Analyzing the impact of remittance, FDI and inflation rate on GDP: A comparative study of Bangladesh, Pakistan and Sri-Lanka using VAR and BEKK-GARCH approach

Abstract

This study examines the impact, conditional correlation and volatility spillover effect of remittances, foreign direct investment and inflation rate on GDP in Bangladesh, Pakistan, and Sri-Lanka, three Asian nations that are particularly vulnerable. While numerous studies have examined the relationship between remittances, FDI and IR on GDP but this paper approaches advanced econometric techniques to capture spillover effect and the dynamic interactions between the variables. For estimation purposes the study employs different econometric techniques such as Augmented Dickey-Fuller (ADF) test, VAR model, Granger causality tests, Impulse Response function, Variance Decomposition and BEKK-GARCH model. Bangladesh and Sri Lanka's REM, FDI and IR have no significant effects on GDP according to the VAR model. BEKK-GARCH demonstrates that three countries have both unidirectional and bidirectional transmissions of volatility, with the exception of Pakistan's REM and Bangladesh's FDI. We find that VAR model may not be adequate in capturing the complex dynamics between variables, which can be better captured by BEKK-GARCH model. Our comparison research shows how these variables affect GDP differently and similarly in each of the three nations, giving policymakers information they can use to create customized policies that encourage economic growth.

1. Introduction

Economic development is a crucial concern particularly for the South Asian vulnerable countries that like Bangladesh, Pakistan and Sri-Lanka. The worst political and economic crisis to affect Sri-Lanka since its independence is currently underway. According to the BBC, other economies in the region are also impacted by the same global challenges including rising inflation and interest rates, declining currencies, high levels of debt and diminishing foreign currency reserves. Circumstances similar to those Sri-Lanka experienced before the economic crisis are being experienced by Pakistan and Bangladesh now. The cost of commodities has skyrocketed in Pakistan, where the annual inflation rate reached a 13-year high of 21.3 % in June 2022 [1]. There are widespread worries that Bangladesh could soon experience an uncontrollable economic crisis similar to that experienced by its neighbors Pakistan and Sri-Lanka as 7.42 percent was an 8-year high for inflation in Bangladesh in May 2022 [2].

A nation's economic development can come from a variety of big and minor areas, one of them being GDP [3]. Governments, economists and companies use it as a crucial indication of a nation's economic performance to determine the state of an economy [4]. Significant external funding and foreign exchange earnings come from remittances and foreign aid [5]. High GDP has been linked in some economies to higher remittances, whereas in others it has been linked to lower remittances. As a result, GDP is different in each economy [6]. Remittance flows may be the source of national funding [7]. According to Chicago University's neoclassical approach, inflation is essentially defined as an economic phenomenon. According to Ref. [8], “Inflation is called by simply the circumstance where a certain amount of production demands higher money to get that".

The GDP of Bangladesh is upward slopping and increasing smoothly, showing significant growth with an average annual growth rate of around 5.5 % [9]. From 2012 through 2020, Bangladesh received an average of $1246.76 million in monthly remittances [10]. FDI inflows into Bangladesh have been increasing, with $2.1 billion received in 2020, a 67 % increase compared to the previous year. The average inflation rate over the period from 1987 to 2021 was 6.4 % per year, with relatively stable rates [11], although some years experienced higher inflation exceeding 10 %. Pakistan experienced low levels of economic growth, averaging around 4 % per year, from the late 1980s to the early 2000s. The World Bank group has observed that the decrease in remittances sent to Pakistan can be attributed to the extensive lockdowns imposed as a result of COVID-19 outbreak [12]. They predict that these financial transfers will decline [13]. Remittances to Pakistan initially grew slowly, averagisng around $1–2 billion per year, but started to grow more rapidly, reaching a peak of over $20 billion in 2019. Pakistan foreign direct investment for 2020 was $2.06B, a 7.92 % decline from 2019. Inflation remained relatively low in the mid-2010s but started to rise again in the late 2010s and early 2020s, reaching a hieght of 10.7 % in 2019. The global financial crisis of Sri-Lanka in 2008–2009 led to a temporary slowdown in GDP growth to around 3–4% per year. In 2019, remittasnces amounted to $6.7 billion, with a significant decrease to $4.9 billion during the COVID-19 pandemic. FDI inflows to Sri-Lanka showed an increasing trend but declined to $450 million in 2020. In 2020, Sri-Lanka was experiencing an inflation rate of 4.6 %, which was higher than that of the previous year.

The current study will critically examine how remittances, foreign direct investment and inflation rate affect GDP. In contrast to earlier researches that exclusively employed VAR models, our methodology integrates the BEKK-GARCH model to account for time-varying volatility, offering a more refined comprehension of the complex interplay between various macroeconomic variables. In order to improve our analysis' accuracy and overcome the limitations of earlier research, this study is crucial. By using BEKK-GARCH in conjunction with VAR, the relationships are seen from a wider perspective and it is shown that BEKK-GARCH is capable of capturing the intricate details of the interdependencies between the variables.

We want to answer important issues like: How do remittances, foreign direct investment and inflation together affect total economic output? What details that conventional VAR models could miss, the BEKK GARCH model might offer? We aim to address the complex linkages and how time-varying volatility influences the impact of these factors by utilizing the BEKK-GARCH model in conjunction with VAR.

The impact of remittances, foreign direct investment and inflation rate on GDP will thus be quantified in this study in a specific sense. The review of several works that are comparable is covered in part two. A few details of the developed model and data were included in the third part which explains the methodological implications in depth. The outcomes of the empirical research are illustrated in detail in part four. Part five is of course, made up of the conclusions and part six policy suggestions.

2. Literature review

2.1. Conceptual review

Drawing from firmly established economic theories which provide theoretical foundations for the links between remittances, foreign direct investment, inflation and GDP is crucial to understanding the complex dynamics of economic development in South Asia.

