Following previous surveyed applied work in formulating the empirical framework of the formal empirical migrant workers’ remittances-FI nexus, this study adapts the following model: FI=β0+β1REMt+Zit β+εt[Eqn 1]

In Equation 1, β₀ refers to the constant term, FI is a multidimensional access to the FI index rather than all other dimensions, because we believe (as defined) that the essence of FI is the ability of a group of people to access the financial services provided by a financial system (banks); therefore, we attempt to study the factors that affect the ability of individuals to access banking goods and services. Migrant workers’ remittances (REM) are measured as the personal remittances per capita (in US$), and Z_itβ is a set of other control variables including FD proxied by the domestic credit to the private sector (%GDP), the trade openness (Open) measured by trade share to gross domestic product (GDP), Inflation as proxied by the Consumer Price Index (CPI), and economic performance proxied by the GDP per capita (GDPC, Constant, measured in US$).

This study uses yearly data covering the period 1990–2022. The current research utilised secondary data gathered from numerous sources, mainly the World Development Indicators (WDI) of the World Bank.

Financial inclusion is measured by quite a few indicators, namely access, usage, and penetration (Immaculate 2018). This article uses the access dimension to proxy FI, which is measured by two indicators; ATMs per 100 000 adults and commercial bank branches per 100 000 adults. It was argued that ATMs are widely used and are more practical in the sense that they are easily accessible and operate beyond banking working hours (Ajefu & Ogebe 2019; Bkwayep & Tsasack 2020; Sami et al. 2020). Moreover, commercial bank branches provide financial services to customers and they are physically separated from the main office (Bkwayep & Tsasack 2020; Inoue & Hamori 2016; Kenneth 2019; Sami et al. 2020). This article follows Immaculate (2018) in constructing the access index (FI) in Jordan, taking into consideration the availability of data for the access FI dimension. The calculation of the proposed procedure comes in two steps. The first step involves the computation of dimension indices (ATMs and bank branches), by applying the following formula for each one: di,tj=(Ai,tj−mjMj−mj)[Eqn 2]

In Equation 2, i and j = 1,2 are the numbers associated with a specific access indicator per country. Ai,tj is the actual value of the variable j, m^j is the minimum value for the variable j and M^j is the maximum observed value for variable j. The computed di,tj lies between 0 and 1. This implies that a maximum value of 1 indicates complete FI. In contrast, the minimum observed value of 0 denotes financial exclusion. The second step involves a combination of dimension index with respect to the access indicator by finding the average sum of the dimension as: d˜1,i,t=d1,i,t+d2,i,t2[Eqn 3]

This section presents the procedure used to examine the asymmetric effects remittance inflow has on financial inclusion in Jordan. The ARDL (Pesaran et al. 2001) model has some advantages over the Johansen approach to cointegration (Johansen & Juselius 1988, 1990) and Engle and Granger’s (1987) approach to cointegration. First, it can be applied regardless of the order of integration of the variables: I(0) or I(1) but not I(2). Furthermore, following Pesaran et al. (2001), the unrestricted ARDL model specification for the standard log-linear functional specification of the long-run and short-run between FI indicators FI_t and remittance inflows of Equation 1 without the asymmetry take the following form: Δ LFIt=α0+∑i=1pγ1iΔ LFIt−i+∑i=0qβ1iΔ LREMt−i   +∑i=0qφ1iΔ Zt−i+γ2i LFIt−i   +β2i LREMt−i+φ2iZt−i+εt[Eqn 4]

Equation 4 also can be viewed as an ARDL of order (p, q) for the multidimensional FI index, where ε_t, and Δ denote the white noise and the first difference terms. The Akaike information criteria are used to determine the appropriate optimal lag length structure, and later the equation is run by order of least squares (Pesaran et al. 2001). Shahbaz and Feridun (2012) indicated that the calculated F-statistic is sensitive to the lag length selection.

The ARDL model requires testing for the existence of long-run equilibrium relationships among the model’s variables. Equation 4 captures only the symmetric effects of remittance inflows on FI under the assumption that the underlying long-run association is described as following a symmetrically linear combination. In addition, it is assumed that there is no difference between the decomposed partial sum processes of negative and positive impacts of these nonstationary stochastic explanatory variables on economic growth (Maxwell & Nicholas 2021).

