Effects of source facies and maturity on individual carbon isotopic compositions of oil

Abstract

Carbon isotopes have been used extensively in tracing the sources of oil. However, primary source facies and secondary alteration controls on oil isotopic compositions have not been well resolved, resulting in application uncertainties. A case study was undertaken for an alkaline lacustrine oil system in a lower Permian formation in the Junggar Basin, NW China. Results indicate that increasing maturity causes the carbon isotopic composition to become heavier for only short–middle-chain compounds, whereas source facies-related carbon assimilation controls the compositions of short-, middle-, and long-chain compounds. In particular, light-carbon assimilation during organic-matter degradation makes the isotopic composition lighter, whereas heavy carbon from the water mass makes it heavier. Accordingly, oils in this study area were divided into Type U and Type N oils based on individual compound carbon isotopic compositions, reflecting the difference in source facies in a highly saline and reducing stratified water environment. The results provide a better understanding of the controls on carbon isotopes in oil in sedimentary basins, reducing the uncertainty in oil–source correlation and addressing the origin of oil.

Keywords

Carbon assimilation / Water stratification / Carbon isotopes / Alkaline lacustrine / Junggar Basin

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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