2.1. Price discovery

Price discovery is about determining which market is more informative. Ward and Schroeder (Reference Ward and Schroeder2002) define price discovery as “…is the process of buyers and sellers arriving at a transaction price for a given quality and quantity of a product at a given time and place” (p. 1) when an asset or similar commodities are traded in different markets such as cattle. A number of different methods have been used to study price discovery. One of the methods is to use bivariate time series analysis with an error correction term and compare speed of adjustment between the two series (Gonzalo and Granger, Reference Gonzalo and Granger1995). The underlying idea behind price discovery can be most easily explained through a graphical example.

Let ${p_1}$ represent the cattle price in region 1 (or market 1) and let ${p_2}$ represent the price of a similar type of cattle in region 2 (or market 2). Suppose that ${p_1}$ is higher than ${p_2}$ as shown in Figure 1 and they move together, that is, cointegrated. Suppose that, for some reasons, a shock occurs in market 1, causing ${p_1}$ to rapidly increase (first panel, Figure 1).Footnote ¹ A number of things can occur after this change. Let us assume that market 1 is the price leader. Although we will likely see a slight downward correction in ${p_1}$ in market 1, we will see a much larger upward change in ${p_2}$ . These changes are demonstrated in the second panel in Figure 1. This occurs because buyers and sellers in market 2 look to market 1 to ascertain the correct price in market 2. In other words, market 1 is more informative; market 2 (follower) adjusts its price more to close the gap when the price gap widens between markets 1 and 2. Let us now assume market 2 is the price leader. In this case, traders in market 2 are much less concerned with market 1. Although there may be a slight increase in ${p_2}$ , market 2 is not heavily influenced by the price change in market 1. Traders in market 1, however, are still very concerned with ${p_2}$ . Although the shock in market 1 causes a temporary increase in ${p_1}$ , the price in market 2 indicates to traders in market 1 that ${p_1}$ is too high. This information is incorporated into market negotiations, and, over time, ${p_1}$ returns to its proper level, slightly higher than ${p_2}$ . This change is shown in the third panel in Figure 1.

Figure 1. Price discovery illustration.

2.2. Vector error correction model and its limitation

Since price discovery is about an identical/similar asset or commodity traded in different markets, a cointegration framework is adopted. The conventional price discovery measure used in the literature has simply compared the speed of adjustment coefficients in the bivariate vector error correction model (VECM) as a share of the total adjustment, as developed in Gonzalo and Granger (Reference Gonzalo and Granger1995) and expanded in Theissen (Reference Theissen2002). Other measures of price discovery used in the literature from a multiple market perspective are information share (IS) of Hasbrouck (Reference Hasbrouck1995) and component share (CS) of Booth et al. (Reference Booth, So and Tse1999) and Chu et al. (Reference Chu, Hsieh and Tse1999). These are discussed at length in the Journal of Financial Market’s special issue on price discovery measurement in 2002 (Baillie et al., Reference Baillie, Booth, Tse and Zabotina2002; De Jong, Reference De Jong2002; Harris et al., Reference Harris, McInish and Wood2002; Hasbrouck, Reference Hasbrouck2002; Lehmann, Reference Lehmann2002). The IS measures each market’s relative contribution to the variance of the efficient price, while the CS decomposes the common efficient price into a weighted average of observed market prices and measures each market’s contribution to the common efficient price. Both IS and CS are based on the reduced-form forecasting errors of a VECM. Yan and Zivot (Reference Yan and Zivot2010) proposed the joint use of the IS and the CS, and De Jong and Schotman (Reference De Jong and Schotman2010) developed the unobserved component approach.

The long-run equilibrium between two prices can be written as:

(1) $${p_{1,t}} = {\beta _2}{p_{2,t}} + c + {u_t}{\rm{\;\;}} \Leftrightarrow {\rm{\;\;}}{u_t} = {p_{1,t}} - {\beta _2}{p_{2,t}} - c$$

where ${p_{1,t}}$ and ${p_{2,t}}$ represent prices in markets 1 and 2 at time $t$ , respectively. The term $c$ (constant term) accounts for average differences in these two markets. The term ${u_t}$ is the (long-run) error, which is zero in equilibrium.

