1 Introduction

During the past 50 years, macroeconomic policy has focused on increasing economic growth, including tax reduction, deregulation, free systems market, and spending on health, education, and infrastructure. However, following Smith and Bentham's ideas, the economic welfare concept has indicated that one should consider not only GDP but also how well-off people are in terms of social welfare. Economic welfare can be appreciated in financial terms on the one hand and includes vital aspects that influence people's welfare, such as income disparity, environmental and employment conditions, and spending patterns (Goossens et al. 2007; Gertner 2010; Bleys 2012; Kalimeris et al. 2020). While GDP has long been used as a determinant of economic welfare, it does not forget the long-term effects, like a lower life expectancy or environmental damage (Van den Bergh and Kallis 2012). As a result, opportunity signs for GDP measures were developed, along with the consumption deflator proposed through Sefton and Weale, which considers present and future consumption possibilities. The evolution of welfare economics started with the utility principle, which aimed to maximise social delight among consumers and manufacturers. Ultimately, economic welfare is the foundation for sustainability and underscores the importance of considering broader social welfare outcomes beyond just economic growth.

Nordhaus and Tobin (1973) introduced the concept of economic well-being, which considers the value of leisure time, unpaid labour, and productivity in the shadow economy to measure economic welfare. To comprehensively measure economic welfare, Stiglitz et al. (2009) recommended the inclusion of people's well-being along with GDP, and Pigou (2017) suggested incorporating components such as education, income, and health. India's population is anticipated to rise to 1.6 billion people by 2040, which will necessitate a significant increase in energy production to keep up with the demand (India Energy Outlook 2021). Currently, coal, oil, and gas contribute over 80% of India's energy needs, and the energy sector contributes around 68.7% of greenhouse gas emissions in the country (India Energy Outlook 2021). Switching to renewable energy sources can improve economic security, encourage development, expand fuel diversity, and reduce greenhouse gas emissions. Additionally, renewable energy requires more labour, offers an alternative to finite fossil fuel reserves, is emission-free, and reduces harm to public health, positively impacting economic welfare. Non-renewable energy production causes land degradation, water and air pollution, unrestricted water loss, biodiversity loss, workplace injuries, global climate change, and related health effects, leading to significant economic costs. Access to reliable and clean energy sources is need for resilient, inclusive, and long-term economic development, and SDGs-7 emphasises the use of accessible, dependable, and clean energy for electricity generation. According to IRENA (2016), two times the proportion of renewable energy by 2030 can lead to a 1.1% increase in world GDP, a 3.7% increase in global well-being, and creating over 24 million new jobs.

A robust financial system is vital to a nation's economic growth and can encourage people to consume renewable goods, improving environmental quality. However, financial development and natural resource exploration can also cause ecological and economic disturbances (Al-Awad and Harb 2005). While financial development has been shown to stimulate economic development and the optimal utilisation of natural resources, it can also lead to increased energy consumption, environmental degradation, and lending at low interest rates (Guo et al. 2019; Hao et al. 2016; Fang et al. 2020). Therefore, it is essential to efficiently use available resources for financial development to benefit the nation's welfare. A stable financial system can also improve the beneficial impacts of resource exploitation on economic development and financial quality by reducing stakeholder volatility and increasing political legitimacy (Hammad Naeem et al. 2023). Additionally, financial development contributes to human well-being by providing people with income to support themselves and fostering technological innovation and human capital development (Nathaniel 2021). Balancing economic growth with environmental protection and efficient resource management is crucial to ensure the benefits of financial development.

Inflation can negatively impact a society's economic, social, and psychological aspects, hindering overall economic development (Naz et al. 2012). It can disrupt the role of finance in promoting growth and stability, preventing people from saving, investing, and engaging in productive activities. High inflation can lead to a collapse of the economy and reduced faith in the currency, causing higher wage demands, promoting child labour, and reducing literacy rates. Additionally, inflation can negatively affect the balance of payments, discouraging exports and hindering economic growth (Yelwa et al. 2015). It can also impact nutrition, health, and education, creating an unsustainable situation. Life expectancy is a crucial indicator of human development, reflecting the effects of public health, happiness, education, income, and environment, regardless of gender. Studies have shown an adverse correlation between a country's inflation rate and its life expectancy, where life expectancy decreases by about 20% for every increase in the inflation rate (Garcia et al. 2019). The average lifespan of a country's population sharply declines with rising inflation rates. However, in India, life expectancy has improved due to better living conditions, healthcare facilities, and nutrition programmes in 2022, influenced by the COVID-19 pandemic. Health is a crucial focus of SDG 3, promoting well-being and healthy lives. Access to necessities such as water, electricity, and housing facilities can increase life expectancy, and higher per capita real income and healthcare expenditure lead to improved life expectancy (Mahumud et al. 2013). The governments have allocated a significant portion of their GDP towards social services, education, health, sanitation, labour welfare, housing, and urban development. Life expectancy benefits economic development in developing nations but is limited by income level (Ngangue and Manfred 2015). While health significantly contributes to real GDP per capita and economic development, in nations with poorer health, longer life expectancies may not have a deterministic impact on economic development (Yıldırım et al. 2020).

