Integration and causality in international freight markets: modeling with error correction and directed acyclic graphs.

1. Introduction

In January 1985, the Baltic Exchange developed the then known Baltic Freight Index (BFI), which, in May 1985, would become the underlying asset of the Baltic International Freight Futures Exchange (BIFFEX) contract, a contract designed to hedge uncertainty associated with volatile freight rates. On the basis of their exposure to the risk of adverse freight-rate fluctuations, ship owners and cargo owners (charterers) could buy or sell BIFFEX contracts to protect their freight-rate revenue or control their freight-rate cost, respectively. In addition to its once use for futures trading, the BFI is also considered to be the leading indicator of the condition in the dry-bulk shipping markets. On a daily basis, it provides accurate information about the level of freight rates across a variety of shipping routes worldwide. This information is extremely valuable for shipping market agents and is an invaluable tool in their decision-making process. This is so because in an industry such as shipping, where trades are being concluded across the globe, there is not a central reporting place to record and monitor the level of activity in the markets.

The principal objective of this article is to assess the degree of interconnectivity between the major shipping routes that constitute the BFI, across different time horizons, reflecting the typical trading patterns in this market. To this end, we employ directed acyclic graphs (DAGs) (Spirtes, Glymour, and Scheines 1993), which enable us to assess this issue in a dynamic manner. Based on an error correction model (ECM) of freight rates, we also develop a framework for estimating forecast error decompositions by employing DAGs. From a practical standpoint, the information provided by innovation accounting techniques, coupled with DAG analysis, provides an intuitive method for assessing the information flows from the shipping markets and allows us to make an assessment on whether the underlying freight index, on which the futures contract is traded, is correctly composed.

The current article, therefore, makes significant contributions to the literature from several angles. To date, research has not estimated the relationships in a truly dynamic manner and has, as such, failed to discuss the implications for derivatives and physical trading. In addition, the application of innovation accounting in this study is different from most other studies in the sense that we use the causal ordering of variables, as indicated by the estimated DAGs, for the orthogonalization of the contemporaneous innovation covariance; as suggested by Swanson and Granger (1997), this step is crucial in providing sound inference in innovation accounting techniques. Finally, investigation of this issue has important implications for the purposes of futures trading, as the BIFFEX contract was eventually delisted in April 2002. In this article, we identify whether the index is correctly composed and whether there are any obvious patterns or redundancies within the index that may have led to the demise of the futures contract. Indeed, our analysis provides a mechanism for deciding upon the correct index and has obvious, yet profound, implications for other research in the area of index construction.

The rest of the article is as follows. Section 2 offers a brief discussion of freight indices. Section 3 outlines the methodologies employed in the article, including an introduction to DAGs, and section 4 describes the underlying properties of the data series. Section 5 presents the empirical results, and section 6 concludes.

2. Freight Indices and Freight Futures

Given uncertainty surrounding freight rates, the benefits of providing a futures market in freight rates has been recognized by market practitioners for a long time. However, the freight futures market was eventually established only in 1985. The reason is that the underlying asset of the market--the service of seaborne transportation--is not a physical commodity that can be delivered at the expiry of the futures contract. This obstacle was overcome with the development of a shipping index, reflecting the level of freight rates across a wide range of shipping routes. This led to the creation, on May 1, 1985, of the BIFFEX contract. The underlying asset, which is delivered at the settlement date, is the cash value of a freight-rate index, originally known as the Baltic Freight Index (BFI), now superseded by the Baltic Panamax Index (BPI). (1) Initially, the index consisted of freight rates across three different vessel sizes: handysize, panamax, and capesize vessels. (2) During the years, the constituent routes of that original index were refined to meet the ever-increasing and changing needs of the physical and derivatives freight markets. Time-charter routes were added in, and gradually, handysize and capesize routes were excluded from the index. (3) The current composition of the underlying index, as well as all the major changes in its composition since its inception, are presented in Table 1; the notes, in the same table, describe some minor amendments to the composition of the index.

These revisions have been driven by the intention to generate an underlying index that promoted the effective functioning of the BIFFEX contract and hence provided a sound basis for effective risk management. The issue of whether the BIFFEX contract was effective in controlling freight-rate risk has been analyzed in the literature by Kavussanos and Nomikos (1999, 2000) and Haigh and Holt (2000). These studies indicated that BIFFEX market participants who use the contract for hedging have miniscule gains in terms of risk reduction. The underlying reason behind this is that the BIFFEX contract provides a cross-hedge against the shipping routes that constitute the underlying index; this induces the presence of basis risk and makes the hedges less effective. The poor hedging performance is also thought to be the primary reason for the low trading activity evidenced in the market. (4) Despite the numerous revisions in the composition of the BPI, trading volume in the market remained at low levels, and, as a result, trading on the BIFFEX contract was terminated in April 2002. (5)

3. Models and Methodologies

Co-integration and Error Correction Framework

The relationship between freight rates is investigated using the following error correction model (Engle and Granger 1987; Johansen 1988, 1991; and Johansen and Juselius 1990):

(1) [DELTA][X.sub.t] = [k-1.summation over i=1][[GAMMA].sub.i][DELTA][X.sub.t-i] + [PI][X.sub.t-1] + [[epsilon].sub.t],