The Quantity Theory of Money [14] is one such theoretical framework that suggests variations in the money supply might affect inflation rates. This theory is especially pertinent to comprehending how inflation affects the functioning of the economy as a whole and how crucial monetary policy is to containing inflationary forces. Remittance inflows can boost aggregate demand which in turn can boost economic activity according to Keynesian theory [15]. Thus, remittances as a part of aggregate demand can have a big impact on how South Asian nations' economies develop. Mundell-Fleming model [16] clarifies the connection between economic growth, exchange rates, and foreign direct investment. According to this concept, foreign direct investment inflows can result in increased capital accumulation, technological advancements and productivity increases, all of which can propel GDP development. The phenomena of Dutch Disease [17] highlights even more how crucial it is to comprehend how FDI and remittances affect economic development. This theory alerts us to the possible drawbacks of relying too much on FDI or remittances, including currency appreciation and a downturn in other export-oriented sectors that could impede hopes for long-term economic growth.

The selection of Bangladesh, Pakistan and Sri-Lanka as representative nations was based on their regional importance in South Asia and their mutual vulnerability to economic challenges such inflation, depreciation of currency values and debt loads. In addition, the 1987–2021 sample period was chosen in order to include noteworthy historical occurrences and economic patterns that have influenced the course of these nations' economies. The selected time includes significant turning points like the COVID-19 pandemic and the global financial crisis of 2008–2009, which had a significant impact on both regional and global economy.

2.2. Empirical literature

The link between remittances, foreign direct investment, inflation rate, and GDP in various situations and locations has been the subject of much research. Here is a summary of a few of these studies [18]. primarily looked at the connections between imports, foreign direct investment, remittances, and Sri Lanka's GDP. Using the multivariant VAR model, a relationship period of 1977–2020 was determined. According to the estimated results, remittances significantly boost economic growth over the long and short terms. However, there is no connection between economic growth and foreign direct investment [19]. used the Granger causality test and the vector autoregression (VAR) approach to investigate the relationship between foreign direct investment (FDI) and its macroeconomic drivers in Bangladesh from 1975 to 2015. According to the VAR results, Bangladesh's GDP growth rate influences foreign direct investment.In order to determine which model best fits Vietnam's economic growth [20], looked into how the inflation rate affects economic growth. For the time-series data from 1996 to 2018, the study used Vector Autoregressive (VAR), cointegration models, and unit root tests to investigate the short- and long-term effects of inflation on economic growth. In the study the VAR model predicts that the inflation rate is positively correlated with GDP.

[21]examines how foreign direct investment (FDI) has affected Pakistan's employment and economic growth. We used secondary data and time series covering the years 1990–2017. Variable relations between one another were described with their lag values using the ADF test, AR root test, and VAR model. Using GDP as the dependent variable, the analysis reveals that FDI has a negative impact on GDP [22]. demonstrates the causal relationship between Bangladesh's GDP, imports, and exports as well as inflation. This work uses co-integration and the Vector Autoregressive Model (VAR) test to investigate the relationship between export, import inflation, and economic growth. The empirical analysis was carried out between 1982 and 2019 utilizing annual secondary data. There is a strong and positive correlation between GDP and inflation [23]. analyses how foreign direct investments (FDI) and remittances affect Albania's economic expansion. We were able to determine through a VAR that there is no correlation between foreign direct investment and economic growth in Albania. However, there is a substantial correlation between remittances and growth [24]. used ADF test, Granger causality test with VAR framework to check causal relationship between remittance and economic growth in Bangladesh, India, and Pakistan using time series data from 1981 to 2015. In Bangladesh, remittances lead to economic growth, while economic growth does not lead to remittance flow. In India, there is a bi-directional significant link between remittances and economic growth. In Pakistan, there is only a one-way causal relationship, where economic growth leads to remittance growth [25]. examined the impact of FDI, Remittances, Inflation, Interest Rate, and Exchange Rate on the GDP Growth Rate of South Asian Economiesbased on the Co-integration analysis. The study found no long-term relationship between FDI and Remittances to have a significant impact on GDP Growth Rate [26]. investigated the impact of foreign remittances and foreign direct investment (FDI) on economic growth in Pakistan using time series data from 1990 to 2018. The ARDL bound test is employed, with GDP as the dependent variable and foreign remittances and FDI as the independent variables. The results from the Auto Regressive Distribution Lag (ARDL) model indicate a long-run relationship between FDI, foreign remittances, and the economic growth of Pakistan [27]. explored the impact of remittances on economic growth in four South Asian emerging countries using panel data from 1977 to 2016. The study finds a negative effect of remittances on economic growth in Bangladesh, Pakistan and Sri-Lanka, while remittances have a positive impact on economic growth in India [28]. studied the relationship between inflation and GDP in Pakistan using time series annual data from 1980 to 2016. The Auto Regressive Distributed Lag (ARDL) Model is employed to estimate and analyze the data. The findings suggest that inflation has a negative impact on the GDP of Pakistan [29]. examined the relationship between Foreign Direct Investment Inflows (FDI) and Gross Domestic Product (GDP) growth in Bangladesh. The study reveals that FDI has a positive and significant impact on both the long-run and short-run GDP growths of Bangladesh.

The lack of BEKK-GARCH in earlier researches on this particular subject highlights a sizable gap in the literature. By including this comprehensive modeling technique into our comparative analysis, we hope to enhance the methodological resources available to macroeconomic scholars and expand our understanding of how FDI, inflation rate, and remittances affect GDP in the chosen South Asian states. While the use of the VAR model in earlier researches has provided insightful information about concurrent relationships, it is unable to adequately represent the complex dynamics of fluctuating volatility and conditional correlations between the variables of interest. By addressing this gap, we aim to contribute to the existing literature by providing more robust and accurate findings, which can have significant implications for policymakers and researchers in the field of international economics and finance.

3. Methodology

3.1. Data sources

The data we have collected are secondary data; most of them are collected from internet. We have taken the time series data from 1987 to 2021. By classifying the variables as independent and dependent, we may justify and organize the selection process. Remittances, foreign direct investment (FDI), and inflation rate are the three variables that are thought to be independent and that affect GDP. The dependent variable is GDP, which represents the total amount of economic output impacted by these variables. The Global Development Index issued by the World Bank, Macrotrends.net, Bangladesh Bank report, International Monetary Fund, various journals etc. are only a few of the sources that were used to gather the relevant time series data for this study. The summary statistics are reported in the below Table 1.

