Introduction

The achievement of environmental sustainability in all aspects of life is a crucial fence for human civilization. In 2012, the UN made the Sustainable Development Goals (SDGs) agenda toward surpassing, prospering, and accomplishing the desired sustainable progress. The SDGs comprise 17 main objectives, 169 strategies, and 232 benchmarks coordinated on a national and global scale (Zafar et al. 2020). Environmental stewardship is critical to achieving these SDGs and has lately attracted scientific attention. Carbon emissions can boost environmental development efficiency in several ways. Controlling carbon emissions may improve health and contribute to the SDGs’ achievement. Furthermore, CO2 emissions are linked to important socio-economic sectors such as agriculture, industry, housing, and shipping, contributing to a country’s ED and residents’ revenue He et al. (2018). Furthermore, Jones and Klenow (2016) revealed that ED is directly associated with contentment in certain ways, such as humanity, scarcity, ingestion, and ease. But, reducing CO2 emissions frequently originates from short-term monetary gain. Furthermore, worries about pollutant emissions are severe environmental issues caused by human global warming and thus become the primary concern of governments and regulations in many countries. Emission factors are required for ED and EU Wang et al. (2020). The ultimate goal of the authorities is to create a balance between SDGs and emissions. Particularly, authorities linger to promote fast economic expansion at the least feasible cost of CO2 pollution. Extensive empirical studies have revealed that growing energy depletion and ED are the primary drivers of emissions (Akadiri et al. 2019). This research investigated a linkage between ED and pollutant emissions, proposing an inverted U-shaped link and bi-directional causation (Sunday et al. 2017). According to Kirikkaleli and Adebayo (2021a); Khan et al. (2022), green energy technologies and renewable energy use help to reduce CO2 emissions.

The World Intellectual Property Organization (WIPO) publishes the “IPC Green Inventory” to uncover ecologically sound innovations based on the IPC categorization. Alternative energy production, transportation, energy conservation, waste management, agriculture/forestry, administrative, regulatory or design issues, and nuclear power generation are examples of green technologies listed under the “IPC Green Inventory.” As shown in Fig. 1, a summary of these classes may be argued as influencing avenues via which green technology advances may affect CO2 emissions. For starters, green technology innovation may reduce CO2 emissions indirectly through improved energy efficiency. Using data from seventeen OECD nations, Abdulrashid (2016) discovered that green innovation reduces energy intensity in industrial sectors. Improving energy efficiency might lower the energy consumed per production unit while reducing carbon dioxide emissions (Acheampong 2018). Second, the energy use structure might act as a possible influencing route. Green innovation can lower the cost of clean energy while increasing the proportion of clean energy in the energy consumption structure (Ahmad and Raza 2020). Renewable energy reduces carbon emissions and is more ecologically friendly than fossil fuels. Third, green technological advances can help to improve industrial structure by encouraging the growth of low-pollution and environmentally friendly industries (Ahmad et al. 2018). Finally, end-of-pipe treatment, such as carbon capture, use, and storage, may considerably reduce carbon dioxide emissions (Ahmed et al. 2021).

Fig. 1
figure 1

The impact of green technology innovation on carbon dioxide emissions

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Despite being one of the world’s seven nuclear powers, Pakistan still confronts major issues, including an energy crisis and environmental degradation that need immediate policy-level measures. The current research incorporates environmental-related green technologies to check the source of contaminant contamination and estimate Pakistan’s economic and environmental sustainability. Environmental technology, also known as clean technology or green technology, is the use of one or more environmental monitoring, green chemistry, environmental science, and electronic procedures to monitor, model, and conserve natural resources and the environment, as well as to reduce the negative impact of human activities. Furthermore, environmentally friendly technology may properly manage garbage generated by human activities.

To execute it, a strong link is required between all local stakeholders who, each for the functions of their own competence, may activate improvement processes with shared aims in the creation of innovation processes and sustainability improvement Akadiri et al. (2019). While changing energy production and limited funds are becoming more important in environmental projects. PPP fetch collected many possibilities, cash, and funds to foster cooperation for green plans and a more ecological environment. Though no one agrees on the concept of PPP, this endeavor collaboration involves régimes, NGOs, and for-profit businesses to produce a more valuable product than feasible if they worked alone. More precisely, it entails long-term cooperation between PPP sectors to provide affordable goods and services (Akam et al. 2021). The private industry invests in the government sector, enabling collaboration standardization, hazard, and cost reduction, and the interchange of technologies and information. It encompasses many short- and long-term bilateral relationships, including organization, divestiture, funding, advancement, and administration. As per Zeeshan et al. (2020), the wide range of such governance systems in public-private partnerships incorporates several specialized forms, including private financing initiatives (PFI), build-operate-transfer (BOT), and reconstruct-operate-transfer (ROT). PPP is used in various projects in Pakistan, notably energy, urban governance, mass transit, and environmental protection. Pakistan is one of South Asia’s most important public-private partnership markets, with over USD 200 billion spent in the energy industry from 1992 to 2018 (Aliyu et al. 2017). With mounting evidence of climate change and a conversion away from FFE toward RE, long-term growth is crucial. Environmental sustainability studies can play an important role in reaching sustainable development goals in Pakistan.

