Homework 7, Problem 1
Andrii Baryshpolets
7 May 2017
Vector Error Correction (VEC) Models
library(Quandl)## Warning: package 'Quandl' was built under R version 3.3.3## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(zoo)
library(xts)
library(dygraphs)
library(knitr)
library(forecast)## Loading required package: timeDate## This is forecast 7.3library(urca)
library(vars)## Warning: package 'vars' was built under R version 3.3.3## Loading required package: MASS## Loading required package: strucchange## Warning: package 'strucchange' was built under R version 3.3.3## Loading required package: sandwich## Warning: package 'sandwich' was built under R version 3.3.3## Loading required package: lmtest## Warning: package 'lmtest' was built under R version 3.3.3library(tseries)
Quandl.api_key('rxVQhZ8_nxxeo2yy4Uz4')
COP <- Quandl("FRED/MCOILWTICO", type="ts")
RCGP <- Quandl("FRED/GASREGCOVM", type="ts")(a)
ly1=100*log(COP)
ly2=100*log(RCGP)
ly1 <- window(ly1, start=c(1995,1), end=c(2017,4))## Warning in window.default(x, ...): 'end' value not changedly2 <- window(ly2, start=c(1995,1), end=c(2017,4))
plot(cbind(ly1,ly2), col=c(2,4), plot.type = "single", xlab="Years 1995-2017")
legend("right", c(expression(Log_Crude_Oil_Prices),expression(Log_Regular_Conventional_Gas_Price)), col=c(2,4), lty=1, bty = "n")
(b)
dl.y1 <- diff(ly1,lag=1)
dl.y2 <- diff(ly2,lag=1)
par(mfrow=c(2,1))
plot(dl.y1, xlab=" Years 1995-2017", ylab="", main="Diff Log Crude Oil Prices")
plot(dl.y2, xlab=" Years 1995-2017", ylab="", main="Diff Log Regular Conventional Gas Price")
adf.test(ly1)##
## Augmented Dickey-Fuller Test
##
## data: ly1
## Dickey-Fuller = -1.6586, Lag order = 6, p-value = 0.7197
## alternative hypothesis: stationarykpss.test(ly1, null="Trend")## Warning in kpss.test(ly1, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: ly1
## KPSS Trend = 0.80651, Truncation lag parameter = 3, p-value = 0.01y1.urkpss <- ur.kpss(ly1, type="tau", lags="short")
summary(y1.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.5563
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216y1.ers1 <- ur.ers(ly1, type="P-test", model="trend")
summary(y1.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 10.1644
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89All three performed tests (ADF, KPSS, ERS) sugeest that y1 is not stationary.
adf.test(dl.y1)## Warning in adf.test(dl.y1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dl.y1
## Dickey-Fuller = -6.63, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationarykpss.test(dl.y1, null="Trend")## Warning in kpss.test(dl.y1, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dl.y1
## KPSS Trend = 0.052193, Truncation lag parameter = 3, p-value = 0.1dl.y1.urkpss <- ur.kpss(dl.y1, type="tau", lags="short")
summary(dl.y1.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.0514
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216dl.y1.ers1 <- ur.ers(dl.y1, type="P-test", model="trend")
summary(dl.y1.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.5881
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89All tests confirm that first difference logarithmic Crude Oil Prices is stationary.
adf.test(ly2)##
## Augmented Dickey-Fuller Test
##
## data: ly2
## Dickey-Fuller = -1.6181, Lag order = 6, p-value = 0.7368
## alternative hypothesis: stationarykpss.test(ly2, null="Trend")## Warning in kpss.test(ly2, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: ly2
## KPSS Trend = 0.7202, Truncation lag parameter = 3, p-value = 0.01y2.urkpss <- ur.kpss(ly2, type="tau", lags="short")
summary(y2.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.5035
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216y2.ers1 <- ur.ers(ly2, type="P-test", model="trend")
summary(y2.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 8.1303
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89All three performed tests (ADF, KPSS, ERS) sugeest that y2 is not stationary.
