Forecasting Carbon Dioxide Emissions in Greece Under Decarbonization: Evidence from an ARIMA Time Series Model

Abstract

Environmental protection and the reduction of carbon dioxide (CO₂) emissions are central priorities within European climate policy. This study analyses and forecasts annual CO₂ emissions in Greece using a univariate time-series framework. Annual data from 1960 to 2024, sourced from Our World in Data, enable the analysis to capture both the historical expansion of emissions and the recent decarbonization phase of the Greek energy system. Using the Box–Jenkins methodology, multiple ARIMA specifications were evaluated based on information criteria and diagnostic tests. To examine the stationarity properties of the series, the Augmented Dickey–Fuller (ADF) unit root test is applied. The findings indicate that the ARIMA (1,1,1) model most accurately represents the stochastic dynamics of the emissions series. The estimated autoregressive and moving-average coefficients, 0.9404 and −0.7165, respectively, are statistically significant at the 1% level. Residual diagnostics confirm the absence of serial correlation, approximate normality, and no significant heteroskedasticity. Forecast evaluation for the 2020–2024 holdout period demonstrates satisfactory predictive performance, with a mean absolute percentage error (MAPE) of approximately 6%. Dynamic forecasts for 2025 to 2030 indicate a gradual decline in national CO₂ emissions, reaching an estimated 45.5 million tonnes by 2030. Overall, the study demonstrates that parsimonious ARIMA models offer a transparent and empirically reliable benchmark for national emissions forecasting. These models provide a reproducible tool for monitoring climate policy outcomes and for supporting evidence-based environmental decision-making. This study contributes to the environmental forecasting literature by providing an updated, diagnostically rigorous univariate benchmark model for Greece’s CO₂ emissions that encompasses both the pre- and post-decarbonization phases of the national energy transition.

1. Introduction

Carbon dioxide (CO₂) emissions have been a major concern in analyses related to climate change, energy transitions, and sustainable development [1,2,3], thus considered a major indicator of air pollution resulting from fossil fuel combustion, which accounts for overall environmental degradation. In this context, the extant literature has largely emphasized carbon dioxide emissions as the key variable for assessing environmental impact. Globally, the increase in CO₂ concentrations is directly associated with anthropogenic energy consumption, industrial processes, and transportation [4,5]. A substantial amount of empirical research highlights that efficient CO₂ emissions monitoring is critical to assessing environmental performance and national decarbonization efforts [6,7,8,9,10], as well as designing evidence-based policies towards sustainable development [11,12,13,14,15]. Time-series forecasting models provide a rigorous framework for representing emissions behavior over large periods and producing transparent data-driven projections based on observed values [16,17,18]. In contrast to structural econometric or hybrid models, ARIMA-based approaches provide a simpler yet statistically rigorous framework demonstrating predictive performance for medium-term projections in both developed and developing economies [2,19].

In Greece, CO₂ emissions monitoring over the past decades has been driven by the long-term restructuring of its power generation system, energy market liberalization, and the country’s compliance with the European climate policy framework [20,21,22,23], which mandated a transition towards renewable energy. This entailed the gradual phase-out of lignite-fired power generation and an increasing reliance on renewable energy technologies [24,25,26,27,28], accompanied by structural reforms reshaping national emission patterns. The scheduled phase-out by 2028, set out in the National Energy and Climate Plan (NECP) [29], has primarily focused on two areas, the Region of Western Macedonia and the Municipality of Megalopolis, major energy and lignite hubs for decades. The specific policy aims at enhancing power system stability and ensuring national power security in line with “The European Green Deal”, a broader European framework for climate action articulated in the Communication issued by the European Commission, in collaboration with the European Parliament, the European Council, the European Economic and Social Committee, and the Committee of the Regions (COM (2019) 640) on 11 November 2019 [30]. The Green Deal aims at climate neutrality by 2050 by gradually reducing greenhouse gas emissions across all sectors of the economy.

In the context of this evolving policy landscape, understanding the statistical properties and forecasting behavior of national CO₂ emissions has become increasingly important. Although multivariate econometric analyses integrating economic activity, energy consumption, and technological change [5,6] have been extensively developed, the stochastic characteristics and univariate forecasting performance of Greek CO₂ emissions remain insufficiently researched [19,31].

The present research addresses the specific empirical gap by applying the Box–Jenkins methodology to a long historical dataset from 1960 to 2024, to identify an appropriate ARIMA model and generating medium-term forecasts extending to 2030. By providing an updated and comprehensive univariate assessment of carbon dioxide (CO₂) emissions in Greece, the research aims to describe both the long-term expansion of fossil-fuel-dependent energy use and the more recent trend towards accelerated decarbonization. Its main contribution lies in offering a transparent, reproducible, and data-driven approach to be used as a benchmark for future comparative research employing hybrid or multivariate forecasting methodologies [1,2,3]. In addition, the study contributes to the energy economics literature by providing a transparent and updated univariate forecasting benchmark for Greece’s CO₂ emissions, using annual data covering the period up to 2024.

