9 Multivariate Time Series Models
9.1 Table 9.1
Load libraries
Spurious regression: OLS involving two independent random walks
##
## Call:
## lm(formula = y ~ x, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.4625 -2.5223 -0.0414 2.1179 8.2879
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.90971 0.24618 15.88 <2e-16 ***
## x -0.44348 0.04733 -9.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.27 on 198 degrees of freedom
## Multiple R-squared: 0.3072, Adjusted R-squared: 0.3037
## F-statistic: 87.8 on 1 and 198 DF, p-value: < 2.2e-169.2 Table 9.4
Unit root tests for log price ratio euro zone versus UK
df <- read_dta("Data/ppp2.dta")
df$ratio <- log(df$cpieuro)-log(df$cpiuk)
urt <- NULL
for (i in c(1:7, 13, 25, 37)) {
urt=rbind(urt, c(ur.df(df$ratio, lag=i-1, type="drift")@testreg$coefficients[2, 3],
ur.df(df$ratio, lag=i-1, type="trend")@testreg$coefficients[2, 3]))
}
rownames(urt) <- paste0(c(rep("ADF(", 10)), c(0:6, 12, 24, 36), ")")
colnames(urt) <- c("Without trend", "With trend")
urt## Without trend With trend
## ADF(0) -2.487359 -2.564232
## ADF(1) -2.532650 -2.622145
## ADF(2) -2.517573 -2.639083
## ADF(3) -2.136895 -2.288173
## ADF(4) -2.069701 -2.228613
## ADF(5) -2.037469 -2.212647
## ADF(6) -2.102550 -2.227121
## ADF(12) -2.989311 -3.040884
## ADF(24) -3.130562 -3.424346
## ADF(36) -2.026738 -1.9749849.3 Table 9.5
OLS results
##
## Call:
## lm(formula = log(fxep) ~ ratio, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.24561 -0.08619 0.01368 0.06477 0.22179
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.344580 0.009013 38.231 < 2e-16 ***
## ratio 1.016570 0.281252 3.614 0.000358 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1045 on 274 degrees of freedom
## Multiple R-squared: 0.04551, Adjusted R-squared: 0.04203
## F-statistic: 13.06 on 1 and 274 DF, p-value: 0.00035819.4 Table 9.6
ADF (cointegration) tests of residuals
DF=ur.df(OLS$residuals, lag=0, type="none")@testreg$coef[1, 3]
ADF1=ur.df(OLS$residuals, lag=1, type="none")@testreg$coef[1, 3]
ADF2=ur.df(OLS$residuals, lag=2, type="none")@testreg$coef[1, 3]
ADF3=ur.df(OLS$residuals, lag=3, type="none")@testreg$coef[1, 3]
ADF4=ur.df(OLS$residuals, lag=4, type="none")@testreg$coef[1, 3]
ADF5=ur.df(OLS$residuals, lag=5, type="none")@testreg$coef[1, 3]
ADF6=ur.df(OLS$residuals, lag=6, type="none")@testreg$coef[1, 3]
ttest <- cbind(DF, ADF1, ADF2,ADF3,ADF4,ADF5,ADF6)
ttest## DF ADF1 ADF2 ADF3 ADF4 ADF5 ADF6
## [1,] -1.500324 -1.4827 -1.436882 -1.486919 -1.638509 -1.531673 -1.4017419.5 Table 9.7
OLS results
##
## Call:
## lm(formula = log(fxep) ~ log(cpieuro) + log(cpiuk), data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.24406 -0.08568 0.01416 0.06590 0.22198
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.3782 0.1893 1.998 0.046703 *
## log(cpieuro) 1.0076 0.2863 3.520 0.000506 ***
## log(cpiuk) -1.0151 0.2819 -3.601 0.000376 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1047 on 273 degrees of freedom
## Multiple R-squared: 0.04562, Adjusted R-squared: 0.03863
## F-statistic: 6.525 on 2 and 273 DF, p-value: 0.0017069.6 Table 9.8
ADF (cointegration) tests of residuals
DF=ur.df(OLS2$residuals, lag=0, type="none")@testreg$coef[1, 3]
ADF1=ur.df(OLS2$residuals, lag=1, type="none")@testreg$coef[1, 3]
ADF2=ur.df(OLS2$residuals, lag=2, type="none")@testreg$coef[1, 3]
ADF3=ur.df(OLS2$residuals, lag=3, type="none")@testreg$coef[1, 3]
