Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data

By Emmanuel Bacry, K. Dayri, Jean-François Muzy. 2011

192 citations recorded by the source corpus.

Abstract We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/0e436b3496de042e977a63580addce2cc835348c

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In Applied Mathematics

Citations in the corpus

Some limit theorems for Hawkes processes and application to financial statistics (2013, unknown): “[13].”
Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes (2015, background): “This function plays an important role in the study of Hawkes processes; see [3].”
Ergodicity and scaling limit of a constrained multivariate Hawkes process (2013, unknown): “[4] in dimension d = 1, the majority of Hawkes processes related publications work in the setting of exponential kernel functions for computational reasons.”
Critical reflexivity in financial markets: a Hawkes process analysis (2013, methodology): “Following [17] the auto-covariance is also discretely sampled at a rate ∆ = h and the Fourier transform is performed in practice with DFT.”
Limit theorems for nearly unstable Hawkes processes (2013, methodology): “In any case, we can remark that both groups find values close to one for ‖φ‖1, which is consistent with the results of [4], where estimations are performed on Bund and Dax futures.”
Market Impacts and the Life Cycle of Investors Orders (2014, methodology): “It has been shown ([Bacry et al., 2012]) that this mean-reversion property is well mimicked using a 2-dimensional Hawkes process using only a single "cross" kernel φ(t) : λ+t = μ + φ ? dJ − s and and λ − t = μ + φ ? dJ + s (10) where φ(t) is a causal (i.”
Learning Triggering Kernels for Multi-dimensional Hawkes Processes (2013, methodology): “The problem of nonparametric estimation of the triggering kernels has been addressed for special cases such as the one-dimensional Hawkes processes (Lewis and Mohler, 2011) or symmetric Hawkes processes (Bacry et al., 2012).”
Hawkes Processes in Finance (2015, methodology): “Let us notice that the existence of non-negligible diagonal and anti-diagonal terms has been confirmed by non-parametric estimations performed thereafter in [2, 7, 6].”
Scaling limits for Hawkes processes and application to financial statistics (2012, unknown): “…7641 and École Polytechnique, 91128 Palaiseau, France †Université Paris Diderot and LPMA CNRS-UMR 7599, Bôıte courrier 7012 75251 Paris, France ‡ENSAE-CREST and LAMA CNRS-UMR 8050, 3, avenue Pierre Larousse, 92245 Malakoff, France §SPE CNRS-UMR 6134 and Université de Corte, 20250 Corte, France”