The novelty of this study lies in providing the first updated univariate ARIMA-based empirical benchmark for forecasting Greece’s CO₂ emissions using the full 1960–2024 dataset. Unlike most previous research, which emphasizes multivariate relationships among emissions, energy consumption, and economic activity, this analysis focuses on the stochastic structure of the emissions series. It offers a transparent forecasting benchmark that reflects both the historical growth in emissions and the recent decarbonization phase of Greece’s energy transition.

2. Literature Review

ARIMA models have a long-established role in energy and environmental economics. Since the seminal work of Box and Jenkins (1976) [32], univariate time-series approaches have been widely applied to forecast energy consumption [33,34,35], emissions, and other environmental indicators [36,37] on account of their structural simplicity and strong empirical performance, particularly when long-term datasets are available. Time-series forecasting, commonly employed for the analysis of energy consumption and environmental indicators, has largely adopted regression-based methods and smoothing techniques [31] in early forecasting research, which, over time, increasingly shifted towards Autoregressive Integrated Moving Average (ARIMA) models, which feature greater flexibility, statistical rigor, and more solid theoretical underpinnings [38,39].

In detail, a substantial body of energy economics research highlights that ARIMA models have been widely used to forecast energy consumption at both national and regional levels. A number of studies suggest that ARIMA-based forecasts perform well, particularly over short- and medium-term periods, even when compared with more sophisticated hybrid paradigms [40,41]. Hybrid extensions integrating ARIMA with neural networks or fuzzy inference systems often demonstrate higher forecasting accuracy; however, they are commonly accompanied by decreased transparency and interpretability [42,43,44,45].

Recent research increasingly focuses on forecasting carbon dioxide (CO₂) emissions, shifting from energy-centered time-series analyses to emissions-specific modeling. Rehman et al. [2] and Sharma et al. [3] use ARIMA-based models with national emissions data and show strong forecasting performance, particularly for short- and medium-term projections. Dritsaki and Dritsaki [5] also apply ARIMA models to European emissions, further confirming their effectiveness for environmental time series. Recent studies explore hybrid approaches that combine ARIMA with machine learning to address non-linear emissions patterns [42,43,44,45,46]. While these hybrid models may improve predictive accuracy, they often reduce interpretability and increase complexity, which can limit their value for policy applications.

Overall, despite the increasing availability of machine learning techniques, recent research generally underscores the value of ARIMA models as fundamental benchmark forecasting tools [46], highlighting interpretability and well-established diagnostic procedures, which are particularly appropriate for policy-oriented analyses, where clarity, replicability, and methodological transparency are critical [47,48,49]. Given this perspective, the present research aligns with relevant studies, which place particular emphasis on transparent and diagnostically rigorous ARIMA-based forecasting frameworks, focusing on the specific case of Greek CO₂ emissions.

Despite the expanding literature on emissions forecasting, several significant gaps remain. First, most existing research focuses on cross-country comparisons or large economies, with limited attention to country-specific, long-term analyses for smaller economies such as Greece. Second, many empirical studies emphasize forecasting performance but do not systematically provide comprehensive diagnostic validation within the Box–Jenkins framework. Third, although hybrid and machine learning methods are increasingly common, few studies provide transparent, reproducible univariate benchmarks as reference points for evaluating more complex models. The present study addresses these gaps by introducing a diagnostically rigorous, fully specified ARIMA-based benchmark for Greece’s CO₂ emissions, using an extended dataset that covers both pre- and post-decarbonization periods.

3. Data and Methodology

This section outlines the methodological framework employed to model and forecast Greece’s CO₂ emissions. Building on the Box–Jenkins approach, the analysis follows the standard stages of ARIMA modelling: identification, estimation, diagnostic checking, and forecasting. Annual CO₂ emissions data for 1960–2024 are examined to determine the stochastic properties of the series, assess stationarity, and identify an appropriate model specification. Diagnostic tests are applied to evaluate residual behaviour and ensure the adequacy of the selected model before generating medium-term forecasts extending to 2030.

By relying on a standard Box–Jenkins framework rather than multivariate or hybrid approaches, the analysis establishes a benchmark focused exclusively on the stochastic properties of emissions, without introducing additional structural assumptions. The combination of rigorous diagnostic testing with ex-post and out-of-sample forecasting enables a direct assessment of the reliability and stability of the selected ARIMA specification under conditions of structural change.

A univariate ARIMA specification is employed as the methodological approach in this study. The objective is to establish a transparent stochastic benchmark that relies exclusively on the internal dynamics of CO₂ emissions, thereby avoiding structural assumptions regarding economic activity, energy composition, or policy variables. This benchmark framework allows subsequent research to evaluate whether the inclusion of exogenous determinants significantly improves forecasting performance relative to a purely time-series specification. However, ARIMA models also present certain limitations, as they rely exclusively on the internal dynamics of the time series and do not explicitly incorporate exogenous economic, technological, or policy variables that may influence emissions.