Table 1.

Summary statistics (1987–2021).

Bangladesh
Variables	Mean	Std.Dev	Variance	Skewness	Kurtosis
GDP	5.37	1.33	1.79	0.27	2.38
REM	5.44	2.59	6.75	0.63	2.01
FDI	0.55	0.51	0.26	0.62	2.32
IR	6.23	2.21	4.88	0.11	2.97
Pakistan
GDP	4.33	1.83	3.34	0.00	2.22
REM	4.63	1.97	3.89	0.25	2.39
FDI	1.00	0.79	0.63	2.24	7.20
IR	8.36	3.82	14.64	0.61	3.87
Sri-Lanka
GDP	4.92	1.83	3.34	0.00	2.22
REM	6.97	1.97	3.89	0.25	2.39
FDI	1.14	0.79	0.63	2.24	7.20
IR	9.08	3.82	14.64	0.61	3.87

In Bangladesh, REM has a higher mean of 5.44 and FDI has a relatively lower mean of 0.55. The standard deviation of REM and IR are quite higher than FDI but it is not so high. The Skewness of GDP being less than zero is negatively skewed and REM, FDI, IR having long-right tail are positively skewed. The kurtosis coefficient of all variables are below 3 that indicates the series is Platykurtic. In Pakisan, IR has the highest mean of 8.36. The standard deviation of IR is quite higher than GDP and REM but it is not so high. The Skewness of GDP being less than zero is negatively skewed and REM, FDI, IR are positively skewed. The kurtosis coefficient of GDP and REM are below 3 that indicates series is Platykurtic. Moreover, the kurtosis of FDI and IR are more than 3, which suggests that the series is fat-tailed and Leptokurtic. In Sri-Lanka, IR has the highest mean of 9.08 and REM has a slightly higher mean of 6.97. The standard deviation of IR is quite higher than GDP and REM but it is not so high. The Skewness of REM being less than zero is negatively skewed and GDP, FDI, IR are positively skewed. The kurtosis coefficient of GDP and REM are below 3 that indicates series as Platykurtic. Moreover, the kurtosis of FDI and IR are more than 3, which suggests that the series is fat-tailed and Leptokurtic.

3.2. Unit root test

Then, to do a stationary test the Unit Root Test has been utilized to determine whether or not the variables are stationary. Beginning with the analysis, this study examines if unit root is available applying the augmented Dickey-Fuller (ADF) and Philips-Perron tests (PP) [30]. Data should be stationary for the analysis's results to be stationary. The test's outcomes will be invalid if the data is not stationary. Hence, the variable's unit root, or non-stationarity, is the null hypothesis (shown in the below Table 2).

Table 2.

Augmented Dickey-Fuller (ADF) Test statistics of three countries.

Bangladesh			Pakistan		Sri-Lanka
Variable	Test statistic	p-value	Test statistic	p-value	Test Statistic	p-value	Remark
GDP	0.11	0.71	−1.35	0.15	−0.65	0.62	Non-Stationary
dGDP	−7.08	0.00	−6.02	0.00	−5.71	0.00	Stationary
REMs	−0.30	0.56	0.65	0.49	0.29	0.76	Non-Stationary
dREM	6.10	0.00	−3.88	0.00	−5.32	0.00	Stationary
FDI	−0.82	0.35	−1.64	0.42	−0.10	0.64	Non-Stationary
dFDI	−5.98	0.00	−3.64	0.00	−7.06	0.00	Stationary
IR	−1.08	0.24	−0.73	0.33	−1.29	0.54	Non-Stationary
dIR	−8.27	0.00	−6.42	0.00	−6.12	0.00	Stationary

Critical values (−2.62, −1.95 and −1.61 at 1 %, 5 % and 10 % significant level respectively).

In Fig. 1, (a), (c), and (e) shows the original data graphs of Bangladesh, Pakistan, and Sri-Lanka are clustered between 1987 and 2021 and (b), (d), and (f) of Bangladesh, Pakistan, and Sri-Lanka shows that transformed data graphs are stationary.

Fig. 1.

Graphical representation of (a), (c), (e) original data, and (b), (d), (f) return data of Bangladesh, Pakistan, and Sri-Lanka, respectively.

3.3. Model specification

3.3.1. VAR model

VAR was selected for this study due to its adaptability in analyzing both short- and long-term dynamics and to determine whether GDP and other independent variables are causally related. The vector autoregression model, or VAR(p), uses the lagged values of these variables from previous p time periods as independent variable to show the linear multivariate autoregressive behavior of a system of K variables in matrix notation (omitting a constant) [31].

$$y_t=A_1{y}_{t-1}+\dots +A_p{y}_{t-p}{+u}_t$$

where, $y_t$ and $y_{t-i}$ with $A_i$ are KxK matrices of coefficients indicating the influence of lagged variables on their current variables, where $i=1,...,p$ are Kx1 vectors carrying values of the K variables for contemporaneous and lagged time periods.

The depiction of the vector moving average (VMA(∞)) in the VAR(p) model,

$$y_t=\sum _{j=0}^{\infty }{\varnothing }_j{u}_{t-j}$$

where ${\varnothing }_o=IK$, $x=j$ when $j<p$, and $x=p$ when $j\ge p$ , are used to recursively generate the ${\varnothing }_j$ matrices from ${\varnothing }_j=\sum _{i=1}^x{\varnothing }_{j-i}{A}_i$. The process with weak stability can be considered as being driven by the error terms, and we may use Impulse Response Functions (IRF) to examine how the jth error propagates

$${IRF}_j\left(h\right)=\frac{{\partial }_{y_{t+h}}}{{\partial }_{u_{jt}}},h=0,1,2,\dots \dots$$

which are provided in this equation's $j^{th}$ column of ${\varnothing }_h\text{.}$.

