# Revisiting Energy Demand Relationship: Theory and Empirical Application

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## Abstract

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## 1. Introduction

## 2. Revisiting the Theory of the Energy Demand Equations

#### Cost Minimization Approach

_{i}’s are prices and Y is the total income. This function (in logs) can be written explicitly as:

## 3. The Energy Demand Equations in Empirical Analysis

- Panels A and D report the results of estimations for Equations (2) and (7), respectively. All the explanatory variables have theoretically expected signs. Apparently, the cost of capital is highly statistically insignificant in both estimations. These are the results from option (i): that is, we impose the theory of energy demand on the data and ignore information coming from data, i.e., the insignificance of the capital cost. Our results are theory-driven only, and, hence, we position ourselves at the upper part of Pagan’s curve [7,11].
- Panels B and E report the results of estimations for Equations (2) and (7) without the cost of capital, respectively. In other words, we follow option (ii), such that we first apply general energy demand specifications to the data and also account for the statistical insignificance of the cost of capital and exclude it from the analysis in the GtSM framework. All the remaining variables have theoretically expected signs and are statistically significant at different levels. In other words, the estimation results are from nesting the theory of energy demand with the data in the GtSM framework, i.e., they are both theory driven and data driven, and, thus, we position ourselves at the middle part of Pagan’s curve [7,11].
- Panels C and F report the results of estimations of Equation (8), where the explanatory variables are only energy price and output/income. In other words, the estimation results are from option (iii). On data, we impose the parsimonious energy demand specification, which omits the theoretically predicted variables of prices of capital, labor, and intermediate consumption based on some assumptions made by default, as discussed in Section 2, without testing the statistical significance of the variables omitted to see whether they can contribute to the explanation of the energy demand pattern. Obviously, we miss some important information, which could come from the prices of labor and intermediate consumption, as the omitted variable tests indicate in Table 2 and Table 3. Apparently, the electricity price has an incorrect sign and is statistically insignificant, most likely due to omitting the important variables (see, e.g., the discussions in [34,35,36,37]). Obviously, if we could follow GtSM and consider both the theoretical coherence and the statistical coherence, we would not end up with such a poor specification of energy demand.

- Sometimes a variable articulated by the theory cannot be exactly measured in practice due to data inaccuracy and unavailability issues, and proxies can provide poor estimates and thus do not help us to approximate the Data Generation Process (DGP) of the variable under interest. This is exactly what we face in our analysis here. The theory in Section 2 articulates the cost of capital as an explanatory variable of energy demand. However, we cannot find the exact cost of capital data for non-oil manufacturing. It can be argued that this is not the case solely for Saudi Arabia, and, even for many developing and developed countries, the cost of capita data is not available for the different branches of industry. Following earlier studies, we proxy it, but it appears that the selected proxy does not contribute to the DGP of industrial energy demand, and it was statistically insignificant.
- Often, theories are vague about variables when it comes to considering the variables in the empirical analysis, and the selected variables may not contribute to the DGP. For example, money demand theories consider income as a scale variable in explaining the behavior of money balance. However, it is not quite clear which income measure should be considered in the empirical analysis. Therefore, GDP, retail turnover, consumption, government expenditure, and industrial production index have all been considered in the empirical analyses of money demand [38].
- All theories are based on certain assumptions, and these assumptions may not be held for the country of interest or for the time period considered (see, [2]).
- Theories do not tell us anything about structural breaks and location shifts, which can play a considerable role in explaining a given process.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Derivation of Energy Demand Equation

## References

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Variable | Notation | Description/definition | Source |
---|---|---|---|

Electricity consumption | $E$ | The demand for electricity in non-oil industrial sector, mtoe.* | IEA [18] |

Output in non-oil manufacturing in real terms | ${Q}_{O}$ | This is the sum of value added and intermediate consumption both in manufacturing (excluding petroleum manufacturing) in million SAR at 2010 prices. | GSTAT [19] and OEGEM [20] |

Value added in non-oil manufacturing in real terms | ${Q}_{V}$ | The value added in manufacturing excluding petroleum manufacturing, in million SAR at 2010 prices. | GSTAT [19] |