where [X.sub.t] is a 7 X 1 vector of the I(1) variables, each representing one of the routes that comprise the BPI, [DELTA] denotes the first difference operator, [[PI].sub.i] is an n X n matrix of parameters, [omega] is a vector of constants, and [[epsilon].sub.t] is a random error term. [PI] determines how many linear combinations of [X.sub.t] are stationary and can be represented as [PI] = [alpha][beta]', where [beta] is the matrix of co-integrating parameters, and the matrix [alpha] is the matrix of weights (also known as the speed of adjustment parameters). All the parameters within the ECM can provide information on both the long-run and the short-run nature of the relationships between the freight prices. First, the long-run structure can be identified by testing the hypothesis associated with [beta] and [alpha]. Similarly, the short-run dependencies among the prices can also be identified through hypothesis testing on [[GAMMA].sub.i]. Additionally, innovation accounting can also be used in order to identify the short-run structure and interdependencies among the freight rates (Swanson and Granger 1997). However, the basic problem of the orthogonalization of residuals from the ECM and the impact on innovation accounting remains unresolved. Most studies employing ECM or vector autoregressions (VARs) have yet to fully address the problem associated with the contemporaneous relationships among variables. Despite this, innovation accounting requires that a causal assumption about contemporaneous correlation be made. Early work in this area employed the Choleski factorization, with more recent applications concentrating on a "structural" factorization suggested by Bernanke (1986) and Sims (1986) simply because in the Choleski factorization the world may not be viewed as being recursive (Cooley and Leroy 1985). However, the problem with both the Bernanke (1986) and Sims (1986) approach is that the correct structural model may not be known to the researcher. Therefore, following Spirtes, Glymour, and Scheines (1993), in this study we examine the contemporaneous relationships among the variables based on the variance-covariance matrix from the innovations (residuals) from the ECM by employing DAGs, which, until now, have been largely ignored in the economics and finance literature. It is to a brief explanation of DAG theory that we now turn.

Directed Acyclic Graphs

For three variables A, B, and C, illustrate a causal fork, A causes B and C, as B [left arrow] A [right arrow] C. The unconditional association between B and C is nonzero (as both B and C have a common cause in A), but the conditional association between B and C, given knowledge of the common cause A, is zero. This is one screening-off property associated with causal relations: Common causes screen-off associations between their joint effects. Illustrate the inverted causal fork, A and C cause B, as A [right arrow] B [left arrow] C. Here, the unconditional association between A and C is zero, but the conditional association between A and C, given the common effect B, is not zero. A second screening-off property associated with causal relations is the following: Common effects do not screen-off association between their joint causes. These screening-off phenomena are captured in the literature of directed acyclic graphs. (6)

A directed acyclic graph is a picture illustrating causal flow between variables with lines with and without arrowheads. Variables connected by a line are said to be adjacent. If we have a set of variables {A,B,C,D,E}: (i) the undirected graph contains only undirected lines (e.g., A--B); (ii) a directed graph contains only directed lines (e.g., B [right arrow] C). A DAG is a graph that contains no directed cyclic paths (an acyclic graph contains no directed path from a variable that returns to itself). Only acyclic graphs are used in the article.

Directed acyclic graphs represent conditional independence as implied by the recursive product decomposition:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where Pr denotes probability. The symbol pal refers to the realization of some subset of the variables that precede (come before in a causal sense) vi in order ([v.sub.1], [v.sub.2], ..., [v.sub.n]). The symbol [PI] refers to the product (multiplication) operator. Pearl (1986) proposes d-separation as a graphical characterization of conditional independence. Verma and Pearl (1990) provide a proof of this proposition, d-separation characterizes the conditional independence relations given by Equation 2. If we formulate a directed acyclic graph in which the variables corresponding to p[a.sub.i] are represented as the parents (direct causes) of [V.sub.i], then the independencies implied by Equation 2 can be read off the graph using the criterion of d-separation (defined in Pearl 1995).

DEFINITION: Let X, Y, and Z be three disjoint subsets of vertices [variables] in a directed acylic graph G, and let p be any path between a vertex [variable] in X and a vertex [variable] in Y, where by "path" we mean any succession of edges, regardless of their directions. Z is said to block p if there is a vertex w on p satisfying one of the following: (i) w has converging arrows along p, and neither w nor any of its descendants are on Z, or (ii) w does not have converging arrows along p, and w is in Z. Further, Z is said to d-separate X from Y on graph G, written [(X [perpendicular to] Y | Z)].sub.G], if and only if Z blocks every path from a vertex [variable] in X to a vertex [variable] in Y.

Geiger, Verma, and Pearl (1990) demonstrate that there is a one-to-one correspondence between the set of conditional independencies, X [perpendicular to] Y | Z, implied by Equation 2 and the set of triples (X, Y, Z) that satisfy the d-separation criterion in graph G. If G is a directed acyclic graph with variable set V, X and Y are in V, and Z is also in V, then G linearly implies the correlation between X and Y conditional on Z being zero if and only if X and Y are d-separated given Z.

Spirtes, Glymour, and Scheines (1993) have applied the notion of d-separation into an algorithm (PC-algorithm) for building directed graphs. PC-algorithm is a sequential set of commands that begin with an unrestricted graph where every variable is connected with every other variable and proceeds step-wise to remove lines between variables and to direct "causal flow." The algorithm is described in detail in Spirtes, Glymour, and Scheines (1993, p. 117).

The algorithm (we will summarize only the generic aspects of PC-algorithm) begins with a complete undirected graph G on the vertex set X. The complete, undirected graph shows an undirected line between every variable of the system (every variable in X). Lines between variables are removed sequentially based on zero correlation or partial correlation (conditional correlation). The conditioning variable(s) on removed lines between two variables is called the sepset of the variables whose line has been removed (for vanishing zero-order conditioning information, the sepset is the empty set). Edges are directed by considering triples X--Y--Z, such that X and Y are adjacent as are Y and Z, but X and Z are not adjacent. Direct lines between triples X--Y--Z as X [right arrow] Y [left arrow] Z if Y is not in the sepset of X and Z. If X [right arrow] Y, Y and Z are adjacent, X and Z are not adjacent, and there is no arrowhead at Y, then orient Y--Z as Y [right arrow] Z. If there is a directed path from X to Y, and a line between X and Y, then direct (X--Y) as X [right arrow] Y.

In applications, Fisher's z may be used to test whether conditional correlations are significantly different from zero. Fisher's z can be applied to test for significance from zero; where

(3) z([rho](i,j | k), n) = [1/2 [square root of n- [absolute value of k] -3]]ln{[absolute value of 1 + [rho](i,j | k)]/1 - [rho](i,j | [absolute vale of k])},

and n is the number of observations used to estimate the correlations, p(i, j | k) is the population correlation between series i and j conditional on series k (removing the influence of series k on each i and j), and [absolute vale of k] is the number of variables in k (that we condition on). If i, j, and k are normally distributed and r(i, j | k) is the sample conditional correlation of i and j given k, the distribution of z([rho](i, j | k),n) - z(r(i, j | k),n) is standard normal. PC-algorithm and its more refined extensions are marketed as the software TETRAD II (Scheines et al. 1994).