However, this approach is found to provide profoundly misleading results in terms of nonlinearity and asymmetries. To address this problem, this article uses the NARDL modeling strategy suggested by Shin et al. (2014) with the belief that the cointegrating association could be asymmetric or nonlinear. The NARDL approach is an asymmetric extension of the linear (ARDL) cointegration model proposed by Pesaran et al. (2001). The linear ARDL approach does not consider the possibility that negative and positive variations of the explanatory variables have different effects on the dependent variable. The NARDL model does not allow detecting the asymmetric effects, permits testing for cointegration in a single equation framework, and works regardless of the order of integration, I(0) or I(1), but not I(2), as it follows the bounds tests F-values.

More importantly, this methodology allows us to test the asymmetrical effects of LREMt+ and LREMt− on FI in Jordan. It provides the possibility to differentiate between the impact of positive (negative) changes in remittance inflows on FI (Maxwell & Nicholas 2021). Thus, this article uses a nonlinear modeling framework, which provides a simple and flexible vehicle for the analysis of joint long-run and short-run asymmetries (Maxwell & Nicholas 2021).

The following is the construction of Shin et al.’s (2014) asymmetric long-run model describing the relationship between remittance inflows and financial inclusion: LFIt=α0+α1xt+α2 LREMt++α3LREMt−+μt[Eqn 5]

In Equation 5, LFI_t is the multidimensional FI indicator, LREM_t is the flow of remittances, x_t is a (k × 1) vector of control variables, and α = (α₀ + α₁ + α₂ + α₃) is a cointegrating vector of long-run parameters. In addition, LREMt+ is the positive effects and LREMt− is the adverse effects on LFI_t, while LREMt+, and LREMt− are partial sums process which accumulate positive and negative changes. The NARDL model is built around the following asymmetric long-run equilibrium relationship where the effect of LREM_t is decomposed into these effects: LREMt=x0+LREMt++LRELMt−[Eqn 6] x₀ is a random initial variable. LREMt+=∑i=1tΔ LREMj+=∑t=1tmax(Δ LREMj,0)[Eqn 7] LREMt+=∑i=1tΔ LREMt+=∑t=1tmax(Δ LREMj,0)[Eqn 8]

Employing positive and negative partial sum decompositions setting, NARDL enables the detection of the long-run and the short-run asymmetric effects by allowing for the joint investigation of the issues of nonstationarity and nonlinearity in the setting of an unrestricted error correction model. From Equation 5, the long-run relationship between FI and remittances inflows is captured by α₂, (α₃) Which is expected to be positive (positive). It is posited that the remittance inflows increase (decrease) will result in higher (lower) long-run changes in the financial inclusion index (i.e. α₂ > α₃). Thus, the long-run association presented by Equation 5 indicates asymmetric long-run remittance inflows pass through to the financial inclusion index.

By extending the ARDL model in Equation 1 and replacing the LREM variable with LREMt−1+ and LREMt−1− the combined short-run and long-run dynamics proposed by Pesaran et al. (2001) are: Δ LFIt=β0++β1LFIt−1+β2Zt−1  +β3LREMt−1++β4LREMt−1−  +∑i=1pγiΔ LFIt−i+∑i=1KθiΔ Zt−i   ∑i=0q(φi+Δ LREMt−1++φi−Δ LREMt−1−)+εt[Eqn 9]

All variables in Equation 9 were described in Equation 1 and K,p, and q are lags order. The long-run coefficients (β3=−φi+/β1,β4=−φi−/β1) are the long-run effects of LREM increase (decrease) on LFI. ∑i=0qφi+(∑i=0qφi+) captures the short-run effect of increases (decreases) in LREM on LFI. The model specification of Equation 9 indicates the asymmetric short-run (long-run) impact of remittance inflows on FI. Next, we test for the presence of asymmetric long-run equilibrium relationships by applying the bounds testing approach of Pesaran et al. (2001) using the Wald F-statistic test: the null of no cointegration H₀:β₁ = β₂ = β₃ = β₄ = 0 against the alternative H₁:β₁ ≠ β₂ ≠ β₃ ≠ β₄ ≠ 0.

