Estimation of conditional mean marks - Financial Models of Interaction Based on Marked Point Pr

= Z

R^d

1B1(t1)f(y(t1, ϕg))ϕg(B2)P_t^!₁(dϕg)Λg(dt1),

whereP_t^! is the Palm distribution of Φ_g. In this case, the calculation of α_f⁽²⁾ is reduced to the theory of unmarked point processes.

3. If, additionally, Φ_g is a Cox process with random intensity measureΞ(·) =^R_·L(t)dt, then

α⁽²⁾_f (B₁×B₂) = Z Z

R^d

1_B₁(t₁)f(y(t₁, ϕ_g))ϕ_g(B₂)P^Φ^g^|^L=λ(dϕ_g)λ(t₁)dt₁P^L(dλ).

Notation: The value of µ⁽²⁾_f (r), r ∈ R^d, represents the mean value of f(y1, y2) over all pairs of points (t₁, y₁),(t₂, y₂) witht₂−t₁ =r. This motivates using the notation

µ⁽²⁾_f (r) =Ef(y(t), y(t+r))t, t+r∈Φg .

In the following, we consider an MPP Φ on the real axis with real-valued marks. As regards the function f, the two functions e(y1, y2) =y1 and v(y1, y2) =y²₁ are employed.

Definition 2.2.6 (Bi-directional second-order statistics). LetΦbe a stationary MPP on R with real-valued marks andW ∈ B such that (2.6) is satisfied forf =eandf =v. Using the above notation, we define

E(r) =µ⁽²⁾_e (r), r ∈R, the bi-directional E-function, and V(r) =µ⁽²⁾_v (r)−(µ⁽²⁾_e (r))², r∈R, the bi-directional V-function.

Note that, in general, E(r) and V(r) are not continuous atr = 0, where the two-point statistics E and V pass into one-point statistics. Similarly toSchlather (2001), E(r) and V(r) can be interpreted as the conditional mean and variance of the mark of a point at an arbitrary timet, given the existence of another pointr units of time away,r∈R. Thus, a negative value of r refers to the existence of another point in the past. E(0) andV(0) are simply the unconditional mean and variance of a mark, respectively.

2.3 Estimation of conditional mean marks

We assume that the Lebesgue density ρ⁽²⁾_f (r) of α⁽²⁾_f (C(·)) exists for r6= 0. Then, it is common to apply a ratio estimator for µ⁽²⁾_f (r) of the form ˆµ⁽²⁾_f (r) = ˆρ⁽²⁾_f (r)/ˆρ⁽²⁾₁ (r) (Stoyan

& Stoyan,2000), where

ρˆ⁽²⁾_f (r) = 1 ν(L)

(t1, y1),(t2, y2)∈Φ

f(y1, y2)1L(t1)K_h((t2−t1)−r)

forr6= 0, a kernel function K_h with bandwidth h, and some observation window Lof the point process. Note that we abstain from an edge correction (e.g.,Stoyanet al., 1995) as the considered values ofr are negligibly small, compared to the size of L. It can be shown that this estimator is asymptotically unbiased, that is, if the bandwidthh converges to zero (cf.

Stoyan & Stoyan,2000). Therein, it is also suggested to use the rectangular kernel instead of the Epanechnikov kernel because of a smaller estimation variance. Furthermore, we replace the bandwidthh by an adaptive bandwidth h_r = min{h,|r|}for the following reason: By using the bi-directional statisticsµ⁽²⁾_f , we explicitly want to take account of the impact of the sign of the distance. So, it would not be reasonable to use a tuple of points with a negative distance for estimation of ˆρ⁽²⁾_f (r) for a positive r, and vice versa. Forr= 0, µ⁽²⁾_f becomes a one-point statistic and we apply ˆρ⁽²⁾_f (0) =^P_(t

1,y1)∈Φf(y1, y1)1L(t1)/ν(L). We end up with the following estimator ofµ⁽²⁾_f :

µˆ⁽²⁾_f (r) = ρˆ⁽²⁾_f (r) ρˆ⁽²⁾₁ (r)









 X

(t1, y1)∈Φ

f(y1, y1)1L(t1) X

(t1, y1)∈Φ

1_L(t₁) , r= 0

(t1, y1),(t2, y2)

∈Φ∩(L×R)

f(y₁, y₂)1_(r−h_r_{, r+h}_r₎(t₂−t₁) X

(t1, y1),(t2, y2)

∈Φ∩(L×R)

1(r−h_r, r+hr)(t2−t1) , r6= 0.

3 Refined analysis of interactions within

high-frequency transaction data through marked point process theory

This chapter is based on the manuscriptMalinowski & Schlather (2011b).

