Interdependency and causality between green technology innovation and consumption-based carbon emissions in Saudi Arabia: fresh insights from quantile-on-quantile and causality-in-quantiles approaches

Abstract

In this paper, we examined the asymmetric dynamics and causality of technological progress––proxied by green technology innovation––on both consumption-based carbon (CCO₂) and territory-based carbon (TCO₂) emissions in Saudi Arabia using quarterly data from 1990Q1 to 2021Q4. Our initial results reject the normality and linearity assumptions of data series and thus emphasize that the observed associations are quantile dependent. We firstly utilized the quantile-on-quantile regression (QQR) approach to draw the interdependency between green technology innovation and both CCO₂ and TCO₂ emissions. We found a strong emission-mitigating impact of green technology innovation only at (extreme) upper emission levels. We also identified a weak positive effect at (extreme) higher emission quantiles. Furthermore, we found that higher emission levels are linked with lower green technology innovation across all emission quantiles whereas a weak positive effect is perceived at lower and medium emission quantiles. We further utilized linear and nonlinear Granger causality-in- quantiles (GCQ) tests to capture an entire picture of the impact of green technology innovation on both CCO₂ and TCO₂ emissions. Under linear specifications of the quantile regression model, we found evidence of strong bidirectional causality between carbon emissions and green technology innovation across lower and upper quantiles. However, we found unidirectional causalities from carbon emissions to green technology innovation at medium quantiles of the conditional distribution. Besides, there is no causality at both extreme lower and extreme upper quantiles. Under nonlinear specifications of the quantile regression model, we found a weak unidirectional causality from green technology innovation to carbon emissions at (extreme) lower quantiles. We also found a weak unidirectional causality from carbon emissions to green technology innovation at medium and extreme upper quantiles. Overall, our findings indicate that green technology innovation helps abate both CCO₂ and TCO₂ emissions in Saudi Arabia. Our study shows policies that target green technology innovation would significantly change carbon emissions.

Introduction

Over the past decade, green technology innovations have attracted the attention of researchers in the fight against excessive carbon emissions. They refer to various forms of innovation activities that significantly improve environmental protection (Razzaq et al. 2021c). Harnessing these innovations can help prevent or significantly reduce the risk of any other negative effects on the environment, pollution, and resource use through the life cycle activities (e.g., Khan et al. 2020c; Khan et al. 2020b; Khan et al. 2020a; Razzaq et al. 2021c).

To formulate relevant climate policies and resolve environmental problems, a reliable measure of carbon emissions must be considered. Liddle (2018a) and Liddle (2018b) distinguished between two types of emissions: consumption-based and territory-based (also called production-based) carbon emissions. Consumption-based emissions consider the impact of trade, including emissions from final domestic consumption and the production of imported products (e.g., Hasanov et al. 2018; Spaiser et al. 2019; Tukker et al. 2020). They can also be defined as emissions of domestic use of fossil fuels plus imports minus exports. Territory-based emissions are those that take place within a country’s territorial boundaries and include exports but omit imports.

Global research on green technology is growing, particularly in Saudi Arabia. On March 27, 2021, the Kingdom of Saudi Arabia launched the Saudi Green Initiative to combat the climate change crisis. This initiative aimed at investing in new products, research and development, and cost-effective technologies to raise vegetation, escalade energy efficiency, increase production capacity, reduce carbon emissions, combat pollution and land degradation, and preserve marine life.

Green technologies could be the future solution to protect the environment from carbon emissions. Notably, they could help countries use specific materials to compensate for depleted natural resources—such as by exploiting factories in the manufacturing of plastics and leathers and employing wind, water, and sunlight energy to generate heat and electricity (Razzaq et al. 2023). Furthermore, green technologies provide easily accessible options and future strategies that could contribute to achieving sustainable goals for smart development, including cleaner production, waste management, and the use of affordable and efficient energy that enables the reduction of toxic carbon emissions and the cost of implementing policies (e.g., Hao et al. 2021; Razzaq et al. 2023; among others).

The current study becomes even more important when it is concerned with analyzing the interdependency and dynamic causality between green technology innovation and carbon emissions in Saudi Arabia. The industrial sector underpins economic development in Saudi Arabia. It includes industries related to oil production and refining, petrochemicals, minerals, and military industries, in addition to cement, construction, equipment, and food industries, among others. Saudi Arabia’s aim in 2030 is to reach 33% of the industrial sector’s contribution to its GDP. The country is also witnessing remarkable development through the transition to industrialization based on innovation, industrial improvement, quality improvement, and the promotion of environmentally friendly integrated technologies. On the other hand, explosive emissions motivate policymakers to make relevant climate policies through higher allocation to research and development, social acceptance, environmental regulation, carbon trading markets, and promotion of environmental technologies (Razzaq et al. 2021c). Therefore, increasing carbon emissions should motivate the government to increase allocations to research and development and environmental technologies (Razzaq et al. 2021c; Maasoumi et al. 2021).

Our study focuses on research dynamics and asymmetric dependency structures between green technology innovation and consumption-based carbon emissions in Saudi Arabia. More specifically, we have tried to answer the following two research questions:

• i. Is there an asymmetric effect in the dependent structure between green technology innovation and consumption-based carbon emissions across quantiles?

• ii. Do the causality effects between green technology innovation and consumption-based carbon emissions significant and change with different market situations?

The two research questions constitute a comprehensive research framework to study the theme of this article. The second research question focuses on the temporal change of the effect of green technology innovation on consumption-related carbon emissions and the asymmetry of the effect under different market conditions, respectively. This question is primarily used to test the validity of the first question. It can also test the predictive power between variables.

Regarding the first question, we utilized the QQR method to depict the dependence structure between green technology innovation and consumption-based carbon emissions. The QQR approach can track the dependence structure under different market situations, including extreme market conditions. Compared with the conventional quantile regression (QR) approach, Sim and Zhou (2015) indicated that QQR can overcome the shortcomings of QR and provide more specific information on independent variable situations and more accurate estimation results in extreme market situations.

To answer the second question, we used the GCQ approach to explore the causality between green technology innovation and consumption-related carbon emissions under several market scenarios. The GCQ method is a functional expansion of standard Granger causality methods and fills in its shortcomings. This approach can be used to test the causality between different markets in the first and second orders simultaneously over various quantiles. It can also show the predictive power between those markets.

The major innovation and contributions of our work, in comparison to previous empirical literature, can be summarized in threefold. (i) We investigated the interdependencies between green technology innovation and consumption-based carbon emissions in Saudi Arabia. To the best of our knowledge, this is the first study that examines these interdependencies in the context of Saudi Arabia. This country possesses abundant oil resources, and due to high oil prices, there is an accumulation of large foreign currency reserves, which enhances Saudi Arabia’s ability to import. Increased import capacity may drive CCO₂ emissions, thus making it attractive to use Saudi Arabia as a case study for empirical assessment. Moreover, Saudi Arabia is the most important exporter of crude petroleum, which is the most carbon-intensive product. (ii) From a methodological perspective, we drew these inter-linkages using the most up-to-date econometric techniques. Interestingly, we firstly used a quantile framework including QQR and linear GCQ methods. Then, we extended the linear GCQ approach to its nonlinear counterpart. More importantly, we utilized a novel nonlinear quantile framework based on Conditional Autoregressive Value-at-Risk (CAViaR) models proposed by Engle and Manganelli (2004) to analyze the asymmetric causal effect between green technology innovation and consumption-based carbon emissions in Saudi Arabia under different market situations analysis, including extreme conditions. To the best of our knowledge, no prior study has employed the nonlinear CAViaR models to investigate the nexus between green technology innovation and consumption-based carbon emissions. We also tested the ability of green technology innovation to predict Saudi Arabia’s consumption-based carbon emissions. (iii) As Saudi Arabia is likely to be one of the world’s fastest-growing emerging economies, our research results provide references and policy implications for emerging countries to develop green technology innovation and prevent risks brought by consumption-based carbon emissions under different market conditions.

The remainder of the paper is organized as follows. The “Literature review” section presents the literature review. The “Data and methodology” section describes the data and discusses the econometric methodology. The “Empirical results and discussions” section analyzes and discusses the empirical results. The “Conclusion” section concludes the paper and offers some policy recommendations.

Literature review

Green technology innovation and carbon emissions

Over the past decade, the various economic sectors of countries have been largely affected by the immense technological progress that continues to increase. Researchers are therefore faced with the obligation to thoroughly assess the effect of technological innovation (also known as eco-innovation, green innovation, environmental innovation, or technological progress)¹ on ecological or environmental sustainability, which is currently considered a vital dilemma. Several researchers have highlighted various factors responsible for the deterioration of the environment. One of the main factors being green innovation has been considered an effective strategy to encounter the global issue of environmental pollution and achieve sustainability goals (e.g., Chang et al. 2023; Sadiq et al. 2023; Hoa et al. 2023; Wang et al. 2023; among others). Therefore, our study highlights green technology innovation as an important determinant for controlling environmental degradation in Saudi Arabia, which is one of the most dynamic emerging economies that will soon have a significant economic share.

The importance of technological innovation in improving the quality of the environment is widely discussed in existing literature. Tao et al. (2021), Hu et al. (2022), and Meng et al. (2022), among others, have shown that technological innovations improve production capacities and therefore combat environmental quality degradation. However, the majority of studies have proposed green technology innovation as a potential remedy to overcome excessive CO₂ emissions. Interestingly, these studies debated the direct or indirect heterogeneous effect of green innovation on CO₂ emissions. For example, Du et al. (2019) considered the effect of green innovation in addition to economic growth on the environment for 71 countries. They showed that green innovations improve the quality of the environment, but in a more effective way in high-income countries than in low-income countries. Similarly, Razzaq et al. (2021b) examined the exact correlation in terms of income between green innovations and CO₂ emissions for the 10 richest countries. They supported the extremely downplaying effect of green technological innovation on the environmental ruin of these high-income and highly polluted countries. Meirun et al. (2021) also reported that Singapore’s fundamental economic growth comes with a high ecological cost and that green technology innovation stabilizes economic growth with the lowest environmental cost.

In addition, some studies have linked green technological innovation to human capital and human development to analyze their effect on carbon emissions. Lin and Ma (2022) have shown that green technological innovation is only effective with a high level of capital and advanced transformation of the industrial structure. Saqib et al. (2022) used an extended version of the environmental Kuznets curve (EKC) for growing industrialized economies (E7) by including human development, the use of renewable energy, and technological innovations for empirical investigation. Their findings reveal that technological modernization helps to mitigate CO₂ emissions. Saqib et al. (2023b) examined the impacts of technological innovation, human capital, and renewable energy sources on the ecological footprints of G7 countries. They showed that technological innovation minimizes the ecological footprint. Their findings also indicate that ecologically sustainable technology improves the quality of the environment.

