Homework 6, Problem 2
Hassan
April 22, 2017
Structural Vector Autoregression (VAR) Models
library(Quandl)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(zoo)
library(xts)
library(dygraphs)
library(knitr)
library(forecast)library(urca)
library(vars)## Loading required package: MASS## Loading required package: strucchange## Loading required package: sandwich## Loading required package: lmtestQuandl.api_key('CEFP3eWxEJwr_uUP9a2D')
RO <- Quandl("FRED/OPHNFB", type="zoo")
HOP <- Quandl("FRED/HOANBS", type="zoo" )(a) ERS unit root test
v1=100*(log(RO))
v2=100*(log(HOP))
par(mfrow=c(1,2), cex=0.6)plot(v1, xlab="years", ylab="value", main="Real Output Per Hour of All Persons")
plot(v2, xlab="years", ylab="value", main="Hours of All Persons")
v1.ers1 <- ur.ers(v1, type="P-test", model="trend")
summary(v1.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 38.597
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89v2.ers1 <- ur.ers(v2, type="P-test", model="trend")
summary(v2.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 6.2319
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89we find the non-stationary results
dv1=diff(v1)
dv2=diff(v2)dv1.ers1 <- ur.ers(dv1, type="P-test", model="trend")
summary(dv1.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.3185
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dv2.ers1 <- ur.ers(dv2, type="P-test", model="trend")
summary(dv2.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.3567
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Both time series are stationary in first differences as found as result of the ERS test.
(b) bivariate reduced form of VAR
y.Q <- cbind(dv1, dv2)
y.Q <- window(y.Q, start="1947 Q2", end="2016 Q4")
VARselect(y.Q, lag.max=8, type="const")## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 2 2 3
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) -1.1343721 -1.2046709 -1.2068439 -1.1847725 -1.1836732 -1.1606766
## HQ(n) -1.1023510 -1.1513022 -1.1321279 -1.0887090 -1.0662622 -1.0219182
## SC(n) -1.0546204 -1.0717513 -1.0207566 -0.9455174 -0.8912503 -0.8150859
## FPE(n) 0.3216246 0.2997932 0.2991468 0.3058307 0.3061794 0.3133203
## 7 8
## AIC(n) -1.1317826 -1.1491281
## HQ(n) -0.9716768 -0.9676748
## SC(n) -0.7330241 -0.6972017
## FPE(n) 0.3225310 0.3170175By using the AIC criteria we have 3 lags.
var1 <- VAR(y.Q, p=3, type="const")
