5 Testing Restrictions on \(\beta\)

This chapter contains the replication of the material of Chapter 10 of Juselius (2006).

5.1 Formulating hypotheses as restrictions on \(\beta\)

1) Long-run money demand relationship:

1.1) Formulated in terms of free parameters

\[\begin{align*} {\bf \beta}_1 &= {\bf \text{H}}_1 {\bf \varphi}_1 \\ &= \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ -1 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & -1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} \begin{bmatrix} \varphi_{11} \\ \varphi_{12} \\ \varphi_{13} \end{bmatrix} \end{align*}\]

H1 <- matrix(c( 1 , 0, 0,
               -1,  0, 0,
                0 , 0, 0,
                0,  1, 0,
                0, -1, 0,
                0,  0, 1), c(6, 3), byrow = TRUE)
H1
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]   -1    0    0
## [3,]    0    0    0
## [4,]    0    1    0
## [5,]    0   -1    0
## [6,]    0    0    1

1.2) Formulated in terms of restricted parameters

\[\begin{align*} {\bf \text{R}}_1^{'} {\bf \beta}_1 &= {\bf 0} \\ \begin{bmatrix} \phantom{-}1 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & \phantom{-}0 \end{bmatrix} \begin{bmatrix} \beta_{11} \\ \beta_{12} \\ \beta_{13} \\ \beta_{14} \\ \beta_{15} \\ \beta_{16} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \end{align*}\]

R1 <- t(matrix(c( 1, 1, 0, 0, 0, 0,
                  0, 0, 1, 0, 0, 0,
                  0, 0, 0, 1, 1, 0), c(3, 6), byrow = TRUE))
t(R1)
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    1    0    0    0    0
## [2,]    0    0    1    0    0    0
## [3,]    0    0    0    1    1    0

1.3) Relationship between specification in terms of free and restricted parameters

  • \({\bf \text{R}}_i = {\bf \text{H}}_{\perp,i}\), i.e., \({\bf \text{R}}_i^{'} {\bf \text{H}}_{i} = {\bf 0}\)
t(R1) %*% H1
##      [,1] [,2] [,3]
## [1,]    0    0    0
## [2,]    0    0    0
## [3,]    0    0    0
  • \({\bf \text{H}}_i\) lies in the left nullspace of \({\bf \text{R}}_i\), i.e., \({\bf \text{H}}_i = {\bf N} \left( {\bf \text{R}}_i^{'} \right)\)
H1x <- MASS::Null(R1)
H1x
##      [,1] [,2] [,3]
## [1,] -0.5 -0.5    0
## [2,]  0.5  0.5    0
## [3,]  0.0  0.0    0
## [4,]  0.5 -0.5    0
## [5,] -0.5  0.5    0
## [6,]  0.0  0.0    1
H1
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]   -1    0    0
## [3,]    0    0    0
## [4,]    0    1    0
## [5,]    0   -1    0
## [6,]    0    0    1

Note, the condition above identifies only the space on which \({\bf \text{H}}_i\) lies. The particular

round(cbind(H1x[,1,drop=FALSE] + H1x[,2,drop=FALSE], H1x[,1,drop=FALSE] - H1x[,2,drop=FALSE], H1x[,3,drop=FALSE]), 3)
##      [,1] [,2] [,3]
## [1,]   -1    0    0
## [2,]    1    0    0
## [3,]    0    0    0
## [4,]    0    1    0
## [5,]    0   -1    0
## [6,]    0    0    1
H1
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]   -1    0    0
## [3,]    0    0    0
## [4,]    0    1    0
## [5,]    0   -1    0
## [6,]    0    0    1

2) Long-run aggregate demand relationship:

2.1) Formulated in terms of free parameters

\[\begin{align*} {\bf \beta}_2 &= {\bf \text{H}}_2 {\bf \varphi}_2 \\ &= \begin{bmatrix} \phantom{-}0 & \phantom{-}0 \\ \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & -1 \\ \phantom{-}0 & \phantom{-}0 \end{bmatrix} \begin{bmatrix} \varphi_{21} \\ \varphi_{22} \end{bmatrix} \end{align*}\]

H2 <- matrix(c( 0,  0,
                1,  0,
                0 , 1,
                0,  0,
                0, -1,
                0,  0), c(6, 2), byrow = TRUE)
H2
##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]    0    0
## [5,]    0   -1
## [6,]    0    0

