# Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models

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## Abstract

**:**

## 1. Introduction

## 2. Review of Stochastic Volatility (SV) Models

- $\left\{{X}_{t}\right\}$ follows a stationary and invertible ARMA(p,q) process given by:$$\varphi \left(L\right){X}_{t}=C+\theta \left(L\right){v}_{t},$$
- $\left\{{X}_{t}\right\}$ follows a stationary and invertible ARFIMA(p,d,q) process given by:$$\varphi \left(L\right){(1-L)}^{d}{X}_{t}=C+\theta \left(L\right){v}_{t},$$

## 3. Gegenbauer ARMA (GARMA) Model

- The power spectrum:$${f}_{X}\left(\omega \right)={\left[4{(\mathrm{cos}\omega -\eta )}^{2}\right]}^{-d}g\left(\omega \right),\phantom{\rule{4pt}{0ex}}-\pi <\omega <\pi ,$$
- The process in (5) is stationary and explains long memory when $\left|\eta \right|<1$ and $0<d<1/2$, or $\left|\eta \right|=1$ and $0<d<1/4$, with the stationary condition on $\varphi \left(L\right)$. From (6), it is clear that the long memory features are characterized by an unbounded spectrum at the Gegenbauer frequency $\omega ={\omega}_{g}=\mathrm{arccos}\left(\eta \right)$ when $\left|\eta \right|<1$, and at $\omega =0$ when $\eta =1,$ in addition to the hyperbolic decay of the autocorrelation function (acf).For later reference, we consider a special case, namely, the class of GARMA$(0,d,0;\eta )$ given by:$${(1-2\eta L+{L}^{2})}^{d}{X}_{t}={v}_{t}.$$Following regularity conditions are useful for further analysis.
- Under the AR regularity conditions:
- (a1)
- $\left|\eta \right|<1$ and $d<1/2$; or
- (a2)
- $\left|\eta \right|=1$ and $d<1/4,$

the Wold representation of (7) is given as:$${X}_{t}=\psi \left(L\right){v}_{t}=\sum _{j=0}^{\infty}{\psi}_{j}{v}_{t-j},$$$${\psi}_{j}=2\eta \left(\right)open="("\; close=")">\frac{d-1+j}{j}{\psi}_{j-2},$$These coefficients, ${\psi}_{j}$, reduce to the corresponding standard long memory (or binomial) coefficients when $\eta =1$, such that ${\psi}_{j}=\frac{\Gamma (2d+j)}{\Gamma (j+1)\Gamma \left(2d\right)}.$ - Under the MA regularity conditions:
- (b1)
- $\left|\eta \right|<1$ and $d>-1/2$; or
- (b2)
- $\left|\eta \right|=1$ and $d>-1/4,$

(6) admits an invertible solution, such that:$${v}_{t}={(1-2\eta L+{L}^{2})}^{d}{X}_{t}=\sum _{j=0}^{\infty}{\pi}_{j}{X}_{t-j},$$

## 4. Generalized Long Memory SV (GLMSV) Models

#### 4.1. Properties of GLMSV

- $E\left({Y}_{t}\right)=0$ and $Var\left({Y}_{t}\right)=\mathrm{exp}[\gamma \left(0\right)/2],$
- ${\gamma}_{Y}\left(k\right)=Cov({Y}_{t},{Y}_{t+k})=0$ for all $k\ne 0,$
- $\left\{{Y}_{t}\right\}$ is a martingale difference.