The functional relationship between GDP, REM, FDI and IR of Bangladesh, Pakistan, Sri-Lanka can be expressed in the following way:

$$GDP=f\left(REM,FDI,IR\right)$$

The model will test the effect of REM, FDI, IR on GDP

$$Y_t={{\beta }_0+{\beta }_1{X}_t+{\beta }_2{Y}_t+{\beta }_{3}Z}_t+{\epsilon }_t$$

$${GDP}_t={{\beta }_0+{\beta }_1{REM}_t+{\beta }_2{FDI}_t+{\beta }_{3}IR}_t+{\epsilon }_t$$

where, GDP = Growth Rate of GDP, REM = Remittances, FDI = Foreign Direct Investment, and IR = Inflation Rate.

In this model ${\beta }_0$ is the current time period of the observation of each variables based on the lag values. ${\beta }_1$ , ${\beta }_2$ and ${\beta }_3$ are the coefficient of all those independent variables. This paper has been conducted on the following hypothesis:

H0 = REM has no significant impact on GDP.

H1= REM has significant impact on GDP.

H0 = FDI has no significant impact on GDP.

H2 = FDI has significant impact on GDP.

H0 IR has no significant impact on GDP.

H3 = IR has significant impact on GDP.

The coefficient of regression, $\beta$ represents how much the dependent variable changes when the independent variable changes by one unit. The Cholesky decomposition of the contemporaneous covariance matrix served as the foundation for the forecast variance decompositions and impulse responses.

3.3.2. Structural analysis by granger causality

In order to investigate the causal relationship among the variables of the system, the linear Granger causality tests should be applied by using the following strategy. Compare the unrestricted models:

$$\Delta Y_t={{\alpha }_1+{\sum }_{i=1}^{m1}{\beta }_{1i}\Delta Y}_{t-i}+{{{\sum }_{j=1}^{m1}{\theta }_{1j}\Delta Y}_{j-i}+e}_{1t}$$

$$\Delta X_t={{\alpha }_2+{\sum }_{i=1}^{m1}{\beta }_{2i}\Delta Y}_{t-i}+{{{\sum }_{j=1}^{m2}{\theta }_{2j}\Delta X}_{j-i}+e}_{2t}$$

with the restricted models:

$$\Delta Y_t={{\alpha }_1+{\sum }_{i=1}^{m1}{\beta }_{1i}\Delta Y}_{t-i}$$

$$\Delta X_t={{\alpha }_2+{\sum }_{i=1}^{m1}{\beta }_{2i}\Delta Y}_{t-i}$$

where, $\Delta Y_t$ and $\Delta X_t$ first order forward differences of the variables; $\alpha ,\beta \text{and}\theta$ are the parameters to be estimated; and $e_1$ and $e_2$ are standard random errors. If both of them are statistically significant, there is a bivariate causal relationship among the variables; if both of them are statistically insignificant, neither the changes in variable $Y$ nor the changes in variable $X$ have any effect over other variables.

3.3.3. BEKK-GARCH model

[32]proposed BEKK-GARCH (1,1) modelis a development of the [33] GARCH model, which has fewer parameters and permits interaction between conditional variances and covariance [34]. [35] introduced the model that is shown below. When extending the univariate GARCH model to an n-dimensional multivariate model, it requires the estimation of n different mean and corresponding variance equations and $\left(n^2-n\right)/2$ covariance equations. The mean equation is defined as:

$$r_t=\mu +{\epsilon }_{t,}{\epsilon }_t|{\mathrm{\Omega }}_{t-1}\sim N\left(0,H_t\right)$$

where $,r_t$ is a vector of appropriately defined returns and μ a (N∗1) is the vector of the parameters that estimates the mean of the return series. The residuals vector is ${\epsilon }_t$ with the corresponding conditional covariance matrix $H_t$ given the available information set ${\varnothing }_{t-1}$. The covariance matrix of the unrestricted BEKK model in bi-variate case is represented as:

$$H_t=C^{\prime }C+A^{\prime }{\epsilon }_{t-1}{A}_{t-1}^{\prime }C+B^{\prime }{H}_{t-1}B$$

where $H_t$ is the conditional covariance matrix and C denotes the 2 × 2 upper triangular matrices. The element A_ij of the 2 × 2 matrix, A indicates the impact of variable i volatility on variable j and reflects the ARCH effect of volatility. The element B_ij of the 2 × 2 matrix, B indicates the persistence of volatility transmission between market i and market j and reflects the GARCH effect of volatility. The total estimated parameters in the bi-variate BEKK-GARCH(1,1) model can be written as:

$$\left[\begin{array}{cc}{h}_{11,t}& h_{12,t}\\ h_{21,t}& h_{22,t}\end{array}\right]=\left[\begin{array}{cc}{c}_{11,t}& c_{12,t}\\ c_{21,t}& c_{22,t}\end{array}\right]\left[\begin{array}{cc}{c}_{11,t}& c_{12,t}\\ c_{21,t}& c_{22,t}\end{array}\right]+\left[\begin{array}{cc}{a}_{11,t}& a_{12,t}\\ a_{21,t}& a_{22,t}\end{array}\right]\left[\begin{array}{cc}{\epsilon }_{1,t-1}^2& {\epsilon }_{1,t-1,}{\epsilon }_{2,t-1}\\ {\epsilon }_{2,t-1}& {\epsilon }_{2,t-1}^2\end{array}\right]\left[\begin{array}{cc}{a}_{11,t}& a_{12,t}\\ a_{21,t}& a_{22,t}\end{array}\right]+\left[\begin{array}{cc}{b}_{11,t}& b_{12,t}\\ b_{21,t}& b_{22,t}\end{array}\right]\left[\begin{array}{cc}{h}_{11,t-1}& h_{12,t-1}\\ h_{21,t-1}& h_{22,t-1}\end{array}\right]\left[\begin{array}{cc}{b}_{11,t}& b_{12,t}\\ b_{21,t}& b_{22,t}\end{array}\right]$$

where C is the parameter matrix is the ARCH effect coefficien. matrix and B is the GARCH effect coefficient matrix. The BEKK model assumes the diagonality of the A and B matrices. All of the off-diagonal elements are therefore equal to zero. The model's log likelihood function is expressed as follows:

$$\ln \mathcal{l}\left(\theta \right)=-\frac{pN}{2}\mathit{ln}2\pi -\frac{1}{2}\sum _{i=1}^p\left(\ln \left|H_t\right|+\frac{1}{2}{\epsilon }_t^{\prime }{H}_t{\epsilon }_t\right)$$

where θ is the unknown parameter of the model. N is the number of variable, in this 4-variable model N = 4. P is the number of observations.