The bivariate VECM contains this long-run equilibrium in equation (1) as:

(2) $$\eqalign{ \Delta {p_{1,t}} = {\alpha _1}\left( {{p_{1,t - 1}} - {\beta _2}{p_{2,t - 1}} - c} \right) + \mathop \sum \limits_{k = 1}^{K - 1} {\gamma _{11,k}}\Delta {p_{1,t - k}} + \mathop \sum \limits_{k = 1}^{K - 1} {\gamma _{12,k}}\Delta {p_{2,t - k}} + {e_{1,t}} \cr \Delta {p_{2,t}} = {\alpha _2}\left( {{p_{1,t - 1}} - {\beta _2}{p_{2,t - 1}} - c} \right) + \sum\limits_{k = 1}^{K - 1} {{\gamma _{21,k}}} \Delta {p_{1,t - k}} + \sum\limits_{k = 1}^{K - 1} {{\gamma _{22,k}}} \Delta {p_{2,t - k}} + {e_{2,t}} \cr} $$

where $\Delta{p_{i,t}}$ represents the differenced prices. Coefficients ${\alpha _1}$ and ${\alpha _2}$ are the adjustment rate parameters, and they represent the speed with which a displaced ${p_1}$ (or ${p_2}$ ) returns to its long-run equilibrium. Using vectors and matrices, equation (2) is simplified as:

(3) $$\Delta {{\bf{p}}_t} = {\bf{\alpha }}\left( {{\bf{\beta^\prime}}{{\bf{p}}_{t - 1}} - {\bf{c}}} \right) + \mathop \sum \limits_{k = 1}^{K - 1} {{\bf{\Gamma }}_k}\Delta {{\bf{p}}_{t - k}} + {{\bf{e}}_t} = {\bf{\Pi }}\left( {{{\bf{p}}_{t - 1}} + {\bf{\mu }}} \right) + \mathop \sum \limits_{k = 1}^{K - 1} {{\bf{\Gamma }}_k}\Delta {{\bf{p}}_{t - k}} + {{\bf{e}}_t}$$

where ${\bf{\alpha '}} = \left[ {{\alpha _1},{\alpha _2}} \right]$ is 2×1 vector of adjustment coefficients (speed of adjustment), ${\bf{\beta '}} = \left[ {1,{\beta _2}} \right]$ is 2×1 cointegrating vector, ${\bf{c}}$ is an intercept, ${\bf{\Pi }} = {\bf{\alpha \beta '}}$ , and ${\bf{\mu }} = - {{\bf{\Pi }}^{ - 1}}{\bf{c}}$ . $\;\;K$ is the lag-length of the VAR and ${{\bf{e}}_t}$ is the reduced-form shock.

Following Gonzalo and Granger (Reference Gonzalo and Granger1995), the relative ratio of the adjustment coefficients is defined by

(4) $${\theta _1} = {{\left| {{\alpha _2}} \right|} \over {\left| {{\alpha _1}} \right| + \left| {{\alpha _2}} \right|}},{\rm{}}{\theta _2} = {{\left| {{\alpha _1}} \right|} \over {\left| {{\alpha _1}} \right| + \left| {{\alpha _2}} \right|}},\;{\rm{and}}\;{\theta _1} + {\theta _2} = 1$$

A high $\theta \;$ indicates a low $\alpha $ (speed of adjustment toward a new long-run equilibrium), which implies that this market slowly responds to an unpredicted shock in the system; therefore, the market is the price discovery reference market. If ${\theta _1} = {\theta _2}=0.5$ , both markets contribute equally to the price discovery process, that is, both markets move at a similar rate toward the long-run equilibrium. Note that ${\theta _i}$ is applicable only for the bivariate case with ${{\bf{\alpha }}_{2 \times 1}}$ . For the multivariate case with one cointegrating vector, we may arrange price discovery according to relative magnitude of ${\alpha _i}$ , but it is not clear how to calculate ${\theta _i}$ . Moreover, if there are multiple cointegrating vectors, comparing price discovery using ${\alpha _i}$ is not applicable.

However, an Achilles heel to price discovery using the speed of adjustment coefficients with the VECM outlined in equations (2) and (4) is its inability to compare more than two variables at a time. This is because there must be one cointegrating relationship between the relevant variables. For the multivariate case with one cointegrating relation, we may arrange price discovery according to relative magnitude of ${\alpha _i}$ but it is not clear how to calculate ${\theta _i}$ . Another shortcoming is that some markets may result in possibly having a negative weight generated from potential collinearity. In addition, in the multivariate case there might be more than one cointegrating relationship. For n time series, there might be as many as n−1 cointegrating relationships. Since there is no way of choosing in this latter case which of these possible cointegrating relationships to use in the model (Kim, Reference Kim2011), one cannot use this approach to compare more than two series at a time. This is a limitation when examining markets of a commodity, which almost always involves more than two markets, such as cattle markets.

