Evaluation of environmental geochemical signatures due to RO rejects on arid agricultural farms and tangible solutions

Abstract

The impact of reverse osmosis (RO) rejects in the groundwater presents a significant challenge in arid regions. This study collected groundwater samples, product water, and reverse osmosis brine (ROB) from evaporation ponds and analyzed them for major ions and trace elements. Test boreholes were drilled near the ROB site along the flow direction, and borehole sediment samples were collected. The samples were predominantly gravelly sand, and the depth to water level fluctuated around 30 m below ground level (bgl), with minerals mainly consisting of calcite, gypsum, and quartz. Data loggers reflected a rise in water level (<22 m bgl) corresponding to higher electrical conductivity (>16 mS/cm) during the cropping period in many locations, confirming the impact of ROB in groundwater. The results were further supported by enriched signatures of δ¹⁸O (∼ +1.5‰) and δ²H (∼ +15‰). The saturation index of the minerals reflected that carbonate minerals (Calcite > Dolomite) were saturated in the ROB relative to the groundwater. The vertical variation of mineral assemblages in the boreholes indicated gypsum precipitation in the capillary zone along with calcite and dolomite. The assemblage varies as the groundwater moves from the disposal site. The speciation of different compounds along the groundwater path indicated higher carbonate and sulfate species (CaCO₃ > CaHCO₃> CaSO₄ > NaSO₄ > MgSO₄) near the disposal site, with variations along the flow direction. Considering the significant variation in temperature in the region (5 to 50 ℃), the water sample composition was modeled using PHREEQC, suggesting that the increase in temperature led to supersaturation of epsomite and gypsum compositions. The ROB was theoretically mixed with groundwater and product water in different proportions, and an optimum composition (10:90) for safe disposal was derived and tested fit for reuse in agriculture.

Keywords

Brine disposal / Groundwater / Geochemistry / Monitoring / Modeling / Recommendation

CRediT authorship contribution statement

Mohammed Musah: Conceptualization, Data curation, Formal analysis, Methodology. Stephen Taiwo Onifade: Conceptualization, Methodology, Supervision, Writing – original draft, Writing – review & editing. Elma Satrovic: Data curation, Writing – original draft, Visualization. Joseph Akwasi Nkyi: Writing – original draft.

Declaration of interests

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.

Appendix.

CS-ARDL model specifications

The study’s CS-ARDL models augmented with the lags of the cross-sectional averages of the input and output variables to account for cross-sectional dependence are specified as;

Interactive Model:(A1)${\mathrm{lnEFP}}_{\mathit{it}}={\mathrm{w}}_{\mathrm{i}}+\sum _{\mathrm{j}=1}^{{\mathrm{p}}_{\mathrm{y}}}{\mathrm{\lambda }}_{\mathit{ij}}{\mathrm{lnEFP}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{1\mathrm{j}}{\mathrm{lnFG}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{2\mathrm{j}}{\mathrm{lnGI}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{3\mathrm{j}}{\mathrm{lnY}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{4\mathrm{j}}{{\mathrm{lnY}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{5\mathrm{j}}{\mathrm{lnNRR}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathrm{\delta }}_{1\mathrm{j}}{\ln \left(\mathrm{FG}\ast \mathrm{GI}\right)}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{1\mathrm{j}}{\overline{\mathrm{lnEFP}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{2\mathrm{j}}{\overline{\mathrm{lnFG}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{3\mathrm{j}}{\overline{\mathrm{lnGI}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{4\mathrm{j}}{\overline{\mathrm{lnY}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{5\mathrm{j}}{\overline{{\mathrm{lnY}}^2}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{6\mathrm{j}}{\overline{\mathrm{lnNRR}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{7\mathrm{j}}{\overline{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}}_{\mathrm{t}-\mathrm{j}}+{\mathrm{\epsilon }}_{\mathrm{i}\mathrm{t}}$${\mathrm{lnEFP}}_{\mathit{it}}={\mathrm{w}}_{\mathrm{i}}+\sum _{\mathrm{j}=1}^{{\mathrm{p}}_{\mathrm{y}}}{\mathrm{\lambda }}_{\mathit{ij}}{\mathrm{lnEFP}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{1\mathrm{j}}{\mathrm{lnFG}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{2\mathrm{j}}{\mathrm{lnGI}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{3\mathrm{j}}{\mathrm{lnY}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{4\mathrm{j}}{{\mathrm{lnY}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{5\mathrm{j}}{\mathrm{lnNRR}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathrm{\delta }}_{1\mathrm{j}}{\ln \left(\mathrm{FG}\ast \mathrm{GI}\right)}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{1\mathrm{j}}{\overline{\mathrm{lnEFP}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{2\mathrm{j}}{\overline{\mathrm{lnFG}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{3\mathrm{j}}{\overline{\mathrm{lnGI}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{4\mathrm{j}}{\overline{\mathrm{lnY}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{5\mathrm{j}}{\overline{{\mathrm{lnY}}^2}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{6\mathrm{j}}{\overline{\mathrm{lnNRR}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{7\mathrm{j}}{\overline{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}}_{\mathrm{t}-\mathrm{j}}+{\mathrm{\epsilon }}_{\mathrm{i}\mathrm{t}}$