summary(var1)##
## VAR Estimation Results:
## =========================
## Endogenous variables: dv1, dv2
## Deterministic variables: const
## Sample size: 276
## Log Likelihood: -602.811
## Roots of the characteristic polynomial:
## 0.7387 0.7387 0.4311 0.4311 0.4069 0.4069
## Call:
## VAR(y = y.Q, p = 3, type = "const")
##
##
## Estimation results for equation dv1:
## ====================================
## dv1 = dv1.l1 + dv2.l1 + dv1.l2 + dv2.l2 + dv1.l3 + dv2.l3 + const
##
## Estimate Std. Error t value Pr(>|t|)
## dv1.l1 -0.06110 0.05921 -1.032 0.30307
## dv2.l1 0.06312 0.07128 0.886 0.37667
## dv1.l2 0.05149 0.05596 0.920 0.35840
## dv2.l2 -0.19658 0.08401 -2.340 0.02002 *
## dv1.l3 0.00524 0.05506 0.095 0.92425
## dv2.l3 -0.19238 0.07313 -2.631 0.00901 **
## const 0.62337 0.07462 8.353 3.56e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.793 on 269 degrees of freedom
## Multiple R-Squared: 0.1176, Adjusted R-squared: 0.09788
## F-statistic: 5.973 on 6 and 269 DF, p-value: 7.068e-06
##
##
## Estimation results for equation dv2:
## ====================================
## dv2 = dv1.l1 + dv2.l1 + dv1.l2 + dv2.l2 + dv1.l3 + dv2.l3 + const
##
## Estimate Std. Error t value Pr(>|t|)
## dv1.l1 0.10659 0.05068 2.103 0.0364 *
## dv2.l1 0.62592 0.06101 10.260 <2e-16 ***
## dv1.l2 0.10479 0.04790 2.188 0.0296 *
## dv2.l2 -0.01811 0.07190 -0.252 0.8014
## dv1.l3 0.08847 0.04713 1.877 0.0616 .
## dv2.l3 -0.04412 0.06259 -0.705 0.4815
## const -0.02740 0.06387 -0.429 0.6683
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6787 on 269 degrees of freedom
## Multiple R-Squared: 0.4387, Adjusted R-squared: 0.4261
## F-statistic: 35.03 on 6 and 269 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## dv1 dv2
## dv1 0.62881 0.07009
## dv2 0.07009 0.46064
##
## Correlation matrix of residuals:
## dv1 dv2
## dv1 1.0000 0.1302
## dv2 0.1302 1.0000(c) Blanchard and Quah approach to obtain SVAR model
apply(y.Q, 2, mean)## dv1 dv2
## 0.5273581 0.3047287y.svar <- sweep(y.Q, 2, apply(y.Q, 2, mean))
apply(y.svar, 2, mean)## dv1 dv2
## -3.886224e-18 1.758137e-17plot(y.svar)
estimating reduced form VAR model
VARselect(y.svar, lag.max=8)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 2 2 3
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) -1.1343721 -1.2046709 -1.2068439 -1.1847725 -1.1836732 -1.1606766
## HQ(n) -1.1023510 -1.1513022 -1.1321279 -1.0887090 -1.0662622 -1.0219182
## SC(n) -1.0546204 -1.0717513 -1.0207566 -0.9455174 -0.8912503 -0.8150859
## FPE(n) 0.3216246 0.2997932 0.2991468 0.3058307 0.3061794 0.3133203
## 7 8
## AIC(n) -1.1317826 -1.1491281
## HQ(n) -0.9716768 -0.9676748
## SC(n) -0.7330241 -0.6972017
## FPE(n) 0.3225310 0.3170175myVAR <- VAR(y.svar, ic="SC", lag.max=8)
summary(myVAR)##
## VAR Estimation Results:
## =========================
## Endogenous variables: dv1, dv2
## Deterministic variables: const
## Sample size: 277
## Log Likelihood: -617.473
## Roots of the characteristic polynomial:
## 0.6086 0.6086 0.2946 0.2946
## Call:
## VAR(y = y.svar, lag.max = 8, ic = "SC")
##
##
## Estimation results for equation dv1:
## ====================================
## dv1 = dv1.l1 + dv2.l1 + dv1.l2 + dv2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## dv1.l1 -0.073370 0.057108 -1.285 0.2000
## dv2.l1 0.076501 0.073177 1.045 0.2968
## dv1.l2 0.103478 0.056191 1.842 0.0666 .
## dv2.l2 -0.334295 0.072971 -4.581 7.05e-06 ***
## const 0.006032 0.049350 0.122 0.9028
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.8213 on 272 degrees of freedom
## Multiple R-Squared: 0.1027, Adjusted R-squared: 0.08949
## F-statistic: 7.782 on 4 and 272 DF, p-value: 5.966e-06
##
##
## Estimation results for equation dv2:
## ====================================
## dv2 = dv1.l1 + dv2.l1 + dv1.l2 + dv2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## dv1.l1 0.110054 0.047441 2.320 0.0211 *
## dv2.l1 0.643941 0.060790 10.593 <2e-16 ***
## dv1.l2 0.110611 0.046679 2.370 0.0185 *
## dv2.l2 -0.046616 0.060619 -0.769 0.4426
## const 0.001089 0.040997 0.027 0.9788
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6823 on 272 degrees of freedom
## Multiple R-Squared: 0.4275, Adjusted R-squared: 0.4191
## F-statistic: 50.78 on 4 and 272 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## dv1 dv2
## dv1 0.67459 0.08419
## dv2 0.08419 0.46554
##
## Correlation matrix of residuals:
## dv1 dv2
## dv1 1.0000 0.1502
## dv2 0.1502 1.0000Blanchard-Quah long run restriction - matrix is going to be restricted to 0
mySVAR <- BQ(myVAR)
summary(mySVAR)##
## SVAR Estimation Results:
## ========================
##
## Call:
## BQ(x = myVAR)
##
## Type: Blanchard-Quah
## Sample size: 277
## Log Likelihood: -622.518
##
## Estimated contemporaneous impact matrix:
## dv1 dv2
## dv1 0.7131 0.4075
## dv2 -0.2457 0.6365
##
## Estimated identified long run impact matrix:
## dv1 dv2
## dv1 0.7833 0.000
## dv2 -0.1809 1.581
##
## Covariance matrix of reduced form residuals (*100):
## dv1 dv2
## dv1 67.459 8.419
## dv2 8.419 46.554(d) contemporaneous impact and the long run impact matrices for the SVAR
1) Contemporaneous impact:
Positive one SD technology shock will increases the Real Output Per Hour of All Persons by 0.7131% and will lowers the Hours of All Persons by 0.2457%.A negative one SD non-technology shock will increase Real Output Per Hour of All Persons by 0.4075%, and will increase the Hours of All Persons by 0.6365%. The result of a positive shock was expected. Though, the result of a negative shock is unexpected as it shows an increase in productivity despite a negative technological shock.