This research encapsulates the critical factors influencing economic welfare in India and highlights their significance. As a primary driver of economic expansion, energy plays a vital role in industrial production, transportation, and household activities. Understanding its impact on economic welfare is crucial for policymakers and stakeholders. Similarly, inflation also affects economic welfare in various stages and the need to manage price stability and promote economic growth with the welfare of the society. On the other hand, financial development helps capital allocation and investment to promote economic growth as well as the welfare of the country's people. on the basis of existing research on energy consumption, inflation, and financial development on economic welfare (Aydin 2014; Apergis and Payne 2012; Behera and Mishra 2020; Fang 2011; Sari et al. 2008; Chang et al. 2009; Monsef and Mehrjardi 2015; Benabou 1988; Lucas 1973; Hercowitz 1987; Diamond 1988; Grossman 1972; Demirguc-Kunt and Klapper 2012; Rajan and Zingales 2003; King and Levine 1993; Demirguc-Kunt and Maksimovic 1998). However, these studies have overlooked the aspect of the economic welfare index, with some considering only GDP as a measure of economic welfare. The study addresses these gaps in the literature by creating an economic welfare index using more comprehensive variables such as employment, human capital, CO2 emissions, gross domestic product, real consumption of households and government, life expectancy, and workforce engagement. To fill these gaps, this study uses advanced econometric tools such as the PCA method, novel dynamic ARDL model, and KRLS approach for investigating the relationship between NREC, REC, INF, FD, and EW in India.

2 Background and literature review

In order to understand the welfare of countries, a thorough review of the literature is essential. This study comprises a three-part review of the existing literature. Firstly, the study examines the theories relevant to economic welfare. Subsequently, the study delves into the conceptual and empirical literature on the components incorporated in this analysis.

2.1 Theoretical highlights

It is not difficult to argue that poverty is prevalent worldwide, and an unequal economic system is responsible for this poverty and other related issues. Despite progress in financial development by developing nations, several factors, including energy demand and environmental problems, contribute to the increase in global poverty and inequality. Income redistribution and structural changes in the world economy are necessary to reduce income divergence and the resulting inequality. Social welfare economics employs theories that evaluate allocation systems using specific types of efficiency. The Pareto efficiency criterion suggests that if no exchange can be made that benefits one person while harming another, the system is said to be Pareto optimal. Alternatively, a resource allocation change is considered Kaldor–Hicks efficient in economics if it results in a net increase in benefits over a net decrease in costs (Stringham 2001). However, bringing about changes without negatively affecting at least one person is challenging. The Pareto efficiency criterion becomes contradictory when combined with the disparities decrease multiplier. It states that one should avoid reducing the wealth of others, but in the long term, it allows for the decline of the wealth of many. Therefore, the Pareto efficiency criterion is unsustainable. However, it could be consistent with situations where Pareto improvements are made, and inequality remains constant or decreases, and in such cases, its viability would be preserved. Additionally, the Pareto efficiency criterion is unrelated to distribution issues. It only aims to define states that are considered efficient similarly, even when some individuals' wealth increases. At the same time, others' wealth decreases (Mishan and Mishan 1967). Political decisions are often expected to complement the Pareto efficiency criterion when making equitable decisions. However, this criterion may not be impartial as different Pareto improvements can have varying effects on inequality, which can impact the sustainability of Pareto optimality in the long run (Van staVErEn 2012). Therefore, it may not add much value to policy options analysis and political decisions and can lead to considering options with different sustainability conditions as equivalent. Moreover, achieving Pareto efficiency is almost impossible in practical economic dynamics. A Pareto efficient transaction for those directly involved may reduce the wealth of consumers who could have paid less for the purchase or suppliers who believe that the purchase should have been at a higher price. Political decisions are often expected to complement the Pareto efficiency criterion when making equitable decisions.

The Kaldor–Hicks efficiency test, which determines if an action or project is desirable if the benefits' value exceeds the cost's monetary value, can also be examined (Krutilla 2005). However, this criterion does not address distribution concerns and would be inadequate in the inequality decrease multiplier paradigm. The compensation test, developed by Kaldor and Hicks, suggests that a policy change is reasonable if winners can pay losers to address the limited applicability of the Pareto principle in making policy decisions (Krutilla 2005). Additionally, the impossibility theorem suggests that combining individuals' preferences into a national social welfare function that complies with a reasonable minimum set of acceptable conditions is impossible, even internationally. From the perspective of justice, Rawls argues that the poorest among us are the least fortunate, and fundamental freedoms and social rights to resources like money and wealth are intertwined and cannot be rationalised separately (Pleasants 2002). In contrast to utilitarianism, Rawls proposed an alternative ethical theory that could support notions of justice based on a shared understanding of what justice is. Regarding the environment, research has indicated a beneficial relationship between emission intensity and energy use by industries and individuals, as well as a high level of eco-efficiency and stock market functioning (Pleasants 2002). Commitment to sustainability reporting has also been demonstrated to affect economic growth favourably. However, economic growth without structural reforms does not necessarily improve environmental quality. Financial innovation is vital in developing nations, where energy-intensive sectors often bestow economic growth. Ignoring financial innovation's effect on energy consumption makes assessing compliance with greenhouse gas emission targets challenging. Environmental conditions are crucial for long-term sustainability and quality of life, impacting human health and environmental services, leading to climatic changes and natural catastrophes that harm impacted populations (Ibarrarán et al. 2009).

