1 Introduction

Child abuse is a widespread problem with serious social consequences [1]. It affects not only the victims but also entire communities [2]. A 2011 report by the Department of Social Policy and Intervention at the University of Oxford estimated that around 40 million children worldwide, under the age of 14, urgently need help and care. In Malaysia, the rising cases of child abuse present a significant challenge to the country’s efforts to achieve its Sustainable Development Goals. According to the Malaysian Ministry of Health [3], child abuse is a multifaceted issue that includes physical and emotional harm, sexual exploitation, neglect, and negligent treatment, highlighting the urgent need for action to protect the country's future generations. Recognising that documented cases represent only a fraction of the actual incidents is significant, as it indicates a more pervasive problem that has gone unnoticed.

Given the extensive reporting on child abuse, it is imperative to acknowledge the depth of this issue. A wealth of research highlights the profound negative impact of child abuse on a child's health and development, with consequences that ripple through society. Disturbingly, individuals who experienced abuse as children are more likely to perpetrate abuse in the future, highlighting a troubling cycle. Furthermore, children who have been subjected to abuse are at a higher risk of engaging in criminal behaviour. Given the wide-ranging effects, it is crucial for researchers to delve deeper into this issue and address its significant implications for society.

The existing literature extensively examines the linear relationship between macroeconomic indicators and child abuse, such as unemployment [4] and GDP per capita [5] However, a gap remains in understanding the non-linear effects of these variables, as previous studies do not employ the non-linear autoregressive distributed lag (ARDL) method. Our research aims to fill this gap by utilising the non-linear ARDL (NARDL) technique to examine the non-linear impact of these variables on child abuse. Recognising that the relationship between these factors may not be a simple linear connection is crucial. The non-linear perspective of the ARDL method allows us to uncover thresholds, turning points, and non-linear patterns that significantly influence child abuse. This comprehensive approach surpasses the limitations of conventional linear ARDL techniques, providing a more complex understanding of the intricate interplay between economic circumstances and child abuse. This in-depth analysis offers a nuanced perspective by delving into the complexities of the subject matter. Additionally, non-linear models better capture the data and account for a larger portion of the observed variation. By employing the non-linear ARDL method, we improve the model’s goodness of fit and enhance the estimation of the link between economic conditions and child mistreatment, generating more robust and reliable results that strengthen the validity of our conclusions as emphasized by Mukhtar et al. [6] and Shin et al. [7], therefore the NARDL method offers greater accuracy in capturing complex dynamics.

Evidence suggests that families living in poverty are at a higher risk of being reported to Child Protective Services (CPS) for neglect. Although the link between poverty and child abuse is well-established, most research examining this connection has primarily focused on developed countries, highlighting a gap in understanding its impact in a developing nation. A recent review conducted by Bywaters et al. [8] underscores this trend, revealing that over 67% of the reviewed literature originates from research conducted in the United States, with less than 2% stemming from studies conducted in Asian countries. This glaring disparity highlights a significant gap in our understanding of the role poverty plays in child abuse, particularly in Asian contexts and specifically in Malaysia.

Recognizing this gap, Yob et al. [9] sought to address this dearth of empirical evidence by investigating the influence of poverty on child abuse within the Malaysian context. However, it is noteworthy that their study only considered a linear relationship between poverty and child abuse. Expanding upon this research, our study not only explores the complex dynamics between poverty, unemployment, GDP, and child abuse in Malaysia but also delves into a non-linear relationship between these factors. By applying the NARDL method, this study uncovers new insights that the traditional linear ARDL approach cannot capture. Specifically, the NARDL method reveals the non-linear impacts of poverty, unemployment, and GDP on child abuse in Malaysia, offering a more comprehensive understanding of these relationships. These findings provide policymakers with valuable information to design targeted and effective policies and programs aimed at reducing child abuse in the Malaysian context. Ultimately, the non-linear ARDL method empowers policymakers to understand the connection between economic difficulties and child abuse. Policymakers can make well-informed decisions regarding resource allocation, early intervention programs, and policy interventions to effectively address child abuse by considering the non-linear dynamics and potential thresholds. This evidence-based approach enables policymakers to implement strategies that mitigate the adverse effects of economic factors on child well-being. Thereby, the specific purpose of this study is to investigate the non-linear relationships between poverty, economic growth, unemployment, and child abuse in Malaysia, utilizing the non-linear ARDL method to provide a nuanced understanding of these dynamics and to inform evidence-based policy interventions.

2 An overview of child abuse in Malaysia

The illustration of cases of child abuse in Malaysia spanning from 1989 to 2019 is shown in Fig. 1. The data reveals a concerning upward trajectory, indicating the distressing prevalence of child abuse within the country. Initially, recorded cases fluctuated between 276 and 1,161 from 1989 to 1999. However, starting in 2000, a significant surge in reported cases occurred, with the number of incidents reaching 6,061 in 2019. This upward trend likely reflects increased public awareness, improved reporting mechanisms, and heightened attention towards child abuse issues. These statistics paint a distressing picture of the scale of child abuse in Malaysia, underscoring the urgent need to address this pressing social problem. They emphasise the importance of conducting further research and studies within Malaysia to identify the underlying factors contributing to child abuse and comprehend its profound impact on children.

Fig. 1
figure 1

Source: Department of Social Welfare, Malaysia

Cases of Child abuse in Malaysia from 1989 to 2019.

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3 Theoretical framework

The relationship between inequality and per capita income is captured by the Kuznet Curve, initially proposed by Simon Kuznet in 1950. Shaari et al. [5] have expanded upon this curve to explore the connection between child abuse and per capita income. According to the curve, these variables demonstrate an inverted U-shaped relationship during periods of economic prosperity. In the early stages of development, as per capita income rises, there may be an increase in child abuse cases due to societal shifts like urbanisation and migration that disrupt established support networks. Economic inequality and imbalances can also contribute to social unrest and elevate the risk of child abuse. However, as economic development progresses and per capita income increases, child abuse tends to decline. This decrease can be attributed to reduced family stress and improved social, medical, and educational access.