ADF4=ur.df(OLS2$residuals, lag=4, type="none")@testreg$coef[1, 3]
ADF5=ur.df(OLS2$residuals, lag=5, type="none")@testreg$coef[1, 3]
ADF6=ur.df(OLS2$residuals, lag=6, type="none")@testreg$coef[1, 3]
ttest <- cbind(DF, ADF1, ADF2,ADF3,ADF4,ADF5,ADF6)
ttest## DF ADF1 ADF2 ADF3 ADF4 ADF5 ADF6
## [1,] -1.511092 -1.493352 -1.445349 -1.493263 -1.643185 -1.537527 -1.4088999.7 Table 9.10
Maximum eigenvalue tests for cointegration
Y <- cbind(s=df$fxep, p=df$cpieuro, pstar=df$cpiuk)
summary(joci.3 <- ca.jo(log(Y), type="eigen", ecdet="const", K=3))##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.602005e-01 5.538624e-02 1.483851e-02 -1.114583e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 2 | 4.08 7.52 9.24 12.97
## r <= 1 | 15.56 13.75 15.67 20.20
## r = 0 | 82.28 19.77 22.00 26.81
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## s.l3 p.l3 pstar.l3 constant
## s.l3 1.000000 1.000000 1.000000 1.000000
## p.l3 -8.945951 -4.492447 6.539565 -0.421128
## pstar.l3 11.432758 4.644277 -6.000746 -1.109746
## constant -13.013509 -1.081750 -2.593314 6.492272
##
## Weights W:
## (This is the loading matrix)
##
## s.l3 p.l3 pstar.l3 constant
## s.d 0.001815465 0.0004296142 -0.0123638920 3.918377e-15
## p.d -0.001592238 0.0007878685 -0.0002595288 -9.948096e-16
## pstar.d -0.001690273 -0.0057697130 -0.0002100942 -1.733606e-159.8 Table 9.11
Maximum eigenvalue tests for cointegration
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 6.240329e-02 3.594226e-02 1.666837e-02 1.387779e-17
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 2 | 4.42 7.52 9.24 12.97
## r <= 1 | 9.63 13.75 15.67 20.20
## r = 0 | 16.95 19.77 22.00 26.81
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## s.l13 p.l13 pstar.l13 constant
## s.l13 1.000000 1.0000000 1.0000000 1.000000
## p.l13 -9.680770 0.9251439 0.7601737 -1.803155
## pstar.l13 10.186738 2.7358162 -0.8346392 -6.111982
## constant -3.019162 -18.6722070 0.1053253 35.439004
##
## Weights W:
## (This is the loading matrix)
##
## s.l13 p.l13 pstar.l13 constant
## s.d 0.006495869 -0.0051660978 -1.612575e-02 6.337177e-16
## p.d -0.000779766 -0.0004467662 1.787728e-03 -3.986908e-16
## pstar.d -0.004050001 -0.0002400787 9.250164e-05 -1.298598e-159.9 Table 9.12
Johansen estimation results
## s.l13 p.l13 pstar.l13 constant
## s.l13 1.000000 1.0000000 1.0000000 1.000000
## p.l13 -9.680770 0.9251439 0.7601737 -1.803155
## pstar.l13 10.186738 2.7358162 -0.8346392 -6.111982
## constant -3.019162 -18.6722070 0.1053253 35.4390049.10 Table 9.13
Univariate cointegrating regressions by OLS
df <- read_dta("Data/money.dta")
colnames(df)=c("date", "infl", "m", "y", "cpr", "tbr")
summary(MD <- lm(m ~ y+tbr, data=df))##
## Call:
## lm(formula = m ~ y + tbr, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.141931 -0.036746 -0.001974 0.044776 0.147458
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.188592 0.122285 26.07 <2e-16 ***
## y 0.422754 0.015903 26.58 <2e-16 ***
## tbr -0.031172 0.001879 -16.59 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.05769 on 161 degrees of freedom
## Multiple R-squared: 0.815, Adjusted R-squared: 0.8127
## F-statistic: 354.6 on 2 and 161 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.118571 -0.008014 0.002469 0.011418 0.069896
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0000664 0.0018236 -0.036 0.97100
