**1. Introduction**

Prices of commodity futures contracts with different maturities are linked through the forward curve. Understanding of the shape and the characteristics of the term structure is of utmost importance for storage decisions, hedging and roll-over strategies as well as calendar spread trading. Several strands of literature address this topic. The theoretical underpinning of the forward curve goes back to Working's (1949) theory of storage that establishes an equilibrium relation between nearby and distant futures contracts and explains storage under backwardation by the concept of convenience yield (Brennan 1958). In contrast, Keynes' theory of normal backwardation decomposes a futures price into an expected future spot price and an expected risk premium that risk-averse hedgers gran<sup>t</sup> to speculators (Fama and French 1987). The statistical modelling of the forward curve has benefitted from Nelson and Siegel's (1987) proposal to describe the term structure parsimoniously in terms of level, slope and curvature. A dynamic version of this model has been introduced by Diebold and Li (2006). Applications to commodity futures can be found in Karstanje et al. (2017). An alternative method uses a set of state variables (factors), particularly spot price, convenience yield, and interest rate, to derive the forward curve under no-arbitrage conditions (Gibson and Schwartz 1990; Schwartz 1997). Applications of this approach to agricultural futures include Geman and Nguyen (2005) and Sorensen (2002), among others.

With the rise of the modern market microstructure, interest has shifted from the estimation of equilibrium relations towards the understanding of price discovery, i.e., the question of how new information is absorbed in asset prices and how this information is transferred along the forward curve. Since there is no explicit market microstructure theory designed for commodity futures with di fferent maturities, most studies in this area are non-structural and try to identify empirical patterns in

data. Mallory et al. (2015) use contemporaneous and time-lagged correlations of nearby and deferred futures contracts for corn to investigate the speed at which liquidity providers revise their beliefs in response to the occurrence of an information event. They find that the correlation of price revisions disappears even for short time lags and conclude that new information to the market is immediately transmitted across all contract maturities. Hu et al. (2017) pursue a similar objective, but instead of simple correlations, they apply co-integration techniques to explore price discovery among nearby and deferred futures contracts of corn and live cattle. They report a larger share of price discovery in nearer to maturity contracts. The dominance of nearer contracts, however, is less pronounced for live cattle than for corn, which is explained by di fferences in the storability of these commodities. Recently, Volkenand et al. (2019) investigate the duration dependence among agricultural futures with di fferent maturities, exploiting the fact that the time between market events (transactions or price changes) carries information (Easley and O'Hara 1992). They apply an autoregressive conditional duration (ACD) model to price durations for corn, wheat, live cattle, and lean hog. The authors report linkages between nearby and deferred futures contracts. They conclude that information is quickly processed along the forward curve.

The aforementioned studies rest on a traditional view of the price discovery mechanism according to which price revisions are driven by the arrival of exogenous information. This view has been challenged by the concept of market reflexivity (Soros 1987) which assumes that trading activity is also endogenously driven by positive feedback mechanisms. Sources of potential endogeneity encompass informational cascades leading to herding, as well as speculation based on technical analysis (e.g., momentum trading) and algorithmic trading (Filimonov et al. 2014). Furthermore, hedging strategies combined with portfolio execution rules can lead to self-excitement of price moves (Kyle and Obizhaev 2019). While this co-existence of exogenous and endogenous price dynamics contradicts the e fficient market hypothesis (Fama 1970), it can be helpful to understand puzzling phenomena on financial markets, such as "flash crashes" or excess volatility (Hardiman et al. 2013). The concept of market reflexivity has originally been introduced in a narrative, non-technical manner, but since then it experienced an underpinning by statistical methods that allow one to disentangle exogenous and endogenous sources of market activities and thus measure the degree of market reflexivity. More specifically, self-exiting Hawkes processes have been proposed as a device to quantify reflexivity (e.g., Filimonov and Sornette 2012). Bacry et al. (2016), for example, find that less than 5% of the price changes in the DAX (German stock index) and BUND (German Bond) futures markets are driven by external sources. In the context of commodity futures markets, Filimonov et al. (2014) find that reflexivity has increased since the mid-2000s to 70%. They trace this back to the increase in automated trading in the course of the transition to an electronic trading environment. In fact, automated trading generated about 40% of the total futures volume traded in the grain and oilseed markets between 2012 and 2014 (Haynes and Roberts 2015).

Despite the increasing interest in market reflexivity as an alternative to the prevalent tenet of market e fficiency and rational expectations, there exists no empirical study applying this concept to the forward curve of commodities. Against this backdrop, our objective is to examine price discovery in nearby and deferred agricultural futures contracts while explicitly taking into account potential market reflexivity. We apply a four-dimensional Hawkes model to storable and non-storable agricultural commodities. The Hawkes model allows us to divide the intensity of the trading activity in a futures contract with a certain maturity into three parts: a reaction on external sources like new information, market reflexivity, and reactions on trading activity in contracts of di fferent maturities. Using this approach, we review previous findings regarding price discovery in nearby and deferred futures contracts. In particular, we examine whether nearby contracts dominate deferred contracts in price discovery while accounting for potential market reflexivity (e.g., Gray and Rutledge 1971). We also explore whether price discovery and potential market reflexivity di ffer between storable and non-storable commodities. In line with Hu et al. (2017), we expect that dominance of nearby contracts in price discovery is more pronounced for storable commodities. Moreover, we conjecture

that commodities with a high share of automated trading, such as grains and oilseeds, show a higher level of endogeneity. Since market microstructure theory emphasizes the importance of the direction of the transactions in the price discovery process (cf. Glosten and Milgrom 1985), we differentiate between buyer- and seller-initiated transactions in our analyses.

The remainder of the paper is organized as follows: Section 2 explains the statistical methods used in the analyses. Section 3 presents the dataset and the results of the empirical application and Section 4 concludes.