Child abuse is influenced by a complex interplay of individual, familial, environmental, and social risk factors, as elucidated by developmental ecology and ecological-transactional models. These models underscore the intricate interactions required to comprehend the factors contributing to and influenced by child abuse. These models highlight the economic circumstances of a family as a critical determinant [10, 11]. Family income serves as an indicator of the financial resources available to meet children’s needs. Parents under increased stress and strain are more susceptible to engaging in abusive behaviours, often driven by inadequate income. Likewise, parents’ employment status greatly influences the family's financial stability and opportunities. Financial strains and disrupted family dynamics stemming from unemployment or precarious employment can heighten the risk of abuse.

As a result, economic pressures such as low incomes, job losses, rising living costs, and economic downturns may be linked to parental stress, inadvertently impacting vulnerable children. According to the family stress hypothesis, stress can be transmitted from parents to children due to economic variables such as poverty, unemployment, inflation, and economic growth. This understanding emphasises the importance of examining economic circumstances and their impact on family dynamics when analysing risk factors for child abuse.

The family stress theory, originally proposed by Hill [12], underscores the challenges families face in response to various stressors. Within this hypothesis, the interparental relationship and its interaction with economic pressure play a significant role in children's well-being. Economic pressure can manifest in various forms, including job losses (unemployment), declining income, and increasing living expenses. Stress arising from financial difficulties is a common precursor to domestic violence, including mistreating children at home [13, 14]. The concept also considers the impact of environmental influences and changes in people's lifestyles. For example, when both partners become more engaged in the workforce and prioritise their careers due to increased job demands, altered work schedules, or an imbalance in work-life equilibrium, it can strain personal relationships and the household as a whole [15]. Family stress theory suggests that these economic, social, and family dynamic shifts can create tension at home, thereby increasing the risk of child abuse. Since the family stress theory focuses on the psychological consequences of inadequate resources, therefore it is very reliable to adopt in the study as the psychological would drive the actions of individuals including child abuse cases [8].

4 Literature review

A limited number of past research using econometric approaches, such as Shaari et al. [5], Shaari et al. [16], Rambeli et al. [17], Shaari et al. [18] and so forth have identified numerous factors contributing to child abuse,. In Malaysia, Shaari et al. [19] first investigated the relationship between macroeconomi variables, such as inflation and unemployment, and child abuse. The study employed the Johansen cointegration, Vector Error Correction Model (VECM) and Granger Causality to examined the linkage among inflation, unemployment and child abuse in Malaysia, covering data period 1982–2011. The findings from the VECM approach show that unemployment can lead to child abuse. The consistent findings also found from the Granger causality which indicate that unemployment has a Granger cause child abuse. In addition to the findings, no connection between inflation and child abuse was found in Malaysia.

Then, Shaari et al. [16] again investigated various macroeconomic factors, including unemployment, inflation and real GDP. The study used a different range of data from 1989 to 2917, employing a different method, namely the ARDL approach, and found that inflation, real GDP and unemployment have no short-run impacts on child abuse. In the long, on the other hand, those factors can result in higher child abuse cases. Shaari et al. [5] extended the study by delving into the linkage between GDP per capita and child abuse in Malaysia. The Kuzents curve was used to serve as a building block for the study, using data from 1988 to 2018. With the employment of the ARDL approach, the study found that there are two stages of the relationship between the two variables: first, when GDP per capita escalates, child abuse increases, and second, when GDP per capita continues to increase, child abuse starts to drop. In addition to the findings, inflation was also found to contribute to child abuse.

Khan et al. [20] examined the relationship between economic growth and child labor in Pakistan, focusing on how factors like exports, household size, and rural population affect child labor. Using data from 1972 to 2021, the study employed the ARDL method. The findings revealed an inverted U-shaped Kuznets curve, showing that child labor initially increases during early economic development due to low household income but decreases in the long run as economic growth raises household income levels. Additionally, exports, larger household sizes, and rural populations were found to contribute positively to higher child labor rates.

Instead of the ARDL approach, Rambeli et al. [17], employed the Ordinary Least Square (OLS) analysis to support the aforementioned findings that income and inflation can be detrimental to child abuse. However, they added another important variable, specifically the female labour force, and found a positive relationship between the variable and child abuse. Shaari et al. [18] also supported the positive connection between women in the workforce and child abuse in Malaysia. Using the ARDL technique and covering 24-year data from 1990 to 2014, they found that as the economy grew, child abuse decreased, likely due to the challenges faced by working women in balancing their personal and professional lives. The study also observed a positive correlation between inflation and child abuse.

Lefebvre et al. [21] used a multi-stage sampling design, selecting 17 child welfare sites from 46 organizations in Ontario. Data were collected from child welfare workers during 5,265 investigations conducted in 2013. Investigations with missing socio-economic data or involving community caregivers were excluded, leaving 3,790 cases for analysis. Weighted descriptive and chi-square analyses were conducted to explore socio-economic conditions, while logistic regression on the unweighted sample examined the relationship between economic hardship and substantiated child maltreatment. The analysis found that 9% of households investigated had run out of money for necessities like food, housing, or utilities in the past six months. Children in these households were more likely to experience developmental and academic challenges, along with caregivers facing mental health or substance use issues. Even after accounting for clinical and case characteristics, economic hardship nearly doubled the likelihood of a substantiated maltreatment investigation.

Wong et al. [22] perspective examined how job loss, income reduction and parenting practices can affect child abuse in Hong Kong during the period when the world experienced the COVID-19 pandemic. A cross-sectional online survey was conducted with 600 randomly selected parents living with children under 10 years old. The survey was completed in 2020. The study revealed that income reduction and job loss were significantly associated with children being physically abused. Parental exposure to intimate partner violence was strongly associated with all forms of child maltreatment.

