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.numeric
library("vars")
## Loading required package: MASS
## Loading required package: strucchange
## Loading required package: sandwich
## Loading required package: urca
## Loading required package: lmtest
library("gdata")
## gdata: read.xls support for 'XLS' (Excel 97-2004) files ENABLED.
##
## gdata: read.xls support for 'XLSX' (Excel 2007+) files ENABLED.
##
## Attaching package: 'gdata'
## The following objects are masked from 'package:xts':
##
## first, last
## The following object is masked from 'package:stats':
##
## nobs
## The following object is masked from 'package:utils':
##
## object.size
library("stargazer")
##
## Please cite as:
## Hlavac, Marek (2015). stargazer: Well-Formatted Regression and Summary Statistics Tables.
## R package version 5.2. http://CRAN.R-project.org/package=stargazer
library("tseries")
library("urca")
library(zoo)
Quandl.auth(" 3HQPzReHC1WfzuSG41gy")
## Warning: 'Quandl.auth' is deprecated.
## Use 'Quandl.api_key' instead.
## See help("Deprecated")
#Problem 2
#Obtain quarterly time series for Disposable Personal Income FRED/DPI and for Personal Consumption Expenditures FRED/PCEC.
#(a) Let y1,t = log P CEt and y2,t = log P DIt. Test these log transformed time series and their first differences for unit root to verify that y1,t and y2,t are I(1).
#(b) The disposable income hypothesis suggest that PCEt = φDPIt where φ is the marginal propensity to consume. Consumption and disposable income should thus be growing at the same rate, and log P CEt −log P DIt −log φ should be I(0). Perform the trace and max eigenvalue tests for cointegration of y1,t and y2,t. Interpret the results.
#(c) Perform the test for the presence of a restricted constant rather than unrestricted constant, interpret the result. If necessary rerun the cointegration test from (b).
#(d) Estimate the unrestricted VEC model, examine significance of variables in the two equations.
#(e) Perform following three tests: (1) β2 = −1, (2) α2 = 0, (3) joint hypothesis β2 = −1 and α2 = 0.
#Interpret the results.
#(f) Convert the VEC model into a VAR model in levels. Create and plot eight quarter ahead forecast.
#a)
D<-Quandl("FRED/DPI",type="zoo")
P<-Quandl("FRED/PCEC",type="zoo")
lD=log(D)
lP=log(P)
a<-cbind(lD,lP)
plot(a,xlab="year", main="1960-Q1-2016-Q1")
dlD<-diff(lD)
dlP<-diff(lP)
par(mfrow=c(1,2))
plot(dlD,xlab = "year",main="1960-2016")
plot(dlP,xlab = "year",main="1960-2016")
#the Unit-Root tests
summary( ur.ers(lD, type="P-test", lag.max=8, model="trend") )
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 83.7733
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89
summary( ur.ers(dlD, type="P-test", lag.max=8, model="trend") )
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.2919
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89
summary( ur.ers(lP, type="P-test", lag.max=8, model="trend") )
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 63.1974
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89
summary( ur.ers(dlP, type="P-test", lag.max=8, model="trend") )