Monte Carlo studies with small sample sizes suggest that TETRAD II works well, if the researcher applies an inverse relationship between sample size and significance level on line removal test. When sample size falls below 100 observations, significance levels as high as 0.20 are recommended (Sprites, Glymour, and Scheines 1993, Chapter 5). As sample size grows above 100, the suggestion is to drop the applied significance level to more traditional values (e.g., 0.10 or 0.05).

As previously suggested, applications of DAGs in economics and finance are not commonplace. Recently, however, Swanson and Granger (1997) suggested a similar procedure to sort-out causal flow on innovations from a vector autoregression (VAR). Their procedure considers only first-order conditional correlation and involves more subjective insight by the researcher to achieve a "structural recursive ordering."

4. Data

Daily data from February 2, 1996, through May 7, 2001, were collected from the Baltic Exchange. The starting date of February 2, 1996, is chosen, because this is the last date that a major change was made to the BPI. (7) Focusing on this period allows us to concentrate on the interactions of the freight rates when there have been no major changes occurring to any of the underlying routes. Data was also collected from May 8, 2001, through September 30, 2002, for an out-of-sample forecasting evaluation (described below).

Summary statistics on the prices are presented in Table 2 along with the correlations among the freight rates. As observed by the coefficient of variation, the spot routes (R1, R2, and R3) show less variability than the time-charter rates show (R1A, R2A, R3A, and R4). This reflects the higher volatility of time-charter routes compared to the underlying spot routes. Turning next to the correlation coefficients, we can note that the highest correlation coefficients are evidenced between the spot and their corresponding time-charter routes (i.e., routes R1 and R1A, routes R2 and R2A, and routes R3 and R3A). This is not surprising given that the definitions of spot routes are very similar to those of the corresponding time-charter routes. Take for instance routes R1 and R1A. Route R1 reflects cargo movements of grain from the U.S. Gulf to Belgium or Holland; route R1A, on the other hand, is a time-charter route for a round-trip voyage from the northwest Continent (Europe) to the Atlantic Coast, North or South America, and back to the northwest Continent. Therefore, route R1A consists of two legs: a ballast leg from Europe to the United States--as there are few dry bulk cargoes originating in Europe--and a laden leg from the United States to Europe. Similarly, route R2 (Grain from the U.S. Gulf to Japan) is similar to route R2A (time-charter route from Continent to the Far East via the U.S. Gulf). R3 and R3A are also linked. R3 represents the shipping freight cost of transporting grain from the North Pacific to Japan, whereas route R3A typically represents cargo flows from Japan to the North Pacific for the transportation of grain and then back to Japan. Finally, route R4 comprises a ballast leg from Japan to the North Pacific to load coal and then back to the European continent. These similarities in the definitions of the BPI routes are manifested by the high values of correlation coefficients evidenced in Table 2.

5. Empirical Results

Unit root tests (Dickey and Fuller 1981) on the levels and first differences of the routes indicate that the series in levels follow unit root processes (these results are excluded to conserve space). As a result, co-integration techniques are used to examine the existence of a long-run relationship between the shipping routes. The lag length (k) in the vector error correction model (VECM) of Equation 1, chosen on the basis of the Schwarz Bayesian Information Criterion (SBIC) (Schwartz 1978), is two. The VECM of Equation 1 is then estimated using the maximum likelihood estimation procedure of Johansen and Juselius (1990). (8) The estimated [[lambda].sub.max] and [[lambda].sub.trace] statistics in Table 3 indicate that there are three co-integrating relationships between the underlying freight rate routes; therefore, for the remainder of the study, an ECM with three co-integrating relationships is modeled.

In order to obtain further insight into the short- and long-run properties of freight rates, we perform a series of tests on the estimated long- and short-run coefficients of the ECM. First, in order to test formally whether freight rates for a route respond to the long-run information generated from all the other freight routes, we perform hypothesis tests on whether an entire row of ct equals zero. Results from these tests indicate that all routes react to shocks in other markets at the 1% level of significance, which is consistent with the efficient market theorem provided by Berg-Andreassen (1997). (9) In general, the greater the number of co-integrating vectors, the more stable the system (recall that we find three), and we test the hypothesis that each of the seven freight rate series is not in the co-integrating space, or, in other words, is not present in any of the three co-integrating vectors. At stringent levels of significance (i.e., 1% level), we find that each of the series belongs to the larger system that links the freight rates together.

Though testing whether each of the series is part of the entire co-integrating system is informative, it is also interesting to test other hypotheses associated with the long-run relationships. That is, to test whether groups of freight prices may belong to one particular vector, whereas other groups may belong to another vector. Here we have shipping routes with common geographical characteristics (e.g., shipping routes originating in the Atlantic or Pacific basins) or contract types (e.g., time-charter vs. spot) or even route coverage (for instance, R2 and R2A). Therefore, we also test for over-identifying restrictions, as outlined in Hansen and Juselius (1995), along these lines within the co-integrating space. Focusing such restrictions on time-charter versus spot and on geographical characteristics resulted in no clear over-identifying restrictions, and therefore, we continue to employ the "unrestricted" co-integrating matrix in the ensuing analysis. This finding indicates that the freight markets are efficient in the sense that available tonnage is moved effectively from market to market, thus stabilizing the freight markets and ensuring that over the longer term, freight rates for different routes move together.

A more detailed insight on the causal relationship between freight rates is obtained by analyzing the decompositions of forecast errors generated from the ECM of Equation 1. Crucial to such analysis is the treatment of contemporaneous innovations in the time series (Sims 1980). In this article, we follow the factorization commonly referred to as the "Bernanke ordering." (10) In the ensuing analysis, we employ the directed graphs algorithms given in Spirtes, Glymour, and Scheines (1993) to achieve a just-identified system in contemporaneous time (a similar suggestion was made by Swanson and Granger [1997]). A DAG is an assignment of causal flow (or the lack thereof) among a set of variables (vertices) based on identifying restrictions in the innovation correlation matrix, [SIGMA]([[epsilon].sub.t]), from the ECM (where we represent the innovations as [[epsilon].sub.it]). The off-diagonal elements of the scaled inverse of the [SIGMA]([[epsilon].sub.t]) (or any other correlation matrix) are, in fact, the negatives of the partial correlation coefficients between the corresponding pair of variables (in our case, freight rates) given the remaining variables in the matrix (Whittaker 1990).

