Homework 5 Problem 2
Direction
Transform the data and test using KPSS tests.
IND <- Quandl("FRED/INDPRO", type="ts")
CPI <- Quandl("FRED/CPIAUCSL", type="ts")Data transformation:
y1t <- log(IND)
y2t <- log(CPI)
dy1t <- diff(y1t)
dy2t <- diff(y2t)Now we have the log of industrial production index \(y_{1,t}\), log of consumer price index \(y_{2,t}\), also the first difference for both \(y_{1,t}\) and \(y_{2,t}\), i.e. \(\Delta y_{1,t} = y_{1,t}\) \(\Delta y_{2,t} = y_{2,t}\)
(a) Unit Root Test
ADF test for nonstationarity for \(y_{1,t}\) and \(y_{2,t}\):
y1t.urdf <- ur.df(y1t, selectlags = "BIC")
summary(y1t.urdf)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.098543 -0.005566 -0.000158 0.005584 0.124578
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 0.0003092 0.0001403 2.204 0.0277 *
z.diff.lag 0.5096825 0.0251805 20.241 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01656 on 1162 degrees of freedom
Multiple R-squared: 0.2711, Adjusted R-squared: 0.2698
F-statistic: 216.1 on 2 and 1162 DF, p-value: < 2.2e-16
Value of test-statistic is: 2.204
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62y2t.urdf <- ur.df(y2t, selectlags = "BIC")
summary(y2t.urdf)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.014131 -0.001435 -0.000081 0.001332 0.016985
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 2.580e-04 2.889e-05 8.928 <2e-16 ***
z.diff.lag 5.930e-01 2.806e-02 21.137 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002864 on 826 degrees of freedom
Multiple R-squared: 0.6008, Adjusted R-squared: 0.5998
F-statistic: 621.4 on 2 and 826 DF, p-value: < 2.2e-16
Value of test-statistic is: 8.9284
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62Comparing the \(\tau\) statististics for both \(y_{1,t}\) and \(y_{2,t}\) that are less that its critical value in absolute term, I conclude \(y_{1,t}\) and \(y_{2,t}\) are non-stationary.
ADF test for\(\Delta y_{1,t}\) and \(\Delta y_{2,t}\):
dy1t.urdf <- ur.df(dy1t, selectlags = "BIC")
summary(dy1t.urdf)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.098018 -0.004722 0.001089 0.006772 0.124297
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 -0.50498 0.02876 -17.557 <2e-16 ***
z.diff.lag 0.04342 0.02926 1.484 0.138
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01659 on 1161 degrees of freedom
Multiple R-squared: 0.244, Adjusted R-squared: 0.2427
F-statistic: 187.3 on 2 and 1161 DF, p-value: < 2.2e-16
Value of test-statistic is: -17.5572
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62dy2t.urdf <- ur.df(dy2t, selectlags = "BIC")
summary(dy1t.urdf)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.098018 -0.004722 0.001089 0.006772 0.124297
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 -0.50498 0.02876 -17.557 <2e-16 ***
z.diff.lag 0.04342 0.02926 1.484 0.138
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01659 on 1161 degrees of freedom
Multiple R-squared: 0.244, Adjusted R-squared: 0.2427
F-statistic: 187.3 on 2 and 1161 DF, p-value: < 2.2e-16
Value of test-statistic is: -17.5572
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62Comparing the \(\tau\) statististics for both \(\Delta y_{1,t}\) and \(\Delta y_{2,t}\) that are greater that its critical value in absolute term, I conclude that \(\Delta y_{1,t}\) and \(\Delta y_{2,t}\) are stationary. We will use \(\Delta y_{1,t}\) and \(\Delta y_{2,t}\) for the VAR estimation.
