### Model specification

The study employs the gravity model to establish a relationship between real effective exchange rate and cross-border trade balance. The gravity model represents the bilateral trade model pioneered by Timbergen [40], Poyhonen [34] and Linneman [29]. Bergstrand [12, 13] demonstrated its empirical robustness. The bilateral trade model adopted in this study follows Kodongo and Ojah [27], Boke and Doganay [14], Bahmani-Oskooee and Fariditavana [7] and Arize et al. [4]. This is represented below

$${\text{NCBT}}_{ij} = \beta_{0} Y_{i}^{\beta 1} Y_{j}^{\beta 2} {\text{REX}}_{i}^{\beta 3} U_{ij}$$

(1)

where \({\text{NCBT}}_{ij}\) is the net cross-border trade between country *i* and *j*. \(i\) is the country under investigation (Nigeria) while \(j\) is the cross-border trading partner country (i.e., Benin Republic). In this paper, our definition of cross-border trade includes the trading activities that take place between two different nationals at the various markets close to the border. Also, any recorded merchandise goods that crosses the border from country \(i\) to country \(j\) and vice versa is taken into consideration. For the purpose of this study, CBT is measured as the value of export of country \(i\) to country \(j\) divided by the value of import from country \(j\) to country \(i\). This is similar to the measure adopted in Hop et al. (2013), Antweiler [2], Bahmani-Oskooee and Fariditavana [7] and Arize et al. [4]. \({\text{RER}}_{i}\) represents the real exchange rate in country \(i,\) proxied by real effective exchange rate (REER). The choice of using real effective exchange rate in place of real exchange rate in this study is justified in the literature as being appropriate for a bilateral trade analysis (see [4]). Real effective exchange rate is a measure of average of exchange rates between two countries and shows the fluctuation in the general value of the currency. Taking natural log of both sides, we have;

$$\ln {\text{NCBT}}_{ij} = \ln \beta_{0} + \beta_{1} \ln Y_{i} + \beta_{2} \ln Y_{j} + \beta_{3} \ln {\text{REER}}_{i} + U_{ij}$$

(2)

We begin the specification of the long-run relationship between real effective exchange rate cross-border trade and, enhanced with real domestic and foreign income by rewriting Eq. (2) in the following way [4];

$$\ln {\text{NCBT}}_{t} - \ln \beta_{0} - \beta_{1} \ln Y_{i} - \beta_{2} \ln Y_{j} - \beta_{3} \ln {\text{REER}}_{i} = U_{t}$$

(3)

$${\text{ncbt}}_{t} - \beta_{0} - \beta_{1} y_{i} - \beta_{2} y_{j} - \beta_{3} {\text{reer}}_{t} = \varepsilon_{t}$$

(4)

where \({\text{ncbt}}_{t}\) represents the natural log of the value of Nigeria’s export divided by the value of its import from cross-border trading partner; \({\text{reer}}_{t}\) measures real effective exchange rate and it is described in a way that its increase would mean effective depreciation; \(y_{i}\) is a series of real domestic income while \(y_{j}\) represents that of trading partner country. \(\varepsilon_{t}\) represents the disturbing variable. All variables in abridged form of Eq. (4) are converted into natural log. This allows their slope coefficients to represent the elasticity of the regressand in line with the regressors. The unobservable term calculates the extent of disequilibrium and is given as;

$${\text{ec}}_{t} = {\text{ncbt}}_{t} - \hat{\beta }_{0} - \hat{\beta }_{1} y_{i} - \hat{\beta }_{2} y_{j} - \hat{\beta }_{3} {\text{reer}}_{t}$$

(5)

We further generate a one-lagged error-correction term to estimate parameters associated with the short-run model as;

$${\text{ec}}_{t - 1} = {\text{ncbt}}_{t - 1} - \hat{\beta }_{0} - \hat{\beta }_{1} y_{it - 1} - \hat{\beta }_{2} y_{jt - 1} - \hat{\beta }_{3} {\text{reer}}_{t - 1}$$

