### Data, variables, and the models

The methodology used in this study was based the standard tools of growth empirics and the sample period spans from 1961 to 2015. The selection of this period was based on the data availability for France. In light of the method's inherent characteristics and the qualities of time series data employed, this study examined two distinct models for dealing with structural breaks. While the study used the ARDL bounds testing technique to cointegration for the long-run and short-run dynamics, this approach was confirmed using the Johansen cointegration test. We detect direction of causation of the forecast variables using the VECM Granger causality test. Using the standard tools of growth empirics, we specify the following functional forms:

$$Y_{t} \, = \,f(EC_{t} ,F_{t} ,K_{t} ,TRD_{t} ),$$

(1)

where \(Y\) denotes the real GDP per capita, \(EC\) is the electric power consumption measured in kWh per capita, \(F\) is the domestic credit to the private sector as the percentage of GDP, \(K\) is the natural real capital stock per capita. \(TRD\) represents trade measured in three ways: first, \(TO\) which is the trade openness, i.e. the sum of total exports and imports as the percentage of GDP. Second, \(EX\) which denotes the total exports per capita, defined as the total exports of goods and services in current USD divided by the total population and \(t\) is the time period. The third way is \(IM\) which is the total import per capita, defined as the total imports of goods and services in current USD divided by the total population. To ensure consistency, the study transformed all the series into natural log specification. This specification is then modelled as follows:

$$\ln Y_{t} \, = \,\beta_{1} + \beta_{2} \ln EC_{t} + \beta_{3} \ln F_{t} + \beta_{4} \ln K_{t} + \beta_{5} \ln TRD_{t} + \mu_{t}$$

(2)

where \(\ln\) denotes the natural logarithms of all the variables at time \(t\) as defined in Eq. (1) with their respective parameters, i.e. \(\beta_{1} - \beta_{5}\) and \(\mu\) is the error term in the model. Furthermore, we consulted the database of the World Bank's World Development Indicators (WDI) for compiling the data for this study, namely: real GDP per capita, domestic credit to the private sector, capital stock per capita, power (electric) consumption (kg of oil equivalent) per capita, exports and imports per capita, and trade openness. All these variables were measured in real terms except domestic credit to the private sector and trade openness, which are not. The period chosen was based on the availability of data for France. For example, electricity consumption and capital stock are available up the period selected for this study.

This paper proceeds to estimate model (2) utilising Pesaran et al. (2001) developed ARDL bounds test approach to cointegration. The study employs this methodology because of its superiority over all other methods of linear regression estimation, including the notable Johansen and Juselius (1990) approach.^{Footnote 1} One of the primary, distinguishing characteristics and superiority of the ARDL model over the Johansen and Juselius method is its capacity to discern between the values of the dependent and independent variables in such a sparse manner. To take this advantage for obtaining a robust solution, Eq. 2 is modelled using the unconditional error correction model (UECM). The following is the specification^{Footnote 2}:

$$\begin{aligned} \Delta \ln Y_{t} = c_{1} + \sum\nolimits_{i = 0}^{p} {d_{11,i} } \Delta \ln Y_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{12,i} } \Delta \ln EC_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{13,i} } \Delta \ln F_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{14,i} } \Delta \ln K_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{15,i} } \Delta \ln TRD_{t - i} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \pi_{11} \ln Y_{t - 1} + \pi_{12} \ln EC_{t - 1} + \pi_{13} \ln F_{t - 1} + \pi_{14} \ln K_{t - 1} + \pi_{15} \ln TRD_{t - 1} + \pi_{1D} DUM_{t} + u_{1t} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{aligned}$$

(3)

$$\begin{aligned} \Delta \ln EC_{t} = c_{2} + \sum\nolimits_{i = 0}^{p} {d_{21,i} } \Delta \ln Y_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{22,i} } \Delta \ln EC_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{23,i} } \Delta \ln F_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{24,i} } \Delta \ln K_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{25,i} } \Delta \ln TRD_{t - i} \\ \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\, + \pi_{21} \ln Y_{t - 1} + \pi_{22} \ln EC_{t - 1} + \pi_{23} \ln F_{t - 1} + \pi_{24} \ln K_{t - 1} + \pi_{25} \ln TRD_{t - 1} + \pi_{2D} DUM_{t} + u_{2t} \,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{aligned}$$