Other studies have instead linked green technological innovation to the country’s pollution level. Lingyan et al. (2022) supported the idea that green innovations effectively reduce carbon emissions only when emissions are high. In contrast, Sun et al. (2022) explored the increased effectiveness of green innovations in OECD countries with lower pollution rates. On the other hand, some studies have considered the direct effect of green innovation on consumption-based energy consumption. Tao et al. (2021) have shown that eco-innovation can achieve the emerging seven or growing industrialized (E7) countries’ goal of carbon neutrality. In addition, Xu et al. (2021) and Qin et al. (2021) validated the increasing effect of green innovation on carbon neutrality for China and G7 countries, respectively.

Another group of studies examined the impact of technological innovation in conjunction with other different factors. Saqib (2022) investigated the role of technical innovation, economic growth, and renewable energy use in the reduction of CO₂ emissions in the world’s 18 most developed countries. Their findings confirm that CO₂ emissions are reduced by positive technological innovation shocks and increased by negative shocks. Sharif et al. (2022) examined the role of green technological innovation and green financing in reducing CO₂ emissions in the G7 countries. Their results show that both green technology innovation and green financing have a significantly negative impact on CO₂ emissions. They also found that social globalization positively moderates the nexus between CO₂ emissions and economic growth but negatively and significantly causes green technology innovation and green financing with CO₂ emissions. Saqib et al. (2023a) investigated the impact of environmental technological innovation, economic complexity, energy productivity, the use of renewable electricity generation, and environmental taxes on CO₂ emissions in the G-10 countries. Their empirical results indicate that the increased use of environment-based technology, economic complexity, and renewable electricity generation has a significant positive impact on CO₂ emission reduction in both short-term and long-term projections. They also found evidence of both unidirectional and bidirectional causality from CO₂ emissions to renewable energy, electrical generation, and environment-based technologies, respectively. Saqib et al. (2023c) explored the link between CO₂ emissions, economic growth, renewable energy supply, development of patents, and gross fixed capital formation in the context of 32 OECD countries. Using a methodology based on the panel quantile regression approach, the authors showed that technological innovation harms CO₂ emissions. However, this impact is found to vary greatly across quantiles. They also explored the possibility of heterogeneity and asymmetry in the moderating effect of technological innovation on economic growth and renewable energy. Their findings indicated that technological innovation exerts a wide range of moderating effects.

Green technology innovation and consumption-based carbon emissions

The abundant empirical literature on the relationship between technological innovation and CO₂ emissions presents two main limitations. First, the aforementioned studies have extensively focused on traditional or territory-based CO₂ emissions data. Second, they have neglected the role of trade-adjusted consumption-based CO₂ (CCO₂) emissions data. Knight and Schor (2014), Liddle (2018a), Liddle (2018b), and Hasanov et al. (2018) considered territory-based and consumption-based CO₂ emissions in their empirical analysis. The CCO₂ emissions database has been developed to calculate the carbon emissions based on the domestic use of fossil fuels plus the embodied emissions from imports minus exports. By comparing the CCO₂ emissions data to the conventionally measured TCO₂ emissions data, Liddle (2018a) and Liddle (2018b) confirmed the usefulness of this new data insofar as it allows them to test the relevance of trade in national emissions.

To the best of our knowledge, the literature on CCO₂ emissions is scant. Interestingly, we found no single study that directly investigates the nexus between green innovation and CCO₂ emissions from the strand of the Saudi Arabian economy. Therefore, the current study fills the literature gap by assessing the direct or indirect impact of green innovation on CCO₂ emissions in Saudi Arabia.

However, we note the existence of a few studies that have investigated the role of green innovation in mitigating CCO₂ emissions by considering individual or groups of countries. Using different dimensions, periods, and econometric approaches, all these studies have indicated evidence of a negative relationship between green innovation and CCO₂ emissions. For example, Khan et al. (2020c) explored the unknown determinants of CCO₂ emissions in G7 countries during the period 1990–2017 by using the second-generation panel cointegration techniques. They found evidence of a significant negative impact of environmental innovation on CCO₂ emissions in both short term and long term. They also showed the existence of unidirectional causality running from environmental innovation to CCO₂ emissions. Besides, Razzaq et al. (2021c) examined the asymmetric inter-linkages between green technology innovation and both CCO₂ and TCO₂ emissions in BRICS countries from 1990 to 2017. Using QQR and GCQ frameworks, they found that green technology innovation (CCO₂ and TCO₂ emissions) mitigates (instigates) both CCO₂ and TCO₂ emissions (green technology innovation) when a country is embodied at higher emissions quantiles. Ding et al. (2021) analyzed the effect of eco-innovation, in addition to other factors, on CCO₂ emissions for G7 countries during the period 1990–2018. Using advanced panel data techniques including the cross-sectional autoregressive distributed lag model (CS-ARDL), the augmented mean group (AMG) estimator, and a panel causality test, the authors found that eco-innovation significantly decreased CCO₂ emissions at both the short run and long run. They also showed the existence of a significant unidirectional causality from eco-innovation to CCO₂ emissions.

Based on the Stochastic Impacts by Regression on Population, Affluence, and Technology (STIRPAT) framework, Jiang et al. (2022) analyzed the impact of environment-related technologies, among other variables, on CCO₂ emissions in BRICS countries over the period 1985–2018. Using the dynamic common correlated effects model, the authors found evidence of a significant negative effect of environment-related technologies on CCO₂ emissions.

Using the dynamic autoregressive-distributed lag (ARDL) simulations and frequency domain Granger causality approaches, Abbasi et al. (2022) examined the impact of technological innovation, among other factors, on both CCO₂ emissions and TCO₂ emissions in Pakistan from 1990Q1 to 2019Q4. They found that technological innovations substantially reduce both CCO₂ and TCO₂ emissions in the long run. However, their findings indicate an absence of causality between technological innovation and carbon emissions at all frequencies. Furthermore, Meng et al. (2022) evaluated the impact of green innovation, among other determinants, on CCO₂ emissions in BRICS countries over the period 1995–2020. Using a CS-ARDL modeling framework, the authors found evidence of a negative relationship between green innovation and CCO₂ emissions. Additionally, Kirikkaleli et al. (2023) explored the linkage between environmental innovation (proxied by patents) and CCO₂ emissions in Denmark by using the nonlinear ARDL, Fourier ARDL, and dynamic OLS approaches. Their results indicate that only positive shocks in environmental innovations have a significantly negative effect on CCO₂ emissions.

The study of Liu et al. (2023) investigates the dynamic effect of green technology innovation (proxied by eco patents as a percentage of total patents) on CCO₂ emissions in BRICS countries from 2000 to 2020. Using the CS-ARDL, augmented mean group (AMG), and common correlated effect mean group (CCEMG) methods for empirical analysis, the authors showed that technology innovation mitigates the CCO₂ emissions in both the short term and long term. They further indicated that the magnitude of long-term estimates is higher than that of the short term. Moreover, Xie et al. (2023) examined the empirical relationship between green innovation (proxied by the development of environment-related technologies as a percentage of all technologies), in addition to other factors, and CCO₂ emissions in a panel of high-income OECD countries over the period 1990–2020. Using the moment quantile regressions (MMQR) method, the authors found a significant negative impact of green innovation on CCO₂ emissions at different quantiles.

In the case of the USA, Razzaq et al. (2023) scrutinized the asymmetric linkage between green technology innovation (measured in the environmental-related technologies as a percentage of all technologies), among other factors, and CCO₂ emissions from 1990Q1 to 2018Q4. Using an econometric methodology based on the Quantile ARDL and GCQ approaches, they found an adverse link between green technology innovation and CCO₂ emissions in the long-term, mainly at lower-(extreme)upper emissions quantiles. Furthermore, their results indicate that the impact in the short run follows a similar direction, with the exception that its magnitude and statistical significance vary across quantiles. The findings about the GCQ reveal the existence of a unidirectional causality between green innovation and CCO₂ emissions across the major quantiles of the conditional distribution.

Data and methodology

Data description and preliminary analysis

In this paper, we investigated the dynamic interlinkages between green technology innovation (GTI) and both CCO₂ and TCO₂ emissions in Saudi Arabia using data from 1990 to 2021. CCO₂ emissions represent a country’s emissions that are adjusted for trade. This is calculated as the production-based emissions minus emissions embedded in exports, plus emissions embedded in imports. TCO₂ emissions, or production emissions, are those that occur within a country’s territorial borders and include exports but omit imports. Data on both CCO₂ and TCO₂ emissions are obtained from the Global Carbon Atlas database.² The data on GTI is described by environmental technologies as a percentage of total technologies and is sourced from the OECD statistics database. The sample period is selected based on the availability of the data. All these data series are firstly available at low frequency (annual) and then converted into high frequency (quarterly) by using the quadratic match-sum method. Therefore, the selected time-period spans from 1990Q1 to 2021Q4. This conversion method solves the problem of seasonal variation by eliminating the corresponding data deviations (e.g., Razzaq et al. 2021c; 2021a; 2021b; Mighri and AlSaggaf 2023; among others). In addition, these authors argued that this technique uses the local quadratic polynomials for all interpretations of the annual series to fill the higher frequency observations that are linked to the sample period. Then, we transformed all variables into their natural logarithmic forms to develop uniform measurement units and normalize outliers (Razzaq et al. 2023). As argued by Razzaq et al. (2020), An et al. (2021), and Razzaq et al. (2023), among others, the log transformation of variables overcomes the distributional properties while offering outputs in the form of elasticities.

The trends of both CCO₂ and TCO₂ emissions in Fig. 1 suggest the highest growth in Saudi Arabia since 2000; however, a declining trend was observed after 2015 with a steady increase after 2020. Interestingly, Saudi Arabia has shown an increasing trend in TCO₂ emissions from 208.497 MtCO₂ in 1990 to 672.3799 MtCO₂ in 2021. This may indicate the failure of the government to adopt a green technology optimization aimed at recycling and relocating polluting industries to other destinations. Except for 2015 and 2016, CCO₂ emissions are lower than TCO₂ emissions for the remaining years. This can be attributed to the use of import data instead of domestic production in the calculation. The average ratio of CCO₂ to TCO₂ emissions is lower than one, suggesting that Saudi Arabia is an exporter of carbon emissions. The same conclusion can be obtained by calculating the difference between CCO₂ and TCO₂ emissions. This difference is found to be negative on average, indicating that Saudi Arabia is a net exporter of carbon emissions. Therefore, to identify the environmental impact of a country, CCO₂ emissions seem to be the best measure of environmental degradation in Saudi Arabia.

