*3.1. Data*

Empirical analyses of market reflexivity can be conducted for different time horizons. Long term analyses are useful if the development of endogeneity and its determinants over time are of interest. Long-term reflexivity is expected to be caused mainly by herding. In this paper, however, we focus on the analysis of short-term reflexivity. Short-term reflexivity is expected to be caused by algorithmic trading (Filimonov et al. 2014) that is supposed to distort the price discovery process in electronic commodity futures markets (cf. Tang and Xiong 2012; Bicchetti and Maystre 2013). For our analysis we utilize the Chicago Mercantile Exchange (CME) Group's best-bid–best-offer (BBO) futures data. We focus on corn, wheat, live cattle and lean hog. These commodities represent the most actively traded storable and non-storable agricultural futures contracts, respectively. The analysis is based on futures contracts of the first two maturities, i.e., the nearest (front contract) and the next to nearest (back contract).<sup>1</sup> We confine our analysis to the daytime trading session because the trading activity is much higher compared to the evening trading session. At the beginning of the daytime trading session, orders that have accumulated in the previous night are processed (cf. Gurgul and Syrek 2017), which can lead to a distorted picture of the price discovery process. Therefore, we exclude the first hour of the daytime trading session. Furthermore, to ensure stationarity of the parameters of the estimated Hawkes process, we follow Filimonov et al. (2014) and base our estimations on intervals of 10 min. Intervals of 10 min are considered to contain a su fficient number of events for reliable estimation. It can be assumed that in short intervals of 10 min, market reflexivity is not caused by behavioral mechanisms, such as herding, but instead is mainly caused by automated trading. The observation period covers all trading days in March 2016, which corresponds to 528 intervals of 10 min. March has been chosen as the observation period, because we want to exclude a rollover of contracts and the e ffect of a new harvest. On the other hand, we are interested in the e ffect of a release of three public reports on the short-term reflexivity: a World Agricultural Supply and Demand Estimates (WASDE) report on world agricultural supply and demand estimations, the United States Department of Agriculture (USDA) report on prospective plantings, and a National Agricultural Statistics Service (NASS) report on grain stocks. BBO data contain top-of-book quotes, transactions, and corresponding time stamps with a resolution of one second. In line with the above-mentioned market microstructure models, we regard the trading activity as the starting point of the price formation process and base our analyses on transactions. However, many transactions share the same time stamp. Therefore, to take all transactions into account, we follow Wang et al. (2016) and simulate sub-second time stamps. In addition, BBO data do not include the direction of transactions. Therefore, to distinguish between buyer- and seller-initiated transactions, we apply the trade classification algorithm proposed by Lee and Ready (1991). To summarize, we analyze a four-dimensional point process whose dimensions are buyer- and seller-initiated transactions in a futures front and back contract for corn, wheat, live cattle and lean hog.

#### *3.2. Descriptive Statistics*

Table 1 contains descriptive statistics for the considered commodities. The total number of buyerand seller-initiated transactions is considerably higher in the front contract than in the back contract for corn and wheat. In contrast, the total number of transactions is similar in the front and back contract for live cattle and lean hog. The number of seller-initiated transactions is higher than the number of buyer-initiated transactions in both the front and back contract for corn. In contrast, the number of seller-initiated transactions is lower than the number of buyer-initiated transactions in both the front

<sup>1</sup> An inclusion of further contracts with longer maturities would be desirable to obtain a clearer picture of price discovery along the entire term structure curve. However, this comes at the cost of increasing the dimension of the Hawkes model. Moreover, the number of transactions becomes rather small for futures contracts with longer maturity, which renders the estimation unreliable.

and back contract for live cattle. The coefficients of variation show that the number of transactions varies more widely over the intervals for storable than for non-storable commodities.



The trading volume is much higher for storable than for non-storable commodities. Moreover, the trading volume is significantly higher in the front contract than in the back contract for corn and for wheat. In contrast, the trading volume is similar in both the front and back contract for live cattle and lean hog. Furthermore, the imbalance between the buy-and-sell volume is larger for corn and wheat than for live cattle and lean hog. A high coefficient of variation indicates that the trading volume varies considerably over the intervals. The coefficient of variation is highest for corn.

To avoid bid–ask bounce effects, our descriptive statistics are based on mid-quotes rather than transaction prices. The mid-quotes are higher in the back contract than in the front contract for corn, wheat, and lean hog. This points to a normal forward curve during the observation period. In contrast, mid-quotes are higher in the front contract than in the back contract for live cattle. This points to an inverted forward curve during the observation period. The standard deviation of the mid-quotes is higher for storable than for non-storable commodities. However, low coefficients of variation for all commodities indicate that prices exhibit little volatility during the observation period.

