# Temporal Aggregation and Long Memory for Asset Price Volatility

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Alternative Volatility Measures

#### 2.1. Stochastic Volatility Model

**Assumption**

**1.**

**Remark**

**1.**

#### 2.2. Temporal Aggregation

#### 2.3. Temporal Aggregation in the Frequency Domain

**Assumption**

**2.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

## 3. Long-Memory Parameter Estimates across Aggregation Levels

#### 3.1. Log-Periodogram Regressions

#### 3.2. Equivalence of Estimates across Aggregation Levels

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Lemma**

**2.**

**Proof.**

## 4. S&P 500 Volatility

#### 4.1. Low Frequency Data

**Remark**

**7.**

#### 4.2. High Frequency Data

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma**

**A1**

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

**Proof of Lemma**

**1.**

**Proof of Lemma**

**2.**

## References

- Bollerslev, Tim, and Jonathan H. Wright. 2000. Semiparametric estimation of long-memory volatility dependencies: The role of high-frequency data. Journal of Econometrics 98: 81–106. [Google Scholar] [CrossRef]
- Chambers, Marcus J. 1998. Long-memory and aggregation in macroeconomic time series. International Economic Review 39: 1053–72. [Google Scholar] [CrossRef]
- Deo, Rohit, Clifford Hurvich, and Yi Lu. 2006. Forecasting realized volatility using a long-memory stochastic volatility model: Estimation, prediction and seasonal adjustment. Journal of Econometrics 131: 29–58. [Google Scholar] [CrossRef]
- Diebold, Francis X., and Atsushi Inoue. 2001. Long memory and regime switching. Journal of Econometrics 105: 131–59. [Google Scholar] [CrossRef] [Green Version]
- Ding, Zhuanxin, Clive W. J. Granger, and Robert F. Engle. 1993. A long memory property of stock market return and a new model. Journal of Empirical Finance 1: 86–106. [Google Scholar] [CrossRef]
- Engle, Robert F., and Aaron D. Smith. 1999. Stochastic permanent breaks. Review of Economics and Statistics 81: 533–74. [Google Scholar] [CrossRef] [Green Version]
- Geweke, John, and Susan Porter-Hudak. 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis 4: 221–38. [Google Scholar] [CrossRef]
- Gourieroux, Christian, and Joann Jasiak. 2001. Memory and infrequent breaks. Economics Letters 70: 29–41. [Google Scholar] [CrossRef]
- Granger, Clive W. J., and Namwon Hyung. 2004. Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance 11: 399–421. [Google Scholar]
- Granger, Clive W. J., and Roselyne Joyeux. 1980. An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1: 15–39. [Google Scholar] [CrossRef]
- Hassler, Uwe. 2011. Estimation of fractional integration under temporal aggregation. Journal of Econometrics 162: 240–47. [Google Scholar] [CrossRef] [Green Version]
- Hosking, Jonathan R. M. 1981. Fractional differencing. Biometrika 68: 165–76. [Google Scholar] [CrossRef]
- Ikeda, Shin S. 2015. Two scale realized kernels: A univariate case. Journal of Financial Econometrics 13: 126–65. [Google Scholar] [CrossRef]
- Lu, Yang K., and Pierre Perron. 2010. Modeling and forecasting stock return volatility using a random level shift model. Journal of Empirical Finance 17: 138–56. [Google Scholar] [CrossRef]
- McCloskey, Adam, and Pierre Perron. 2013. Memory parameter estimation in the presence of level shifts and deterministic trends. Econometric Theory 29: 1196–237. [Google Scholar] [CrossRef] [Green Version]
- Ohanissian Arek, Jeffrey R. Russell, and Ruey S. Tsay. 2008. True or spurious long memory? a new test. Journal of Business and Economic Statistics 26: 161–75. [Google Scholar] [CrossRef]
- Perron, Pierre. 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57: 1361–401. [Google Scholar] [CrossRef]
- Perron, Pierre. 1990. Testing for a unit root in a time series regression with a changing mean. Journal of Business and Economic Statistics 8: 153–62. [Google Scholar]
- Perron, Pierre, and Zhongjun Qu. 2007. An Analytical Evaluation of the Log-Periodogram Estimate in the Presence of Level Shifts. Working paper. Boston: Boston University. [Google Scholar]
- Perron, Pierre, and Zhongjun Qu. 2010. Long-memory and level shifts in the volatility of stock market return indices. Journal of Business and Economic Statistics 28: 275–90. [Google Scholar] [CrossRef]
- Robinson, Peter M. 1995. Log periodogram regression of time series with long range dependence. Annals of Statistics 23: 1048–72. [Google Scholar] [CrossRef]
- Souza, Leonardo Rocha. 2005. A note on Chambers’s “Long-memory and aggregation in macroeconomic time series”. International Economic Review 46: 1059–62. [Google Scholar] [CrossRef] [Green Version]
- Souza, Leonardo Rocha. 2007. Temporal aggregation and bandwidth selection in estimating long memory. Journal of Time Series Analysis 28: 701–22. [Google Scholar] [CrossRef]
- Souza, Leonardo Rocha. 2008. Why aggregate long memory time series? Econometric Reviews 27: 298–316. [Google Scholar] [CrossRef]
- Schwert, G. William. 1990. Indexes of United States stock prices, 1802–1987. Journal of Business 63: 399–426. [Google Scholar] [CrossRef]
- Varneskov, Rasmus T., and Pierre Perron. 2018. Combining long memory and level shifts in modeling and forecasting the volatility of asset returns. Quantitative Finance 18: 371–93. [Google Scholar] [CrossRef]
- Wei, William W. S. 1978. Some consequences of temporal aggregation in seasonal time series models. In Seasonal Analysis of Economic Time Series; Edited by A. Zeller. Washington, DC: US Department of Commerce, Bureau of Census. [Google Scholar]