Fig. 1

Trends of CCO₂ emissions, TCO₂ emissions, and GTI in Saudi Arabia

In addition, Fig. 1 depicts the trend of GTI in Saudi Arabia. Overall, it shows a time-varying pattern in GTI as a percentage of total technologies. The global ratio of GTI increases but at a steady rate. The share of GTI appears to be slower because other technologies are growing faster than environmental technologies. Consequently, the relative share of environmental technologies varies over time and seems to have an asymmetric effect on both CCO₂ and TCO₂ emissions in Saudi Arabia.

Before conducting further analysis, we made a preliminary statistical analysis by testing the normality, linearity, stationarity, and other time series properties of the selected variables. Table 1 reports the descriptive statistics of GTI and both CCO₂ and TCO₂ emissions in Saudi Arabia. Table 1 indicates that all variables have positive mean values. The highest standard deviation is observed for GTI whereas the data is negatively skewed in all series. The Jarque–Bera statistics are statistically significant at the 5% level, suggesting that GTI, CCO₂, and TCO₂ emissions deviate from the normal distribution and thus have an asymmetric pattern. Usually, the characteristics of non-normality are due to structural changes, crises, and political and institutional factors across countries (e.g., Aziz et al. 2020; Razzaq et al. 2020; Razzaq et al. 2021c; among others).

Table 1 Descriptive statistics

As a part of the preliminary analysis, we also calculated the Pearson correlation coefficients to understand the association between the series. The bivariate Pearson correlations summarized in Table 1 indicate the existence of a significantly weak positive correlation between GTI and CCO₂ emissions, as well as between TCO₂ emissions and GTI. These results suggest that an increase in GTI causes a weak increase in both CCO₂ and TCO₂ emissions in Saudi Arabia.

We also identified which of the series is nonlinear and conducted unit root tests using more advanced and powerful methods to detect the order of integration more precisely. Interestingly, we examined the linear properties of each series by using the linearity tests of Harvey and Leybourne (2007) and Harvey et al. (2008) at the 1%, 5%, and 10% significance levels. The results reported in Table 2 show the null of linearity is rejected in all cases using the ${W}_{\lambda }$ test of Harvey et al. (2008), which displays better size and more power than the ${W}^{*}$ test of Harvey and Leybourne (2007). Therefore, the results provide strong evidence that all series are nonlinear.

Table 2 Linearity test results

Time series are often characterized by unit root problems which may lead to spurious regressions. Therefore, we identified the order of integration of the series by carrying out the conventional linear unit root tests namely the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP), and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests. The results of these three-unit root tests are reported in Table 3. The main conclusion drawn from Table 3 is that GTI is stationary at level, i.e., I(0), while both CCO₂ and TCO₂ emissions variables are stationary at the first difference, i.e., I(1). Moreover, the results indicate that none of the series is I(2).

Table 3 Conventional linear unit root test results

However, we note that standard linear unit root tests lose power and provide misleading results not only because of the existence of nonlinear structures but also due to the evidence of structural breaks. These tests may deceptively prove the variables to be I(1) or I(2) when there are one or more structural breaks in the series. To consider the issue of power loss along with structural breaks, we utilized the one-break LM unit root test of Zivot and Andrews (1992) and the two breaks ADF unit root test of Narayan and Popp (2010). These tests have the advantage of accounting for one and two endogenous structural breaks in intercept and both the intercept and trend terms, respectively. Using these tests allows us to ascertain structural break(s) in the data series and to confirm the exclusion of I(2) variables.

The results of these unit root tests with structural break(s) are summarized in both Table 4 and 5. The results of both tests confirm time breaks in the data and also ascertain that none of the variables in the model is I(2). All the recognized break dates are generally related to both global and local events. Nevertheless, all these unit root tests are based on conditional means and do not provide the possibility to check the stationarity of each series at different quantiles of the conditional distribution. Therefore, the obtained results may again lead to biased estimates. Given data features, the quantile-based analysis seems to be the most appropriate procedure that simultaneously accounts for non-normality, structural changes, and non-linearity issues.

Table 4 One break ADF test results (Zivot and Andrews 1992)

Table 5 Two breaks ADF unit root test results (Narayan and Popp 2010)

Methodology

Quantile autoregressive (QAR) unit root test

Due to the non-normal distribution of our variables with a couple of non-linearities, the conventional unit root tests are unsuitable for inspecting the order of integration of the series. Therefore, we employed the QAR unit root test, which was initially proposed by Koenker and Xiao (2004) and then extended by Galvao (2009). The first advantage of the QAR unit root test deals with the non-normality or asymmetric distribution of the series. The extended version of this test allows the series to be examined with covariate stationarity and a linear time trend. The second advantage of the QAR test over the conventional tests is that it proves the stationarity or existence of a unit root not only on the conditional mean but also at each quantile of the conditional distribution. Therefore, this test examines the local stationarity of the series at different quantiles, while the conventional tests provide a global result. Given the distinct properties of the QAR unit root test, we estimated the $\tau$-th conditional quantile function of the time series process ${Y}_{t}$ based on the following linear QAR model:

$${Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)={\mu }_{1}\left(\tau \right)+{\mu }_{2}\left(\tau \right)t+\alpha \left(\tau \right){Y}_{t-1}+\sum \nolimits_{j=1}^{p}{\alpha }_{j}\left(\tau \right)\Delta {Y}_{t-j}+{F}_{u}^{-1}\left(\tau \right)$$

(1)

where ${Y}_{t}$ is a time series process; ${\mathcal{F}}_{t-1}^{Y}\equiv {\left({Y}_{t-1},\dots ,{Y}_{t-s}\right)}^{\prime}\in {\mathbb{R}}^{s}$ denotes the past information set of ${Y}_{t}$; ${Q}_{\tau }^{Y}\left(.\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)$ denotes the $\tau$-th quantile conditional value of the conditional distribution function of ${Y}_{t}$ (i.e., ${F}_{Y}\left(.\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)$); ${\mu }_{1}\left(\tau \right)$ is a drift term; $t$ is a linear trend; $\alpha \left(\tau \right)$ is the persistence parameter; and ${F}_{u}^{-1}$ stands for the inverse conditional distribution function of the errors, for each quantile $\tau \in {\mathcal{T}}_{n}$, where ${\mathcal{T}}_{n}={\left\{{\tau }_{j}\right\}}_{j=1}^{n}\subset [\mathrm{0,1}]$ is a grid of $n$ equidistributed points. Next, we estimated $\widehat{\alpha }\left(\tau \right)$ for each quantile $\tau$ of ${F}_{Y}\left(.\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)$. We then examined the quantile stationarity by testing the null hypothesis ${H}_{0}:\alpha \left(\tau \right)=1$ for $\tau \in {\mathcal{T}}_{n}$. The decision rule is to compare the estimated critical value and the estimated $t$-statistic proposed by Koenker and Xiao (2004) and Galvao (2009) at different quantiles of the conditional distribution.

Quantile cointegration test

Xiao (2009) extended the conventional cointegration model of Engle and Granger (2012) by introducing the quantile cointegration model with quantile-varying coefficients. This model allows capturing systematic effects of conditioning variables on the location, scale, and shape of the conditional distribution of the response variable.

To deal with the issues of endogeneity in the conventional cointegration model, Xiao (2009) decomposed the errors of the cointegrating equation into lead-lag terms and a pure innovation component.

$${Q}_{\tau }^{Y}\left({Y}_{t}\left|{I}_{t-1}^{Y},{I}_{t-1}^{X}\right.\right)=\alpha \left(\tau \right)+{\beta \left(\tau \right)}^{\prime}{X}_{t}+{\gamma \left(\tau \right)}^{\prime}{X}_{t}^{2}+\sum \nolimits_{j=-K}^{K}\Delta {X}_{t-j}^{\prime}{\Pi }_{j}+\sum \nolimits_{j=-K}^{K}\Delta {X}_{t-j}^{2^{\prime}}{\Gamma }_{j}+{F}_{u}^{-1}\left(\tau \right)$$

(2)

where ${Y}_{t}$ and ${X}_{t}$ are two nonstationary time series, and $\beta (\tau )$ is a vector of constants.

In the next step, Xiao (2009) developed a test statistic for the stability of the cointegrating coefficients in Eq. (2). Under ${H}_{0}: \beta \left(\tau \right)=\beta$ over all quantiles $\tau \in {\mathcal{T}}_{n}$, Xiao (2009) introduced the test statistic ${{\text{sup}}}_{\tau }\left|{\widehat{V}}_{n}\left(\tau \right)\right|$, which is based on the supremum norm of the absolute value of the difference ${\widehat{V}}_{n}\left(\tau \right)=\left(\widehat{\beta }\left(\tau \right)-\widehat{\beta }\right)$. To calculate the critical values of the test statistic ${{\text{sup}}}_{\tau }\left|{\widehat{V}}_{n}\left(\tau \right)\right|$, we performed 1000 Monte Carlo simulations.

QQR approach

Sim and Zhou (2015) proposed the QQR method to deal with various shortcomings of the conventional QR approach (Koenker 2005). First, the QQR approach combines the local linear regression with the nonparametric properties of QR. Second, it provides slope coefficients of the independent variables on different quantiles of the dependent variable together with the approximate impacts of each quantile of the independent variables on the dependent variable, using the OLS technique. Third, this procedure allows the independent variable to be clustered as a quantile distribution, thus addressing the issue of interconnectedness or reverse causality. Lastly, the QQR method is a good fit for evaluating data properties series characterized by low skewness and high excess kurtosis.

Let ${X}_{t}$ and ${Y}_{t}$ two-time series, whose relationship is given by the following equation:

$${Y}_{t}={\beta }^{\theta }\left({X}_{t}\right)+{\varepsilon }_{t}^{\theta }$$

(3)

where $\theta$ is the quantile of the dependent variable, ${\beta }^{\theta }$ denotes the slope of the relationship between ${X}_{t}$ and ${Y}_{t}$, and ${\varepsilon }_{t}^{\theta }$ is the quantile error term.

Let ${x}_{t}^{\tau }$ be the $\tau$-quantile of ${X}_{t}$. Following Sim and Zhou (2015), we approximated Eq. (3) using the first-order Taylor expansion of ${\beta }^{\theta }\left({X}_{t}\right)$ around ${X}_{t}^{\tau }$:

$${\beta }^{\theta }\left({X}_{t}\right)\approx {\beta }_{1}\left(\tau ,\theta \right)+{\beta }_{2}\left(\tau ,\theta \right)\left({X}_{t}-{X}_{t}^{\tau }\right)$$

(4)

Then, Eq. (3) can be rewritten as follows:

$${Y}_{t}={\beta }_{1}\left(\tau ,\theta \right)+{\beta }_{2}\left(\tau ,\theta \right)\left({X}_{t}-{X}_{t}^{\tau }\right)+{\varepsilon }_{t}^{\theta }$$

(5)

For each value of $\tau$, we used the quantile regression method to estimate Eq. (5). Formally, we estimated $\beta \left(\tau ,\theta \right)$ as follows:

$$\widehat{\beta }\left(\tau ,\theta \right)=\underset{\beta \left(\tau ,\theta \right)}{{\text{argmin}}}\sum \nolimits_{t=1}^{T}{\rho }_{\theta }\left({Y}_{t}-{\beta }_{1}\left(\tau ,\theta \right)-{\beta }_{2}\left(\tau ,\theta \right)\left({X}_{t}-{X}_{t}^{\tau }\right)\right)$$

(6)

where ${\rho }_{\theta }\left(.\right)$ denotes the check function.