2.2) Formulated in terms of restricted parameters

\[\begin{align*} {\bf \text{R}}_2^{'} {\bf \beta}_2 &= {\bf 0} \\ \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} \begin{bmatrix} \beta_{21} \\ \beta_{22} \\ \beta_{23} \\ \beta_{24} \\ \beta_{25} \\ \beta_{26} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{align*}\]

R2 <- t(matrix(c( 1, 0, 0, 0, 0, 0,
                  0, 0, 1, 0, 1, 0,
                  0, 0, 0, 1, 0, 0,
                  0, 0, 0, 0, 0, 1), c(4, 6), byrow = TRUE))
t(R2)
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    0    0    0    0    0
## [2,]    0    0    1    0    1    0
## [3,]    0    0    0    1    0    0
## [4,]    0    0    0    0    0    1

2.3) Relationship between specification in terms of free and restricted parameters

  • \({\bf \text{R}}_i = {\bf \text{H}}_{\perp,i}\), i.e., \({\bf \text{R}}_i^{'} {\bf \text{H}}_{i} = {\bf 0}\)
t(R2) %*% H2
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    0    0
  • \({\bf \text{H}}_i\) lies in the left nullspace of \({\bf \text{R}}_i\), i.e., \({\bf \text{H}}_i = {\bf N} \left( {\bf \text{R}}_i^{'} \right)\)
H2x <- MASS::Null(R2)
H2x
##            [,1]       [,2]
## [1,]  0.0000000  0.0000000
## [2,] -0.7071068 -0.7071068
## [3,] -0.5000000  0.5000000
## [4,]  0.0000000  0.0000000
## [5,]  0.5000000 -0.5000000
## [6,]  0.0000000  0.0000000
H2
##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]    0    0
## [5,]    0   -1
## [6,]    0    0

Note, the condition above identifies only the space on which \({\bf \text{H}}_i\) lies. The particular

round(cbind((H2x[,1,drop=FALSE] + H2x[,2,drop=FALSE]) / (H2x[2,1] + H2x[2,2]), (H2x[,2,drop=FALSE] - H2x[,1,drop=FALSE])), 3)
##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]    0    0
## [5,]    0   -1
## [6,]    0    0
H2
##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]    0    0
## [5,]    0   -1
## [6,]    0    0

3) Long-run term structure relationship:

3.1) Formulated in terms of free parameters

\[\begin{align*} {\bf \beta}_3 &= {\bf \text{H}}_3 {\bf \varphi}_3 \\ &= \begin{bmatrix} \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 \\ \phantom{-}1 & \phantom{-}0 \\ -1 & -1 \\ \phantom{-}0 & \phantom{-}1 \end{bmatrix} \begin{bmatrix} \varphi_{31} \\ \varphi_{32} \end{bmatrix} \end{align*}\]

H3 <- matrix(c( 0,  0,
                0,  0,
                0 , 0,
                1,  0,
               -1,  0,
                0,  1), c(6, 2), byrow = TRUE)
H3
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    1    0
## [5,]   -1    0
## [6,]    0    1

3.2) Formulated in terms of restricted parameters

\[\begin{align*} {\bf \text{R}}_2^{'} {\bf \beta}_2 &= {\bf 0} \\ \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & \phantom{-}0 \end{bmatrix} \begin{bmatrix} \beta_{21} \\ \beta_{22} \\ \beta_{23} \\ \beta_{24} \\ \beta_{25} \\ \beta_{26} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{align*}\]

R3 <- t(matrix(c( 1, 0, 0, 0, 0, 0,
                  0, 1, 0, 0, 0, 0,
                  0, 0, 1, 0, 0, 0,
                  0, 0, 0, 1, 1, 0), c(4, 6), byrow = TRUE))
t(R3)
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    0    0    0    0    0
## [2,]    0    1    0    0    0    0
## [3,]    0    0    1    0    0    0
## [4,]    0    0    0    1    1    0

3.3) Relationship between specification in terms of free and restricted parameters

  • \({\bf \text{R}}_i = {\bf \text{H}}_{\perp,i}\), i.e., \({\bf \text{R}}_i^{'} {\bf \text{H}}_{i} = {\bf 0}\)
t(R3) %*% H3
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    0    0
  • \({\bf \text{H}}_i\) lies in the left nullspace of \({\bf \text{R}}_i\), i.e., \({\bf \text{H}}_i = {\bf N} \left( {\bf \text{R}}_i^{'} \right)\)
H3x <- MASS::Null(R3)
H3x
##            [,1] [,2]
## [1,]  0.0000000    0
## [2,]  0.0000000    0
## [3,]  0.0000000    0
## [4,] -0.7071068    0
## [5,]  0.7071068    0
## [6,]  0.0000000    1
H3
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    1    0
## [5,]   -1    0
## [6,]    0    1