Conrad-Hiebner and Byram [23] used a systematic review to analyze the relationship between familial economic insecurity and child maltreatment based on 26 longitudinal studies published between 1970 and 2016, focusing on variables such as income, unemployment, material hardship, and income transfers. The review found that economically insecure children experience 3–9 times more maltreatment than their secure counterparts, with income losses, cumulative material hardship, and housing hardship emerging as the most consistent predictors. While economic insecurity is linked to physical abuse, neglect, and psychological maltreatment, the evidence lacks causal clarity and consistent replication.

Lindo 1t al. [24, 25] examined the relationship between economic downturns and child abuse in California using county-level data from 1996 to 2009. The study investigated how deviations in economic conditions influence reported child abuse rates while controlling for statewide annual shocks. The findings revealed that overall economic conditions are not strongly linked to child abuse rates. However, a deeper analysis shows opposing effects based on the gender of layoffs: male layoffs increase child abuse rates, whereas female layoffs reduce them. These findings align with family-time-use models, suggesting that variations in caregiver presence impact child abuse risks.

.Yob et al. [9] examined the relationship between economic and social factors (divorce, unemployment, poverty, inflation, and economic growth) and child abuse cases in Malaysia using data from 1989 to 2019 and applying the ARDL approach. The findings revealed that, in the long run, unemployment, inflation, and economic growth significantly influence child abuse cases, while divorce and poverty do not. In the short run, divorce, unemployment, and economic growth increase reported child abuse cases, whereas poverty shows a negative relationship. The results suggest that family stress caused by economic and social challenges can increase the risk of child abuse.

McLeigh et al. [26] examined how neighborhood poverty influences child abuse and neglect rates in South Carolina, focusing on whether social cohesion (trust and shared expectations among neighbors) mediates this relationship. Using data from a survey of 483 caregivers with children under 10, substantiated reports of child maltreatment, and Census block group data, the study found that poverty directly increases rates of child abuse and neglect. Further analysis revealed that social cohesion mediates the link between poverty and abuse rates, but not neglect rates.

Lindo and Schaller [27] explored how economic factors, such as income, employment, overall economic conditions, and welfare, influence child maltreatment rates. The study reviewed two common research approaches: experimental methods that examine how changes in family income affect the likelihood of individual maltreatment, and area studies that assess how local economic conditions impact overall maltreatment rates. The results suggested that economic stability, both at the family and community level, plays a crucial role in reducing child abuse and neglect. Kim et al. [28] also conducted a systematic review to observe the link between poverty and child maltreatment in the United States, using data from 27 studies. About 40% of the studies measured poverty based on income, while 71% relied on administrative data to assess child maltreatment. The findings consistently show that higher income reduces the risk of child maltreatment, while families living in poverty face a higher risk.

5 Methodology

5.1 Variable description and data source

Table 1 provides a detailed overview of the key variables examined in the study, elucidating their definitions, proxies, and the respective sources of data. The first variable, child abuse, is measured by the number of reported child abuse cases, with data sourced from the Department of Social Welfare in Malaysia. This variable serves as a critical metric for understanding the prevalence of child abuse in the country. Poverty, the second variable, is represented by the incidence of absolute poverty, offering insights into the proportion of the population living below the defined poverty line. Data for poverty is collected from the Department of Statistics in Malaysia, ensuring accuracy and consistency in measurement. The third variable, unemployment, is gauged by the number of unemployed people, with data sourced from the World Bank, providing a global perspective on unemployment trends in Malaysia. Lastly, Economic Growth is measured by GDP per capita, with the World Bank serving as the data source. This study analyzes data spanning from 1989 to 2020, offering a comprehensive examination of trends and patterns over three decades. The 31-year data period was chosen based on data availability. Narayan [29] demonstrated that a 30-year dataset is sufficient for cointegration analysis, as it calculates critical values specific to a sample size of 31 observations and shows they are significantly higher than those reported for larger sample sizes in previous studies, validating the robustness of the bounds testing approach to cointegration for small datasets. According to Daly et al. [30], the ARDL method is very flexible and works well even with small data sets. This study offers insights into the overall economic health and standard of living in Malaysia. The meticulous documentation in Table 1 not only enhances the transparency and reliability of the study but also provides a comprehensive reference for researchers, policymakers, and readers seeking to understand the impacts of poverty, unemployment, and economic growth on child abuse in the Malaysian context.

Table 1 Variable descriptions
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Figure 2 shows our research framework for this study aims to explore the intricate relationships between key economic indicators and child abuse. The independent variables under scrutiny include poverty, unemployment, and economic growth, while the dependent variable is child abuse. Visually represented in the conceptual model, arrows connect each independent variable to child abuse, signifying the anticipated directional influences. The framework is theoretically grounded, drawing on relevant theories that elucidate how economic factors may impact child abuse dynamics. It takes into account contextual factors, recognizing potential variations influenced by societal, cultural, or regional nuances. The overarching research questions inquire into the correlation between increased poverty, unemployment rates, economic growth, and higher instances of child abuse. Methodologically, the framework hints at the data collection and analysis approaches, including potential control variables or moderating factors. If the study involves a temporal element, the framework may incorporate a sequential aspect to capture changes over time. In essence, this research framework serves as a visual guide, providing a structured foundation for investigating the complex relationships between economic indicators and child abuse.

Fig. 2
figure 2

Research framework

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5.2 Research technique

Common methods for estimating symmetric correlations in economics include Granger causality, OLS (ordinary least squares), and quantile regression. However, these approaches may miss some instances of asymmetry. To overcome this shortcoming, the non-linear autoregressive distributed lag (ARDL) method has grown in favour. In addition to generating coefficient estimates for both short- and long-term relationships, the non-linear ARDL method is also suitable for use with relatively small sample numbers [30]. Using data from 1989 to 2020, this study uses the non-linear ARDL model to examine the connection between economic growth, unemployment, and child abuse. Child abuse (CA), poverty (POV), unemployment (U), and gross domestic product (GDP) per capita data were obtained from the Malaysian Department of Social Welfare, countryeconomy.com, and the Department of Statistics Malaysia.