Sim and Zhou (2015) recognized the need to correctly weight the function rather than estimating this model due to the impact applied locally by the $\tau$-quantile of ${X}_{t}$ on ${Y}_{t}$. Otherwise, the impact will not be included in the neighborhood of $\tau$. To smooth out undesirable impacts that could affect the findings, Sim and Zhou (2015) chose the normal kernel function. Therefore, the generated weights are inversely linked to the distance between ${X}_{t}$ and ${X}_{t}^{\tau }$ or, equivalently, between the empirical distribution of ${X}_{t}$, $F\left({X}_{t}\right)$ and $\tau$. Then, the estimator can be respecified as follows:

$$\widehat\beta\left(\tau,\theta\right)=\underset{\beta\left(\tau,\theta\right)}{\mathrm{argmin}}\sum\nolimits_{t=1}^T\rho_\theta\left(Y_t-\beta_1\left(\tau,\theta\right)-\beta_2\left(\tau,\theta\right)\left(X_t-X_t^\tau\right)\right)K\left(\frac{X_t-X_t^\tau}h\right)$$

(7)

where $h$ denotes the bandwidth.

When dealing with a non-parametric approach, we should select an appropriate bandwidth to get good results. To do so, we chose the Silverman optimal bandwidth given by:

$$h=\alpha \sigma {N}^{-1/3}$$

(8)

where $\sigma ={\text{min}}\left(\frac{IQR}{1.34},std\left(X\right)\right)$, with IQR denoting the interquartile range, $N$ representing the sample size, and $\alpha =3.49$.

GCQ tests

Following Troster (2018), Troster et al. (2018), Troster et al. (2019), and Ahmed et al. (2022), we considered two stationary time series namely ${Y}_{t}$ and ${X}_{t}$. Let ${X}_{t}={{\text{GTI}}}_{t}$ be the green technology innovation measure, and ${Y}_{t}={{\text{CCO}}}_{2,t}$ is the consumption-based carbon emission measure. Let ${\mathcal{F}}_{t}\equiv {\left({\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right)}^{\prime}\in {\mathbb{R}}^{d}; d=s+q$ be the past information sets of ${Y}_{t}$ and ${X}_{t}$ with ${\mathcal{F}}_{t}^{Y}\equiv {\left({Y}_{t-1},\dots ,{Y}_{t-s}\right)}^{\prime}\in {\mathbb{R}}^{s}$ and ${\mathcal{F}}_{t}^{X}\equiv {\left({X}_{t-1},\dots ,{X}_{t-q}\right)}^{\prime}\in {\mathbb{R}}^{q}$. The null of Granger non-causality in distribution from ${X}_{t}$ to ${Y}_{t}$ (Granger 1969; Granger 1980) is defined as follows:

$$\begin{array}{cc}{H}_{0}^{X\nRightarrow Y}:{F}_{Y}\left(y\left|{\mathcal{F}}_{t}^{Y},{I}_{t}^{X}\right.\right)={F}_{Y}\left(y\left|{\mathcal{F}}_{t}^{Y}\right.\right);& \forall y\in {\mathbb{R}}\end{array}$$

(9)

where ${F}_{Y}\left(y\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)$ and ${F}_{Y}\left(y\left|{\mathcal{F}}_{t}^{Y}\right.\right)$ denote the conditional distribution functions of ${Y}_{t}$ given $\left({\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right)$ and ${\mathcal{F}}_{t}^{Y}$, respectively. However, most studies test for Granger-non-causality in the mean.

Therefore, the null hypothesis is redefined as follows:

$$\begin{array}{cc}{H}_{0}^{X\nRightarrow Y}: {\text{E}}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)={\text{E}}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y}\right.\right);& \forall Y\in {\mathbb{R}}\end{array}$$

(10)

where ${\text{E}}\left(.\right)$ denotes the conditional mean operator.

In the steps, we firstly tested for Granger non-causality in the mean by utilizing a linear F-test on the estimated parameters of bivariate vector autoregressive (VAR) models. Formally, we tested ${H}_{0}:{\beta }_{j}=0, \forall j=1,\dots ,p$ based on the following VAR models:

$${Y}_{t}=\sum \nolimits_{j=1}^{p}{\alpha }_{j}{Y}_{t-j}+\sum \nolimits_{j=1}^{p}{\beta }_{j}{X}_{t-j}+{\varepsilon }_{t}$$

(11)

$${X}_{t}=\sum \nolimits_{j=1}^{p}{\alpha }_{j}^{*}{Y}_{t-j}+\sum \nolimits_{j=1}^{p}{\beta }_{j}^{*}{X}_{t-j}+{\varepsilon }_{t}^{*}$$

(12)

where the optimal lag orders $p$ are selected based on BIC, considering a lag length of up to 12, while ${\varepsilon }_{t}$ and ${\varepsilon }_{t}^{*}$ denote two serially uncorrelated white noise processes.

Following Troster et al. (2019) and Ahmed et al. (2022), we considered the heteroscedasticity issues in ${\varepsilon }_{t}$ and ${\varepsilon }_{t}^{*}$ by applying the robust heteroscedasticity-consistent covariance matrix estimator of the residuals of the VAR model (MacKinnon and White 1985). Moreover, we checked for serial correlation in residuals using the test proposed by Breusch (1978) and Godfrey (1978). To consider the potential nonlinearities in the data series, we applied three parameter stability test statistics (SupF, AveF, and ExpF) on the estimated coefficients of the VAR model proposed by Andrews (1993), Andrews and Ploberger (1994), and Hansen (1997). We further tested the null hypothesis that the residuals of the VAR model are independent and identically distributed (iid) versus the alternative hypothesis of nonlinearity by using the BDS test (Broock et al. 1996). If the null hypothesis of linearity is rejected, then we may consider the nonlinear tests for Granger non-causality in the mean. Interestingly, we employed the nonlinear tests proposed by Hiemstra and Jones (1994) and Diks and Panchenko (2006) on the standardized residuals of the VAR model.

Otherwise, the Granger non-causality in the mean test in Eq. (10) overlooks potential tail-dependence or causality at other moments of the conditional distribution (Troster 2018; Troster et al. 2018, 2019; Ahmed et al. 2022). We followed Troster (2018) by further testing causality at each quantile of the conditional distribution. The quantile causality test entirely determines the distribution of the series while verifying whether the relationship between ${X}_{t}$ and ${Y}_{t}$ is symmetric or asymmetric over the conditional distribution of ${Y}_{t}$. Therefore, quantile causality analysis allows detecting possibly nonlinear or asymmetric links across the distribution. Let ${Q}_{\tau }^{Y,X}\left(.\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)$ and ${Q}_{\tau }^{Y}\left(.\left|{\mathcal{F}}_{t}^{Y}\right.\right)$ denote the $\tau$-quantiles of ${F}_{Y}\left(.\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)$ and ${F}_{Y}\left(.\left|{\mathcal{F}}_{t}^{Y}\right.\right)$, respectively. Following Troster (2018), Eq. (9) can be equally tested as follows:

$$\begin{array}{cc}{H}_{0}^{QC: X\nRightarrow Y}:{ Q}_{\tau }^{Y,X}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)={Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y}\right.\right);& {\text{a}}.{\text{s}}. \forall \tau \in \mathcal{T}\end{array}$$

(13)

where $\mathcal{T}\subset [\mathrm{0,1}]$ and the conditional $\tau$-quantiles of ${Y}_{t}$ satisfy the following two restrictions:

$$\begin{array}{cc}{\text{Prob}}\left[{Y}_{t}\le {Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y}\right.\right)\left|{\mathcal{F}}_{t}^{Y}\right.\right]\equiv \tau ;& {\text{a}}.{\text{s}}. \forall \tau \in \mathcal{T}\end{array}$$

(14)

$${\text{Prob}}\left[{Y}_{t}\le { Q}_{\tau }^{Y,X}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right]\equiv \tau ; {\text{a}}.{\text{s}}. \forall \tau \in \mathcal{T}$$

(15)

Otherwise, it is known that

$${\text{Prob}}\left[{Y}_{t}\le {Q}_{\tau }\left({Y}_{t}\left|{\mathcal{F}}_{t}\right.\right)\left|{\mathcal{F}}_{t}\right.\right]={\text{E}}\left\{\mathbf{I}\left[{Y}_{t}\le {Q}_{\tau }\left({Y}_{t}\left|{\mathcal{F}}_{t}\right.\right)\left|{\mathcal{F}}_{t}\right.\right]\right\}$$

(16)

where $\mathbf{I}\left(.\right)$ is an indicator function. Therefore, Eq. (13) is equivalent to:

$$\begin{array}{c}{\text{E}}\left\{\mathbf{I}\left[{Y}_{t}\le { Q}_{\tau }^{Y,X}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right]\right\}\\ ={\text{E}}\left\{\mathbf{I}\left[{Y}_{t}\le { Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y}\right.\right)\right]\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right\}; {\text{a}}.{\text{s}}.\forall \tau \in \mathcal{T}\end{array}$$

(17)

where the left-hand-side of Eq. (17) is equal to the $\tau$-quantile of ${F}_{Y}\left(.\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right)$.