3.1 Introduction

In the classical context of low-frequency data, asset prices are usually modeled as geometric Brownian motions or, more generally, as (jump) diffusion processes, that is, solutions to stochastic differential equations, with a possibly time-varying and random underlying volatility. Barndorff-Nielsen & Shephard (2001), for example, propose a sophisticated stochastic volatility model whose underlying volatility is given by a Lévy driven Ornstein-Uhlenbeck process. While inhomogeneity of volatility in those models is often seen to be caused by the flow of new information, the focus of this chapter is on volatility effects on a high-frequency scale caused by temporal proximity of past or future trades.

Excited by the seminal work ofEngle & Russell (1998) on modeling financial data at its highest level of disaggregation as marked point processes (MPPs), a plenty of MPP models for high-frequency data have been developed in recent econometric literature (seeBauwens

& Hautsch(2009) for a broad survey). As financial transactions occur irregularly spaced in time, a standard procedure in this setting is to consider time stamps as the points of a point process, marked by the according (log) prices or other characteristics. MPPs turned out to be a well-suited tool for modeling temporal (and spatial) dependencies between marks as well as interactions between marks and locations of the points.

Two main classes of MPP models for transaction data are commonly used: The class of dynamic duration and autoregressive conditional duration (ACD) models (e.g., Ghysels &

Jasiak,1998;Engle,2000; Bauwens & Giot, 2001;Hautsch, 2004), see Engle & Russell(2009) orPacurar (2008) for a survey, and the broad class of models based on doubly stochastic Poisson processes (DSPP) or Cox processes, where the underlying intensity can also be specified dynamically (e.g.,Hawkes,1971; Russell, 1999;Hautsch,2004; Bauwens & Hautsch, 2006;Rydberg & Shephard,2000;Frey,2000;Centanni & Minozzo,2006).

The main contribution of this chapter is the analysis of new statistics for temporal MPPs, namely thebi-directional E- and V-function. These functions can be interpreted as conditional expectations and variances, respectively. While the definition here is tailored to the temporal context, a slightly different definition is already known from MPPs in the field of spatial statistics (Schlather,2001). The new statistics will be used to detect and to model interaction phenomena within high-frequency financial data and, based on this method, we

will discuss an extension of the commonly used ultra-high frequency GARCH model ofEngle (2000) that includes certain effects of interaction. When applied to the UHF-GARCH model, our statistics serve as useful additional model fitting criteria, besides classical model fitting criteria.

Note that it is a well-known fact that the underlying (time-varying) variance of a price process generally increases in times of high trading intensity (e.g.,Easley & O’Hara,1992).

Though, the bi-directional V-function will be able to distinguish between the influence of past and future transactions and therefore allows for a more profound insight into the latent volatility process and into interaction effects.

Our analysis is based on high-frequency transaction data. Due to increased automatization in financial markets and the fast development in computing power and storage capacity, (financial) databases today provide high-frequency data for a wide range of markets. Si-multaneously, many econometric tools like ARCH, GARCH and related models have been developed enabling an analysis of the market’s behavior at the fine scale of transaction data (e.g., Goodhart & O’Hara (1997), Ghysels & Jasiak (1998), Engle & Russell (1998, 2009), Engle(2000),Zhang et al.(2001),Racicot et al.(2008), and the references therein).

Other approaches for modeling transaction price processes include, for example, the probit regression model ofHausmanet al.(1992) or the approach of decomposition of price changes inRydberg & Shephard(2003).

Compared to classical time-series analysis, high-frequency data pose some specific chal-lenges. The most important feature is that financial transactions are not equally spaced so that the standard theory of time series, which is based on fixed time interval analysis, cannot be applied. One approach is aggregating returns to equally-spaced intervals but such aggregation will either lose information (if the new intervals are large) or create noise due to interpolation issues, or both. Aït-Sahalia et al. (2005), for example, show that, when microstructure noise is included into the model, it is reasonable to keep transaction data at their original ultimate frequency level. To avoid the disadvantages of temporal aggregation, various point process methods have been developed that are tailored to the irregular spacing of transaction level data. SeeBauwens & Hautsch (2009) for a survey.

Further, inter-trade durations are usually clustered. That means, the autocorrelation function of the durations is significantly positive with a slow decay, which can be associated with long-memory properties of the process. Besides clustering, important properties are the discreteness of the price process and diurnal or periodical patterns, e.g., volatility of prices, traded volume and frequency of transactions exhibit a U-shaped pattern over the course of the day.

All those features can have substantial implications on temporal dependencies and on measuring volatility or other characteristics on a small scale. Especially, sequent inter-transaction returns are not free of correlation as it is often assumed for low-frequency data.

The remainder of this chapter is organized as follows: Section 3.2.1 starts with a detailed example that shows the usefulness and importance of the bi-directional E-function and its relatives. Section 3.2.2 provides an intuitive definition of the second-order statistics E and V as well as appropriate estimators. In Section 3.3, we briefly review the ACD

Im Dokument Financial Models of Interaction Based on Marked Point Processes and Gaussian Fields (Seite 21-25)