## z.lag.1 -0.1302751 0.0411833 -3.163 0.00189 **
## z.diff.lag1 0.3548291 0.0799944 4.436 1.77e-05 ***
## z.diff.lag2 -0.3485983 0.0834526 -4.177 5.01e-05 ***
## z.diff.lag3 0.4370226 0.0846448 5.163 7.66e-07 ***
## z.diff.lag4 -0.1496378 0.0889194 -1.683 0.09450 .
## z.diff.lag5 0.3508476 0.0795820 4.409 1.98e-05 ***
## z.diff.lag6 -0.0910213 0.0828410 -1.099 0.27365
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02275 on 149 degrees of freedom
## Multiple R-squared: 0.2758, Adjusted R-squared: 0.2418
## F-statistic: 8.106 on 7 and 149 DF, p-value: 2.419e-08##
## Call:
## lm(formula = infl ~ tbr, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8222 -1.4433 -0.2281 1.1153 6.9737
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.15799 0.33322 3.475 0.000655 ***
## tbr 0.55827 0.05278 10.578 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.98 on 162 degrees of freedom
## Multiple R-squared: 0.4085, Adjusted R-squared: 0.4049
## F-statistic: 111.9 on 1 and 162 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.2310 -0.8866 -0.0235 0.7727 4.2806
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.05534 0.11477 -0.482 0.63036
## z.lag.1 -0.13929 0.07270 -1.916 0.05729 .
## z.diff.lag1 -0.60433 0.09990 -6.049 1.12e-08 ***
## z.diff.lag2 -0.38192 0.11005 -3.471 0.00068 ***
## z.diff.lag3 -0.26250 0.11221 -2.339 0.02065 *
## z.diff.lag4 -0.09450 0.10831 -0.872 0.38435
## z.diff.lag5 -0.04630 0.10043 -0.461 0.64547
## z.diff.lag6 -0.05611 0.08079 -0.695 0.48840
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.437 on 149 degrees of freedom
## Multiple R-squared: 0.3607, Adjusted R-squared: 0.3307
## F-statistic: 12.01 on 7 and 149 DF, p-value: 4.236e-12##
## Call:
## lm(formula = cpr ~ tbr, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.85555 -0.23869 -0.07406 0.12861 2.47128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.46126 0.06533 7.06 4.63e-11 ***
## tbr 1.03795 0.01035 100.31 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3882 on 162 degrees of freedom
## Multiple R-squared: 0.9842, Adjusted R-squared: 0.9841
## F-statistic: 1.006e+04 on 1 and 162 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.74347 -0.14938 -0.04633 0.13622 1.48284
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0001915 0.0232992 0.008 0.9935
## z.lag.1 -0.3855820 0.0953090 -4.046 8.35e-05 ***
## z.diff.lag1 0.1697979 0.1007128 1.686 0.0939 .
## z.diff.lag2 -0.1257426 0.0979529 -1.284 0.2012
## z.diff.lag3 0.1154392 0.0956164 1.207 0.2292
## z.diff.lag4 0.0852857 0.0932637 0.914 0.3620
## z.diff.lag5 0.0535072 0.0832304 0.643 0.5213
## z.diff.lag6 -0.0878361 0.0813895 -1.079 0.2822
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2919 on 149 degrees of freedom
## Multiple R-squared: 0.2546, Adjusted R-squared: 0.2195
## F-statistic: 7.269 on 7 and 149 DF, p-value: 1.721e-079.11 Figure 9.1
Residuals of money demand regression.
plot(MD$residuals, type="l", xlab="", ylab="", col="blue", ylim=c(-0.15, 0.15), sub="Figure 9.1 Residuals of money demand regression.", cex.sub=.8)
abline(h=0, lty=2)
9.12 Figure 9.2
Residuals of Fisher regression.
plot(FE$residuals, type="l", xlab="", ylab="", col="brown", ylim=c(-4, 8), sub="Figure 9.2 Residuals of Fisher regression.", cex.sub=.8)
abline(h=0, lty=2)
9.13 Figure 9.3
Residuals of risk premium regression.