(b) VAR Estimation
In estimating the reduced form VAR model, we’ll let vars to automatically choose the appropriate lags using post-estimation info criterions. We can estimate the VAR model using the following code:
y <- cbind(dy1t, dy2t)
y <- na.trim(y)
var <- VAR(y, lag.max=8, ic="AIC", type="const")
var
VAR Estimation Results:
=======================
Estimated coefficients for equation dy1t:
=========================================
Call:
dy1t = dy1t.l1 + dy2t.l1 + dy1t.l2 + dy2t.l2 + dy1t.l3 + dy2t.l3 + dy1t.l4 + dy2t.l4 + dy1t.l5 + dy2t.l5 + dy1t.l6 + dy2t.l6 + dy1t.l7 + dy2t.l7 + const
dy1t.l1 dy2t.l1 dy1t.l2 dy2t.l2 dy1t.l3
0.3119228703 0.2181774249 0.0952428204 -0.0930330560 0.0809581628
dy2t.l3 dy1t.l4 dy2t.l4 dy1t.l5 dy2t.l5
-0.0006986961 0.0404475882 -0.1110367505 -0.0773211659 -0.0249262474
dy1t.l6 dy2t.l6 dy1t.l7 dy2t.l7 const
0.0133328809 -0.2069329954 0.0297572787 -0.1086330658 0.0021726752
Estimated coefficients for equation dy2t:
=========================================
Call:
dy2t = dy1t.l1 + dy2t.l1 + dy1t.l2 + dy2t.l2 + dy1t.l3 + dy2t.l3 + dy1t.l4 + dy2t.l4 + dy1t.l5 + dy2t.l5 + dy1t.l6 + dy2t.l6 + dy1t.l7 + dy2t.l7 + const
dy1t.l1 dy2t.l1 dy1t.l2 dy2t.l2 dy1t.l3
-0.0058262613 0.4433806900 0.0290656163 0.0439567059 0.0067641397
dy2t.l3 dy1t.l4 dy2t.l4 dy1t.l5 dy2t.l5
0.0681148385 -0.0021679389 0.0404206993 -0.0199103348 0.0548126410
dy1t.l6 dy2t.l6 dy1t.l7 dy2t.l7 const
0.0241330113 0.0598067743 0.0038612591 0.0932928562 0.0004596373 summary(var)
VAR Estimation Results:
=========================
Endogenous variables: dy1t, dy2t
Deterministic variables: const
Sample size: 822
Log Likelihood: 6452.474
Roots of the characteristic polynomial:
0.9028 0.7865 0.7371 0.7371 0.7223 0.7223 0.6597 0.6597 0.6512 0.6512 0.6193 0.6193 0.4736 0.4736
Call:
VAR(y = y, type = "const", lag.max = 8, ic = "AIC")
Estimation results for equation dy1t:
=====================================
dy1t = dy1t.l1 + dy2t.l1 + dy1t.l2 + dy2t.l2 + dy1t.l3 + dy2t.l3 + dy1t.l4 + dy2t.l4 + dy1t.l5 + dy2t.l5 + dy1t.l6 + dy2t.l6 + dy1t.l7 + dy2t.l7 + const
Estimate Std. Error t value Pr(>|t|)
dy1t.l1 0.3119229 0.0351873 8.865 < 2e-16 ***
dy2t.l1 0.2181774 0.1141490 1.911 0.05632 .
dy1t.l2 0.0952428 0.0367932 2.589 0.00981 **
dy2t.l2 -0.0930331 0.1245860 -0.747 0.45544
dy1t.l3 0.0809582 0.0369912 2.189 0.02891 *
dy2t.l3 -0.0006987 0.1242586 -0.006 0.99551
dy1t.l4 0.0404476 0.0370572 1.091 0.27538
dy2t.l4 -0.1110368 0.1239133 -0.896 0.37048
dy1t.l5 -0.0773212 0.0368772 -2.097 0.03633 *
dy2t.l5 -0.0249262 0.1232141 -0.202 0.83973
dy1t.l6 0.0133329 0.0368413 0.362 0.71752
dy2t.l6 -0.2069330 0.1208461 -1.712 0.08721 .