(6)

According to Arize et al. [4], the effect of the exchange rate fluctuation on trade does not seem to have a priori direction or is multi-faceted, that is, \(\hat{\beta }_{3}\) might be either negative or positive. If \({\text{reer}}_{t}\) falls, competitiveness in prices increases in the local country, which might result in more export with less import (volume effect). On the other hand, an increase in \({\text{reer}}_{t}\) increases the values of each import units (import value effect), which is capable of reducing the trade balance. In testing for nonlinear readjustment or the presence of asymmetry in the relationship between exchange rate and bilateral/cross-border trade in Nigeria, we begin by incorporating likelihood of asymmetric nonlinear re-adjustments to equilibrium. In achieving this, this study employs alternative functional form of *f* (\({\text{ec}}_{t}\)) to model the likely nonlinear outcome of \({\text{ec}}_{t}\) on the net cross-border trade as follows;

$$f({\text{ec}}_{t} ) = \left\{ {\begin{array}{*{20}l} {\delta {\text{ec}}_{t - 1}^{3} + \delta {\text{ec}}_{t - 1}^{4} } \hfill \\ {\delta {\text{ec}}_{t - 1}^{2} + \delta {\text{ec}}_{t - 1}^{4} } \hfill \\ {\delta {\text{ec}}_{t - 1}^{2} + \delta {\text{ec}}_{t - 1}^{3} + \delta {\text{ec}}_{t - 1}^{4} } \hfill \\ \end{array} } \right.$$

(7)

The Eq. (7) above agrees with the error correction cum autoregressive distributed lag (ARDL) model discussed in Eq. (10). Equation (7) is a nonlinear model based on Sollis [39]. It is not the model we have estimated but it can be used to capture how a nonlinear relationship can be eventually derived from an asymmetric processes. It suggests that a deviation from equilibrium is initially an asymmetric phenomenon. This means that an asymmetric error process is the main source of any possible nonlinearity. Equation (7) merely helps us to acknowledge the presence of asymmetry in the speed of reversion within positive and negative deviations or asymmetry at the point which positive and negative reverts back to the mean (see [4]). This is not to conclude that any nonlinear process is completely asymmetric but two variables with supposedly nonlinear relationship initially exhibit asymmetric behavior (see [10]).

As mentioned in “Methods” section, the nonlinear cointegrating model popularized recently by Shin et al. [38] has the undoubted benefit of giving room for short- and long-run asymmetry. The nonlinear autoregressive distributed lag (NARDL) technique supports a dynamic error correction illustration linked to asymmetric long-run cointegrating regression, and the technique permits a relationship to display long-run asymmetry, short-run asymmetry and combined short- and long-run asymmetries. It similarly supports the probability of quantifying the successive responses of the regressands to negative and positive shocks of the regressors from the asymmetric dynamic multipliers.

From Schodert [36], Shin et al. [38], Arize et al. [4], we begin the NARDL technique by building new variables that explain instances of appreciation and depreciation. The simple idea involves disintegrating the time series,\({\text{reer}}_{t}\), into two, namely \(\left( {{\text{reer}}_{t}^{ + } } \right)\) and \(\left( {{\text{reer}}_{t}^{ - } } \right)\) as follows;

$${\text{reer}}_{t}^{ + } = \mathop \sum \limits_{j = 1}^{t} \Delta {\text{reer}}_{t}^{ + } = \mathop \sum \limits_{j = 1}^{t} \hbox{max} \left( {\Delta {\text{reer}}_{t} , 0} \right)$$

(8)

$${\text{reer}}_{t}^{ - } = \mathop \sum \limits_{j = 1}^{t} \Delta {\text{reer}}_{t}^{ - } = \mathop \sum \limits_{j = 1}^{t} \hbox{min} \left( {\Delta {\text{reer}}_{t} , 0} \right)$$

