**2. Methodology**

The price discovery process has been mainly examined using two measures: The Hasbrouck information share (Hasbrouck 1995) and the Harris–McInish–Wood component share (Harris et al. 2002). Both measures are derived from a reduced form vector error correction model that is estimated based on equidistant time intervals. However, Easley and O'Hara (1992) show that liquidity providers also consider the time between market events within the price setting process. Since the timing of transactions and the frequency in which they occur have information value of their own, fixed-interval aggregation schemes lead to a loss of information (cf. Bauwens and Hautsch 2007). Taking into account the irregular occurrence of transactions requires one to consider the data as a point process. The simplest type of point process is the homogeneous Poisson process. Since the homogeneous Poisson process assumes independently distributed events, it is not suited to describe well-known structures such as correlations and clustering of transactions. The ACD model (Engle and Russell 1998), in contrast, accounts for correlation structures in the data and can be used to model the time between transactions. However, in a multivariate framework, the asynchronous arrival of transactions renders the application of the ACD model difficult and dynamic intensity models are preferable. In autoregressive conditional intensity (ACI) models (Russell 1999), the intensity is directly modeled in terms of an autoregressive process. On the other hand, Hawkes processes (Hawkes 1971) describe the intensity in terms of an additive structure and can be regarded as clusters of Poisson processes. According to this view, all events belong to one of two classes—immigrants and descendants. The exogenous immigrants can trigger clusters of descendants, each of whom in turn can trigger own descendants. In this branching process, the so-called branching ratio is defined as the average number of daughter events per mother event. Hautsch (2004), Bowsher (2007), and Large (2007) confirm that Hawkes processes model the dynamics in financial point processes remarkably well. Since the linear structure of the Hawkes model allows one to separate external influences on the process from internal feedback mechanisms, it is well suited to examine price discovery and potential market reflexivity. Technically speaking, Hawkes processes refer to a class of models for stochastic self-exciting and mutually exciting point processes (Hawkes 1971). These can be regarded as non-homogeneous Poisson processes whose intensity depends on both time *t* and the history of the process. The intensity function <sup>λ</sup>*i*(*t*) of a Hawkes process is defined as:

$$\lambda\_i(t) = \mu\_i + \sum\_{j=1}^{D} \sum\_{t\_k^j < t} \phi\_{ij} \binom{t - t\_k^j}{t} \,\forall i, j \in [1 \dots D] \tag{1}$$

where *D* is the number of dimensions in the process. The non-negative parameter μ is the baseline intensity and commonly assumed to be constant. The baseline intensity describes the arrival rate of events triggered by external sources. In our analysis, we use μ to examine how futures contracts of different maturities react to new information. The non-negative kernel function, φ, describes the arrival rate of events that are triggered by previous events within the process. Various kernel functions can be found in the literature. The most widely used are power-law and exponential parameterizations of the kernel function. In our analysis, we follow Bacry et al. (2017) and choose the following exponential parametrization of the kernel functions:

$$\|\phi\|\|\_{ij}(t) = \alpha\_{ij}\beta\_{ij}\exp\left(-\beta\_{ij}t\right)\mathbf{1}\_{t\geq 0} \tag{2}$$

where α and β > 0. In this parametrization, α describes the degree of influence of past points on the intensity process and β determines the time decay of the influence of past points on the intensity process. From the chosen parametrization, it follows

$$\int\_0^\infty \phi(t)dt = \alpha = \|\phi\|\_1\tag{3}$$

where ||φ||1 is a matrix of kernel norms. Each matrix element describes the total impact that events of the type defined by a column of the matrix has on events of the type defined by a row of the matrix. According to the population representation of a Hawkes process (Hawkes and Oakes 1974), the process is considered stable if ||φ||1 < 1. For a stable Hawkes process, a kernel norm ||φ*ii*||1 stands for the average number of events of type *i* that is directly triggered by a past event of the same type *i*. In our analysis, we use ||φ*ii*||1 to measure market reflexivity in futures contracts with different maturities. On the other hand, a kernel norm ||φ*ij*||1 with *i* - *j* stands for the average number of events of type *i* that is directly triggered by an event of a different type *j*. We use ||φ*ij*||1 to measure price discovery between futures contracts with different maturities. Furthermore, following Bacry et al. (2016), the ratio between the baseline intensity μ*i* and the average intensity γ*i* describes an exogeneity ratio, i.e., the ratio between the number of events that is triggered by external sources and the total number of events of type *i*:

$$R\_i = \frac{\mu\_i}{\mathcal{Y}\_i} \tag{4}$$

where the average intensities can be derived by

$$\gamma = \left( I - \|\phi\|\|\_{1} \right)^{-1} \mu \tag{5}$$

with *I* as the identity matrix.

Various methods to estimate Hawkes processes have been proposed in the literature. Estimation procedures include maximum likelihood estimation (Ogata 1998) and the resolution of a Wiener–Hopf system (Bacry et al. 2016). In our analysis, we follow Bacry et al. (2017) and estimate the Hawkes process with least-squares. To assess the goodness-of-fit of the estimated Hawkes model, we carry out a residual analysis according to Ogata (1989). Ogata's residual analysis of point process data is based on the random time change theorem by Meyer (1971). The random time change theorem states that a point process is transformed into a homogeneous Poisson process by its compensator <sup>Λ</sup>(*t*). The compensator is determined by the Doob–Meyer decomposition of a point process and is described by the following monotonically increasing function:

$$
\Lambda(t) = \int\_0^t \lambda(t)dt. \tag{6}
$$

In accordance with Ogata (1989), we use the compensator with the conditional intensity λ ˆ of the estimated Hawkes model to transform the observed data and regard the resulting process as a residual process. In line with the random time change theorem, if the residual process behaves like a homogeneous Poisson process, then the conditional intensity λ ˆ of the estimated Hawkes model is a good approximation to the true intensity λ of the observed point process. To check whether the residual process behaves similar to a homogeneous Poisson process, we apply Kolmogorov–Smirnov and Ljung–Box tests. On the one hand, the Kolmogorov–Smirnov test examines the null hypothesis that the distribution of the residuals is a homogeneous Poisson distribution. On the other hand, the Ljung–Box test examines the null hypothesis that the residuals are independently distributed. If the null hypothesis of both the Kolmogorov–Smirnov test and the Ljung–Box test cannot be rejected at the 5% significance level, we conclude that the estimated Hawkes process is a good approximation of the observed point process.

#### **3. Empirical Application**