3.1. Data

To model and forecast annual CO₂ emissions in Greece, the analysis relies on annual CO₂ emissions data drawn from Our World in Data [50]. The data consist of annual CO₂ emissions measured in million tonnes for the period 1960–2024, produced exclusively by industrial activities and fossil-fuel combustion of coal, oil, and natural gas used for transport, electricity generation, heating, and industrial energy use. They also include emissions from gas flaring during oil and gas extraction, as well as from specific industrial processes, such as cement and steel production, where CO₂ is released through chemical reactions. Emissions related to land use change, such as deforestation, forest degradation, ecosystem conversion, peatland drainage, and carbon uptake from vegetation regrowth (LULUCF- Land Use, Land-Use Change and Forestry), are excluded from the specific dataset, and thus fall outside the scope of the present analysis. The length of the data series allows reliable estimation and application of time-series techniques. Figure 1 illustrates the total carbon dioxide (CO₂) emissions in Greece over the period 1960–2024. Notably, from 1960 to 2009, the series follows a marked upward trend, transitioning thereafter into a steady and continuous downward trend to the present, without signs of recovery.

An examination of the descriptive statistics in

Table 1. Descriptive statistics of annual fossil CO₂ emissions (million metric tonnes CO₂).

CO2 (Mt)
Mean	64,316,524
Median	62,576,332
Maximum	115,000,000
Minimum	9,391,531
Std. Dev.	31,248,601
Skewness	−0.134552
Kurtosis	1.930817
Jarque–Bera	3.292165
Probability	0.192804
Observations	65

reveals that average annual CO₂ emissions account for approximately 64.3 million metric tonnes of CO₂ (Mt CO₂), whereas the highest observed value is roughly 115 million metric tonnes of CO₂ (Mt CO₂). The series distribution exhibits mild negative skewness, thus suggesting a slight concentration of observations toward higher values; in contrast, the kurtosis value (1.93) is slightly below the normal benchmark of 3, indicating a relatively flatter distribution with no evidence of pronounced extreme observations. In addition, the Jarque–Bera statistic equals 3.29 with an associated p-value of 0.193 (Table 1). Since the p-value exceeds conventional significance levels (1%, 5%, and 10%), the null hypothesis of normality cannot be rejected.

3.2. Methodological Framework

The empirical analysis follows the Box–Jenkins methodological framework, which involves four successive steps: testing for stationarity, identifying the appropriate model structure, estimating parameters, and performing diagnostic evaluation. Within this framework, Autoregressive Integrated Moving Average (ARIMA) models are the conventional and widely endorsed tool for time-series forecasting [51,52]. Econometric and time-series approaches are widely employed in environmental and ecological economics to analyse emission dynamics and environmental degradation processes, providing empirically grounded tools for policy-oriented environmental assessment and forecasting [53].

ARIMA models are well-suited for environmental and energy time series. They capture stochastic persistence, gradual adjustment, and path-dependent dynamics often seen in emissions data. Unlike structural or multivariate models, ARIMA uses only the time series’ internal structure. This enables robust forecasting even when drivers such as economic or technological trends are complex or only partially observed. This feature matters for national CO₂ emissions, which reflect policy changes, energy transitions, economic cycles, and new technologies. ARIMA models also provide statistically optimal linear forecasts for integrated processes. They remain interpretable, transparent, and replicable. These qualities are vital for policy-oriented environmental analysis, where clear methods are paramount.

In ARIMA modelling, the coefficient of determination (R²) is not considered a primary criterion for assessing goodness of fit. In contrast to static regression models, where R² quantifies the proportion of variance explained by contemporaneous explanatory variables, ARIMA models are estimated on transformed, typically differenced, time series and are designed to capture the stochastic dependence structure of the data rather than contemporaneous explanatory power. Consequently, the concept of explained variance is less meaningful, particularly when the model is specified in first differences, as the variance of the transformed series does not directly correspond to the variance of the original process. Furthermore, ARIMA models are estimated using likelihood-based methods, and model selection is guided by information criteria such as the Akaike Information Criterion (AIC) and Schwarz Criterion (SC), which balance model fit and parsimony. As a result, R² does not provide a reliable or theoretically consistent measure of model adequacy in this context and may be misleading if interpreted as in regression analysis. In accordance with the standard time-series econometrics literature [54,55], model performance in this study is evaluated using information criteria and residual diagnostic tests rather than R².

An ARIMA (p, d, q) model integrates autoregressive components of order p, differencing of order d to ensure stationarity, and moving average components of order q. Once the series has been rendered stationary through the appropriate differencing, the model represents the current value of the series as a linear function of its past values and past innovations [54,55]. The general ARIMA (p, d, q) is formulated as follows:

$$\varphi \left(L\right)(1-L{)}^d{y}_t=\theta \left(L\right){\epsilon }_t$$

(1)

where $y_t$ denotes CO₂ emissions, $L$ is the lag operator, $d$ represents the order of differencing required to achieve stationarity, and ${\epsilon }_t$ is a white-noise error term. The polynomials $\varphi \left(L\right)$ and $\theta \left(L\right)$ correspond to the autoregressive and moving average components of orders $p$ and $q$, respectively. All estimations are carried out using EViews 7 (version released in 2009), a widely used econometric software for time-series and econometric analysis.