4. Results and discussions

4.1. Optimal lag selection

Lag length refers to the number on past observation that are used to predict the current value of the variable. Sequential modified LR test, Final Prediction Errors (FPE), and Akaike Information Criterion (AIC) were used to determine the ideal latency. This analysis shows that the order two lag length yielded the minimum AIC and LR value when compared to other order lag lengths. The outcomes of the selection criterion for three countries are displayed in Table 3 below.

Table 3.

VAR(p) model lag selection criterion.

Bangladesh				Pakistan				Sri-Lanka
Lag	LR	FPE	AIC	Lag	LR	FPE	AIC	Lag	LR	FPE	AIC
0 1 2 3	NA 26.76* 18.21 7.06	0.0002* 0.0002 0.0002 0.0005	2.84 2.84* 3.05 3.68	0 1 2 3 4	NA 13.65 13.83 34.82 26.87*	0.0005 0.001 0.0016 0.0007 0.0003*	3.93 4.45 4.86 3.87 2.87*	0 1 2 3	NA 24.55 25.79 14.29	0.0001* 0.0002 0.0001 0.0002	2.75 2.84 2.69* 2.94

We have used lag 1 for Bangladesh, lag 4 for Pakistan and lag 2 for Sri-Lanka.

4.2. Estimation of VAR model

VAR model is used to predict the relationship that affects each other. T statistics value in square bracket reports the result. T statistics value for five percent level of significance is 1.96. If this value is less than 1.96 the variable at lag 1,2,3 and 4 is insignificant. If this value is greater than 1.96 the variable at lag 1,2,3 and 4 is significant at that lag length.

According to the results in Table 4, Bangladesh REM, FDI and IR have insignificant impact on GDP. This result is similar to the [36] study; it estimated a short- and long-term relationship between remittances and Saudi Arabia's GDP using ARDL-ECM. The major finding is that remittance outflows have no appreciable impact on GDP, either in the short or long term. Pakistan's REM and FDI have significant impact on GDP, but inflation has insignificant impact on GDP. The results align with the factual information that [28,37] presented regarding Pakistan's inflation and GDP. Sri-Lanka REM, FDI and IR have insignificant impact on GDP [38]. has previously suggested an inverse link between Sri-Lanka's remittances and GDP. Employing information from Sri-Lanka using the multivariant VAR model [18], finds no correlation between FDI and GDP.

Table 4.

Estimated results of VAR model.

Bangladesh		Pakistan		Sri-Lanka
	GDP		GDP		GDP
GDP(-1)	−0.53 [-2.98]	GDP(-1)	−1.18 [-4.44]	GDP(-1)	−0.47 [-2.38]
REM(-1)	−0.17 [-0.49]	GDP(-2)	−1.45 [-3.87]	GDP(-2)	−0.09 [-0.47]
FDI(-1)	−0.03 [-0.81]	GDP(-3)	−1.00 [-2.61]	REM(-1)	0.69 [0.45]
IR(-1)	0.06 [0.69]	GDP(-4)	−0.04 [-0.13]	REM(-2)	−0.48 [-0.29]
		REM(-1)	−0.80 [-1.47]	FDI(-1)	−0.04 [-0.25]
		REM(-2)	0.26 [0.55]	FDI(-2)	−0.34 [-1.87]
		REM(-3)	0.60 [1.80]	IR(-1)	−0.10 [-0.62]
		REM(-4)	1.44 [2.31]	IR(-2)	0.03 [0.21]
		FDI(-1)	0.53 [1.29]
		FDI(-2)	0.02 [0.07]
		FDI(-3)	−0.70 [-1.98]
		FDI(-4)	−0.86 [-2.10]
		IR(-1)	−0.11 [-0.38]
		IR(-2)	−0.56 [-2.06]
		IR(-3)	−0.66 [-1.95]
		IR(-4)	−0.20 [-0.70]

4.3. Auto regression (AR) root

The VAR system must be stationary to become stable. The estimate VAR is stable if every root of the characteristic AR polynomial has a modulus of less than one and lies inside the unit circle. If any of the estimated roots have a modulus greater than one and are outside the unit circle the estimated VAR is not stable.

No root lies outside the unit circle (in Fig. 2 a. Bangladesh, b. Pakistan, c. Sri-Lanka). Therefore, VAR satisfies the stability condition.

Fig. 2.

Inverse Root of AR Characteristic Polynomial of a. Bangladesh, b. Pakistan, and c. Sri-Lanka.

4.4. Result of granger causality test

The structures of the causal relationships between variables were analyzed through the Granger causality approach [39,40]. F-Statistic is employed to evaluate the Granger causality test's overall significance. It contrasts the fit of a model with and without lagged variables. F statistics are linked to probability values (p-values) in order to determine the statistical significance of the observed F values. If probability value is less than any $\alpha$ level, then the hypothesis would be rejected at that level. H₀ is X does not have the granger cause Y. If the p-value is greater than 0.05, accept H₀. It means no causality exists. If the p value is less than 0.05, reject H₀. It means causality exists.

The empirical results of our model (in Table 5) takes GDP as the dependent and REM, FDI, IR as the independent variables. The test exhibits the presence of a possible causal relationship that exists between the selected variables. From the test results of the table, no bidirectional relationship found among the variables. No causal relationship between the variables could be identified according to the test results in case of Bangladesh and Sri-Lanka. For Pakistan (GDP→REM) and (GDP→FDI) p-value is less than and equal to 0.05 respectively. So, we can reject null hypothesis that means causality exists. There is existence of a unidirectional relationship between REM, FDI and GDP of Pakistan. This study confirms [41] assumption, which holds that there is a unidirectional causal relationship between GDP and FDI and remittances of Pakistan. According to our analysis, there is no Granger causal relationship between GDP and inflation rates in Bangladesh, Pakistan and Sri-Lanka. Similarly, according to Ref. [42] research, there is no Granger-causal relationship between inflation rates and per capita GDP in the Nigerian economy.

