6 Identification of the long-run structure
This chapter contains the replication of the material of Chapter 12 of Juselius (2006).
6.1 Just-identified long-run structures
Load the function ca_jo_jus06_hr3_fun() from GitHub.
source("https://raw.githubusercontent.com/mmoessler/juselius-2006/main/R/ca_jo_jus06_hr3_fun.R")Note, the function ca_jo_jus06_hr3_fun() is based on the function bh6lrtest() from the library urca (see also Pfaff (2008)) and extended such that restriction can be imposed on each of the three cointegrating relationships individually.
Thus, the switching algorithm proposed by Johansen and Juselius (1992) is applied by iterating over all three cointegrating relationships. This allows us to estimate just-identified and over-identified long-run structures as in Chapter 12.
Check the code on GitHub (mmoessler/juselius-2006/main/R/ca_jo_jus06_fun.R) for more information.
6.1.1 \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006)
Construct the design matrices \(H_i\) for \(i=1,2,3\).
H1 <- matrix(c( 1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = TRUE)
H2 <- matrix(c( 0, 0, 0, 0,
1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = TRUE)
H3 <- matrix(c( 0, 0, 0, 0,
1, 0, 0, 0,
0, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = TRUE)Estimate the (just) identified cointegrating and adjustment coefficients.
z <- ca.jo.res.02
# z$P <- 6
b.s1.res <- ca_jo_jus06_hr3_fun(z = z,
H1 = H1, H2 = H2, H3 = H3,
nor.id1 = 1, nor.id2 = 3, nor.id3 = 4)Compare results for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006))
beta.norm <- b.s1.res$V
round(beta.norm, 2)## [,1] [,2] [,3]
## [1,] 1.00 0.00 0.00
## [2,] -0.94 0.04 0.01
## [3,] 0.00 1.00 0.00
## [4,] 0.00 0.00 1.00
## [5,] 3.04 0.20 -0.63
## [6,] -0.27 0.01 -0.01Compare results for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006))
alpha.norm <- b.s1.res$W
round(alpha.norm, 2)## [,1] [,2] [,3]
## Lm3rC.d -0.22 -0.56 2.98
## Lyr.d 0.05 -0.29 -1.84
## Dpy.d -0.01 -0.82 -0.47
## Rm.d 0.00 0.03 -0.09
## Rb.d 0.00 0.02 0.13Compute standard error for \(\widehat{\beta}\)
H <- matrix(rep(0, (3*6)*(3*4-3)), nrow = (3*6), ncol = (3*4-3))
H[( 0+1): 6,(0+1):3] <- H1[,-1]
H[( 6+1):12,(3+1):6] <- H2[,-2]
H[(12+1):18,(6+1):9] <- H3[,-2]Sig.e <- b.s1.res$z$DELTA
S11 <- b.s1.res$z$SKK
TT <- nrow(b.s1.res$z$R0)cov.beta <- H %*% solve( t(H) %*% ( (t(alpha.norm) %*% solve(Sig.e) %*% alpha.norm) %x% S11 ) %*% H ) %*% t(H) / TTCompare with standard error for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006))
matrix(round(c(beta.norm)/sqrt(diag(cov.beta)), 2), nrow = 6, ncol = 3)## [,1] [,2] [,3]
## [1,] Inf NaN NaN
## [2,] -6.55 3.24 2.06
## [3,] NaN Inf NaN
## [4,] NaN NaN Inf
## [5,] 1.51 1.16 -7.03
## [6,] -8.08 5.11 -5.12Compute standard error for \(\widehat{\alpha}\)
cov.alpha <- Sig.e %x% solve(t(beta.norm) %*% S11 %*% beta.norm) / TTCompare with standard error for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006))
round(alpha.norm/matrix(sqrt(diag(cov.alpha)), nrow = 5, ncol = 3, byrow = TRUE), 2)## [,1] [,2] [,3]
## Lm3rC.d -4.78 -1.31 2.66
## Lyr.d 1.91 -1.16 -2.78
## Dpy.d -0.72 -5.51 -1.21
## Rm.d -0.72 1.50 -1.83
## Rb.d 0.86 0.75 1.906.1.2 \(\mathcal{H}_{S.2}\) in Table 12.1 of Juselius (2006)
Construct the design matrices \(H_i\) for \(i=1,2,3\).