The empirical methodology employs the standard Box–Jenkins framework, widely used in environmental and energy economics to model stochastic processes in emissions and energy indicators. Stationarity is assessed using unit root procedures, such as the Augmented Dickey–Fuller test, which is essential for valid ARIMA modeling and statistical inference. Model identification relies on autocorrelation and partial autocorrelation functions, parameter estimation is conducted via maximum likelihood, and diagnostic checking is performed using residual tests. These procedures are well-established in time-series econometrics. Such techniques are frequently applied in environmental forecasting studies, especially for emissions and energy-related variables, due to their transparency, interpretability, and empirical reliability [18,31,54,55,56,57].

In particular, stationarity was tested using the Augmented Dickey–Fuller (ADF) unit root test, while model adequacy was evaluated through standard diagnostic procedures including the Ljung–Box Q test for residual autocorrelation, the Jarque–Bera test for normality, and the ARCH LM test for conditional heteroskedasticity.

3.3. The Box–Jenkins Procedure

The Box–Jenkins methodology, a widely recognized approach, follows a systematic sequence of steps, ensuring its consistent and replicable application in time-series modeling [56,57]. It comprises the following stages:

• Stationarity assessment: Stationarity is examined through visual inspection, correlogram analysis, and formal unit root tests. In the case of non-stationary time series, appropriate transformations, such as first-order differencing, are applied to achieve stationarity and ensure the data are appropriate for subsequent econometric modeling.
• Model identification: Model identification involves using the autocorrelation function (ACF) and the partial autocorrelation function (PACF) to determine the appropriate orders of the autoregressive and moving average components. Once stationarity is achieved, the correlograms of the stabilized series are examined, where a sharp cutoff in the PACF after lag p suggests a potential AR (p) specification; a cutoff in the ACF after lag q indicates a potential MA (q) structure.
• Parameter estimation: Once the model structure is defined by the values of $p$, $d$, and $q,$ the parameters of the ARIMA model are estimated, typically using maximum likelihood methods.
• Diagnostic checking: Diagnostic tests ensure that the residuals of the estimated model exhibit white noise behavior. To assess model adequacy, the residuals are subjected to a series of diagnostic tests, primarily examining the absence of serial correlation (white noise), while additional tests for normality and heteroskedasticity are also considered to support statistical inference and forecast interval interpretation. If the specific diagnostic criteria are satisfied, the model is considered statistically reliable and appropriate for forecasting applications.
• Forecasting: After the adequacy of the selected ARIMA model has been verified through diagnostic checking, the model is employed to generate out-of-sample forecasts. Forecasts are produced recursively, drawing on the estimated parameters and past series observations, to project future values over the specified forecast period, thus ensuring that forecasts remain fully consistent with the model underlying stochastic structure.

The structured Box–Jenkins approach provides a systematic framework for model selection, estimation, diagnostic validation, and forecasting, thereby supporting methodological transparency and the interpretability of the empirical results.

4. Results

4.1. Stationarity Analysis and Test

Assessing stationarity is essential in empirical research within environmental and energy economics, particularly for analyses involving carbon emissions and long-term environmental dynamics. Establishing the integration properties of emissions series is required to support valid inference and forecast projections. Unit root testing methods, such as the Augmented Dickey–Fuller test, are frequently employed in environmental time-series studies to determine whether emissions follow persistent stochastic trends or represent stable processes [58,59].

The results of the Augmented Dickey–Fuller (ADF) unit root tests (

Table 2. Augmented Dickey–Fuller (ADF) Unit Root Test Results.

Variable	Specification	ADF Statistic	p-Value	1% CV	5% CV	10% CV	Conclusion
CO2	Constant, Trend	1.393	1.000	−4.108	−3.482	−3.169	Non-stationary
DCO2	Constant	−4.939	0.0001	−3.538	−2.908	−2.592	Stationary

Note: The null hypothesis of the ADF test shows the presence of a unit root. Critical values are based on MacKinnon (1996) [60]. D denotes the first difference operator.

) indicate that CO₂ emissions are non-stationary in their level form. The critical values used for the test follow the response surface approximations proposed by MacKinnon [60]. The ADF statistic for the level series (1.393) is greater than the critical values at the 1%, 5%, and 10% significance levels (−4.108, −3.482, and −3.169), which suggests that the null hypothesis of a unit root cannot be rejected. The positive ADF statistic in levels reflects the strong upward trend and non-stationary behaviour of the series, which is consistent with the presence of a unit root. Such outcomes may occur in trending macroeconomic or environmental time series and do not indicate a calculation error. Following first differencing, the ADF statistic (−4.939) is less than the corresponding critical values (−3.538, −2.908, and −2.592), leading to rejection of the null hypothesis and confirming that the differenced series is stationary.

