# Multifractal Analysis of Market Efficiency across Structural Breaks: Implications for the Adaptive Market Hypothesis

^{*}

## Abstract

**:**

## 1. Introduction

- To assess the impact of structural breaks on the long-memory in the price, volume, and price-volume relationship;
- To detect the possible change in the price-volume relationship among these breaks.

## 2. Literature Review

#### 2.1. Multifractality and Price-Volume Relationship

#### 2.2. MFDCCA and the Adaptive Market Hypothesis

## 3. Methodology

#### 3.1. Structural Breaks

#### 3.2. Detrended Fluctuation Analysis

#### 3.3. MFDFA

- a
- The detrended residuals are calculated using the following equation:$$\epsilon \left(i\right)={X}_{i}-\left(\widehat{X}\right)$$
- b
- The fluctuation function is calculated as the RMS of the detrended residuals:$${\left[{F}_{v}\left(s\right)\right]}^{2}=\frac{1}{s}\sum _{j=1}^{s}{[\epsilon \left((\nu -1)s\right)+j]}^{2}$$
- c
- The ${q}^{th}$ order of the fluctuation function is calculated for a given value of q.$${\left[{F}_{v}\left(s\right)\right]}^{q}=\frac{1}{s}\sum _{j=1}^{s}{[\epsilon \left((\nu -1)s\right)+j]}^{q}$$
- d
- The power law relation is then described by the equation:$${F}_{q}\left(s\right)\sim {s}^{{h}_{\left(q\right)}}$$

#### 3.4. MFDCCA

## 4. Data and Analysis

#### 4.1. Data

#### 4.2. Analysis of Structural Breaks

`strucchange`package (Zeileis et al. 2002) in the R software (R Core Team 2020). The breakpoints were calculated using the method prescribed in Zeileis et al. (2003) based on the procedure of Bai and Perron (2003). The optimal number of structural breaks was chosen at the lowest BIC value. The lowest BIC was obtained at four breaks (i.e., five segments), as shown in Figure 2.

#### 4.3. Multifractal Analysis

#### Interpretation of the Graphs

## 5. Results and Discussion

#### 5.1. Entire Data

#### 5.2. Segment 1–Segment 5

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AMH | Adaptive Market Hypothesis |

BIC | Bayesian Information Criteria |

EMH | Efficient Market Hypothesis |

MFDFA | Multifractal Detrended Fluctuation Analysis |

MFDCCA | Multifractal Detrended Cross-Correlation Analysis |

MDM | Mixture of Distribution Model |

RSS | Residual Sum of Squares |

SIF | Sequential arrival of Information Flow |

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Price | Volume | Return | volChange | |
---|---|---|---|---|

Number of observations | 5957 | 5957 | 5957 | 5957 |

Minimum | 2600.12 | 423 | −0.11 | −5.68 |

Maximum | 40,267.6 | 1,166,709 | 0.17 | 5.29 |

Median | 13,156.7 | 148,410 | 0 | 0 |

Mean | 14,153.1 | 167,683 | 0 | 0 |

SE of mean | 136.58 | 1513.21 | 0 | 0 |

CI of mean 0.95 | 267.74 | 2966.44 | 0 | 0.01 |

Standard deviation | 10,541.3 | 116,792 | 0.01 | 0.29 |

Price Segments | [1995-07-17, 2002-07-10] | [2002-07-10, 2006-01-25] | [2006-01-25, 2009-09-07] | [2009-09-07, 2014-05-30] | [2014-05-30, 2019-08-06] |
---|---|---|---|---|---|

Number of observations | 1710 | 893 | 895 | 1178 | 1280 |

Minimum | 2600.12 | 2834.41 | 8160.4 | 15,175.1 | 22,951.83 |

Maximum | 5933.56 | 9648.08 | 20,873.3 | 24,716.9 | 40,267.62 |

Median | 3575.04 | 5358.35 | 13,799.5 | 18,342 | 28,798.47 |

Mean | 3742.69 | 5341.6 | 13,614.1 | 18,536.4 | 30,533.3 |

SE of mean | 15.11 | 60.62 | 98.07 | 49.89 | 124.99 |

CI of mean 0.95 | 29.64 | 118.97 | 192.47 | 97.89 | 245.22 |

Standard deviation | 624.87 | 1811.44 | 2933.86 | 1712.46 | 4471.93 |

Segment | Fluctuation Type | Price Is Persistent | Volume Is Persistent | Price-Volume |
---|---|---|---|---|

Cross-Correlation | ||||

Is Persistent | ||||

Entire Data | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No | |

1 | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No | |

2 | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No | |

3 | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No | |

4 | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No | |

5 | Large fluctuations | No | No | No |

Small fluctuations | Yes | No | No |

Segment | Multifractal Strength | Intensity and Complexity | ||||
---|---|---|---|---|---|---|

Price | Volume | Price-Volume Correlation | Price | Volume | Price-Volume Correlation | |

All Data | 0.322 | 0.714 | 0.493 | 0.506 | 0.991 | 0.752 |

1 | 0.518 | 0.749 | 0.273 | 0.749 | 1.030 | 0.430 |

2 | 0.412 | 0.430 | 0.238 | 0.601 | 0.597 | 0.387 |

3 | 0.367 | 0.811 | 0.527 | 0.548 | 1.063 | 0.710 |

4 | 0.363 | 0.550 | 0.231 | 0.518 | 0.717 | 0.356 |

5 | 0.212 | 0.426 | 0.251 | 0.354 | 0.617 | 0.416 |

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**MDPI and ACS Style**

Patil, A.C.; Rastogi, S. Multifractal Analysis of Market Efficiency across Structural Breaks: Implications for the Adaptive Market Hypothesis. *J. Risk Financial Manag.* **2020**, *13*, 248.
https://doi.org/10.3390/jrfm13100248

**AMA Style**

Patil AC, Rastogi S. Multifractal Analysis of Market Efficiency across Structural Breaks: Implications for the Adaptive Market Hypothesis. *Journal of Risk and Financial Management*. 2020; 13(10):248.
https://doi.org/10.3390/jrfm13100248

**Chicago/Turabian Style**

Patil, Ashok Chanabasangouda, and Shailesh Rastogi. 2020. "Multifractal Analysis of Market Efficiency across Structural Breaks: Implications for the Adaptive Market Hypothesis" *Journal of Risk and Financial Management* 13, no. 10: 248.
https://doi.org/10.3390/jrfm13100248