2.3. Cluster analysis and tournament

As noted, when there are multiple markets, this technique, equation (4), may not be applicable. The tournament approach using cluster analysis provides a way to overcome this problem as demonstrated in Kim, Tejeda, and Wright (Reference Kim, Tejeda and Wright2016). Cluster analysis divides cattle price data into groups (clusters) that are meaningful, useful, or both (Tan, Steinbach, and Kumar, Reference Tan, Steinbach and Kumar2006). It implements a hierarchy using various techniques, for example, pairwise distance between all data points. Once the cluster analysis provides the resulting cluster(s), the sequential VECM approach (similar to a tournament) is applied and we can identify the price discovery region(s) for each hierarchy. The main contribution of this paper is the development of a technique that allows the researcher to compare many markets at once.

Early attempts at overcoming this problem included a round robin tournament approach. This means using the ECM to compare each series to every other series. This method may be inefficient and labor intensive, as growth in the number of markets being studied involves a substantial increase in the number of comparisons. An approximation to a round robin tournament approach was a study by Oellermann, Brorsen, and Farris (Reference Oellermann, Brorsen and Farris1989) which considered four markets: labeled feeder futures, feeder cash, live cattle cash, and live cattle futures. The authors compared four pairs of variables: (1) feeder cash vs. feeder futures, (2) feeder cash vs. live cattle futures, (3) feeder cash vs. live cattle cash, and (4) feeder futures vs. live cattle futures. Nonetheless, it was technically not a round robin tournament since possible combinations such as (5) live feeder futures vs. live cattle futures and (6) live cattle futures vs. live cattle cash were not included.

Another, more efficient, approach is to use a single elimination tournament. Our paper uses a single elimination tournament. Using this approach in a study with $n$ markets allows us to reduce the number of comparisons to $n - 1$ . For example, if one were to do a study including prices from 30 different markets, one would only have to run the 29 bivariate VECMs. A downside to a single elimination tournament is we cannot say who the second strongest competitor is. Just because a competitor loses in the final round does not mean it is second strongest. The second, third, and fourth strongest competitors may have already been matched against the strongest competitor and eliminated by the time the final round is played.

Sports tournaments often address this issue through the use of double and triple elimination tournaments. In a double elimination tournament, one must lose twice before being eliminated. The tournament is designed so that if two teams play each other once, they do not play each other again until the final round. This ensures the second strongest team makes it to the final round before being eliminated. The triple elimination tournament works on the same principle except it ensures the top three teams make it to the final stages of the tournament. Unfortunately, a double elimination tournament requires twice as many matches as a single elimination tournament. A triple elimination tournament requires three times as many matches, etc. Rather than use a double or triple elimination tournament, along with the cumbersome load it would entail, we will use a single elimination tournament by incorporating a cluster analysis. We will use this data mining tool to segregate our variables into groups or clusters, allowing us to identify “winners” in subcategories of the U.S. cattle market.

To perform a cluster analysis means nothing more than dividing data into groups. This is usually done to make the data more meaningful, useful, or both (Tan, Steinback, and Kumar, Reference Tan, Steinbach and Kumar2006, p. 487). During this process of grouping, a set of data or other objects are sorted into multiple groups so that objects within a cluster have high similarity, but are less similar to objects in other clusters. Sameness can be based on whatever factors the researcher considers relevant. Quite often, differences involve distance measures. Regardless of difference criteria, the goal of this process is to create clusters where the objects within a group are similar to one another and different from the objects in the other groups. Greater similarity within a group and larger differences between groups are usually preferred, since this leads to more distinct clustering.

