# Modelling the Effects of Oil Prices on Global Fertilizer Prices and Volatility

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## Abstract

**:**

## Introduction

## 2. Data

## 3. Model Specifications

#### 3.1 Minimum LM unit root test with two endogenous breaks

#### 3.2 Autoregressive Distributed Lag Model

#### 3.3 Conditional Mean and Conditional Volatility Models

## 4. Empirical Results

#### 4.1 Minimum LM unit root test with one and two breaks

#### 4.2 Granger Causality Test

#### 4.3 ARDL and Volatility Models for Crude Oil and Global Fertilizer Prices

#### 4.4 Alternative Volatility Models for Crude Oil and Six Global Fertilizer Prices

## 5. Concluding Remarks

## Acknowledgments

## References

- A. Banerjee, R.L. Lumsdaine, and J.H. Stock. “Recursive and Sequential Tests for a Unit Root: Theory and International Evidence.” Journal of Business and Economic Statistics 10 (1992): 271–287. [Google Scholar]
- J. Benten, and T. Engsted. “A revival of the autoregressive distributed lag model in estimating energy demand relationships.” Energy 26 (2001): 45–55. [Google Scholar] [CrossRef]
- T. Bollerslev. “Generalized Autoregressive Conditional Heteroscedasticity.” Journal of Econometrics 31 (1986): 307–327. [Google Scholar] [CrossRef]
- P. Bougerol, and N. Picard. “Stationarity of GARCH Processes and of Some Non-Negative Time Series.” Journal of Econometrics 52 (1992): 115–127. [Google Scholar] [CrossRef]
- F. Boussama. “Asymptotic Normality for the Quasi-Maximum Likelihood Estimator of a GARCH Model.” Comptes Rendus de l’Academie des Sciences Série I 331 (2000): 81–84. (in French). [Google Scholar]
- F. Chan, D. Marinova, and M. McAleer. “Modelling Thresholds and Volatility in US Ecological Patents.” Environmental Modelling and Software 20 (2005): 1369–1378. [Google Scholar] [CrossRef]
- F. Chan, D. Marinova, and M. McAleer. “Rolling Regressions and Conditional Correlations of Foreign Patents in the USA.” Environmental Modelling and Software 20 (2005): 1413–1422. [Google Scholar] [CrossRef]
- S.T. Chen, H.I. Kuo, and C.C. Chen. “Modeling the Relationship between the Oil Price and Global Food Prices.” Applied Energy 87 (2010): 2517–2525. [Google Scholar] [CrossRef]
- D.A. Dickey, and W.A. Fuller. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root.” Journal of the American Statistical Association 74 (1979): 427–431. [Google Scholar]
- D.A. Dickey, and W.A. Fuller. “Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root.” Econometrica 49 (1981): 1057–1072. [Google Scholar] [CrossRef]
- J. Dimitropoulos, L.C. Hunt, and G. Judge. “Estimating underlying energy demand trends using UK annual data.” Applied Economics Letters 12 (2005): 239–244. [Google Scholar] [CrossRef] [Green Version]
- L. Elie, and T. Jeantheau. “Consistency in Heteroskedastic Models.” Comptes Rendus de l’Académie des Sciences Série I 320 (1995): 1255–1258. (in French). [Google Scholar]
- R.F. Engle. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (1982): 987–1007. [Google Scholar] [CrossRef]
- FAO. “Current World Fertilizer Trends and Outlook to 2012.” 2008. Url: ftp://ftp.fao.org/agl/agll/docs/cwfto12.pdf.
- FAPRI UMC report. “Fertilizer and Fuel Prices and Cost of Production.” 2004. Url: http://www.fapri.missouri.edu/outreach/publications/2004/FAPRI_UMC_Report_10_04.pdf.
- L. Glosten, R. Jagannathan, and D. Runkle. “On the Relation Between the Expected Value and Volatility of Nominal Excess Return on Stocks.” Journal of Finance 46 (1992): 1779–1801. [Google Scholar] [CrossRef]
- C.W.J. Granger. “Investigation Causal Relations by Econometric Models and Cross-spectral Methods.” Econometrica 37 (1969): 424–438. [Google Scholar] [CrossRef]
- D.F. Hendry. Dynamic Econometrics: Advanced Text in Econometrics. Oxford: Oxford University Press, 1995. [Google Scholar]
- S. Hoti, M. McAleer, and L. Pauwels. “Modelling Environmental Risk.” Environmental Modelling and Software 20 (2005): 1289–1298. [Google Scholar] [CrossRef]
- L.C. Hunt, G. Judge, and Y. Ninomiya. “Underlying trends and seasonality in UK energy demand: a sectoral analysis.” Energy Ecnomics 25, 1 (2003): 93–118. [Google Scholar] [CrossRef]
