# Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices

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## Abstract

**:**

## 1. Introduction

## 2. The Hybrid Recalibration Process

#### 2.1. Overview of the Methodology

#### 2.2. Implementation of the Proposed Methodology

#### 2.3. Generalized Additive Model for Location, Scale, and Shape

#### 2.4. Density Selection

#### 2.5. Selection of the Regressors

#### 2.6. Performance Analysis

## 3. Combinations for Increased Accuracy

- (1)
- As in the previous combination approach, for each proposed parametric distribution function $w$ (MEQ-QR-SEP2, MEQ-QR-SEP3, and MEQ-QR-ST3), the corresponding quantile functions ${\widehat{y}}_{h,w}^{\mathsf{\alpha}}$ are derived for the usual target percentiles, so that ${\widehat{y}}_{h,w}^{\mathsf{\alpha}}={F}_{h,w}^{-1}\left(\mathsf{\alpha}\right)$. Note as the distribution functions vary with time (i.e., as the information set of explanatory variables evolves) so will the quantile functions.
- (2)
- The predicted quantile functions ${\widehat{y}}_{h,w}^{\mathsf{\alpha}}$ are combined for each percentile $\mathsf{\alpha}$ using the asymmetric absolute loss function to yield the LAD regression. The LAD regression may be viewed as a particular case of quantile regression and intuitively, this method assigns specific weights for each percentile depending on the inverse of the absolute deviation error, so that larger weights are given to models that show smaller deviation error during the in-sample data set. Recall that the weights are sequentially updated after each additional moving window. As constructed quantile functions are sample unbiased, then we might expect that the weights’ sum to unity and there is strong intuitive appeal for omitting the constant (see [46]).

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Density Functions

## Appendix A.1. Box-Cox Power Exponential

**Figure A1.**The PDF from a BCPE distribution for specific parameter values. Upper-left: BCPE(5, 0.2, 6, 10), upper-right: BCPE(5, 0.2, −4, 10), lower-left: BCPE(5, 0.2, 1, 2) and lower-right: BCPE(5, 0.2, 6, 0.1).

## Appendix A.2. Skew Exponential Power Type 2

**Figure A2.**The PDF from a SEP2 distribution for specific parameter values. Upper-left: SEP2(0, 1, 10, 3), upper-right: SEP2(0, 1, 10, 1), lower-left: SEP2(0, 1, −10, 1), and lower-right: SEP2(0, 1, 0, 3).

## Appendix A.3. Skew Exponential Power Type 3

**Figure A3.**The PDF from a SEP3 distribution for specific parameter values. Upper-left: SEP3(0, 1, 7, 7), upper-right: SEP3(0, 1, 1, 7), lower-left: SEP3(0, 1, 1, 0.1), and lower-right: SEP3(0, 1, 1, 1).

## Appendix A.4. Skew t Type 3

**Figure A4.**The PDF from a ST3 distribution for specific parameter values. Upper-left: ST3(0, 1, 7, 7), upper-right: ST3(0, 1, 7, 0.1), lower-left: ST3(0, 1, 0.1, 0.1), and lower-right: ST3(0, 1, 1, 1).

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**Figure 3.**Comparison between the predicted PDF using SEP2 model and the actual price during the period from 1 January 2014 to 30 June 2014.

Variable | $\mathsf{\mu}$ | $\mathsf{\sigma}$ | $\mathsf{\upsilon}$ | $\mathsf{\tau}$ |
---|---|---|---|---|

MEQ Price P50 | + | |||

Net Demand P50 | + | |||

MEQ Price P1 | - | + | - | |

MEQ Price Mean | + | |||

MEQ Price P30 | - | |||

MEQ Price P99 | + | |||

Intercept | - | - | + | + |

Variable | $\mathsf{\mu}$ | $\mathsf{\sigma}$ | $\mathsf{\upsilon}$ | $\mathsf{\tau}$ |
---|---|---|---|---|