adf.test(dl.y2)## Warning in adf.test(dl.y2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dl.y2
## Dickey-Fuller = -7.9022, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationarykpss.test(dl.y2, null="Trend")## Warning in kpss.test(dl.y2, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dl.y2
## KPSS Trend = 0.037877, Truncation lag parameter = 3, p-value = 0.1dl.y2.urkpss <- ur.kpss(dl.y2, type="tau", lags="short")
summary(dl.y2.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.0403
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216dl.y2.ers1 <- ur.ers(dl.y2, type="P-test", model="trend")
summary(dl.y2.ers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0533
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89All tests confirm that first difference logarithmic Regular Conventional Gas Price is stationary.
(c)
y <- cbind(ly1, ly2)
y <- window(y, start=c(1995,1), end=c(2010,12))
VARselect(y, type="const", lag.max=12)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 6 6 2 6
##
## $criteria
## 1 2 3 4 5
## AIC(n) 7.353484 7.050363 7.000602 7.015203 6.968809
## HQ(n) 7.396638 7.122286 7.101294 7.144664 7.127039
## SC(n) 7.459916 7.227750 7.248944 7.334499 7.359059
## FPE(n) 1561.637976 1153.310834 1097.380193 1113.619041 1063.280429
## 6 7 8 9 10
## AIC(n) 6.908009 6.931080 6.921235 6.926159 6.945912
## HQ(n) 7.095008 7.146848 7.165772 7.199464 7.247987
## SC(n) 7.369214 7.463239 7.524349 7.600227 7.690935
## FPE(n) 1000.758328 1024.392235 1014.716198 1020.183056 1041.114353
## 11 12
## AIC(n) 6.956168 6.983254
## HQ(n) 7.287012 7.342867
## SC(n) 7.772146 7.870186
## FPE(n) 1052.556306 1082.326306Schwarz information criterion suggests using 2 lags.
y.CA <- ca.jo(y, ecdet="const", type="trace", K=2, spec="transitory", season=12)
summary(y.CA)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 1.980704e-01 9.957529e-03 -3.903128e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 1.90 7.52 9.24 12.97
## r = 0 | 43.84 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ly1.l1 ly2.l1 constant
## ly1.l1 1.000000 1.0000000 1.0000000
## ly2.l1 -1.587125 -0.1377993 0.2504144
## constant -275.337070 -429.0868103 -322.0681760
##
## Weights W:
## (This is the loading matrix)
##
## ly1.l1 ly2.l1 constant
## ly1.d 0.04824429 -0.008302568 1.915537e-16
## ly2.d 0.24802732 -0.003053918 9.713652e-16y.CA <- ca.jo(y, ecdet="const", type="eigen", K=2, spec="transitory", season=12)
summary(y.CA)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 1.980704e-01 9.957529e-03 -3.903128e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 1.90 7.52 9.24 12.97
## r = 0 | 41.94 13.75 15.67 20.20
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ly1.l1 ly2.l1 constant
## ly1.l1 1.000000 1.0000000 1.0000000
## ly2.l1 -1.587125 -0.1377993 0.2504144
## constant -275.337070 -429.0868103 -322.0681760
##
## Weights W:
## (This is the loading matrix)
##
## ly1.l1 ly2.l1 constant
## ly1.d 0.04824429 -0.008302568 1.915537e-16
## ly2.d 0.24802732 -0.003053918 9.713652e-16(d)
y.VEC <- cajorls(y.CA, r=1)
y.VEC## $rlm
##
## Call:
## lm(formula = substitute(form1), data = data.mat)
##
## Coefficients:
## ly1.d ly2.d
## ect1 0.04824 0.24803
## sd1 6.08302 4.57391
## sd2 2.31774 2.22917
## sd3 8.03228 6.21726
## sd4 4.09514 5.92019
## sd5 5.27266 6.26258
## sd6 4.71141 4.85558
## sd7 4.94215 3.27466
## sd8 5.02513 4.23428
## sd9 3.40821 4.01838
## sd10 1.76899 -0.01183
## sd11 -0.20628 -0.33635
## ly1.dl1 0.27002 0.16276
## ly2.dl1 -0.11007 0.18754
##
##
## $beta
## ect1
## ly1.l1 1.000000
## ly2.l1 -1.587125
## constant -275.337070summary(y.VEC$rlm)## Response ly1.d :
##
## Call:
## lm(formula = ly1.d ~ ect1 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 +
## sd7 + sd8 + sd9 + sd10 + sd11 + ly1.dl1 + ly2.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -26.112 -4.685 1.140 6.214 19.117
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.04824 0.08132 0.593 0.55375