Table 5.

Granger causality test.

Bangladesh
Null Hypothesis	Direction of Causality	F- statistic	Probability
REM doesn't Granger Cause GDP	REM→GDP	50.32	0.57
GDP doesn't Granger Cause REM	GDP→REM	0.78	0.38
FDI doesn't Granger Cause GDP	FDI→GDP	1.02	0.32
GDP doesn't Granger Cause FDI	GDP→FDI	0.86	0.35
IR doesn't Granger Cause GDP	IR→GDP	0.57	0.45
GDP doesn't Granger Cause IR	GDP→IR	0.43	0.51
Pakistan
REM doesn't Granger Cause GDP	REM→GDP	1.01	0.48
GDP doesn't Granger Cause REM	GDP→REM	4.06	0.02
FDI doesn't Granger Cause GDP	FDI→GDP	1.04	0.46
GDP doesn't Granger Cause FDI	GDP→FDI	3.08	0.05
IR doesn't Granger Cause GDP	IR→GDP	0.68	0.69
GDP doesn't Granger Cause IR	GDP→IR	0.64	0.72
Sri-Lanka
REM doesn't Granger Cause GDP	REM→GDP	0.32	0.57
GDP doesn't Granger Cause REM	GDP→REM	0.78	0.38
FDI doesn't Granger Cause GDP	FDI→GDP	1.02	0.32
GDP doesn't Granger Cause FDI	GDP→FDI	0.86	0.35
IR doesn't Granger Cause GDP	IR→GDP	0.57	0.45
GDP doesn't Granger Cause IR	GDP→IR	0.43	0.51

4.5. Impulse response functions

A shock's impact on a series' behavior is tracked over time using an impulse response function [43]. Period is indicated on the X axis, and percentage variation is indicated by the Y axis. The black line represents the impulse response function while the red line represents the interval.

4.5.1. Impulse response function of Bangladesh

In the first graph of Fig. 3 a. Bangladesh, we see that unexpected increase in remittances growth leads to increase in GDP growth rate in second period, then it decreases and becomes insignificant in period 3. After that its increase fades away over 5 years. Shock on FDI reduced GDP in the third period but after that it increased up to the fourth period and its impact became insignificant in period 6. In the third graph a shock on IR has asymmetric effect on GDP and fades away in period 8. This outcome is consistent with the discovery of [44].

Fig. 3.

Impulse Response Function of a. Bangladesh, b. Pakistan, and c. Sri-Lanka.

4.5.2. Impulse response function of Pakistan

In the second graph of Fig. 3 b. Pakistan, a shock on REM to GDP initially has no noticeable impact on GDP in 1 and 2. The response gradually declines in the fifth period and remains in the negative region. The shock on FDI reduced GDP remains steady state value from 3 to 5. FDI rises above in the 8th period and becomes negative until the end of the period. After the 6th period, IR starts to decrease and becomes close to 0 but remains negative.

4.5.3. Impulse response function of Sri-Lanka

In the third graph of Fig. 3 c. Sri-Lanka, a shock on REM decreases GDP and becomes insignificant in period 4. After that it’s increase fades away over 6 years. Shock on FDI has asymmetric effect on GDP and comes near 0 in period 10. The shock of IR has a positive impact on inflation rate up to period 2 and then decreases and again increases. It remains positive until the end of the period.

4.6. Variance decomposition analysis

A method for determining the percentage of a time series' forecast error variance that can be assigned to the shocks or innovations from each variable is called variance decomposition. The variance decomposition standard errors (S.E.) provide an indication of the degree of uncertainty surrounding the estimations of the contributions of the different variables. The result of variance decomposition shows (in Table 6) that in the long run, the variation of GDP growth not only depends on shocks of GDP growth but also the shocks of other variables.

Table 6.

Variance decomposition of three countries.

	Period	S.E.	GDP	REM	FDI	IR
Bangladesh	1	0.218	100.000	0.000	0.000	0.000
	3	0.292	89.926	1.713	2.366	5.993
	5	0.301	87.172	3.882	2.634	6.310
	7	0.311	87.533	3.813	2.735	5.916
	9	0.312	87.237	4.137	2.717	5.907
Pakistan	1	0.478	100.000	0.000	0.000	0.000
	3	0.792	77.731	11.236	6.528	4.503
	5	0.856	70.081	11.936	12.013	5.968
	7	0.955	63.399	18.867	11.978	5.754
	9	0.984	61.215	20.227	13.084	5.471
Sri-Lanka	1	0.465	100.000	0.000	0.000	0.000
	3	0.548	86.933	7.565	3.813	1.687
	5	0.581	81.140	11.498	4.957	2.402
	7	0.591	80.604	11.670	5.284	2.440
	9	0.592	80.532	11.699	5.292	2.475

The variance decomposition of GDP shows that at the first period, GDP forecasts 100 % error variance by itself, so the other variables in the model do not have strong influence on GDP. In the long run the contribution of other variable increased. Pakistan's GDP in the long run is influenced by other most.

4.7. Summary result of BEKK-GARCH

The shock spillover and volatility spillover between the dependent variable DGDP and the independent variables DREM, DFDI, and DIR are respectively captured by the off-diagonal elements of matrices A and B. The BEKK-GARCH results are shown in Table 7.

Table 7.

BEKK-GARCH results.

