# Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Long-Range-Dependent Models

#### 2.1. The Fractional Difference Operator

#### 2.2. Cross-Sectional Aggregation

**Proposition**

**1.**

**Proof.**

## 3. Nonfractional Long-Range Dependence Generation

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Nonfractional Long-Range Dependence and the Antipersistent Property

**Theorem**

**2.**

**Proof.**

## 5. Nonfractional Long-Range Dependence Estimation

**Theorem**

**3.**

**Proof.**

## 6. Application

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs for Lemmas 1 and 2

**Proof for Lemma 1.**

**Proof for Lemma 2.**

## Appendix B. Codes for Long-Range Dependence Generation by Cross-Sectional Aggregation

csadiff <- function(x, a, b){ |

iT <- length(x) |

n <- nextn(2*iT - 1, 2) |

k <- 0:(iT-1) |

coefs <- (beta(a+k,b)/beta(a,b))^(1/2) |

csax <- fft(fft(c(x, rep(0, n - iT))) * |

fft(c(coefs, rep(0, n - iT))), inverse = T) / n; |

return(Re(csax[1:iT])) |

} |

function [csax] = csa_diff(x,a,b) |

iT = size(x,1); |

n = 2.^nextpow2(2*iT-1); |

coefs = ( beta(a+(0:iT-1),b) ./ beta(a,b) ).^(1/2); |

csax = ifft(fft(x, n).*fft(coefs’, n)); |

csax = cx(1:iT, :); |

end |

## References

- Adenstedt, Rolf K. 1974. On Large-Sample Estimation for the Mean of a Stationary Random Sequence. The Annals of Statistics 2: 1095–107. [Google Scholar] [CrossRef]
- Altissimo, Filippo, Benoit Mojon, and Paolo Zaffaroni. 2009. Can Aggregation Explain the Persistence of Inflation? Journal of Monetary Economics 56: 231–41. [Google Scholar] [CrossRef]
- Andrews, Donald W. K., and Patrik Guggenberger. 2003. A Bias-Reduced Log-Periodogram Regression Estimator For The Long-Memory Parameter. Econometrica 71: 675–712. [Google Scholar] [CrossRef]
- Baillie, Richard T. 1996. Long Memory Processes and Fractional Integration in Econometrics. Journal of Econometrics 73: 5–59. [Google Scholar] [CrossRef]
- Baillie, Richard T., Fabio Calonaci, Dooyeon Cho, and Seunghwa Rho. 2019. Long Memory, Realized Volatility and Heterogeneous Autoregressive Models. Journal of Time Series Analysis, 1–20. [Google Scholar] [CrossRef]
- Baillie, Richard T., and Sang Kuck Chung. 2002. Modeling and forecasting from trend-stationary long memory models with applications to climatology. International Journal of Forecasting 18: 215–26. [Google Scholar] [CrossRef]
- Balcilar, Mehmet. 2004. Persistence in Inflation: Does Aggregation Cause Long Memory? Emerging Markets Finance and Trade 40: 25–56. [Google Scholar] [CrossRef]
- Beran, Jan, Yuanhua Feng, Sucharita Ghosh, and Rafal Kulik. 2013. Long-Memory Processes: Probabilistic Theories and Statistical Methods. Berlin/Heidelberg, Germany: Springer. [Google Scholar] [CrossRef]
- Bhardwaj, Geetesh, and Norman R. Swanson. 2006. An Empirical Investigation of the Usefulness of ARFIMA Models for Predicting Macroeconomic and Financial Time Series. Journal of Econometrics 131: 539–78. [Google Scholar] [CrossRef][Green Version]
- Chevillon, Guillaume, Alain Hecq, and Sébastien Laurent. 2018. Generating Univariate Fractional Integration Within a Large VAR(1). Journal of Econometrics 204: 54–65. [Google Scholar] [CrossRef]
- Cooley, James W., Peter A. W. Lewis, and Peter D. Welch. 1969. The Fast Fourier Transform and its Applications. IEEE Transactions on Education 12: 27–34. [Google Scholar] [CrossRef][Green Version]
- Davidson, Russell, and James G. MacKinnon. 2004. Econometric Theory and Methods. Oxford: Oxford University Press. [Google Scholar]
- Diebold, Francis X., and Glenn D. Rudebusch. 1989. Long Memory and Persistence in Agregate Output. Journal of Monetary Economics 24: 189–209. [Google Scholar] [CrossRef][Green Version]
- Ergemen, Yunus Emre, Niels Haldrup, and Carlos Vladimir Rodríguez-Caballero. 2016. Common long-range dependence in a panel of hourly Nord Pool electricity prices and loads. Energy Economics 60: 79–96. [Google Scholar] [CrossRef]
- 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]
- Gil-Alana, Luis A. 2005. Statistical modeling of the temperatures in the Northern Hemisphere using fractional integration techniques. Journal of Climate 18: 5357–69. [Google Scholar] [CrossRef]
- GISTEMP. 2020. GISS Surface Temperature Analysis (GISTEMP), Version 4. Available online: https://data.giss.nasa.gov/gistemp/ (accessed on 30 August 2021).