Directed graphs provide an algorithm for removing edges between markets (similar to that previously described) and also for directing the causal flow of information between the markets. The algorithm starts with a complete undirected graph (like the one shown in the top panel of Figure 1) where innovations in every market are connected with innovations in every other market. The algorithm removes edges based on vanishing correlation and partial correlation, the latter measured based on the scaled inverse correlation matrix. Edges between the variables are sequentially removed based on either vanishing-zero-order-correlation (unconditional correlations) or vanishing conditional correlations, where conditioning is done on all possible sets with members 1, 2, ..., k - 2, where k is the number of variables studied (seven in this case).

[FIGURE 1 OMITTED]

The middle panel of Figure 1 gives the final directed graph based on the innovations from the ECM (Eqn. 1). We see two undirected edges in panel B: R1-R2 and R4-R3A, where the program is not able to direct the edges between the markets. Nevertheless, some other interesting and intuitively pleasing patterns emerge. (11) The first observation is that there are no complete "sinks" whereby a particular route only receives information from other routes but does not generate any information to other routes; this leads us to conclude that no routes are redundant in terms of generating information in contemporaneous time. However, some routes seem to "receive" more information from other markets rather than generate information. For instance, the graph illustrates that R1 is led in contemporaneous time by R1A and R2A but does not "lead" any other route; that is, we do not have a directed edge away from that route. Additionally, we can note that R1 and R2 are clearly linked together although we are not able to distinguish the direction of causality between them. Despite this undirected edge, it is clear that R1 does not seem to be leading other shipping routes in terms of information discovery.

Turning next to R1A, we can note that though it leads R1, it is also influenced by R2 and R4, the Eastern Hemisphere time-charter route, which does ultimately connect Japan, Australia, and the European continent (the same destination as R1A). On the other hand, R2, which is considered by shipping practitioners as being the benchmark route, is clearly a dominant route in contemporaneous time. This is verified by observing the number of directed edges leaving that route and influencing other routes. As can be seen, R2 leads R1A, R2A, and R3 in contemporaneous time and is also linked with R1, although we cannot determine the direction of causation in this case. (12) Focusing our attention on the time-charter routes, we see that R2A "leads" R1 in contemporaneous time, but it is clearly influenced (being led) by both R2 and R3A. This is in contrast to R2A's spot equivalent, R2, which is a more dominant route in terms of price leadership.

Not surprisingly, routes R4, R3A, and R3 are all linked together. The common characteristic of these routes is that they reflect trading in the Pacific Basin. Interestingly, this group of routes is also connected via R2A to the U.S. Gulf region. For instance, route R3A leads R2A, but both these routes have the common characteristic that they are routes that ultimately head for Japan]South Korea. R4 also seems to lead route R1A. Intuitively, these routes originate in very different parts of the globe but have the common feature that their final destination is northern Europe.

Forecast error decompositions based on the DAGs are provided in Table 4. (13) In this study, we provide horizons from 1, 2, 3, and 5 days (the very short run) to intermediate run (10 days) to the long run (30 and 60 days). The maximum forecast horizon is set to 60 days, because this is the typical duration of hire in the time-charter routes of the BPI. Looking at the forecast decompositions for R1, we can note that this route is quite heavily influenced by R1A and R2, which combined, explain almost 61% of the uncertainty in R1 after just 1 day, and their impact is even stronger when we consider the longer term. This finding is not surprising given the results from the DAGs as well as the relatively low level of physical trading activity on this route. Indeed, as suggested by the DAG analysis, R1 acts as a "near sink," as it is quite unimportant in generating information affecting other markets. Finally, recall that DAGs suggest we cannot assign the direction of causation between R1 and R2 in contemporaneous time. However, we can see from the decompositions that R2 explains up to 49.00% (after 10 days) of the variation in R1, whereas R1 explains at most 0.81% of the variation in R2. Therefore, R2 dominates R1 in terms of information discovery across all forecast horizons.

For R1A, we see that it is heavily influenced by R2 in the short run and continues to be influenced by this widely regarded influential market in the much longer term, although at that time the Far Eastern routes (R3, R3A, and R4) also help explain some of the variation. After one day, R4 influences R1A, accounting for 15.9%. This complements the findings from DAGs that R1A is affected by R4 in contemporaneous time. We can also note that R1A, unlike its "spot" counterpart (R1), does influence other markets particularly in the longer term (explaining 27.113%, 12.109%, and 28.313% in R1, R2, and R2A, respectively, after 60 days). These findings indicate that R1A is more influential, in terms of information dissemination, compared to R1. This is expected given the large scope of trading reflected in route R1A. Therefore, R1A represents the wider North and South Atlantic to Continent trade, as opposed to the U.S. Gulf-Continent trade only reflected in R1, and hence reflects more accurately the trading conditions in the Atlantic Basin. As a result, it should exert greater influence on other routes, compared to R1.

The importance of R2 in trade and in leading other routes in terms of price discovery is also confirmed by its error decompositions. We can note that R2 is highly exogenous in the short run, meaning that it explains 100% of its own variation after 1 day and continues to explain more than 90% of its own variation after 5 days. In the intermediate to long run, R2 is affected by other markets, most notably R1A and R3A; however, their combined effect accounts for only 24.3% of the total variation after 60 days. Interestingly, it takes quite a long time for these markets to influence R2, but this is commensurate to the period of time taken for the ships to move between these regions so as to exploit any differences in the level of freight rates. Surprisingly however, is that, even though R2 seems to have the greatest influence on other markets, in contemporaneous time, short-, intermediate-, and the long-term, its weighting in the BPI is identical to R2A (12.5%), which seems to be less influential. Indeed, R2A (the time-charter equivalent of R2) is most heavily influenced by R2 in the short run but is also influenced by R1A particularly for the longer horizons. This is because R1A and R2A link the Atlantic and Pacific trades and a degree of substitutability exists between these routes. For instance, vessels that are available at the Continent will choose to trade either in the Atlantic Basin (R1A) or the Pacific Basin (R2A) depending on the level of freight rates prevailing in these two regions. Any imbalance in the relative level of rates between these regions will be ironed out by an adjustment of the supply of tonnage in each region. Given the time scales involved in the time-charter routes, these adjustments take more than 30 days, explaining why the impact of R1A on R2A is more over longer horizons.

