# Log Periodic Power Analysis of Critical Crashes: Evidence from the Portuguese Stock Market

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Model: The Log-Periodic Power Law (LPPL)

#### The LPPL Equation

## 4. Data and Methodology

## 5. Results and Discussion

#### 5.1. Analysis of the 1998 Crash

#### Sensitivity Analysis of the Critical Times (tc) for 1996–1998: The Impact of Tlast

#### 5.2. Analysis of the 2007 Crash

#### Sensitivity Analysis of the Critical Times (tc) for 2007: The Impact of Tlast

#### 5.3. Analysis of the 2015 Crash

#### Sensitivity Analysis of the Critical Times (tc) for 2015: The Impact of Tlast

#### 5.4. Robustness Analysis

#### 5.4.1. Log-Periodic Analysis of the Data Minimizing the Root Mean Squared Deviation

#### 5.4.2. Artificial Series and Critical Time Sensitivity

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Fitting of the Log-Periodic Linear Model for the 1996–1998 period, for samples with different ending points (tlast).

A | B | C | β | φ | tc | ω | λ | SRSD | SRSD/NDP | Tlast | Tlast Natural | Nosc | NDP | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 9.41 | −0.68 | 0.14 | 0.51 | 199.91 | 98.18 | 7.35 | 2.35 | 12.57 | 0.02 | 98.08 | 29 January 1998 | 3.60 | 542 |

2 | 9.41 | −0.68 | 0.14 | 0.51 | 199.91 | 98.18 | 7.35 | 2.35 | 12.74 | 0.02 | 98.12 | 13 February 1998 | 4.23 | 553 |

3 | 9.41 | −0.69 | 0.14 | 0.51 | 199.91 | 98.18 | 7.42 | 2.33 | 12.88 | 0.02 | 98.17 | 2 March 1998 | 6.05 | 564 |

4 | 9.69 | −0.93 | 0.09 | 0.42 | 199.42 | 98.28 | 8.91 | 2.02 | 13.53 | 0.02 | 98.21 | 17 March 1998 | 4.84 | 575 |

5 | 11.78 | −2.86 | −0.02 | 0.20 | 74.86 | 98.64 | 14.09 | 1.56 | 13.69 | 0.02 | 98.25 | 1 April 1998 | 4.27 | 586 |

6 | 11.79 | −2.86 | −0.02 | 0.20 | 74.91 | 98.64 | 13.96 | 1.57 | 14.09 | 0.02 | 98.29 | 16 April 1998 | 4.49 | 597 |

7 | 10.94 | −1.99 | 0.03 | 0.30 | 109.37 | 98.66 | 15.00 | 1.52 | 14.83 | 0.02 | 98.33 | 1 May 1998 | 4.95 | 608 |

8 | 10.94 | −1.99 | 0.03 | 0.30 | 109.36 | 98.66 | 15.00 | 1.52 | 15.36 | 0.02 | 98.38 | 18 May 1998 | 5.33 | 619 |

9 | 10.87 | −1.92 | 0.03 | 0.30 | 109.35 | 98.66 | 15.00 | 1.52 | 16.12 | 0.03 | 98.42 | 2 June 1998 | 5.70 | 630 |

10 | 10.61 | −1.65 | 0.04 | 0.35 | 109.26 | 98.68 | 15.00 | 1.52 | 17.93 | 0.03 | 98.46 | 17 June 1998 | 5.97 | 641 |

11 | 10.57 | −1.60 | 0.04 | 0.36 | 109.13 | 98.70 | 15.00 | 1.52 | 20.63 | 0.03 | 98.50 | 2 July 1998 | 6.21 | 652 |

12 | 9.61 | −0.71 | −0.08 | 0.74 | 194.99 | 98.55 | 15.00 | 1.52 | 22.95 | 0.03 | 98.54 | 17 July 1998 | 12.93 | 663 |

13 | 9.61 | −0.68 | −0.08 | 0.77 | 194.77 | 98.60 | 15.00 | 1.52 | 24.27 | 0.04 | 98.59 | 3 August 1998 | 13.05 | 674 |

**Table 2.**Fitting of the Log-Periodic Linear Model for the 2003–2007 period, for samples with different ending points (tlast).

A | B | C | β | φ | tc | ω | λ | SRSD | SRSD/NDP | Tlast | Tlast Natural | Nosc | NDP | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 9.33 | −0.25 | 0.16 | 0.76 | −20.35 | 107.02 | 11.59 | 1.72 | 23.00 | 0.02 | 106.96 | 15 December 2006 | 7.65 | 994 |

2 | 9.33 | −0.25 | 0.16 | 0.75 | −20.29 | 107.01 | 11.50 | 1.73 | 23.09 | 0.02 | 107.00 | 1 January 2007 | 11.11 | 1005 |

3 | 9.35 | −0.27 | 0.16 | 0.72 | −20.64 | 107.05 | 12.23 | 1.67 | 23.32 | 0.02 | 107.04 | 16 January 2007 | 12.55 | 1016 |

4 | 9.37 | −0.28 | 0.15 | 0.70 | −20.90 | 107.10 | 12.57 | 1.65 | 23.87 | 0.02 | 107.09 | 1 February 2007 | 11.54 | 1028 |

5 | 9.52 | −0.35 | 0.11 | 0.64 | −22.52 | 107.44 | 14.48 | 1.54 | 25.43 | 0.02 | 107.13 | 15 February 2007 | 6.02 | 1038 |

6 | 9.54 | −0.36 | 0.11 | 0.63 | −22.71 | 107.47 | 14.72 | 1.53 | 25.79 | 0.02 | 107.16 | 1 March 2007 | 6.18 | 1048 |

