# Equity Return Dispersion and Stock Market Volatility: Evidence from Multivariate Linear and Nonlinear Causality Tests

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data and Methodology

#### 3.1. Data

#### 3.2. Methodology

#### 3.2.1. Bivariate Linear Causality Tests

#### 3.2.2. Nonlinearity Tests

#### 3.2.3. Multivariate Granger Causality tests

#### Multivariate Linear Causality

- (1)
- ${H}_{0}^{1}:{A}_{xy}\left(L\right)=0$,
- (2)
- ${H}_{0}^{2}:{A}_{yx}\left(L\right)=0$, and,
- (3)
- both ${H}_{0}^{1}:{A}_{xy}\left(L\right)=0$ and ${H}_{0}^{2}:{A}_{yx}\left(L\right)=0$,

#### Multivariate Nonlinear Causality

## 4. Empirical Results

#### 4.1. Descriptive Statistics and Stationarity Test

#### 4.2. Bivariate Causality Tests

#### 4.3. Multivariate Granger Causality Tests

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Connolly, R.; Stivers, C. Information content and other characteristics of the daily cross-sectional dispersion in stock returns. J. Empir. Financ.
**2006**, 13, 79–112. [Google Scholar] [CrossRef] - Angelidis, T.; Sakkas, A.; Tessaromatis, N. Stock market dispersion, the business cycle and expected factor returns. J. Bank. Financ.
**2015**, 59, 265–279. [Google Scholar] [CrossRef] [Green Version] - Maio, P. Cross-sectional return dispersion and the equity premium. J. Financ. Mark.
**2016**, 29, 87–109. [Google Scholar] [CrossRef] - Hamilton, J.D.; Lin, G. Stock market volatility and the business cycle. J. Appl. Econ.
**1996**, 11, 573–593. [Google Scholar] [CrossRef] - Schwert, G.W. Stock Volatility during the Recent Financial Crisis. Eur. Financ. Manag.
**2011**, 17, 789–805. [Google Scholar] [CrossRef] - Choudhry, T.; Papadimitriou, F.I.; Shabi, S. Stock market volatility and business cycle: Evidence from linear and nonlinear causality tests. J. Bank. Financ.
**2016**, 66, 89–101. [Google Scholar] [CrossRef] [Green Version] - Bai, Z.D.; Wong, W.K.; Zhang, B.Z. Multivariate linear and nonlinear causality tests. Math. Comput. Simul.
**2010**, 81, 5–17. [Google Scholar] [CrossRef] - Bai, Z.D.; Li, H.; Wong, W.K.; Zhang, B.Z. Multivariate Causality Tests with Simulation and Application. Stat. Probab. Lett.
**2011**, 81, 1063–1071. [Google Scholar] [CrossRef] - Bai, Z.; Hui, Y.; Jiang, D.; Lv, Z.; Wong, W.K.; Zheng, S. A new test of multivariate nonlinear causality. PLoS ONE
**2018**, 13, e0185155. [Google Scholar] [CrossRef] - Chow, S.C.; Cunado, J.; Gupta, R.; Wong, W.K. Causal Relationships between Economic Policy Uncertainty and Housing Market Returns in China and India: Evidence from Linear and Nonlinear Panel and Time Series Models. Stud. Nonlinear Dyn. Econom.
**2018**, 22. [Google Scholar] [CrossRef] - Christie, W.; Huang, R. Equity Return Dispersions; Work Paper; Vanderbilt University: Nashville, TN, USA, 1994. [Google Scholar]
- Duffee, G.R. Asymmetric Cross-Sectional Dispersion in Stock Returns: Evidence and Implications; U.C. Berkeley: Berkeley, CA, USA, 2001. [Google Scholar]
- Loungani, P.; Rush, R.; Tave, W. Stock market dispersion and unemployment. J. Monet. Econ.
**1990**, 25, 367–388. [Google Scholar] [CrossRef] - Stivers, C.T. Firm-level return dispersion and the future volatility of aggregate stock market returns. J. Financ. Mark.
**2003**, 6, 389–411. [Google Scholar] [CrossRef] - Stivers, C.; Sun, L. Cross-Sectional Return Dispersion and Time Variation in Value and Momentum Premiums. J. Financ. Quant. Anal.
**2010**, 45, 987–1014. [Google Scholar] [CrossRef] - Bhootra, A. Are momentum profits driven by the cross-sectional dispersion in expected stock returns? J. Financ. Mark.
**2011**, 14, 494–513. [Google Scholar] [CrossRef] - Jiang, X. Return dispersion and expected returns. Financ. Mark. Portfol. Manag.
**2010**, 24, 107–135. [Google Scholar] [CrossRef] - Demirer, R.; Jategaonkar, S. The Conditional Relation Between Dispersion and Return. Rev. Financ. Econ.
**2013**, 22, 125–134. [Google Scholar] [CrossRef] - Demirer, R.; Jategaonkar, S.P.; Khalifa, A.A. Oil price risk exposure and the cross-section of stock returns: The case of net exporting countries. Energy Econ.