plot(RP$residuals, type="l", xlab="", ylab="", col="orange", ylim=c(-1, 3), sub="Figure 9.3 Residuals of risk premium regression.", cex.sub=.8)
abline(h=0, lty=2)
9.14 Table 9.14
Trace and maximum eigenvalue tests for cointegration
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.458156e-01 1.731705e-01 1.027893e-01 5.100304e-02 2.239700e-02
## [6] 5.575823e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 3.60 7.52 9.24 12.97
## r <= 3 | 11.93 17.85 19.96 24.60
## r <= 2 | 29.17 32.00 34.91 41.07
## r <= 1 | 59.41 49.65 53.12 60.16
## r = 0 | 104.26 71.86 76.07 84.45
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## infl.l5 m.l5 y.l5 cpr.l5 tbr.l5 constant
## infl.l5 1.000000 1.00000 1.0000000 1.0000000 1.0000000 1.000000
## m.l5 36.656697 29.74807 54.0899909 3.8165096 -5.0106647 -51.349675
## y.l5 -15.856453 -18.56765 -30.5983755 -2.5538370 7.1138867 6.304776
## cpr.l5 -2.571899 -34.04671 0.9079543 -1.1598350 0.3472115 1.646984
## tbr.l5 3.638924 36.01204 1.1681748 0.1777462 -1.0667505 -2.343884
## constant -120.335303 -27.47772 -117.5286189 -1.2478775 -24.9127841 275.375048
##
## Weights W:
## (This is the loading matrix)
##
## infl.l5 m.l5 y.l5 cpr.l5 tbr.l5
## infl.d -0.076905806 -4.476970e-02 -0.0503732918 0.0451856048 -0.0520655925
## m.d -0.000249533 1.600878e-04 -0.0004854944 -0.0003452946 0.0001517958
## y.d -0.001449370 7.724998e-05 -0.0001018686 0.0004575570 0.0001025061
## cpr.d 0.064149061 9.989139e-03 -0.0289892535 0.0623279416 0.0026641115
## tbr.d 0.047738044 -1.682065e-04 -0.0294019158 0.0532235550 0.0139059927
## constant
## infl.d -6.471404e-14
## m.d -3.077914e-16
## y.d -6.298405e-16
## cpr.d 1.378946e-14
## tbr.d 1.232139e-15##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.690240e-01 1.971095e-01 1.299891e-01 7.253771e-02 1.732417e-02
## [6] -1.882436e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 2.76 7.52 9.24 12.97
## r <= 3 | 14.66 17.85 19.96 24.60
## r <= 2 | 36.66 32.00 34.91 41.07
## r <= 1 | 71.35 49.65 53.12 60.16
## r = 0 | 120.86 71.86 76.07 84.45
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## infl.l6 m.l6 y.l6 cpr.l6 tbr.l6 constant
## infl.l6 1.000000 1.00000 1.000000 1.0000000 1.0000000 1.00000
## m.l6 36.334894 204.19217 57.751754 1.6133149 -10.1762449 -200.70476
## y.l6 -15.962519 -74.54249 -31.016102 -2.8609857 7.2899859 -27.84829
## cpr.l6 -4.676546 98.22571 -3.901578 -0.1247101 0.7361456 28.87986
## tbr.l6 5.764312 -94.35876 6.197234 -0.6761898 -1.3263551 -25.04501
## constant -115.861539 -832.43682 -134.719135 13.3510670 5.0054743 1448.32325
##
## Weights W:
## (This is the loading matrix)
##
## infl.l6 m.l6 y.l6 cpr.l6 tbr.l6
## infl.d -9.120064e-02 8.038836e-03 -1.090757e-01 0.0438439893 -4.825389e-02
## m.d -9.171581e-05 -9.180989e-05 -2.395308e-04 -0.0004508571 1.371533e-04
## y.d -1.475957e-03 -5.898253e-05 4.127615e-05 0.0006668840 -1.867595e-05
## cpr.d 8.931169e-02 -3.201404e-03 -1.888907e-02 0.0863241017 -1.684758e-03
## tbr.d 4.960367e-02 -1.101318e-03 -2.911223e-02 0.0784889361 8.438562e-03
## constant
## infl.d -1.763205e-14
## m.d -2.650060e-16
## y.d -2.364325e-16
## cpr.d -5.776105e-15
## tbr.d -8.371206e-15##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.458156e-01 1.731705e-01 1.027893e-01 5.100304e-02 2.239700e-02
## [6] 5.575823e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 3.60 7.52 9.24 12.97
## r <= 3 | 8.32 13.75 15.67 20.20
## r <= 2 | 17.25 19.77 22.00 26.81
## r <= 1 | 30.23 25.56 28.14 33.24
## r = 0 | 44.86 31.66 34.40 39.79
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## infl.l5 m.l5 y.l5 cpr.l5 tbr.l5 constant