Nonlinear Model:(A2)${\mathrm{lnEFP}}_{\mathit{it}}={\mathrm{w}}_{\mathrm{i}}+\sum _{\mathrm{j}=1}^{{\mathrm{p}}_{\mathrm{y}}}{\mathrm{\lambda }}_{\mathit{ij}}{\mathrm{lnEFP}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{1\mathrm{j}}{\mathrm{lnFG}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{2\mathrm{j}}{\mathrm{lnGI}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{3\mathrm{j}}{\mathrm{lnY}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{4\mathrm{j}}{{\mathrm{lnY}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{5\mathrm{j}}{\mathrm{lnNRR}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathrm{\pi }}_{1\mathrm{j}}{{\mathrm{lnFG}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{1\mathrm{j}}\overline{\mathrm{lnEFP}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{2\mathrm{j}}\overline{\mathrm{lnFG}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{3\mathrm{j}}\overline{\mathrm{lnGI}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{4\mathrm{j}}\overline{\mathrm{lnY}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{5\mathrm{j}}\overline{{\mathrm{lnY}}^2}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{6\mathrm{j}}{\overline{\mathrm{lnNRR}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{7\mathrm{j}}\overline{{\mathrm{lnFG}}^2}}_{\mathrm{t}-\mathrm{j}}+{\mathrm{\epsilon }}_{\mathrm{i}\mathrm{t}}$${\mathrm{lnEFP}}_{\mathit{it}}={\mathrm{w}}_{\mathrm{i}}+\sum _{\mathrm{j}=1}^{{\mathrm{p}}_{\mathrm{y}}}{\mathrm{\lambda }}_{\mathit{ij}}{\mathrm{lnEFP}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{1\mathrm{j}}{\mathrm{lnFG}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{2\mathrm{j}}{\mathrm{lnGI}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{3\mathrm{j}}{\mathrm{lnY}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{4\mathrm{j}}{{\mathrm{lnY}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathit{\beta }}_{5\mathrm{j}}{\mathrm{lnNRR}}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{{\mathrm{p}}_{\mathrm{x}}}{\mathrm{\pi }}_{1\mathrm{j}}{{\mathrm{lnFG}}^2}_{\mathit{it}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{1\mathrm{j}}\overline{\mathrm{lnEFP}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{2\mathrm{j}}\overline{\mathrm{lnFG}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{3\mathrm{j}}\overline{\mathrm{lnGI}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{4\mathrm{j}}\overline{\mathrm{lnY}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{5\mathrm{j}}\overline{{\mathrm{lnY}}^2}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{\mathrm{\gamma }}_{6\mathrm{j}}{\overline{\mathrm{lnNRR}}}_{\mathrm{t}-\mathrm{j}}+\sum _{\mathrm{j}=0}^{\mathrm{p}}{{\mathrm{\gamma }}_{7\mathrm{j}}\overline{{\mathrm{lnFG}}^2}}_{\mathrm{t}-\mathrm{j}}+{\mathrm{\epsilon }}_{\mathrm{i}\mathrm{t}}$where $\overline{\mathrm{lnEFP}}$$\overline{\mathrm{lnEFP}}$, $\overline{\mathrm{lnFG}}$$\overline{\mathrm{lnFG}}$, $\overline{\mathrm{lnGI}}$$\overline{\mathrm{lnGI}}$, $\overline{\mathrm{lnY}}$$\overline{\mathrm{lnY}}$, $\overline{{\mathrm{lnY}}^2},\overline{\mathrm{lnNRR}}$$\overline{{\mathrm{lnY}}^2},\overline{\mathrm{lnNRR}}$, $\overline{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}$$\overline{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}$, and $\overline{{\mathrm{lnFG}}^2}$$\overline{{\mathrm{lnFG}}^2}$ are the cross-sectional means of the endogenous and exogenous series correspondingly.