MEQ Price Mean | + | |||

Net Demand Mean | + | |||

Exports Mean | - | |||

MEQ Price P1 | - | + | ||

MEQ Price Mean | + | |||

Wind Mean | + | |||

MEQ Price P30 | + | |||

Net Demand P99 | - | |||

Intercept | + | + | - | + |

(%) | P1 | P5 | P30 | P50 | P70 | P95 | P99 |
---|---|---|---|---|---|---|---|

BCPE | 93.43 | 81.95 | 55.50 | 34.63 | 19.86 | 0.18 | 0.02 |

SEP2 | 96.20 | 90.27 | 56.29 | 38.48 | 11.61 | 0.42 | 0.05 |

SEP3 | 91.82 | 85.84 | 56.64 | 39.00 | 22.38 | 3.34 | 0.11 |

ST3 | 91.93 | 84.71 | 58.17 | 40.57 | 22.77 | 4.02 | 0.16 |

MEQ | 77.97 | 70.06 | 48.20 | 37.34 | 28.01 | 12.34 | 7.65 |

MEQ-QR | 93.14 | 80.43 | 61.81 | 36.76 | 30.92 | 8.20 | 2.39 |

(%) | P1 | P5 | P30 | P50 | P70 | P95 | P99 |
---|---|---|---|---|---|---|---|

MEQ-QR-BCPE | 94.54 | 82.19 | 56.22 | 32.18 | 21.93 | 4.19 | 0.53 |

MEQ-QR-SEP2 | 95.30 | 93.04 | 64.85 | 47.06 | 28.91 | 6.84 | 2.57 |

MEQ-QR-ST3 | 95.19 | 93.29 | 77.19 | 58.45 | 31.65 | 4.72 | 0.91 |

MEQ-QR-SEP3 | 95.59 | 94.74 | 79.23 | 58.61 | 32.45 | 3.06 | 0.27 |

BCPE | 93.43 | 81.95 | 55.50 | 34.63 | 19.86 | 0.18 | 0.02 |

SEP2 | 96.20 | 90.27 | 56.29 | 38.48 | 11.61 | 0.42 | 0.05 |

SEP3 | 91.82 | 85.84 | 56.64 | 39.00 | 22.38 | 3.34 | 0.11 |

ST3 | 91.93 | 84.71 | 58.17 | 40.57 | 22.77 | 4.02 | 0.16 |

MEQ-QR | 93.14 | 80.43 | 61.81 | 36.76 | 30.92 | 8.20 | 2.39 |

(%) | P1 | P5 | P30 | P50 | P70 | P95 | P99 |
---|---|---|---|---|---|---|---|

QRA | 95.14 | 90.46 | 74.52 | 52.38 | 31.80 | 3.53 | 1.97 |

EWC | 95.36 | 93.61 | 72.94 | 54.57 | 30.73 | 5.80 | 1.59 |

MEQ-QR-SEP2 | 95.30 | 93.04 | 64.85 | 47.06 | 28.91 | 6.84 | 2.57 |

MEQ-QR-ST3 | 95.19 | 93.29 | 77.19 | 58.45 | 31.65 | 4.72 | 0.91 |

MEQ-QR-SEP3 | 95.59 | 94.74 | 79.23 | 58.61 | 32.45 | 3.06 | 0.27 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bello, A.; Bunn, D.; Reneses, J.; Muñoz, A. Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices. *Energies* **2016**, *9*, 959.
https://doi.org/10.3390/en9110959

**AMA Style**

Bello A, Bunn D, Reneses J, Muñoz A. Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices. *Energies*. 2016; 9(11):959.
https://doi.org/10.3390/en9110959

**Chicago/Turabian Style**

Bello, Antonio, Derek Bunn, Javier Reneses, and Antonio Muñoz. 2016. "Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices" *Energies* 9, no. 11: 959.
https://doi.org/10.3390/en9110959