(9)

where \(\Delta {\text{reer}}_{t}^{ + }\) and \(\Delta {\text{reer}}_{t}^{ - }\) represent fractional sums of depreciations and appreciations, respectively. Following Shin et al. [38], the trade balance equality is denoted by the asymmetric ARDL model in Eq. (10);

$$\begin{aligned} \Delta {\text{ncbt}}_{t} & = \omega_{0} + \omega_{1} {\text{ncbt}}_{t - 1} + \omega_{2}^{+} {\text{reer}}_{t - 1}^{+} + \omega_{2}^{-} {\text{reer}}_{t - 1}^{-} + \omega_{3} y_{i - 1} + \omega_{4} y_{j - 1} \\ & \quad + \mathop \sum \limits_{t = 1}^{p} \rho_{i} \Delta {\text{ncbt}}_{t - i} + \mathop \sum \limits_{t = 0}^{r} \rho_{i}^{+} \Delta {\text{reer}}_{t - i}^{+} + \mathop \sum \limits_{t = 0}^{r} \epsilon_{i}^{-} \Delta {\text{reer}}_{t - i}^{-} + \mathop \sum \limits_{t = 0}^{q} \epsilon_{i} \Delta y_{{i\left({t - 1} \right)}} \\ & \quad + \mathop \sum \limits_{t = 0}^{s} \epsilon_{i} \Delta y_{{j\left({t - 1} \right)}}^{{}} + \mu_{t} \\ \end{aligned}$$

(10)

where for instance, \(- \frac{{\omega_{2}^{ + } }}{{\omega_{1} }}, - \frac{{\omega_{2}^{ - } }}{{\omega_{1} }} ,\) give coefficients of long-run real effective exchange rate. Equation (10) allows the possibility that the process being modeled can exhibit asymmetries in the short run, the long run, long run and short run only. The level terms in Eq. (10) represent the long-run association. Short-run asymmetries are then captured by the lags of asymmetric exchange rate terms that show in the first-differenced setup. We can now estimate the autoregressive distributed lag (ARDL) model in Eq. (10) and subsequently test for the presence of asymmetric nonlinear (cointegrating) long-run association. This could be done using two techniques: one, conduct a *t* test on the coefficient of the error-correction term (ECT) where the null is set as \(\omega_{1} = 0\), and the alternative hypothesis as \(\omega_{1} \ne 0\). If the null hypothesis cannot be rejected, Eq. (10) will reduce to the linear regression with only first differences signifying that there is no long-run relationship among the levels of \({\text{ncbt}},{\text{reer}}_{t}^{ + } ,{\text{reer}}_{t}^{ - } , y_{t - 1} \;{\text{and}}\;y_{j - 1} .\)

The critical value of the said t test of \(\omega_{1} = 0\) is stated in Shin, Pesaran et al. [33]. The other test is known as F_{PSS}. We base this on F-test for the joint null hypothesis that coefficients of all variables are jointly equal to 0 \(\left( {H_{0} :\omega_{1} = \omega_{2}^{ + } = \omega_{2}^{ - } = \omega_{3} = \omega_{4} = 0 } \right)\) against the alternative hypothesis that the coefficients of lagged leveled variables are not equal to 0 jointly \(\left( {H_{1} :\omega_{1} \ne \omega_{2}^{ + } \ne \omega_{2}^{ - } \ne \omega_{3} \ne \omega_{4} \ne 0 } \right)\) or \(\left( {H_{1} :\omega_{1} \;{\text{or}}\;\omega_{2}^{ + } \;{\text{or}}\;\omega_{2}^{ - } \;{\text{or}}\;\omega_{3} \;{\text{or}}\;\omega_{4} \ne 0} \right).\) Also as reported in Pesaran et al. [33], the critical values are reported. The long-run coefficients of real effective exchange rate are obtained thus; \(L_{\text{r}}^{ + } = - \frac{{\omega_{2}^{ + } }}{{\omega_{1} }}\) and \(L_{\text{r}}^{ - } = - \frac{{\omega_{2}^{ - } }}{{\omega_{1} }}\). The long-run asymmetry can be investigated using the popular Wald test, where symmetry implies that;

$$L_{\text{r}}^{ + } = L_{\text{r}}^{ - }$$

### Results

Table 3 in the appendix shows the short- and long-run estimate of the nonlinear relationship that exists between exchange rate movements and cross-border trade in Nigeria. The essence of the analysis is premised not only on the unending insignificant relationships that exist between exchange rate and bilateral trade as reported by many authors but also on the high probability of model misspecification.