Therefore, CO₂ emissions are integrated of order one, I (1), and the differencing order is set to $d=1$.

Establishing stationarity is essential for ARIMA modelling. This directly supports the study’s primary objective: developing a statistically valid forecasting framework for CO₂ emissions. Non-stationary time series may produce misleading inferences and unstable forecasts. Using first differencing ensures that the underlying stochastic process keeps stable statistical properties over time. Identifying the series as integrated of order one aligns with the gradual evolution of national emissions, which are influenced by long-term energy transitions and economic dynamics. The Augmented Dickey–Fuller test, within the Box–Jenkins framework, provides a robust empirical basis for model specification, estimation, and forecasting.

4.2. Model Identification and Selection

The model identification stage both serves a technical function and directly contributes to the study’s primary objective: uncovering the short-run stochastic dynamics governing CO₂ emissions in Greece and establishing an empirically valid forecasting framework. Analysis of the autoregressive and moving-average components facilitates the identification of persistence patterns and adjustment mechanisms in emissions changes. Consequently, evaluating alternative ARIMA specifications provides insight into the evolution of shocks to CO₂ emissions and clarifies whether emissions exhibit systematic short-run dependence.

For model identification, the analysis employs the correlogram, with special emphasis on the autocorrelation (ACF) and partial autocorrelation (PACF) methods as the primary diagnostic tools. The specific functions demonstrate the dependence structure of the time series and inform the selection of appropriate autoregressive and moving-average orders within the Box–Jenkins methodology.

An examination of the ACF and PACF of the first-differenced CO₂ emissions series (Figure 2) demonstrates pronounced short-term autoregressive behavior. The autocorrelation function exhibits a gradual decline, whereas the PACF displays a sharp cutoff after the first lag. This pattern suggests a first-order autoregressive process in the differenced series, indicating that ARIMA (1,1,0) including a constant term is an appropriate baseline model to describe the short-run dynamics of CO₂ emissions.

Although the autoregressive coefficient was statistically significant, the constant term was not, suggesting the absence of a deterministic drift in the first-differenced series. Accordingly, the constant term was considered redundant, rendering the initial specification over-parameterized relative to the available data. A more efficient ARIMA (1,1,0) model without a constant term was, therefore, estimated. Under this specification, the autoregressive parameter remained statistically significant, and the model satisfied the stability condition, indicating that a driftless autoregressive structure adequately describes the fundamental short-run dynamics of the researched series.

To further explore the presence of additional short-run dynamics arising from temporary (transitory) disturbances, an ARIMA (1,1,1) model was also estimated. Thus, the moving-average component, which accounts for the influence of past random errors (innovations) on current values, was introduced to describe the effects not fully explained by the autoregressive coefficient alone. The results demonstrate that both the autoregressive and moving-average coefficients are statistically significant at the 1% level.

Beyond parameter significance, the stability of the ARIMA (1,1,1) specification was also evaluated by examining the inverted roots of the autoregressive and moving-average polynomials. The stability of the ARIMA (1,1,1) specification was evaluated by examining the inverted roots of the autoregressive and moving-average polynomials. In ARIMA modeling, both stability and invertibility require all roots to be located within the unit circle. The estimation results indicate that this condition is satisfied, confirming that the process is dynamically stable and that the model provides a valid statistical representation of the underlying time-series dynamics.

4.3. Model Estimation and Specification

After completing model identification and selection, the research employed the ARIMA (1,1,1) model to estimate annual CO₂ emissions data for Greece from 1960 to 2024. Given the absence of deterministic drift in the first-differenced series, the model was estimated without a constant term. The results are shown in Figure 3 below.

Overall, the results demonstrate that both the autoregressive and moving-average components are statistically significant at the 1% level of significance. The positive autoregressive coefficient reflects strong short-run persistence in changes in CO₂ emissions; in contrast, the negative moving-average coefficient describes the partial correction of past disturbances in the emissions growth process. The magnitude and signs of the estimated components are economically meaningful and consistent with standard time-series observed in environmental and energy-related variables [61].

The stochastic dynamics of CO₂ emissions are modeled using an ARIMA (1,1,1) process. In its most general specification, the model can be formulated as follows:

$$\Delta {\mathrm{CO}}_{2,t}={\varphi }_1\Delta {\mathrm{CO}}_{2,t-1}+{\epsilon }_t+{\theta }_1{\epsilon }_{t-1}$$

(2)

where $\Delta {\mathrm{CO}}_{2,t}={\mathrm{CO}}_{2,t}-{\mathrm{CO}}_{2,t-1}$ implies the first difference in CO₂ emissions, ${\epsilon }_t$ is a white-noise error term with zero mean and constant variance, and ${\varphi }_1$ and ${\theta }_1$ represent the autoregressive and moving-average components, respectively.

