Cointegration of mean years of schooling, income per capita and national education spending

Johansen Cointegration and VECM for Education and Learning

Field: Education and LearningLevel: Doctoral dissertation (S3)Method: Johansen Cointegration and VECM

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Testing Johansen cointegration and long-run equilibrium among mean years of schooling, GDP per capita and education spending with a VECM, illustrated with synthetic data. The Johansen cointegration test checks whether several non-stationary series share a long-run equilibrium, and the VECM separates short-run dynamics from the correction towards that equilibrium. It suits theses on prices, exchange rates, budgets or economic indicators that move together over the long run.

Cointegration of mean years of schooling, income per capita and national education spending: result sheet

Project info

Field of studyEducation and Learning
LevelDoctoral dissertation (S3)
Method typeTime Series
Estimated build time± 7 working days
Toolsurca::ur.df, urca::ca.jo, urca::cajorls, vars::vec2var, vars::irf, ggplot2
PriceUS$28 - US$178

Research goal

Testing Johansen cointegration and long-run equilibrium among mean years of schooling, GDP per capita and education spending with a VECM, illustrated with synthetic data.

Variables

Role	Variable	Unit
Main	Mean years of schooling	years
Supporting 1	GDP per capita	IDR million
Supporting 2	Education spending	IDR trillion

Example result

1Cointegration rank
44.385Trace stat (r = 0)
34.915% critical (r = 0)
1.105Long-run GDP per capita
-0.533Adjustment alpha
0.004Alpha p-value

ADF tests show that 3 of 3 series are I(1). The Johansen trace test (statistic 44.39 vs 5% critical 34.91) finds 1 cointegrating relation. In the long run, a one-unit rise in GDP per capita is associated with a 1.105 change in Mean years of schooling; the adjustment coefficient is -0.533 (p = 0.004), indicating error correction towards equilibrium.

Computed from illustrative synthetic data, not from a real study.

About Johansen Cointegration and VECM

The Johansen cointegration test checks whether several non-stationary series share a long-run equilibrium, and the VECM separates short-run dynamics from the correction towards that equilibrium. It suits theses on prices, exchange rates, budgets or economic indicators that move together over the long run.

Steps

1Run ADF unit root tests (urca::ur.df) on levels and first differences to confirm that all series are I(1)
2Run the Johansen test (urca::ca.jo, trace statistic, K = 2) and decide the number of cointegrating vectors
3Estimate the VECM with cajorls to obtain the cointegrating vector (beta) normalised on Y
4Read the adjustment coefficients (alpha) and test their significance
5Compute the error-correction term (ECT) and check that it is stationary around zero
6Convert the VECM to a level VAR (vec2var) and compute impulse responses

Assumptions to check

All series are integrated of order one, I(1)
At least one linear combination is stationary (cointegration)
VECM residuals are free of autocorrelation and the lag length is adequate
The deterministic terms (constant or trend) match the data pattern

Reading and reporting the results

If the trace statistic for r = 0 exceeds the 5% critical value, the series are cointegrated; report the rank used. The beta coefficients describe the long-run relation, while a negative and significant alpha in the Y equation shows that Y adjusts back to equilibrium after a deviation.

Example R code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

Show code

# Contoh kode R: Johansen Cointegration and VECM
suppressPackageStartupMessages({
  library(ggplot2)
  library(patchwork)
  library(urca)
  library(vars)
})
set.seed(42)
# ---- Gaya grafik ----
pal <- c("#1f6feb", "#e8590c", "#2f9e44", "#ae3ec9", "#f59f00")
warna <- list(bg = "#f4f6fa", panel = "#ffffff", teks = "#1f2933", redup = "#5f6b78", grid = "#e3e8ee", bar = "#1d3557")
tata <- 1
theme_set(theme_minimal(base_size = 11) + theme(legend.position = "bottom"))