Considering the facts stated above, the research contributes in many ways: First, the study emphasizes the relevance of green innovations, energy use, and public-private partnerships in energy investment to decrease carbon emissions while taking economic development and power pricing into account. Second, the study employs novel econometric approaches to provide firm evidence that may help policymakers to achieve SDG-7 such as “provide widespread access to cheap, dependable, and advanced energy services by 2030” and SDG-13 “with legislative will and technological measures, it is still conceivable to restrict the rise in global mean temperature to two degrees Celsius over pre-industrial levels and thereby prevent the worst impacts of climate change.” Third, for analysis, Bayer and Hanck’s test applies for co-integration also dynamic ARDL method inspects the long-and short-run connections among the variables, and FDC confirms the short-medium and long-term causality. To verify the robustness of the ARDL estimate, the authors employ DOLS, FMOLS, and CCR techniques. Fourth, considering the paucity of studies on this novel method in the environmental literature, this study will undoubtedly contribute to the literature in terms of econometrically. Finally, the research’s innovative results expand existing research on green innovation, energy use, and PPP in energy investment crucial for national policymakers.

The remainder of the research is ordered as tracks: “Literature review” section determined the reviewed literature. “Methodology and procedure” section labels the data collecting and methodological procedure. “Results and discussion” section contains a summary of the results and comments. “Conclusions and policy implications” section recapitulates the findings and makes policy recommendations.

Literature review

A literature review focuses on the explanatory factors related to the study’s response variable. Several econometric techniques for single or panel countries have been used in diverse studies to assess environmental factors in Pakistan. For the sake of brevity, we separated the literature evaluation into two sections: “Green innovations and PPP’s impact on the environment” and “Economic development, energy use, and environment.”

Green innovations, PPP, and the environment

The link between GI and carbon emissions has been a popular topic, with research primarily split into two classifications: the favorable effects of GI on combating environmental change and the beneficial impact of GI in publicizing the emergence of innovative energy. On the one hand, technical advancements that increase technology and R&D investment may successfully cut carbon emissions. GI and environmental legislation have lowered CO2 emissions and improved environmental integrity. Several studies have investigated the link between technical breakthroughs and carbon emanations; however, the empirical results have been inconsistent. For example, Ganda (2018) analyzed the environmental factors for OECD economies between 2000 and 2014. The statistical evidence reveals that investing in renewables (green energy) boosts environmental protection and boosts energy savings. Ganda (2019) inspected the rapport between technological innovation (TI) and carbon emissions in OECD nations using the GMM method from 2000 to 2014. Their outcome confirmed that TI has a considerable adverse impact on the environment and proposed that OECD nations enhance environmental quality via investment in TI. Similarly, Zeeshan et al. (2020) inspected the effect of public-private partnership investment in energy and technology on China’s consumption-based carbon emissions (CBE) from 1990 to 2017. The unit root test was based on generalized least squares (GLS); the Maki co-integration test, FMOLS, DOLS, CCR, and the FDC test were used. The study’s findings revealed that the cointegrating link between PPP, TI, REC, and CBE is proven and heads to higher CBE over time. Additionally, Fernández Fernández et al. (2018) applied an OLS technique to scrutinize the influence of R&D expenditure on carbon dioxide emissions in the US, the European Union, and China from 1994 to 2013. According to their results, R&D expenditures favor reducing emissions for the US and the European Union. Contrary, R&D spending in China is expanding carbon emissions. Their empirical research proposes that policymakers increase R&D spending, especially private and governmental expenditures, to promote innovation since innovations effectively alleviate environmental change. Koçak and Ulucak (2019) evaluated the relationship between R&D spending and the environment in 19 OECD nations from 2003 to 2015. Their research included three distinct methodologies: STIRPAT, OLS, and GMM. They discovered that R&D expenses had a large and favorable influence on CO2 emissions lessening. Their study suggested that controls in energy R&D expenditure should be strictly enforced to reduce energy intensity, FFE, and carbon emanations. Kirikkaleli et al. (2023) advocated for environmentally friendly technology that may cut carbon emissions in Denmark. As a result, we develop the following hypotheses:

  • H1: Green innovation may considerably help Pakistan reduce CO2 emissions.

The necessity of PPP investment for energy sector transformation is irrefutably vital, mainly for states with unsatisfied investment demands, like Pakistan. Conversely, it is grave to contain research that addresses the impact of PPP on environmental stewardship. There has been a trivial empirical study on the rapport amid PPP and CO2 emissions; most earlier studies related energy innovation funding to carbon emission functions. Shahbaz et al. (2020) explored the PPP investment in energy with carbon emissions for China during 1984–2018. Utilizing the bootstrap ARDL co-integration technique, research analysis shows that the link between PPP and carbon emission is favorable, resulting in a deterioration of environmental stewardship. Lorente and Álvarez-Herranz (2016) used a limited inverted V-lag distribution model to investigate the impact of energy innovation (EI), as assessed by public-energy budgets, in handling climate change in OECD nations during 1990–2014. They observed that EI enhances the ecological environment and recommend that politicians increase public spending, especially in the energy sector, to promote innovation and reduce CO2 emissions. Álvarez-Herránz et al. (2017) studied the impact of public expenditure on energy, R&D, and RE on carbon pollution in 17 OECD nations from 1990 to 2012. Their outcome revealed that public funding for R&D in the energy industry diminishes carbon emissions and enhances environmental protection.

