ECO 5316 H6 Q1
Raymond Kim
I first obtain monthly for federal funds rate FRED/FEDFUNDS and for Moody’s AAA rated corporate bonds FED/RIMLPAAAR_N_M. These two time series should be co-integrated.
First we load the “Quandl” packages and etc.
library("Quandl")
library("vars")
library("gdata")
library("stargazer")A) Plot Time Series and Conduct Unit Root Test
We plot both series in the same plot, and also plot their first differences.
ff<-Quandl("FRED/FEDFUNDS", type="zoo")
mood<-Quandl("FED/RIMLPAAAR_N_M", type="zoo")
ff<-window(ff,start="Jan 1960", end="Jan 2016")
mood<-window(mood,start="Jan 1960", end="Jan 2016")
ffd<-diff(ff,1)
moodd<-diff(mood,1)
par(mfrow=c(1,2))
plot(ff,xlab="Year", ylab="Interest Rate", col="red", main="Fed Funds and AAA Bonds")
lines(mood, col="blue")
legend("topleft", c("Fed Funds","AAA Bonds"),lty=c(1,1), col=c("red", "blue"))
par(mfrow=c(1,2))
plot(ffd,xlab="Year", ylab="Interest Rate", col="red", main="First Diff Fed Funds")
plot(moodd,xlab="Year", ylab="Interest Rate", col="blue", main="First Diff AAA Bonds")
Now we conduct an Elliot Rothenberg and Stock (ERS) unit root test on all four of our variables. The null hypothesis is: \[H_o=Stationary\]
summary( ur.ers(ff, type="P-test",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: 9.0253
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89summary( ur.ers(mood, type="P-test",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: 44.5105
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89summary( ur.ers(ffd, type="P-test",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: 0.2257
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89summary( ur.ers(moodd, type="P-test",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: 0.19
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89From the summary statistics, we can see that the Federal Funds rate and AAA ratings are not stationary and have a unit root. The first differences of both do not reject the null hypothesis, so the first differences are consistent with being stationary.
B) Perform Trace and Max Eigenvalue Tests to Determine Cointegration
I conduct the Johansen’s methodology to perform the trace test and the max eigenvalue test.
two<-cbind(ff,mood)
trace.two<- ca.jo(two, ecdet="const", type="trace", K=2, spec="longrun")
summary(trace.two)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 5.918327e-02 5.939614e-03 9.557320e-19
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.00 7.52 9.24 12.97
## r = 0 | 44.93 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ff.l2 mood.l2 constant
## ff.l2 1.000000 1.00000 1.00000
## mood.l2 -1.208595 -73.82892 3.03463
## constant 3.582429 519.39052 -171.64079
##
## Weights W:
## (This is the loading matrix)
##
## ff.l2 mood.l2 constant
## ff.d -0.03729059 1.570362e-04 -2.502138e-20
## mood.d 0.01701971 6.886164e-05 -5.617668e-20eigen.two<- ca.jo(two, ecdet="const", type="eigen", K=2, spec="longrun")
summary(eigen.two)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 5.918327e-02 5.939614e-03 9.557320e-19
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.00 7.52 9.24 12.97
## r = 0 | 40.94 13.75 15.67 20.20
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ff.l2 mood.l2 constant
## ff.l2 1.000000 1.00000 1.00000
## mood.l2 -1.208595 -73.82892 3.03463
## constant 3.582429 519.39052 -171.64079
##
## Weights W:
## (This is the loading matrix)
##
## ff.l2 mood.l2 constant
## ff.d -0.03729059 1.570362e-04 -2.502138e-20
## mood.d 0.01701971 6.886164e-05 -5.617668e-20The value of the trace test statistic 44.93 is greater than the critical value for \(H_o: rank(r)=0\) but it does not reject \(H_o:rank(r)=1\). Looking at the eigen test, the test statistic 40.94 is greater than the critical value for \(H_o: rank(r)=0\) but it does not reject \(H_o:rank(r)=1\). According to both tests, they suggest that Federal Funds Rate and AAA Bonds are in fact cointegrated with one relationship.
C) Perform the Test for the Presence of a Restricted Constant
I perform the lltest on the presence of a restricted constant in the model. After running an unrestricted test
two.VEC <- lttest(eigen.two, 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 0.02 0.88two.VEC## test statistic p-value
## LR test 0.02 0.88I can see that the constant value is 3.58. From our equation:
\[f(FedFunds)=BondRate\]
Let us assume that the function \(f\) is a linear transformation represented by \(\alpha\) and lets log the equation.
\[log(FedFunds)-log(BondRate)+log(\alpha)\]
I can assign \(B_1,B_2,B_3\) a matrix. I first restrict the constant \(B_3=-B_2\). So the test hypothesis is \(H_o: B_3=-B_2\).
mat<- matrix(c(1,0,0,0,1,-1), c(3,2))
eigen.two.beta <- blrtest(eigen.two, H=mat, r=1)
summary(eigen.two.beta)##
## ######################
## # 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,] 1 0
## [2,] 0 1
## [3,] 0 -1
##
## Eigenvalues of restricted VAR (lambda):
## [1] 0.0492 0.0011
##
## The value of the likelihood ratio test statistic:
## 7.08 distributed as chi square with 1 df.
## The p-value of the test statistic is: 0.01
##
## Eigenvectors, normalised to first column
## of the restricted VAR:
##
## [,1] [,2]
## [1,] 1.0000 1.0000
## [2,] -0.8774 -8.8074
## [3,] 0.8774 8.8074
##
## Weights W of the restricted VAR:
##
## [,1] [,2]
## ff.d -0.0359 2e-04
## mood.d 0.0114 1e-04The test statistic is significant, so we reject the restriction of \(B_3=-B_2\).
D) Test Restriction of \(\beta_2=-1\)
I conduct the same test as in C) but with a different matrix. The null hypothesis is \(H_o:\beta_2=-1\).
mat2<- matrix(c(1,-1,0,0,0,1), c(3,2))
eigen.two.beta2 <- blrtest(eigen.two, H=mat2, r=1)
summary(eigen.two.beta2)##
## ######################
## # 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,] 1 0
## [2,] -1 0
## [3,] 0 1
##
## Eigenvalues of restricted VAR (lambda):
## [1] 0.0552 0.0000
##
## The value of the likelihood ratio test statistic:
## 2.81 distributed as chi square with 1 df.
## The p-value of the test statistic is: 0.09
##
## Eigenvectors, normalised to first column
## of the restricted VAR:
##
## [,1] [,2]
## [1,] 1.0000 1.0000
## [2,] -1.0000 -1.0000
## [3,] 2.0669 -117.8762
##
## Weights W of the restricted VAR:
##
## [,1] [,2]
## ff.d -0.0385 0
## mood.d 0.0139 0The results of the test is that we cannot strongly reject the null hypothesis. So it seems that the restriction of \(\beta_2\) of -1 is consistent with the unrestricted tests, and close to the real value of \(\beta_2\).
E) Plot the Error Term \(B'y_t\) and show the Deviations from the LR Equilibrium
Plot the residuals from the two restrictions.
plot(eigen.two)
plotres(eigen.two)
car<-cajorls(eigen.two.beta2)