Variable	Bangladesh		Pakistan		Sri-Lanka
	Coefficient	Probability	Coefficient	Probability	Coefficient	Probability
A(1,1)	−0.01	0.95	1.20	0.00	0.26	0.00
A(1,2)	−0.06	0.04	−0.02	0.37	0.00	0.03
A(1,3)	0.02	0.08	−0.01	0.01	−0.02	0.00
A(1,4)	−0.02	0.00	−0.07	0.02	−0.02	0.05
A(2,1)	2.02	0.17	0.04	0.94	−31.53	0.00
A(2,2)	0.63	0.00	−0.57	0.00	0.51	0.00
A(3,1)	−0.56	0.76	−0.50	0.00	3.73	0.15
A(3,3)	0.25	0.01	0.59	0.00	1.25	0.00
A(4,1)	0.51	0.87	2.98	0.00	−6.49	0.00
A(4,4)	0.38	0.00	0.18	0.00	0.03	0.04
B(1,1)	0.40	0.02	0.01	0.57	0.83	0.00
B(1,2)	−0.01	0.44	0.00	0.84	0.00	0.32
B(1,3)	‵-0.02	0.06	0.00	0.00	0.00	0.00
B(1,4)	0.00	0.02	−0.01	0.01	0.00	0.02
B(2,1)	3.55	0.00	−1.78	0.06	41.27	0.00
B(2,2)	0.95	0.00	0.64	0.00	0.40	0.00
B(3,1)	6.21	0.08	−0.21	0.20	2.65	0.14
B(3,3)	0.65	0.00	0.15	0.01	0.01	0.17
B(4,1)	−7.94	0.09	0.10	0.47	−1.38	0.31
B(4,4)	0.72	0.00	0.66	0.00	0.42	0.00

4.7.1. Result analysis of Bangladesh

From the coefficients A(1,2) and A(2,1) (in Table 8) represent the cross-volatility spillover effects between the two variables in the model. Similarly, the coefficients B(1,2) and B(2,1) represent the cross-autocorrelation effects. A(1,2) is significant but A(2,1) is not significant, which suggests that there is a unidirectional volatility spillover effect from REM to GDP but not vice versa. This means that shocks in REM have a significant impact on the volatility of GDP, but shocks in GDP do not significantly affect the volatility of REM. A(1,3) and A(3,1) are not significant; so there is no spillover effect between GDP and FDI. This means that shocks in FDI and GDP have no significant impact on the volatility of other. A(1,4) is significant but A(4,1) is not significant, which suggests that there is a unidirectional volatility spillover effect from IR to GDP, but not vice versa. This means that shocks in IR have a significant impact on the volatility of GDP, but shocks in GDP do not significantly affect the volatility of IR.

Table 8.

BEKK-GARCH result of Bangladesh.

A(1,1)A(1,2) = 0.00	shock of GDP have positive effect on next day's REM covariance
A(2,1)A(2,2) = 1.27	shock of REM have positive effect on next day's GDP covariance
A(1,1)A(1,3) = - 0.00	shock of GDP have negative effect on next day's FDI covariance
A(3,1)A(3,3) = - 0.14	shock of FDI have negative effect on next day's GDP covariance
A(1,1)A(1,4) = 0.00	shock of GDP have positive effect on next day's IR covariance
A(4,1)A(4,4) = 0.19	shock of IR have positive effect on next day's GDP covariance

B(1,2) is significant and B(2,1) is not significant, which suggests that the conditional correlation of GDP is affected by the lagged residuals of REM, but the conditional correlation of variable REM is not affected by the lagged residuals of GDP. This type of cross effect is also known as a unidirectional spillover effect.But in terms of correlation, B(1,3) and B(3,1) are not significant, which suggests that the conditional correlation of GDP and FDI is not affected by the lagged residuals of each other. B(1,4) is significant and B(4,1) is not significant, which suggests that the conditional correlation of GDP is affected by the lagged residuals of IR, but the conditional correlation of variable IR is not affected by the lagged residuals of GDP.

4.7.2. Result analysis of Pakistan

From the coefficients A(1,2) and A(2,1) (in Table 9) represent the cross-volatility spillover effects between the two variables in the model. Similarly, the coefficients B(1,2) and B(2,1) represent the cross-autocorrelation effects. A(1,2) and A(2,1) are not significant, which suggests that there is no volatility spillover effect from REM to GDP, vice versa. This means that shocks in REM have no significant impact on the volatility of GDP, shocks in GDP have no significantly affect the volatility of REM. A(1,3) and A(3,1) are significant. So their bidirectional effect between GDP and FDI.A(1,4) and A(4,1) are significant, which suggests that there is a bidirectional volatility spillover effect from GDP to IR.

Table 9.

BEKK-GARCH result of Pakistan.

A(1,1)A(1,2) = - 0.02	shock of GDP have negative effect on next day's REM covariance
A(2,1)A(2,2) = 0.02	shock of REM have positive effect on next day's GDP covariance
A(1,1)A(1,3) = - 0.01	shock of GDP have negative effect on next day's FDI covariance
A(3,1)A(3,3) = - 0.29	shock of FDI have nagative effect on next day's GDP covariance
A(1,1)A(1,4) = - 0.08	shock of GDP have negative effect on next day's IR covariance
A(4,1)A(4,4) = 0.26	shock of IR have positive effect on next day's GDP covariance

B(1,2) and B(2,1) are not significant, suggesting that the conditional correlation of REM is not affected by the lagged residuals of GDP and the conditional correlation of variable GDP is not affected by the lagged residuals of REM. B(1,3) is significant but B(3,1) is not significant, suggesting that the conditional correlation of GDP is affected by the lagged residuals of FDI. B(1,4) is significant but B(4,1) is not significant, suggesting that the conditional correlation of GDP is affected by the lagged residual of IR.

4.7.3. Result analysis of Sri-Lanka

From (Table 10) the coefficients A(1,2) and A(2,1) represent the cross-volatility spillover effects between the two variables in the model. Similarly, the coefficients B(1,2) and B(2,1) represent the cross-autocorrelation effects. A(1,2) and A(2,1) are significant, suggesting that there is a bidirectional volatility spillover effect from REM to GDP and vice versa. This means that shocks in REM have a significant impact on the volatility of GDP and shocks in GDP significantly affect the volatility of REM. A(1,3) is significant but A(3,1) is not significant; so there is unidirectional effect between GDP and FDI. A(1,4) is significant but A(4,1) is significant, which suggests that there is a bidirectional volatility spillover effect from GDP to IR. This means that shocks in GDP and IR have significant impact on the volatility of each other.

Table 10.

BEKK-GARCH result of Sri-Lanka.