Prior to data analysis, we conduct descriptive statistics to gain insights into the data's characteristics. Following this, the unit root test is conducted to determine the order of integration of the variables. Then, the Ordinary Least Squares (OLS) method is utilized in the non-linear Autoregressive Distributed Lag (ARDL) method to estimate the cointegration relationship and identify the optimal lag order. This method allows for various orders of integration, including I(0), I(1), and mixed orders. The ARDL and non-linear ARDL (NARDL) approaches are particularly valuable for small datasets with limited observations. The NARDL method enables accurate long-term approximations and econometric forecasting, incorporating short-term corrections to maintain long-run data relationships. These distinctive features enhance the NARDL method's efficacy in empirical analysis.

The model utilised in this study intends to perform a relationship analysis and is represented as follows:

$${\text{CA}} = \, \smallint \left( {{\text{POV}}_{{\text{t}}} ,{\text{ U}}_{{\text{t}}} ,{\text{ GDP}}_{{\text{t}}} } \right)$$
(1)

where: CA represents child abuse, POV and U represent poverty and unemployment, respectively, and GDP denotes GDP per capita.

All variables are transformed using the natural logarithm. Logarithmic transformations simplify the interpretation and analysis of variable relationships by converting complex patterns into linear connections. They also address skewness in variables, achieving a more normal distribution, and stabilize variances, reducing overall variability. Furthermore, logarithmic transformations can help manage potential measurement errors from demographic changes over time. Hence, Eq. (1) is logarithmically transformed into:

$$LNCA={\beta }_{0}+{\beta }_{1}{LNPOV}_{t}+{\beta }_{2}{LNU}_{t}+{\beta }_{3}{LNGDP}_{t}+{\varepsilon }_{t}$$
(2)

where; LNCA, LNPOV, LNU AND LNGDP represent the logarithm form of child abuse and the explanatory factors, namely poverty, unemployment, and GDP respectively. This relationship between these variables is examined through regression analysis. The regression equation reveals the unbalanced long-term relationship between child abuse and the mentioned explanatory variables. β0 is the intercept and \({\varepsilon }_{t}\) denotes the error term. The coefficients β1, β2, and β3 represent the correlations between poverty, unemployment, and GDP with child abuse, respectively. These coefficients reflect the relationships and magnitudes of influence between the explanatory factors and child abuse in the regression model.

Following the linear ARDL model in Eq. (2), changes in the dependent variable are expected to respond symmetrically to fluctuations in independent variables. However, after evaluating the integration sequence, we opt to employ the NARDL approach developed by Shin et al. [7] to integrate the directional asymmetrical immediate and long-term effects of poverty, unemployment and economic growth on child abuse. Thereby expanding Eq. (2) to a simplified NARDL specification as:

$$LNCA= {\uptheta }_{0}+{\uptheta }_{1}{LNPOV}_{t}^{+}+ {\uptheta }_{2}{LNPOV}_{t}^{-}+ {\uptheta }_{3}{LNU}_{t}^{+}+ {\uptheta }_{4}{LNU}_{t}^{-}+{\uptheta }_{5}{LNGDP}_{t}^{+}+{\uptheta }_{6}{LNGDP}_{t}^{-}+ {\varepsilon }_{t}$$
(3)

where \({\uptheta }_{0}\) is the intercept, \({\uptheta }_{1},{\uptheta }_{3}\), and \({\uptheta }_{5}\) are the coefficients of the positive changes in poverty, unemployment and GDP, respectively, and \({\uptheta }_{2, }{\uptheta }_{4}\), and \({\uptheta }_{6}\) are the coefficients of the negative changes of poverty, unemployment and GDP respectively while \({\varepsilon }_{t}\) is the white noise.

Subsequently, in Eqs. (4) and (5), the shocks in the independent variables (LNPOV, LNU, and LNGDP) are disaggregated into positive and negative effects on the dependent variable (LNCA). These effects are represented as \({Z}_{t}^{+}\) and \({Z}_{t}^{-}\), respectively. The impact of the shock from the independent variables, categorised as both positive and negative, on the explained variable is given as follows:

$${Z}_{t}^{+}= \sum_{i=1}^{t}\Delta {Z}_{t}^{+}= \sum_{i=1}^{t}{max\Delta Z}_{i},0$$
(4)
$${Z}_{t}^{-}= \sum_{i=1}^{t}\Delta {Z}_{t}^{-}= \sum_{i=1}^{t}{min\Delta Z}_{i},0$$
(5)

Where Z represents the explanatory variables (LNPOV, LNU, and LNGDP), the asymmetric autoregressive distributed lag (ARDL) model is employed after validating co-integration. This non-linear ARDL model analyses the connection between the dependent and independent variables using positive and negative coefficients. The coefficients below describe how the independent variables react to positive and negative inputs.

Considering the lagged values or changes in these variables, Eq. (3) is expanded to provide a more comprehensive NARDL model that incorporates lagged variables and their changes to capture the dynamic effects and time dependencies in the data and it is represented as:

$$\begin{aligned} \Delta LNCA_{t} = & \uptheta _{0} + \uptheta _{1} LNCA_{{t - 1}} + \uptheta _{2} LNPOV_{{t - 1}}^{ + } + \uptheta _{3} LNPOV_{{t - 1}}^{ - } + \uptheta _{4} LNU_{{t - 1}}^{ + } + \uptheta _{5} LN{\text{U}}_{{t - 1}}^{ - } \\ & + \uptheta _{6} LNGDP_{{t - 1}}^{ + } + \uptheta _{7} LNGDP_{{t - 1}}^{ - } + \sum\limits_{{i = 1}}^{m} {\delta _{{1i}} } \Delta LNCA_{{t - i}} \\ & + \sum\limits_{{i = 0}}^{n} {\delta _{{2i}} } \Delta LNPOV_{{t - i}}^{ + } + \sum\limits_{{i = 0}}^{o} {\delta _{{3i}} } \Delta LNPOV_{{t - i}}^{ - } + \sum\limits_{{i = 0}}^{p} {\delta _{{4i}} } \Delta LNU_{{t - i}}^{ + } \\ & + \sum\limits_{{i = 0}}^{q} {\delta _{{5i}} } \Delta LNU_{{t - i}}^{ - } + \sum\limits_{{i = 0}}^{r} {\delta _{{6i}} } \Delta LNGDP_{{t - i}}^{ + } + \sum\limits_{{i = 0}}^{s} {\delta _{{7i}} } \Delta LNGDP_{{t - i}}^{ - } + \mu _{i} \\ \end{aligned}$$
(6)

