Vector Error Correction (VECM) Model
Ibikunle Gabriel
2026-03-26
What is a VECM Model? A VECM is a special case of VAR that is used when variables are non-stationary but cointegrated.
Cointegration means variables share a long-run equilibrium even if each one individually trends over time. VECM captures both: Short-term dynamics (like VAR) Long-term equilibrium (error correction term) Error Correction Term: ΔYt=α(Yt−1−βXt−1)+short-term terms+ϵtY_t = ({t-1} - X{t-1}) + + _tΔYt​=α(Yt−1​−βXt−1​)+short-term terms+ϵt​
- If variables are cointegrated → use VECM.
- If variables are stationary → plain VAR is enough.
Set and confirm the working directory
getwd()## [1] "C:/Users/Hp/Documents/ARDL"Load required packages
Load the data into the R environment
data <- read.csv("C:\\Users\\Hp\\Documents\\ARDL\\ECON DEVELOPMENT DATA.csv")Select the Variables needed
data_var <- data[, c("ECNDEV","INF", "EDU", "K", "HEA", "UNEMP")]Do some data exploration
head(df)##
## 1 function (x, df1, df2, ncp, log = FALSE)
## 2 {
## 3 if (missing(ncp))
## 4 .Call(C_df, x, df1, df2, log)
## 5 else .Call(C_dnf, x, df1, df2, ncp, log)
## 6 }tail(df)##
## 1 function (x, df1, df2, ncp, log = FALSE)
## 2 {
## 3 if (missing(ncp))
## 4 .Call(C_df, x, df1, df2, log)
## 5 else .Call(C_dnf, x, df1, df2, ncp, log)
## 6 }glimpse(df)## function (x, df1, df2, ncp, log = FALSE)dim(df)## NULLConvert to time series object
data_var <- ts(data_var)Select optimal lag length
lag_select <- VARselect(data_var, lag.max = 10, type = "const")
lag_select$selection## AIC(n) HQ(n) SC(n) FPE(n)
## 4 4 4 4Perform Johansen Cointegration Test
johansen_test <- ca.jo(data_var,
type = "trace", # or "eigen"
ecdet = "const", # deterministic trend
K = 2) # lag = p
summary(johansen_test)##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 9.418956e-01 8.373230e-01 6.054274e-01 4.617373e-01 2.754849e-01
## [6] 2.063891e-01 7.771561e-16
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 5 | 6.47 7.52 9.24 12.97
## r <= 4 | 15.50 17.85 19.96 24.60
## r <= 3 | 32.84 32.00 34.91 41.07
## r <= 2 | 58.88 49.65 53.12 60.16
## r <= 1 | 109.73 71.86 76.07 84.45
## r = 0 | 189.40 97.18 102.14 111.01
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ECNDEV.l2 INF.l2 EDU.l2 K.l2 HEA.l2
## ECNDEV.l2 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
## INF.l2 2.384042e-02 -6.533930e-04 1.909929e-03 2.035639e-03 -3.739982e-04
## EDU.l2 -3.258790e-02 4.734721e-03 3.298484e-02 -2.230834e-02 5.325055e-03
## K.l2 -1.549129e-05 -6.649319e-07 8.761501e-07 -8.013002e-06 -2.299670e-06
## HEA.l2 7.653749e-02 -2.805510e-02 -8.155767e-02 1.247992e-01 -1.325824e-02
## UNEMP.l2 -2.726798e-01 -1.019339e-01 -3.541020e-02 1.142133e-01 1.132103e-02
## constant 1.399781e+00 -1.010100e-01 -1.382035e+00 -9.552031e-01 -6.489344e-01
## UNEMP.l2 constant
## ECNDEV.l2 1.000000e+00 1.000000e+00
## INF.l2 3.334640e-03 3.799670e-04
## EDU.l2 1.001677e-01 -9.457907e-03
## K.l2 -6.380153e-06 2.623403e-06
## HEA.l2 -4.258175e-02 1.760955e-03
## UNEMP.l2 -2.521259e-01 -8.258689e-02
## constant -3.997089e+00 2.409528e-01
##
## Weights W:
## (This is the loading matrix)
##
## ECNDEV.l2 INF.l2 EDU.l2 K.l2 HEA.l2
## ECNDEV.d -7.121601e-04 -8.768154e-02 -0.1095208 -0.03499773 -0.1809679
## INF.d -2.508586e+01 1.016641e+02 -27.0498918 12.79407731 -12.7552889
## EDU.d -5.281632e-02 -9.429869e-01 -1.2112211 -1.09774652 1.2936498
## K.d -3.540611e+03 -1.215327e+05 16982.5081788 142.50761496 19290.3602536
## HEA.d 5.880631e-02 1.161045e+00 3.7810039 -3.60644498 5.4506441
## UNEMP.d 1.380530e-01 -5.146009e-01 -0.9514587 2.24010606 5.1919656
## UNEMP.l2 constant
## ECNDEV.d 0.01818742 -3.585833e-14
## INF.d -1.32622944 -2.794687e-12
## EDU.d -0.24557811 9.466956e-13
## K.d 534.89303335 -4.532365e-08
## HEA.d 0.25770717 -1.057715e-11
## UNEMP.d 0.50879884 1.894373e-12Estimate VECM
vecm_model <- cajorls(johansen_test, r = 1) # r = number of cointegrating equations
summary(vecm_model$rlm)## Response ECNDEV.d :
##
## Call:
## lm(formula = ECNDEV.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 +
## K.dl1 + HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.044843 0.000052 0.002060 0.005358 0.009370
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -7.122e-04 5.823e-03 -0.122 0.904
## ECNDEV.dl1 -1.121e-01 2.161e-01 -0.518 0.610
## INF.dl1 6.444e-05 2.509e-04 0.257 0.800
## EDU.dl1 4.738e-03 7.709e-03 0.615 0.545
## K.dl1 4.339e-07 6.471e-07 0.671 0.510
## HEA.dl1 -3.089e-03 6.343e-03 -0.487 0.631
## UNEMP.dl1 1.652e-03 6.151e-03 0.269 0.791
##
## Residual standard error: 0.011 on 21 degrees of freedom
## Multiple R-squared: 0.1069, Adjusted R-squared: -0.1908
## F-statistic: 0.359 on 7 and 21 DF, p-value: 0.9158
##
##
## Response INF.d :
##
## Call:
## lm(formula = INF.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 +
## K.dl1 + HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5092 -2.7816 -0.5785 1.5357 7.8756
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -2.509e+01 2.421e+00 -10.361 1.04e-09 ***
## ECNDEV.dl1 -5.553e+01 8.987e+01 -0.618 0.5433
## INF.dl1 -5.559e-01 1.043e-01 -5.329 2.77e-05 ***
## EDU.dl1 7.391e+00 3.206e+00 2.306 0.0314 *
## K.dl1 -1.345e-03 2.691e-04 -4.997 6.03e-05 ***
## HEA.dl1 1.608e+00 2.637e+00 0.610 0.5487
## UNEMP.dl1 5.367e+00 2.557e+00 2.098 0.0481 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.573 on 21 degrees of freedom
## Multiple R-squared: 0.8382, Adjusted R-squared: 0.7843
## F-statistic: 15.54 on 7 and 21 DF, p-value: 5.228e-07
##
##
## Response EDU.d :
##
## Call:
## lm(formula = EDU.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 +
## K.dl1 + HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.36276 -0.05292 -0.01031 0.09743 0.26650
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -5.282e-02 7.871e-02 -0.671 0.5095
