R code in this document provides a time series analysis of the relationship between per capita GDP and CO₂ emissions.

# Load necessary libraries
library(readxl)
library(writexl)
library(tidyverse)
library(ggplot2)
library(tseries)
library(urca) # for cointegration test
library(forecast) # for auto.arima
library(vars)

(1) Import and Preprocess

# Set working directory
setwd("C:/Users/PC - MSI/OneDrive - The University of Liverpool/An - 2025/2024 - UoL/Sem 1/ECON 705 Data Mgt/Exercise/FINAL")
# Import data from Excel
rawdata_co2_emissions <- read_excel("ECON705_Final_Exam_Data.xlsx", sheet= "annual-co2-emissions")
rawdata_co2_emissions_captia <- read_excel("ECON705_Final_Exam_Data.xlsx", sheet= "co-emissions-per-capita")
rawdata_consumption_co2_gdp <- read_excel("ECON705_Final_Exam_Data.xlsx", sheet= "consumption-co2-per-capita-vs-g")
rawdata_continents <- read_excel("ECON705_Final_Exam_Data.xlsx", sheet= "continents-according-to-our-wor")

# Merge data
raw_data <- left_join(rawdata_co2_emissions, rawdata_co2_emissions_captia, by = c("Entity","Year")) %>%
  left_join(rawdata_consumption_co2_gdp, by = c("Entity","Year")) %>%
  rename (
    CO2_emissions = "Annual CO2 emissions",
    CO2_emissions_capita = "Annual CO2 emissions per capita",
    Consumption_CO2_emissions_capita = "Per capita consumption-based CO2 emissions",
    GDP_capita = "GDP per capita, PPP (constant 2017 international $)",
    Population = "Population (historical)"
  ) %>%
  dplyr::select(Entity, Code, Year,CO2_emissions, CO2_emissions_capita
         ,Consumption_CO2_emissions_capita,GDP_capita, Population)

# Subset: World data from 1990 onwards
data <- raw_data %>%
  filter(Year >= 1990, Entity == "World") %>%
  mutate(
    GDP = GDP_capita * Population,
    CO2_emissions = CO2_emissions / 10^6,  # Convert to million tonnes
    lgdp = log(GDP),
    lco2 = log(CO2_emissions)
  ) %>%
  drop_na()

(2) Pre-estimation checks

# Scatterplot of log(GDP) vs log(CO2)
ggplot(data) +
  geom_point(aes(x = lgdp, y = lco2)) +
  theme_minimal() +
  labs(title = "Scatter plot of Log GDP and Log CO2 Emissions", 
       x = "log(GDP)", y = "log(CO2Emissions)") 
# ADF Test for Stationarity
adf.test(data$lgdp)
adf.test(data$lco2)

# Since both log gdp and log co2 emissions are found to be nonstationary
# we need Cointegration test before proceeding to proper time seires modelling

# Johansen Cointegration Test
gdp_co2_matrix <- cbind(data$lgdp, data$lco2)
colnames(gdp_co2_matrix) <- c("lgdp", "lco2")
coint_test <- ca.jo(gdp_co2_matrix, type = "trace", ecdet = "const", K = 2)
summary(coint_test)

# No cointegration → transform data to achcieve stationarity, first-differencing

# First differences of logs
data <- data %>%
  mutate(
    diff_lgdp = c(NA, diff(lgdp)),
    diff_lco2 = c(NA, diff(lco2))
  )%>%
  drop_na()

# Line plot of differenced series
ggplot(data, aes(x = Year)) +
  geom_line(aes(y = diff_lgdp, colour = "Log Difference of GDP")) +
  geom_line(aes(y = diff_lco2, colour = "Log Difference of CO2 Emissions")) +
  theme_minimal() +
  labs(title = "Log Differences of GDP and CO2 Emissions",
       x = "Year", y = "Log Difference") +
  scale_colour_manual(values = c("Log Difference of GDP" = "blue", "Log Difference of CO2 Emissions" = "red"))

(3) ARMA estimation and Diagnostic checks

# Choose lags by auto.arima and fit model
auto_model <- auto.arima(data$diff_lco2, xreg = data$diff_lgdp, allowmean = TRUE, ic = "aic")
summary(auto_model)

# Residuals and squared residuals
residuals <- residuals(auto_model)
squared_residuals <- residuals^2
# Plot residuals
plot(residuals, main = "Residuals of ARMA Model", ylab = "Residuals")
abline(h = 0, col = "red")
# ACF/PACF of residuals
acf(residuals, main = "ACF of Residuals")
pacf(residuals, main = "PACF of Residuals")
# ACF/PACF of squared residuals 
acf(squared_residuals, main = "ACF of Squared Residuals")
pacf(squared_residuals, main = "PACF of Squared Residuals")

If the squared residuals exhibit patterns or significant autocorrelations, it might indicate heteroskedasticity suggesting that the variance of the residuals changes over time. In such cases, models that account for changing variance (like GARCH) could be considered. The absence of significant autocorrelation in both residuals and squared residuals suggests that the current ARMA model adequately captures the dynamics between percentage changes in GDP and CO2 emissions.