H1 <- matrix(c( 1, 0, 0, 0,
-1, 0, 0, 0,
0, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = T)
H2 <- matrix(c( 0, 0, 0, 0,
1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0,-1, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = T)
H3 <- matrix(c( 0, 0, 0, 0,
1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 0,
0, 0, 0, 1), nrow = 6, ncol = 4, byrow = T)Estimate the (just) identified cointegrating and adjustment coefficients.
z <- ca.jo.res.02
b.s2.res <- ca_jo_jus06_hr3_fun(z = z,
H1 = H1, H2 = H2, H3 = H3,
nor.id1 = 1, nor.id2 = 3, nor.id3 = 3)Compare results for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.2}\) in Table 12.1 of Juselius (2006))
beta.norm <- b.s2.res$V
round(beta.norm, 2)## [,1] [,2] [,3]
## [1,] 1.00 0.00 0.00
## [2,] -1.00 0.03 0.04
## [3,] 0.00 1.00 1.00
## [4,] -4.70 -0.54 0.32
## [5,] 5.99 0.54 0.00
## [6,] -0.24 0.02 0.01Compare results for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.2}\) in Table 12.1 of Juselius (2006))
alpha.norm <- b.s2.res$W
round(alpha.norm, 2)## [,1] [,2] [,3]
## Lm3rC.d -0.22 -2.47 1.91
## Lyr.d 0.05 1.75 -2.04
## Dpy.d -0.01 0.31 -1.12
## Rm.d 0.00 0.12 -0.09
## Rb.d 0.00 -0.15 0.17Compute standard error for \(\widehat{\beta}\)
H <- matrix(rep(0, (3*6)*(3*4-3)), nrow = (3*6), ncol = (3*4-3))
H[( 0+1): 6,(0+1):3] <- H1[,-1]
H[( 6+1):12,(3+1):6] <- H2[,-2]
H[(12+1):18,(6+1):9] <- H3[,-2]Sig.e <- b.s2.res$z$DELTA
S11 <- b.s2.res$z$SKK
TT <- nrow(b.s2.res$z$R0)cov.beta <- H %*% solve( t(H) %*% ( (t(alpha.norm) %*% solve(Sig.e) %*% alpha.norm) %x% S11 ) %*% H ) %*% t(H) / TTCompare with standard error for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006))
matrix(round(c(beta.norm)/sqrt(diag(cov.beta)), 2), nrow = 6, ncol = 3)## [,1] [,2] [,3]
## [1,] Inf NaN NaN
## [2,] -Inf 3.81 4.80
## [3,] NaN Inf Inf
## [4,] -1.44 -4.53 2.99
## [5,] 2.40 4.53 NaN
## [6,] -7.46 6.58 5.14Compute standard error for \(\widehat{\alpha}\)
cov.alpha <- Sig.e %x% solve(t(beta.norm) %*% S11 %*% beta.norm) / TTCompare with standard error for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.1}\) in Table 12.1 of Juselius (2006)
round(alpha.norm/matrix(sqrt(diag(cov.alpha)), nrow = 5, ncol = 3, byrow = TRUE), 2)## [,1] [,2] [,3]
## Lm3rC.d -4.78 -2.02 1.49
## Lyr.d 1.91 2.43 -2.70
## Dpy.d -0.72 0.73 -2.54
## Rm.d -0.72 2.31 -1.70
## Rb.d 0.86 -2.13 2.286.2 Over-Identified structures
6.2.1 \(\mathcal{H}_{S.3}\) in Table 12.3 of Juselius (2006)
Construct the design matrices \(H_i\) for \(i=1,2,3\).