- L.C. Hunt, and Y. Ninomiya. “Primary Energy Demand in Japan: An Empirical Analysis of Long-term Trends and Future CO2 Emissions.” Energy Policy 33, 11 (2005): 1409–1424. [Google Scholar] [CrossRef]
- T. Jeantheau. “Strong Consistency of Estimators for Multivariate ARCH Models.” Econometric Theory 14 (1998): 70–86. [Google Scholar] [CrossRef]
- C.T. Jones. “A Single-equation Study of U.S. Petroleum Consumption: the Role of Model Specification.” Southern Economic Journal 59, 4 (1993): 687–700. [Google Scholar] [CrossRef]
- J. Lee, and M.C. Strazicich. “Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks.” Review of Economics and Statistics 85 (2003): 1082–1089. [Google Scholar] [CrossRef]
- S.W. Lee, and B.E. Hansen. “Asymptotic Theory for the GARCH(1,1) Quasi-Maximum Likelihood Estimator.” Econometric Theory 10 (1994): 29–52. [Google Scholar] [CrossRef]
- S. Ling, and W.K. Li. “On Fractionally Integrated Autoregressive Moving-Average Models with Conditional Heteroskedasticity.” Journal of the American Statistical Association 92 (1997): 1184–1194. [Google Scholar] [CrossRef]
- S. Ling, and M. McAleer. “Stationarity and the Existence of Moments of a Family of GARCH Processes.” Journal of Econometrics 106 (2002): 109–117. [Google Scholar] [CrossRef]
- S. Ling, and M. McAleer. “Necessary and Sufficient Moment Conditions for the GARCH(r,s) and Asymmetric Power GARCH(r,s) Models.” Econometric Theory 18 (2002): 722–729. [Google Scholar] [CrossRef]
- S. Ling, and M. McAleer. “Asymptotic Theory for a Vector ARMA-GARCH Model.” Econometric Theory 19 (2003): 278–308. [Google Scholar] [CrossRef]
- S. Ling, and M. McAleer. “On Adaptive Estimation in Nonstationary ARMA Models with GARCH Errors.” Annals of Statistics 31 (2003): 642–674. [Google Scholar]
- R. Lumsdaine, and D. Papell. “Multiple Trend Breaks and the Unit Root Hypothesis.” Review of Economics and Statistics 79 (1997): 212–218. [Google Scholar] [CrossRef]
- D. Marinova, and M. McAleer. “Modelling Trends and Volatility in Ecological Patents in the USA.” Environmental Modelling and Software 18 (2003): 195–203. [Google Scholar] [CrossRef]
- M. McAleer, and F. Chan. “Modelling Trends and Volatility in Atmospheric Carbon Dioxide Concentrations.” Environmental Modelling and Software 20 (2006): 1273–1279. [Google Scholar] [CrossRef]
- M. McAleer, F. Chan, and D. Marinova. “An Econometric Analysis of Asymmetric Volatility: Theory and Application to Patents.” Journal of Econometrics 139 (2007): 259–284. [Google Scholar] [CrossRef]
- M. McAleer. “Automated Inference and Learning in Modeling Finanical Volatility.” Econometric Theory 21 (2005): 232–261. [Google Scholar] [CrossRef]
- C.R. Nelson, and C.I. Plosser. “Trends and random walks In Macroeconomic Time Series.” Journal of Monetary Economics 10 (1982): 139–162. [Google Scholar] [CrossRef]
- D.B. Nelson. “Conditional Heteroscedasticity in Asset Returns: A New Approach.” Econometrica 59 (1991): 347–370. [Google Scholar] [CrossRef]
- D.B. Nelson. “Stationarity and Persistence in the GARCH(1,1) Model.” Econometric Theory 6 (1990): 318–334. [Google Scholar] [CrossRef]
- P. Perron. “The Great Crash, the Oil Price Shock and the Unit Root Hypothesis.” Econometrica 57 (1989): 1361–1401. [Google Scholar] [CrossRef]
- P.C.B. Phillips, and P. Perron. “Testing for a unit root in time series regression.” Biometrika 75 (1988): 335–346. [Google Scholar] [CrossRef]
- S. Said, and D. Dickey. “Testing for unit roots in autoregressive moving average models with unknown order.” Biometrika 71 (1984): 599–607. [Google Scholar] [CrossRef]
- N. Shephard. “Statistical Aspects of ARCH and Stochastic Volatility.” In Time Series Models in Econometrics, Finance, and Other Fields. Edited by D.P. Cox, O.E. Barndorff-Nielsen and D.V. Hinkley. London: Chapman and Hall, 1996, pp. 1–67. [Google Scholar]
- E. Zivot, and D.W.K. Andrews. “Further Evidence on Great Cash, the Oil Price Shock and the Unit Root Hypothesis.” Journal of Business and Economic Statistics 10: 251–270.