CCEMG model specifications

In the CCEMG procedure, the estimated model is augmented with cross-sectional averages of the regressors to account for cross-sectional correlations. Our study’s augmented CCEMG models to control for cross-sectional dependence are specified as;

Interactive Model:(A3)${\mathrm{lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0+{\mathit{\beta }}_1{\mathrm{lnFG}}_{\mathit{it}}{+{\mathit{\beta }}_2{\mathrm{lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{lnY}}_{\mathit{it}}+\mathit{\beta }}_4{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{lnNRR}}_{\mathit{it}}+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+{\mathrm{\psi }}_1\overline{{\mathrm{lnFG}}_{\mathit{it}}}+{\mathrm{\psi }}_2\overline{{\mathrm{lnGI}}_{\mathit{it}}}+{\mathrm{\psi }}_3\overline{{\mathrm{lnY}}_{\mathit{it}}}+{\mathrm{\psi }}_4\overline{{{\mathrm{lnY}}^2}_{\mathit{it}}}+{\mathrm{\psi }}_5\overline{{\mathrm{lnNRR}}_{\mathit{it}}}+{\mathrm{\psi }}_6\overline{{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}}{+\mathrm{\epsilon }}_{\mathit{it}}$${\mathrm{lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0+{\mathit{\beta }}_1{\mathrm{lnFG}}_{\mathit{it}}{+{\mathit{\beta }}_2{\mathrm{lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{lnY}}_{\mathit{it}}+\mathit{\beta }}_4{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{lnNRR}}_{\mathit{it}}+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+{\mathrm{\psi }}_1\overline{{\mathrm{lnFG}}_{\mathit{it}}}+{\mathrm{\psi }}_2\overline{{\mathrm{lnGI}}_{\mathit{it}}}+{\mathrm{\psi }}_3\overline{{\mathrm{lnY}}_{\mathit{it}}}+{\mathrm{\psi }}_4\overline{{{\mathrm{lnY}}^2}_{\mathit{it}}}+{\mathrm{\psi }}_5\overline{{\mathrm{lnNRR}}_{\mathit{it}}}+{\mathrm{\psi }}_6\overline{{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}}{+\mathrm{\epsilon }}_{\mathit{it}}$