2) Long run impact:
The long run cumulative effect of any non-technology shock on Real Output Per Hour of All Persons is found as 0. The long run cumulative effect of a single positive one SD technology shock on Real Output Per Hour increases it by 0.7833% and decreases Hours of All Persons by 0.1809%. These results are align with the prevailing theories.
(e) Ploting
myIRF.c <- irf(mySVAR, n.ahead=12, ci=.9, cumulative=TRUE)
myIRF <- irf(mySVAR, n.ahead=12, ci=.9)
myIRF.c$irf[[2]] <- -myIRF.c$irf[[2]]
myIRF.c$Lower[[2]] <- -myIRF.c$Lower[[2]]
myIRF.c$Upper[[2]] <- -myIRF.c$Upper[[2]]
myIRF.c$irf[[2]] <- -myIRF.c$irf[[2]]
myIRF.c$Lower[[2]] <- -myIRF.c$Lower[[2]]
myIRF.c$Upper[[2]] <- -myIRF.c$Upper[[2]]
par(mfrow=c(2,2), cex=.6, mar=c(4,4,2,1))
source("plotIRF.R")
plotIRF(myIRF.c, lwd=2.5, ask=FALSE, vnames="dv1", vlabels="Real Output Per Hour", slabels=c("technology shock","non-technology shock"))
plotIRF(myIRF.c, lwd=2.5, ask=FALSE, vnames="dv2", vlabels="Hours of All", slabels=c("technology shock","non-technology shock"))
From IRF graphs we see that when a positive technological shock is applied, it’d have a long-term positive effect on labor productivity, however, a negative effect on number of working hours in a country. The most important thing is that this effect does not dampens over time. At first, productivity increases immediately, goes a little down and then gets to the initially increased level and stays stable at this level over time. We see that the number of hours shows the same behavior and stabilizes over time at the initially changed level. Although, a non-technological shock increases productivity immediately, but it dampens to the pre-shock level after the 5th period. Unlike productivity, labor hours do not get back to the pre-shock level over time. It increases immediately and then gradually keeps increasing until reaching a stable “high” level over time. #(f) IRF and Gali (1999) Our IRFs look similarly to Gali (1999) with some variations. The IRF for technological shock on productivity is the same, although Gali`s non-technological shock brings productivity even to the lower level than pre-shock. However, eventually it comes to the initial level as in our case. Both tech and non-tech shocks appear to be the same as in our model, with minor discrepancies.
(g) FEVD for SVAR
par(mar=c(4,5,2,1))
plot( fevd(mySVAR, n.ahead=40) ,addbars=3 )
mySVAR.fevd <- fevd(mySVAR, n.ahead=10)
mySVAR.fevd[[1]][c(1,3,6,10),]## dv1 dv2
## [1,] 0.7538244 0.2461756
## [2,] 0.7437308 0.2562692
## [3,] 0.7180470 0.2819530
## [4,] 0.7173526 0.2826474mySVAR.fevd[[2]][c(1,3,6,10),]## dv1 dv2
## [1,] 0.12966358 0.8703364
## [2,] 0.08723805 0.9127619
## [3,] 0.08738878 0.9126112
## [4,] 0.08757486 0.9124251As we can see from FEVDs, the difference between long-run and short-run explanations of shocks do not differ much for both shocks. ##The productivity shock in SR: productivity fluctuations is explained for 75% by its own variable and for 25% by labor hours. ## In LR this ratio changes to 71% and 29% respectively. ##The hours shock in SR: labor hours fluctuations are explained for 87% by its own variable and for 13% by productivity. ##Beginning from the 2nd period the ratio changes to 91% and 9% respectively and keeps at this proportion in the LR