Statistics | MAP (US$ /metric ton) | Urea (US$ /metric ton) | Ammonia (US$ /metric ton) | Acid (US$ /metric ton) | Rock (US$ /metric ton) | MOP (US$ /metric ton) | Poil (Price of Oil. US$/Bale) |
---|---|---|---|---|---|---|---|

Sample | 254 | 254 | 254 | 254 | 254 | 254 | 254 |

Mean | 258.07 | 225.80 | 280.72 | 428.30 | 78.46 | 206.18 | 48.29 |

Medium | 237 | 234.50 | 278.25 | 445.00 | 79.50 | 210.00 | 51.56 |

Maximum | 582.5 | 357.5 | 357.5 | 566.25 | 121.5 | 392.5 | 88.32 |

Minimum | 142.5 | 50.5 | 176 | 338.5 | 58 | 126 | 22.97 |

Std. Dev. | 89.39 | 55.51 | 53.89 | 70.01 | 18.97 | 57.95 | 17.23 |

Series | ADF tests | |||
---|---|---|---|---|

With constant | With constant and trend | Critical values | ||

With trend | With constant and trend | |||

Poil | -1.326(1) | -0.493(1) | -3.457 (1%) -2.873 (5%) -2.573 (10%) | -3.995 (1%) -3.428 (5%) -3.137 (10%) |