## sd1 6.08302 3.00078 2.027 0.04416 *
## sd2 2.31774 3.05064 0.760 0.44842
## sd3 8.03228 2.99738 2.680 0.00807 **
## sd4 4.09514 3.07874 1.330 0.18520
## sd5 5.27266 3.14608 1.676 0.09552 .
## sd6 4.71141 3.17557 1.484 0.13969
## sd7 4.94215 3.14817 1.570 0.11825
## sd8 5.02513 3.07894 1.632 0.10445
## sd9 3.40821 3.04442 1.119 0.26446
## sd10 1.76899 3.03716 0.582 0.56101
## sd11 -0.20628 2.96477 -0.070 0.94461
## ly1.dl1 0.27002 0.11374 2.374 0.01867 *
## ly2.dl1 -0.11007 0.14766 -0.745 0.45702
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.325 on 176 degrees of freedom
## Multiple R-squared: 0.134, Adjusted R-squared: 0.06508
## F-statistic: 1.945 on 14 and 176 DF, p-value: 0.02469
##
##
## Response ly2.d :
##
## Call:
## lm(formula = ly2.d ~ ect1 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 +
## sd7 + sd8 + sd9 + sd10 + sd11 + ly1.dl1 + ly2.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.3100 -2.3394 0.3539 3.5036 11.8312
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.24803 0.04799 5.169 6.35e-07 ***
## sd1 4.57391 1.77075 2.583 0.010606 *
## sd2 2.22917 1.80017 1.238 0.217249
## sd3 6.21726 1.76874 3.515 0.000559 ***
## sd4 5.92019 1.81675 3.259 0.001343 **
## sd5 6.26258 1.85649 3.373 0.000913 ***
## sd6 4.85558 1.87389 2.591 0.010367 *
## sd7 3.27466 1.85772 1.763 0.079681 .
## sd8 4.23428 1.81687 2.331 0.020912 *
## sd9 4.01838 1.79650 2.237 0.026556 *
## sd10 -0.01183 1.79221 -0.007 0.994740
## sd11 -0.33635 1.74950 -0.192 0.847763
## ly1.dl1 0.16276 0.06712 2.425 0.016318 *
## ly2.dl1 0.18754 0.08713 2.152 0.032732 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.913 on 176 degrees of freedom
## Multiple R-squared: 0.4769, Adjusted R-squared: 0.4353
## F-statistic: 11.46 on 14 and 176 DF, p-value: < 2.2e-16(e)
adjustment parameters are estimated as \(\alpha\)= (0.04824, 0.24803). \(\alpha_1\) is not significant and \(\alpha_2\) is significant at the level of 1%. No, their signs are not consistent with error correction mechanism that moves the system back to the long run equilibrium as they are both positive.