With time series data, like that used in this study, the most popular distance measure is Euclidean distance. Let ${p_{i,t}}$ be the (cattle) price in market $i\,=\,1, \cdots ,n$ . The Euclidean distance between markets $i$ and $j$ is defined as follows:

(5) $$d\left( {i,j} \right) = \sqrt {\mathop \sum \limits_{t = 1}^T {{\left( {{p_{i,t}} - {p_{j,t}}} \right)}^2}} $$

Note that the Euclidean distance is nonnegative, the distance of a market to itself is 0, that is, $d\left( {i,i} \right) = 0$ and the distance is symmetric, that is, $d( {i,j}) = d({j,i})$ . Equation (5) is converted to the root mean square deviation (RMSD) if the expression inside the square root is divided by $T$ . A limitation of using equation (5) for the cattle market is of not accounting for different economic factors on how the market prices are determined. Prices may vary based on transportation costs, seasonality, product transformation, etc. (i.e. heterogeneous). These factors may create large differences between the prices of markets being considered as centers of price discovery. Thus, the differences in prices may reflect many potential economic factors that may not be price discovery related. Future work considers methods that account for these economic factors within the price equilibrium relationships.

Clustering comes in different types, including nested and unnested, that is, partitional or hierarchical. In a partitional clustering, the set of data objects is sorted into non-overlapping subsets (clusters) so that each object is in one, and only one, subset. To obtain a hierarchical clustering, we permit clusters to have subclusters, which are a set of nested clusters. Graphical representations of clusters resemble a tree.

A hierarchical clustering method works by grouping markets into a tree of subclusters. We can further divide hierarchical clustering methods into two types: (i) agglomerative hierarchical clustering and (ii) divisive hierarchical clustering. The first type, agglomerative hierarchical clustering, is based on a bottom-up strategy. It usually starts by letting one object form a cluster and forms successively larger clusters around the original one, until all objects have been sorted. The single cluster is the starting point for the hierarchy. A divisive hierarchical clustering method works in the opposite direction. It starts by placing all the objects in one group. The group is then split into several smaller groups. These smaller groups are then each partitioned into even smaller groups. This process is repeated until objects within each group are similar enough to each other. Figure 2 shows an agglomerative hierarchical clustering method and a divisive hierarchical clustering method, on a data set of five objects, $\left\{ {a,\;b,\;c,\;d,\;e} \right\}$ . Initially, the agglomerative method places each object into a cluster of its own. The clusters are then merged step-by-step according to some criterion. For example, clusters ${C_1}$ and ${C_2}$ may be merged if an object in ${C_1}$ and an object in ${C_2}$ form the minimum Euclidean distance between any two objects from different clusters. This is a single-linkage approach in that each cluster is represented by all the objects in the cluster, and the similarity between two clusters is measured by the similarity of the closest pair of data points belonging to different clusters.

Figure 2. Dendrogram representation for hierarchical clustering. (Reproduced from fig. 10.6 in Han, Kamber, and Pei (Reference Han, Kamber and Pei2012), p. 460).

The cluster-merging process repeats until all the objects are eventually merged to form one cluster. The divisive method proceeds in the contrasting way. All the objects are used to form one initial cluster. The cluster is split according to some principle such as the maximum Euclidean distance between the closest neighboring objects in the cluster. The cluster-splitting process repeats until, eventually, each new cluster contains only a single object. A tree structure, called a dendrogram, is commonly used to represent the process of hierarchical clustering. It shows how objects are grouped together (in an agglomerative method) or partitioned (in a divisive method) step-by-step. Figure 2 shows a dendrogram for the five objects presented in Figure 2, where $l\,=\,0$ shows the five objects as singleton clusters at level 0. At $l\,=\,1$ , objects $a$ and $b$ are grouped together to form the first cluster, and they stay together at all subsequent levels. We can also use a vertical axis to show the similarity scale between clusters. For example, when the similarity of two groups of objects $\left\{ {a,\;b} \right\}$ and $\left\{ {c,\;d,\;e} \right\}$ is roughly 0.16 (similarity scale on the right in Figure 3), they are merged together to form a single cluster.

Figure 3. Cluster analysis dendrogram.

Notes: (1) State_fed_S = fed cattle steer in state; State_fed_H = fed cattle heifer in state; State Name_feeder_S = feeder cattle steer in state; State Name_feeder_H = feeder cattle heifer in state. (2) Vertical axis, Height, is the Euclidean distance between markets; height of 100 is approximately $3.5/cwt difference between markets (see equation (5)). (3) Boxes cut and visualize the dendrogram to five clusters (authors’ choice).