Note, the condition above identifies only the space on which \({\bf \text{H}}_i\) lies. The particular

round(cbind(H3x[,1,drop=FALSE] / H3x[4,1], H3x[,2,drop=FALSE]), 3)
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    1    0
## [5,]   -1    0
## [6,]    0    1
H3
##      [,1] [,2]
## [1,]    0    0
## [2,]    0    0
## [3,]    0    0
## [4,]    1    0
## [5,]   -1    0
## [6,]    0    1

5.2 Same restrictions on all \(\beta\)

Load the function ca_jo_jus06_hr3_fun() from GitHub.

source("https://raw.githubusercontent.com/mmoessler/juselius-2006/main/R/blrtest_fun.R")

Note, the function blrtest_fun() is based on the function blrtest() from the library urca (see also Pfaff (2008)) and edited such that it is compatible with the results from the function ca_jo_jus06_fun.

5.2.1 \(\mathcal{H}_1\)

\(\mathcal{H}_1\): \(\beta_{trend}=0\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

# exclusion of trend in cir
H1 <- matrix(c(1,0,0,0,0,0,
               0,1,0,0,0,0,
               0,0,1,0,0,0,
               0,0,0,1,0,0,
               0,0,0,0,1,0,
               0,0,0,0,0,1,
               0,0,0,0,0,0), nrow=7, ncol=6, byrow=T)

b.h01.res <- blrtest_fun(z = z, H = H1, r = r)

# test results
b.h01.res$teststat
## [1] 0.9233936
b.h01.res$pval
## [1] 0.819779 3.000000
# restricted cointegrating vectors
round(b.h01.res$V[,1]/b.h01.res$V[3,1], 2)
## [1]  0.07 -0.03  1.00 -0.29  0.59  0.00  0.00
round(b.h01.res$V[,2]/b.h01.res$V[1,2], 2)
## [1]   1.00  -1.22  -3.71 -10.39   8.82  -0.25   0.00
round(b.h01.res$V[,3]/b.h01.res$V[4,3], 2)
## [1]  0.00  0.02  0.01  1.00 -0.64 -0.01  0.00

5.2.2 \(\mathcal{H}_2\)

\(\mathcal{H}_2\): \(\beta_{D_S831}=0\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

# exclusion of shift in cir
H2 <- matrix(c(1,0,0,0,0,0,
               0,1,0,0,0,0,
               0,0,1,0,0,0,
               0,0,0,1,0,0,
               0,0,0,0,1,0,
               0,0,0,0,0,0,
               0,0,0,0,0,1), nrow=7, ncol=6, byrow=T)

b.h02.res <- blrtest_fun(z = z, H = H2, r = r)

# test results
b.h02.res$teststat
## [1] 19.08055
b.h02.res$pval
## [1] 0.0002631122 3.0000000000
# restricted cointegrating vectors
round(b.h02.res$V[,1]/b.h02.res$V[3,1], 2)
## [1]  0.06 -0.03  1.00 -0.32  0.56  0.00  0.00
round(b.h02.res$V[,2]/b.h02.res$V[1,2], 2)
## [1]   1.00  -1.72  -3.41 -42.17  28.60   0.00   0.00
round(b.h02.res$V[,3]/b.h02.res$V[4,3], 2)
## [1] 0.08 0.14 0.09 1.00 2.11 0.00 0.00

5.2.3 \(\mathcal{H}_3\)

\(\mathcal{H}_3\): \(\beta_{m^{r}}=-\beta_{y^{r}}\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

# exclusion of shift in cir
H3 <- matrix(c( 1,0,0,0,0,0,
               -1,0,0,0,0,0,
                0,1,0,0,0,0,
                0,0,1,0,0,0,
                0,0,0,1,0,0,
                0,0,0,0,1,0,
                0,0,0,0,0,1), nrow=7, ncol=6, byrow=T)
H3
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    0    0    0    0    0
## [2,]   -1    0    0    0    0    0
## [3,]    0    1    0    0    0    0
## [4,]    0    0    1    0    0    0
## [5,]    0    0    0    1    0    0
## [6,]    0    0    0    0    1    0
## [7,]    0    0    0    0    0    1
b.h03.res <- blrtest_fun(z = z, H = H3, r = r)