**2015**, 49, 132–140. [Google Scholar] [CrossRef] - Lillo, F.; Mantegna, R.N. Variety and Volatility in Financial Markets. Phys. Rev. E
**2000**, 62, 6126–6134. [Google Scholar] [CrossRef] - Solnik, B.; Roulet, J. Dispersion as cross-sectional correlation. Financ. Anal. J.
**2000**, 56, 54–61. [Google Scholar] [CrossRef] - Baur, D. Multivariate market association and its extremes. J. Int. Financ. Markets Inst. Money
**2006**, 16, 355–369. [Google Scholar] [CrossRef] [Green Version] - Statman, M.; Scheid, J. Global Diversification. J. Invest. Manag.
**2005**, 3, 55–63. [Google Scholar] [CrossRef] - Statman, M.; Scheid, J. Correlation, Return Gaps, and the Benefits of Diversification. J. Portfol. Manag.
**2008**, 34, 132–139. [Google Scholar] [CrossRef] - Demirer, R. Can Advanced Markets Help Diversify Risks in Frontier Markets? Evidence from Gulf Arab Stock Markets. Res. Int. Bus. Financ.
**2013**, 29, 77–98. [Google Scholar] [CrossRef] - Eiling, E.; Gerard, B. Dispersion, Equity Return Correlations and Market Integration; University of Toronto: Toronto, ON, Canada, 2011. [Google Scholar]
- Goyal, A.; Santa-Clara, P. Idiosyncratic risk matters. J. Financ.
**2003**, 58, 975–1006. [Google Scholar] [CrossRef] - Bali, T.; Cakici, N.; Yan, X.; Zhang, Z. Does idiosyncratic risk really matter? J. Financ.
**2005**, 60, 905–929. [Google Scholar] [CrossRef] - Pollet, J.; Wilson, M. Average correlation and stock market returns. J. Financ. Econ.
**2010**, 96, 364–380. [Google Scholar] [CrossRef] - Garcia, R.; Mantilla-Garcia, D.; Martellini, L. A model-free measure of aggregate idiosyncratic volatility and the prediction of market returns. J. Financ. Quant. Anal.
**2014**, 49, 1133–1165. [Google Scholar] [CrossRef] - Chichernea, D.C.; Holder, A.D.; Petkevich, A. Does return dispersion explain the accrual and investment anomalies? J. Account. Econ.
**2015**, 60, 133–148. [Google Scholar] [CrossRef] - Demirer, R.; Omay, T.; Yuksel, A.; Yuksel, A. Global Risk Aversion and Emerging Market Return Comovements. Econ. Lett.
**2018**, 173, 118–121. [Google Scholar] [CrossRef] - Aruoba, S.; Diebold, F.; Scotti, C. Real-time measurement of business condition. J. Bus. Econ. Stat.
**2009**, 27, 417–427. [Google Scholar] [CrossRef] - Granger, C.W. Investigating causal relations by econometric models and cross-spectral methods. Econometrica
**1969**, 37, 424–438. [Google Scholar] [CrossRef] - Hui, Y.C.; Wong, W.K.; Bai, Z.D.; Zhu, Z.Z. A New Nonlinearity Test to Circumvent the Limitation of Volterra Expansion with Application. J. Korean Stat. Soc.
**2017**, 46, 365–374. [Google Scholar] [CrossRef] - Denker, M.; Keller, G. On U-statistics and v. mise’statistics for weakly dependent processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete
**1983**, 64, 505–522. [Google Scholar] [CrossRef] - Dickey, D.A.; Fuller, W.A. Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc.
**1979**, 74, 427–431. [Google Scholar] - Lam, K.; Liu, T.S.; Wong, W.K. A pseudo-Bayesian model in financial decision making with implications to market volatility, under- and overreaction. Eur. J. Oper. Res.
**2010**, 203, 166–175. [Google Scholar] [CrossRef] - Lam, K.; Liu, T.S.; Wong, W.K. A New Pseudo Bayesian Model with Implications to Financial Anomalies and Investors’ Behaviors. J. Behav. Financ.
**2012**, 13, 93–107. [Google Scholar] [CrossRef] - Fung, E.S.; Lam, K.; Siu, T.K.; Wong, W.K. A New Pseudo Bayesian Model for Financial Crisis. J. Risk Financ. Manag.
**2011**, 4, 42–72. [Google Scholar] [CrossRef] - Guo, X.; McAleer, M.; Wong, W.K.; Zhu, L.X. A Bayesian approach to excess volatility, short-term underreaction and long-term overreaction during financial crises, North American. J. Econ. Financ.
**2017**, 42, 346–358. [Google Scholar]

**Figure 1.**Daily equity return dispersion (RD) and stock market volatility (SV). Note: RD and SV are equity return dispersion and stock market volatility, respectively.