#### 4.2. Identification of GLMSV and LMSV

**Lemma**

**1.**

**Proof.**

**Spectral Densities**

- Standard LMSV when $\eta =1:$The sdf of $\left\{{U}_{t}\right\}$ is given by:$${f}_{U}{\left(\omega \right)\sim [2\left(\mathrm{sin}(\omega /2)\right]}^{-4d}\frac{{\sigma}_{v}^{2}}{2\pi},\phantom{\rule{4pt}{0ex}}-\pi <\omega <\pi ,$$
- GLMSV when $\left|\eta \right|<1:$The sdf of $\left\{{U}_{t}\right\}$ is given by:$${f}_{U}\left(\omega \right)\sim {[4{(\mathrm{cos}\omega -\eta )}^{2}]}^{-d}\frac{{\sigma}_{v}^{2}}{2\pi},\phantom{\rule{4pt}{0ex}}-\pi <\omega <\pi ,$$

## 5. Estimation and Forecasting

#### 5.1. Spectral-Likelihood Estimator

#### 5.2. Finite Sample Properties

#### 5.3. Estimating and Forecasting Volatility

**X**is given by:

**R**is the $h\times n$ matrix of covariances between ${\mathit{U}}_{h}$ and

**U**. Using ${\tilde{\mathit{X}}}_{h}={\tilde{\mathit{U}}}_{h}-(\mu +c){\mathbf{1}}_{h}$, the predictions of ${\sigma}_{n+j}^{2}$$(j=1,\dots ,h)$ are given by exponentiating the elements of ${\tilde{\mathit{X}}}_{h}$, and multiplying by ${\tilde{\sigma}}_{\tilde{Y}}^{2}$.

## 6. Empirical Analysis

#### 6.1. Data and Preliminary Results

#### 6.2. Estimates and Forecasts for the GLMSV Model

## 7. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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DGP | Parameters | |||||
---|---|---|---|---|---|---|

$\mathit{\mu}$ | ${\mathit{\sigma}}_{\mathit{\u03f5}}$ | $\mathit{\sigma}$ | $\mathit{\varphi}$ | $\mathit{d}$ | $\mathit{\eta}$ | |