Price of electricity consumed in industry in real terms | ${P}_{E}$ | ${P}_{E}=\frac{{P}_{N}}{{P}_{MAN}}*100$ ${P}_{N}$ is the nominal price of the electricity consumed in industry in SAR/toe. ${P}_{MAN}$ is the deflator of the non-oil manufacturing value added, which is calculated as below: ${P}_{MAN}=\frac{{Q}_{N}}{Qv}*100$ ${Q}_{N}$ is the nominal value added in non-oil manufacturing, in million SAR. | Own calcula tion using GSTAT [19] data |

Cost of capital in real terms | ${P}_{K}$ | This is the United States Seven-Year Treasury note yield, at constant maturity, adjusted for the US inflation rate, %. | OEGEM [20] |

Average annual wage in real terms | ${P}_{L}$ | ${P}_{L}=\frac{{W}_{N}}{CPI}*100$ ${W}_{N}$ is the average annual wage in nominal term, which is calculated as below: ${W}_{N}=\frac{ER}{ET}$ $ER$ is the total earnings in thousand SAR. $ET$ is total employment in thousand persons. $CPI$ is Consumer Price Index, 2010 = 100. | Own calcula tion using GSTAT [19] data |

Price of intermediate consumption in real terms | ${P}_{M}$ | ${P}_{M}=\frac{{P}_{MAN}}{{P}_{NOIL}}*100$ ${P}_{NOIL}$ is the deflator of the non-oil value added, which is calculated as below: ${P}_{NOIL}=\frac{Q{N}_{NOIL}}{Q{R}_{NOIL}}*100$ $Q{N}_{NOIL}$ is the nominal value added in non-oil manufacturing, in million SAR. $Q{R}_{NOIL}$ is the real value added in non-oil manufacturing, in million SAR at 2010 prices. | Own calcula tion using GSTAT [19] data |

**Note:**IEA = International Energy Agency; GSTAT = General Authority for Statistics of Saudi Arabia; OEGEM = Oxford Economics Global Economic Model database. We conclude that the variable is electricity consumption in the non-oil industrial sector based on our understanding of the IEA definitions for industry and energy industry own use.

Panel A. Estimation of Equation (2) | Panel B. Estimation of Equation (2) without ${\mathit{p}}_{\mathit{k}}$ | Panel C. Estimation of Equation (8), where total output is used | ||||
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Estimated long-run elasticities | ||||||