Additionally, Waqih et al. (2019) examined the association between private investment and carbon emanations employing data from the SAARC group using FDI as a metric of PPP. They discovered that emissions follow PPP, and carbon emanations fall at a certain investment amount, indicating an inverted-U connection between the parameters. Hence, we form the following hypothesis:

  • H2: Public-private partnership investment in energy may reduce carbon emissions in Pakistan.

Economic development, energy use, and environment

Economic expansion contributes to environmental sustainability Destek and Sarkodie (2019). ED substantially impacts CO2 emissions Ren et al. (2021). Initially, Panayotou (1994) proposed the well-known EKC idea, motivating research on the link between GDP and carbon emanations. In this approach, EG is anticipated to have an inverted U-shape linking with pollutant emissions. Their fallouts showed that economic expansion throughout industrialization might impact the climate by raising emissions in any place. Though, when a country enters the (post-industrial) period, ED will assist it in minimizing environmental deterioration by lowering emissions. Destek et al. (2018) used second-generation panel data procedures to examine annual data from 1980 to 2013. The analysis shows a U-shaped bond amid real income and carbon impact, whereas FFE adds to environmental degradation in EU countries. Moreover, Erdoğan et al. (2019) investigated the relationship amid CO2 and ED in BRICS countries during 1992–2016. According to the findings, ED causes an increase in emissions. Abbasi (2021) investigated EG and environment using CS-ARDL and DH causality. Long- and short-run interactions were shown to be desirable. Radulescu et al. (2022) stated that OECD officials should consider banking and economic growth as a “green energy promoting mechanism” when creating policies to support environmentally friendly energy sources. Rehman et al. (2022) used the ARDL approach in combination with FMOLS and CCR. Long-run and short-run data indicate that fossil fuels have a positive relationship with economic growth, but renewable energy have an adversative relationship with economic growth. Many earlier researchers, notably Selden and Song (1994) and Holtz-Eakin and Selden (1995), attempted to inspect rapport amid CO2 emissions and EG to assess the accuracy of the EKC theory. After examining the harmful effects of pollutant emissions, most of the queries focused on GDP. The statistical findings for the EKC connection were contradictory; moreover, it has been demonstrated that EG increases pollution, for example, Tan et al. (2020); Bilal et al. (2021). Meanwhile, several researches have shown that EG is adversely connected to pollution intensity Acheampong (2018). Similarly, several scholars validated the EKC relationship, whereas others obtained conflicting conclusions (Balsalobre-Lorente et al. 2018; Abbasi et al. 2020; Supachart et al. 2020; Iqbal et al. 2021); Yu et al. 2022). Comprehensively, we found no single research that explored GI, PPP, EU, ED, and PP on CO2 emissions in Pakistan. This might be connected to sample selection (country/region), periods, variables, and analytical procedures. Furthermore, empirical data revealed different outcomes in the literature, as mentioned above. However, Khan et al. (2019); Mehar (2020) focused on Pakistan’s economic growth, pollutant emissions, and other issues. Contrary to these associations, some research demonstrated CO2 emission linked to EG strategies and waste to the energy transition. They discovered that R&D investments had a large and favorable influence on CO2 emanations reduction. They suggested that controls on energy R&D expenditure should be strictly enforced to reduce energy efficiency, conventional energy utilization, and CO2 emissions. Based on previous research, we formulate the following hypothesis:

  • H3: Economic development and energy use may increase carbon emissions in Pakistan.

  • H4: Pakistan’s energy usage may raise carbon emissions.

The literature specifically led us to understand that green innovation, public-private partnerships, energy use, economic development, and power prices are the main determinants of sustainable development; even so, the dearth of systematic studies on the connection of these robust, distinct factors for Pakistan, has inspired us to fill this space. Yet, it could be considered a rare study to use novel econometric approaches, including dynamic ARDL, FDC, DOLS, FMOLS, and CCR.

Methodology and procedure

Data sources and variable selection

The current empirical research focuses on green innovation, public-private partnerships, energy use, economic development, and the effect of power pricing on CO2 emissions. We utilize green patent counts as a measure of green technological innovations, according to Xu et al. (2021). The number of patent applications, rather than the number of patents granted, is used in this article. Because there is a time lag in patent authorization owing to monitoring, payment of yearly fees, and market environment, the number of patent applications accurately reflects the year’s green innovation activities, according to Lin and Long (2021). Hence, green innovations may have a lower influence on environmental performance by introducing energy breakthroughs and energy-efficient equipment \({\alpha }_{i}^{1}=\frac{{\partial CE}_{it}}{{\partial GI}_{it}}<0.\) According to Buso and Stenger (2018), public-private partnerships are not only advantageous to domestic production via investment, but they are also critical to the country’s ecosystem. They went on to argue that transition via decentralization of energy production through PPP investment in the energy sector might have a negative effect on environmental sustainability by impacting emission \({\alpha }_{i}^{2}=\frac{{\partial CE}_{it}}{{\partial PPP}_{it}}<0\). Also Zeeshan et al. (2020), economic development is the fundamental cause of high pollutant emissions since economic development is based on high energy use, which steadily degrades the environmental quality \({\alpha }_{i}^{\mathrm{3,4}}=\frac{{\partial CE}_{it}}{{\partial EU, ED}_{it}}>0\). Conversely, for oil-importing countries, rising power prices are expected to result in lower CO2 emissions Shan et al. (2021). This is due to the fact that a rise in power prices reduces energy use \({\alpha }_{i}^{5}=\frac{{\partial CE}_{it}}{{\partial PP}_{it}}<0\). The data is choosen based on availablity and the significance of research objective and converted into a logarithmic form to remove heteroscedasticity. The data sources and their unit measures are listed in Table 1.