In Eq. (6), the coefficients \({\uptheta }_{1}\) to \({\uptheta }_{7}\) and \({\delta }_{1}\) to \({\delta }_{7}\) represent the long-run and short-run impacts of the independent variables (LNPOV, LNU, and LNGDP) on child abuse, respectively. The lag orders are denoted by values from m to s. In the NARDL framework, including lagged values is particularly important because the model aims to capture the asymmetric effects of variables, which may be better captured by considering how past values of variables influence current outcomes.

To evaluate the asymmetric response (LNCA) of the dependent variable to positive and negative shocks in the explanatory variables like LNPOV, LNU, and GDP, we must first build the short-term and long-term connections in the non-linear ARDL model. Equations (7), (8), and (9) are used to complete this evaluation; we utilise the asymmetric dynamic multiplier technique to derive the results.

$${P}_{q}^{+}\left(LNPOV\right)=\sum_{j=0}^{q}\frac{{\partial LN\text{CA}}_{t+1}}{{\partial LNPOV}_{t-1}^{+}}, {P}_{q}^{-}\left(LNPOV\right)=\sum_{j=0}^{q}\frac{{\partial LNCA}_{t+1}}{{\partial LNPOV}_{t-1}^{-}}$$
(7)
$${P}_{q}^{+}\left(LNU\right)=\sum_{j=0}^{q}\frac{{\partial LN\text{CA}}_{t+1}}{{\partial LNU}_{t-1}^{+}}, {P}_{q}^{-}\left(LNU\right)=\sum_{j=0}^{q}\frac{{\partial LNCA}_{t+1}}{{\partial LNU}_{t-1}^{-}}$$
(8)
$${P}_{q}^{+}\left(LNGDP\right)=\sum_{j=0}^{q}\frac{{\partial LN\text{CA}}_{t+1}}{{\partial LNGDP}_{t-1}^{+}}, { P}_{q}^{-}\left(LNPOV\right)=\sum_{j=0}^{q}\frac{{\partial LNCA}_{t+1}}{{\partial LNGDP}_{t-1}^{-}}$$
(9)

In the final stage of the study, diagnostic tests are employed to evaluate the accuracy of the non-linear ARDL model's analysis of short- and long-term associations. Different tests serve various purposes in this evaluation. The utilization of the CUSUM as well as CUSUM of square tests helps to assess the model stability. These tests help determine if the model exhibits any instability over time. The Breusch-Godfrey serial correlation LM test is employed to detect the presence of serial correlation in the model, which can affect the reliability of the results.

To assess the normality of the data, the Jarque–Bera test is applied. This test examines whether the data follows a normal distribution. The Breusch-Pagan-Godfrey test is suitable to identify heteroskedasticity, which is the presence of non-constant variance in the data. The Ramsey RESET test is employed to detect any specification errors in the regression, ensuring the model is correctly specified. These diagnostic tests are crucial for ensuring the reliability and accuracy of the conclusions drawn from the non-linear ARDL model. They help identify potential issues or biases in the model's estimation and provide confidence in the robustness of the results.

6 Results

6.1 Descriptive statistics results

The descriptive results of the variables before employing the NARDL approach reveal key insights into the characteristics of the dataset as presented in Table 2. The dataset consists of 32 observations for four variables: LNCA, LNPOV, LNU, and LNGDP. In terms of real LNGDP, the mean and median values stand at 10.2205 and 10.2223, respectively, with a slightly negatively skewed distribution. LNPOV exhibits a mean of 11.8693 and a median of 11.7822, showing a slightly positively skewed distribution. LNCA has a mean of 7.5096 and a median of 7.4122, displaying a distribution that is slightly negatively skewed. LNU has a mean and median of 12.7718 and 12.8120, respectively, with a slightly negatively skewed distribution. Across all variables, there are variations in the range, standard deviation, skewness, and kurtosis, suggesting diverse patterns within the dataset. These descriptive statistics provide a foundational understanding of the dataset's central tendencies and distributional characteristics, laying the groundwork for the subsequent application of the ARDL approach.

Table 2 Descriptive statistics
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6.2 Unit root test results

The Augmented Dickey-Fuller (ADF) test was employed in this study to assess the stationarity of the data. We have considered two sets of the ADF unit root test at intercept as well as at intercept and trend. Performing these two sets of tests on a sequence provides a more thorough analysis of the data's stationarity properties. This can help in accurately modelling and forecasting the data, as well as ensuring the validity of statistical inferences drawn from the analysis.

Table 3 presents the results of the unit root tests conducted on the relevant variables namely; lnCA, lnPOV, lnU, and lnGDP. The results indicate that at the level, lnCA, lnPOV, lnU, and lnGDP either have a unit root or are non-stationary. However, when an intercept is included, they either become stationary or exhibit a unit root at the first difference. This suggests that these variables have a first-order integration.

Table 3 Unit root test results
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Furthermore, when an intercept and trend are considered, lnCA and lnU are stationary at the level, but all variables become stationary at the first difference. This indicates a mixed order of integration of the variables. It is crucial to note that the ARDL technique relies on stationary variables to establish reliable long-run correlations between the variables of interest. Therefore, these findings hold significance. Based on the unit root test outcomes, which validate the stationarity characteristics of the variables, we can confidently utilise the ARDL methodology to investigate the non-linear connection between GDP per capita and child abuse.