AR(1) | ||||||

True | 0 | 2.221 | 0.199 | 0.98 | 0 | 1 |

$n=1024$ | 0.0091 | 2.1201 | 0.4863 | 0.9693 | −0.1086 | 0.8972 |

(0.2967) | (0.1901) | (0.4455) | (0.0305) | (0.3847) | (0.0830) | |

[0.2967] | [0.2152] | [0.5297] | [0.0323] | [0.3993] | [0.1320] | |

$n=2048$ | 0.0011 | 2.1666 | 0.4047 | 0.9753 | −0.0869 | 0.8986 |

(0.2170) | (0.1137) | (0.3361) | (0.0119) | (0.3590) | (0.0797) | |

[0.2168] | [0.1261] | [0.3938] | [0.0128] | [0.3691] | [0.1289] | |

ARFIMA(1,2d,0) | ||||||

True | 0 | 2.221 | 0.572 | 0.30 | 0.2 | 1 |

$n=1024$ | 0.0034 | 2.0796 | 0.5907 | 0.8551 | 0.1004 | 0.8901 |

(0.4011) | (0.3942) | (0.6551) | (0.2127) | (0.3359) | (0.0824) | |

[0.4007] | [0.4185] | [0.6547] | [0.5944] | [0.3500] | [0.1373] | |

$n=2048$ | 0.0063 | 2.2014 | 0.4303 | 0.8939 | 0.1568 | 0.9003 |

(0.3394) | (0.1618) | (0.4617) | (0.1983) | (0.2928) | (0.0870) | |

[0.3391] | [0.1629] | [0.4826] | [0.6261] | [0.2957] | [0.1322] | |

GARMA(1,d,0), Case 1 | ||||||

True | 0 | 2.221 | 0.520 | 0.30 | 0.4 | 0.7 |

$n=1024$ | 0.0027 | 1.9444 | 0.9212 | 0.0988 | 0.3301 | 0.6984 |

(0.0684) | (0.5856) | (0.5432) | (0.3501) | (0.1072) | (0.0307) | |

[0.0684] | [0.6473] | [0.6749] | [0.4035] | [0.1280] | [0.0307] | |

$n=2048$ | −0.0017 | 2.0965 | 0.7608 | 0.1693 | 0.3572 | 0.7005 |

(0.0564) | (0.3353) | (0.4068) | (0.3143) | (0.0797) | (0.0052) | |

[0.0564] | [0.3575] | [0.4724] | [0.3401] | [0.0904] | [0.0053] | |

GARMA(1,d,0), Case 2 | ||||||

True | 0 | 2.221 | 0.675 | 0.70 | 0.3 | 0.3 |

$n=1024$ | 0.0037 | 2.0348 | 0.8180 | 0.6441 | 0.2668 | 0.3016 |

(0.0871) | (0.5725) | (0.5207) | (0.2018) | (0.1481) | (0.0937) | |

[0.0871] | [0.6016] | [0.5395] | [0.2092] | [0.1516] | [0.0936] | |

$n=2048$ | −0.0014 | 2.1928 | 0.7022 | 0.6847 | 0.2905 | 0.3006 |

(0.0696) | (0.2076) | (0.2269) | (0.1099) | (0.0747) | (0.0459) | |

[0.0696] | [0.2094] | [0.2283] | [0.1109] | [0.0752] | [0.0458] |

Data | Mean | Std. Dev. | Skewness | Kurtosis |
---|---|---|---|---|

YEN/USD | 0.0028 | 0.6617 | −0.3225 | 8.1747 |

EUR/USD | −0.0045 | 0.6383 | 0.1717 | 5.9683 |

GBP/USD | −0.0060 | 0.6163 | −0.3377 | 9.6188 |

Parameters | YEN/USD | EUR/USD | GBP/USD |
---|---|---|---|

w | 0.0045 | 0.0010 | 0.0018 |

(0.0010) | (0.0005) | (0.0008) | |

$\alpha $ | 0.0342 | 0.0329 | 0.0394 |

(0.0036) | (0.0046) | (0.0056) | |

$\beta $ | 0.9565 | 0.9647 | 0.9556 |

(0.0047) | (0.0044) | (0.0065) |

Parameters | YEN/USD | EUR/USD | GBP/USD | |||
---|---|---|---|---|---|---|

FIEGARCH | GIEGARCH | FIEGARCH | GIEGARCH | FIEGARCH | GIEGARCH | |

$\mu $ | −0.7736 | −0.8589 | −0.7916 | −0.9170 | −0.8771 | −1.0338 |

(0.0474) | (0.0427) | (0.0715) | (0.0776) | (0.0809) | (0.0760) | |

$\varphi $ | −0.1084 | 0.9749 | −0.2401 | 0.9854 | −0.2006 | 0.9881 |

(0.0278) | (0.0034) | (0.0509) | (0.0014) | (0.0379) | (0.0019) | |

${\gamma}_{1}$ | −1.1991 | −0.0415 | −0.2439 | −0.0119 | −0.7873 | −0.0229 |

(0.3571) | (0.0064) | (0.1325) | (0.0036) | (0.2147) | (0.0042) | |