Nonlinear Model:(A4)${\mathrm{lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0+{\mathit{\beta }}_1{\mathrm{lnFG}}_{\mathit{it}}{+{\mathit{\beta }}_2{\mathrm{lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{lnY}}_{\mathit{it}}+\mathit{\beta }}_4{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{lnNRR}}_{\mathit{it}}+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+{\mathrm{\Omega }}_1\overline{{\mathrm{lnFG}}_{\mathit{it}}}+{\mathrm{\Omega }}_2\overline{{\mathrm{lnGI}}_{\mathit{it}}}+{\mathrm{\Omega }}_3\overline{{\mathrm{lnY}}_{\mathit{it}}}+{\mathrm{\Omega }}_4\overline{{{\mathrm{lnY}}^2}_{\mathit{it}}}+{\mathrm{\Omega }}_5\overline{{\mathrm{lnNRR}}_{\mathit{it}}}+{\mathrm{\Omega }}_6\overline{{{\mathrm{lnFG}}^2}_{\mathit{it}}}{+\mathrm{\epsilon }}_{\mathit{it}}$${\mathrm{lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0+{\mathit{\beta }}_1{\mathrm{lnFG}}_{\mathit{it}}{+{\mathit{\beta }}_2{\mathrm{lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{lnY}}_{\mathit{it}}+\mathit{\beta }}_4{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{lnNRR}}_{\mathit{it}}+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+{\mathrm{\Omega }}_1\overline{{\mathrm{lnFG}}_{\mathit{it}}}+{\mathrm{\Omega }}_2\overline{{\mathrm{lnGI}}_{\mathit{it}}}+{\mathrm{\Omega }}_3\overline{{\mathrm{lnY}}_{\mathit{it}}}+{\mathrm{\Omega }}_4\overline{{{\mathrm{lnY}}^2}_{\mathit{it}}}+{\mathrm{\Omega }}_5\overline{{\mathrm{lnNRR}}_{\mathit{it}}}+{\mathrm{\Omega }}_6\overline{{{\mathrm{lnFG}}^2}_{\mathit{it}}}{+\mathrm{\epsilon }}_{\mathit{it}}$From the above equations $\overline{{\mathrm{lnFG}}_{\mathit{it}}}$$\overline{{\mathrm{lnFG}}_{\mathit{it}}}$, $\overline{{\mathrm{lnGI}}_{\mathit{it}}}$$\overline{{\mathrm{lnGI}}_{\mathit{it}}}$, $\overline{{\mathrm{lnY}}_{\mathit{it}}}$$\overline{{\mathrm{lnY}}_{\mathit{it}}}$, $\overline{{{\mathrm{lnY}}^2}_{\mathit{it}},}$$\overline{{{\mathrm{lnY}}^2}_{\mathit{it}},}$ $\overline{{\mathrm{lnNRR}}_{\mathit{it}}}$$\overline{{\mathrm{lnNRR}}_{\mathit{it}}}$, $\overline{{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}}$$\overline{{\ln \left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}}$, and $\overline{{{\mathrm{lnFG}}^2}_{\mathit{it}}}$$\overline{{{\mathrm{lnFG}}^2}_{\mathit{it}}}$ denote the cross-sectional averages of the input variables.

AMG model specifications

The AMG approach follows a two-staged process. At the first stage, Eqs. (1), (2), and (3) are specified in a T−1 dummies and first differenced form as;

Interactive Model:(A5)${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0{+\mathit{\beta }}_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\mathit{\beta }}_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\mathit{\beta }}_4\mathrm{\Delta }{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\sum _{\mathrm{t}=2}^{\mathrm{T}}{\mathrm{\varnothing }}_{\mathrm{t}}\left({\mathrm{\Delta }\mathrm{D}}_{\mathrm{t}}\right){+\mathrm{\epsilon }}_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\mathit{\alpha }}_0{+\mathit{\beta }}_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\mathit{\beta }}_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\mathit{\beta }}_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\mathit{\beta }}_4\mathrm{\Delta }{{\mathrm{lnY}}^2}_{\mathit{it}}+{\mathit{\beta }}_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\sum _{\mathrm{t}=2}^{\mathrm{T}}{\mathrm{\varnothing }}_{\mathrm{t}}\left({\mathrm{\Delta }\mathrm{D}}_{\mathrm{t}}\right){+\mathrm{\epsilon }}_{\mathit{it}}$

Nonlinear Model:(A6)${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_0{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\sum _{t=2}^T{\mathrm{\varnothing }}_t\left({\mathrm{\Delta }D}_t\right){+\epsilon }_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_0{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\sum _{t=2}^T{\mathrm{\varnothing }}_t\left({\mathrm{\Delta }D}_t\right){+\epsilon }_{\mathit{it}}$From the equations above, ${\mathrm{\Delta }\mathrm{D}}_{\mathrm{t}}$${\mathrm{\Delta }\mathrm{D}}_{\mathrm{t}}$ is the first difference order of T−1 dummies, and ${\mathrm{\varnothing }}_{\mathrm{t}}$${\mathrm{\varnothing }}_{\mathrm{t}}$ is its coefficient.