As one might expect, once we turn our attention to the markets that are related to the Pacific Basin (R3, R3A, and R4), we see a similar geographical divide in the short run that was found in the U.S. Gulf. According to the forecast error decompositions, after one day, R3 is only influenced by itself (41.563%) and by R3A (54.371%). Its effect on other routes is fairly small, over the long run. Focusing on R3A, this time-charter route represents cargo voyages between Japan and the U.S. West Coast (or British Columbia) and back, or between Japan and Australia and back, thus, in description, being quite different from the other routes, as it may never link to the United States. Not surprisingly, the short-run error decompositions indicate that the route is exogenous in the short run as it explains 100% of its forecast error for the 1-step ahead forecast; for longer horizons, however, it tends to be influenced by other routes, most notably R2. Within 60 days, about 23% of R3A's variability is explained by R2, reflecting once again that arbitrage should link the freight markets together. We can also note that R3A has a clear influence on other markets. Focusing on the longer term, R3A explains 13.32% of the variation in R1, 12.85% in R2A, 50.34% of the variation in R3, and almost 50% of the variation in R4. This is consistent with the results from DAG analysis showing that there is an obvious consistency between the methodologies. Indeed, in contemporaneous time, R3A and R4 are connected (but not directed), and R3A causes both R3 and R2A. R3A is an important route (as important as R2) in terms of price discovery.

Finally, R4 typically comprises a ballast leg from Japan to the North Pacific to load coal and then back to the Continent. The short-run error decompositions indicate that it explains about 87% of its own variation after 1 day but is also affected by R3A (about 11.9%), which also is linked to Japan. The DAG suggests that we cannot sign the causation in contemporaneous time, but results show that R3A affects R4 in the longer term (50.34%) much more than R4 affects R3A in the long term (3.21%). R3 has a little effect on R4 in the short, intermediate, and longer terms even though, as seen in the DAG, there is a contemporaneous causality suggesting R3 causes R4. Like other routes, R2 affects R4 as time passes, accounting for about 11.36% after 60 days.

Some leading routes, like R2 and R3A, seem to dominate the other routes in terms of information dissemination. However, whereas R3A seems to be appropriately weighted within the BPI, R2 is under-weighted relative to its importance. To further illustrate the importance of R2 and to demonstrate out-of-sample forecasting ability, we fit an ECM through December 17, 1999, and forecasted recursively, through the out-of-sample data set, 1- step to 5-step horizons. At each data point in the out-of-sample period, we re-estimated the ECM before forecasting the new 1- to 5-step ahead horizons. Table 5 presents results given in root mean squared errors and the Theil U-statistics over the out-of-sample period. The Theil U is the ratio of the mean squared forecast error of the ECM to that from a random walk forecast ([P.sub.t] = [P.sub.t-1]). Theil U's greater than 1.0 indicate relatively poor forecasting ability; whereas values less than 1.0 indicate improved forecasting ability (relative to the random walk). Notice from the table that Theil U's are generally less than 1.0, except for the R2 market. For this market, the Theil U's are greater than 1.0 at all horizons, indicating that the information in the ECM does not help in forecasting the R2 market. All of these results combined (DAG, innovation accounting, and forecasting) suggest that information is created in R2 (it is the most difficult to forecast), and then the information is passed around via the ECM to other markets, which explains why their Theil U-statistics are less than at 1.0 at various horizons. The important information from R2 is helpful in forecasting other markets at horizons 1 to 5 steps ahead.

In addition, it seems that the information provided by R1 is already reflected in other routes (like R1A and R2) and could conceivably be ignored as a means of providing new information not captured in other markets. Indeed, R1 follows rather than leads over all time frames. The same seems to be true for R2A. These two routes together comprise almost a quarter of the weighting of the BPI (22.5%), yet their influence is trivial. To illustrate this point, we exclude routes R1 and R2A from the estimation process and re-estimate the ECM of Equation 1 using the modeling procedure described in section 2. The ensuing graph is presented in Figure 1, panel C. We can clearly see that innovations in most freight rates are linked between each other. In fact, with the exception of the R1A-R3, R2-R4, and R2-R3A pairs of routes, the remaining combinations of pairs are connected between them. This indicates that the flow of information between the routes has increased following the exclusion of the two redundant routes from the system. In addition, the fact that none of the connected pairs are directed indicates that there is a balance in the flow of information within each pair of routes, as none of the routes leads any of the other routes. Therefore, these findings suggest that R1 and R2A do not contribute any new information to the system of freight rates and, as a result, should not be included in the calculation of the index. From a practitioner's standpoint, this implies that the current index may have been diluted by redundant routes, thus deterring hedging activity in a way suggested by Haigh and Holt (2000) and Kavussanos and Nomikos (2000).

6. Conclusions

Though there have been several attempts in recent years to study the level of interconnectivity and linkages within the volatile dry-bulk ocean freight industry, to date no study has conducted an analysis in a truly dynamic nature. In this article, we employ DAGs to assess causation and linkages among the world's major shipping routes that compose the Baltic Panamax Index (the index on which freight futures trading was based). The DAG analysis also allows us to address issues surrounding the causal ordering on innovations from a VAR or an error correction model from which we generate familiar forecast error decompositions and impulse responses.