(f)
rest.alpha <- matrix(c(1,0), c(2,1))
y.CA.ralpha <- alrtest(y.CA, A=rest.alpha, r=1)
summary(y.CA.ralpha)##
## ######################
## # 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]
## [1,] 1
## [2,] 0
##
## Eigenvalues of restricted VAR (lambda):
## [1] 0.082 0.000 0.000
##
## The value of the likelihood ratio test statistic:
## 25.68 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]
## RK.ly1.l1 1.0000
## RK.ly2.l1 -1.5589
## RK.constant -278.3270
##
## Weights W of the restricted VAR:
##
## [,1]
## [1,] -0.2446
## [2,] 0.0000VEC with restriction on alpha_2
y.VEC.rest1 <- cajorls(y.CA.ralpha, r=1)
y.VEC.rest1## $rlm
##
## Call:
## lm(formula = substitute(form1), data = data.mat)
##
## Coefficients:
## ly1.d ly2.d
## ect1 0.01989 0.22771
## sd1 6.16105 4.65599
## sd2 2.35313 2.25799
## sd3 8.05228 6.27209
## sd4 4.01057 5.86157
## sd5 5.08712 6.11692
## sd6 4.39434 4.58973
## sd7 4.58845 2.97139
## sd8 4.73235 3.97373
## sd9 3.18491 3.80830
## sd10 1.57007 -0.20095
## sd11 -0.32171 -0.44939
## ly1.dl1 0.29063 0.18202
## ly2.dl1 -0.12807 0.17142
##
##
## $beta
## ect1
## ly1.l1 1.00000
## ly2.l1 -1.55894
## constant -278.32704summary(y.VEC.rest1$rlm)## Response ly1.d :
##
## Call:
## lm(formula = ly1.d ~ ect1 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 +
## sd7 + sd8 + sd9 + sd10 + sd11 + ly1.dl1 + ly2.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25.946 -4.719 1.290 6.271 18.953
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.01989 0.08040 0.247 0.80487
## sd1 6.16105 3.00238 2.052 0.04165 *
## sd2 2.35313 3.05307 0.771 0.44189
## sd3 8.05228 2.99965 2.684 0.00796 **
## sd4 4.01057 3.08111 1.302 0.19473
## sd5 5.08712 3.14682 1.617 0.10776
## sd6 4.39434 3.17109 1.386 0.16758
## sd7 4.58845 3.14109 1.461 0.14586
## sd8 4.73235 3.07381 1.540 0.12546
## sd9 3.18491 3.04160 1.047 0.29648
## sd10 1.57007 3.03530 0.517 0.60562
## sd11 -0.32171 2.96558 -0.108 0.91374
## ly1.dl1 0.29063 0.11263 2.580 0.01069 *
## ly2.dl1 -0.12807 0.14716 -0.870 0.38534
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.332 on 176 degrees of freedom
## Multiple R-squared: 0.1325, Adjusted R-squared: 0.06354
## F-statistic: 1.921 on 14 and 176 DF, p-value: 0.02696
##
##
## Response ly2.d :
##
## Call:
## lm(formula = ly2.d ~ ect1 + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 +
## sd7 + sd8 + sd9 + sd10 + sd11 + ly1.dl1 + ly2.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.4165 -1.8241 0.8595 3.8376 11.9833
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.22771 0.04790 4.754 4.12e-06 ***
## sd1 4.65599 1.78847 2.603 0.010020 *
## sd2 2.25799 1.81866 1.242 0.216049
## sd3 6.27209 1.78684 3.510 0.000569 ***
## sd4 5.86157 1.83537 3.194 0.001665 **
## sd5 6.11692 1.87451 3.263 0.001323 **
## sd6 4.58973 1.88896 2.430 0.016114 *
## sd7 2.97139 1.87110 1.588 0.114070
## sd8 3.97373 1.83102 2.170 0.031329 *
## sd9 3.80830 1.81183 2.102 0.036984 *
## sd10 -0.20095 1.80807 -0.111 0.911631
## sd11 -0.44939 1.76655 -0.254 0.799493
## ly1.dl1 0.18202 0.06709 2.713 0.007331 **
## ly2.dl1 0.17142 0.08766 1.955 0.052119 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.963 on 176 degrees of freedom
## Multiple R-squared: 0.4661, Adjusted R-squared: 0.4236
## F-statistic: 10.97 on 14 and 176 DF, p-value: < 2.2e-16(g)
It is not possible to achieve the long run equilibrium, if we do not restrict one of the alphas to be equal to 0.