Based on the empirical estimation, the fitted ARIMA (1,1,1) model is given by:

$$\Delta {\mathrm{CO}}_{2,t}=0.9404\Delta {\mathrm{CO}}_{2,t-1}+{\epsilon }_t-0.7165{\epsilon }_{t-1}$$

(3)

Both coefficients are statistically significant at the 1% level. Furthermore, the inverted roots of the autoregressive and moving-average polynomials, as indicated in the model output, are located within the unit circle. This finding satisfies the standard ARIMA stability and invertibility condition |root| < 1 and demonstrates that the estimated model is dynamically stable.

In addition, the Durbin–Watson statistic approximates its theoretical value of two, indicating no evidence of first-order residual autocorrelation. Higher-order serial correlation is further examined using the residual correlogram and Ljung–Box Q statistics (Figure 4). The findings demonstrate that the model provides an adequate and statistically robust representation of the stochastic dynamics underlying annual CO₂ emissions in Greece, as evidenced by the significance of the estimated parameters and the results of diagnostic tests on the residuals.

4.4. Diagnostic Tests

The adequacy of the estimated ARIMA specification was evaluated through a series of standard diagnostic tests applied to the model residuals. These tests examine potential violations of classical time-series assumptions, including residual autocorrelation, non-normality, conditional heteroskedasticity, and structural instability. Results are presented in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 and interpreted using established test statistics, probability values, and graphical criteria commonly employed in time-series diagnostics. Ιn particular, the correlogram and Ljung–Box statistics are used to detect residual autocorrelation, the Jarque–Bera statistic evaluates normality, the ARCH LM test examines potential heteroskedasticity, and the CUSUM test assesses parameter stability over time.

Residual diagnostic testing constitutes an essential aspect of ARIMA modelling because valid forecasts depend on residuals that approximate white noise. Tests for serial correlation (Ljung–Box Q), normality (Jarque–Bera), and conditional heteroskedasticity (ARCH LM) are widely recommended in the time-series literature to assess model adequacy. These procedures are routinely applied in environmental and energy forecasting to verify that estimated models capture the underlying stochastic structure of emissions processes and yield statistically reliable projections [54,55,56,57].

To verify the adequacy of the ARIMA (1,1,1) model, the analysis employed a series of diagnostic tests. Examination of the residual correlograms and the Ljung–Box Q-statistics revealed no evidence of residual serial correlation, thus demonstrating that the model successfully accounts for the dependence structure in the data. Residual normality was assessed using the Jarque–Bera test, and the null hypothesis of normality could not be rejected, implying that the residuals conform reasonably well to a normal distribution. The ARCH LM test, employed to detect potential conditional heteroskedasticity, demonstrated no significant evidence of time-dependent volatility at conventional significance levels. Breakpoint tests, also performed to assess parameter stability, revealed some variation across subperiods, particularly in the late 2000s, which may suggest underlying economic and energy-market changes during the researched period. Despite the specific variations, the ARIMA (1,1,1) model remains a suitable and reliable univariate benchmark for describing the stochastic dynamics of CO₂ emissions.

The identification of localized parameter variation around 2008 aligns with significant macroeconomic and energy-sector shifts during that period. Nevertheless, the primary objective of this study is to establish a stable stochastic benchmark for medium-term forecasting, rather than to structurally model regime changes.

• Residual Autocorrelation—Correlogram and Ljung–Box Test

Residual diagnostic tests confirm that the ARIMA (1,1,1) model provides an adequate statistical representation of the series. The correlogram of standardized residuals (Figure 4) indicates no evidence of remaining autocorrelation, and the Ljung–Box Q-statistics do not reject the null hypothesis of no serial correlation at conventional significance levels. These results suggest that the residuals follow a white-noise process, thus corroborating the assumption that the model is appropriately specified.

• Residual Normality—Jarque–Bera test

Residual normality is assessed as part of the diagnostic checking process, as Gaussian innovations are significant for standard statistical inference and for the interpretation of forecast uncertainty bands derived from model-based standard errors. To assess residual normality, the Jarque–Bera (JB) test was applied, which is widely used in time-series analysis, and enables direct evaluation of deviations from normality through residual skewness and kurtosis. As shown in Figure 5, the JB statistic is 0.720 with an associated p-value of 0.698, demonstrating that the null hypothesis of normality cannot be rejected at conventional significance levels. Although the Jarque–Bera test does not reject the null hypothesis of normality, the residual distribution exhibits mild deviations from perfect normality according to the descriptive statistics (skewness and kurtosis). However, these deviations remain limited and do not affect the validity of the model diagnostics or the reliability of the forecast intervals.

• Heteroskedasticity Test—ARCH LM test

The ARCH LM test (Figure 6) was employed to examine the presence of conditional residual heteroskedasticity. The results demonstrated that the null hypothesis of no ARCH effects cannot be rejected at the 5% significance level, despite the test statistics lying close to the 10% rejection threshold. In the auxiliary regression of the ARCH test, C denotes the constant term, while RESID²(−1) represents the lagged squared residuals used to detect potential ARCH effects in the variance of the residual process. Overall, the results provide no clear evidence of residual volatility, supporting the assumption of homoskedastic disturbances in the analysis.