2.2 Empirical highlights

2.2.1 Economic welfare (EW) and non-renewable energy consumption (NREC)

Apergis and Payne (2012) studied the relationship between NREC and REC in emerging market economies from 1990 to 2007. They found that all variables positively impact GDP, except renewable energy consumption. The study also found a short-term, unidirectional causal linkage between REC and GDP in the Central American nations. The findings suggest a causal relationship between NREC, REC, and GDP over the long and short term. Their study concluded that renewable energy acts as a burden on the economies. Aydin (2014) employed Granger causality for 26 OECD countries between 1998 and 2015 to investigate the link between REC, NREC, and GDP. The empirical findings indicated that bidirectional causation exists between NREC and GDP, as well as between GDP, REC, and NREC. They suggested that there is a bidirectional relationship between NREC and GDP, meaning that NREC and GDP affect each other simultaneously. When the NREC of the countries increases, the growth in GDP is also affected. Behera and Mishra (2020) examined the correlation between NREC, REC, and economic growth in the G7 nations from 1990 to 2015. They applied a panel ARDL model to investigate short and long-term relationships. The model confirmed that capital stock, energy costs, and labour availability have a beneficial effect on the long-term economic growth of the G7 nations. Additionally, they found that capital, energy prices, and labour boost economic growth, while NREC discourages the growth of the countries. The reason behind this is that countries do not use efficient technologies for the extraction of non-renewable energy sources.

2.2.2 Economic welfare (EW) and renewable energy consumption (REC)

Using Cobb–Douglas production functions, Fang (2011) determined how much renewable energy consumption contributed to the nation’s well-being. Multivariate OLS and SPSS software were used to conduct this analysis for China from 1978 to 2008. According to the findings, real GDP increased by 0.120%, rural households of per capita income increased by 0.444%, GDP per person increased by 0.162%, and urban households of per capita income increased by 0.368% with a 1% increase in REC, respectively. However, the share of RECs that contributes to economic welfare is minimal, and the growth of economic welfare is only slightly adversely affected by an increase in the share of RECs. Sari et al. (2008) examined the link between using all energy sources, industrial production, and employment in the USA using the ARDL approach. They discovered that employment and industrial outputs were the main factors influencing fossil fuel consumption and renewable energy sources but had little to no influence on the consumption of wood energy and natural gas in the long term. For the OECD countries, Chang et al. (2009) applied a panel threshold regression approach to simulate the effect of energy prices on the growth of renewable energy underneath the variety of economic development rates between 1997 and 2006. They stated that there is no connection between gross domestic product and the proportion of the total energy supply from renewable sources. They determined that a nation’s economic growth rate influenced REC's ability to cope with fluctuations in oil prices. Low-economic growth nations could not use renewable energy sources to reduce the results of a negative price shock, whereas nations with high economic growth could.

2.2.3 Economic welfare (EW) and inflation (INF)

Inflation can positively affect markets by reducing the number of firms and widening price ranges, thus benefiting buyers with lower search costs. However, its impact on monopolistic pricing efficiency depends on market structure and preferences. In some countries, inflation can harm life expectancy, firm search, and exit limit pricing variation, thereby limiting these effects. A study by Monsef and Mehrjardi (2015) showed that inflation significantly and adversely affects life expectancy in 136 countries. Benabou (1988) demonstrated that inflation may raise price dispersion in a market where price adjustment is expensive. This increase in competition lowers real prices and raises welfare, ultimately boosting the economic welfare of countries. Further, Lucas (1973) and Hercowitz (1981) examined the deviation instead of the trend of cumulative prices to examine the effects of unexpected inflation on prices and welfare. Diamond (1988) assumed search and price adjustment technologies for mild inflation; similar results were obtained; however, as inflation rises further, the “thin-market” externality worsens, and welfare declines. Grossman (1972) examined the relationship between health and inflation. The study found that inflation has a adverse relationship with life expectancy. Furthermore, the study concluded that the rising inflation rate has an adverse impact on household welfare.

2.2.4 Financial Development (FD) and Economic Welfare (EW)

FD has been extensively studied as a critical economic growth and development driver. The literature suggests that FD can promote EW by improving access to credit, promoting financial stability, and reducing poverty (Demirguc-Kunt and Klapper 2012; Subhan et al. 2023; Rajan and Zingales 2003). Furthermore, FD can facilitate resource allocation to more productive uses, leading to higher economic growth rates (King and Levine 1993). Additionally, Bencivenga and Smith (1991) underpin the assumption that the financial system can contribute to economic growth. By easily accessing financial facilities like credit and banking systems, households and firms can purchase productive equipment and enhance the living standard of the people, which boosts the welfare of the country (Demirguc-Kunt and Maksimovic 1998). It is critical to mention that FD has advantages and disadvantages. FD can create over-debt financing and also contribute to inequality if it benefits people and firms with higher incomes. Therefore, the relationship between FD and EW is compounded and depended on the deeper economic and social prospects (Rajan and Zingales 2003). Last but not least, the existing research reveals a favourable relationship between FD and EW, although the magnitude of this relationship may depend on different factors. Thus, authorities should focus on adopting measures that boost FD while ensuring that financial systems remain steady and equitable.