${\gamma}_{2}$ | 0.7254 | 0.0321 | 0.5071 | 0.0140 | 0.4779 | 0.0108 |

(0.2696) | (0.0043) | (0.1496) | (0.0023) | (0.1727) | (0.0027) | |

d | 0.1491 | 0.3350 | 0.2368 | 0.4988 | 0.2495 | 0.4996 |

(0.0345) | (0.0750) | (0.0365) | (0.0854) | (0.0431) | (0.0624) | |

$\eta $ | 1 | 0.3892 | 1 | 0.8583 | 1 | 0.8570 |

(0.0026) | (0.0014) | (0.0006) | ||||

${\omega}_{g}$ | 0 | 1.1710 | 0 | 0.5388 | 0 | 0.5414 |

Parameters | YEN/USD | EUR/USD | GBP/USD |
---|---|---|---|

$\mu $ | −1.2366 | −1.2030 | −1.3069 |

(0.0579) | (0.0544) | (0.0538) | |

${\sigma}_{\u03f5}$ | 2.5173 | 2.3482 | 2.2844 |

(0.0414) | (0.0384) | (0.0368) | |

$\sigma $ | 0.0868 | 0.1621 | 0.0974 |

(0.0350) | (0.0378) | (0.0311) | |

$\varphi $ | 0.9872 | 0.9939 | 0.9980 |

(0.0066) | (0.0042) | (0.0039) | |

d | 0.3173 | 0.4702 | 0.4987 |

(0.1475) | (0.1029) | (0.1869) | |

$\eta $ | 0.8032 | 0.9597 | 0.8400 |

(0.0001) | (0.0003) | (0.0009) | |

${\omega}_{g}$ | 0.6381 | 0.2849 | 0.5735 |

Data | Parameter | GARCH | FIEGARCH | GIGARCH | GLMSV |
---|---|---|---|---|---|

YEN/USD | a | 0.0859 | 0.0715 | 0.0404 | −0.2356 |

(0.0676) | (0.0660) | (0.0687) | (0.0898) | ||

b | 0.5887 | 0.5906 | 0.7116 | 1.5406 | |

(0.1951) | (0.1765) | (0.1933) | (0.2596) | ||

S.E. | 0.6496 | 0.6482 | 0.6467 | 0.6335 | |

${R}^{2}$ | 0.1609 | 0.1644 | 0.1682 | 0.2020 ^{†} | |

EUR/USD | a | 0.0329 | −0.0493 | −0.0631 | 0.1012 |

(0.0448) | (0.0652) | (0.0563) | (0.0360) | ||

b | 0.8955 | 1.0968 | 1.1251 | 0.2704 | |

(0.1116) | (0.1729) | (0.1438) | (0.0307) | ||

S.E. | 0.5624 | 0.5748 | 0.5640 | 0.5560 | |

${R}^{2}$ | 0.3226 | 0.2922 | 0.3187 | 0.3379 ^{†} | |

GBP/USD | a | 0.0314 | 0.0464 | 0.0052 | −0.4764 |

(0.0317) | (0.0435) | (0.0398) | (0.1018) | ||

b | 0.7768 | 0.5747 | 0.7346 | 1.4038 | |

(0.1445) | (0.1704) | (0.1520) | (0.2140) | ||

S.E. | 0.3090 | 0.3143 | 0.3107 | 0.3050 | |

${R}^{2}$ | 0.2950 | 0.2708 | 0.2875 | 0.3134 ^{†} |

Data | Loss Function | GARCH | FIEGARCH | GIGARCH | GLMSV |
---|---|---|---|---|---|

YEN/USD | $c=2$ (MSE) | 0.2129 | 0.2137 | 0.2106 | 0.1594 ^{†} |

$c=1$ | 0.4661 | 0.5006 | 0.4812 | 0.4232 ^{†} | |

$c=0$ (QLIKE) | 284.45 ^{†} | 342.79 | 299.66 | 320.26 | |

EUR/USD | $c=2$ (MSE) | 0.1578 | 0.1648 | 0.1588 | 0.0849 ^{†} |

$c=1$ | 0.4565 ^{†} | 0.5245 | 0.5020 | 0.5787 | |

$c=0$ (QLIKE) | 362.20 | 497.04 | 476.01 | 354.85 ^{†} | |

GBP/USD | $c=2$ (MSE) | 0.0479 | 0.0514 | 0.0502 | 0.0089 ^{†} |

$c=1$ | 0.2995 | 0.3804 | 0.3773 | 0.1983 ^{†} | |

$c=0$ (QLIKE) | 35,926 ^{†} | 41,066 | 37,279 | 36,204 |

Data | VaR | GARCH | FIEGARCH | GIEGARCH | GLMSV | ||||
---|---|---|---|---|---|---|---|---|---|

5% | 1% | 5% | 1% | 5% | 1% | 5% | 1% | ||

YEN/USD | PV | 0.036 | 0.008 | 0.034 | 0.004 | 0.036 | 0.004 | 0.0040 | 0.010 |

UC | 0.6540 | 0.3168 | 0.9752 | 0.5989 | 0.5029 | 1.5034 | 0.0733 | 1.2558 | |

[0.4187] | [0.5735] | [0.3234] | [0.4390] | [0.4782] | [0.2201] | [0.7866] | [0.2625] | ||