Table 1 Data and quantification
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Model specification

The use of ARDL time series approaches is especially widespread in the social sciences. For instance, Rafindadi and Ozturk (2017) assessed how the green EC affected the expansion of the German economy. A co-integration strategy is necessary for the ARDL model. The co-integration testing approach, in contrast to the usual “two-step” procedure, is more confined to limited data and used in social sciences. Conventional statistical procedures do not employ this co-integration metric Engle et al. (1987). Derivatives of the control variable(s), specifically short and long-run divergences, are problematic to evaluate since the ARDL model has unique standard lags and initial inconsistencies. Except for error correction, Pesaran et al. (2001) designed a flexible command that allows running ARDL models automatically. Also, the dynamic ARDLS approach is efficient for inevitably approximating, accelerating, and estimating the graphic representation of the real shift in control and its impact on the controlled variable, whereas the extra parameters remain stable. Given the novel advantages currently, the dynamic ARDLS model gained high attention from researchers, econometricians, and policymakers for more concrete and reliable empirical evidence as Ulucak (2020) explored the link of biomass energy with CO2 emissions in China Abbasi et al. (2021a, b, c) investigated natural resources depletion for the UK, Abbasi et al. (2021a) inspected energy depletion for Thailand, and Khan et al. (2019) examined macroeconomic factors for Pakistan. The present study employs a novel dynamic ARDLS approach to analyze the relevant indicators in the controlled variable owing to a control variable given the empirical significance of the model. Figure 2 illustrates the processes of the methodological approach.

Fig. 2
figure 2

Methodological framework

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Unit root tests

Beforehand performing dynamic ARDLS, we ascertain the stationarity amid the factors. Because of its adaptability, the approach may be applied independent of variable action, that is, I(0), I(1), or a combination of both, and it is a highly efficient approach. Furthermore, Abbasi et al. (2021c) observed that if any factor is non-stationary, this might lead to misleading regression. To ensure that none of the parameters is I(2), Initially, we use ADF, PP, and KPSS unit roots tests. Next, employ the modernized unit root test with two structural breaks recommended by Paresh Kumar Narayan (2013). The Monte Carlo models revealed that this unit root test has stable and accurately discerns the breaches’ real periods. As a result, we decided to use this unit root test to analyze the order of each series. They suggested two approaches for determining the order of integration. Model (1) allows just structural breaks in the intercept, but Model (2) permits structural breaks in both the trend feature’s intercept and slope. To investigate the association between the factors, the following model expression is postulated:

$$\begin{array}{c}\mathrm{Model}-1:w_t=K+\alpha_1w_{t-1}+\alpha_2t+\varphi_1{D\left(T_B\right)}_{1,t}+\varphi_2{D\left(T_B\right)}_{2,t}+\\\varnothing_1{DU}_{1,t-1}+\varnothing_2{DU}_{2,t-1}+\sum_{i=1}^P\rho_i{\Delta w}_{t-i}+e_{1t}\end{array}$$
(1)
$$\begin{array}{c}M\mathrm{odel}-2:w_t=K+\alpha_1w_{t-1}+\alpha_2t+\varphi_1{D\left(T_B\right)}_{1,t}+\varphi_2{D\left(T_B\right)}_{2,t}+\\\varphi_1{DU}_{1,t-1}+\varphi_2{DU}_{2,t-1}+\vartheta_1{DT}_{1,t-1}+\vartheta_2{DT}_{2,t-1}+\sum_{i=1}^P\rho_i{\Delta w}_{t-i}+e_{2t}\end{array}$$
(2)

where wt denotes the factors under consideration, ln (n + g + δ) t), Δ is the 1st difference machinist, (\({w}_{t}-{w}_{t-1}\)) and (\({e}_{1t,}{e}_{2t}\)) are the shocks term implied by being normal distributions and white noise. To avoid serial correlation, the first difference lagged response variable \({DU}_{i,t}=1(\mathrm{t}>{T}_{B,i})\) and \({DT}_{i,t}=1(\mathrm{t}>{T}_{B,i}) (t-{T}_{B,i})\), is employed in the model if present. The general to specific t-sig approach determines the best lag order (p). This strategy has a greater efficacy trend than information-based techniques, including the SBC and AIC. \({T}_{B1}\) and \({T}_{B2}\). The probable break dates \({T}_{B1}\) and \({T}_{B2}\) are selected using the network test or consecutive approach. Lastly, the t-statistic of \({w}_{t-1}\) is exercised to test the H0 versus H1 assumption.

