# Non-Parametric Estimation of Intraday Spot Volatility: Disentangling Instantaneous Trend and Seasonality

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## Abstract

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## 1. Introduction

## 2. Intraday Seasonality Dynamics

- ${T}_{n}$ and ${s}_{n}$ representing the trend and seasonality in the volatility,
- ${h}_{n}$ the intraday volatility component,
- ${w}_{n}$ the white noise,
- and ${q}_{n}{I}_{n}$ the discretized finite activity counting process.

#### 2.1. The Adaptive Seasonality Model

**Definition 1**

**.**For fixed choices of $0<\u03f5\ll 1$ and $\u03f5\ll {c}_{1}<{c}_{2}<\infty $, the space ${\mathcal{A}}_{\u03f5}^{{c}_{1},{c}_{2}}$ of intrinsic mode functions consists of functions $f:\mathbb{R}\to \mathbb{R}$, $f\in {C}^{1}(\mathbb{R})\cap {L}_{\infty}(\mathbb{R})$ having the form

**Definition 2**

**.**The space ${\mathcal{C}}_{\u03f5}^{{c}_{1},{c}_{2}}$ of superpositions of intrinsic mode functions consists of functions f having the form

#### 2.2. The Adaptive Trend Model

**Definition 3**

**.**For fixed choices of $0<\u03f5\ll 1$ and $\u03f5\ll {c}_{1}<\infty $, the space ${\mathcal{T}}_{\u03f5}^{{c}_{1}}$ consists of functions $T:\mathbb{R}\to \mathbb{R}$, $T\in {C}^{1}(\mathbb{R})$ so that ${F}_{T}$ exists in the distribution sense, and

#### 2.3. The Adaptive Volatility Model

- ${s}_{n}$, discretely sampled from $s(t)\in {\mathcal{C}}_{\u03f5}^{{c}_{1},{c}_{2}}$, is the volatility seasonality,
- ${T}_{n}$, discretely sampled from $T(t)\in {\mathcal{T}}_{\u03f5}^{{c}_{1}}$, is the volatility trend,
- and ${z}_{n}=log\left(\right)open="|"\; close="|">{h}_{n}{w}_{n}+{q}_{n}{I}_{n}{e}^{-({T}_{n}+{s}_{n})}$ is an additive noise process that satisfies $E({z}_{n})<\infty $ and $\mathrm{var}({z}_{n})<\infty $.

**Remark 1.**

**Remark 2.**

#### 2.4. Inference

## 3. Simulation Study

- ${q}_{n}={e}^{{T}_{n}+{s}_{n}}{j}_{n}$, where ${j}_{n}\sim N(0,{\sigma}_{j})$ with ${\sigma}_{j}\in \left(\right)open="\{"\; close="\}">4{\sigma}_{w},10{\sigma}_{w}$, and with ${\sigma}_{w}$ the standard deviation of the estimated i.i.d. GED(ν),
- and ${I}_{n}\sim B(1,\lambda )$ data) with $\lambda \in \left(\right)open="\{"\; close="\}">1/288,1/1440$, that is, one jump per day or week on average.

- As observed in Figure 1, the jumps do not affect the results, but the choice of process for ${h}_{n}{w}_{n}$ does. The effect of this choice is comparatively more pronounced on the AISE, which is a measure of the total estimator’s variance.
- There is no relationship between the seasonality dynamics and the trend’s estimate; whether the seasonality is with constant amplitude or with amplitude modulations, the AISE and AIAE for the trend are similar. However, for the seasonality, the AISE and AIAE are higher with amplitude modulations.

**Figure 1.**Trend and seasonality preliminary simulation results. In each panel: mean estimate (black line), true quantity (red line) and estimated 95% confidence intervals (shaded area). (

**a**) Trend for the entire (simulated) data; (

**b**) Seasonality for the first week of the (simulated) data.

**Table 1.**Trend and seasonality complete simulation results. In each cell: trend (left) and seasonality (right).

Fourier Flexible Form | GED | EGARCH(1,1) | GARCH(1,1) | |

No jumps | AISE | $1.50/0.30$ | $1.97/0.49$ | $1.83/0.44$ |

AIAE | $11.36/5.38$ | $13.25/6.80$ | $12.71/6.47$ | |

$\lambda =1/288$, ${\sigma}_{j}=4{\sigma}_{w}$ | AISE | $1.48/0.30$ | $1.95/0.49$ | $1.81/0.44$ |