Mean | Stdev | Skewness | Kurtosis | J-B | ADF Test | |
---|---|---|---|---|---|---|

$R{D}_{t}$ | 0.627 *** | 0.275 | 4.057 *** | 35.136 *** | 732,071.353 *** | −8.9165 *** |

$S{V}_{t}$ | 1.000 *** | 1.778 | 8.723 | 101.069 *** | 5,921,984.155 *** | −9.4998 *** |

$M{P}_{t}$ | 0.025 *** | 0.988 | −0.508 *** | 15.633 *** | 138,157.710 *** | −82.3209 *** |

$AD{S}_{t}$ | −0.018 *** | 0.876 | −1.206 *** | 3.921 *** | 11,929.823 *** | −7.7651 *** |

Panel A: The Predictive Power of Equity Return Dispersion | ||||

$R{D}_{t}$→$S{V}_{t}$ | $R{D}_{t}$→$M{P}_{t}$ | $R{D}_{t}$→$S{V}_{t}$|$AD{S}_{t}$ | $R{D}_{t}$→$M{P}_{t}$|$AD{S}_{t}$ | |

Lags | 15 | 9 | 16 | 16 |

F-Stat | 188.760 *** | 3.196 *** | 9.716 × 10^{−7} | 1.136 × 10^{−8} |

$R{D}_{t}$→$S{V}_{t}$|$ADS{1}_{t}$ | $R{D}_{t}$→$S{V}_{t}$|$ADS{2}_{t}$ | $R{D}_{t}$→$M{P}_{t}$|$ADS{1}_{t}$ | $R{D}_{t}$→$M{P}_{t}$|$ADS{2}_{t}$ | |

Lags | 9 | 9 | 9 | 9 |

F-Stat | 1.729 × 10^{−6} | 1.714 × 10^{−6} | 1.744 × 10^{−8} | 1.749 × 10^{−8} |

Panel B: The Predictive Power of Business Conditions | ||||

$ADS{1}_{t}$→$S{V}_{t}$ | $ADS{2}_{t}$→$S{V}_{t}$ | $ADS{1}_{t}$→$M{P}_{t}$ | $ADS{2}_{t}$→$M{P}_{t}$ | |

Lags | 16 | 16 | 9 | 9 |

F-Stat | 1.146 | 3.579 *** | 0.738 | 1.768 |

$AD{S}_{t}$→$R{D}_{t}$ | $ADS{1}_{t}$→$R{D}_{t}$ | $ADS{2}_{t}$→$R{D}_{t}$ | ||

Lags | 9 | 9 | 9 | |

F-Stat | 4.068 *** | 0.513 | 5.967 *** |

$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$ | $\mathit{A}\mathit{D}\mathit{S}{1}_{\mathit{t}}$ | $\mathit{A}\mathit{D}\mathit{S}{2}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$ | $\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{M}{\mathit{P}}_{\mathit{t}}$ | |
---|---|---|---|---|---|---|

Lags | 11 | 10 | 16 | 10 | 15 | 2 |

T-Stat | 7.734 *** | 7.845 *** | 7.893 *** | 8.970 *** | 3.574 *** | 8.547 *** |

Panel A: The Predictability of Stock Market Volatility | ||||

Lags | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$|$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$|$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$|$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$ |

1 | 7.879 *** | 7.8190 *** | 7.758 *** | 7.824 *** |

2 | 7.718 *** | 7.665 *** | 7.533 *** | 7.525 *** |

3 | 7.637 *** | 7.659 *** | 7.533 *** | 7.621 *** |

4 | 7.908 *** | 7.871 *** | 7.745 *** | 7.772 *** |

5 | 7.461 *** | 7.484 *** | 7.309 *** | 7.449 *** |

6 | 7.155 *** | 7.207 *** | 7.141 *** | 7.279 *** |

7 | 6.770 *** | 6.813 *** | 6.611 *** | 6.662 *** |

8 | 6.617 *** | 6.721 *** | 6.461 *** | 6.535 *** |

9 | 5.984 *** | 6.169 *** | 5.741 *** | 5.884 *** |

10 | 5.918 *** | 6.067 *** | 5.646 *** | 5.742 *** |

Panel B: The Predictability of Equity Market Premium | ||||

Lags | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$|$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$|$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$|$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$ |