ARDL bounds

The bounds test was evaluated to ascertain the long-term rapport between the elements. The following technique is created to assess ARDL bounds:

$$\begin{array}{c}{\Delta CE}_t=\varphi_0+\varphi_1{CE}_{t-i}+\varphi_2{GI}_{t-i}+\varphi_3{PPP}_{t-i}+\varphi_4{EU}_{t-i}+\varphi_5{ED}_{t-i}+\varphi_6{PP}_{t-i}+\\\sum_{i=1}^q\beta_1{\Delta EQ}_{t-i}+\sum_{i=1}^q\beta_2{\Delta GI}_{t-i}+\sum_{i=1}^q\beta_3{\Delta PPP}_{t-i}+\sum_{i=1}^q\beta_4{\Delta EU}_{t-i}+\sum_{i=1}^q\beta_5{\Delta ED}_{t-i}+\\\sum_{i=1}^q\beta_5{\Delta PP}_{t-i}+\varepsilon_t\end{array}$$
(3)

where CE, GI, PPP, EU, ED, and PP represent carbon dioxide emissions, green innovation, public-private partnership in energy, energy use, economic development, and power prices, respectively. Furthermore, t − i reveals the optimum lags selected by AIC, and ß examines the long-run connection. According to K. R. Abbasi et al., (2021c), if there is a long-term link amid the parameters, we can derive the ARDL for both the short and long terms.

The research also employs Bayer and Hanck (2013) as a robust co-integration check, a mixture of the co-integration tests developed by Banerjee et al. (1998); Engle et al. (1987); Johansen (1991). The Bayer and Hanck (2013) co-integration test, as per [18], is based on merging several test approaches to avoid the common difficulties of previous co-integration tests. Bayer and Hanck (2013) used Fisher’s formula to develop the co-integration test to enhance it. Alola et al. (2019) showed Fisher’s equation as follows:

$$EG-JOH=-2[\mathrm{ln}\left(PEG\right)+\mathrm{ln}\left(PJOH\right)]$$
(4)
$$EG-JPH-BO-BDM=-2[\mathrm{ln}\left(PEG\right)+\mathrm{ln}\left(PJOH\right)+\mathrm{ln}\left(PBO\right)+\mathrm{ln}\left(PBDM\right)]$$
(5)

where PEG, PJOH, PBO, and PBDM denote the level of significance for the co-integration tests developed by Engle and Granger (1987), Johansen (1991), Boswijk (1994), and Banerjee et al. (1998), respectively.

Dynamic ARDLS method

Due to the ARDL model’s inability to anticipate short- and long-term, Jordan and Philips (2018) developed dynamic ARDLS that forecasted and instantly exhibited the predicted change on the controlled variables while keeping other control factors constant. To use the ARDLS model, the sample data must be integrated at I(1) or I(0); this study meets requirements and performs 5000 simulations for the parameter vector. The plots are frequently used to evaluate the true shock on the regressor and its influence on the outcome variable. The innovative ARDL bounds approach, as per Jordan and Philips (2018), specified:

$$\begin{array}{c}{\Delta CE}_{y}={\alpha }_{0}+{\theta }_{0}{CE}_{t-1}+{\beta }_{1}{\Delta GI}_{t}+{\theta }_{1}{GI}_{t-1}+{\beta }_{2}{\Delta PPP}_{t}+{\theta }_{2}{PPP}_{t-1}+\\ {\beta }_{3}{\Delta EU}_{t}+{\theta }_{3}{EU}_{t-1}+{\beta }_{4}{\Delta ED}_{t}+{\theta }_{4}{ED}_{t-1}+{\beta }_{5}{\Delta PP}_{t}+{\theta }_{5}{PP}_{t-1}+{\varepsilon }_{t}\end{array}$$
(6)

Spectral Granger causality

The research also seeks to ascertain the relationship between green innovation, public-private partnerships in energy, energy use, economic development, and power prices on CO2 emissions in Pakistan. As a result, the Breitung and Candelon (2006) causality tests were applied in this investigation. As opposed to the usual Granger causality, the estimating procedure allows the controlled factor to be computed at certain intervals; this theoretical context was created by Geweke 1982; Hosoya (1991). The major difference amid the period perception and the frequency domain archetypal denotes a specific change within a time series. The spectral FDC computes the extent of a given change in a dataset (Zeeshan et al., 2020).

The FDC can discover nonlinearities, causal disparities, and seasonality fluctuations in the short sample. It also indicates that earlier alterations may be recognized, and procedural action is taken. The mechanism, though, can only reproduce for a certain duration. Breitung and Candelon (2006) differentiate long-, medium-, and short-term causation. This study’s FDC approach fits the standards provided by Hosoya (1991). A horizontal vector \(\left[{x}_{t},{y}_{t}\right]\) with \({d}_{t}\) at time-period \(t=1,\dots ..T\) where \({d}_{t}\) compute a pre-defined vector autoregression role \(\theta \left(L\right){d}_{t}={\varepsilon }_{t},\) where \(\theta \left(L\right)=1-{\theta }_{1}L-\dots - {\theta }_{p}{L}^{p}({L}^{k}{d}_{t-k,}k\) is a lag polynomial), and \({\varepsilon }_{t}\) signifies the error term. The VAR construction of \({x}_{t}\) is specified as:

$${x}_{t}={\alpha }_{1}{x}_{t-1}+\dots +{\alpha }_{p}{x}_{t-p}+{\beta }_{1}{y}_{t-1}\dots +{\beta }_{p}{y}_{t-p}+{\varepsilon }_{1t}$$
(7)

There are two projected renderings, t (time), p (lag), and \({\varepsilon }_{t}\) (error term). Our study also uses OLS CUSUM to assess model robustness. The LM test verifies serial correlation, while the BG test determines heteroscedasticity and the Jarque-Bera test also detects residual normalization. Lastly, the Ramsey RESET test checks the model stability.