# ---- Data sintetis: tiga deret I(1) yang berkointegrasi (tren stokastik bersama) ----
nt <- 120
w1 <- cumsum(rnorm(nt, 0, 0.6))                                  # tren stokastik (random walk) pertama
w2 <- cumsum(rnorm(nt, 0, 0.6))                                  # tren stokastik kedua
b1 <- runif(1, 0.6, 1.1); b2 <- runif(1, 0.3, 0.7) * sample(c(-1, 1), 1)
u <- as.numeric(arima.sim(list(ar = runif(1, 0.2, 0.5)), n = nt, sd = runif(1, 0.45, 0.8)))  # galat keseimbangan stasioner
zx1 <- w1 + rnorm(nt, 0, 0.1)
zx2 <- w2 + rnorm(nt, 0, 0.1)
zy <- b1 * zx1 + b2 * zx2 + u                                   # hubungan jangka panjang Y = b1 X1 + b2 X2 + u
sk <- function(v) (v - mean(v)) / sd(v)
d <- data.frame(periode = seq(as.Date("2015-01-01"), by = "month", length.out = nt),
                mean_years_of_schooling = round(pmax(100 + 15 * sk(zy), 0), 3),
                gdp_per_capita = round(pmax(50 + 10 * sk(zx1), 0), 3),
                education_spending = round(pmax(30 + 6 * sk(zx2), 0), 3))
write.csv(d, "data_contoh.csv", row.names = FALSE)

# ---- Uji akar unit ADF: level vs diferensi pertama ----
dv <- d[, c("mean_years_of_schooling", "gdp_per_capita", "education_spending")]
adf_tau <- function(v, tipe) { u_ <- ur.df(v, type = tipe, lags = 4, selectlags = "AIC"); c(u_@teststat[1], u_@cval[1, 2]) }
adf <- t(sapply(dv, function(v) c(adf_tau(v, "drift"), adf_tau(diff(v), "drift"))))
colnames(adf) <- c("tau_level", "kritis5_level", "tau_diff", "kritis5_diff")
print(round(adf, 3))                                            # tolak H0 akar unit bila tau < nilai kritis
i1 <- sum(adf[, "tau_level"] > adf[, "kritis5_level"] & adf[, "tau_diff"] < adf[, "kritis5_diff"])

# ---- Uji kointegrasi Johansen (trace) ----
cj <- ca.jo(dv, type = "trace", ecdet = "const", K = 2, spec = "transitory")
print(summary(cj))
stat <- rev(cj@teststat); krit <- rev(cj@cval[, 2])             # urutan r = 0, r <= 1, r <= 2
rank_uji <- 0
while (rank_uji < 3 && stat[rank_uji + 1] > krit[rank_uji + 1]) rank_uji <- rank_uji + 1
rank_pakai <- min(max(rank_uji, 1), 2)                          # cadangan: VECM butuh 1 <= r <= 2

# ---- VECM: vektor kointegrasi (beta) dan koefisien penyesuaian (alpha) ----
vecm <- cajorls(cj, r = 1)                                      # vektor pertama, dinormalisasi pada Y
print(vecm$rlm); print(vecm$beta)
beta <- vecm$beta[, 1]
alpha <- coef(vecm$rlm)["ect1", ]
lr_x1 <- -unname(beta[2]); lr_x2 <- -unname(beta[3])            # Y = lr_x1 X1 + lr_x2 X2 + konstanta
ect <- as.numeric(as.matrix(cbind(dv, 1)) %*% beta)             # galat koreksi (ECT)
p_alpha <- summary(vecm$rlm)[[1]]$coefficients["ect1", 4]       # uji t alpha pada persamaan dY

# ---- IRF dari representasi VAR level ----
vv <- vec2var(cj, r = rank_pakai)
ir <- irf(vv, impulse = c("gdp_per_capita", "education_spending"), response = "mean_years_of_schooling", n.ahead = 20, boot = TRUE, runs = 100)