## *3.3. Results*

Table 2 shows the estimated baseline intensities μ, the derived average intensities γ, and exogeneity ratios *R*. All figures for μ, γ, and *R* are mean values over 528 intervals of 10 min. As a robustness check, we conducted additional calculations with a di fferent time interval length (30 min) for corn and lean hog (see Tables A1 and A2, Figure A1 in the Appendix A). The qualitative findings are similar and, thus, we focus on the discussion of the results of the 10-min intervals in what follows.


**Table 2.** Estimated baseline intensities, average intensities, and exogeneity ratios.

The baseline intensity μ is by far the highest for the front contract of corn. This means that in absolute terms, the number of transactions that is triggered by external sources is highest for the front contract of corn. In addition, μ is of similar size for buyer- and seller-initiated transactions in a contract of a certain maturity for all commodities. It is striking that μ is noticeably larger in the front contract than in the back contract for corn and wheat. In contrast, μ is of similar size in the front contract and back contract for live cattle and lean hog. This suggests that price discovery takes place in the front contract for storable commodities, but not for non-storable commodities. These estimation results for the baseline intensity μ are therefore in line with Working (1949) and Gray and Rutledge (1971). Compared to the baseline intensity, the derived average intensity γ is higher for all examined contracts. Similar to the baseline intensity, γ is considerably larger in the front contract than in the back contract for corn and wheat. In contrast, γ is of similar size in the front and back contract for live cattle and lean hog. The derived exogeneity ratio *R* is higher for non-storable than for storable commodities. About 75% (65%) of all observed transactions in the lean hog (corn) futures are triggered by external sources. The results also find that *R* is of similar size for buyer- and seller-initiated transactions in a commodity futures contract of a certain maturity. Moreover, *R* is slightly higher in the back contract than in the front contract for all commodities. Apparently, the derived exogeneity ratio is considerably higher for the examined agricultural futures contracts than for the DAX and BUND futures contracts examined by Bacry et al. (2016). Moreover, the derived exogeneity ratios are higher than the corresponding figures in Filimonov et al. (2014).

Figure 1 presents the estimated matrices of kernel norms ||φ||1. Each kernel norm describes the total impact that transactions of a certain contract type have on transactions of the same or on other contracts. Figure 1 has to be interpreted column-wise. A darker background color of a kernel norm

illustrates a larger impact than a lighter color. A kernel norm ||φ*ii*||<sup>1</sup> on the diagonals of the matrices describes the impact that transactions of a certain type have on transactions of the same type. A kernel norm ||φ*ij*||1 on the o ff-diagonals of the matrices describes the impact that transactions of a certain type have on transactions of a di fferent type. In general, the kernel norm values are higher for storable than for non-storable commodities. According to the population representation of the Hawkes process, each buyer-initiated (seller-initiated) transaction directly triggers 0.3102 buyer-initiated transactions (0.3147 seller-initiated) on average in the front contract for corn. The respective values for other commodities are slightly smaller. Apparently, the kernel norm values on the diagonals of the matrix are considerably higher compared to those on the o ff-diagonals for all commodities. This indicates that a substantial part of the observed transactions is endogenously driven and points to the presence of market reflexivity for all commodities. In addition, the kernel norm values on the diagonals of the matrix are higher in the front contract than in the back contract for corn and wheat. In contrast, the kernel norm values on the diagonals are of similar size in the front and back contract for live cattle and lean hog. As expected, the estimation results for the matrices of kernel norms ||φ||1 show that di fferences between front and back contracts are less pronounced for non-storable than for storable commodities.


The kernel norms on the o ff-diagonals show that in the cases of corn and wheat, the impact of front contracts on back contracts is larger than vice versa. This relationship, however, is not prevalent for live cattle and lean hog. Thus, our results are in line with Hu et al. (2017), who show that nearby futures contracts dominate deferred futures contracts in price discovery for storable commodities, but not for non-storable commodities. No di fferences can be identified between buyerand seller-initiated transactions.

Table 3 contains the results of the residual analyses of the estimated Hawkes processes. For more than about 80% of the 10-min intervals, the Kolmogorov–Smirnov test does not reject the null hypothesis that the distribution of the residuals is homogeneous Poisson at the 5% significance level for the considered commodities. Except for the front contracts of corn and wheat, for more than approximately 90% of the 10-min intervals, the Ljung–Box test does not reject the null hypothesis that the residuals are independently distributed at the 5% significance level. We can therefore conclude that the estimated Hawkes process with the chosen parameterization of the kernel function is a good approximation of the observed point process.