Results and discussion

Table 2 shows the results of the descriptive statistics; the peak reveals Kurtosis, and the Jarque-Bera test value confirms that CE, GI, EU, ED, and PP are normally distributed. The mean values of PPP and GI are (6.70 and 21.32, respectively), higher than the other variables’ mean values. On the other hand, the aggregate findings suggest that CE, EU, ED, and PP show a positive trend.

Table 2 Descriptive measurements
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The outcomes of four distinct unit root assessments are presented in Table 3. Before using the unique dynamic ARDLS model, it is critical to ensure that the study’s parameters are stationary and integrated to the highest order of I(0) or I(1); otherwise, the results will be erroneous. Three unit root tests were performed to determine the integration through ADF-PP and KPSS. All parameters in this research are stationary and integrated to the highest order of I(0) and I(1), indicating the viability of a unique DARDLS approach.

Table 3 Unit root testing
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Standard unit root tests containing ADF, PP, and KPSS do not account for Structural Breaks (SBs) and may provide insufficient results K. R. Abbasi et al. (2021c). Considering these reservations, we employ Narayan and Popp’s (2010) unit root check of two endogenous breakdowns. The advantage of this strategy is that SBs do not affect the results. Table 3 shows that all parameters are stationary at the 5% level, implying that all factors are discovered as the I(0) procedure. As a consequence, the findings are suitable for immediately applying co-integration estimates.

The dynamic ARDLS approach allows different lags for research parameters. We utilized SIC for lag determination; given to SIC, lag one is the finest match for the theory, followed by the lowest. After completing the requirements, the next phase has been to investigate the relationships presented by the current background. The short-term disruptions indicate an imbalanced series; this imbalance is rectified when the series recovers to a viable long-term connection. However, because not all linkages between I(1) time series confirm the presence of co-integration, it is necessary to validate a co-integration. The table demonstrates the consequences of the ARDL bounds validating co-integration, comprising permissible critical values and calculated p-values depending on apparent conclusion (Kripfganz and Schneider (2019). The ARDL bounds verifying co-integration investigation demonstrate (f) and (t) values of factors, whereas projected model are greater than the upper critical bound, backed by significant empirical p-values. Consequently, H0 is rejected and confirmed that all series in the model are cointegrated (Table 4).

Table 4 ARDL bounds and Bayer-Hanck’s test
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After completing the above tests, the research uses the Bayer and Hanck combined co-integration technique, which is used to investigate the robustness of long-term connections. Table 5 depicts the combined co-integration tests, including the EG-JOH and EG-JOH-BO-BDM. The study revealed that Fisher’s statistics for the EG-JOH and EG-JOH-BO-BDM tests exceeded the critical values at the 5% significant level. The null hypothesis of no co-integration was rejected. In Pakistan, we may infer a long-run link between variables from 1980 to 2019.

Table 5 Dynamic ARDLS
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Table 5 delineates the ARDL models’ results. The impact of GI in the short and long term is clearly described as negative, and significant green innovation plays a vital role in reducing carbon emissions, which asserts that technological progress in Pakistan is in its developing stage and is attempting to improve the quality of the environment. A 1% increase in green innovation reduces carbon emissions by 0.03 and 0.01 percent, respectively. It may look low due to the limited investment in renewable energies from 1980 to 2019. There is more space in the industrial sector where inefficient energy-saving measures are implemented, as well as transitioning to natural gas in automobiles and enforcing energy efficiency requirements for home appliances. The outcomes are consistent with Bandyopadhyay et al. (2022); Bandyopadhyay and Rej (2021); Shahbaz et al. (2020), demonstrating that technological advancements enhance environmental quality by lessening carbon emissions. Paradoxically, PPP energy investment is positively related to environmental quality. Infers that a 1% boost in PPP investment in energy generates a 0.1% boost in carbon emissions in the short run. In line with Ahmad and Raza (2020), the outcome stated that energy-based PPP benefits Brazil’s environmental degradation.

Furthermore, the research revealed that energy usage is positively linked to CE, and this relationship is statistically substantial at the 1% level. Consequently, a 1% escalation in energy use in Pakistan results in a 0.61 and 0.90 percent upsurge in CE in the short and long term, correspondingly. It is demonstrated that energy usage increases CE at high speed from the short to long run because the energy usage coefficient becomes thinner with time and is unitary in the long run. In this case, one may consider the direct alternative of reducing energy consumption to minimize CO2 emissions; however, this energy drop may harm Pakistan’s economic development. The solution is to bring cleaner energy sources such as renewables, biofuels, and nuclear energy Das et al. (2022). The outcome supports Abdulrashid (2016) for Nigeria, Ahmad et al. (2018) for China, and Khan et al. (2019) for Pakistan.