7 | 9.54 | −0.36 | 0.11 | 0.63 | −22.72 | 107.47 | 14.75 | 1.53 | 25.89 | 0.02 | 107.21 | 16 March 2007 | 6.52 | 1059 |

8 | 9.54 | −0.35 | 0.11 | 0.63 | −22.88 | 107.50 | 14.97 | 1.52 | 26.00 | 0.02 | 107.25 | 2 April 2007 | 6.85 | 1070 |

9 | 9.54 | −0.35 | 0.11 | 0.64 | −22.92 | 107.51 | 15.00 | 1.52 | 26.25 | 0.02 | 107.29 | 17 April 2007 | 7.20 | 1081 |

10 | 9.54 | −0.35 | 0.11 | 0.64 | −22.95 | 107.51 | 15.00 | 1.52 | 26.33 | 0.02 | 107.33 | 2 May 2007 | 7.62 | 1092 |

11 | 9.53 | −0.35 | 0.11 | 0.64 | −22.94 | 107.51 | 15.00 | 1.52 | 26.47 | 0.02 | 107.38 | 17 May 2007 | 8.32 | 1103 |

**Table 3.**Fitting of the Log-Periodic Linear Model for the 2012–2015 period, for samples with different ending points (tlast).

A | B | C | β | φ | tc | ω | λ | SRSD | SRSD/NDP | Tlast | Tlast Natural | Nosc | NDP | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 8.37 | 0.31 | 0.56 | 0.20 | 31.84 | 115.20 | 5.64 | 3.05 | 34.15 | 0.05 | 115.09 | 2 February 2015 | 2.88 | 685 |

2 | 8.37 | 0.31 | 0.55 | 0.20 | 31.67 | 115.25 | 5.92 | 2.89 | 34.61 | 0.05 | 115.13 | 16 February 2015 | 2.99 | 695 |

3 | 8.37 | 0.31 | 0.55 | 0.20 | 31.54 | 115.28 | 6.19 | 2.76 | 34.90 | 0.05 | 115.17 | 2 March 2015 | 3.13 | 705 |

4 | 8.40 | 0.28 | 0.60 | 0.25 | 0.17 | 115.28 | 6.23 | 2.74 | 35.05 | 0.05 | 115.21 | 16 March 2015 | 3.63 | 715 |

5 | 8.63 | 0.01 | −21.10 | 0.23 | 53.27 | 115.29 | 5.25 | 3.31 | 36.56 | 0.05 | 115.25 | 2 April 2015 | 3.57 | 728 |

6 | 8.30 | 0.39 | 0.40 | 0.20 | −13.07 | 115.47 | 8.41 | 2.11 | 36.86 | 0.05 | 115.29 | 16 April 2015 | 3.81 | 736 |

7 | 8.65 | 0.01 | −20.07 | 0.23 | 52.95 | 115.39 | 6.10 | 2.80 | 37.54 | 0.05 | 115.34 | 4 May 2015 | 3.95 | 747 |

8 | 8.42 | 0.26 | 0.57 | 0.20 | −13.45 | 115.54 | 8.53 | 2.09 | 37.86 | 0.05 | 115.38 | 18 May 2015 | 4.03 | 757 |

9 | 8.42 | 0.27 | 0.53 | 0.20 | 30.04 | 115.65 | 9.72 | 1.91 | 38.77 | 0.05 | 115.42 | 2 June 2015 | 4.09 | 768 |

10 | 8.45 | 0.24 | 0.59 | 0.20 | −13.96 | 115.65 | 9.55 | 1.93 | 39.31 | 0.05 | 115.46 | 16 June 2015 | 4.29 | 778 |

11 | 8.66 | 0.01 | −19.07 | 0.27 | 52.50 | 115.51 | 7.21 | 2.39 | 40.29 | 0.05 | 115.50 | 2 July 2015 | 6.70 | 790 |

Parameters | 1998 | 2007 | 2015 |
---|---|---|---|

A | 10.58 | 9.55 | 8.55 |

B | −1.58 | −0.37 | 0.13 |

C | 0.04 | −0.11 | 1.20 |

β | 0.36 | 0.61 | 0.20 |

φ | 108.75 | 11.45 | 30.48 |

tc | 98.75 | 107.51 | 115.54 |

ω | 15.00 | 15.43 | 7.76 |

RMSD | 0.04 | 0.03 | 0.06 |

tlast | 98.50 | 107.29 | 115.42 |

tlast natural | 2 July 1998 | 17 April 2007 | 2 June 2015 |

NDP | 652 | 1081 | 768 |

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**MDPI and ACS Style**

Gonçalves, T.C.; Borda, J.V.Q.; Vieira, P.R.; Matos, P.V.
Log Periodic Power Analysis of Critical Crashes: Evidence from the Portuguese Stock Market. *Economies* **2022**, *10*, 14.
https://doi.org/10.3390/economies10010014

**AMA Style**

Gonçalves TC, Borda JVQ, Vieira PR, Matos PV.
Log Periodic Power Analysis of Critical Crashes: Evidence from the Portuguese Stock Market. *Economies*. 2022; 10(1):14.
https://doi.org/10.3390/economies10010014

**Chicago/Turabian Style**

Gonçalves, Tiago Cruz, Jorge Victor Quiñones Borda, Pedro Rino Vieira, and Pedro Verga Matos.
2022. "Log Periodic Power Analysis of Critical Crashes: Evidence from the Portuguese Stock Market" *Economies* 10, no. 1: 14.
https://doi.org/10.3390/economies10010014