**Table 3.** Residual analyses.

Figure 2 displays the evolution of the average intensity γ over the whole observation period. The figure is based on daily mean values of the average intensity. In general, the evolution of γ for the front contract is similar to the evolution of γ for the back contract for all commodities. In particular, γ is higher for the front contract than for the back contract throughout the whole observation period for corn and wheat. In contrast, for live cattle and lean hog, γ is higher for the front contract than for the back contract in the first half of the observation period only. In addition, the evolution of the average intensity of buyer-initiated transactions is similar to the evolution of the average intensity of seller-initiated transactions in a contract of a certain maturity for all commodities.

In general, the average intensity shows a relatively constant progression for all commodities; however, γ increases around certain days where public information became available. For instance, the average intensity increases for all commodities on 9 March 2016 when the USDA released its monthly WASDE report (USDA 2016b). Furthermore, γ increases for live cattle on 22 March 2016 following a USDA cattle on feed report (USDA 2016c). Most notably, γ increases considerably towards the end of the observation period for corn and wheat on 31 March 2016. On this day (11:00 a.m.), the USDA reported an unexpected increase in the corn acreage (USDA 2016a). On the same day, the USDA published its quarterly grain stocks reports (USDA 2016d). This may explain why the increase in average intensities is rather pronounced at that time. Overall, Figure 2 documents that the publication of USDA reports a ffects not only volatility and trading volume, but also the intensity of trading activity.

**Figure 2.** Evolution of average intensities (*y*-axis) in both buy and sell transactions for corn, wheat, live cattle, and lean hog in March 2016.

Figure 3 depicts the evolution of the baseline intensity μ on 31 March 2016 for corn. The figure is based on 24 intervals of 10 min. The baseline intensity μ increases considerably at the time of the publication of the USDA report for transactions in both the front and back contract. Moreover, the evolution of μ for seller-initiated transactions is similar to the evolution of μ for buyer-initiated transactions at the time of the publication of the USDA report for the front contract. In contrast, for the back contract, the evolution of μ for seller-initiated transactions differs from the evolution of μ for buyer-initiated transactions at the time of the publication of the USDA report.

**Figure 3.** Evolution of the baseline intensity (*y*-axis) for corn on 31 March 2016.

The evolution of the baseline intensity for corn on 31 March 2016 suggests that new information is quickly incorporated into market prices through trading activity. Moreover, it is becoming clearer that new information does not only affect nearby futures contracts, but also affects deferred futures contracts.

Figure 4 presents the estimated matrix of kernel norms ||φ||1 for corn on 31 March 2016. The figures are mean values over 24 intervals of 10 min.


**Figure 4.** Estimated matrix of kernel norms for corn on 31 March 2016.

Compared to the whole observation period, the kernel norm values ||φ*ii*||1 on the diagonal of the matrix are considerably higher for corn on 31 March 2016. This indicates an increase in market reflexivity on the publication day of the USDA report. Moreover, compared to the whole observation period, the kernel norm values ||φ*ij*||1 on the off-diagonal of the matrix are higher for the back contract for corn on 31 March 2016. This indicates an increase in the impact that transactions in the back contract have on transactions in both the front and back contract on the publication day of the USDA report.

To gain an overall picture of the price formation process in agricultural futures contracts with different maturities, Figure 5 illustrates the composition of the estimated intensity of the trading activity in both the front and back contract exemplarily for corn and live cattle. The figure is based on mean values of 528 intervals of 10 min. It shows the relative contribution of the estimated baseline intensity μ*i* and the kernel norms ||φ*ii*||1 and ||φ*ij*||1 to the total intensity λ ˆ *i*.

**Figure 5.** Composition of the estimated intensity for corn and live cattle.

The relative contribution of the baseline intensity μ*i* to the total intensity λ ˆ *i* is largest for the front contract of corn. It amounts to approximately 30%. This means that external sources, such as new information, account for approximately 30% of the total intensity of both buyer-initiated and seller-initiated transactions for the front contract of corn. Moreover, the relative contribution of μ*i* is noticeably larger in the front contract than in the back contract for corn. In contrast, the relative contribution of μ*i* is of similar size in the front and back contract for live cattle. Therefore, in line with Working Working (1949) and Gray and Rutledge (1971), the composition of the estimated intensity suggests that nearby futures contracts dominate deferred futures contracts in price discovery for storable commodities, but not for non-storable commodities.