To estimate the $\tau$-quantile of ${F}_{Y}\left(.\left|{\mathcal{F}}_{t}\right.\right)$, Troster (2018) assumed that ${Q}_{\tau }\left(.\left|\mathcal{F}\right.\right)$ is correctly specified by a parametric QAR model $m\left(.,\theta \left(\tau \right)\right)\in \mathcal{M}=\left\{m\left(.,\theta \left(\tau \right)\right)\left|\theta \left(.\right):\tau \to \theta \left(\tau \right)\in\Theta \subset {\mathbb{R}}^{p}, \forall \tau \in \mathcal{T}\right.\right\}$, where $\mathcal{M}$ is a family of functions. Under ${H}_{0}^{QC: X\nRightarrow Y}$ in Eq. (13), the conditional quantile ${Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t}^{Y}\right.\right)$ is correctly specified by a parametric QAR model $m\left({\mathcal{F}}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)$ for some ${\theta }_{0}\in \mathcal{B}\subset \mathcal{M}$, using only ${\mathcal{F}}_{t}^{Y}$. Then, the testing problem in Eq. (13) can be redefined as follows:

$${H}_{0}^{X\nRightarrow Y}:\mathrm{ E}\left\{\mathbf{I}\left[{Y}_{t}\le m\left({I}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)\right]\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right\}=\tau ; {\text{a}}.{\text{s}}.\forall \tau \in \mathcal{T}$$

(18)

versus

$${H}_{0}^{X\nRightarrow Y}:{\text{E}}\left\{\mathbf{I}\left[{Y}_{t}\le m\left({I}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)\right]\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right\}\ne \tau ; {\text{a}}.{\text{s}}. \mathrm{for\;some\;} \tau \in \mathcal{T}$$

(19)

where $m\left({\mathcal{F}}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)$ is the only element of $\mathcal{M}$ that correctly specifies the true ${Q}_{\tau }^{Y}\left(.\left|{\mathcal{F}}_{t}^{Y}\right.\right)$ for all $\tau \in \mathcal{T}$.

The null hypothesis in Eq. (18) can be rewritten as follows:

$${H}_{0}^{X\nRightarrow Y}:{\text{E}}\left\{\left(\mathbf{I}\left[{Y}_{t}-m\left({I}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)\le 0\right]-\tau \right)\left|{\mathcal{F}}_{t}^{Y},{\mathcal{F}}_{t}^{X}\right.\right\}=0; {\text{a}}.{\text{s}}.\forall \tau \in \mathcal{T}$$

(20)

${H}_{0}^{X\nRightarrow Y}$ in Eq. (20) is then characterized by the following sequence of conditional moment restrictions:

$${\text{E}}\left\{\left(\mathbf{I}\left[{Y}_{t}-m\left({I}_{t}^{Y},{\theta }_{0}\left(\tau \right)\right)\le 0\right]-\tau \right){\text{exp}}\left(i{{\varvec{\omega}}}^{\prime}{I}_{t}\right)\right\}=0; \forall \tau \in \mathcal{T}$$

(21)

where ${\text{exp}}\left(i{{\varvec{\omega}}}^{\mathrm{^{\prime}}}{I}_{t}\right)\equiv {\text{exp}}\left[i\left({\omega }_{1}{\left({Y}_{t-1},{X}_{t-1}\right)}^{\mathrm{^{\prime}}}+\dots +{\omega }_{r}{\left({Y}_{t-r},{X}_{t-r}\right)}^{\mathrm{^{\prime}}}\right)\right]$ is a weighting function $\forall {\varvec{\omega}}\in \mathcal{W}\subset {\mathbb{R}}^{r}$ with $r\le d$ and ${i}^{2}=-1$.

The test statistic proposed by Troster (2018) is based on the sample analog of the moment restriction of Eq. (21), i.e.,

$${v}_{T}\left({\varvec{\omega}},\tau \right)\equiv \frac{1}{\sqrt{T}}\sum \nolimits_{t=1}^{T}\left[\mathbf{I}\left({Y}_{t}-m\left({I}_{t}^{Y},{\theta }_{T}\left(\tau \right)\right)\le 0\right)-\tau \right]{\text{exp}}\left(i{{\varvec{\omega}}}^{\prime}{I}_{t}\right)$$

(22)

where ${\theta }_{T}$ is a $\sqrt{T}$-consistent estimator of ${\theta }_{0}\left(\tau \right)$, $\forall \tau \in \mathcal{T}$.

Given the sample ${\left\{\left({Y}_{t},{X}_{t}\right)\right\}}_{t\in {\mathbb{Z}}}$, Troster (2018) defined ${v}_{T}\left({\varvec{\omega}},\tau \right)$ as the quantile marked-residual process, $\forall {\varvec{\omega}}\in {\mathbb{R}}^{d}$ and $\forall \tau \in \mathcal{T}$. Their proposed test statistic ${S}_{T}$ is a Cramér-von Mises (CvM) functional norm of ${v}_{T}\left({\varvec{\omega}},\tau \right)$ defined as follows:

$${S}_{T}\equiv {\int }_{\mathcal{T}}{\int }_{\mathcal{W}}{\left|{v}_{T}\left({\varvec{\omega}},\tau \right)\right|}^{2}d{F}_{\omega }\left(\omega \right)d{F}_{\tau }\left(\tau \right)$$

(23)

where ${F}_{\omega }\left(.\right)$ is the conditional distribution function of a $d$-variate standard normal random vector, ${F}_{\tau }\left(.\right)$ follows a uniform discrete distribution over $\mathcal{T}$, and the vector of weights ${\varvec{\omega}}\in {\mathbb{R}}^{d}$ is drawn form a standard normal distribution.

To estimate ${S}_{T}$ in Eq. (23), Troster (2018) used its sample counterpart. The author considered a $T\times n$ matrix $\Psi$ with elements ${\psi }_{i,j}={\Psi }_{\tau j}\left({Y}_{i}-m\left({\mathcal{F}}_{i}^{Y},{\theta }_{T}\left({\tau }_{j}\right)\right)\right)$, where ${\Psi }_{\tau j}\left(.\right)$ is the function ${\Psi }_{\tau j}\left(\varepsilon \right):=I\left(\varepsilon \le 0\right)-{\tau }_{j}$ for all $\tau \in \mathcal{T}$, and $m\left({\mathcal{F}}_{i}^{Y},{\theta }_{T}\left({\tau }_{j}\right)\right)$ is a parametric QAR model for the conditional $\tau$-quantile of ${Y}_{i}$. Finally, the form of the proposed test statistic ${S}_{T}$ is given as follows:

$${S}_{T}=\frac{1}{Tn}\sum \nolimits_{j=1}^{n}\left|{\psi }_{.j}^{\prime}{\varvec{W}}{\psi }_{.j}\right|$$

(24)

where ${\varvec{W}}$ is a $T\times T$ matrix comprising the elements ${w}_{t,s}={\text{exp}}\left[-0.5{\left({I}_{t}-{I}_{s}\right)}^{2}\right]$, and ${\psi }_{.j}{\prime}$ is the j-th column of $\Psi$.

To calculate the critical values for ${S}_{T}$ in Eq. (24), Troster (2018) suggested using a subsampling method by following the approach of Sakov and Bickel (2000). By considering a time series $\left\{{Z}_{t}=\left({Y}_{t},{X}_{t}\right)\right\}$ with sample size$T$, Troster (2018) generated $B=\left(T-b+1\right)$ subsamples of optimal size $b=[k{T}^{2/5}]$ of the form$\left\{{Z}_{i},\dots ,{Z}_{i+b-1}\right\}$, where $k$ is a constant parameter and $\left[.\right]$ is the integer part of a number.³ Then, Troster (2018) defined the cumulative distribution function of the ${S}_{T}$ test statistic as$G\left(x\right)\equiv {\text{Prob}}\left({S}_{T}\le x\right)$. They estimated $G\left(x\right)$ by$\widehat{G}\left(x\right)={B}^{-1}\sum_{i=1}^{B}\left({S}_{b,i}\le x\right)$, where ${S}_{b,i}\equiv {\int }_{\mathcal{T}}{\int }_{\mathcal{W}}{\left|{v}_{b,i}\left({\varvec{\omega}},\tau \right)\right|}^{2}d{F}_{\omega }\left(\omega \right)d{F}_{\tau }\left(\tau \right)$ denotes the test statistic calculated with the i-th subsample $\left\{{Z}_{i},\dots ,{Z}_{i+b-1}\right\}$ of size $b$ and with ${v}_{b,i}\left({\varvec{\omega}},\tau \right)\equiv {v}_{b,i}\left({Z}_{i},\dots ,{Z}_{i+b-1},\tau \right)$ denotes the inference process computed for each i-th subsample$i\le B$. Afterwards, Troster (2018) obtained the critical values as the $\left(1-\tau \right)$-th quantile of$\widehat{G}\left(.\right)$, i.e.,${c}_{T,b}\left(1-\tau \right)={\widehat{G}}^{-1}\left(1-\tau \right)$. Lastly, the rule of decision consists in rejecting ${H}_{0}^{X\nRightarrow Y}$ in Eq. (18) if${S}_{T}>{c}_{T,b}\left(1-\tau \right)$. Following Troster (2018), Troster et al. (2018), and Troster et al. (2019), we can also use the corresponding p-values for ${S}_{T}$ in Eq. (24), which are obtained by averaging the subsample test statistics over the $B$ subsamples to decide on the rejection or acceptance of the null hypothesis ${H}_{0}^{X\nRightarrow Y}$ in Eq. (18).

To apply the ${S}_{T}$ test statistic in Eq. (24), we followed Troster (2018), Troster et al. (2018), and Troster et al. (2019) by specifying the following three parametric QAR models for modeling the conditional quantiles of ${Y}_{t}$ for all $\tau \in \mathcal{T}$:

$$QAR\left(1\right): {m}^{1}\left({\mathcal{F}}_{t}^{Y},{\theta }_{T}\left(\tau \right)\right)={\mu }_{0}\left(\tau \right)+{\mu }_{1}\left(\tau \right){Y}_{t-1}+{\sigma }_{t}{\Phi }_{\varepsilon }^{-1}\left(\tau \right)$$

(25)

$$QAR\left(2\right): {m}^{2}\left({\mathcal{F}}_{t}^{Y},{\theta }_{T}\left(\tau \right)\right)={\mu }_{0}\left(\tau \right)+{\mu }_{1}\left(\tau \right){Y}_{t-1}+{\mu }_{2}\left(\tau \right){Y}_{t-2}+{\sigma }_{t}{\Phi }_{\varepsilon }^{-1}\left(\tau \right)$$

(26)

$$QAR\left(3\right): {m}^{3}\left({\mathcal{F}}_{t}^{Y},{\theta }_{T}\left(\tau \right)\right)={\mu }_{0}\left(\tau \right)+{\mu }_{1}\left(\tau \right){Y}_{t-1}+{\mu }_{2}\left(\tau \right){Y}_{t-2}+{\mu }_{3}\left(\tau \right){Y}_{t-3}+{\sigma }_{t}{\Phi }_{\varepsilon }^{-1}\left(\tau \right)$$

(27)

where the parameters $\theta \left(\tau \right)={\left({\mu }_{0}\left(\tau \right),{\mu }_{1}\left(\tau \right),{\mu }_{2}\left(\tau \right),{\mu }_{3}\left(\tau \right),{\sigma }_{t}\right)}{\prime}$ are estimated using the maximum likelihood method for all $\tau \in \mathcal{T}$, and ${\Phi }_{\varepsilon }^{-1}\left(\tau \right)$ denotes the inverse of the standard normal distribution function.