AIAE | $11.27/5.39$ | $13.16/6.80$ | $12.64/6.47$ | |

$\lambda =1/1440$, ${\sigma}_{j}=4{\sigma}_{w}$ | AISE | $1.49/0.30$ | $1.97/0.49$ | $1.82/0.44$ |

AIAE | $11.34/5.38$ | $13.23/6.80$ | $12.69/6.47$ | |

$\lambda =1/288$, ${\sigma}_{j}=10{\sigma}_{w}$ | AISE | $1.47/0.31$ | $1.94/0.49$ | $1.81/0.44$ |

AIAE | $11.24/5.41$ | $13.14/6.82$ | $12.63/6.49$ | |

$\lambda =1/1440$, ${\sigma}_{j}=10{\sigma}_{w}$ | AISE | $1.49/0.30$ | $1.96/0.49$ | $1.82/0.44$ |

AIAE | $11.33/5.38$ | $13.22/6.80$ | $12.69/6.47$ | |

Synchrosqueezed Transform | GED | EGARCH(1,1) | GARCH(1,1) | |

No jumps | AISE | $1.45/0.48$ | $1.93/0.68$ | $1.79/0.63$ |

AIAE | $11.20/6.66$ | $13.14/7.95$ | $12.60/7.64$ | |

$\lambda =1/288$, ${\sigma}_{j}=4{\sigma}_{w}$ | AISE | $1.43/0.48$ | $1.91/0.68$ | $1.77/0.63$ |

AIAE | $11.11/6.67$ | $13.06/7.96$ | $12.53/7.65$ | |

$\lambda =1/1440$, ${\sigma}_{j}=4{\sigma}_{w}$ | AISE | $1.45/0.48$ | $1.93/0.68$ | $1.78/0.63$ |

AIAE | $11.18/6.66$ | $13.12/7.95$ | $12.58/7.65$ | |

$\lambda =1/288$, ${\sigma}_{j}=10{\sigma}_{w}$ | AISE | $1.42/0.49$ | $1.90/0.68$ | $1.77/0.63$ |

AIAE | $11.09/6.68$ | $13.04/7.97$ | $12.52/7.66$ | |

$\lambda =1/1440$, ${\sigma}_{j}=10{\sigma}_{w}$ | AISE | $1.44/0.48$ | $1.92/0.68$ | $1.78/0.63$ |

AIAE | $11.17/6.66$ | $13.12/7.96$ | $12.58/7.65$ |

## 4. Application to Foreign Exchange Data

- Use $\widehat{T}(n\tau )$ and ${\widehat{f}}_{k}(n\tau )$ to recover ${\widehat{z}}_{n}={y}_{n}-\widehat{T}(n\tau )-{\sum}_{k=1}^{K}{\widehat{f}}_{k}(n\tau )$ for $n\in \left(\right)open="\{"\; close="\}">1,\cdots ,N$.
- Bootstrap B samples ${z}_{n}^{b}$ for $b\in \left(\right)open="\{"\; close="\}">1,\cdots ,B$ and $n\in \left(\right)open="\{"\; close="\}">1,\cdots ,N$.
- Add back $\widehat{T}(n\tau )$ and ${\widehat{f}}_{k}(n\tau )$ to ${z}_{n}^{b}$ in order to obtain ${y}_{n}^{b}={z}_{n}^{b}+\widehat{T}(n\tau )+{\sum}_{k=1}^{K}{\widehat{f}}_{k}(n\tau )$ for $b\in \left(\right)open="\{"\; close="\}">1,\cdots ,B$ and $n\in \left(\right)open="\{"\; close="\}">1,\cdots ,N$.
- Estimate ${\widehat{T}}^{b}(n\tau )$ and ${\widehat{f}}_{k}^{b}(n\tau )$ for $b\in \left(\right)open="\{"\; close="\}">1,\cdots ,B$ and $n\in \left(\right)open="\{"\; close="\}">1,\cdots ,N$ to compute pointwise confidence intervals.