H1 <- matrix(c( 1, 0,
-1, 0,
0, 0,
0, 0,
0, 0,
0, 1), nrow = 6, ncol = 2, byrow = TRUE)
H2 <- matrix(c( 0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 0, 0,
0, 0, 0,
0, 0, 1), nrow = 6, ncol = 3, byrow = TRUE)
H3 <- matrix(c( 0, 0, 0,
0, 0, 0,
1, 0, 0,
-1, -1, 0,
0, 1, 0,
0, 0, 1), nrow = 6, ncol = 3, byrow = TRUE)Estimate the (over) identified cointegrating and adjustment coefficients.
z <- ca.jo.res.02
b.s3.res <- ca_jo_jus06_hr3_fun(z = z,
H1 = H1, H2 = H2, H3 = H3,
nor.id1 = 1, nor.id2 = 3, nor.id3 = 4, df = 4)Compare results for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.3}\) in Table 12.3 of Juselius (2006))
beta.norm <- b.s3.res$V
round(beta.norm, 2)## [,1] [,2] [,3]
## [1,] 1.00 0.00 0.00
## [2,] -1.00 0.03 0.00
## [3,] 0.00 1.00 -0.20
## [4,] 0.00 0.00 1.00
## [5,] 0.00 0.00 -0.80
## [6,] -0.34 0.01 -0.01Compare results for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.3}\) in Table 12.3 of Juselius (2006))
alpha.norm <- b.s3.res$W
round(alpha.norm, 2)## [,1] [,2] [,3]
## Lm3rC.d -0.21 0.24 3.38
## Lyr.d 0.06 -0.44 -1.40
## Dpy.d 0.00 -0.84 -0.29
## Rm.d 0.00 0.02 -0.07
## Rb.d 0.00 0.05 0.13Compare value for LR-test on page 220 of Juselius (2006)
round(b.s3.res$teststat, 2)## [1] 4.05round(b.s3.res$pval, 2)## [1] 0.4 4.0Compute standard error for \(\widehat{\beta}\)
H <- matrix(rep(0, (3*6)*(2+3+3-4)), nrow = (3*6), ncol = (2+3+3-4))
H[( 0+1): 6,(0+1):1] <- H1[,-1]
H[( 6+1):12,(1+1):3] <- H2[,-2]
H[(12+1):18,(3+1):4] <- H3[,-c(1,2)]Sig.e <- b.s3.res$z$DELTA
S11 <- b.s3.res$z$SKK
TT <- nrow(b.s3.res$z$R0)cov.beta <- H %*% solve( t(H) %*% ( (t(alpha.norm) %*% solve(Sig.e) %*% alpha.norm) %x% S11 ) %*% H ) %*% t(H) / TTCompare with standard error for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.3}\) in Table 12.3 of Juselius (2006))
matrix(round(c(beta.norm)/sqrt(diag(cov.beta)), 2), nrow = 6, ncol = 3)## [,1] [,2] [,3]
## [1,] Inf NaN NaN
## [2,] -Inf 3.7 NaN
## [3,] NaN Inf -Inf
## [4,] NaN NaN Inf
## [5,] NaN NaN -Inf
## [6,] -13.67 5.5 -10.71Compute standard error for \(\widehat{\alpha}\)
cov.alpha <- Sig.e %x% solve(t(beta.norm) %*% S11 %*% beta.norm) / TTCompare with standard error for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.3}\) in Table 12.3 of Juselius (2006)
round(alpha.norm/matrix(sqrt(diag(cov.alpha)), nrow = 5, ncol = 3, byrow = TRUE), 2)## [,1] [,2] [,3]
## Lm3rC.d -4.78 0.54 3.24
## Lyr.d 2.35 -1.64 -2.29
## Dpy.d -0.11 -5.36 -0.80
## Rm.d -0.28 1.01 -1.57
## Rb.d 0.69 1.89 2.056.2.2 \(\mathcal{H}_{S.4}\) in Table 12.3 of Juselius (2006)
Construct the design matrices \(H_i\) for \(i=1,2,3\).