The F-test considers the stationarity properties of the variables. Specifically, Pesaran et al. (2001) compute lower and upper bounds for the critical values at any significance level. The lower (upper) bound assumes that the variable is I(0). If the F-statistic is higher (lower) than the upper (lower) bound critical value, the null is rejected (accepted). If the statistic lies between the two bounds, the inference remains inconclusive.

Finally, testing for the presence of long-run (short-run) asymmetry between FI and remittance inflows required applying the Wald test, where the null hypothesis of no asymmetry H0:β3=β4(H0:φi+=φi−) is tested against the alternative hypothesis of the existence of asymmetry Hα:β3≠β4(Hα:φi+≠φi−)

It is highly recommended that the linearity feature of the series should be examined before performing the unit root tests. Using a unit root test for exploring the structure of the series is very important and crucial for the reliability and interpretation of the estimation results (Mehmet 2019). Among several tests for linearity available in the literature, this study uses the BDS linearity test developed by Brock et al. (1996) and the Jarque–Bera on the individual series tests were applied to test for the presence of nonlinearity in the relationship between remittances and access to financial inclusion.

The application of the bounds test for cointegration requires that the variables under review should not be integrated of order two, I(2), otherwise the conclusion implies that employing the ordinary estimation would yield inconsistent and spurious results (Maxwell & Nicholas 2021; Yusuf, Yahya & Nasiru 2012). To account for this problem, the commonly used Augmented Dickey-Fuller (ADF; Dickey & Fuller 1979) unit root test approach was applied. If we fail to reject the nonlinearity feature of the series, the nonlinear unit root test is applied for determining the order of integration. In this study, the Kruse (2011) nonlinear unit root test was used for this task, where the null hypothesis states that the series has a unit root against the alternative where the series does not have a unit root. Kruse (2011) proposes a τ test by comparing the critical values presented by Kruse with the statistical values of the estimated parameter in question, and in case the null hypothesis is rejected, then the series is said to be stationary.

The Jarque–Bera on the individual series test was applied to test for the presence of nonlinearity in the relationship between remittances and access to FI. The normality test results presented in Table 1 provide support for the rejection of normality, thus indicating the nonlinearity dependence of the series. Furthermore, the BDS test for individual time series rejects the null hypothesis that they are independent and identically distributed. As a result, both Jarque–Bera and the BDS test indicate that the series are nonlinear. The nonlinearity results imply estimating the paramaters of these series in nonlinear models, otherwise the researcher encounters the misspecification problem.

Furthermore, the BDS test was applied to the residuals of the fitted linear model. The results of the BDS test approach proposed by Brock et al. (1996) reported in Table 2 reveal that the test statistics is overwhelmingly at a 1% significant level, thus providing support for the rejection of the null hypothesis that the residuals are linearly dependent at all possible dimensions at all levels of significance. The rejection of linearity dependence implies that the relationship is nonlinear, hence paving the way for using nonlinear modeling. Therefore, the result provides strong evidence of nonlinearity in the relationship between access to FI and remittances which is preferred in detecting the asymmetric effects.

The application of the bounds test for cointegration requires that the variables under review should not be integrated of order two, I(2); otherwise, this conclusion implies that employing the ordinary estimation would yield inconsistent and spurious results (Maxwell & Nicholas 2021; Yusuf et al. 2012). The ADF (Dickey & Fuller 1979) unit root test approach was applied. Table 3 reports the ADF results for all unit root tests, where they demonstrate that none of the series under review is I(2). They are indeed either integrated of order zero (at level) or one (at the first difference), hence justifying the application of linear and nonlinear ARDL models.

As indicated in Table 1, the time series lack the linearity feature; hence, the nonlinear unit root test is implemented to explore the features of unit roots. Table 4 presents the Kruse (2011) nonlinear unit root test results of the three cases of raw, demeaned, or detrended series. The results of rejecting the null hypothesis are mixed in the three cases. Therefore, these results indicate that the series is stationary and nonstationary depending on the test type (raw, demeaned, or detrended).