3 Data and methodology

3.1 Data description

The present research investigates how NREC, REC, INF, and FD affect EW in India. This research used time-series data from 1990 to 2019 on the basis of the availability of the data sources. Table 1 depicts the sources and measurement of the variables considered for the analysis based on an extensive review of the pertinent literature. The function of the model is reported in Eq. 1:

Table 1 Proxy of variables, sources, and measurements
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$$EW=f\left(NREC, REC, INF, FD\right)$$
(1)

EW is an abbreviation for economic welfare, which indicates the general well-being or level of living in a society at any given time. Several factors include employment, human capital, GDP, real consumption, and life expectancy. In order to make large datasets more manageable, PCA is used to split them up into more minor variables. When viewed from the economic well-being perspective, it can pinpoint the main determinants of human welfare. The results of PCA are presented in Table 2 and visualised through a biplot Fig. 1. The correlation matrix's eigenvalues describe how much variance each component could capture in the original data. PC1, which explains 96.85% of the variation in the data, is chosen because it is significantly higher than the noise level and has an eigenvalue of 7.74. Based on these results, one can conclude that a single dominant factor drives the eight measures of economic welfare used in the study. PCA is useful for simplifying large datasets and identifying the most critical factors contributing to economic welfare.

Table 2 Correlation matrix of the eigenvalues
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Fig. 1
figure 1

Biplot of principle component analysis

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Inflation (INF) measures the mean price of a bundle of goods and services consumed by households, typically referred to as the consumer price index (CPI), increasing over a period of time. The domestic credit to private sector banks (DCPSB) as a percentage (%) of GDP is often deployed as a proxy for FD. This indicator measures the amount of credit banks that provide to the private sector as a part of the country’s overall economic output. NREC refers to the total energy consumed from non-renewable sources such as oil, gas, and coal. This is typically measured in exajoules (EJ), a unit of energy equivalent to one quintillion (10^18) joules. The total energy of the country is a measure of primary energy consumption, including REC and NREC. We get REC by subtracting the NREC (oil, gas, coal, and nuclear) from primary energy consumption.

3.2 Model specification and modelling

This research has employed the novel DYARDL model to examine the relationship between NREC, REC, INF, FD, and EW. This model is frequently used for time-series data. It can test multiple hypotheses when the dependent variable is analysed at either its level or first difference (Rafindadi and Ozturk 2017). The cointegration test is essential before the estimation of ARDL in the error correction method. The ARDL cointegration method introduced by Pesaran et al. (2001a, b) is more accurate in small samples and better than the simple two-step Johansen (1988) cointegration technique. However, this cointegration technique is not widely accepted in econometric practices. Besides the fact that the ARDL model in error correction mode can have critical constraints that include multiple lags, lagging, and anomalies in data, it is hard to examine the effect of shifts in the independent variables in the short and long term. Jordan and Philips (2018) solved all these problems that arise in the traditional ARDL method by introducing novel DYARDL estimation. This method is applicable to estimating several ARDL models, including the EC mechanism. The graphical presentation of empirical analysis is shown in Fig. 1. Before estimating the DYARDL model, it is required to check the stationarity of each variable at level and first difference. If a variable is not stationary at its level, it is considered to have a unit root. If the variable can become stationary in the first differences, it is considered to have no unit root. Those variables are considered for DYARDL simulations, which are stationarity at level and first difference. This research has employed two unit root tests for stationarity, ADF, and PP, to investigate the relationship between NREC, REC, INF, FD, and EW using the following Eq. 2 as given:

$${EW}_{t}= {\alpha }_{0}+{\beta }_{1}\left({NREC}_{t}\right)+{\beta }_{2}\left({REC}_{t}\right)+{\beta }_{3}\left({INF}_{t}\right)+{\beta }_{4}\left({FD}_{t}\right)+{\mu }_{t}$$
(2)

where “t” represents time, the equation includes a constant term, \({\alpha }_{0}\) and coefficients, \({\beta }_{1}\) to \({\beta }_{4}\), as well as an error term \({\mu }_{t}\). The flow chart of the empirical analysis is presented in Fig. 2

Fig. 2
figure 2

Steps of empirical methodology

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3.2.1 Cointegration technique