Regressor | Coef. | P-value | Coef. | P-value | Coef. | P-value |

${p}_{k}$ | 0.097 | 0.643 | − | − | − | − |

${p}_{l}$ | 1.453 | 0.000 | 1.505 | 0.000 | − | − |

${p}_{m}$ | 1.980 | 0.086 | 1.493 | 0.087 | − | − |

${p}_{e}$ | −0.224 | 0.200 | −0.285 | 0.075 | 0.630 | 0.392 |

${q}_{o}$ | 0.608 | 0.001 | 0.532 | 0.000 | 0.892 | 0.021 |

$SER$ | 0.171098 | 0.168028 | 0.604501 | |||

${R}_{ADJ}^{2}$ | 0.969779 | 0.970854 | 0.622769 | |||

Post-estimation test results | ||||||

Q | 0.368 | 0.544 | 0.316 | 0.574 | 0.809 | 0.000 |

$JB$ | 0.369 | 0.832 | 0.849 | 0.654 | 6.330 | 0.042 |

$F$ for ${p}_{k}$ | − | − | 0.200 | 0.658 | 0.567 | 0.456 |

$F$ for ${p}_{l}$ | − | − | − | − | 507.355 | 0.000 |

$F$ for ${p}_{m}$ | − | − | − | − | 51.613 | 0.000 |

Cointegration test results | ||||||

$E{G}_{\tau}^{a}$ | −5.492 | 0.027 | −5.545 | 0.011 | −2.623 | 0.440 |

$E{G}_{z}^{a}$ | −34.067 | 0.024 | −34.239 | 0.009 | −5.674 | 0.852 |

**Notes:**$e$ is the dependent variable in the estimations. $SER$ is standard error of regression.${R}_{ADJ}^{2}$ is adjusted R-Squared. Q is the Q-statistic of the first order auto-correlation coefficient with the null hypothesis that the residuals are not correlated.$JB$ is the Jarque–Bera statistic of normality test with the null hypothesis that the residuals are normally distributed.$F$ is the F-statistics of the omitted variable test with the null hypothesis that a tested variable can be omitted.$E{G}_{\tau}^{a}$ and $E{G}_{z}^{a}$ are the degree of freedom adjusted Engle–Granger tau- and z-statistics. Coef. and P-value mean the coefficient and its probability value. For simplicity, intercepts are not reported. Estimation period: 1980–2018.

Panel D. Estimation of Equation (7) | Panel E. Estimation of Equation (7) without ${\mathit{p}}_{\mathit{k}}$ | Panel F. Estimation of Equation (8), where income is used | ||||
---|---|---|---|---|---|---|

Estimated long-run elasticities | ||||||

Regressor | Coef. | P-value | Coef. | P-value | Coef. | P-value |

${p}_{k}$ | 0.030 | 0.885 | − | − | − | − |

${p}_{l}$ | 1.368 | 0.000 | 1.439 | 0.000 | − | − |

${p}_{m}$ | 1.838 | 0.111 | 1.621 | 0.056 | − | − |

${p}_{e}$ | −0.218 | 0.217 | −0.249 | 0.105 | 0.364 | 0.727 |

${q}_{v}$ | 0.540 | 0.002 | 0.499 | 0.000 | 1.019 | 0.029 |

$SER$ | 0.170074 | 0.169650 | 0.608458 | |||

${R}_{ADJ}^{2}$ | 0.970140 | 0.970289 | 0.672940 | |||

Post-estimation test results | ||||||

Q | 0.594 | 0.441 | 0.669 | 0.413 | 31.490 | 0.000 |

$JB$ | 0.164 | 0.921 | 0.922 | 0.631 | 0.687 | 0.709 |

$F$ for ${p}_{k}$ | − | − | 0.029 | 0.865 | 0.808 | 0.375 |

$F$ for ${p}_{l}$ | − | − | − | − | 399.451 | 0.000 |

$F$ for ${p}_{m}$ | − | − | − | − | 67.250 | 0.000 |

Cointegration test results | ||||||

$E{G}_{\tau}^{a}$ | −5.351 | 0.035 | −5.378 | 0.015 | −1.853 | 0.795 |

$E{G}_{z}^{a}$ | −33.126 | 0.031 | −33.258 | 0.013 | −5.137 | 0.881 |

**Notes:**$e$ is the dependent variable in the estimations. $SER$ is standard error of regression.${R}_{ADJ}^{2}$ is adjusted R-Squared. Q is the Q-statistic of the first order auto-correlation coefficient with the null hypothesis that the residuals are not correlated. $JB$ is the Jarque–Bera statistic of normality test with the null hypothesis that the residuals are normally distributed.$F$ is the F-statistics of the omitted variable test with the null hypothesis that a tested variable can be omitted.$E{G}_{\tau}^{a}$ and $E{G}_{z}^{a}$ are the degree of freedom adjusted Engle–Granger tau- and z-statistics. Coef. and P-value mean the coefficient and its probability value. For simplicity, intercepts are not reported. Estimation period: 1980–2018.

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**MDPI and ACS Style**

Hasanov, F.J.; Mikayilov, J.I.
Revisiting Energy Demand Relationship: Theory and Empirical Application. *Sustainability* **2020**, *12*, 2919.
https://doi.org/10.3390/su12072919

**AMA Style**

Hasanov FJ, Mikayilov JI.
Revisiting Energy Demand Relationship: Theory and Empirical Application. *Sustainability*. 2020; 12(7):2919.
https://doi.org/10.3390/su12072919

**Chicago/Turabian Style**

Hasanov, Fakhri J., and Jeyhun I. Mikayilov.
2020. "Revisiting Energy Demand Relationship: Theory and Empirical Application" *Sustainability* 12, no. 7: 2919.
https://doi.org/10.3390/su12072919