1 | 11.365 *** | 11.379 *** | 11.363 *** | 11.302 *** |

2 | 12.910 *** | 13.079 *** | 12.904 *** | 12.877 *** |

3 | 12.878 *** | 13.053 *** | 12.867 *** | 12.928 *** |

4 | 13.357 *** | 13.643 *** | 13.364 *** | 13.428 *** |

5 | 13.275 *** | 13.693 *** | 13.272 *** | 13.420 *** |

6 | 12.519 *** | 12.931 *** | 12.527 *** | 12.694 *** |

7 | 11.823 *** | 12.206 *** | 11.844 *** | 12.038 *** |

8 | 11.805 *** | 12.155 *** | 11.807 *** | 12.048 *** |

9 | 11.716 *** | 11.996 *** | 11.695 *** | 11.950 *** |

10 | 11.104 *** | 11.405 *** | 11.068 *** | 11.321 *** |

Panel C: The Predictive Power of Business Conditions | ||||

Lags | $\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$→$\mathit{R}{\mathit{D}}_{\mathit{t}}$ | $\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$→$\mathit{R}{\mathit{D}}_{\mathit{t}}$ | $\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$→$\mathit{R}{\mathit{D}}_{\mathit{t}}$ | |

1 | −1.122 | −5.676 *** | 1.755 * | |

2 | −1.366 | −6.626 *** | 1.808 * | |

3 | −1.352 | −6.930 *** | 2.627 ** | |

4 | −2.015 * | −6.917 *** | 2.317 * | |

5 | −0.820 | −4.650 *** | 2.711 ** | |

6 | −1.718 * | −5.231 *** | 2.311 * | |

7 | −2.147 * | −5.412 *** | 2.425 ** | |

8 | −2.148 * | −4.913 *** | 1.708 * | |

9 | −1.919 * | −4.669 *** | 1.928 * | |

10 | −1.987 * | −4.427 *** | 1.053 |

Panel A: The Predictability of Stock Market Volatility | |||

$\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | |

Lags | 10 | 9 | 9 |

LR | 535.909 *** | 560.136 *** | 573.599 *** |

Panel B: The Predictability of Equity Market Premium | |||

$\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | |

Lags | 10 | 9 | 9 |

LR | 37.812 | 37.456 | 39.096 |

Panel A: The Predictability of Stock Market Volatility | |||

Lags | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$→$\mathit{S}{\mathit{V}}_{\mathit{t}}$ |

1 | 7.706 *** | 7.661 *** | 7.614 *** |

2 | 7.529 *** | 7.454 *** | 7.217 *** |

3 | 7.140 *** | 7.286 *** | 7.037 *** |

4 | 6.736 *** | 7.496 *** | 6.565 *** |

5 | 6.321 *** | 6.954 *** | 5.967 *** |

6 | 5.818 *** | 6.610 *** | 5.694 *** |

7 | 5.380 *** | 6.107 *** | 4.963 *** |

8 | 5.447 *** | 6.016 *** | 4.969 *** |

9 | 4.731 *** | 5.387 *** | 4.095 *** |

10 | 4.665 *** | 5.168 *** | 4.108 *** |

Panel B: The Predictability of Equity Market Premium | |||

Lags | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}{\mathit{S}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{1}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ | $\mathit{R}{\mathit{D}}_{\mathit{t}}$+$\mathit{A}\mathit{D}\mathit{S}{\mathbf{2}}_{\mathit{t}}$→$\mathit{M}{\mathit{P}}_{\mathit{t}}$ |

1 | 11.271 *** | 11.296 *** | 11.260 *** |

2 | 12.523 *** | 12.655 *** | 12.280 *** |

3 | 12.557 *** | 12.594 *** | 12.370 *** |

4 | 13.092 *** | 12.988 *** | 12.846 *** |

5 | 12.590 *** | 12.727 *** | 11.980 *** |

6 | 11.753 *** | 11.797 *** | 11.288 *** |

7 | 10.6764 *** | 10.733 *** | 10.478 *** |

8 | 10.749 *** | 10.493 *** | 10.584 *** |

9 | 10.662 *** | 10.577 *** | 10.141 *** |

10 | 10.075 *** | 9.923 *** | 9.601 *** |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Demirer, R.; Gupta, R.; Lv, Z.; Wong, W.-K.
Equity Return Dispersion and Stock Market Volatility: Evidence from Multivariate Linear and Nonlinear Causality Tests. *Sustainability* **2019**, *11*, 351.
https://doi.org/10.3390/su11020351

**AMA Style**

Demirer R, Gupta R, Lv Z, Wong W-K.
Equity Return Dispersion and Stock Market Volatility: Evidence from Multivariate Linear and Nonlinear Causality Tests. *Sustainability*. 2019; 11(2):351.
https://doi.org/10.3390/su11020351

**Chicago/Turabian Style**

Demirer, Riza, Rangan Gupta, Zhihui Lv, and Wing-Keung Wong.
2019. "Equity Return Dispersion and Stock Market Volatility: Evidence from Multivariate Linear and Nonlinear Causality Tests" *Sustainability* 11, no. 2: 351.
https://doi.org/10.3390/su11020351