This research has used the ARDL bounds test to check whether cointegration exists between the variables in the order of integration I(0) and I(1) in the long term. The bound test for cointegration Eq. 3 is given below:

$${\Delta EW}_{t}= {\varphi }_{0}+{{\varphi }_{1}EW}_{t-i}+{\varphi }_{2}\left({NREC}_{t-i}\right)+{\varphi }_{3}\left({REC}_{t-i}\right)+{\varphi }_{4}\left({INF}_{t-i}\right)+{\varphi }_{5}\left({FD}_{t-i}\right)+\sum_{i=1}^{r}{{\beta }_{1}EW}_{t-1}+\sum_{i=1}^{r}{{\beta }_{2}NREC}_{t-1}+\sum_{i=1}^{r}{{\beta }_{3}REC}_{t-1}+\sum_{i=1}^{r}{{\beta }_{4}INF}_{t-1}+\sum_{i=1}^{r}{{\beta }_{5}FD}_{t-1}++{\mu }_{t}$$
(3)

where \({\varphi }_{1}\) to \({\varphi }_{5}\) and \({\beta }_{1}\) to \({\beta }_{5}\) are presented the long-run and short-run coefficients of the independent variables for this ARDL bounds test which can be used by Khan et al. (2019), and this approach formulates the null and alternative hypotheses of the research for the cointegration of the variables in Eqs. 4 and 5.

$${H}_{0}={\varphi }_{1}={\varphi }_{2}={\varphi }_{3}={\varphi }_{4}={\varphi }_{5}$$
(4)
$${H}_{1}={\varphi }_{1}\ne {\varphi }_{2}\ne {\varphi }_{3}\ne {\varphi }_{4}\ne {\varphi }_{5}$$
(5)

The null hypothesis is accepted or rejected on the basis of F-statistics values. When the F-statistics value is greater than the upper bound limit, as noted by Pesaran et al. (2001a, b), this suggests rejecting the null hypothesis and indicating a long-term relationship between the variables. Conversely, F-statistics values are lower than the lower bounds limit, which suggests that there is no long-term relationship among the variables. If the F-statistics values fall within the lower and upper bounds, there are uncertain results.

3.2.2 ARDL estimation

This research has employed the ARDL model given by Pesaran et al. (2001a, b), which is more suitable for small sample sizes and variables cointegrated with the orders I(0) and I(1), and the ARDL bound test allows for multiple lag selection criteria. The long-run model employed in Eq. 6.

$$EW={\alpha }_{0}+\sum_{i=1}^{r}{\sigma }_{1}{EW}_{t-i}+\sum_{i=1}^{r}{\sigma }_{2}{NREC}_{t-i}+\sum_{i=1}^{r}{\sigma }_{3}{REC}_{t-i}+\sum_{i=1}^{r}{\sigma }_{4}{INF}_{t-i}+\sum_{i=1}^{r}{\sigma }_{5}{FD}_{t-i}+ {\mu }_{t}$$
(6)

Equation 6 represents the long-term fluctuation of the parameters with \(\sigma\). The AIC estimates the lags for each parameter on the basis of the maximum number. The short-term ARDL model was estimated employing the EC method and is presented in Eq. 7 as follows:

$${EW}_{t}={\alpha }_{0}+\sum_{i=1}^{r}{\gamma }_{1}{\Delta EW}_{t-i}+\sum_{i=1}^{r}{\gamma }_{2}{\Delta NREC}_{t-i}+\sum_{i=1}^{r}{\gamma }_{3}{\Delta REC}_{t-i}+\sum_{i=1}^{r}{\gamma }_{4}{\Delta INF}_{t-i}+\sum_{i=1}^{r}{\gamma }_{5}{\Delta FD}_{t-i}+{\varphi ECT}_{t-i}+{\varepsilon }_{t}$$
(7)

In Eq. 6, \(\gamma\) symbolises short-term variation, and ECT refers to the error correction term, demonstrating how the system returns to stability after an interruption. The value of ECM must lie between 0 and 1, and its coefficient should be negative and significant, which suggests that the model was adjusted in the subsequent cycle.

3.2.3 Novel dynamic ARDL estimation

The novel DYARDL model was developed by Jordan and Philips (2018); this model solved the weaknesses and difficulties of the ARDL model. This model forecasts the real-time effect of changes in NREC, REC, INF, and FD on EW while other things being constant. It should require cointegration among the variables in orders I(0) and I(1). This research has used the DYARDL model, which included 5000 simulations of multivariable distributed vectors. For the reliability and validity of the model, diagnostic tests have used, including the Breusch–Godfrey Lagrange multiplier (LM) test for serial correlation, Cameron and Trivedi’s (White’s test) test for heteroscedasticity, and the Jarque–Bera test for residual normality. The DYARDL error correction form elaborated in Eq. 8 is given below:

$${\Delta EW}_{t}= {\varphi }_{0}+{\theta }_{0}\left({EW}_{t-1}\right)+{\beta }_{1}\Delta \left({NREC}_{t}\right)+{\theta }_{1}\left({NREC}_{t-1}\right)+{\beta }_{2}\Delta \left({REC}_{t}\right)+{\theta }_{2}\left({REC}_{t-1}\right)+{\beta }_{3}\Delta \left({INF}_{t}\right)+{\theta }_{3}\left({INF}_{t-1}\right)+{\beta }_{4}\Delta \left({FD}_{t}\right)+{\theta }_{4}\left({FD}_{t-1}\right)+{\gamma ECT}_{t-1}+{\varepsilon }_{t}$$
(8)