- Granger, Clive W. J. 1966. The Typical Spectral Shape of an Economic Variable. Econometrica 34: 150–61. [Google Scholar] [CrossRef]
- Granger, Clive W. J. 1980. Long Memory Relationships and the Aggregation of Dynamic Models. Journal of Econometrics 14: 227–38. [Google Scholar] [CrossRef]
- Granger, Clive W. J. 1999. Aspects of Research Strategies for Time Series Analysis. In Presentation to the Conference on New Developments in Time Series Economics. New Haven: Yale University. [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: 15–29. [Google Scholar] [CrossRef]
- Haldrup, Niels, and J. Eduardo Vera-Valdés. 2017. Long Memory, Fractional Integration, and Cross-Sectional Aggregation. Journal of Econometrics 199: 1–11. [Google Scholar] [CrossRef][Green Version]
- Hassler, Uwe, and Barbara Meller. 2014. Detecting multiple breaks in long memory the case of U.S. inflation. Empirical Economics 46: 653–80. [Google Scholar] [CrossRef][Green Version]
- Hosking, Jonathan R. M. 1981. Fractional Differencing. Biometrika 68: 165–76. [Google Scholar] [CrossRef]
- Hurvich, Clifford M., Rohit Deo, and Julia Brodsky. 1998. The Mean Squared Error of Geweke and Porter-Hudak’s Estimator of the Memory Parameter of a Long-Memory Time Series. Journal of Time Series Analysis 19: 19–46. [Google Scholar] [CrossRef]
- Jensen, Andreas Noack, and Morten Ørregaard Nielsen. 2014. A Fast Fractional Difference Algorithm. Journal of Time Series Analysis 35: 428–36. [Google Scholar] [CrossRef][Green Version]
- Künsch, Hans. 1987. Statistical Aspects of Self-Similar Processes. Bernouli 1: 67–74. [Google Scholar]
- Lenssen, Nathan J. L., Gavin A. Schmidt, James E. Hansen, Matthew J. Menne, Avraham Persin, Reto Ruedy, and Daniel Zyss. 2019. Improvements in the GISTEMP Uncertainty Model. Journal of Geophysical Research: Atmospheres 124: 6307–26. [Google Scholar] [CrossRef]
- Linden, Mikael. 1999. Time Series Properties of Aggregated AR(1) Processes with Uniformly Distributed Coefficients. Economics Letters 64: 31–36. [Google Scholar] [CrossRef]
- Mills, Terence C. 2007. Time series modeling of two millennia of northern hemisphere temperatures: Long memory or shifting trends? Journal of the Royal Statistical Society. Series A: Statistics in Society 170: 83–94. [Google Scholar] [CrossRef]
- Oppenheim, Alan V., and Ronald W. Schafer. 2010. Discrete-Time Signal Processing. London: Pearson. [Google Scholar]
- Oppenheim, Georges, and Marie Claude Viano. 2004. Aggregation of Random Parameters Ornstein-Uhlenbeck or AR Processes: Some Convergence Results. Journal of Time Series Analysis 25: 335–50. [Google Scholar] [CrossRef]
- Osterrieder, Daniela, Daniel Ventosa-Santaulària, and J. Eduardo Vera-Valdés. 2019. The VIX, the Variance Premium, and Expected Returns. Journal of Financial Econometrics 17: 517–58. [Google Scholar] [CrossRef][Green Version]
- Phillips, Peter C. B. 2009. Long Memory and Long Run Variation. Journal of Econometrics 151: 150–58. [Google Scholar] [CrossRef][Green Version]
- Portnoy, Stephen. 2019. Edgeworth’s Time Series Model: Not AR(1) but Same Covariance Structure. Journal of Econometrics 213: 281–88. [Google Scholar] [CrossRef]
- Robinson, Peter M. 1978. Statistical Inference for a Random Coefficient Autoregressive Model. Scandinavian Journal of Statistics 5: 163–68. [Google Scholar] [CrossRef]
- Robinson, Peter M. 1995a. Gaussian Semiparametric Estimation of Long Range Dependence. The Annals of Statistics 23: 1630–61. [Google Scholar] [CrossRef]
- Robinson, Peter M. 1995b. Log-Periodogram Regression of Time Series with Long Range Dependence. The Annals of Statistics 23: 1048–72. [Google Scholar] [CrossRef]
- Veitch, Darryl, Anders Gorst-Rasmussen, and Andras Gefferth. 2013. Why FARIMA Models are Brittle. Fractals 21: 1–12. [Google Scholar] [CrossRef][Green Version]
- Vera-Valdés, J. Eduardo. 2021a. Temperature Anomalies, Long Memory, and Aggregation. Econometrics 9: 9. [Google Scholar] [CrossRef]
- Vera-Valdés, J. Eduardo. 2021b. The persistence of financial volatility after COVID-19. Finance Research Letters. [Google Scholar] [CrossRef]
- Vera-Valdés, J. Eduardo. 2020. On long memory origins and forecast horizons. Journal of Forecasting 39: 811–26. [Google Scholar] [CrossRef][Green Version]
- Zaffaroni, Paolo. 2004. Contemporaneous Aggregation of Linear Dynamic Models in Large Economies. Journal of Econometrics 120: 75–102. [Google Scholar] [CrossRef]