• Stability/Structural breaks

Parameter stability was initially assessed using the Chow forecast test (Figure 7) to examine the predictive performance stability of the ARIMA (1,1,1) model over the post-2008 period. The test results suggest the presence of forecast instability within the examined period, which implies that the underlying dynamics of CO₂ emissions have likely changed when compared with previous periods. Given that the Chow test relies on a pre-specified breakpoint, parameter stability was further examined using the Quandt–Andrews test for unknown structural breaks.

The results (Figure 8), providing evidence of localized parameter variation with stronger indications in the late 2000s and more moderate signals in subsequent years, are broadly consistent with major macroeconomic and energy-related changes over the sample period.

Overall, the diagnostic test evidence corroborates the adequacy of the ARIMA (1,1,1) model for the objectives of the present research. The absence of residual autocorrelation, as indicated by the residual correlograms and the Ljung–Box Q-statistics, suggests that the model effectively describes the serial dependence in the data. The Jarque–Bera test fails to reject the null of normality, indicating that the residuals appear to follow an approximately normal distribution, thereby reinforcing the appropriateness of standard inference and the reliability of model-based forecast uncertainty estimates. The ARCH LM test results provide no strong evidence of conditional heteroskedasticity at conventional significance levels; thus, time-varying volatility is not a dominant feature of the residual process. In addition, although stability tests identify localized parameter variations, particularly in the late 2000s, in line with major macroeconomic and energy-sector developments, the overall dynamic coherence of the model is not undermined. Overall, the diagnostic test results confirm the appropriateness of the ARIMA (1,1,1) model for describing the stochastic and long-run behaviour of CO₂ emissions.

Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 collectively summarize the diagnostic checks performed on the residuals of the estimated ARIMA model, including tests for serial correlation, normality, conditional heteroskedasticity, and parameter stability.

4.5. Forecasting Performance and Results

To evaluate the predictive performance of the ARIMA (1,1,1) model, forecasting performance is examined in two stages. First, to assess predictive accuracy over a recent period with available observed data, an ex-post forecast evaluation is conducted with a holdout sample. In the second stage, the validated model generates out-of-sample forecasts to explore the future evolution of CO₂ emissions.

• Ex-post Forecast Evaluation (2020–2024)

To assess predictive accuracy, an ex-post forecast evaluation was carried out using the 2020–2024 holdout period. Forecast performance was measured using standard metrics, including root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and Theil’s inequality coefficient. The results (Figure 9) demonstrate satisfactory forecasting performance, with a mean absolute percentage error of approximately 6% and a Theil’s U value well below unity, indicating that the model outperforms a naïve benchmark. Decomposition of Theil’s inequality coefficient reveals that a significant portion of forecast error is due to systematic deviations during this period, consistent with the significant economic and energy-related disruptions observed between 2020 and 2024. Overall, the ex-post evaluation confirms the adequacy of the ARIMA (1,1,1) model as a reliable univariate benchmark for forecasting.

• Out-of-Sample Forecasts (2025–2030)

Given the satisfactory ex-post forecasting performance, the ARIMA (1,1,1) model is employed to generate dynamic out-of-sample forecasts for 2025–2030 (Figure 10). The projected trajectory suggests a gradual decline in annual CO₂ emissions in Greece throughout the forecast period, consistent with recent decarbonization trends. Forecast uncertainty increases when the projection period expands, as reflected by the widening confidence intervals. The analysis indicates a decline in emissions across the forecast horizon, consistent with recent decarbonization trends.

Table 3. Dynamic forecasts of Greece’s annual CO₂ emissions for 2025–2030 with ±2 S.E. confidence intervals.

Year	Forecasted CO2 Emissions (Mt)	Lower Bound (−2 S.E.)	Upper Bound (+2 S.E.)
2025	51.8	45.0	58.5
2026	50.5	39.8	60.2
2027	49.2	34.5	63.8
2028	48.0	29.8	66.5
2029	46.8	24.5	69.0
2030	45.5	20.0	71.2

below illustrates the dynamic forecasts of Greece’s annual CO₂ emissions for 2025–2030, with the corresponding confidence intervals. The results demonstrate a continuous downward trajectory in emissions, with the widening confidence bands signaling the increasing uncertainty as the projection period extends.

The downward trend projected by the model forecasts (Figure 11) complies with recent energy-sector developments in Greece. Over the past seven years, the rapid phase-out of lignite-fired power generation, the significant increase in the use of renewable energy sources and natural gas, and the gradual adoption of electric vehicles have contributed to reducing CO₂ emissions. In conjunction with wider environmental and climate policies, the above considerations account for the projected decline in emissions described by the model. Notably, Greece’s exit from economic adjustment programs and the subsequent economic recovery have partially offset the decline, yielding a more moderate trend. The balance between decarbonization measures and economic growth is evident in the gradual downward trajectory of the model.