b.h03.res$teststat
## [1] 3.364939
b.h03.res$pval
## [1] 0.3387061 3.0000000
b.h03.res$V
##             [,1]          [,2]          [,3]        [,4]          [,5]
## [1,]  1.00000000  1.0000000000  1.000000e+00  1.00000000   1.000000000
## [2,] -1.00000000 -1.0000000000 -1.000000e+00 -1.00000000  -1.000000000
## [3,] 16.50417371 -3.4601214048 -3.557886e+00  1.75219052   1.127624813
## [4,] -9.43854258 -8.0670685093 -1.082881e+02 39.89698866  35.199973941
## [5,]  9.71465325  7.8188372616  8.229685e+01 17.88461844 -39.097781341
## [6,] -0.01695032 -0.2459345769  6.593580e-01 -0.19343299   0.151984555
## [7,]  0.00212673 -0.0007979788 -3.345322e-03  0.01264414  -0.004088109
##              [,6]
## [1,]  1.000000000
## [2,] -1.000000000
## [3,] -0.638387946
## [4,] -6.596251960
## [5,] 16.547016757
## [6,]  0.458790040
## [7,] -0.008307031
round(b.h03.res$V[,1]/b.h03.res$V[3,1], 3)
## [1]  0.061 -0.061  1.000 -0.572  0.589 -0.001  0.000
round(b.h03.res$V[,2]/b.h03.res$V[1,2], 3)
## [1]  1.000 -1.000 -3.460 -8.067  7.819 -0.246 -0.001
round(b.h03.res$V[,3]/b.h03.res$V[4,3], 3)
## [1] -0.009  0.009  0.033  1.000 -0.760 -0.006  0.000

5.2.4 \(\mathcal{H}_4\)

\(\mathcal{H}_4\): \(\beta_{R_{m}}=-\beta_{R_{b}}\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

# exclusion of shift in cir
H4 <- matrix(c( 1,0,0, 0,0,0,
                0,1,0, 0,0,0,
                0,0,1, 0,0,0,
                0,0,0, 1,0,0,
                0,0,0,-1,0,0,
                0,0,0, 0,1,0,
                0,0,0, 0,0,1), nrow=7, ncol=6, byrow=T)
H4
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    0    0    0    0    0
## [2,]    0    1    0    0    0    0
## [3,]    0    0    1    0    0    0
## [4,]    0    0    0    1    0    0
## [5,]    0    0    0   -1    0    0
## [6,]    0    0    0    0    1    0
## [7,]    0    0    0    0    0    1
b.h04.res <- blrtest_fun(z = z, H = H4, r = r)

b.h04.res$teststat
## [1] 4.903692
b.h04.res$pval
## [1] 0.1789867 3.0000000
b.h04.res$V
##               [,1]          [,2]          [,3]         [,4]         [,5]
## [1,]  1.000000e+00  1.0000000000   1.000000000   1.00000000  1.000000000
## [2,] -7.433982e-01 -0.9970751757  -1.008996255   6.72003608 -0.906214286
## [3,]  1.590373e+01 -3.4500717550  -2.030723284   0.10736226 -0.487137651
## [4,] -1.024963e+01 -7.6066125054 -60.313807582 -32.72588057 -2.889621902
## [5,]  1.024963e+01  7.6066125054  60.313807582  32.72588057  2.889621902
## [6,] -7.229269e-03 -0.2484915620   0.385226931   0.26368433  0.451542662
## [7,]  9.109952e-04 -0.0007686502   0.001221223  -0.03337794 -0.009797564
##               [,6]
## [1,]   1.000000000
## [2,]   2.025171972
## [3,]   3.480962389
## [4,]  58.668047047
## [5,] -58.668047047
## [6,]  -0.245900308
## [7,]  -0.006510077
round(b.h04.res$V[,1]/b.h04.res$V[3,1], 3)
## [1]  0.063 -0.047  1.000 -0.644  0.644  0.000  0.000
round(b.h04.res$V[,2]/b.h04.res$V[1,2], 3)
## [1]  1.000 -0.997 -3.450 -7.607  7.607 -0.248 -0.001
round(b.h04.res$V[,3]/b.h04.res$V[4,3], 3)
## [1] -0.017  0.017  0.034  1.000 -1.000 -0.006  0.000

5.2.5 \(\mathcal{H}_5\)

\(\mathcal{H}_5\): \(\beta_{m^{r}}=-\beta_{y^{r}}\) and \(\beta_{trend}=0\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