dy1t.l7 0.0297573 0.0350855 0.848 0.39661
dy2t.l7 -0.1086331 0.1116065 -0.973 0.33067
const 0.0021727 0.0004912 4.423 1.11e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.008703 on 807 degrees of freedom
Multiple R-Squared: 0.1932, Adjusted R-squared: 0.1792
F-statistic: 13.8 on 14 and 807 DF, p-value: < 2.2e-16
Estimation results for equation dy2t:
=====================================
dy2t = dy1t.l1 + dy2t.l1 + dy1t.l2 + dy2t.l2 + dy1t.l3 + dy2t.l3 + dy1t.l4 + dy2t.l4 + dy1t.l5 + dy2t.l5 + dy1t.l6 + dy2t.l6 + dy1t.l7 + dy2t.l7 + const
Estimate Std. Error t value Pr(>|t|)
dy1t.l1 -0.0058263 0.0108062 -0.539 0.58993
dy2t.l1 0.4433807 0.0350559 12.648 < 2e-16 ***
dy1t.l2 0.0290656 0.0112994 2.572 0.01028 *
dy2t.l2 0.0439567 0.0382611 1.149 0.25095
dy1t.l3 0.0067641 0.0113602 0.595 0.55173
dy2t.l3 0.0681148 0.0381606 1.785 0.07464 .
dy1t.l4 -0.0021679 0.0113805 -0.190 0.84897
dy2t.l4 0.0404207 0.0380545 1.062 0.28847
dy1t.l5 -0.0199103 0.0113252 -1.758 0.07912 .
dy2t.l5 0.0548126 0.0378398 1.449 0.14785
dy1t.l6 0.0241330 0.0113142 2.133 0.03323 *
dy2t.l6 0.0598068 0.0371126 1.611 0.10746
dy1t.l7 0.0038613 0.0107750 0.358 0.72017
dy2t.l7 0.0932929 0.0342750 2.722 0.00663 **
const 0.0004596 0.0001509 3.047 0.00239 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002673 on 807 degrees of freedom
Multiple R-Squared: 0.4078, Adjusted R-squared: 0.3975
F-statistic: 39.69 on 14 and 807 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
dy1t dy2t
dy1t 7.575e-05 7.511e-07
dy2t 7.511e-07 7.144e-06
Correlation matrix of residuals:
dy1t dy2t
dy1t 1.00000 0.03229
dy2t 0.03229 1.00000(c) SVAR Estimation
Implementing Blanchard - Quah approach in R using “BQ”:
svar <- BQ(var)
summary(svar)
SVAR Estimation Results:
========================
Call:
BQ(x = var)
Type: Blanchard-Quah
Sample size: 822
Log Likelihood: 6437.336
Estimated contemporaneous impact matrix:
dy1t dy2t
dy1t 0.007721 0.004017
dy2t -0.001156 0.002410
Estimated identified long run impact matrix:
dy1t dy2t
dy1t 0.01706 0.00000
dy2t -0.00277 0.01228
Covariance matrix of reduced form residuals (*100):
dy1t dy2t
dy1t 7.575e-03 7.511e-05
dy2t 7.511e-05 7.144e-04(d) Interpretation
In the long run, the cummulative effect of shocks in demand side to industrial production is zero, while the long run effect of 1 standard deviation shocks in productivy will increase industrial production by 0.01706%.
(e) Impulse Response Function for SVAR
svar.irfs <- irf(svar, n.ahead=40)
svar.irfs.c <- irf(svar, n.ahead=40, ci=.9, cumulative=TRUE)
svar.irfs$irf[[2]] <- -svar.irfs$irf[[2]]
svar.irfs$Lower[[2]] <- -svar.irfs$Lower[[2]]
svar.irfs$Upper[[2]] <- -svar.irfs$Upper[[2]]
svar.irfs.c$irf[[2]] <- -svar.irfs.c$irf[[2]]
svar.irfs.c$Lower[[2]] <- -svar.irfs.c$Lower[[2]]
svar.irfs.c$Upper[[2]] <- -svar.irfs.c$Upper[[2]]
setwd("C:/Users/evanu/OneDrive/Time Series/Homework 5")
source("plotIRF.R")
par(mfrow=c(2,2), cex=.9, mar=c(4,4,2,1))
plotIRF(svar.irfs.c, vnames="dy1t", vlabels="Industrial Production", slabels=c("Productivity Shock","Demand shock"))
plotIRF(svar.irfs.c, vnames="dy2t", vlabels="CPI", slabels=c("Productivity Shock","Demand shock"))
```
(f) FEVD for SVAR
plot( fevd(svar, n.ahead=40) ,addbars=10 )
We can see that the majority of the variance of industrial production is due to the changes in its own value, and it is also holds in the long-run. The other way around for consumer price index, the majority of the variance in consumer price index is contributed by the variability in industrial production, and it is also holds in the long run.