In this research, the hclust function in R software is utilized. This program uses the maximum distance, ${d_{max}}\left( {{C_i},{C_j}} \right) = \mathop {\max }_{{p \in {C_i},p' \in {C_j}}} \left\{ \left| {p - p'} \right|\right\} $ , where $\left| {p - p'} \right|$ is the distance between two markets, to measure the distance between clusters. It is sometimes called a complete-linkage algorithm.

3.1. Cattle price data

This study uses weekly cattle price data from May 2001 through October 2016. All cattle cash and futures price data are obtained from the Livestock Marketing Information Center (LMIC), which compiled the U.S. Department of Agriculture Agricultural Marketing Service (USDA-AMS) Livestock Mandatory Price Reporting (LMR) report data. The fed cattle price data were collected from all Free on Board (FOB) negotiated trade prices within a region. Regions studied include Texas–New Mexico–Oklahoma (TX–NM–OK), Kansas (KS), Nebraska (NE), Colorado (CO), and Iowa–Minnesota (IA-MN). A weekly fed cattle price for each region represents the weighted average from the steers and heifers prices. All weights and grades of cattle were included in this calculation. For feeder cattle price, only calves from 500 to 900 pounds were included. Data from all the auctions in the relevant geographical area were included. Regions studied include CO, KS, Missouri (MO), Montana (MT), NE, OK, South Dakota (SD), and TX. The raw data gave an average price for steers and heifers, divided into categories by hundred-pound increments. Head counts from these auctions were not available at the time of this study. A simple, unweighted average was calculated for both steers and heifers considering the specified weight ranges.Footnote ²

Unfortunately, both the fed cattle price and the feeder cattle price series contained missing observations. The data set was made complete by taking an average of the observations above and below the missing point and using this estimate of the missing observation. If there was more than one missing observation in a row, the missing points were estimated to be contained in a line between the closest observable points (linear interpolation). In addition, an extensive number of typos were present in the raw feeder cattle price data. Prices greater than $3/cwt were removed from the data, and these missing observations were estimated using linear interpolation. The price level of $3/cwt was chosen to maximize the amount of data retained, while still eliminating outliers and probable mistakes in the raw data.

Boxed beef cutout price is also included and it is provided by the weekly average beef cutout prices for Choice 600–900 lbs. from the “National Weekly Boxed Beef Cutout and Boxed Beef Cuts—Negotiated Sales” report by the Agricultural Marketing Service of the USDA and obtained from LMIC. The raw boxed beef cutout data had no missing observations and were used as provided. In addition to these prices, the feeder cattle index is included. The feeder cattle index is a 7-day weighted average of the total dollars sold divided by the total pounds during the same period. The feeder cattle index is included to gauge the level of the potential price discovery type of information that the index could possibly provide. The USDA-AMS issues daily reports which contain cattle eligible for the index. Cattle futures prices are collected from LMIC as well, which was compiled from Chicago Mercantile Exchange. The futures prices are of nearby contracts referring to the contracts’ close price that is closest to expiration. It rolls over on the first date of the new month after the preceding contracts end. The weekly averages are calculated by taking the previous Monday–Friday for the week ending on a Friday data. In total, we have 30 cattle price series which are the combinations of regions, feeder/fed, steer/heifer, and futures prices. Table 1 presents basic statistics.

Table 1. Basic statistics (dollars per cwt)

Source: Livestock Marketing Information Center (LMIC).

State Name_fed_S = fed cattle steer in state; State Name_fed_H = fed cattle heifer; State Name_feeder_S = feeder cattle steer; State Name_feeder_H = feeder cattle heifer.

3.2. Stationary tests and cointegration tests

Before moving to cointegration tests to check if there exists one cointegrating relationship among pairs of series, stationarity tests for each price series were conducted. This study utilizes the Phillips Perron test to check for stationarity of each variable (Phillips and Perron, Reference Phillips and Perron1988). As expected, all the cattle prices are non-stationary (results are not reported to save space). All matched series were tested for cointegration using the Johansen’s trace test, and all were found to be cointegrated. The fact that all these pairs are cointegrated makes them suitable for analysis with the ECM. The optimal lag of VECM is determined using the Schwarz information criterion. The number of cointegration relationship in the bivariate case is at most one cointegrating vector and can be investigated based on likelihood ratio (LR)-type tests (Lütkepohl and Krätzig, Reference Lütkepohl and Krätzig2004). Johansen (Reference Johansen1995) provides critical values for the LR test, which is known as the trace test.