6.3 Bound test results

The test statistics and critical values at different levels of significance are listed in Table 4. The F-statistic value is compared to the crucial values at each significance level to assess the outcomes. We can conclude that there is a long-term correlation if the estimated F-statistic is greater than the crucial values. The estimated F-statistic of 14.6090 exceeds the thresholds established by the various levels of significance. Therefore, the long-term link between the variables is significant, and the alternative hypothesis is accepted. The results of the Bound test indicate a statistically significant relationship between GDP per capita and the incidence of child abuse in Malaysia from 1989 to 2019. The ARDL method's usefulness in estimating and analysing the non-linear relationship between these variables is further supported by the Bound test results.

Table 4 Bound test results
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6.4 Long-run estimation results

Table 5 displays the findings of the non-linear association between socioeconomic status and child abuse. These results provide insights into how macroeconomic factors affect the prevalence of child abuse. LNGDP_POS has a coefficient of 4.2660, which measures expansions in GDP. This indicates that a 1% rise in GDP per capita is connected with an estimated 4.266% rise in child abuse. The p-value of 0.0000 indicates a highly significant positive correlation between GDP growth and increased incidences of child abuse. This highly significant relationship suggests that economic expansion, while generally beneficial, might inadvertently increase the stress on families or amplify societal dynamics that contribute to child abuse, such as shifting family roles, greater expectations, or reporting improvements. In addition, the coefficient for LNGDP_NEG, which stands for GDP declines, is 3.9977. This suggests a negative correlation between GDP per capita and child abuse, with a decrease of 1% in GDP per capita being related to a 3.99% increase in child abuse. There is a statistically significant inverse correlation between economic downturns and an increase in the occurrence of child abuse (p = 0.0220). This highlights that economic contractions also exacerbate the prevalence of child abuse, potentially due to increased financial stress, reduced access to resources, or deteriorating mental health among caregivers. Interestingly, while both positive and negative GDP shocks increase child abuse rates, the magnitude of the effect is slightly higher for economic expansion. This suggests that in the long run, the stresses and societal changes tied to growth might have a more pronounced influence than the struggles during downturns.

Table 5 Long-run estimations
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Looking at the unemployment (lnU) variable, we find a positive association between long-term increases in child abuse and increases in unemployment (coefficient for LNU_POS = 1.1177). Job losses may directly strain families and create environments where abuse becomes more likely. This relationship is statistically significant, with a p-value of 0.0000, emphasizing the importance of addressing unemployment to reduce child abuse risks. On the other hand, negative changes in unemployment are represented by a coefficient of -2.9350 for LNU_NEG. This indicates that a long-term decrease in unemployment of 1 percentage point is connected with an estimated 2.935% drop in child abuse. With a p-value of 0.0000, there is a statistically significant link between unemployment and child abuse. This negative relationship highlights the protective effect of stable employment, which can provide families with financial security and emotional stability. The greater magnitude of the reduction compared to the increase suggests that job creation and economic stability can have a substantial impact on reducing child abuse over time.

Regarding poverty (lnPOV), the coefficient for LNPOV_POS, which represents positive changes in poverty, is − 0.1813. This implies that a 1% increase in poverty is associated with an estimated 0.1813% decrease in child abuse over time. This relationship can be interpreted through the lens of opportunity cost for working parents. When both parents work to cope with the rising cost of living, they may face challenges in balancing work and family responsibilities. However, if one parent, such as the mother, chooses to leave the workforce to focus on caregiving, the household may lose additional income, but the children could receive more direct attention, reducing the likelihood of neglect and abuse. This is consistent with the findings of Paxson and Waldfogel [31], which indicate that an increase in the number of children with absent fathers or working mothers is associated with a rise in cases of child mistreatment. The inverse association is statistically significant at the 5% level (p = 0.0423), suggesting that measures aimed at reducing poverty may positively affect child abuse rates. However, the coefficient for LNPOV_NEG, representing negative changes in poverty, is 0.0874, but it is not statistically significant. Therefore, no definitive conclusions can be drawn regarding the impact of decreasing poverty on the prevalence of child abuse. This indicates that while poverty alleviation is essential, its direct long-term impact on child abuse may be more complex and influenced by additional factors like social support systems, community resources, and family dynamics.

6.5 Short-run estimation results

Our results reported in Table 6 reveal that an increase in GDP (lnGDP_POV) is associated with an estimated 8.4902 percentage point surge in child abuse rates in the short run. The short-term influence of economic expansion on child abuse rates is highlighted by the statistical significance of this link at the 1% level (p = 0.0027). We estimate a direct 8.4902 percentage point increase in child abuse rates for every percentage point increase in GDP per capita. This finding highlights how economic growth, while generally beneficial, can sometimes bring challenges like increased stress on families or more frequent reporting of child abuse. As people work longer hours or face greater job demands during economic booms, family relationships can become strained, making it harder to provide stable and nurturing environments for children. At the same time, economic growth often improves systems for detecting and reporting child abuse, which can make the problem appear more widespread even if actual incidents have not increased. These factors show that while economic expansion brings opportunities, it also requires attention to its social impacts to ensure families and children are supported. However, the correlation for LNGDP_NEG is − 5.4257, indicating that a drop in GDP per capita is linked to an immediate 5.4257% drop in child abuse rates. This correlation is statistically significant at the 1% level (p = 0.0033), highlighting the short-term negative impact of economic downturns on child abuse rates. Short-term reductions in child abuse rates of 5.4257% are predicted for every 1% drop in GDP per capita. This suggests that during economic downturns, there might be a temporary decline in reported child abuse cases, but this doesn’t necessarily mean that actual incidents decrease. One possible reason is underreporting—families facing financial stress may become less likely to report abuse due to fear of losing resources, stigma, or focusing on immediate survival needs. Schools, healthcare providers, and other institutions that often identify and report abuse might also have reduced interactions with children during recessions, especially if parents withdraw children from school or delay medical visits to save money.

