Nonparametric estimation of trawl processes: Theory and applications

TL;DR

This paper develops and analyzes a nonparametric estimator for the trawl function in continuous-time trawl processes, establishing consistency and feasible central limit theorems under a double in-fill/long-span asymptotic regime. By linking the derivative of the autocovariance to the trawl function, it delivers practical estimators for the trawl function, its ABI/AVAR, and slices of the trawl set, along with feasible inference procedures. The work also demonstrates substantial finite-sample performance through extensive simulations and provides three applications: model misspecification testing, forecasting high-frequency limit-order-book spreads, and estimating busy-time distributions in queues. Overall, the methodology offers a robust, data-driven alternative to parametric trawl models with strong theoretical guarantees and versatile applications. The paper underlines the importance of nonparametric trawl estimation for capturing complex short- and long-range dependence in infinitely divisible, stationary processes.

Abstract

Trawl processes belong to the class of continuous-time, strictly stationary, infinitely divisible processes; they are defined as Levy bases evaluated over deterministic trawl sets. This article presents the first nonparametric estimator of the trawl function characterising the trawl set and the serial correlation of the process. Moreover, it establishes a detailed asymptotic theory for the proposed estimator, including a law of large numbers and a central limit theorem for various asymptotic relations between an in-fill and a long-span asymptotic regime. In addition, it develops consistent estimators for both the asymptotic bias and variance, which are subsequently used for establishing feasible central limit theorems which can be applied to data. A simulation study shows the good finite sample performance of the proposed estimators. The new methodology is applied to model misspecification testing, forecasting high-frequency financial spread data from a limit order book and to estimating the busy-time distribution of a stochastic queue.

Figures (29)

• Figure 1: Graphical representation of the increments $X_{(k+1)\Delta_{n}}-X_{k\Delta_{n}}$.
• Figure 2: QQ plot of the debiased and standardised estimation error $\sqrt{n\Delta_{n}}\frac{\left(\hat{a}(1)-a(1)-\frac{1}{2}\Delta_n a'(1)\right)}{\sqrt{\sigma_a^2(1)}}$ based on $5000$ Monte Carlo repetitions. In our simulations, the Lévy seed followed a standard normal distribution and we used a supGamma trawl $a(s)=(1+s)^{-H}$ with $H=1.5$. Furthermore, the distance between observations was set to $\Delta_n=n^{-1/3}$ with $n$ denoting the sample size, where $n=5000$ in Figure \ref{['fig:qq1']} and $n=10000$ in Figure \ref{['fig:qq2']}.
• Figure 3: Comparison of two trawl functions $a(x)$, specified in \ref{['eq:trawlspec']}, and their associated ACFs $\rho(x)=\Gamma(x)/\Gamma(0)$, where $\Gamma(x)=\int_x^{\infty}a(s)ds$. While the piecewise exponential specification exhibits a clear breakpoint, the corresponding ACFs are visually nearly indistinguishable. This illustrates why parametric ACF-based methods can fail to detect model misspecification.
• Figure 4: Trawl function estimation and specification testing for the Poisson trawl process with piecewise-exponential trawl function. Test statistics for specification tests across lags $0:60$. Dashed lines indicate $\alpha=0.05$ critical values; dotted lines show the Bonferroni-corrected thresholds ($\alpha/61\approx0.0008$).
• Figure 5: Using the maximum of the test statistic (comparing with an exponential trawl function) to identify the breakpoint in a hybrid trawl estimation. The hybrid estimator uses nonparametric estimates $\hat{a}(t)$ up to the detected breakpoint $\hat{c}_1$ (identified as the lag maximizing $|T(t)|$), then switches to an exponential tail $\hat{a}(\hat{c}_1) \exp(-\hat{\lambda}(t - \hat{c}_1))$ where $\hat{\lambda}$ is fitted via log-linear regression on the next 30 lags.

Theorems & Definitions (30)

• Definition 1
• Theorem 1: Consistency
• Example 1
• Theorem 2
• Remark 3.1
• Theorem 3
• Remark 3.2
• Proposition 1
• Theorem 4
• Remark 3.3