The ${S}_{T}$ test statistic in Eq. (24) allows us to test for quantile causality on the distribution of ${Y}_{t}$ using different linear specifications of the QAR model under ${H}_{0}^{X\nRightarrow Y}$ in Eq. (18). Furthermore, the modeling framework proposed by Troster (2018) allows for any $\sqrt{T}$-consistent estimator of ${\theta }_{T}\left(\tau \right)$ such as the CAViaR estimator by Engle and Manganelli (2004). Under ${H}_{0}^{X\nRightarrow Y}$ in Eq. (21), Troster et al. (2019) specified nonlinear QAR models using CAViaR specifications. As argued by Troster et al. (2019) and Ahmed et al. (2022), the nonlinear CAViaR models specify an AR process of the conditional quantiles, which avoids distributional assumptions and provides reliable performance compared to alternative models (Bao et al. 2006; Yu et al. 2010; Chen et al. 2012; Jeon and Taylor 2013). Following Troster et al. (2019) and Ahmed et al. (2022), we considered two specifications of the nonlinear CAViaR models, namely the symmetric absolute value (SAV) and the asymmetric slope (AS) under ${H}_{0}^{X\nRightarrow Y}$ in Eq. (21) as follows:

$$SAV: { Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)=\alpha \left(\tau \right)+\beta \left(\tau \right){ Q}_{\tau }^{Y}\left({Y}_{t-1}\left|{\mathcal{F}}_{t-2}^{Y}\right.\right)+\gamma \left(\tau \right)\left|{Y}_{t-1}\right|$$

(28)

$$AS: { Q}_{\tau }^{Y}\left({Y}_{t}\left|{\mathcal{F}}_{t-1}^{Y}\right.\right)=\alpha \left(\tau \right)+\beta \left(\tau \right){ Q}_{\tau }^{Y}\left({Y}_{t-1}\left|{\mathcal{F}}_{t-2}^{Y}\right.\right)+\gamma \left(\tau \right){\left({Y}_{t-1}\right)}^{+}+\delta \left(\tau \right){\left({Y}_{t-1}\right)}^{-}$$

(29)

where ${\left({Y}_{t-1}\right)}^{+}={\text{max}}\left\{{Y}_{t-1},0\right\}$ and ${\left({Y}_{t-1}\right)}^{-}=-{\text{min}}\left\{{Y}_{t-1},0\right\}$.

Empirical results and discussions

Quantile unit root test results

We examined the persistence of unit roots in a grid of 19 quantiles ${\mathcal{T}}_{n}={\left\{{\tau }_{j}\right\}}_{j=1}^{n}=\left[0.05;0.95\right]$ using the QAR unit root test on the level series. The non-normality of the variables is the main reason for using the quantile unit root tests instead of conventional ones. Advantageously, the quantile unit root tests exclude the possibly biased results and provide a more robust inference. Interestingly, we tested the null hypothesis of a unit root in each quantile, i.e., ${H}_{0}:\alpha \left(\tau \right)=1$. Following Troster et al. (2018), we included ten lags of the difference of the dependent variable to avoid the serial correlation of the residuals. To check whether there is a unit root in quantiles, we compared the critical values with the estimated $t$-statistics values. For each quantile, the null hypothesis ${H}_{0}:\alpha \left(\tau \right)=1$ cannot be rejected if the estimated $t$-statistics value is numerically greater than the corresponding estimated critical value at the 5% level of significance.

The results of the QAR unit root test summarized in Table 6 indicate that all variables are non-stationary, i.e., I(1), at the 5% significance level across all quantiles of the conditional distribution. The only exception is observed for the GTI variable, which is level stationary, i.e., I(0), at only the 85th quantile. The presence of model parameters referring to dissimilar integration orders guarantees the applicability of the QQR technique.

Table 6 QAR unit root test results

Quantile cointegration test results

After validating the non-stationarity assumption of all variables, we checked for the existence of cointegration (long-run) relationships between CCO₂ (TCO₂) emissions and GTI across quantiles. To test whether these cointegration relationships change over the conditional distribution, we employed the quantile cointegration test developed by Xiao (2009), which is based on the QAR model in Eq. (2). The test statistic of this model is calculated using an equally spaced grid of 19 quantiles ${\mathcal{T}}_{n}=\left[0.05;0.95\right]$. Furthermore, we included two lags and two leads of $\left(\Delta {Z}_{t},\Delta {Z}_{t}^{2}\right)$ in the QAR model. The results are reported in Table 7, confirming the existence of a non-linear cointegration relationship at the 1% level of significance between model parameters. Both test statistics ($\beta$ and $\gamma$) reject the null of linear cointegration between CCO₂ (TCO₂) emissions and GTI at the 1% significance level.

Table 7 Quantile cointegration test results

Then, we estimated the cointegration coefficients $\widehat{\alpha }\left(\tau \right)$, $\widehat{\beta }\left(\tau \right)$ and $\widehat{\gamma }\left(\tau \right)$ of the QAR model in Eq. (2). The findings are summarized in Table 8. Regarding the non-linear cointegration relationship between GTI and CCO₂ emissions, the estimated coefficients are significantly negative at the extremely low quantile $\left(\tau =0.05\right)$, while they become significantly positive for all quantiles greater than or equal to $\tau =0.25$. Meanwhile, the estimated coefficients of the QAR model between GTI and TCO₂ emissions are significantly positive for all quantiles greater than or equal to $\tau =0.2$.

Table 8 Estimated coefficients of the QAR model in Eq. (2)

QQR results

After confirming the existence of a long-run cointegration relationship between CCO₂ (TCO₂) emissions and green innovation, we employed the QQR approach to identify these associations in Saudi Arabia. The non-normality and non-linearity of the data series are two further conditions that validate the use of the QQR technique. Interestingly, we checked the causative relationship between dependent and independent variables within the bivariate modeling framework under the QQR approach. In this regard, we plotted in Figs. 2 and 3 the QQR slope coefficients in three-dimensional graphs. The plot in Figs. 2a and 3a visualizes the impact of the $\tau$-th equidistant quantile of GTI on $\theta$-th equidistant quantile of CCO₂ emissions (TCO₂ emissions), as indicated by the slope parameters ${\beta }_{2}\left(\tau ,\theta \right)$. Meanwhile, the plot in Figs. 2b and 3b show the reverse effect of the $\tau$-th equidistant quantile of CCO₂ emissions (TCO₂ emissions) on the $\theta$-th equidistant quantile of GTI. It should stressed that lower quantiles denote lower values of variables while upper quantiles reflect higher values of the respective series. In addition, the color bar represents the slope coefficients on the $z$-axis. Notably, the lowest (highest) value of the slope coefficient is marked with a dark or gloomy blue (yellow) color. The yellow and green colors highlight strong positive and normal positive coefficients, while the bright blue color shows moderately positive coefficients. The gloomy blue color shows trivial negative coefficients. Overall, the plots of slope coefficients of QQR estimation show that the relationship between green innovation and carbon emissions is not symmetrical. Alternatively, the shape of all four three-dimensional quantile curves varies across the different levels of quantiles and takes the general W form.

Fig. 2

QQR estimates for GTI and CCO₂ emissions (3-dimensions)

Fig. 3

QQR estimates for GTI and TCO₂ emissions (3-dimensions)

Figure 2a shows the impact of GTI on CCO₂ emissions in Saudi Arabia. Both dim and dark blue contours of the graph characterize the strong negative influence of GTI on CCO₂ emissions. GTI greatly diminishes CCO₂ emissions at upper quantiles (0.65–0.90) as well as at extreme upper quantiles of CCO₂ emissions, which are distributed across all quantiles of GTI. This finding indicates that the emissions-lowering impact of GTI is greater for both upper and extremely upper quantiles of CCO₂ emissions. However, a weak positive effect of GTI on CCO₂ emissions is also observed at higher quantiles (0.6–0.7 and 0.8–0.9) as well as at extreme upper quantiles of CCO₂ emissions. This positive association spreads from medium to upper quantiles (0.5–0.9) of GTI. Besides, there is a peak showing the weak positive relationship between GTI and CCO₂ emissions at the higher quantiles (0.6–0.8) of CCO₂ emissions and GTI. In general, Fig. 2a seems to show W-shape visuals where the left wing is dominated by a strong negative effect of GTI on CCO₂ emissions, while the right-wing appears dominated by a weak positive impact of GTI on CCO₂ emissions. Overall, our findings indicate that the emissions-mitigating impact of GTI on CCO₂ emissions is highest and persistent across the higher levels of emissions. There are two main reasons through which the results obtained in this study can be explained. First, at the primary (lower) level of emissions, the government’s focus on environmental technologies appears to be rather weak. Thus, the marginal effect of GTI appears positive but weak because emissions and technologies grow at the initial level of development. Second, industrialization, exports, rapid growth, and construction depend mainly on high rates of resource extraction and downstream emissions. Therefore, the net increase in emissions is relatively higher compared to GTI.

Since the adoption of Saudi Arabia’s Vision 2030, the government has focused on adopting policies aimed at confronting the challenges of energy and climate change through innovative solutions that include the circular carbon economy and diversification of energy sources. Accordingly, the implementation of environmental technologies, carbon trading markets, and environmental regulations and legislations will stimulate GTI which in turn will help mitigate emissions at the highest level of emissions. In addition, environmental technologies have inevitable effects on decay and consumption; Consequently, knowledge leakage and diffusion across regions and industries vary. Also, the costs of disclosing technology to competitors in some industries are much higher than the benefits brought by a patent application, which makes companies more inclined to keep the technology secret rather than apply for patents (Bai et al. 2020). According to Razzaq et al. (2021c), these effects will lead to different technological impacts on emissions reduction.

Our findings are somewhat reverberated by previous results, where the emissions-mitigating impact of GTI is confirmed (e.g., Razzaq et al. 2021c; Meng et al. 2022). Nevertheless, this nexus does not linearly hold, which is the basic proposal of the current paper. Such a result is consistent with previous findings of Razzaq et al. (2021c) who showed that the nexus between GTI and CCO₂ emissions does not linearly hold. As argued by Razzaq et al. (2021c) and Bai et al. (2020), technological innovations materialize through a decadence and consumption effect, indicating a lag effect between technological innovation and technology diffusion/adoption, and the marginal benefits of these technologies are diminished by continued technical development. However, our findings contradict the results of Meng et al. (2022) who reported that GTI linearly mitigates CCO₂ emissions.