**Figure 1.**Computational times at several sample sizes for a MATLAB implementation of the algorithms. Axes are logarithmic. The reported times are the average of 100 replications for all sample sizes for the linear convolution and discrete Fourier transform algorithms and for sample sizes up to 1000 for the $AR\left(1\right)$ aggregation algorithm. For larger sample sizes, the $AR\left(1\right)$ aggregation algorithm was computed once due to computational restrictions.

**Figure 2.**Autocorrelation functions for an $I(-0.4)$ process and a $CSA(0.075,2.8)$ one. The right plot shows lags 100 to 150.

**Figure 3.**Mean periodograms of the $I\left(d\right)$ and $CSA(0.2,2(1-d\left)\right)$ processes for long-range dependence parameters $d=0.4$ (

**left**) and $d=-0.4$ (

**right**). A sample size of $T={10}^{3}$ was used and ${10}^{4}$ replications.

**Figure 4.**Autocorrelation function for a $CSA(a,b)$ processes for different values of the parameter a while having the same asymptotic behavior.

**Figure 5.**White noise series, ${\epsilon}_{t}$, and filtered processes using cross-sectional aggregation, $CSA(0.28,1.2)$, and the fractional difference operator, $I\left(0.4\right)$ (

**left**). Autocorrelation functions for the white noise series and filtered processes (

**right**).

**Figure 6.**London temperature anomalies obtained from GISTEMP (

**top left**) and its autocorrelation function (

**top right**). Residuals from fitted $CSA$ and $I\left(d\right)$ models to the series (

**bottom**).

**Table 1.**Computational times in seconds of the MATLAB implementation of the different algorithms to generate long-range dependence. $LC$ and $DFT$ stand for Linear Convolution and Discrete Fourier Transform, respectively. The reported times are the average of 100 replications for all sample sizes for the $LC$ and $DFT$ algorithms and for sample sizes up to 1000 for the $AR\left(1\right)$ aggregation algorithm. For larger sample sizes, the $AR\left(1\right)$ aggregation algorithm was computed once due to computational restrictions.

$\mathit{T}={10}^{2}$ | $\mathit{T}={10}^{3}$ | $\mathit{T}={10}^{4}$ | $\mathit{T}=5\times {10}^{4}$ | $\mathit{T}={10}^{5}$ | |
---|---|---|---|---|---|

$AR\left(1\right)$ Agg. | $2.02\times {10}^{-3}$ | $1.70\times {10}^{-1}$ | $8.08\times {10}^{1}$ | $9.23\times {10}^{3}$ | $8.29\times {10}^{4}$ |

$LC$ | $1.00\times {10}^{-5}$ | $1.10\times {10}^{-4}$ | $5.51\times {10}^{-3}$ | $1.86\times {10}^{-1}$ | $8.60\times {10}^{-1}$ |

$DFT$ | $4.00\times {10}^{-5}$ | $9.00\times {10}^{-5}$ | $9.30\times {10}^{-4}$ | $7.04\times {10}^{-3}$ | $8.46\times {10}^{-3}$ |