H1 <- matrix(c( 1, 0, 0,
-1, 0, 0,
0, 0, 0,
0, 1, 0,
0,-1, 0,
0, 0, 1), nrow = 6, ncol = 3, byrow = TRUE)
H2 <- matrix(c( 0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 0, 0,
0, 0, 0,
0, 0, 1), nrow = 6, ncol = 3, byrow = TRUE)
H3 <- matrix(c( 0, 0, 0,
0, 0, 0,
0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 0, 1), nrow = 6, ncol = 3, byrow = TRUE)Estimate the (over) identified cointegrating and adjustment coefficients.
z <- ca.jo.res.02
b.s4.res <- ca_jo_jus06_hr3_fun(z = z,
H1 = H1, H2 = H2, H3 = H3,
nor.id1 = 1, nor.id2 = 3, nor.id3 = 4,
conv.val = 1e-10, max.iter = 100,
df = 3)Compare results for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.4}\) in Table 12.3 of Juselius (2006))
beta.norm <- b.s4.res$V
round(beta.norm, 2)## [,1] [,2] [,3]
## [1,] 1.00 0.00 0.00
## [2,] -1.00 0.03 0.00
## [3,] 0.00 1.00 0.00
## [4,] -13.32 0.00 1.00
## [5,] 13.32 0.00 -0.81
## [6,] -0.15 0.01 -0.01Compare results for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.4}\) in Table 12.3 of Juselius (2006))
alpha.norm <- b.s4.res$W
round(alpha.norm, 2)## [,1] [,2] [,3]
## Lm3rC.d -0.23 -0.54 -0.08
## Lyr.d 0.05 -0.16 -0.31
## Dpy.d -0.01 -0.79 -0.34
## Rm.d 0.00 0.03 -0.08
## Rb.d 0.00 0.02 0.15Compare value for LR-test on page 221 of Juselius (2006)
round(Re(b.s4.res$teststat), 2)## [1] 2.84round(b.s4.res$pval, 2)## [1] 0.42 3.00Compute standard error for \(\widehat{\beta}\)
H <- matrix(rep(0, (3*6)*(3*3-3)), nrow = (3*6), ncol = (3*3-3))
H[( 0+1): 6,(0+1):2] <- H1[,-1]
H[( 6+1):12,(2+1):4] <- H2[,-2]
H[(12+1):18,(4+1):6] <- H3[,-1]Sig.e <- b.s4.res$z$DELTA
S11 <- b.s4.res$z$SKK
TT <- nrow(b.s4.res$z$R0)cov.beta <- H %*% solve( t(H) %*% ( (t(alpha.norm) %*% solve(Sig.e) %*% alpha.norm) %x% S11 ) %*% H ) %*% t(H) / TTCompare with standard error for \(\widehat{\beta}\) (see \(\mathcal{H}_{S.4}\) in Table 12.3 of Juselius (2006))
matrix(round(c(beta.norm)/sqrt(diag(cov.beta)), 2), nrow = 6, ncol = 3)## [,1] [,2] [,3]
## [1,] Inf NaN NaN
## [2,] -Inf 4.08 NaN
## [3,] NaN Inf NaN
## [4,] -5.73 NaN Inf
## [5,] 5.73 NaN -10.63
## [6,] -5.16 5.31 -4.80Compute standard error for \(\widehat{\alpha}\)
cov.alpha <- Sig.e %x% solve(t(beta.norm) %*% S11 %*% beta.norm) / TTCompare with standard error for \(\widehat{\alpha}\) (see \(\mathcal{H}_{S.4}\) in Table 12.3 of Juselius (2006)
round(alpha.norm/matrix(sqrt(diag(cov.alpha)), nrow = 5, ncol = 3, byrow = TRUE), 2)## [,1] [,2] [,3]
## Lm3rC.d -4.90 -1.28 -0.10
## Lyr.d 1.92 -0.66 -0.62
## Dpy.d -0.35 -5.42 -1.20
## Rm.d -0.73 1.79 -2.32
## Rb.d 0.46 0.94 3.07