## ECNDEV.dl1 -4.330e-01 2.922e+00 -0.148 0.8836
## INF.dl1 -4.994e-03 3.391e-03 -1.473 0.1556
## EDU.dl1 9.419e-01 1.042e-01 9.039 1.1e-08 ***
## K.dl1 4.821e-06 8.747e-06 0.551 0.5873
## HEA.dl1 -1.199e-02 8.574e-02 -0.140 0.8901
## UNEMP.dl1 1.525e-01 8.314e-02 1.834 0.0809 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1487 on 21 degrees of freedom
## Multiple R-squared: 0.8666, Adjusted R-squared: 0.8221
## F-statistic: 19.48 on 7 and 21 DF, p-value: 7.438e-08
##
##
## Response K.d :
##
## Call:
## lm(formula = K.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 + K.dl1 +
## HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6750 -1080 -143 978 12012
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -3540.6110 2227.6405 -1.589 0.1269
## ECNDEV.dl1 22634.4567 82685.9137 0.274 0.7870
## INF.dl1 -74.7834 95.9723 -0.779 0.4445
## EDU.dl1 3776.1077 2949.3676 1.280 0.2144
## K.dl1 0.5227 0.2476 2.112 0.0469 *
## HEA.dl1 -618.6025 2426.4558 -0.255 0.8012
## UNEMP.dl1 -954.3335 2353.0815 -0.406 0.6892
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4207 on 21 degrees of freedom
## Multiple R-squared: 0.5817, Adjusted R-squared: 0.4423
## F-statistic: 4.172 on 7 and 21 DF, p-value: 0.005044
##
##
## Response HEA.d :
##
## Call:
## lm(formula = HEA.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 +
## K.dl1 + HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.17514 -0.02036 0.02056 0.08065 1.08749
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 5.881e-02 2.032e-01 0.289 0.775
## ECNDEV.dl1 -5.247e+00 7.541e+00 -0.696 0.494
## INF.dl1 -8.194e-04 8.752e-03 -0.094 0.926
## EDU.dl1 3.423e-01 2.690e-01 1.273 0.217
## K.dl1 -6.151e-06 2.258e-05 -0.272 0.788
## HEA.dl1 -4.701e-02 2.213e-01 -0.212 0.834
## UNEMP.dl1 3.586e-02 2.146e-01 0.167 0.869
##
## Residual standard error: 0.3837 on 21 degrees of freedom
## Multiple R-squared: 0.1116, Adjusted R-squared: -0.1846
## F-statistic: 0.3768 on 7 and 21 DF, p-value: 0.9056
##
##
## Response UNEMP.d :
##
## Call:
## lm(formula = UNEMP.d ~ ect1 + ECNDEV.dl1 + INF.dl1 + EDU.dl1 +
## K.dl1 + HEA.dl1 + UNEMP.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.81262 -0.11473 -0.00509 0.14447 0.88711
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 1.381e-01 1.688e-01 0.818 0.4227
## ECNDEV.dl1 6.453e+00 6.266e+00 1.030 0.3148
## INF.dl1 5.722e-03 7.273e-03 0.787 0.4403
## EDU.dl1 1.319e-01 2.235e-01 0.590 0.5613
## K.dl1 -3.853e-05 1.876e-05 -2.054 0.0527 .
## HEA.dl1 2.044e-02 1.839e-01 0.111 0.9125
## UNEMP.dl1 4.369e-01 1.783e-01 2.450 0.0231 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3188 on 21 degrees of freedom
## Multiple R-squared: 0.4884, Adjusted R-squared: 0.3178
## F-statistic: 2.863 on 7 and 21 DF, p-value: 0.02902Convert to VAR (for IRF, Forecast, etc.)
vecm_var <- vec2var(johansen_test, r = 1)Impulse Response Function (IRF)
irf_result <- irf(vecm_var, n.ahead = 10, boot = TRUE)## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs produced## Warning in chol.default(SKK, pivot = TRUE): the matrix is either rank-deficient
## or not positive definite## Warning in log(1 - lambda[x + 1]): NaNs producedplot(irf_result)
Forecast
forecast_vecm <- predict(vecm_var, n.ahead = 10)
plot(forecast_vecm)