**Table 2.**Mean and standard deviation (in parentheses) of estimated long-range dependence parameters by the $GPH$, $BR$, and $LW$ methods for the $CSA(a,b)$ and $I\left(d\right)$ processes where $b=2(1-d)$ so that they show the same degree of long-range dependence. Furthermore, the parameter a was selected following (12) below with $k=10$, and only a quadratic term was added for the bias-reduced method. We used the $MSE$ optimal bandwidth of ${T}^{4/5}$ (see Hurvich et al. (1998)) and a sample size of $T={10}^{3}$ with ${10}^{4}$ replications.

$\mathit{d}=0.4$ | $\mathit{d}=0.2$ | $\mathit{d}=-0.2$ | $\mathit{d}=-0.4$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{CSA}(\mathit{a},\mathit{b})$ | $\mathit{I}\left(\mathit{d}\right)$ | $\mathit{CSA}(\mathit{a},\mathit{b})$ | $\mathit{I}\left(\mathit{d}\right)$ | $\mathit{CSA}(\mathit{a},\mathit{b})$ | $\mathit{I}\left(\mathit{d}\right)$ | $\mathit{CSA}(\mathit{a},\mathit{b})$ | $\mathit{I}\left(\mathit{d}\right)$ | |

$GPH$ | 0.425 | 0.391 | 0.260 | 0.195 | 0.177 | −0.194 | 0.211 | −0.387 |

(0.042) | (0.043) | (0.043) | (0.042) | (0.042) | (0.042) | (0.042) | (0.043) | |

$BR$ | 0.434 | 0.402 | 0.264 | 0.201 | 0.159 | −0.198 | 0.172 | −0.395 |

(0.066) | (0.066) | (0.066) | (0.066) | (0.066) | (0.065) | (0.065) | (0.067) | |

$LW$ | 0.424 | 0.390 | 0.258 | 0.194 | 0.178 | −0.196 | 0.213 | −0.389 |

(0.033) | (0.034) | (0.034) | (0.033) | (0.033) | (0.033) | (0.034) | (0.034) |

**Table 3.**$MLE$ estimates of $CSA(a,b)$ processes. Standard deviations are shown in brackets. We used ${10}^{3}$ replications, and all random vector were sampled from an $\mathcal{N}(0,{\sigma}^{2})$ distribution.

$(\mathit{a},\phantom{\rule{4pt}{0ex}}\mathit{b},\phantom{\rule{4pt}{0ex}}{\mathit{\sigma}}^{2})$ | $\mathit{T}=50$ | $\mathit{T}={10}^{2}$ | $\mathit{T}={10}^{3}$ |
---|---|---|---|

$(0.2,1.2,1)$ | $(0.403,1.772,0.870)$ | $(0.344,1.599,0.873)$ | $(0.247,1.239,0.896)$ |

$[0.369,0.722,0.187]$ | $[0.229,0.577,0.124]$ | $[0.049,0.140,0.042]$ | |

$(0.4,1.8,0.5)$ | $(0.575,2.089,0.440)$ | $(0.517,1.954,0.443)$ | $(0.404,1.673,0.447)$ |

$[0.481,0.755,0.095]$ | $[0.320,0.661,0.063]$ | $[0.089,0.230,0.021]$ | |

$(1.2,2.2,1.5)$ | $(0.993,1.864,1.365)$ | $(1.233,2.155,1.336)$ | $(1.202,2.219,1.351)$ |

$[0.842,0.730,0.291]$ | $[0.691,0.676,0.193]$ | $[0.265,0.341,0.066]$ | |

$(0.8,2.4,0.2)$ | $(0.690,1.977,0.181)$ | $(0.855,2.233,0.178)$ | $(0.812,2.278,0.179)$ |

$[0.649,0.728,0.039]$ | $[0.492,0.667,0.025]$ | $[0.171,0.336,0.009]$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vera-Valdés, J.E. Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation. *Econometrics* **2021**, *9*, 39.
https://doi.org/10.3390/econometrics9040039

**AMA Style**

Vera-Valdés JE. Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation. *Econometrics*. 2021; 9(4):39.
https://doi.org/10.3390/econometrics9040039

**Chicago/Turabian Style**

Vera-Valdés, J. Eduardo. 2021. "Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation" *Econometrics* 9, no. 4: 39.
https://doi.org/10.3390/econometrics9040039