**Figure 1.**The periodogram of the squared daily returns (

**left**), the 5-periods aggregation of the squared daily returns divided by 5 (

**middle**), and the 20-periods aggregation of the squared daily returns divided by 20 (

**right**) for the daily S&P 500 returns data. The sample period is from 13 August 1928 to 30 December 2011.

**Figure 2.**The periodogram of the realized volatility obtained from 1-min returns (

**left**), the squared 1-min returns multiplied by 330 (

**middle**), and the difference between them (

**right**) for the high-frequency S&P 500 returns data. The sample period is from 7 October 1986 to 2 March 2007.

**Figure 3.**Log squared daily returns (

**left**), log realized volatility obtained from 1-min returns (

**middle**) and the difference (

**right**) for the high-frequency S&P 500 returns data. The sample period is from 7 October 1986 to 2 March 2007.

**Figure 4.**The periodogram of the log squared daily returns (

**left**), the log realized volatility obtained from 1-min returns (

**middle**), and the difference between them (

**right**) for the high-frequency S&P 500 returns data. The sample period is from 7 October 1986 to 2 March 2007.

**Figure 5.**The periodogram of the log realized volatility obtained from 110-min returns (

**left**), 30-min returns (

**middle**), and 5-min returns (

**right**) for the high-frequency S&P 500 returns data. The sample period is from 7 October 1986 to 2 March 2007.

${\left[log\left({\mathit{r}}_{\mathit{t}}^{2}\right)\right]}^{\left(\mathit{S}\right)}$ | $log\left({\mathit{r}}_{\mathit{t}}^{2}\right)$ | $log\left\{{\left[{\mathit{r}}_{\mathit{t}}^{2}\right]}^{\left(\mathit{S}\right)}\right\}$ | ||||
---|---|---|---|---|---|---|

$\mathit{S}$ | SLP | TLP | SLP | TLP | SLP | TLP |

$S=1$ | 0.5659 | 0.0941 | 0.5659 | 0.0941 | 0.5659 | 0.0941 |

$S=5$ | 0.6313 | 0.2570 | 0.6317 | 0.2317 | 0.6236 | 0.3092 |

$S=10$ | 0.7770 | 0.3544 | 0.7762 | 0.3200 | 0.7983 | 0.4359 |

$S=20$ | 0.7440 | 0.4457 | 0.7427 | 0.3915 | 0.7554 | 0.4689 |

${\left[log\left({\mathit{r}}_{\mathit{t},\mathit{n}}^{\left(\mathit{k}\right)2}\right)\right]}^{\left(\mathit{S}\right)}$ | $log\left[{\mathit{r}}_{\mathit{t},\mathit{n}}^{\left(\mathit{k}\right)2}\right]$ | $log\left\{{\left[{\mathit{r}}_{\mathit{t},\mathit{n}}^{\left(\mathit{k}\right)2}\right]}^{\left(\mathit{S}\right)}\right\}$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=\mathit{T}$ | $\mathit{N}=\mathit{T}$ | $\mathit{N}=\mathit{TS}$ | $\mathit{N}=\mathit{T}$ | |||||

$\mathit{k}$ | SLP | TLP | SLP | TLP | SLP | TLP | SLP | TLP |

$k=1$ | 0.6231 | 0.3949 | 0.6218 | 0.3765 | 0.3964 | 0.0676 | 0.6768 | 0.4842 |

$k=5$ | 0.6336 | 0.3432 | 0.6323 | 0.3260 | 0.4595 | 0.0603 | 0.6497 | 0.4269 |

$k=30$ | 0.6439 | 0.2421 | 0.6443 | 0.2203 | 0.5275 | 0.0679 | 0.6637 | 0.3072 |

$k=330$ | 0.4835 | 0.0415 | 0.4835 | 0.0415 | 0.4835 | 0.0415 | 0.4835 | 0.0415 |

© 2020 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**

Perron, P.; Shi, W.
Temporal Aggregation and Long Memory for Asset Price Volatility. *J. Risk Financial Manag.* **2020**, *13*, 182.
https://doi.org/10.3390/jrfm13080182

**AMA Style**

Perron P, Shi W.
Temporal Aggregation and Long Memory for Asset Price Volatility. *Journal of Risk and Financial Management*. 2020; 13(8):182.
https://doi.org/10.3390/jrfm13080182

**Chicago/Turabian Style**

Perron, Pierre, and Wendong Shi.
2020. "Temporal Aggregation and Long Memory for Asset Price Volatility" *Journal of Risk and Financial Management* 13, no. 8: 182.
https://doi.org/10.3390/jrfm13080182