## infl.l5 1.000000 1.00000 1.0000000 1.0000000 1.0000000 1.000000
## m.l5 36.656697 29.74807 54.0899909 3.8165096 -5.0106647 -51.349675
## y.l5 -15.856453 -18.56765 -30.5983755 -2.5538370 7.1138867 6.304776
## cpr.l5 -2.571899 -34.04671 0.9079543 -1.1598350 0.3472115 1.646984
## tbr.l5 3.638924 36.01204 1.1681748 0.1777462 -1.0667505 -2.343884
## constant -120.335303 -27.47772 -117.5286189 -1.2478775 -24.9127841 275.375048
##
## Weights W:
## (This is the loading matrix)
##
## infl.l5 m.l5 y.l5 cpr.l5 tbr.l5
## infl.d -0.076905806 -4.476970e-02 -0.0503732918 0.0451856048 -0.0520655925
## m.d -0.000249533 1.600878e-04 -0.0004854944 -0.0003452946 0.0001517958
## y.d -0.001449370 7.724998e-05 -0.0001018686 0.0004575570 0.0001025061
## cpr.d 0.064149061 9.989139e-03 -0.0289892535 0.0623279416 0.0026641115
## tbr.d 0.047738044 -1.682065e-04 -0.0294019158 0.0532235550 0.0139059927
## constant
## infl.d -6.471404e-14
## m.d -3.077914e-16
## y.d -6.298405e-16
## cpr.d 1.378946e-14
## tbr.d 1.232139e-15##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.690240e-01 1.971095e-01 1.299891e-01 7.253771e-02 1.732417e-02
## [6] -1.882436e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 4 | 2.76 7.52 9.24 12.97
## r <= 3 | 11.90 13.75 15.67 20.20
## r <= 2 | 22.00 19.77 22.00 26.81
## r <= 1 | 34.69 25.56 28.14 33.24
## r = 0 | 49.51 31.66 34.40 39.79
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## infl.l6 m.l6 y.l6 cpr.l6 tbr.l6 constant
## infl.l6 1.000000 1.00000 1.000000 1.0000000 1.0000000 1.00000
## m.l6 36.334894 204.19217 57.751754 1.6133149 -10.1762449 -200.70476
## y.l6 -15.962519 -74.54249 -31.016102 -2.8609857 7.2899859 -27.84829
## cpr.l6 -4.676546 98.22571 -3.901578 -0.1247101 0.7361456 28.87986
## tbr.l6 5.764312 -94.35876 6.197234 -0.6761898 -1.3263551 -25.04501
## constant -115.861539 -832.43682 -134.719135 13.3510670 5.0054743 1448.32325
##
## Weights W:
## (This is the loading matrix)
##
## infl.l6 m.l6 y.l6 cpr.l6 tbr.l6
## infl.d -9.120064e-02 8.038836e-03 -1.090757e-01 0.0438439893 -4.825389e-02
## m.d -9.171581e-05 -9.180989e-05 -2.395308e-04 -0.0004508571 1.371533e-04
## y.d -1.475957e-03 -5.898253e-05 4.127615e-05 0.0006668840 -1.867595e-05
## cpr.d 8.931169e-02 -3.201404e-03 -1.888907e-02 0.0863241017 -1.684758e-03
## tbr.d 4.960367e-02 -1.101318e-03 -2.911223e-02 0.0784889361 8.438562e-03
## constant
## infl.d -1.763205e-14
## m.d -2.650060e-16
## y.d -2.364325e-16
## cpr.d -5.776105e-15
## tbr.d -8.371206e-159.15 Table 9.15
ML estimates of cointegrating vectors
md = data.frame(df$m, df$infl, df$y, df$tbr)
md.6 = ca.jo(md, type="eigen", ecdet="const", K=6)
summary(md.6)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.430975e-01 1.230206e-01 6.779006e-02 1.956202e-02 -2.710379e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 3 | 3.12 7.52 9.24 12.97