where in Eq. 8, the difference operator \((\Delta )\) shows the short-term dynamic relationship, while the intercept term \({(\varphi }_{0})\) represents the constant value that shows the beginning point of this relationship. The long-term coefficients denoted by \({(\theta }_{1}, {\theta }_{2}, {\theta }_{3}, {\theta }_{3})\) and short-term coefficients denoted by \({(\beta }_{1}, {\beta }_{2}, {\beta }_{3}, {\beta }_{4})\). The white noise error term represented by \({\varepsilon }_{t}\) manages the random fluctuations in the model, and ECT is an error correction term. We then used the KRLS machine learning algorithm, as provided in the works of Ferwerda et al. (2017), to investigate the causal relationship.

3.3 Result and discussion

A thorough statistical analysis is performed before conducting the time-series econometric analysis. The descriptive statistics presented in Table 3 include the mean values and standard deviations (SDs) for EW and NREC, REC, INF, and FD. The average of EW is 9.33E-07 with a SD of 2.599549, the average of NREC is 16.90094 with a SD of 7.365708, the average of REC is 1.251836 with a SD of 0.66465, the average of INF is 7.28976 with a SD of 3.220232, and the average of FD is 37.72791 with a SD of 12.07083. The shape of the variables is shown through skewness, with all the variables being right-skewed except FD, which is negatively skewed. The kurtosis statistic indicates that each variable has a platykurtic distribution (lower peak or short-tailed), and the outcomes of the Jarque–Bera test show that each variable residual is normally distributed.

Table 3 Descriptive statistics
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It can be challenging to make accurate predictions and get trustworthy results from time-series data that are not stationary. Thus, it is essential to use stationary time-series data to avoid errors in regression analysis. In this study, we did not consider the assumption of stationarity after the second difference for I(2) variables, which could result in biased findings, as noted by Jordan and Philips (2018) and Pesaran et al. (2001a, b). The results of the PP and ADF tests demonstrated that each variable is stationary at the first difference presented in Table 4.

Table 4 Unit root test result
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Different time lags for the independent and dependent variables are allowed by the ARDL approach. The results of multiple test approaches for identifying the ideal lag length are shown in Table 5. LR, FPE, AIC, SBIC, and HQIC are some of these tests. These testing have led us to choose a fourth lag. We first determine the stochastic characteristic of the components and the optimal lag length for the method. Then, we analyze the cointegration relationship applying bound test methods with critical and comparative p values from Kripfganz and Schneider (2020). These p values are suited for bounds analysis with zero- or one-order variables. For small sample sizes, Kripfganz and Schneider’s (2020) linear regression model provides a reliable critical value for the bounds test.

Table 5 Optimal lag selection
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Determining if a cointegration relationship exists is crucial because not all relationships between I(1) data series indicate cointegration. A bounds test was used to test for cointegration between EW, NREC, REC, INF, and FD. The findings shown in Table 6 suggest that the T- and F-statistics have a higher level of significance in comparison with the upper limit critical values. Therefore, at 5% significance, the null hypothesis of no cointegration is rejected. This indicates that EW, NREC, REC, INF and FD in India have a cointegrating connection, and all the calculated variables have affirmed the existence of cointegration among the variables.

Table 6 Pesaran et al. (2001a, b) test for cointegration
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This research used two models to ensure the estimation results was robust. Tables 7 and 8 reveal the results of the ARDL and DYARDL, respectively. The findings of both models are similar in the context of the sign of the coefficients, but the magnitude of the coefficients may differ from each other.

Table 7 ARDL estimation
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Table 8 Dynamic ARDL result
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Table 8 depicts the findings of the DYARDL model. The DYARDL estimation is unique because it can accurately calculate the direction and size of variable changes and their short-term and long-term relationships. The DYARDL simulation revealed that NREC has a positive and statistically significant effect on India’s EW at a 10% significance level in the long run and insignificant in the short run. Specifically, a 1% rise in NREC leads to a 0.110% and 0.253% rise in economic welfare in the short and long term, respectively. NREC sources such as coal, oil, and natural gas can boost EW in India. Hereby, NREC sources have advantages such as high energy density, cost-effectiveness, job creation, and significant tax revenues. However, they also have significant environmental impacts, such as pollution and contributing to climate change. Therefore, considering economic and environmental factors, a balanced approach is necessary for energy production and use. These findings are similar to those of Menegaki and Tugcu (2017) and Adams et al. (2018), who found that NREC has a favourable effect on sustainable EW and gross domestic product when adjusted for purchasing power parities.