Figure 2b displays the reverse effect of CCO₂ emissions on GTI in Saudi Arabia. Like the patterns displayed in Fig. 2a, those of Fig. 2b reflect a W-shape relationship between CCO₂ emissions and GTI, where the left-wing appears dominated by a weak positive impact of CCO₂ emissions on GTI, while the right wing is dominated by a strong negative effect of CCO₂ emissions on GTI. Figure 2b shows a strong negative impact of CCO₂ emissions on GTI across all emissions quantiles, while a weak positive effect is detected across lower and medium emissions quantiles (0.1–0.3 and 0.4–0.6). This positive relationship is found to only persist across medium quantiles of GTI. Overall, our findings suggest that higher CCO₂ emissions imply less GTI. Therefore, the higher emissions failed to motivate GTI in Saudi Arabia. On the contrary, at higher emissions levels, GTI significantly contributes to CCO₂ emissions reduction in Saudi Arabia.

Since the global warming environment is changing with heavy global carbon emissions, most countries have the incentive to invest in green innovation (e.g., Saunila et al. 2018; Razzaq et al. 2021c; among others). Saudi Arabia topped the 2021 list of the largest Arab countries that emit harmful emissions from the energy sector, as its total CO₂ emissions grew by ~ 1.4%. Despite topping the list of Arab countries that emit harmful emissions, the Kingdom’s emissions from combustion decreased in 2021 to 5.6 million tons, compared to 5.7 million tons in 2020. Saudi Arabia’s plan launched in October 2021 is aimed at reducing its emissions and reaching carbon neutrality by 2060. The Kingdom’s plan includes a package of investments amounting to 700 billion riyals (US $186.62 billion) to develop a green economy, as well as energy initiatives that seek to reduce carbon emissions by about 278 million tons annually by 2030.

Our results somewhat support those found by Razzaq et al. (2021c) for BRICS countries. The negative effect of CCO₂ emissions on GTI at lower quantiles of CCO₂ can be explained by the fact that at the lower level of emissions, there is insufficient incentive for the government to implement GTI as a remedial measure. More importantly, the government has fewer social incentives and pressures to implement its green reform agenda. Lack of clarity in economic vision also prompts governments to divert money towards physical infrastructure and public goods. Despite increasing social and economic responsibilities, the Kingdom of Saudi Arabia adopted a green growth strategy by launching the Saudi Green Initiative in 2021 to combat climate change, raise the quality of life, and protect the planet for future generations.

The negative impacts of CCO₂ emissions on GTI at both middle and upper quantiles of CCO₂ indicate that the government is still lagging behind its potential due to heavy reliance on traditional energy sources and less stringent environmental laws. Otherwise, the positive effect of CCO₂ emissions on GTI across lower and middle quantiles, even if it is weak, shows that rising emissions in Saudi Arabia motivate policymakers to introduce green technologies as a countermeasure. Therefore, the government must take appropriate steps to address emissions levels by providing the necessary funds to GTI, which are generally expensive and require a large amount of research and development as well as fixed capital investment.

For comparison and robustness analysis, we further examined the effect of GTI on TCO₂ emissions in Saudi Arabia. Figure 3a portrays the effect of GTI on TCO₂ emissions, suggesting that GTI strongly reduces TCO₂ emissions mainly at upper (0.60–0.90) and extremely upper quantiles of TCO₂ emissions, while a weak positive effect is captured at higher quantiles (0.80–0.90) of TCO₂ emissions. Overall, these findings are consistent with those examining the impact of GTI on CCO₂ emissions. Figure 3b indicates that higher TCO₂ emissions provoke less GTI across all quantiles of TCO₂ emissions and at lower quantiles of GTI. As compared to CCO₂ emissions, TCO₂ emissions get a more positive effect, indicating that higher territorial emissions cause higher GTI across medium to upper (0.40–0.80) quantiles of TCO₂ emissions and GTI.

Quantile causality analysis

Before applying the GCQ tests, we firstly reported in Table 9 the results of the conventional linear Granger causality tests in mean (Granger 1969; Granger 1980) based on the bivariate VAR models of Eqs. (11) and (12). The outcomes indicate that all VAR residuals are serially uncorrelated at the 5% significance level. We uncovered weak causality by utilizing the Granger causality tests in the mean. We found evidence that TCO₂ emissions lead GTI at the 10% significance level.

Table 9 Linear Granger-causality tests in mean

Second, we reported in Table 10 the results of the nonlinearity tests. Regarding the results of the BDS test (Broock et al. 1996), we rejected the null hypothesis that the VAR residuals are iid for all bivariate VAR models. Based on the outcomes of the VAR parameter stability tests (SupF, AveF, and ExpF), we rejected the null hypothesis of stable parameters for all bivariate VAR specifications. Therefore, the results of both BDS and parameter stability tests suggest applying the nonlinear Granger causality tests in mean based on the bivariate VAR models.

Table 10 Nonlinearity tests

In the next step, we reported in Table 9 the results of the nonlinear Granger causality tests in the mean of Hiemstra and Jones (1994) and Diks and Panchenko (2006). The results of both tests indicate the nonexistence of significant nonlinear Granger causality in the mean between CCO₂ emissions and GTI and between TCO₂ emissions and GTI.

The conventional Granger causality tests in mean are generally restricted since they consider only causality at the mean of the conditional distribution. Under this restriction, the findings presented in Table 9, 10, and 11 still provide little evidence to support causality between changes in carbon emissions and changes in GTI. To overcome these limitations, we applied the Granger causality in quantiles tests, which also helped in improving our analysis of the QQR method.

Table 11 Nonlinear Granger-causality tests in mean

Table 12 summarizes the results of the Granger causality test in an equally spaced grid of 19 quantiles ${\mathcal{T}}_{{\text{n}}}=\left[0.05;0.95\right]$. We employed the ${{\text{S}}}_{{\text{T}}}$ test statistic in Eq. (24), which tests the null hypothesis of Granger non-causality between carbon emissions and GTI in a given quantile $\tau$ for three different lag orders ${\text{L}}\in \left\{\mathrm{1,2},3\right\}$. To calculate ${S}_{T}$, we implemented a subsample size $b=\left[k{T}^{2/5}\right]\approx 35$, where [⋅] is the integer part of a number, $T=128$, and $k=5$ is a constant parameter.⁴ We considered three quantile intervals, namely [0.05, 0.20], [0.40, 0.60], and [0.80, 0.95], to characterize the lower quantiles, middle quantiles, and upper quantiles, respectively.

Table 12 Linear GCQ test results

The estimated subsampling p-values clearly show evidence of significant quantile causality. The probability values reported in the first row denote the causal relationship between both covariates at the mean, validating the overall causality conditional on mean values of underlying variables. Alternatively, GTI is found to significantly Granger-cause both CCO₂ and TCO₂ emissions in a large number of quantiles, mainly at both lower and upper quantiles. However, the reported probability values fail to reject the null hypothesis of no Granger causality at the median or mean quantile. These results are robust to different specifications of the QAR model under the null of Granger non-causality. Therefore, large positive or negative fluctuations in GTI led to extreme changes in carbon emissions in Saudi Arabia.

Table 12 also displays the subsampling p-values for Granger causality in quantiles running from carbon emissions to GTI. Both carbon emissions variables are found to significantly Granger cause GTI at almost all quantiles. These findings are also robust to different specifications of the QAR model. Accordingly, large positive or negative variations in carbon emissions Granger-cause extreme variations in GTI. Overall, our empirical findings indicate the existence of strong bidirectional linkages between carbon emissions and GTI in Saudi Arabia under linear specifications of the quantile regression model, which are found to be equally spread across innovative lower and upper quantiles. Except for the three medium quantiles $\tau =\left(0.45;0.50;0.55\right)$, we found a significant unidirectional relationship running from CCO₂ emissions (TCO₂ emissions) to GTI. For both extreme lower and extreme upper quantiles, the results indicate no evidence of linear causality between CCO₂ emissions (TCO₂ emissions) and GTI. The p-values reported in the last row denote the causal link between GTI and carbon emissions at the mean, validating the overall casualty conditional on the mean values of both covariates. Therefore, the results of the linear Granger causality-in-quantiles test support the conclusion on the interdependency between GTI changes and both CCO₂ and TCO₂ emissions in Saudi Arabia. Our results suggest that any policy intervention regarding green innovation significantly affects carbon emissions and is therefore of greater policy importance. Likewise, any political intervention to control environmental deprivation also leads to green technological innovation in the country. Our results are found to be consistent with those found by Razzaq et al. (2021c) who have shown evidence of a bidirectional Granger causality between GTI and CCO₂ emissions across all BRICS countries across all considered quantiles except the median quantile. However, our findings contradict those obtained by Razzaq et al. (2023) who found evidence of unidirectional causality from GTI to CCO₂ emissions in the USA across lower, medium, and higher quantiles.

We further applied the ${S}_{T}$ test statistic in Eq. (24) under both nonlinear SAV and SA specifications. To emphasize the benefits of employing these nonlinear CAViaR models, we compared quantile predictions for both CCO₂ and TCO₂ emissions using linear and nonlinear specifications of the QAR models. Interestingly, we provided additional incentive to use nonlinear SAV and SA models under ${H}_{0}^{X\nRightarrow Y}$ in Eq. (21) by comparing the root mean squared error (RMSE) of 30 recursive quarterly out-of-sample forecasts of the conditional distribution. This allows us to verify whether SAV and/or SA nonlinear quantile specifications improve the out-of-sample predictability of forecasts across different quantiles of the conditional distribution. For the predictability exercise, the first forecast is performed on July 1, 2014, and our in-sample period is increased at each quarterly forecast. To do so, we considered the grid of quantiles ${\mathcal{T}}_{n}=\left[0.05;0.95\right]$.

Table 13 presents the RMSE of the recursive quantile forecasts. For both CCO₂ and TCO₂ emissions covariates, the results show that nonlinear CAViaR models provide lower RMSE for $\tau \in [0.05;0.2]$ than linear QAR models. Therefore, nonlinear CAViaR models have a better specification for the (extreme) lower quantiles of carbon emissions series. At both medium and upper quantile forecasts, the outcomes show mixed results. Interestingly, for CCO₂ emissions, the CAViaR models provide a lower RMSE for the extreme upper-tail quantile $\tau =0.95$ than linear QAR models. For the remaining quantiles, the nonlinear CAViaR models generally provide lower RMSE than the linear QAR models. Overall, our results indicate that nonlinear CAViaR models provide a better specification for the (extreme) lower and (extreme) upper quantiles of both CCO₂ and TCO₂ emissions.