5. Conclusions

The empirical findings of this study align with recent research using time-series methods to analyze and forecast carbon emissions. The literature consistently indicates that ARIMA-based models offer a robust framework for capturing the stochastic dynamics of CO₂ emissions and generating reliable short-term forecasts in environmental and energy-related time series. For example, Kour [61] demonstrates that ARIMA models effectively represent long-term emission trends and produce accurate forecasts of national CO₂ trajectories. Similarly, Kumari and Singh [62] highlight the usefulness of time-series forecasting approaches for modelling emissions dynamics and supporting environmental policy analysis, while Li and Zhang [63] emphasize that statistical time-series models provide an interpretable and reliable benchmark for analysing emissions patterns during energy transitions. However, beyond confirming the general findings of previous studies, the present analysis provides additional empirical and methodological evidence by implementing a fully specified Box–Jenkins framework with comprehensive diagnostic validation. In particular, model adequacy is explicitly verified through unit root testing, information-criterion-based model selection, residual autocorrelation diagnostics, normality and heteroskedasticity testing, and parameter stability assessment. Furthermore, the ex-post forecast evaluation for the 2020–2024 holdout period, yielding a mean absolute percentage error (MAPE) of approximately 6%, offers direct evidence of satisfactory predictive performance. This level of forecast accuracy is comparable to, and in several cases lower than, the error ranges typically reported in similar ARIMA-based environmental forecasting studies, thereby reinforcing the robustness of the obtained results. Consequently, the contribution of this study lies not only in confirming that parsimonious ARIMA specifications can capture the stochastic behaviour of national CO₂ emissions, but also in providing a more transparent, statistically validated, and policy-relevant forecasting benchmark.

The present research examines the dynamic behavior of annual carbon dioxide (CO₂) emissions in Greece using a univariate time-series framework based on the Box–Jenkins methodology. Drawing on annual data from 1960 to 2024, the analysis demonstrated that the CO₂ emissions series is integrated of order one, and that, following a comparison of alternative model specifications, the ARIMA (1,1,1) model provides the most appropriate representation of the underlying data-generating process, outperforming simpler autoregressive models based on information criteria and diagnostic tests.

The primary contribution of this study is empirical and methodological rather than technical. While ARIMA modelling is not new, its application in the present paper provides an updated univariate forecasting benchmark for Greece’s CO₂ emissions using annual data covering the full 1960–2024 period. To the best of the author’s knowledge, the recent literature has focused either on broader energy and environmental forecasting applications, cross-country analyses, or related energy variables in the Greek case, rather than on a diagnostically validated univariate ARIMA benchmark specifically for Greece’s national CO₂ emissions over a period encompassing both the pre- and post-decarbonization phases of the energy transition. In this respect, the present study provides a transparent, policy-oriented time-series framework tailored to the Greek case that may serve as a useful empirical benchmark for future multivariate or hybrid forecasting research.

In addition, residual diagnostic checks confirmed the absence of serial correlation and indicated that the residuals behave as white noise, corroborating that the ARIMA (1,1,1) model effectively describes the short-run dynamics of emissions.

Dynamic forecasts for 2025–2030 demonstrate a gradual decline in annual CO₂ emissions, reflecting recent trends in the time-series data. Although forecast uncertainty increases over the projection period, the downward trajectory remains stable. The specific results underscore the value of time-series forecasting models as analytical tools for monitoring emission trends and informing medium-term assessments of environmental performance.

The empirical application of ARIMA models demonstrates that they can deliver informative and forecast projections of national CO₂ emissions, providing a transparent, up-to-date, data-driven perspective on the emissions trajectory in Greece. By drawing on the most recent available data and a diagnostically reliable univariate framework, the present research establishes a clear empirical benchmark for assessing medium-term progress towards climate neutrality objectives.

The projections also carry direct implications for climate policy assessment, enabling policymakers to monitor shifts in emissions and assess whether observed dynamics remain aligned with existing mitigation strategies. Beyond their practical relevance, the methodological approach and results contribute to the broader environmental forecasting literature by demonstrating that efficient time-series models can perform well even under conditions of structural change [64,65]. In addition, they provide policymakers with a timely, evidence-based benchmark for short- and medium-term progress assessments toward national and EU climate targets, enabling an evaluation of whether current CO₂-mitigation policies are consistent with observed emission patterns.

It should be noted that the present analysis adopts a univariate time-series framework focusing on the stochastic behaviour of CO₂ emissions. While this approach provides a transparent benchmark for forecasting, future research may further extend the analysis by incorporating additional economic or energy-related variables.

Future research could extend the present analysis by incorporating selected exogenous variables within an ARIMAX framework to examine whether conditioning on economic activity or energy system indicators enhances forecasting performance. Given indications of localized parameter instability over the sample period, further research could explore more flexible specifications capable of accommodating evolving dynamics. Finally, hybrid forecasting patterns combining ARIMA with neural network components may offer a complementary approach for describing both linear and non-linear dynamics in emissions behavior.