MAP | -2.154(9) | -2.248(9) | ||

Urea | -2.439(3) | -3.125(3) | ||

Ammonia | -1.089(9) | -2.301(9) | ||

Rock | -2.372(0) | -2.681(0) | ||

Acid | -2.179(0) | -1.926(0) | ||

MOP | 3.280(0) | 1.327(0) |

Series | LM_{τ} | k | TB1 | TB2 |
---|---|---|---|---|

Poil | -6.017*** | 8 | 20071129 | 20080327 |

MAP | -8.239*** | 8 | 20071108 | 20080327 |

Urea | -8.264*** | 8 | 20071220 | 20080424 |

Ammonia | -5.775** | 7 | 20080320 | |

Rock | -7.926*** | 8 | 20070412 | 20080313 |

Acid | -15.920*** | 0 | 20071220 | 20080410 |

MOP | -9.549*** | 8 | 20071213 | 20080424 |

Dependent Variable | Period | ||
---|---|---|---|

Period 1 | Period 2 | Period 3 | |

MAP | 4.030* | 4.381* | 4.958** |

Urea | 4.099* | 4.743** | 5.195** |

Ammonia | 3.429* | 3.576* | |

Rock | 0.336 | 1.086 | 0.477 |

Acid | 4.040* | 3.378* | 3.622* |

MOP | 3.492* | 3.183* | 3.654* |

Period | 2003/01/09-2007/11/22 | 2007/11/29-2008/03/20 | 2008/03/27-2008/12/04 |
---|---|---|---|

Series (Poil) | ARMA (3,2) | ARMA (2,1) | ARMA (3,3) |

GJR (1,1) | GJR (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.519 (0.062) | 0.393 (0.016) | 0.617 (0.030) |

AR (2) | 0.154 (0.007) | 0.280 (0.002) | 0.199 (0.010) |

AR (3) | -0.181 (0.061) | 0.032 (0.087) | |

MA (1) | 0.473 (0.064) | -0.268 (0.065) | 0.323 (0.011) |

MA (2) | -0.753 (0.050) | -0.293 (0.013) | |

MA (3) | 0.012 (0.077) | ||

Variance Equation | |||

ω | 0.527 (0.178) | 0.372 (0.164) | 0.007 (0.014) |

α | 0.133 (0.034) | 0.238 (0.085) | 0.282 (0.031) |

β | 0.235 (0.108) | 0.207 (0.199) | 0.485 (0.079) |

γ | -0.108 (0.075) | 0.147 (0.096) | |

Log moment | -0.819 | -0.598 | -0.156 |

Second moment | 0.421 | 0.519 | 0.768 |

Short run persistence | 0.079 | 0.311 | 0.282 |

Long run persistence | 0.314 | 0.519 | 0.768 |

BIC | 2.491 | 3.814 | 4.601 |

Period | 2003/01/09-2007/11/01 | 2007/11/08-2008/03/20 | 2008/03/27-2008/12/04 |
---|---|---|---|

Series (MAP) | ARMA (2,1) | ARMA (1,1) | ARMA (1,0) |

GARCH (1,1) | GARCH (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.633 (0.212) | 0.848 (0.115) | 0.819 (0.056) |

AR (2) | -0.284 (0.122) | ||

MA (1) | 0.137 (0.064) | -0.228 (0.092) | |

Oil Price (-1) | 0.236 (0.107) | 0.636 (0.217) | 0.613 (0.225) |

Oil Price (-2) | 0.280 (0.303) | ||

Variance Equation | |||

ω | 0.768 (0.363) | 0.015 (0.712) | 0.032 (0.700) |

α | 0.108 (0.042) | 0.288 (0.104) | 0.387 (0.113) |

β | 0.275 (0.057) | 0.266 (0.086) | 0.469 (0.150) |

γ | |||

Log moment | -0.478 | -0.373 | -0.105 |

Second moment | 0.385 | 0.554 | 0.856 |

Short run persistence | 0.108 | 0.288 | 0.387 |

Long run persistence | 0.385 | 0.554 | 0.856 |

BIC | 5.465 | 8.169 | 7.610 |

Period | 2003/01/09-2007/12/13 | 2007/12/20-2008/04/17 | 2008/04/24-2008/12/04 |
---|---|---|---|

Series (Urea) | ARMA (1,1) | ARMA (1,1) | ARMA (1,1) |

GARCH (1,1) | GARCH (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.675 (0.018) | 0.756 (0.052) | 0.779 (0.047) |