Our results suggest that over the longer term, all freight prices are interconnected, verifying the suggestion that the dry-bulk shipping market is efficient and bulk tonnage is moved effectively between markets stabilizing freight rates. However, through our unique application of DAG analysis, we find a strong geographical pattern to information linkages and that some routes are dominant in terms of "price leadership." The results also indicate that from a futures contract design standpoint, some routes are "redundant" in terms of information flow and could have been effectively dropped from the index, as information from these markets is already captured in other markets. These redundant routes have therefore diluted the index, making the hedging strategy less effective, and thus helping to explain the demise of the freight futures contract. Importantly, while the DAG analysis provides guidance as to what should be the appropriate compilation and weighting of the index and might help explain optimal futures contract design, it may have further obvious, yet profound implications in the general area of index construction.

Table 1. Baltic Freight Index: Changes in Its Composition since Its
Inception

       Vessel (dwt)    Size Cargo                   Route

R1        55,000       Light Grain    U.S. Gulf to ARA (a)
R1A       70,000       T/C            Trans-Atlantic round (duration
                                        45-60 days)
R2        52,000       HSS            U.S. Gulf to South Japan
R2A       70,000       T/C            Skaw Passero to Taiwan (Japan)
                                        (50-60 days)
R3        52,000       HSS            U.S. Pacific Coast to South Japan
R3A       70,000       T/C            Trans-Pacific Round (35-50 days)
R4        21,000       HSS            U.S. Gulf to Venezuela
R5        35,000       Barley         Antwerp to Jeddah (Saudi Arabia)
          38,000       T/C            South America to Far East
R6       120,000       Coal           Hampton Roads (U.S.) to South
                                        Japan
R7        65,000       Coal           Hampton Roads (U.S.) to ARA
         110,000       Coal           Hampton Roads (U.S.) to ARA
RS       130,000       Coal           Queensland (Australia) to
                                        Rotterdam
R9        55,000       Coke           Vancouver (Canada) to Rotterdam
          70,000       T/C            Japan-Korea to Skaw Passero (50-
                                        60 days)
R10       90,000       Iron Ore       Monrovia (Liberia) to Rotterdam
         150,000       Iron Ore       Tubarao (Brazil) to Rotterdam
R11       25,000       Pig Iron       Vitoria (Brazil) to China
          25,000       Phosphate      Casablanca (Morocco) to West
                                        Coast India
R12       20,000       Potash         Hamburg (Germany) to West Coast
                                        India
          14,000       Phosphate      Aqaba (Jordan) to West Coast
                                        India
R13       14,000       Phosphate      Aqaba (Jordan) to West Coast
                                        India
R14      140,000       Iron Ore       Tubarao (Brazil) to Beilun and
                                        Baoshan (China)
R15      140,000       Coal           Richards Bay (South Africa) to
                                        Rotterdam

                       4/01/85-    4/1l/88-    6/08/90-    5/02/91-
       Vessel (dwt)    3/11/88     3/08/90     4/02/91     4/02/93

R1        55,000        20%         20%         10%         10%
R1A       70,000                                10%         10%
R2        52,000        20%         20%         20%         10%
R2A       70,000                                            10%
R3        52,000        15%         15%          7.50%       7.50%
R3A       70,000                                 7.50%       7.50%
R4        21,000         5%          5%          5%          5%
R5        35,000         5%          5%
          38,000                                 5%          5%
R6       120,000         5%          7.50%       7.50%       7.50%
R7        65,000         5%          5%          5%
         110,000                                             5%
RS       130,000         5%          5%          5%          5%
R9        55,000         5%          5%          5%          5%
          70,000
R10       90,000         5%          5%          5%
         150,000                                             5%
R11       25,000         5%
          25,000                     2.50%       2.50%       2.50%
R12       20,000         2.50%
          14,000                     5%          5%          5%
R13       14,000         2.50%
R14      140,000
R15      140,000

                       5/02/93-    3/11/93-    6/05/98-      From
       Vessel (dwt)    2/11/93     5/05/98     29/10/99    1/11/99

R1        55,000        10%         10%         10%         10%
R1A       70,000        10%         10%         10%         20%
R2        52,000        10%         10%         10%         12.5%
R2A       70,000        10%         10%         10%         12.5%
R3        52,000         7.50%      10%         10%         10%
R3A       70,000         7.50%      10%         10%         20%
R4        21,000         5%
R5        35,000
          38,000         5%
R6       120,000         7.50%       7.50%
R7        65,000
         110,000         5%          7.50%       7.50%
RS       130,000         5%          7.50%
R9        55,000
          70,000         5%         10%         10%         15%
R10       90,000
         150,000         5%          7.50%       7.50%
R11       25,000
          25,000         2.50%
R12       20,000
          14,000         5%
R13       14,000
R14      140,000                                 7.50%
R15      140,000                                 7.50%

The following minor amendments of the Index are not presented in
Table 1.

(1.) As of 6 May 1998. Routes 2 and 3 refer to a 54,000 dwt panamax
vessel.

(2.) Routes 1A, 2A, 3A, and 9 were based on a 64,000 dwt panamax vessel
for the period up to 2 February 1996.

(3.) Route 5 was 20,000 dwt barley from Antwerp to Red Sea for the
period 1 January 1985 to 4 February 1986.

(4.) Route 7 was based on a 100,000 dwt vessel for the period 5
February 1991 to 4 February 1993.

(5.) Route 8 was based on a 110,000 dwt vessel for the period 1 January
1985 to 5 February 1992.

(6.) Route 10 was based on a 135,000 dwt vessel for the period 5
February 1991 to 2 August 1995.

(7.) Route 11 was 20,000 dwt sugar from Recife (Brazil) to U.S. East
Coast for the period 1 January 1985 to 8 May 1986.