(h)
Rolling Scheme one-month ahead forecast
ly1=100*log(COP)
ly2=100*log(RCGP)
y.fcst <- cbind(ly1, ly2)
y.fcst <- na.trim(y.fcst)
y.fcst <- window(y.fcst, start=1995+0, end=2017+3/12)## Warning in window.default(x, ...): 'end' value not changedfstM <- 1995+0
lstM <- 2010+11/12
y.1 <- window(y.fcst, end=lstM)
y.2 <- window(y.fcst, start=lstM+1/12)
y.VAR.f1 <-data.frame()
y.VAR.f2 <-data.frame()
for(i in 1:length(y.2[,2]))
{
y <- window(y.fcst, end = lstM + (i-1)/12)
y.CA <- ca.jo(y, ecdet="const", type="eigen", K=3, spec="transitory")
y.VAR <- vec2var(y.CA, r=1)
y.VAR.updt <- predict(y.VAR, n.ahead=1)
y.VAR.f1 <-rbind(y.VAR.f1, as.ts(y.VAR.updt$fcst$ly1))
y.VAR.f2 <-rbind(y.VAR.f2, as.ts(y.VAR.updt$fcst$ly2))
}
y.VAR.f1 <- ts(y.VAR.f1, start=2011, frequency=12)
y.VAR.f2 <- ts(y.VAR.f2, start=2011, frequency=12)
y.pgas <-as.ts(y.fcst[,2])
y.pgas1 <-as.ts(y.1[,2])
y.pgas2 <-as.ts(y.2[,2])
par(mfrow=c(1,1))
plot(y.pgas, type="l", pch=16, lty="solid", main="One month ahead forecast for gas price", xlim=c(2010,2017), ylim=c(45,140))
lines(y.pgas1, type="o", pch=16, lty="solid")
lines(y.VAR.f2[,1], type="o", pch=16, lty="dashed", col="blue")
VEC.f.rol <- as.ts(y.pgas2)
accuracy(VEC.f.rol, y.VAR.f2[,1])## ME RMSE MAE MPE MAPE ACF1
## Test set -1.41565 4.930924 3.547289 -1.667412 3.922441 0.05211672
## Theil's U
## Test set 0.5858428RMSE is 4.930924.
(i)
par(mfrow=c(1,2))
acf(dl.y2, type="correlation", lag=48, xlab="Lag", ylab="ACF", main="Gas Price")
acf(dl.y2, type="partial", lag=48, xlab="Lag", ylab="PACF", main="Gas Price")
Considering the ACF and PACF, we can suggest different ARMA models. For further analysis we choose the simpliest possible ARMA(2,2).
arma22 <- Arima(y.pgas1, order=c(2, 0, 2))
tsdiag(arma22, gof.lag=36)
(j)
rol.f <-zoo()
for(i in 1: length(y.2[,2]))
{
y <- window(dl.y2, end = lstM+(i-1)/12)
rol.updt <- arima(y, order=c(2,0,2))
rol.f <- c(rol.f, forecast(rol.updt, 1)$mean)
}
rol.f <-as.ts(rol.f)
par(mfrow=c(1,1))
plot(rol.f, type="o", pch=16, xlim=c(2010, 2017), main="One month ahead forecast for ARMA (2,2)")
lines(rol.f, type="p", pch=16, lty="solid", col="red")
lines(dl.y2, type="l", pch=16, lty="solid")
lines(ly1, type="o", pch=16, lty="dashed")
accuracy(rol.f, dl.y2)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set -0.5683734 5.221744 3.814195 Inf Inf -0.06078359 NaNRMSE is 5.221744.
(k-l) Conclusion
RMSE is lower for VEC model. This result suggests that a VEC model is preferable to a univariate model in terms of forecasting gas prices.