The empirical results specify that the ED has a favorable effect on CE. This result reveals that a 1% increase in ED has a 0.41 percent influence on CE in the short run. The factor demonstrates the underlying cause of the CE in Pakistan. Our results are similar to the previous research by Hussain et al. (2020), which found that economic expansion boosts CO2 emissions. Furthermore, Festus Fatai Adedoyin (2020) discovered that the GDP considerably improved the UK’s environment in the short term. Additionally, Jamel and Maktouf (2017); Jardón et al. (2017) claimed that EG impacts the environment in Latin American nations. Also, power prices are more unlikely to reduce CO2 emissions, as indicated by a strong negative coefficient of power prices. A 1% rise in PP results in a 0.14 and 0.19 percent decrease in CE. The outcomes are steady with Ike et al. (2020) indicating that energy prices are more useful in lowering emissions.

After all, ECT estimates the speediness variation and ensures an adverse and substantial coefficient. The ECT suggests 92 percent of the imbalance is addressed in the long run. Abbasi et al. (2020); (Xu et al. (2023); Yi et al. (2023); Zhang et al. (2023) confirmed the results, stating that both value and scale are negative, showing that ECT is substantial. According to the R2 statistic, control factors account for 67% of the variability in the dependent variables. The anticipated p-value validated the provided model’s appropriateness.

The DARDL simulations immediately exhibit forecasts for genuine regressor move and its influence on the controlled factor through fixing the control factors constant. Change in control factors would be predicted by 10% positive and adverse change in GI, PPP, EU, ED, PP, and its impact on CE for the short and long term. Figure 3 exhibits the impulse response graph, illustrating the association between GI, PPP, and the EU regarding carbon dioxide emissions. A ten-percent increase or decrease in GI and PPP signifies a negative short and long-run effect on CE.

Fig. 3
figure 3

The impulse-response plot (GI, PPP, and EU). The dots signify the average forecast value, while the dark color line symbolizes (75, 90, and 95%)

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The impulse-response plot is shown in Fig. 4 to determine the link between ED and PP with CE. The EU graph shows that a 10% upsurge has a long useful effect while decreasing environmental pollution in the short run, but a 10% reduction has a detrimental long-term and a positive short-term impact on emissions. The plot shows that a 10% positive or negative shift in PP reduces long- and short-term emissions.

Fig. 4
figure 4

The impulse-response plot (ED and PP). The dots signify the average forecast value, while the dark color line symbolizes (75, 90, and 95%)

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After several co-integration tests, long-run coefficients were determined. In contrast to the dynamic ARDL test, we employed the DOLS, FMOLS, and CCR estimation methods to evaluate the robustness. All estimate parameters produced almost comparable results. As a result, the study’s outcomes could be considered robust (Table 6).

Table 6 DOLS, FMOLS, and CCR estimation
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Diagnostic inspection

Numerous diagnostic approaches were used to verify the associated model, as indicated in Table 7, and found no serial correlation or heteroscedasticity in the model. The Jarque-Bera/Ramsey RESET estimations likewise ruled out the null hypothesis, suggesting that the estimated distribution is normal, and the model is accurately assessed. By and large, the model accurately predicts policy motives.

Table 7 Diagnostic examination
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Stabilization assessments were carried out using the (OLS_CUSUM) draw to maintain the long and short-term criteria. The CUSUM measure graph in Fig. 5 demonstrates that the values are within the critical bounds at a 5% significance level, indicating that the parameters remain consistent.

Fig. 5
figure 5

OLS CUSUM plot

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Spectral Granger causality

Additionally, this research uses the FDC test (Breitung and Candelon 2006) to determine the crucial link between GI, PPP, EU, ED, PP, and carbon emissions for the short-medium and long term. Figure 6 exemplifies that blue, green, and pink colors reveal the long-short and medium-run outcomes. The conclusive evidence indicates that GI, PPP, EU, ED, and PP Granger are responsible for changes in long-term frequency distributions. Furthermore, the GI, EU, and PP exhibit the only medium-run CO2 emissions trend. Finally, the EU validated the short-term environmental impact. In other words, the conclusion implies that the GI, PPP, EU, ED, and PP are critical indicators of Pakistan’s long-, medium- and short-run sustainable development. These findings are corroborated by Mendonça et al., (2020), who found that energy usage and GDP benefit the environment, while Cheng et al. (2021); Kirikkaleli and Adebayo (2021a, b) indicated that investment in energy through public-private partnerships degrades environmental quality by rising carbon emissions. Also, Rej et al. (2022); Emran Hossain et al. (2023) confirmed that renewable energy and technological innovation help to reduce CO2 emissions while showing a bidirectional link. On the contrary, Shahbaz et al. (2020) reported that GI negatively influences carbon dioxide emissions. The outcomes infer the reinforcement by asserting that public-private partnerships are critical for domestic production and Pakistan’s climate change mitigation efforts. The energy production change via green innovation and PPP investment in the energy sector may improve environmental quality by lowering carbon emissions.

Fig. 6
figure 6

Source: authors’ estimation. Note: blue, green, and pink colors uncover the long-short and medium-run outcomes

Frequency domain causality pictorial outcome.