A(1,1)A(1,2) = 0.00	shock of GDP have positive effect on next day's REM covariance
A(2,1)A(2,2) = - 16.08	shock of REM have negative effect on next day's GDP covariance
A(1,1)A(1,3) = 0.00	shock of GDP have positive effect on next day's FDI covariance
A(3,1)A(3,3) = 4.66	shock of FDI have positive effect on next day's GDP covariance
A(1,1)A(1,4) = - 0.01	shock of GDP have negative effect on next day's IR covariance
A(4,1)A(4,4) = 1.62	shock of IR have positive effect on next day's GDP covariance

B(1,2) is significant and B(2,1) is not significant, suggesting that the conditional correlation of GDP is affected by the lagged residuals of REM, but the conditional correlation of variable REM is not affected by the lagged residuals of GDP. B(1,3) is significant but B(3,1) is not significant, indicating that the conditional correlation of GDP is affected by the lagged residuals of FDI. B(1,4) is significant but B(4,1) is insignificant, indicating that the conditional correlation of GDPis affected by the lagged residuals of IR.

4.8. Key findings

Using VAR models supplemented by the BEKK-GARCH model, our research deviates from earlier research and provides new insights into the economic dynamics of Bangladesh, Pakistan and Sri-Lanka. The VAR model revealed that, REM, FDI and IR had no noticeable impact on GDP in Bangladesh and Sri-Lanka. This result is consistent with research [36] which also found an insignificant impact of remittance outflows on GDP and contrasts [19,20,23]. On the other hand, Pakistan exhibited notable long-term GDP effects from REM and FDI. This result is in line with accurate data about Pakistan's GDP and inflation trends that have been provided in earlier research [24]. According to our research, there is no Granger causal relationship between GDP and inflation rates in Bangladesh, Pakistan or Sri-Lanka. These results are consistent with earlier research on the Nigerian economy [42]. Additionally, BEKK-GARCH models shed light on the spillover effects of volatility, showing how REM, FDI and IR shocks affect GDP volatility. By using the BEKK-GARCH model, our analysis covers a significant gap in the literature and provides a deeper understanding of the intricate relationships between FDI, inflation rate, remittances and GDP in South Asian nations.

5. Conclusion

Our research examined how the GDP of three economically vulnerable Asian nations were affected by remittances, foreign direct investment and inflation rate using Vector Autoregression (VAR) models and we extended our study by including the BEKK-GARCH model to account for the time-varying volatility. The empirical results have revealed different influence patterns in each of the three countries. Surprisingly, according to VAR model REM, FDI and IR did not significantly affect GDP in Bangladesh and Sri-Lanka, indicating that their economic systems are robust. Pakistan, on the other hand, showed the long-term effects of REM and FDI on GDP, suggesting a higher dependence on outside sources for steady economic growth. Fascinating dynamics were shown by Granger causality tests, which showed that Pakistan demonstrated a unidirectional relationship, highlighting the influence of REM and FDI on its GDP. Variance decomposition showed that, at first, GDP primarily predicted its own error variance, suggesting that other variables had little effect. But with time, other factors became more significant, especially in Pakistan where a number of factors affected GDP. Further information on the volatility spillover effects among the variables is provided by the BEKK-GARCH models. While GDP shocks have no discernible effect on REM volatility, REM volatility is found to have a considerable impact on GDP volatility in Bangladesh and Sri-Lanka. Similar to this, IR shocks influence Pakistan's GDP volatility, but the opposite is not seen.The analysis's result indicates that although REM and FDI are key factors influencing economic performance in these nations, their effects differ depending on the situation. The study also emphasizes how vulnerable economies that depend too much on FDI or REM are prone to sudden drops in these inflows, which can upset macroeconomic equilibrium. In order to gain a thorough understanding of the factors influencing economic growth in developing countries, future research alternatives could investigate the GDP contributions of the informal sector and conduct a comparative analysis across a wider spectrum of South Asian nations.

6. Policy recommendations and implications

Several significant policy recommendations and implications arise from our thorough examination of the effects of inflation rate (IR), foreign direct investment (FDI) and remittances (REM) on the GDP of Bangladesh, Pakistan and Sri-Lanka.

According to VAR model, in addition to remittances and foreign direct investment, policymakers in Bangladesh and Sri-Lanka should think about expanding their economic drivers. Although our analysis did not find any substantial effects of these factors on GDP, relying too much on them could lead to vulnerabilities. Policymakers should create risk mitigation strategies because these economies are vulnerable to external shocks, especially in the case of Pakistan. To improve general economic stability, emergency plans should be created to deal with sudden drops in remittances or foreign direct investment and steps to lessen reliance on particular economic contributors should be taken. Sustained economic growth can be achieved by expanding institutions, making investments in technology and creating an atmosphere that is business-friendly. On account of the volatility and spillover effects noted in the BEKK-GARCH model, financial resilience needs to be strengthened. The possible negative effects of economic shocks on GDP can be lessened by creating strong financial instruments, enhancing regulatory frameworks and implementing risk management techniques. Policymakers should keep a careful eye on the substantial unidirectional volatility spillover effect that exists in Bangladesh between REM and GDP as well as between IR and GDP, and control shocks in these sectors to lessen their negative effects on GDP volatility. Since there are no significant volatility spillover effects between GDP and REM in Pakistan, policymakers should concentrate on making each of these sectors more resilient to shocks from outside the country. In order to mitigate GDP-driven fluctuations in foreign direct investment (FDI), Sri-Lanka needs to attract foreign investments, manage the volatility of the IR sector and stabilize bidirectional volatility spillovers. Bangladesh, Pakistan, and Sri-Lanka can develop inclusive and resilient economies that can withstand external shocks and achieve long-term prosperity by putting these policy ideas into practice.

Funding sources

There are no funding sources for this research.

Data availability statement

Data will be available on request.

Transparency

The authors confirm that the manuscript is an accurate, honest and transparent account of the study.

CRediT authorship contribution statement

Fariea Nazim Jui: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Md. Jamal Hossain: Writing – review & editing, Visualization, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Anwesha Das: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Data curation. Nazia Sultana: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis. Md. Kamrul Islam: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.