068785 0.025128 2.737 0.00685 **
## Unemployment.l5 0.038701 0.074718 0.518 0.60516
## const 0.012796 0.050080 0.256 0.79863
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.3006 on 171 degrees of freedom
## Multiple R-Squared: 0.3665, Adjusted R-squared: 0.3295
## F-statistic: 9.893 on 10 and 171 DF, p-value: 5.479e-13
##
##
##
## Covariance matrix of residuals:
## Income Unemployment
## Income 0.7985 -0.05280
## Unemployment -0.0528 0.09033
##
## Correlation matrix of residuals:
## Income Unemployment
## Income 1.0000 -0.1966
## Unemployment -0.1966 1.0000Hay ciertos lags de las variables que no son significativos al modelo, por lo que se podria considerar en eliminarlos del modelo.
Pruebas del Modelo
serial.test(modelo1, lags.pt=10, type="PT.asymptotic")##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object modelo1
## Chi-squared = 31.773, df = 20, p-value = 0.04577El modelo no presenta autocorrelación porque según el resultado de la prueba de PortManteu Test > 0.05 Autocorrelación. Básicamente el p-value tiene que ser mayor para pasar la prueba.
roots(modelo1)## [1] 0.7775660 0.7775660 0.7660914 0.7660914 0.7488061 0.7488061 0.6720236
## [8] 0.6720236 0.6364359 0.5646411El modelo no tiene raices unitarias por lo que es estacionaria. Para pasar esta prueba todos los números tienen que ser < 1.
normality.test(modelo1, multivariate.only=FALSE)## $Income
##
## JB-Test (univariate)
##
## data: Residual of Income equation
## Chi-squared = 195.51, df = 2, p-value < 2.2e-16
##
##
## $Unemployment
##
## JB-Test (univariate)
##
## data: Residual of Unemployment equation
## Chi-squared = 33.197, df = 2, p-value = 6.184e-08
##
##
## $JB
##
## JB-Test (multivariate)
##
## data: Residuals of VAR object modelo1
## Chi-squared = 216.79, df = 4, p-value < 2.2e-16
##
##
## $Skewness
##
## Skewness only (multivariate)
##
## data: Residuals of VAR object modelo1
## Chi-squared = 7.8916, df = 2, p-value = 0.01934
##
##
## $Kurtosis
##
## Kurtosis only (multivariate)
##
## data: Residuals of VAR object modelo1
## Chi-squared = 208.9, df = 2, p-value < 2.2e-16Si en la prueba de Jarque-Bera el p-value < 0.05 presenta normalidad en los residuos. Para este modelo el resultado presenta normalidad.
stab <-stability(modelo1, type = "OLS-CUSUM")
par(mar=c(1,1,1,1))
plot(stab)
El modelo no presenta quiebres estructurales, ambas gráficas no salen de los parámetros tolerados.
Causalidad de granger
granger < 0.05 para que exista causalidad o que tiene relación
GrangerIncome <-causality(modelo1, cause = 'Income')
GrangerIncome## $Granger
##
## Granger causality H0: Income do not Granger-cause Unemployment
##
## data: VAR object modelo1
## F-Test = 2.8057, df1 = 5, df2 = 342, p-value = 0.01687
##
##
## $Instant
##
## H0: No instantaneous causality between: Income and Unemployment
##
## data: VAR object modelo1
## Chi-squared = 6.7734, df = 1, p-value = 0.009253GrangerUnemployment <-causality(modelo1, cause = 'Unemployment')
GrangerUnemployment## $Granger
##
## Granger causality H0: Unemployment do not Granger-cause Income
##
## data: VAR object modelo1
## F-Test = 3.7264, df1 = 5, df2 = 342, p-value = 0.002681
##
##
## $Instant
##
## H0: No instantaneous causality between: Unemployment and Income
##
## data: VAR object modelo1
## Chi-squared = 6.7734, df = 1, p-value = 0.009253Ambas variables presentan causalidad.
Respuesta de impulso
UnemploymentIRF <- irf(modelo1, impulse = "Income", response="Unemployment", n.ahead = 20, boot = T )
plot(UnemploymentIRF, ylab = "Unemployment", main = "Shock desde Income")
IncomeIRF <- irf(modelo1, impulse = "Unemployment", response="Income", n.ahead = 20, boot = T )
plot(IncomeIRF, ylab = "Income", main = "Shock desde Unemployment")
Muestra como ambas variables en sus gráficas se estabilizan y vuelven a la normalidad después del shock.
Descomposición de la varianza
FEVD1 <- fevd(modelo1, n.ahead = 10)
plot(FEVD1)
En el ingreso se observa que aunque en menor medida tiene relación con el desempleo. De igual forma para el desempleo.
Predicción
fore<-predict(modelo1, n.ahead = 10, ci=0.95)
fanchart(fore)