## r <= 2 | 11.09 13.75 15.67 20.20
## r <= 1 | 20.74 19.77 22.00 26.81
## r = 0 | 44.01 25.56 28.14 33.24
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## df.m.l6 df.infl.l6 df.y.l6 df.tbr.l6 constant
## df.m.l6 1.00000000 1.00000000 1.0000000 1.00000000 1.00000000
## df.infl.l6 0.02522691 0.01302547 0.5800574 -0.06852558 -0.01314323
## df.y.l6 -0.43486245 -0.51786656 -1.1256148 -0.42825578 0.73439460
## df.tbr.l6 0.02697521 0.03907833 -0.4986329 0.02561792 -0.05203717
## constant -3.29379438 -2.51378290 3.3856274 -2.80155676 -11.86397447
##
## Weights W:
## (This is the loading matrix)
##
## df.m.l6 df.infl.l6 df.y.l6 df.tbr.l6 constant
## df.m.d -0.02556870 -0.0222033813 -0.0007883538 -0.0032729768 -1.266765e-14
## df.infl.d -2.30607010 -3.8089602070 0.0646000111 0.9813842141 -1.296247e-12
## df.y.d -0.06005143 -0.0007372116 0.0013358677 -0.0007555107 -1.755969e-14
## df.tbr.d 1.78182874 -2.2913872495 0.1266841367 -0.1123889704 -1.300095e-14rp = data.frame(df$cpr, df$infl, df$y, df$tbr)
rp.6 = ca.jo(rp, type="eigen", ecdet="const", K=6)
summary(rp.6)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 2.033097e-01 1.152047e-01 4.507482e-02 2.911010e-02 3.207203e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 3 | 4.67 7.52 9.24 12.97
## r <= 2 | 7.29 13.75 15.67 20.20
## r <= 1 | 19.34 19.77 22.00 26.81
## r = 0 | 35.91 25.56 28.14 33.24
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## df.cpr.l6 df.infl.l6 df.y.l6 df.tbr.l6 constant
## df.cpr.l6 1.0000000 1.00000000 1.0000000 1.0000000 1.000000000
## df.infl.l6 -0.1372333 -0.01129626 0.5335812 -1.0340607 0.005299823
## df.y.l6 -0.3586022 0.16036590 3.5488249 1.2646617 11.287621216
## df.tbr.l6 -0.9315895 -1.01073363 -2.3294569 -0.7822223 -1.424690005
## constant 3.1739647 -2.21538165 -22.7777993 -7.3701421 -87.989126221
##
## Weights W:
## (This is the loading matrix)
##
## df.cpr.l6 df.infl.l6 df.y.l6 df.tbr.l6 constant
## df.cpr.d -0.500577200 -0.356905834 0.0375056148 -0.0043169541 1.049438e-15
## df.infl.d -0.067177872 0.587825534 0.0586485985 0.1239229209 -1.159858e-14
## df.y.d 0.002984518 -0.005698168 0.0005505938 0.0000244601 3.412406e-17
## df.tbr.d -0.360211378 -0.075795367 0.0443907971 -0.0165170383 -1.323401e-159.16 Table 9.16
df$ecm1 = as.matrix(md)%*%md.6@V[1:ncol(md), 1]
df$ecm2 = as.matrix(rp)%*%rp.6@V[1:ncol(rp), 1]
df <- df%>%
mutate(dm = m - lag(m, 1),
dinfl = infl - lag(infl, 1),
dcpr = cpr - lag(cpr, 1),
dy = y - lag(y, 1),
dtbr = tbr - lag(tbr, 1),
ecm1t = lag(ecm1, 1),
ecm2t = lag(ecm2, 1)
)
summary(lm(ecm1t ~ dm + dinfl + dcpr + dy + dtbr, data = df))##
## Call:
## lm(formula = ecm1t ~ dm + dinfl + dcpr + dy + dtbr, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.141479 -0.033344 -0.000882 0.036622 0.136619
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.1973739 0.0056192 569.007 < 2e-16 ***
## dm -3.6105953 0.4079460 -8.851 1.73e-15 ***
## dinfl -0.0142605 0.0025745 -5.539 1.25e-07 ***
## dcpr 0.0003991 0.0133508 0.030 0.97619
## dy -1.7000807 0.5343018 -3.182 0.00176 **
## dtbr 0.0089148 0.0154692 0.576 0.56524
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.05464 on 157 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.4698, Adjusted R-squared: 0.453
## F-statistic: 27.83 on 5 and 157 DF, p-value: < 2.2e-16