(a) ARA-Amsterdam, Rotterdam and Antwerp

Table 2. Descriptive Statistics and Correlation Analysis on Freight
Prices

                                Descriptive Statistics

                        R1         R1A         R2         R2A

Mean                  12.265     9063.49     20.693    10545.81
Median                12.397     9338        21.775    11049.5
Standard deviation     2.336     2262.55      3.875     2840.83
CV                     0.190        0.25      0.187        0.269
[m.sub.3]             -0.947       -0.887    -0.553       -0.530
[m.sub.4]             -0.240       -0.345    -0.692       -0.308
Min                    7.6       4106        12.314     4143
Max                    7.6      13071        26.929    16386

                                     Correlations

                        R1         R1A         R2         R2A

R1                     1
R1A                    0.942        1
R2                     0.832        0.929     1
R2A                    0.744        0.889     0.957        1
R3                     0.847        0.813     0.742        0.613
R3A                    0.819        0.841     0.815        0.739
R4                     0.656        0.555     0.369        0.210

                           Descriptive Statistics

                        R3         R3A          R4

Mean                  13.213     9204.88      7656.52
Median                13.173     9492.5       7003.0
Standard deviation     2.330     2364.16      2374.07
CV                     0.176        0.257        0.310
[m.sub.3]             -1.188       -1.023       -0.762
[m.sub.4]              0.004       -0.297        0.458
Min                    8.883     3757         3206
Max                   17.693    13250        12883

                                Correlations

                        R3         R3A          R4

R1
R1A
R2
R2A
R3                     1
R3A                    0.953        1
R4                     0.865        0.759        1

Summary statistics are presented for daily freight prices for the
period 2 February 1996 to 7 May 2001 (1330 observations). CV represents
the coefficient of variation, and [m.sub.3] and [m.sub.4] represent
sample skewness and kurtosis, respectively. R1, R1A, R2, R2A, R3, R3A,
and R4 represent the freight prices (described in Table 1) for the
routes that comprise the BPI.

Table 3. Co-integration Analysis of Freight Rates

                                                       Johansen (1988)
                                                        Tests for the
                                                        Number of Co-
                                                         integrating
                                                         Vectors (a)

                                                       [[lambda].sub.
                                                           trace]
[[lambda].sub.trace]                                      Critical
Test Statistic                    H0 (b)                    Value

204.96                             r = 0                   124.25
133.29                  r [less than or equal to] 1         95.18
 75.26                  r [less than or equal to] 2         70.60
 40.27                  r [less than or equal to] 3         48.28
 21.05                  r [less than or equal to] 4         31.52
  7.82                  r [less than or equal to] 5         17.95
  3.50                  r [less than or equal to] 6          8.18

                         Johansen (1988) Tests for the Number of Co-
                                   integrating Vectors (a)

                                                       [[lambda].sub.
                        [[lambda].sub.max]                  max]
[[lambda].sub.trace]           Test                       Critical
Test Statistic              Statistic          H0          Value

204.96                        71.67           r = 0        44.91
133.29                        58.03           r = 1        39.43
 75.26                        34.99           r = 2        33.32
 40.27                        19.22           r = 3        27.14
 21.05                        13.23           r = 4        21.07
  7.82                         4.32           r = 5        14.90
  3.50                         3.50           r = 6         8.18

[[lambda].sub.max] (r, r + 1) = -T ln(1 - [[lambda].sub.r+1]) and
[[lambda].sub.trace](r) = -T[[summation of].sup.n.sub.i=r+1] ln(1 -
[[lambda].sub.i] are the estimated (ordered from largest to smallest)
eigenvalues on II matrix in Equation 2. Critical values for the
[[lambda].sub.max] and [[lambda].sub.trace] and statistics (at the 5%
level) are from Osterwald-Lenum (1992). The optimal lag length (k) is
based on the Schwartz Bayesian Criterion (1978). The sample size (M is
equal to 1330.

(a) r represents the number of co-integrating vectors.

(b) [H.sub.0] represents the null hypothesis.

Table 4. Error Decompositions

Steps Ahead    Standard Error    R1         R1A           R2

R1
   1               0.0203        28.902     20.515       40.429
   2               0.0371        25.107     21.132       43.231
   3               0.0531        22.653     21.312       45.245
   5               0.0824        19.718     21.374       47.654
  10               0.1410        16.625     21.810       49.001
  30               0.2685        13.712     25.158       43.957
  60               0.3864        12.295     27.113       39.967
R1A
   1               0.0143         0.0000    48.577       33.140
   2               0.0300         1.0546    40.745       41.681
   3               0.0460         1.8502    36.347       46.069
   5               0.0775         2.6051    31.901       50.245
  10               0.1443         2.9650    28.772       52.426
  30               0.2923         2.3324    30.656       47.559
  60               0.4245         1.8644    32.303       43.644
R2
   1               0.0090         0.0000     0.0000    100.00
   2               0.0178         0.0762     0.5911     98.641
   3               0.0270         0.2400     1.4888     96.594
   5               0.0453         0.5562     3.1409     92.775
  10               0.0833         0.8124     5.7059     86.349
  30               0.1598         0.3958     9.9138     73.306
  60               0.2224         0.2150    12.109      66.197
R2A
   1               0.0134         0.0000    16.060      50.729
   2               0.0279         0.6208    16.120      57.653
   3               0.0432         1.1550    16.200      60.660
   5               0.0734         1.7442    16.516      62.802
  10               0.1366         2.0830    17.829      62.441
  30               0.2691         1.4750    23.788      54.861
  60               0.3859         1.0501    28.313      50.244
R3
   1               0.0089         0.0000     0.0000      4.0662
   2               0.0168         0.2200     0.0003      6.5251
   3               0.0246         0.5701     0.0002      8.7793
   5               0.0400         1.2856     0.0100     12.537
  10               0.0748         2.4786     0.1428     18.109
  30               0.1655         3.8610     1.3715     22.732
  60               0.2434         4.7773     4.0919     26.258
R3A
   1               0.0140         0.0000     0.0000      0.0000
   2               0.0265         0.7625     0.0000      1.2285
   3               0.0395         1.5544     0.0011      3.0032
   5               0.0656         2.5992     0.0261      6.3272
  10               0.1247         3.7159     0.3151     11.647
  30               0.2671         4.7364     3.0890     17.694
  60               0.3839         5.5585     8.5386     23.763
R4
   1               0.0116         0.0000     0.0000      0.0942
   2               0.0227         0.8488     0.0065      1.8365
   3               0.0344         1.6375     0.0090      3.6396
   5               0.0584         2.5584     0.0042      6.3741
  10               0.1134         3.3543     0.0802      9.6344
  30               0.2446         3.6945     1.2365     10.229
  60               0.3447         4.1600     4.0203     11.3607