Compared to the relative contribution of μ*i*, the relative contribution of the kernel norms ||φ*ii*||1 to the total intensity is considerably larger. More specifically, the relative contribution of ||φ*ii*||1 to the total intensity is largest for the back contract of corn. It amounts to roughly 70%, which means that market reflexivity accounts for about 70% of the total intensity of both buyer-initiated and seller-initiated transactions for the back contract of corn. Moreover, the relative contribution of ||φ*ii*||1 is clearly larger in the back contract than in the front contract for corn. In contrast, the relative contribution of ||φ*ii*||1 is of similar size in the front and back contract for live cattle.

As expected, the composition of the estimated intensity shows that differences between front and back contracts in the degree of market reflexivity are less pronounced for live cattle than for corn. Compared to the relative contribution of ||φ*ii*||1 , the relative contribution of the kernel norms ||φ*ij*||1 to the total intensity is considerably smaller. More specifically, the relative contribution of ||φ*ij*||1 to the total intensity is higher for live cattle than for corn. It is shown that buyer-initiated transactions in the back contract account for almost 20% of the total intensity of buyer-initiated transactions in the front contract for live cattle. On the other hand, buyer-initiated transactions in the back contract account for approximately 10% of the total intensity of buyer-initiated transactions in the front contract for corn. Therefore, in line with Kendall (1982), the composition of the estimated intensity suggests that interdependencies of current and subsequent supply might link futures contracts of different maturities to non-storable commodities.

## **4. Conclusions**

This paper is the first to examine price discovery in nearby and deferred agricultural commodities futures contracts while explicitly considering potential market reflexivity. For this purpose, we apply a four-dimensional Hawkes model with an exponential parametrization of the kernel functions to storable and non-storable agricultural commodities. We find market reflexivity for both storable and non-storable commodities. Reflexivity accounts for about 50 to 70% of the total trading activity, irrespective of whether transactions have been buyer- or seller-initiated. Storable commodities show a higher level of reflexivity than non-storable commodities and differences between nearby and deferred contracts are less pronounced for non-storable commodities than for storable ones. We conjecture that the degree of reflexivity is related to the trading intensity and the amount of automated trading. Endogeneity results from backward-looking trading strategies, which exploit information from previous trading activities. Most likely, this source of endogeneity is more prevalent in liquid markets than in thin markets. Storable commodities are more frequently traded compared with non-storable commodities, but this holds only for the front contract (c.f. Table 1). Moreover, grains show a higher share of automated trading than livestock contracts (e.g., Haynes and Roberts 2015; Couleau et al. 2019). It appears that these differences are reflected in the endogeneity of the price formation process.

What are the implications of these findings for the price discovery process in agricultural commodity futures? The presence of market reflexivity seems to contradict the efficient market hypothesis, and Filimonov et al. (2014) argue that reflexivity is likely to result in a less efficient price discovery process. Though our analysis does not allow an evaluation of the presence of reflexivity, we can at least compare our results with previous empirical findings on price discovery in agricultural commodity futures markets and screen them for irregularities. Regarding the relation between nearby and deferred futures contracts, our results are in line with Mallory et al. (2015), who find that new information to the market is immediately transmitted across all contract maturities. Moreover, we can replicate the finding of Hu et al. (2017) that nearby contracts are dominant in price discovery for storable commodities, but not for non-storable commodities. Finally, we observe an increased baseline intensity at the time of a USDA publication, suggesting that external information is quickly incorporated into market prices through trading activity despite the presence of market reflexivity. From these findings, we conclude that the co-existence of exogenous and endogenous price dynamics does not change qualitative characteristics of the price discovery process that have been observed earlier without the consideration of market reflexivity.

Some aspects could be incorporated in further empirical analyses to obtain a clearer picture about price discovery in agricultural commodity futures in the presence of market reflexivity. The empirical basis could be broadened by including additional commodities. Moreover, price discovery and market reflexivity should be examined over a longer observation period covering, for example, markets in normal backwardation and in contango. From a methodological point of view, the Hawkes model

can be extended and amended in several ways. Covariates such as order imbalances between buyerand seller-initiated transactions can be included into the intensity function. Other parameterizations of the kernel function such as power-law parameterizations or other estimation methods such as non-parametric estimation techniques can be applied to check the robustness of the results. Finally, methods allowing the assessment of the direct impact of market reflexivity on the price formation process need to be explored.

**Author Contributions:** Conceptualization, S.V., G.F. and M.O.; data curation, S.V. and G.F.; formal analysis, S.V.; methodology, S.V. and G.F.; software, S.V.; supervision, M.O.; validation, S.V. and G.F.; visualization, S.V. and G.F.; writing—original draft, S.V., G.F. and M.O.; writing—review and editing, S.V., G.F. and M.O. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We thank three anonymous reviewers for helpful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