Table 13 RMSE of recursive out-of-sample $\tau$-quantile forecasts

Table 14 shows the p-values of the nonlinear GCQ tests. Contrary to the results of the linear GCQ test depicted in Table 13, we found significant weak causality from GTI to CCO₂ emissions (TCO₂ emissions) only at a few lower quantiles under the nonlinear specifications of the quantile regression model. Therefore, investment in GTI may contribute to environmental performance in Saudi Arabia. Otherwise, the outcomes indicate evidence of significant nonlinear causality from CCO₂ emissions (TCO₂ emissions) to GTI at the medium quantiles $(\tau =0.40;0.50;0.60)$ as well as at the extreme upper quantile $(\tau =0.95)$. These results are also not in line with those shown in Table 13 under the linear specifications of the quantile regression model. Overall, the nonlinear specifications of the quantile regression model disclose different patterns of causal linkages from GTI to CCO₂ emissions (TCO₂ emissions) and from CCO₂ emissions (TCO₂ emissions) to GTI. However, the findings outlined in Table 14 generally fail to support the Granger causality in quantiles between carbon emissions and GTI under nonlinear specifications. Despite all that, our results insist on the obligation to consider nonlinear specifications of the quantile regression model because the linear tests generally neglect some detailed patterns of causality.

Table 14 Nonlinear GCQ test results

Robustness checks

In this paper, we utilized the QQR technique to shed light on the asymmetric inter-linkages between GTI and CCO₂ emissions (TCO₂ emissions) in Saudi Arabia. To validate the results of the QQR approach, we applied the conventional QR approach. Figure 4 shows the validity evaluation graphs of both QQR and QR estimates. We found that the average values of the slope coefficients of the QQR approach are approximately similar to those of the standard QR technique. Such findings validate and prove the robustness of the previous QQR results.

Fig. 4

Comparison of QQR and QR estimates. Notes: The graphs show the estimates of the averaged QQR slope coefficients (solid blue line) and the QR slope parameters (solid brown line) at different quantiles. a Impact of GTI on CCO₂ emissions. b Impact of CCO₂ emissions on GTI. c Impact of GTI on TCO₂ emissions. d Impact of TCO₂ emissions on GTI

For the robustness analysis of the Troster (2018) approach, we further applied the symmetric version of the Sup-Wald quantile-causality test proposed by Koenker and Machado (1999). To do so, we estimated a linear quantile regression model for ${Y}_{t}$, that depends on ${X}_{t-1}$. Then, we tested whether ${H}_{0}: \beta \left(\tau \right)=0$ over a grid of quantiles ${\mathcal{T}}_{n}=\left[0.05;0.95\right]$ as follows:

$${Q}_{\tau }^{{Y}^{*}}\left({Y}_{t}\left|{\mathcal{F}}_{t-1}^{X}\right.\right)=\alpha \left(\tau \right)+\beta \left(\tau \right){X}_{t-1}$$

(30)

In Fig. 5a–d, we plotted against $\tau$ the QR estimates of $\beta \left(\tau \right)$ in dashed lines as well as their 95% confidence intervals in shaded areas. More precisely, the figures depict the plot of the Wald test statistic proposed by Koenker and Machado (1999). The solid red line refers to the critical value at the 5% significance level. Figure 5a–b shows the estimated quantile-regression coefficients of lagged GTI. The QR estimates of $\beta \left(\tau \right)$ vary with quantiles and exhibit a remarkable pattern. First, the QR estimates are positive at both lower and middle quantiles, while they become negative at upper quantiles. Second, the magnitude of QR estimates decreases as $\tau$ moves from 0.05 to 0.95. Third, these estimates are statistically significant at both lower and medium quantiles but insignificant at upper quantiles. Figure 5c and d displays the estimated quantile-regression coefficients of lagged CCO₂ emissions (TCO₂ emissions). The estimation results for both CCO₂ and TCO₂ emissions are completely different. The QR estimates of $\beta \left(\tau \right)$ are heterogeneous across $\tau$ and are insignificantly negative at lower quantiles but become positive and significant at both medium and upper quantiles. Moreover, the magnitude of these estimates increases as $\tau$ moves towards 0 and 1. Koenker and Machado’s (1999) test results corroborate and prove the robustness of the linear GCQ test results displayed in Table 12. Therefore, they represent a convincing argument in favor of using Troster’s (2018) linear quantile causality approach.

Fig. 5

Estimated quantile-regression coefficients, $\beta (\tau )$, of ${GTI}_{t-1}$, ${CCO}_{2,t-1}$, and ${TCO}_{2,t-1}$

To emphasize the usefulness of our results, we further conducted a second robustness analysis to check whether the observed GTI-CCO₂ emissions (TCO₂ emissions) causality improves the out-of-sample predictability to CCO₂ emissions (TCO₂ emissions). Note that the in-sample predictability of GTI does not necessarily imply improvements in out-of-sample predictions. We used the grid of quantiles ${\mathcal{T}}_{n}=\left[0.05;0.95\right]$ and performed 30 recursive quarterly out-of-sample predictions on all considered quantiles using a QAR(1) model––including an expanded model that incorporates lagged values of GTI. The first forecast date was performed on July 01, 2014, and our in-sample period was increased at each quarterly forecast. We compared the forecasting models using the modified test of Diebold and Mariano (1995) developed by Harvey et al. (1997) and Clark and West’s (2007) test of equal predictive accuracy. Under the null hypothesis, both tests assume the equality of the quantile forecast errors of both QAR(1) and expanded QAR(1) forecast models.

Tables 15, 16, and 17 summarize the out-of-sample predictability test results at lower, medium, and upper quantiles. The outcomes indicate that the observed nonlinear causality from $\Delta \left(GTI\right)$ to $\Delta \left({CCO}_{2}\right)$ helps to improve the predictability to $\Delta \left({CCO}_{2}\right)$ at (extreme) lower quantiles $\left(\tau \in \left[0.05, 0.20\right]\right)$. In addition, the observed $\Delta \left(GTI\right)$-$\Delta \left({TCO}_{2}\right)$ nonlinear causality at lower quantiles $\left(\tau =0.20\right)$ as well as at the higher quantiles $\left(\tau =0.80; 0.85; 0.90\right)$ improves the out-of-sample predictability to $\Delta \left({TCO}_{2}\right)$ at 5% significance level. These findings somewhat corroborate the nonlinear GCQ test results presented in Table 14, where the nonlinear causality from $\Delta \left(GTI\right)$ to $\Delta \left({CCO}_{2}\right)$ and from $\Delta \left(GTI\right)$ to $\Delta \left({TCO}_{2}\right)$ is statistically significant at $\tau =0.30$ and $\tau =(0.20;0.25)$, respectively. The reverse nonlinear causality from $\Delta \left({CCO}_{2}\right)$ to $\Delta \left(GTI\right)$ enhances the out-of-sample predictability to $\Delta \left(GTI\right)$ at extreme (lower) quantiles $\left(\tau =0.05; 0.10; 0.25\right)$. The observed $\Delta \left({TCO}_{2}\right)$-$\Delta \left(GTI\right)$ nonlinear causal relationship helps to improve the out-of-sample predictability to $\Delta \left(GTI\right)$ at lower quantiles $\left(\tau =0.05; 0.10\right)$ as well as at upper quantiles $\left(\tau =0.90\right)$.

Table 15 Out-of-sample forecast performance tests (lower quantiles)

Table 16 Out-of-sample forecast performance tests (middle quantiles)

Table 17 Out-of-sample forecast performance tests (upper quantiles)

Policy recommendations

To track and mitigate the overall environmental burden of domestic production and consumption, the government should consider using a benchmark based on consumption-based carbon emissions rather than relying on terrestrial-based carbon emissions that underestimate national emissions due to higher imports from other countries.

The results obtained through this study indicate that green innovations are necessary to transform the traditional production function, leading to environmental stewardship based on an ethical code that indicates the responsibility to use and protect the natural environment through conservation and sustainability practices. This can be done by simultaneously ensuring higher economic growth by replacing traditional products and processes with more efficient ones. Accordingly, the government should implement the green growth agenda by providing higher allocations in the research and development budget, which would translate into advanced environmental innovation, leading to resource conservation, efficient energy consumption, and consequent lower emissions.

It is important to know that green innovation is more responsive to emission control at higher emission levels, but at lower emission levels, legislatures do not have enough incentive and social burden to take sequestration measures. Therefore, the government and decision-makers must promote and adopt environmental technologies in companies, homes, and means of transportation, which would contribute to reducing emissions at the source. It is worth noting that companies usually face financial constraints to fully modernize their production system in developing industrial economies. Besides enhancing R&D allocations, the government should also help improve the adoption of these environmental technologies in companies by removing financial constraints. Therefore, the government must allocate soft financial credits to companies and industries to produce, import, and replace traditional equipment with new, efficient technologies. The government must also help reduce the household carbon footprint by using alternative solutions based on renewable energies.

Last but not least, the government and policymakers should consider imposing a carbon tax on imports. Taxing the carbon content of imports could help reduce the upward trend in global greenhouse gas emissions. However, such mechanisms need to be implemented in the right way by focusing on consumption-based emissions rather than just domestic production.

Conclusion

Although the impact of green technology innovation on consumption-based CO₂ emissions remains a debatable research topic, it has not been analyzed for an oil-exporting country like Saudi Arabia. In this paper, we investigated the interdependency and linear as well as nonlinear GCQ between green technology innovation and consumption-based CO₂ emissions in Saudi Arabia. To the best of our knowledge, this causal relationship has not been explored extensively in practice. We examined this causal effect in detail using recently advanced econometric methods including both linear and nonlinear frameworks. Using quantile regression-based methods, we found that green innovation fails to slow down CCO₂ emissions at lower emissions quantiles, while its strong mitigating impact is embodied at (extreme) higher emissions quantiles. Meanwhile, the linear GCQ test results advocate the interdependency between green innovation changes and CCO₂ emissions in Saudi Arabia.

The current study has one further major contribution. To the best of our knowledge, we introduced for the first time the nonlinear CAViaR models to evaluate nonlinear causalities and possible causal relations in all conditional quantiles between green innovation and CCO₂ emissions. We found evidence of a weak nonlinear causality from green technology innovation to CCO₂ emissions at only the lower quantiles of the distribution. However, we found weak reverse nonlinear causality from CCO₂ emissions to green innovation at medium and extreme upper quantiles.

The results of this study may provide useful implications for decision-makers in shaping their green innovation-CCO₂ emissions-related policies in Saudi Arabia. We believe that our research will inspire future studies on this association in other developed and developing countries, as there is a considerable gap in this area of research.

Lastly, we note that the current study has some limitations. The major one is that the investigation has focused on interdependency and causality to evaluate the impact of green innovation on consumption-based (trade-adjusted) carbon emissions. For future research, the study can use the quantile ARDL model to determine more detailed insights into the association. Additionally, future research should consider the importance of national-level recycling and green technology innovation in tackling environmental woes across different industries and sectors in Saudi Arabia. Furthermore, future research can be conducted for other blocs like ASEAN and MINT countries.