Similarly, this study also revealed that FD has a favourable and statistically significant impact on EW in India, both long-term and short-term, at 1% and 5% significance levels, respectively. A 1% rise in financial development brings a 0.014% and 0.029% rise in EW in the short and long term, respectively. Thus, FD positively impacts economic welfare (EW) through various channels. One channel is increasing the availability of credit and financing to households and businesses, allowing them to invest in new projects, expand existing ones, and increase their consumption. This can increase economic activity, job creation, and economic welfare. Another channel is improving the financial sector's efficiency by introducing new financial products and services, developing more sophisticated financial markets, and increasing competition among financial institutions. These results are consistent with the findings of Musabeh et al. (2020), who found that FD has a positive impact on economic growth in the four countries. They suggested that enhancements in the financial system were positively reflected in national welfare.

Conversely, this study demonstrated that the REC has a significant favourable impact on EW in the short run but a significant adverse impact on EW in the long run in India, based on a 5% significance in the short run and a 1% significance in the long run. The findings indicate that a 1% rise in REC results in a 0. 109% rise in EW in the short term and a 0. 474% decrease in EW in the long term. This relationship between REC and EW is similar to the findings reported by Fang (2011), although the specific magnitudes of the relationships may differ. However, these results contradict those reported by Inglesi-Lotz (2016). This contradiction may have occurred due to differences in the measures of the economic welfare index. Inglesi-Lotz (2016) used economic growth as a measure of economic welfare, while in our study, we used an index of economic welfare composed of deeper variables that reflect the actual picture of economic welfare. Thereby, economic welfare can be negatively affected by the high cost of technology, which produces renewable energy, limitations in infrastructure, and resistance from stakeholders. However, authorities should provide subsidies and more investment in renewable energy to help improve infrastructure and technology related to renewable energy.

Furthermore, the EW is adversely impacted by inflation in the short run at the 1% significance level, while in the long run, it is insignificant. This relationship can be happened by reducing consumer purchasing power and increasing business costs through inflation. Accordingly, increasing the prices of goods and services that consumers cannot afford and finally reducing the demand for goods and services may lead to an economic slowdown. Additionally, businesses may have to pay more for inputs, such as raw materials and labour, which can be passed on to consumers at higher prices, further reducing economic welfare. These findings are consistent with those of Bashir (2002). He found that inflation reduces welfare and hampers the growth of the country.

Additionally, the ECT suggests a long-term link between NREC, REC, INF, FD, and EW and the load capacity factor. The ECT shows that any temporary disruptions to environmental conditions are quickly corrected, with 63% of the necessary adjustments occurring within the first year. As a result, the long-term equilibrium is maintained. The R2 value of 0.81 indicates that the model's independent variables can explain 81% of the changes in the dependent variable. This is a common way to calculate the goodness of fit of a linear regression model, and a high R-squared value indicates that the model fits the data well. Finally, the results of diagnostic tests are presented in Table 9. The Breusch–Godfrey (BG) and Cameron and Trivedi’s (White’s test) have confirmed that there is no serial correlation and heteroscedasticity issue in the model, respectively. The skewness and kurtosis also confirm that estimated residuals are normally distributed.

Table 9 Diagnostic analysis
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The DYARDL estimation has also provided the positive and negative shifting in the NREC, REC, INF, and FD and explanatory factors and their impact on EW through the graphical representation. Figure 3A, B shows the positive and negative changes in NREC and their effect on EW. Figure 3A demonstrates that the 10% positive change in NREC leads to a boost EW in the short and long term. Figure 3B shows the 10% negative change in NREC leading to a decline in EW in the short and long term. Conversely, Fig. 4A, B illustrates the relationship between REC and EW in India. Figure 4A displays the relationship between an increase in REC and its impact on EW. It shows that a 10% rise in REC leads to a drop in EW in both the short term and the long term. Conversely, Fig. 4B shows the connection between a decrease in REC and its effect on EW. A 10% reduction in REC increases EW in the short and long term. Furthermore, Fig. 5A, B illustrates the relationship between INF and EW in India. Figure 5A displays the relationship between an increase in INF and its impact on EW. It shows that a 10% rise in INF leads to a decrease in EW in both the short term and the long term. Conversely, Fig. 5B shows the connection between a decrease in INF and its effect on EW. A 10% reduction in INF increases EW in the short and long term. Additionally, Fig. 6A, B displays the relationship between changes in FD and their effect on EW. The average estimated values are depicted as dots in the plot. Figure 6A shows that a 10% rise in FD results in a corresponding rise in EW. Conversely, Fig. 6B shows that a 10% decrease in FD leads to a decrease in EW.

Fig. 3
figure 3

A, B The graphs demonstrate the effect of NREC on EW, depicting both upward and downward trends. The line from dark blue to light blue indicates a confidence range of 75%, 90%, and 95%, and the dots symbolise the average expected value

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Fig. 4
figure 4

A, B The graphs demonstrate the effect of REC on EW, depicting both upward and downward trends. The line from dark blue to light blue indicates a confidence range of 75%, 90%, and 95%, and the dots symbolise the average expected value

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Fig. 5
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A, B The graphs demonstrate the effect of INF on EW, depicting both upward and downward trends. The line from dark blue to light blue indicates a confidence range of 75%, 90%, and 95%, and the dots symbolise the average expected value