# ---- Grafik ----
nm3 <- make.unique(c("mean_years_of_schooling", "gdp_per_capita", "education_spending"), sep = " ")
dl <- data.frame(periode = rep(d$periode, 3), nilai = c(d$mean_years_of_schooling, d$gdp_per_capita, d$education_spending),
                 seri = factor(rep(nm3, each = nt), levels = nm3))
p1 <- ggplot(dl, aes(periode, nilai, colour = seri)) +
  geom_line(linewidth = 0.7, show.legend = FALSE) +
  facet_wrap(~seri, ncol = if (tata == 1) 1 else 3, scales = "free_y") +
  scale_colour_manual(values = pal[1:3]) +
  labs(title = "Levels of the three series (I(1))", x = NULL, y = NULL)
de <- data.frame(periode = d$periode, ect = ect - mean(ect))
p2 <- ggplot(de, aes(periode, ect)) +
  geom_hline(yintercept = 0, colour = warna$redup) +
  geom_hline(yintercept = c(-2, 2) * sd(de$ect), linetype = "dashed", colour = pal[2]) +
  geom_area(fill = pal[4], alpha = 0.25) +
  geom_line(colour = pal[4], linewidth = 0.7) +
  labs(title = "Error-correction term (deviation from LR)", x = NULL, y = "ECT (centred)")
kej <- c("gdp_per_capita", "education_spending")
lab_s <- make.unique(c("Shock: gdp_per_capita", "Shock: education_spending"), sep = " ")
di <- do.call(rbind, lapply(1:2, function(i) data.frame(
  h = 0:20, irf = ir$irf[[kej[i]]][, 1], lo = ir$Lower[[kej[i]]][, 1], hi = ir$Upper[[kej[i]]][, 1],
  guncangan = lab_s[i])))
di$guncangan <- factor(di$guncangan, levels = lab_s)
p3 <- ggplot(di, aes(h, irf)) +
  geom_hline(yintercept = 0, colour = warna$redup) +
  geom_ribbon(aes(ymin = lo, ymax = hi, fill = guncangan), alpha = 0.2, show.legend = FALSE) +
  geom_line(aes(colour = guncangan), linewidth = 1, show.legend = FALSE) +
  facet_wrap(~guncangan, ncol = 2) +
  scale_colour_manual(values = pal[2:3]) + scale_fill_manual(values = pal[2:3]) +
  labs(title = "IRF of mean_years_of_schooling from VECM (vec2var)", x = "Horizon", y = "Response")
p_hasil <- if (tata == 1) p1 | (p2 / p3) else p1 / (p2 | p3)

print(p_hasil)

The project file you can download

The original project file is available. It is not just the picture above: you get the working R project. It can be downloaded after payment, and we send it by WhatsApp or email.

ZIP: analisis.R + data_contoh.csv + BACA-DULU.txt

What is inside the ZIP

The complete analysis script (analisis.R), commented step by step, ready to run in RStudio.
A sample dataset (data_contoh.csv) so the script runs straight away.
A short guide (BACA-DULU.txt): which packages to install, how to run it and how to swap in your own data.

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

Package 1

Code and Explanation

US$28

excl. 11% VAT

Duration: 7 working days

Output: R files + result figures

Deliverables

Fully commented R code (.R) for Johansen Cointegration and VECM
Example data and step-by-step run instructions
Ready-to-use result figures and metrics table
Short explanation of the method, its assumptions and how to read the results

Choose Package 1

Package 2 · most chosen

Your Data Analysis + Interpretation

US$81

excl. 11% VAT

Duration: 13 working days

Output: Code + analysis + interpretation notes

Includes all of Package 1, plus

Adapting the code to your own research dataset (CSV or Excel)
Assumption checks, result validation and data cleaning
Tables and figures formatted for your manuscript
Interpretation notes for the methods and discussion sections, for you to write up in your own words

Choose Package 2

Package 3

Full Guidance

US$178

excl. 11% VAT

Duration: 18-21 working days

Output: Code + analysis + guidance until ready to present

Includes all of Packages 1 & 2, plus

Comparison against one alternative method with the reasoning for the choice
Two 60-minute online coaching sessions so you fully understand the code and results
Revisions following your supervisor feedback (up to 3 rounds)
Practice Q&A to prepare for your seminar and defence
Full documentation (README, R package list (sessionInfo)) so the results can be reproduced

Choose Package 3

Payment: 50% down payment before work starts, balance after the deliverables are handed over
Tax: Prices exclude 11% VAT
Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions

Do I get the real project file?

Yes. The original project file is a ZIP containing the complete analisis.R script, a data_contoh.csv sample dataset and a BACA-DULU.txt guide. It can be downloaded after payment, and we send it by WhatsApp or email. For your own data (Package 2 and 3) the script is adapted to your dataset.

Is this suitable for a Doctoral dissertation (S3)?

Yes. Johansen Cointegration and VECM is commonly used at this level in Education and Learning. We adapt the analysis to your research question, your own data and your supervisor's requirements.

Is the result sheet real data?

No. The sheet is computed from illustrative synthetic data so you can see the kind of output R Studio gives. For your order we run the same method on your own data.

Will I understand the code?

That is the point of the packages. Every script is commented, and Package 3 includes two coaching sessions plus practice questions so you can explain the method and the results at your seminar and defence.

How is the price decided?

Each project has three packages: Package 1 (code and explanation) US$28, Package 2 (your data analysis and interpretation) US$81 and Package 3 (full guidance) US$178, excluding 11% VAT. Larger datasets, extra methods or tight deadlines adjust the quote after a short consultation.

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