In the short run, exchange rate depreciation increases cross-border trade by 23% but not statistically significant. Also, worthy of note is that there is a positive relationship between foreign GDP (YBN) and cross-border trade in the short-run. In the long-run where most models of trade usually draw inferences from; it is observed that when real effective exchange rate appreciates by a percentage, cross-border trade volume decreases by 42%. The unique thing about this relationship is that it is significant at 5% level. This is contrary to the results obtained by various authors who have worked with linear models (see [11, 14]). Also, one percent increase in domestic GDP (YNG) and foreign GDP (YBN) will increase cross-border trade by 3 and 9%, respectively. This result indicates that Beninese GDP has more impact on cross-border trade in Nigeria than does Nigeria’s GDP itself. The above findings are in the same line of thought with Hoffman and Melly [24], Kodongo and Ojah [20], Hashim and Meagher [27] where it was discovered that Beninese GDP determines cross-border trade more than that of any of her regional cross-border trading partners. Further, our error-correction term (ECT), as an important long-run phenomenon, assumes the expected negative sign and it shows that 96% of the error in the short period disequilibrium is corrected in the long run. This value suggests that there is a considerable very high speed of adjustment to the last period’s disequilibrium.

#### Cointegration test

A cointegration test approach to ARDL analysis is necessary here mainly because our analysis is based on the nonlinear ARDL Model. It is therefore very important to test for asymmetric cointegration. We have employed the Wald Test to achieve this simple but all important task. Table 1 in the appendix presents the results from the asymmetric cointegration test. Here, a null hypothesis of no cointegration is set against the alternative hypothesis of presence of cointegration. It is evident from Table 1 that the value of the F-cal is greater than that of F-tab using the critical value based on the appropriate degree of freedom in the Pesaran et al. [33] statistical table (i.e., 6.083 > 4.35). The result confirms that asymmetric cointegration exists among the variables in the nonlinear model. This is similar to the findings of Arize et al. [4] but contrary to Bahmani-Oskooee and Fariditavana [7].

#### Testing for the presence of asymmetry

In this study, for us to conclude that the exchange rate of equal changes (either appreciation or depreciation) will have different effect on cross-border trade, we necessarily have to discover the presence of asymmetry in the nonlinear model. Table 2 presents the results from the test for the presence or absence of asymmetry using the Wald Test. This results exhibit the computation of the long-run coefficients of the real effective exchange rate as demonstrated in Eq. (10) back in chapter three. Table 1 shows that there is long-run asymmetry in the relationship that exists between real effective exchange rate (appreciation and depreciation) and cross-border trade in Nigeria.

Following Verheyen [41] and Arize et al. [4], for us establish the absence of asymmetry based on the Wald Test, the values of the *t*-stat and *F*-stat must be equal but in this case, they are not equal. Also, Bahmani-Oskooee and Fariditavana [7] posit that asymmetry occurs when long-run coefficient of real effective exchange rate appreciation (\(L_{\text{R}}^{ - }\)) is not equal to the long-run coefficient of real exchange rate depreciation (\(L_{\text{R}}^{ + }\)). That is;

$$(L_{\text{R}}^{ - } \ne L_{\text{R}}^{ + } )$$

From our analysis, \(L_{\text{R}}^{ + }\) is found to be 43.9 while (\(L_{\text{R}}^{ - }\)) is 40.2. This further confirms the presence of asymmetry in the relationship that exists between these two variables that form the center of this study. The fresh findings can go a long way in clearing the air on the debate on whether exchange rate movement has any impact on bilateral trade in the literature.