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Fig. 6
figure 6

A, B The graphs demonstrate the effect of FD on EW, depicting both upward and downward trends. The line from dark blue to light blue indicates a confidence range of 75%, 90%, and 95%, and the dots symbolise the average expected value

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Table 10 presents the KRLS model's pointwise derivatives that accurately predict future EW with high accuracy and a projecting power of 0.997. The model was analysed at different percentiles to evaluate heterogeneous marginal effects, with no variation found, validating the robustness of the results. The mean marginal effect of NREC, REC, INF, and FD on EW is 0.17%, 0.89%, − 0.05%, and 0.01%, respectively, showing these factors' significance in maintaining India's economic welfare. The current study further explores the long-term relationship between these variables and economic welfare by analyzing how their changes impact economic welfare and vice versa. Figure 7 suggests a relationship between NREC and EW. Initially, an increase in NREC results in economic welfare up to a certain point, beyond which a further increase in NREC leads to decreasing returns and declining economic welfare. Furthermore, Fig. 8 shows that a higher level of REC initially increases EW until marginal returns decline. Beyond this point, further increases in REC result in a decline in EW. Additionally, the relationship between INF and EW is depicted in Fig. 9. An increase in INF initially results in a slight decline in EW due to decreasing marginal returns. This suggests that INF has a negative effect on EW up to a specific limit. Beyond that, further increases in INF result in a more significant decline in EW. Lastly, Fig. 10 suggests a connection between FD and EW. Initially, FD is below the EW baseline, but as it increases, economic welfare improves with increasing returns to scale. However, beyond a certain point, FD and EW decline at almost constant marginal returns. This suggests that FD can play a positive role in promoting EW up to a specific limit, but beyond that, the benefits of further FD may diminish. The saturation point is likely influenced by factors such as market saturation, regulatory barriers, and the efficiency of the financial sector. It is noteworthy that various factors, including the economy's structure, the stage of economic development, and the effectiveness of financial institutions and markets, typically influence the relationship between FD and EW.

Table 10 Pointwise derivatives using KRLS
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Fig. 7
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Pointwise marginal effect of NREC.

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Fig. 8
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Pointwise marginal effect of REC

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Fig. 9
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Pointwise marginal effect of INF

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Fig. 10
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Pointwise marginal effect of FD

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3.4 Conclusion and policy implications

The current study examines the dynamic linkages between India's NREC, REC, INF, FD, and EW. The researchers used the newly established DYARDL technique and analysed data from 1990 to 2019. The findings demonstrate that NREC has a favourable and significant effect on EW over the long term; however, there is no significant effect in the short term. The research also revealed that REC significantly affected EW in India, with positive short-term and adverse long-term effects. In addition, this research found that whereas INF had a significant short-term deleterious impact on EW in India, it did not have a significant long-term detrimental impact. On the other hand, it found that FD had statistically significant long-term and short-term effects on EW.

Based on the study's results, numerous policy implications are suggested, such as the effect of NREC on EW in India, which is highlighted by the study's findings and points out the necessity of a balanced energy-use approach. As a result, when making decisions on energy use, the Indian government should take both economic and environmental factors into account. Consequently, the government may advise encouraging a balanced approach to energy consumption by financing both NREC and REC sources. Further, the Indian government should invest more and enhance the allocation of funds for research and development in renewable energy generation while also offering subsidies on renewable energy-related projects. These are creating more jobs and enhancing economic welfare without compromising environmental concerns. The government can also make policies related to renewable projects and boost economic welfare. However, the most precious concern related to REC affects EW negatively in the short term. This can happen when traditional energy producers do not adopt renewable energy technology because of high prices, limitations of infrastructure, and inadequacies. The Indian government should resolve these issues by improving infrastructure, offering financial incentives to traditional energy players, who shift to renewable energy, and also giving financial support to consumers to purchase green products. For inflation, the government should develop policies to reduce the adverse effects of inflation on EW in the short term. The government can also take measures to reduce these effects by reducing government spending, controlling general prices, and monetary growth. Last but not least, the government should enhance financial inclusion through various measures like promoting financial sector efficiency, minimising the financial instability risk, and introducing programmes more effectively like Pradhan Mantri Jan Dhan Yojana. All these measures enhance the welfare of societies. We conclude that more investment and consumption will improve economic welfare and also reduce the risk of financial crises.

The present research provides crucial findings on the effect of NREC, REC, INF, and FD on EW. However, any research cannot be completed without limitations. This research also considered some limitations when discussing the findings. Firstly, this research, which is mainly focused on the Indian context, might not apply the findings to developed countries. Secondly, the research findings may not capture the most recent developments, especially after the COVID-19 pandemic, because this research considers the data period from 1990 to 2019 due to the availability of the data sources. Lastly, the research has considered a limited set of variables; other relevant factors affecting economic welfare have been overlooked, such as trade, institutional quality, and technological innovation. On the basis of these limitations, research has mentioned the direction of future research. Researchers can conduct panel studies, and regional comparisons will provide a better understanding of this relationship between the examined variables in another context and include another relevant variable. They can also investigate non-linearity or threshold effects among the variables, and extending the data period may be available to allow potential shifts in trends in the studied variables.