# exclusion of shift in cir
H5 <- matrix(c( 1,0,0, 0,0,
               -1,0,0, 0,0,
                0,1,0, 0,0,
                0,0,1, 0,0,
                0,0,0, 1,0,
                0,0,0, 0,1,
                0,0,0, 0,0), nrow=7, ncol=5, byrow=T)
H5
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    0    0    0    0
## [2,]   -1    0    0    0    0
## [3,]    0    1    0    0    0
## [4,]    0    0    1    0    0
## [5,]    0    0    0    1    0
## [6,]    0    0    0    0    1
## [7,]    0    0    0    0    0
b.h05.res <- blrtest_fun(z = z, H = H5, r = r)

b.h05.res$teststat
## [1] 9.360627
b.h05.res$pval
## [1] 0.1542893 6.0000000
b.h05.res$V
##               [,1]       [,2]        [,3]       [,4]         [,5]
## [1,]   1.000000000  1.0000000   1.0000000  1.0000000    1.0000000
## [2,]  -1.000000000 -1.0000000  -1.0000000 -1.0000000   -1.0000000
## [3,]  11.889873745 -4.5623725  -5.0812664  1.7878800    2.9687509
## [4,] -15.432956845 -1.8076511 -71.3860686 18.9798079   88.1327580
## [5,]  10.778425924  5.2091465  62.0106339  8.5201907 -115.4979626
## [6,]  -0.007434647 -0.3148795   0.3171851  0.2002191   -0.2237345
## [7,]   0.000000000  0.0000000   0.0000000  0.0000000    0.0000000
round(b.h05.res$V[,1]/b.h05.res$V[3,1], 3)
## [1]  0.084 -0.084  1.000 -1.298  0.907 -0.001  0.000
round(b.h05.res$V[,2]/b.h05.res$V[1,2], 3)
## [1]  1.000 -1.000 -4.562 -1.808  5.209 -0.315  0.000
round(b.h05.res$V[,3]/b.h05.res$V[4,3], 3)
## [1] -0.014  0.014  0.071  1.000 -0.869 -0.004  0.000

5.2.6 \(\mathcal{H}_6\)

\(\mathcal{H}_6\): \(\beta_{m^{r}}=-\beta_{y^{r}}\), \(\beta_{R_{m}}=-\beta_{R_{b}}\) and \(\beta_{trend}=0\)

ca.jo.res.01$P <- 6
z <- ca.jo.res.01
r <- 3

H6 <- matrix(c( 1,0, 0, 0,
               -1,0, 0, 0,
                0,1, 0, 0,
                0,0, 1, 0,
                0,0,-1, 0,
                0,0, 0, 1,
                0,0, 0, 0), nrow=7, ncol=4, byrow=T)
H6
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]   -1    0    0    0
## [3,]    0    1    0    0
## [4,]    0    0    1    0
## [5,]    0    0   -1    0
## [6,]    0    0    0    1
## [7,]    0    0    0    0
b.h06.res <- blrtest_fun(z = z, H = H6, r = r)

b.h06.res$teststat
## [1] 16.51752
b.h06.res$pval
## [1] 0.05683002 9.00000000
b.h06.res$V
##            [,1]        [,2]       [,3]         [,4]
## [1,]  1.0000000   1.0000000   1.000000   1.00000000
## [2,] -1.0000000  -1.0000000  -1.000000  -1.00000000
## [3,]  8.4404556  -7.2716468   1.780715   2.06799810
## [4,] -7.5031367 -11.3584231 -64.755312  55.57540805
## [5,]  7.5031367  11.3584231  64.755312 -55.57540805
## [6,] -0.0664285  -0.3227279   0.602895  -0.02917173
## [7,]  0.0000000   0.0000000   0.000000   0.00000000
round(b.h06.res$V[,1]/b.h06.res$V[3,1], 3)
## [1]  0.118 -0.118  1.000 -0.889  0.889 -0.008  0.000
round(b.h06.res$V[,2]/b.h06.res$V[1,2], 3)
## [1]   1.000  -1.000  -7.272 -11.358  11.358  -0.323   0.000
round(b.h06.res$V[,3]/b.h06.res$V[4,3], 3)
## [1] -0.015  0.015 -0.027  1.000 -1.000 -0.009  0.000

References

Juselius, Katarina. 2006. The Cointegrated VAR Model: Methodology and Applications. Oxford University Press.
Pfaff, Bernhard. 2008. Analysis of Integrated and Cointegrated Time Series with r. Springer Science & Business Media.