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Conclusions and policy implications

Pakistan has been designated one of the world’s top ten nations most impacted by climate change. Whereas the government drives and executes the country’s environmental sustainability initiative, all stakeholders, along with the public and commercial sectors and people, are responsible for protecting the environment and its resources. Given the present situation of the environment, this research contributes to the effort to halt climate change. To explore the factors, the current study identifies the relationships between green innovation (GI), public-private partnership in energy (PPP), energy use (EU), economic development (ED), and power prices (PP) on carbon emissions in Pakistan from 1980 to 2019. The investigation used a dynamic ARDLS model to identify the short- and long-term connections and anticipate positive and negative changes in the dependent variables. In addition, the frequency domain-causality (FDC) approach was used to determine short-, medium-, and long-term causality among the research variables. We confirmed the stationarity of each series before utilizing the dynamic ARDL simulations to guarantee that not any series is I(2); else, outcomes would be incorrect. To accomplish this, ADF, PP, KPSS, and NP (2010) unit root tests were used. As a result, all tests verified that the sample data are integrated into order one, i.e., I(1), and no series discovered I(2). The bounds test, Bayer-Hanck, confirms the presence of a co-integration connection between carbon emissions and its factors and CCR co-integration approaches. The dynamic ARDL simulation model results show that the TI coefficient is negative, implying a reduction in carbon pollution for the short and long-term, while PPP and ED increased CO2 emissions in the short term. But energy use boosted environmental pollution in both the short and long term. Besides, PP has detrimental short- and long-term effects on the environment. In addition, the research used DOLS and FMOLS, which verified the robustness of the findings. Furthermore, the FDC model’s results confirmed the causality in the short, medium, and long-term estimates; based on this arduous fact, our study highlighted the importance of developing and implementing serious policies through which the direct and positive contribution of factors are convinced that green innovation, public-private partnership, energy use, economic development, and power prices are critical drivers in formulating Pakistan’s environmental quality.

Possible policy inferences

The empirical investigation has significant policy inferences for environmental stewardship in many major aspects of sustainable development, as specified.

Because of the positive effects of green innovation on the environment, it is recommended that by developing a long-term green R&D strategy, the government should support green technological innovation projects throughout a period of economic prosperity. Also, Pakistan actively encourages research and development of low-carbon techniques, which are critical for Pakistan’s decarbonization economy; evolves technologies for coal power production and application, as well as carbon dioxide, encapsulate, and storage; establishes a green economy, creates a circular mechanism for all sectors, and vigorously encourages recycling.

Pakistan is in fiscal distress due to its huge foreign debt, limited tax collection, and expanding fiscal imbalance Mehar (2020). In this context, recommending public sector investment in environmental development is challenging. In 2017, parliament approved the “Public Corporate Partnership Act” to stimulate private investment in environmental development, including constructing a fast transit system. This legislation offers means for private participation in environmental advancement; nonetheless, it is important to remember that the feasibility of a PPP model for a sustainable environment in Pakistan is closely tied to contractual management capabilities.

Yet, fossil fuels, industrialization, and economic progress drive Pakistan’s energy consumption which causes CO2 emissions. To reduce carbon emissions, energy protection, and living standards, the government must promote carbon-efficient technologies and make sure users to utilize while making it simpler to use energy effectively. Given economic development’s positive effects, it is advocated that national and local authorities collaborate to create and execute effective methods for elevating the energy usage structure and advancing the transition of the economic development paradigm. Policy actions aimed at achieving a low-carbon economy would incentivize green investments in clean technologies, averting increasing carbonization of the nation’s economic infrastructure. Once the actions are implemented, the economic system may be decarbonized progressively. Pakistan should prioritize economic, social, and environmental changes at the same time to achieve long-term growth.

Energy prices and green technologies breakthroughs may be used to set a goal for moderate emissions. Fossil fuels are currently more expensive in Pakistan than renewable energy. As a consequence, Pakistan may levy a carbon tax to discourage the use of fossil fuels and use the profits to fund renewable energy projects such as wind farms, solar farms, and biomass plants. Carbon pricing and diverting capital to infrastructural, manufacturing systems, and technologies that enable the delivery of goods and services at a significantly lower environmental price (reduced material and energy efficiency) are technically feasible and commercially viable solutions. Additionally, environmentally friendly technical innovation should be promoted via exchanges and collaborations with industrialized nations since technological advances in certain countries are projected to spur high-quality growth.

To reach this aim, enough financing for environmental projects should be provided, as well as a medium- and long-term assessment of the official and unceremonious financial consequences of developing local and global accords. Incorporating a variety of economic measures into the legislative framework could increase the expenditure of Pakistan’s climate change policy. There are several chances to enhance policy mechanisms, such as inducement taxes for water contamination and extraction, deposit-refund schemes, and taxes on manufacturing effluents, mining operations, and pesticide usage. A unified emission reduction plan should include more than just pollutant discharges from the major emission spectrum.

Study limitations and future direction

However, the study was restricted in time and scope to investigate Pakistani environmental elements. This research also proposes other areas for academics and politicians to explore new perspectives and dynamics around the world regarding macroeconomic stability, energy usage, industrial significance, environmental transactions, mass migration, and sustainable development. Future research might build on this model to examine factors that affect the quality of the environment in emerging and developed nations. Pakistan might be expanded by including a few additional explanatory factors relevant to this economy, such as population growth, globalization, and financial development, to assess their influence on carbon emissions.