**Figure 2.**First week of returns and log-volatilities in January 2010. Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (

**a**) Return ${r}_{n}$; (

**b**) Log-volatility ${y}_{n}=2log\left(\right)open="|"\; close="|">{r}_{n}-\widehat{\mu}$.

**Figure 3.**Autocorrelation ${\gamma}_{y}(l)$ and power spectrum ${P}_{y}(\omega )$ of the log-volatility. ${\gamma}_{y}(l)=\widehat{\mathbb{E}}\left(\right)open="["\; close="]">\left(\right)open="\{"\; close="\}">{y}_{n}-\widehat{\mathbb{E}}({y}_{n})\left(\right)open="\{"\; close="\}">{y}_{n-l}-\widehat{\mathbb{E}}({y}_{n})$ and ${P}_{y}(\omega )={\left(\right)}^{{\widehat{F}}_{y}}2$. Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (

**a**) Autocorrelation ${\gamma}_{y}(l)$. The lag l is from 1 to 1440 and the x axis ticks are divided by 288; (

**b**) Power spectrum ${P}_{y}(\omega )$. The frequency ω is from 0 to 5.

**Figure 4.**Trend reconstruction results. In each panel: reconstructed trend (black line), realized volatility (red line) and bipower variation (blue line). In Panel 4b: 95% confidence intervals for the reconstructed trend (shaded area). Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (

**a**) Trend reconstruction for 2010–2013; (

**b**) Zoom during the summer of 2011.

**Figure 5.**Amplitude modulations reconstruction results. In each panel: reconstructed amplitude modulations of the first component (black line) with 95% confidence intervals (shaded area), Fourier flexible form (dashed line) and rolling Fourier flexible form (red line). (Top) CHF/USD (left) and EUR/USD (right); (Bottom) GBP/USD (left) and JPY/USD (right).

**Figure 6.**Seasonality reconstruction results. In each panel: reconstructed seasonality (black line) with 95% confidence intervals (shaded area) and rolling Fourier flexible form (red line). Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (

**a**) Second week of July 2011; (

**b**) Second week of August 2011.

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix. Implementation Details

## References

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^{1}[3,4] consider the addition of dummy variables to capture weekday effects or particular events such as holidays in particular markets, in addition to unemployment reports, retail sales figures, etc. While the periodic model captures most of the seasonal patterns, their dummy variables allow the quantification of the relative importance of calendar effects and announcement events.^{2}${a}_{k}(t)\in {C}^{2}(\mathbb{R})$ and ${sup}_{t\in \mathbb{R}}\left|{a}_{k}^{\prime \prime}(t)\right|\le \u03f5{c}_{2}$ for all $k=1,\dots ,K$, and $T\in {C}^{2}(\mathbb{R})$ so that $\left(\right)open="|"\; close="|">\int {T}^{\u2033}(t)\frac{1}{\sqrt{a}}\psi (\frac{t-b}{a})\mathrm{d}t\le {C}_{T}\u03f5$ for all $b\in \mathbb{R}$ and $a\in (0,\frac{1+\Delta}{{c}_{1}}]$^{3}Although not reported here, the pointwise biases and variances for the trend and seasonality were also studied. While the biases are again indistinguishable (and basically negligible for the trend) between the GED or GARCH with or without jumps, the variances are higher for the GARCH, and show no visible from the jumps.

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**MDPI and ACS Style**

Vatter, T.; Wu, H.-T.; Chavez-Demoulin, V.; Yu, B.
Non-Parametric Estimation of Intraday Spot Volatility: Disentangling Instantaneous Trend and Seasonality. *Econometrics* **2015**, *3*, 864-887.
https://doi.org/10.3390/econometrics3040864

**AMA Style**

Vatter T, Wu H-T, Chavez-Demoulin V, Yu B.
Non-Parametric Estimation of Intraday Spot Volatility: Disentangling Instantaneous Trend and Seasonality. *Econometrics*. 2015; 3(4):864-887.
https://doi.org/10.3390/econometrics3040864

**Chicago/Turabian Style**

Vatter, Thibault, Hau-Tieng Wu, Valérie Chavez-Demoulin, and Bin Yu.
2015. "Non-Parametric Estimation of Intraday Spot Volatility: Disentangling Instantaneous Trend and Seasonality" *Econometrics* 3, no. 4: 864-887.
https://doi.org/10.3390/econometrics3040864