MA (1) | -0.238 (0.088) | -0.183 (0.086) | 0.050 (0.012) |

Oil Price (-1) | 0.806 (0.294) | 3.114 (0.719) | 2.897 (0.225) |

Oil Price (-2) | 0.531 (0.248) | 1.958 (0.735) | 1.493 (0.188) |

Oil Price (-3) | 0.574 (0.163) | ||

Variance Equation | |||

ω | 0.452 (0.313) | 0.647 (0.609) | 0.094 (0.826) |

α | 0.059 (0.023) | 0.364 (0.109) | 0.312 (0.107) |

β | 0.272 (0.088) | 0.279 (0.133) | 0.595 (0.168) |

γ | |||

Log moment | -0.506 | -0.259 | -0.067 |

Second moment | 0.331 | 0.643 | 0.907 |

Short run persistence | 0.059 | 0.364 | 0.312 |

Long run persistence | 0.331 | 0.643 | 0.907 |

BIC | 6.485 | 6.853 | 6.305 |

Period | 2003/01/09-2008/03/13 | 2008/03/20-2008/12/04 |
---|---|---|

Series (Ammonia) | ARMA (2,1) | ARMA (1,0) |

GARCH (1,1) | GARCH (1,1) | |

Mean Equation | ||

AR (1) | 0.883 (0.022) | 0.788 (0.180) |

AR (2) | -0.299 (0.022) | |

MA (1) | 0.216 (0.040) | |

Oil Price (-1) | 1.085 (0.318) | 2.364 (0.489) |

Oil Price (-2) | 0.447 (0.212) | 1.402 (0.315) |

Variance Equation | ||

ω | 0.113 (2.494) | 0.214 (1.130) |

α | 0.066 (0.025) | 0.387 (0.112) |

β | 0.290 (0.038) | 0.512 (0.245) |

γ | ||

Log moment | -0.472 | -0.174 |

Second moment | 0.356 | 0.899 |

Short run persistence | 0.066 | 0.387 |

Long run persistence | 0.356 | 0.899 |

BIC | 7.238 | 7.568 |

Period | 2003/01/09-2007/04/05 | 2007/04/12-2008/03/06 | 2008/03/13-2008/12/04 |
---|---|---|---|

Series (Rock) | ARMA (2,1) | ARMA (1,1) | ARMA (3,2) |

GARCH (1,1) | GARCH (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.334 (0.061) | 0.963 (0.054) | 0.703 (0.263) |

AR (2) | 0.248 (0.009) | -0.149 (0.107) | |

MA (1) | 0.371 (0.061) | -0.223 (0.027) | 0.279 (0.080) |

MA (2) | 0.106 (0.051) | ||

Variance Equation | |||

ω | 0.005 (0.004) | 0.121 (0.164) | 0.160 (0.191) |

α | 0.109 (0.022) | 0.262 (0.084) | 0.369 (0.095) |

β | 0.327 (0.196) | 0.359 (0.105) | 0.442 (0.034) |

γ | |||

Log moment | -0.579 | -0.436 | -0.127 |

Second moment | 0.436 | 0.621 | 0.811 |

Short run persistence | 0.109 | 0.262 | 0.369 |

Long run persistence | 0.436 | 0.621 | 0.811 |

BIC | 1.751 | 2.611 | 2.558 |

Period | 2003/01/09-2007/12/10 | 2007/12/17-2008/03/31 | 2008/04/07-2008/12/04 |
---|---|---|---|

Series (Acid) | ARMA (1.0) | ARMA (2,1) | ARMA (3,2) |

GARCH (1,1) | GARCH (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.648 (0.043) | 0.695 (0.340) | 0.793 (0.190) |

MA (1) | 0.113 (0.052) | 0.101 (0.023) | |

Oil Price (-1) | 0.214 (0.103) | 1.053 (0.304) | 0.628 (0.274) |

Oil Price (-2) | 0.131 (0.062) | 0.325 (0.112) | |

Variance Equation | |||

ω | 0.401 (0.326) | 0.038 (0.550) | 0.329 (1.063) |

α | 0.059 (0.016) | 0.203 (0.098) | 0.298 (0.107) |

β | 0.257 (0.113) | 0.227 (0.126) | 0.463 (0.176) |

γ | |||

Log moment | -0.574 | -0.323 | -0.176 |

Second moment | 0.316 | 0.430 | 0.694 |

Short run persistence | 0.059 | 0.203 | 0.298 |

Long run persistence | 0.316 | 0.430 | 0.694 |

BIC | 7.222 | 7.475 | 7.202 |

Period | 2003/01/09-2007/12/06 | 2007/12/13-2008/04/17 | 2008/04/24-2008/12/04 |
---|---|---|---|

