Hw7
Abdullah Al Shammre
5/8/2017
Introduction
Using VEC model. First, obtain monthly time series for Crude Oil Prices: West Texas Intermediate (WTI), FRED/MCOILWTICO, and the monthly time series for US Regular Conventional Gas Price, FRED/GASREGCOVM.We compare the predictability of VEC model with those of forecasts using AR/MA/ARMA model.
Import data:
Use the monthly time series for Crude Oil Prices: West Texas Intermediate (WTI), FRED/MCOILWTICO.
library(Quandl)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(xts)
library(zoo)
library(dygraphs)
library(knitr)
library(forecast)library(urca)
library(vars)## Loading required package: MASS## Loading required package: strucchange## Loading required package: sandwich## Loading required package: lmtestlibrary(tseries)
Quandl.api_key('3fnnc4EE2uzFb1a44mAQ')
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")
The prices is different but the movement of them is similar.
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 6.89From all results 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 6.89Frome all results 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 6.89Also here from all test 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 6.89Here also shoes that 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 1082.326306y.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-16d.
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-16e.
a1 is not significant a1= (0.04824, 0.24803) and a2 is significant at the level of 1 percent.
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,] 0.0000y.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-16g.
When we do not restrict one of the alphas to be equal to 0. So, it impossible to achieve the long run equilibrium.
h.
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 0.5858428The RMSE for the forecast of the gas price 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")
arma22 <- Arima(y.pgas1, order=c(2, 0, 2))
tsdiag(arma22, gof.lag=36)
we choose the simpliest possible ARMA(2,2).
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")
k.
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 0.5858428accuracy(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 NaNThe RMSE based on the VEC model is 0.04930924 and it is smaller than those based on ARMA(2,6), 5.221744
l.
Gitting best result with smaller RMSE using a VEC model when variables are cointegrated. VEC model includes the error correction term and reduce the deviation from the equilibrium. When variables are cointegrated,
Conclusions
Since RMSE is lower for VEC model, VEC model is preferable to a univariate model in terms of forecasting gas prices.