Table 6 Short-run estimations
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A rise in unemployment (lnU_POV) is insignificantly associated with an increase in the child abuse rate. However, the coefficient for LNU_NEG is − 2.4748 which is statistically significant at the 1% level (p = 0.0023). This implies that child abuse rates are predicted to drop by an estimated 2.4748% for every 1% drop in the unemployment rate (expressed as a percentage of the population under 18). This result shows that lower unemployment can have a protective effect on families, as it often leads to greater household stability and improved access to resources. When parents are employed, they typically have a more stable income, which reduces financial stress and allows them to provide better for their children’s needs. Stable jobs also offer a sense of security and routine, which can positively influence family relationships and reduce the likelihood of conflict or abusive behaviours.

The coefficient for LNPOV_POS is − 0.3530 which indicates that a 0.3530 percentage drop in child abuse rates follows an immediate rise in poverty. At the 5% level of significance (p = 0.0192), the coefficient indicates that alleviating poverty can quickly reduce child abuse rates. Although this finding might seem counterintuitive, it could reflect short-term changes in how families behave or how child abuse is reported during times of economic stress. When poverty increases, families may become more focused on immediate survival, such as securing food, shelter, and other basic needs. This focus might temporarily reduce the likelihood of abusive behaviours or even alter the dynamics within the household, as family members work together to navigate tough financial times. We estimate a 0.35 percentage drop in child abuse rates for every percentage increase in poverty. However, the correlation for LNPOV_NEG is − 0.3553, indicating an instantaneous increase in child abuse rates related to a reduction in poverty. However, the p-value for this correlation is insignificant (0.1412), suggesting that a negative shock in poverty may have little effect on fluctuations in child abuse over the near term. This insignificance suggests that reductions in poverty might not have an immediate or predictable effect on child abuse rates in the short run.

6.6 Diagnostic tests results

The diagnostic tests conducted in this study, as presented in Table 7, were aimed at assessing the adequacy of the model and the fulfilment of specific statistical assumptions. The results indicate that the model is properly specified and meets the criteria of no significant serial correlation, no significant heteroskedasticity, reasonably normal error terms, and no clear evidence of model misspecification. These findings support the accuracy and reliability of the estimated relationships and coefficients in the model.

Table 7 Diagnostic tests results
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6.7 Statibility tests results

The stability of the estimated model coefficients is supported by the results of the CUSUM and CUSUM of Squares tests, as illustrated in Fig. 3. The blue line representing the cumulative sum or cumulative sum of squares of the coefficients falls within the red lines, indicating that the association between the variables remains valid throughout the entire study period. This consistency and stability in the coefficients, as confirmed by the CUSUM and CUSUM of Squares tests, enhance the reliability of the regression analysis results and provide further evidence of the accuracy of the estimated correlations.

Fig. 3
figure 3

CUSUM and CUSUM of squares

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Figure 4 presents the key findings from each test, including unit root tests, the bounds test, long-run estimation tests, short-run estimation tests, diagnostic tests, and stability tests.

Fig. 4
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The key findings of each test

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7 Discussions

Our findings indicate that higher economic growth is associated with an increase in child abuse cases, a trend that is observable in Malaysia's economic cycles. For instance, during the economic downturn in 2020, Malaysia saw a decline in child abuse cases. However, as the economy recovered, incidents of child abuse began to rise. This pattern highlights the complex relationship between economic growth and family well-being. One reason for the rise in child abuse cases during periods of positive economic growth is that such growth is often driven by a workforce intensely focused on their respective industries. Justice and Duncan [32] found that working mothers, burdened by the dual demands of their jobs and household responsibilities, often struggle to balance these roles, leading to significant stress and challenges in managing family life. Additionally, husbands, particularly those in professional roles, who become overly focused on their work often neglect their wives, contributing to family tensions and strained relationships.

The phenomenon of rapid economic growth has far-reaching consequences that extend beyond financial prosperity, impacting societal structures and expectations. As economies expand, societal norms often shift, and the pressure to achieve and maintain an elevated standard of living intensifies. This growing pressure can place considerable strain on families, particularly in urban areas of Malaysia, where households face escalating living costs. In this environment, families may struggle to keep up with the pace of economic changes, leading to significant financial stress.

This stress acts as a catalyst for exacerbating pre-existing tensions within families, forming a volatile mix that can contribute to increased child abuse. The theory of family stress, which suggests that families experience heightened strain when faced with economic challenges, provides a useful lens for understanding this phenomenon. According to this theory, economic strain disrupts family functioning, leading to frustration, emotional distress, and, in some cases, abusive behaviours. As living costs rise and parents struggle to meet financial demands, they may become overwhelmed, and this frustration can manifest in negative ways, including child abuse.

The inability to meet rising societal and financial expectations can create a sense of desperation, which further exacerbates familial tensions. In this context, the rapid pace of economic growth, while bringing financial opportunities to some, can also increase the vulnerability of families already under strain. Policymakers must recognize these dynamics and address the broader social repercussions of economic growth, ensuring that support systems—such as mental health resources, financial aid, and parenting programs—are in place to help families navigate the pressures of economic change. By doing so, we can mitigate the negative impact of economic growth on family stability and child well-being.

Understanding these complex dynamics is imperative for developing targeted support systems and policies that address not only the economic facets but also the intricate interplay between societal shifts, financial pressure, and the well-being of families. Our results are consistent with the finding of Yob et al. [9]. Our research builds upon and extends the insights provided by Shaari et al. [5], particularly in the examination of short-run autoregressive distributed lag (ARDL) estimates. This analysis delves into the immediate effects of fluctuations in macroeconomic variables on child abuse rates in Malaysia, thereby illuminating the temporal intricacies of the relationship between GDP per capita and instances of child abuse. In consonance with their findings, this alignment in immediate impact underscores the consistency and robustness of the observed relationship, as corroborated by both studies. However, while Yob et al. [9].exclusively utilized the ARDL framework, our research expands the analytical scope by incorporating situations where economic growth experiences a decline and its consequential impact on child abuse rates. By embracing both sides of the economic spectrum, our study offers a more comprehensive understanding of the dynamics between economic conditions and child abuse rates. This unique facet contributes valuable insights to the existing body of literature, enriching the discourse on the nuanced interplay between economic factors and child welfare.