Series (MOP) | ARMA (2,1) | ARMA (1,1) | ARMA (1,0) |

GARCH (1,1) | GARCH (1,1) | GARCH (1,1) | |

Mean Equation | |||

AR (1) | 0.830 (0.133) | 0.899 (0.256) | 0.896 (0.101) |

AR (2) | -0.245 (0.107) | ||

MA (1) | -0.122 (0.053) | -0.271 (0.129) | |

Oil Price (-1) | 0.108 (0.036) | 0.958 (0.273) | 0.707 (0.234) |

Oil Price (-2) | 0.062 (0.020) | 0.294 (0.151) | |

Variance Equation | |||

ω | 0.027 (0.028) | 0.602 (0.476) | 0.330 (0.571) |

α | 0.096 (0.032) | 0.438 (0.163) | 0.285 (0.116) |

β | 0.142 (0.014) | 0.234 (0.114) | 0.600 (0.266) |

γ | |||

Log moment | -0.738 | -0.365 | -0.101 |

Second moment | 0.238 | 0.672 | 0.885 |

Short run persistence | 0.096 | 0.438 | 0.285 |

Long run persistence | 0.238 | 0.672 | 0.885 |

BIC | 4.755 | 8.722 | 7.563 |

The percentage change in each fertilizer price | a 1% changes in the crude oil price | |||||
---|---|---|---|---|---|---|

Period 1 | Period 2 | Period 3 | ||||

MAP | Oil(-1) | 1.252% | Oil(-1) | 4.912% | Oil(-1) | 6.416% |

Oil(-2) | 2.931% | |||||

Urea | Oil(-1) | 3.789% | Oil(-1) | 15.445% | Oil(-1) | 23.324% |

Oil(-2) | 2.496% | Oil(-2) | 9.711% | Oil(-2) | 11.497% | |

Oil(-3) | 3.435% | |||||

Ammonia | Oil(-1) | 6.265% | Oil(-1) | 13.834% | ||

Oil(-2) | 2.581% | Oil(-2) | 8.205% | |||

Acid | Oil(-1) | 1.902% | Oil(-1) | 7.929% | Oil(-1) | 11.412% |

Oil(-2) | 1.075% | Oil(-2) | 6.530% | |||

MOP | Oil(-1) | 0.461% | Oil(-1) | 4.914% | Oil(-1) | 6.431% |

Oil(-2) | 0.264% | Oil(-2) | 2.674% |

## Share and Cite

**MDPI and ACS Style**

Chen, P.-Y.; Chang, C.-L.; Chen, C.-C.; McAleer, M.
Modelling the Effects of Oil Prices on Global Fertilizer Prices and Volatility. *J. Risk Financial Manag.* **2012**, *5*, 78-114.
https://doi.org/10.3390/jrfm5010078

**AMA Style**

Chen P-Y, Chang C-L, Chen C-C, McAleer M.
Modelling the Effects of Oil Prices on Global Fertilizer Prices and Volatility. *Journal of Risk and Financial Management*. 2012; 5(1):78-114.
https://doi.org/10.3390/jrfm5010078

**Chicago/Turabian Style**

Chen, Ping-Yu, Chia-Lin Chang, Chi-Chung Chen, and Michael McAleer.
2012. "Modelling the Effects of Oil Prices on Global Fertilizer Prices and Volatility" *Journal of Risk and Financial Management* 5, no. 1: 78-114.
https://doi.org/10.3390/jrfm5010078