(4)

$$\begin{aligned} \Delta \ln F_{t} = c_{3} + \sum\nolimits_{i = 0}^{p} {d_{31,i} } \Delta \ln Y_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{32,i} } \Delta \ln EC_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{33,i} } \Delta \ln F_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{34,i} } \Delta \ln K_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{35,i} } \Delta \ln TRD_{t - i} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, + \pi_{31} \ln Y_{t - 1} + \pi_{32} \ln EC_{t - 1} + \pi_{33} \ln F_{t - 1} + \pi_{34} \ln K_{t - 1} + \pi_{35} \ln TRD_{t - 1} + \pi_{3D} DUM_{t} + u_{3t} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{aligned}$$

(5)

$$\begin{aligned} \Delta \ln K_{t} = c_{4} + \sum\nolimits_{i = 0}^{p} {d_{41,i} } \Delta \ln Y_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{42,i} } \Delta \ln EC_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{43,i} } \Delta \ln F_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{44,i} } \Delta \ln K_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{45,i} } \Delta \ln TRD_{t - i} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \pi_{41} \ln Y_{t - 1} + \pi_{42} \ln EC_{t - 1} + \pi_{43} \ln F_{t - 1} + \pi_{44} \ln K_{t - 1} + \pi_{45} \ln TRD_{t - 1} + \pi_{4D} DUM_{t} + u_{4t} \,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{aligned}$$

(6)

$$\begin{aligned} \Delta \ln TRD_{t} = & c_{5} + \sum\nolimits_{i = 0}^{p} {d_{51,i} } \Delta \ln Y_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{52,i} } \Delta \ln EC_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{53,i} } \Delta \ln F_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{54,i} } \Delta \ln K_{t - i} + \sum\nolimits_{i = 0}^{p} {d_{55,i} } \Delta \ln TRD_{t - i} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & + \pi_{51} \ln Y_{t - 1} + \pi_{52} \ln EC_{t - 1} + \pi_{53} \ln F_{t - 1} + \pi_{54} \ln K_{t - 1} + \pi_{55} \ln TRD_{t - 1} + \pi_{5D} DUM_{t} + u_{5t} .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{aligned}$$

(7)

In models 3 to 7 above, the first part of each equation denotes the short-term dynamics while the second part captures the long-run dynamics. \(\Delta \mathrm{and} {c}_{1} to {c}_{5}\) signify first differences and drifting elements, respectively. The lag length is given by \(p\) and \(u\) is the white noise residuals.^{Footnote 3} DUM refers to the dummy variables required to capture the break dates in the series.^{Footnote 4} The long-run parameters are obtained by \({\beta }_{i}={\pi }_{ii }/(1-{\sum }_{k=1}^{p}{d}_{ii,k}), i=1,..,5\) while \({d}_{ij}\) represents the short-run parameters. According to Pesaran et al (2001), two critical stages must be followed in order to proceed with the estimating process of the ARDL model, namely: (1) doing an F-test to determine the combined significance of the lagged variables. (2) Determining the null hypothesis for the absence of a long-run relationship by comparing it to the alternative hypothesis, i.e. \({H}_{0}:{\pi }_{i1}={\pi }_{i2}={\pi }_{i3}={\pi }_{i4}={\pi }_{i5}=0\), against \({H}_{1}:{\pi }_{ij}\ne 0, j=1,..,5\).

Following the completion of these two stages, the lower and upper critical boundaries for the F-test are established in accordance with Pesaran et al (2001). These guidelines established that, because the lower bound's critical values are I(0) and the upper bound's critical values are I(1), the rule indicates that whenever the F-statistics are statistically greater than zero, the null hypothesis is rejected.^{Footnote 5} However, if the F-statistics are less than the lower bound, the null hypothesis of no long-run relationship is not rejected in this situation.

