# 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 |

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**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.

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

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

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

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

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

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

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

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

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

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**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