IND | 1.7393 | 1.0694 | 8.1759 | 0.9347 | 2.1775 | 1.1999 | 5.2835 | 1.6877 | |

[0.7836] | [0.8991] | [0.0853] | [0.9195] | [0.732] | [0.8781] | [0.2594] | [0.7930] | ||

CC | 5.4979 | 3.3522 | 76.331 * | 2.0605 | 5.4058 | 3.7080 | 7.2375 | 1.6877 | |

[0.3582] | [0.6459] | [0.0000] | [0.8407] | [0.3684] | [0.5922] | [0.2036] | [0.8905] | ||

EUR/USD | PV | 0.036 | 0.008 | 0.058 | 0.010 | 0.054 | 0.008 | 0.044 | 0.018 |

UC | 0.6540 | 0.3168 | 20866 | 0.3457 | 0.3174 | 0.0097 | 0.6177 | 2.0001 | |

[0.4187] | [0.5735] | [0.1486] | [0.5566] | [0.5732] | [0.9214] | [0.4319] | [0.1573] | ||

IND | 1.7393 | 1.0694 | 1.3895 | 1.7610 | 0.0726 | 0.2397 | 3.5391 | 0.9180 | |

[0.7836] | [0.8991] | [0.8460] | [0.7796] | [0.9994] | [0.9934] | [0.4720] | [0.9220] | ||

CC | 5.4979 | 3.3522 | 3.0081 | 1.7610 | 0.3424 | 0.1625 | 6.6769 | 2.8541 | |

[0.3582] | [0.6459] | [0.6987] | [0.8811] | [0.9968] | [0.9995] | [0.2458] | [0.7225] | ||

GBP/USD | PV | 0.068 | 0.016 | 0.048 | 0.004 | 0.050 | 0.004 | 0.052 | 0.012 |

UC | 2.7886 | 2.8064 | 0.0193 | 04137 | 0.0281 | 2.0368 | 1.7100 | 2.3767 | |

[0.0949] | [0.0939] | [0.8894] | [0.5201] | [0.8669] | [0.1535] | [0.1910] | [0.1232] | ||

IND | 1..3485 | 1.7954 | 18.555 * | 0.8417 | 1.0782 | 1.2781 | 7.1894 | 2.7581 | |

[0.8531] | [0.7733] | [0.0010] | [0.9328] | [0.8977] | [0.8651] | [0.1262] | [0.5991] | ||

CC | 2.9997 | 4.0075 | 23.801 * | 1.7850 | 1.0782 | 4.6977 | 3.9607 | 3.8607 | |

[0.7000] | [0.5483] | [0.0002] | [0.8780] | [0.9560] | [0.4539] | [3.9607] | [3.8607] |

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**MDPI and ACS Style**

Peiris, S.; Asai, M.; McAleer, M.
Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models. *J. Risk Financial Manag.* **2017**, *10*, 23.
https://doi.org/10.3390/jrfm10040023

**AMA Style**

Peiris S, Asai M, McAleer M.
Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models. *Journal of Risk and Financial Management*. 2017; 10(4):23.
https://doi.org/10.3390/jrfm10040023

**Chicago/Turabian Style**

Peiris, Shelton, Manabu Asai, and Michael McAleer.
2017. "Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models" *Journal of Risk and Financial Management* 10, no. 4: 23.
https://doi.org/10.3390/jrfm10040023