Our findings reveal a significant association between rising unemployment rates and an increase in the incidence of child abuse over the long term. This aligns with the work of Brown and De Cao [33], who found that every unit increase in the unemployment rate corresponded with a rise in various forms of child abuse, including physical abuse and neglect. Using a non-linear ARDL methodology, we explored the potential effects of reducing unemployment. Notably, our results indicate that as unemployment decreases, incidents of child abuse also decline. This strengthens the nuanced connection between fluctuations in unemployment and child abuse, highlighting the need for policymakers to consider strategies that address the broader social consequences of unemployment.

In the Malaysian context, this issue is particularly pronounced among certain groups. A substantial portion of child abuse cases involve unemployed women, especially in rural or economically disadvantaged areas, where unemployment rates tend to be higher, and access to social support or mental health services is limited. For example, the case in Kampung Menunggui, Kota Belud, Sabah, underscores this connection: an unemployed mother, likely experiencing significant financial strain and stress, was involved in a child abuse incident. This case illustrates the vulnerability of families in areas with higher unemployment, where limited access to support systems exacerbates the risks of child abuse.

The theory of family stress offers a useful framework for understanding these dynamics. According to this theory, economic stress—such as unemployment—can disrupt family functioning, leading to heightened tension and emotional strain. This stress, if not managed effectively, can contribute to negative outcomes, including child abuse. In this context, the stress of unemployment, coupled with the lack of financial resources and social support, can overwhelm parents, particularly mothers, leading to frustration and potentially abusive behaviours toward children.

The findings suggest that efforts to reduce unemployment could have a dual benefit: improving economic stability while also decreasing the incidence of child abuse. Policymakers should therefore prioritize addressing unemployment through job creation and social welfare programs, particularly in rural and disadvantaged areas, to reduce the economic pressures that contribute to family stress and, by extension, child abuse. Additionally, strengthening access to mental health services and community support networks could help mitigate the effects of unemployment on family dynamics, offering families the tools to manage stress in healthier ways.

Some past studies asserted that poverty is a causal factor for child abuse [9, 33], we however find a link between poverty and reduced instances of child abuse. The pressure to escape poverty, especially in urban Malaysia where the cost of living is high and rising, can exacerbate stress. This stress might come from working multiple jobs, long hours, and the anxiety about providing for one's children. If this stress is not managed properly, it could lead to negative consequences, including the potential for abuse. In the Malaysian context, the drive for upward mobility is strong, but it may come with mental health challenges, especially for those who are juggling multiple responsibilities to improve their socioeconomic standing.

These results highlight the complex relationship between economic factors and child abuse. While economic growth may increase the risk of abuse due to various stressors, factors associated with poverty, such as social support and community cohesion, may act as protective factors. Understanding these dynamics is crucial for developing effective interventions to prevent child abuse in Malaysia and other similar contexts. The implication of our results underscores the necessity for policymakers to adopt a comprehensive approach when formulating policies. While acknowledging the association between poverty and child abuse, it is equally essential to recognize that a mere reduction in poverty may not be a panacea for mitigating child abuse. Policymakers should consider both situations—addressing the challenges posed by poverty and understanding the nuanced dynamics surrounding poverty reduction initiatives. This nuanced understanding is critical for the development of effective policies that comprehensively address the complex interplay between socioeconomic factors and child welfare.

8 Conclusions and policy implications

Over the period between 1989 and 2020, this study looks into the correlation between poverty, GDP growth, unemployment, and child abuse in Malaysia. The results of this non-linear ARDL analysis show that GDP, unemployment, and poverty all substantially impact the prevalence of child abuse. Child abuse rates tend to climb in tandem with GDP growth over the long term, and they may rise again during economic downturns.

To effectively address child abuse from an economic development perspective, policymakers must account for the complex relationship between economic factors and abuse rates. This study highlights that economic prosperity, as measured by GDP growth, is linked to a short-term increase in child abuse rates, while economic downturns are associated with a decrease in abuse. Unemployment, on the other hand, contributes to long-term increases in child abuse, although reductions in unemployment significantly lower abuse rates in the short term. Interestingly, poverty shows an indirect relationship with child abuse, suggesting that it is not poverty itself but the stress associated with economic fluctuations and unemployment that aggravates the issue. To mitigate these effects, targeted interventions are crucial. Employers must provide consistent support to working parents through flexible work arrangements and access to mental health services, with policymakers incentivizing family-friendly workplace policies to reduce stress-induced abusive behaviour. For unemployed parents, reducing strain through inclusive social narratives, improved access to mental health and social support services, and job creation initiatives, especially in underserved areas, is essential. Vocational training programs can also help alleviate economic pressures on families. Comprehensive poverty alleviation programs should incorporate community-based support systems that provide emotional and financial resources to families, ensuring these efforts are sustainable and address broader social determinants of child welfare. Additionally, public awareness campaigns and accessible support services must be emphasized to help families know where to seek assistance, with partnerships between local communities and NGOs enhancing the reach and impact of these services. While these interventions address immediate stressors faced by parents, it is equally important to acknowledge the limitations of this study and explore further research to strengthen these findings. Future research may look into geographical peculiarities in the frequency of child abuse and the effects of specific economic problems. Thirdly, the study looks at the total rates of child abuse without making a distinction between other types of abuse, including physical, sexual abuse or emotional abuse. Future studies could focus on these areas to better understand the connection between these particular abuse types and financial worries.