Finally, the error correction model (ECM) technique is used to estimate the variables' long- and short-run dynamics. The vector error correction model (VECM) and Granger causality analysis can be used to detect the causational link between capital all the vaiables considered in this study as empirically described in Eq. 8 below:

$$\begin{aligned} \left( \begin{gathered} \Delta \ln Y_{t} \hfill \\ \Delta \ln \,EC_{t} \hfill \\ \Delta \ln F_{t} \hfill \\ \Delta \ln K_{t} \hfill \\ \Delta \ln TRD_{t} \hfill \\ \end{gathered} \right) = & \left( \begin{gathered} c_{1} \hfill \\ c_{2} \hfill \\ c_{3} \hfill \\ c_{4} \hfill \\ c_{5} \hfill \\ \end{gathered} \right) + \left( \begin{gathered} d_{11,1} \,\,\, \cdots \,\,\,\,d_{15,1} \hfill \\ \,\,\,\, \vdots \,\,\,\,\,\, \ddots \,\,\,\,\,\,\,\, \vdots \hfill \\ d_{51,1} \,\,\, \cdots \,\,\,\,d_{55,1} \hfill \\ \end{gathered} \right)\left( \begin{gathered} \Delta \ln Y_{t - 1} \hfill \\ \Delta \ln \,EC_{t - 1} \hfill \\ \Delta \ln F_{t - 1} \hfill \\ \Delta \ln K_{t - 1} \hfill \\ \Delta \ln TRD_{t - 1} \hfill \\ \end{gathered} \right) + \cdots + \left( \begin{gathered} d_{11,p} \,\,\, \cdots \,\,\,\,d_{15,p} \hfill \\ \,\,\,\, \vdots \,\,\,\,\,\, \ddots \,\,\,\,\,\,\,\, \vdots \hfill \\ d_{51,p} \,\,\, \cdots \,\,\,\,d_{55,p} \hfill \\ \end{gathered} \right)\left( \begin{gathered} \Delta \ln Y_{t - p} \hfill \\ \Delta \ln \,EC_{t - p} \hfill \\ \Delta \ln F_{t - p} \hfill \\ \Delta \ln K_{t - p} \hfill \\ \Delta \ln TRD_{t - p} \hfill \\ \end{gathered} \right) & \\ & + \left( \begin{gathered} \gamma_{1} ECM_{1t - 1} \hfill \\ \gamma_{2} ECM_{2t - 1} \hfill \\ \gamma_{3} ECM_{3t - 1} \hfill \\ \gamma_{4} ECM_{4t - 1} \hfill \\ \gamma_{5} ECM_{5t - 1} \hfill \\ \end{gathered} \right)\left( \begin{gathered} \mu_{1t} \hfill \\ \mu_{2t} \hfill \\ \mu_{3t} \hfill \\ \mu_{4t} \hfill \\ \mu_{5t} \hfill \\ \end{gathered} \right). \\ \end{aligned}$$

(8)

In model 8, all the variables remain as previously defined. \({TRD}_{t}\) is the measure of trade in different ways as earlier defined. \(\Delta\) is the difference operator and \(\ln\) is the natural logarithms of variables. Note that the notations in the matrices denote the parameter for each of the five variables both in the long run and the associated short run as well as their error terms, \({\mu }_{i}\), \(i=1,..,5.\) Furthermore, the \({ECM}_{it-1}\) is the error correction model that is obtained from the estimated results of the long-run relationship among the variable. The \({ECM}_{it-1}\) is an indication of how significant and valid the results of the long-run dynamics are. It suggests the validity, strength, and the fitness of the model. It also validates the acceptability or otherwise of the long-run coefficient results. The *T*-test and *F*-test statistics are used to determine first-differences, while lagged independent variables are used to determine the variables' short-run causal link. In the case of France, this study used time series data from 1961 to 2015.