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.7594
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89
# both variables are non-stationary in levels. but the first differences of the variables are stationary.
#B)trace Test
DP <- ca.jo(a, ecdet="trend", type="trace", K=2, spec="transitory")
summary(DP)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , with linear trend in cointegration
##
## Eigenvalues (lambda):
## [1] 1.169941e-01 3.065947e-02 9.504492e-19
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 8.56 10.49 12.25 16.26
## r = 0 | 42.78 22.76 25.32 30.45
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## lD.l1 lP.l1 trend.l1
## lD.l1 1.0000000000 1.000000000 1.00000000
## lP.l1 -0.9467561512 -1.120563179 -1.80151062
## trend.l1 -0.0008356929 0.003153558 0.01371773
##
## Weights W:
## (This is the loading matrix)
##
## lD.l1 lP.l1 trend.l1
## lD.d 0.01960847 -0.024034218 3.575926e-15
## lP.d 0.12974403 -0.007937361 7.403759e-15
DP <- ca.jo(a, ecdet="trend", type="eigen", K=2, spec="transitory")
summary(DP)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , with linear trend in cointegration
##
## Eigenvalues (lambda):
## [1] 1.169941e-01 3.065947e-02 9.504492e-19
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 8.56 10.49 12.25 16.26
## r = 0 | 34.22 16.85 18.96 23.65
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## lD.l1 lP.l1 trend.l1
## lD.l1 1.0000000000 1.000000000 1.00000000
## lP.l1 -0.9467561512 -1.120563179 -1.80151062
## trend.l1 -0.0008356929 0.003153558 0.01371773
##
## Weights W:
## (This is the loading matrix)
##
## lD.l1 lP.l1 trend.l1
## lD.d 0.01960847 -0.024034218 3.575926e-15
## lP.d 0.12974403 -0.007937361 7.403759e-15
#Part C
lttest(DP, r=1)
## LR-test for no linear trend
##
## H0: H*2(r<=1)
## H1: H2(r<=1)
##
## Test statistic is distributed as chi-square
## with 1 degress of freedom
## test statistic p-value
## LR test 9.9 0
#Part D
DP.VEC <- cajorls(DP, r=1)
DP.VEC
## $rlm
##
## Call:
## lm(formula = substitute(form1), data = data.mat)
##
## Coefficients:
## lD.d lP.d
## ect1 0.019608 0.129744
## constant 0.001245 -0.039420
## lD.dl1 -0.029967 0.175551
## lP.dl1 0.461941 0.042985
##
##
## $beta
## ect1
## lD.l1 1.0000000000
## lP.l1 -0.9467561512
## trend.l1 -0.0008356929
summary(DP.VEC$rlm)
## Response lD.d :
##
## Call:
## lm(formula = lD.d ~ ect1 + constant + lD.dl1 + lP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.051913 -0.004746 -0.000611 0.004612 0.058281
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.019608 0.027290 0.719 0.473
## constant 0.001245 0.010533 0.118 0.906
## lD.dl1 -0.029967 0.067041 -0.447 0.655
## lP.dl1 0.461941 0.071736 6.439 5.44e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01011 on 271 degrees of freedom
## Multiple R-squared: 0.7314, Adjusted R-squared: 0.7274
## F-statistic: 184.5 on 4 and 271 DF, p-value: < 2.2e-16
##
##
## Response lP.d :
##
## Call:
## lm(formula = lP.d ~ ect1 + constant + lD.dl1 + lP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.036571 -0.004066 -0.000287 0.004549 0.059174
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.129744 0.023428 5.538 7.23e-08 ***
## constant -0.039420 0.009042 -4.360 1.85e-05 ***
## lD.dl1 0.175551 0.057555 3.050 0.00251 **
## lP.dl1 0.042985 0.061585 0.698 0.48579
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.008681 on 271 degrees of freedom
## Multiple R-squared: 0.7853, Adjusted R-squared: 0.7821
## F-statistic: 247.8 on 4 and 271 DF, p-value: < 2.2e-16
#Part E
rest.betta <- matrix(c(0,-1,0,0,0,1), c(3,2))
r.rb1 <- blrtest(DP, H=rest.betta , r=1)
summary(r.rb1)
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Estimation and testing under linear restrictions on beta
##
## The VECM has been estimated subject to:
## beta=H*phi and/or alpha=A*psi
##
## [,1] [,2]
## [1,] 0 0
## [2,] -1 0
## [3,] 0 1
##
## Eigenvalues of restricted VAR (lambda):
## [1] 0.0385 0.0024
##
## The value of the likelihood ratio test statistic:
## 23.41 distributed as chi square with 1 df.
## The p-value of the test statistic is: 0
##
## Eigenvectors, normalised to first column
## of the restricted VAR:
##
## [,1] [,2]
## [1,] NaN NaN
## [2,] -Inf -Inf
## [3,] Inf Inf
##
## Weights W of the restricted VAR:
##
## [,1] [,2]
## lD.d NaN NaN
## lP.d NaN NaN
#Part F
DP.VAR <- vec2var(DP, r=1)
DP.VAR.fcst <- predict(DP.VAR, n.ahead=8)
par( mar=c(3,3,1,1), cex=0.75)
plot(DP.VAR.fcst)