In the second stage, common dynamic processes are formed by transforming ${\mathrm{\varnothing }}_{\mathrm{t}}$${\mathrm{\varnothing }}_{\mathrm{t}}$ to ${\mathrm{P}}_{\mathrm{t}}$${\mathrm{P}}_{\mathrm{t}}$(${\mathrm{\varnothing }}_{\mathrm{t}}$${\mathrm{\varnothing }}_{\mathrm{t}}$ = ${\mathrm{P}}_{\mathrm{t}}$${\mathrm{P}}_{\mathrm{t}}$) as;

Interactive Model:(A7)${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_{\mathrm{i}}{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+P_t\left(d_t\right){+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\epsilon }_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_{\mathrm{i}}{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+P_t\left(d_t\right){+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\epsilon }_{\mathit{it}}$(A8)${\mathrm{\Delta lnEFP}}_{\mathit{it}}-P_t\left(d_t\right)={\alpha }_i{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}{+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\epsilon }_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}-P_t\left(d_t\right)={\alpha }_i{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}{+{\mathrm{\delta }}_1\ln {\left(\mathrm{GI}\ast \mathrm{FG}\right)}_{\mathit{it}}+\epsilon }_{\mathit{it}}$

Nonlinear Model:(A9)${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_i{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+P_t\left(d_t\right){+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\epsilon }_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}={\alpha }_i{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}+P_t\left(d_t\right){+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\epsilon }_{\mathit{it}}$(A10)${\mathrm{\Delta lnEFP}}_{\mathit{it}}-P_t\left(d_t\right)={\alpha }_{\mathrm{i}}{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}{+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\epsilon }_{\mathit{it}}$${\mathrm{\Delta lnEFP}}_{\mathit{it}}-P_t\left(d_t\right)={\alpha }_{\mathrm{i}}{+\beta }_1{\mathrm{\Delta lnFG}}_{\mathit{it}}+{\beta }_2{\mathrm{\Delta lnGI}}_{\mathit{it}}+{\beta }_3{\mathrm{\Delta lnY}}_{\mathit{it}}+{\beta }_4\mathrm{\Delta ln}{{\mathrm{Y}}^2}_{\mathit{it}}+{\beta }_5{\mathrm{\Delta lnNRR}}_{\mathit{it}}{+{{\mathrm{\pi }}_1{\mathrm{lnFG}}^2}_{\mathit{it}}+\epsilon }_{\mathit{it}}$where ${\mathrm{d}}_{\mathrm{t}}$${\mathrm{d}}_{\mathrm{t}}$ is the dynamic process.

After the transformations, the elasticities of the predictors are respectively computed as;

Interactive Model:(A11)${\mathit{\beta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{1\mathrm{i}},{\mathit{\beta }}_{2,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{2\mathrm{i}},{\mathit{\beta }}_{3,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{3\mathrm{i}},{\mathit{\beta }}_{4,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{4\mathrm{i}},{\mathit{\beta }}_{5,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{5\mathrm{i}},{\mathrm{\delta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{\delta }}_{1\mathrm{i}},$${\mathit{\beta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{1\mathrm{i}},{\mathit{\beta }}_{2,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{2\mathrm{i}},{\mathit{\beta }}_{3,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{3\mathrm{i}},{\mathit{\beta }}_{4,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{4\mathrm{i}},{\mathit{\beta }}_{5,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{5\mathrm{i}},{\mathrm{\delta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{\delta }}_{1\mathrm{i}},$

Nonlinear Model:(A12)${\mathit{\beta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{1\mathrm{i}},{\mathit{\beta }}_{2,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{2\mathrm{i}},{\mathit{\beta }}_{3,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{3\mathrm{i}},{\mathit{\beta }}_{4,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{4\mathrm{i}},{\mathit{\beta }}_{5,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{5\mathrm{i}},{\mathrm{\pi }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{\pi }}_{1\mathrm{i}},$${\mathit{\beta }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{1\mathrm{i}},{\mathit{\beta }}_{2,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{2\mathrm{i}},{\mathit{\beta }}_{3,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{3\mathrm{i}},{\mathit{\beta }}_{4,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{4\mathrm{i}},{\mathit{\beta }}_{5,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathit{\beta }}_{5\mathrm{i}},{\mathrm{\pi }}_{1,\mathrm{A}\mathrm{M}\mathrm{G}}=\frac{\mathrm{I}}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{\pi }}_{1\mathrm{i}},$

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