Steps Ahead      R2A        R3          R3A        R4

R1
   1            1.8600     0.0744      1.4924     6.7271
   2            1.6568     0.1207      1.6654     7.0867
   3            1.5325     0.1744      1.9251     7.1583
   5            1.3434     0.2986      2.5893     7.0228
  10            0.9419     0.6729      4.5551     6.3936
  30            0.3193     2.1928     10.227      4.4344
  60            0.1910     3.6523     13.324      3.4577
R1A
   1            0.0000     0.1761      2.1780    15.929
   2            0.2286     0.3243      2.3998    13.567
   3            0.4108     0.4406      2.6297    12.252
   5            0.5706     0.6521      3.2145    10.811
  10            0.5177     1.2116      4.9756     9.1321
  30            0.1714     3.4123      9.5108     6.3582
  60            0.0921     5.4644     11.480      5.1525
R2
   1            0.0000     0.0000      0.0000     0.0000
   2            0.1778     0.0167      0.2518     0.2453
   3            0.3257     0.0717      0.7090     0.5705
   5            0.4016     0.2689      1.8440     1.0138
  10            0.2216     0.9813      4.8064     1.1235
  30            0.6776     4.1273     11.091      0.4887
  60            1.3070     7.3926     12.196      0.5840
R2A
   1           24.633      0.0582      3.2548     5.2662
   2           16.860      0.1786      3.1389     5.4288
   3            12.939     0.2817      3.3410     5.4225
   5            9.0904     0.4773      4.1175     5.2520
  10            5.5126     1.0090      6.5804     4.5463
  30            2.1760     3.1103     12.285      2.3048
  60            1.1591     5.1275     12.851      1.2555
R3
   1            0.0000    41.563      54.371      0.0000
   2            0.0027    35.892      57.101      0.2595
   3            0.0103    31.566      58.308      0.7664
   5            0.0242    25.536      58.700      1.9064
  10            0.0150    18.107      57.501      3.6463
  30            0.6352    11.295      56.173      3.9324
  60            2.0626     9.3679     50.341      3.1016
R3A
   1            0.0000     0.0000    100.00       0.0000
   2            0.0973     0.3828     96.384      1.1451
   3            0.1633     0.9106     91.880      2.4871
   5            0.1964     1.7501     84.801      4.3004
  10            0.1197     2.6791     75.806      5.7174
  30            0.3127     3.5431     66.049      4.5759
  60            0.8175     4.4259     53.682      3.2139
R4
   1            0.0000     0.9624     11.901     87.042
   2            0.0134     2.0379     22.339     72.917
   3            0.0191     2.8000     29.472     62.423
   5            0.0131     3.6387     37.609     49.803
  10            0.0371     4.2400     45.460     37.194
  30            1.4830     4.4368     52.600     26.321
  60            3.5180     4.7344     49.382     22.824

The decompositions may not sum to 100 in each row due to rounding. R1,
R1A, R2, R2A, R3, R3A, and R4 represent freight prices (described in
Table 1) that comprise the BPI.

Table 5. Root Mean Squared Errors and Theil U-Statistics on
Out-of-Sample Forecasts, at Horizons of 1 to 5 Days Ahead, of Freight
Rates Based on an Error Correction Model with Three Co-integrating
Vectors

Horizon    No. Observations     RMSE      Theil-U

R1
  1              352             0.135     0.979
  2              351             0.173     0.869
  3              350             0.223     0.829
  4              349             0.278     0.819
  5              348             0.332     0.820
R1A
  1              352            59.773     0.553
  2              351           126.945     0.607
  3              350           205.407     0.673
  4              349           286.439     0.722
  5              348           368.137     0.762
R2
  1              352             0.814     1.307
  2              351             0.888     1.317
  3              350             0.977     1.324
  4              349             1.029     1.266
  5              348             1.085     1.227
R2A
  1              352           190.706     1.082
  2              351           249.426     1.005
  3              350           313.507     0.951
  4              349           381.406     0.932
  5              348           446.460     0.920
R3
  1              352             0.061     0.636
  2              351             0.117     0.649
  3              350             0.182     0.699
  4              349             0.253     0.755
  5              348             0.324     0.800
R3A
  1              352            80.140     0.615
  2              351           167.342     0.649
  3              350           265.418     0.699
  4              349           367.339     0.755
  5              348           470.238     0.800
R4
  1              352           161.383     0.919
  2              351           256.022     0.919
  3              350           322.118     0.889
  4              349           396.660     0.893
  5              348           479.075     0.917

RMSE is the root mean squared error from forecasts of freight rates on
markets R1, R1A, R2, R2A, R3, R3A, and R4 at horizons 1, 2, 3, 4, and 5
days ahead. The column labeled "Theil-U" gives the ratio of the RMSE of
the error correction forecst to the RMSE of a no-change forecast.
Values of this last statistic less than 1.0 are associated with the
error correction forecasts that are "better" than a "naive," no-change
forecast.

Valuable comments and suggestions were provided by journal co-editor Kent Kimbrough, an anonymous reviewer, and by seminar participants at the U.S. Securities and Exchange Commission in February 2002, at the U.S. Commodity Futures Trading Commission in April 2002, and at the NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, St. Louis, Missouri, April 2002. The views, analysis, and conclusions do not represent the opinions of the United States Commodity Futures Trading Commission.

(1) The BPI is calculated every London business day by the Baltic Exchange, from data supplied by a panel of 14 international shipbrokers, and is reported in the market at 1 P.M. London time. Each panelist submits its view of that day's rate on each of the BPI constituent routes. These rates are based either on actual shipping fixtures concluded in the market or, in the absence of an actual fixture, reflect the panelist's expert view of what the rate would be on that day if a fixture had been concluded.

(2) These are the three major classes of vessels that are used for the transportation of different dry-bulk commodities across different parts of the world. Capesize vessels (around 140,000 dead-weight tons [dwt]) transport iron ore mainly from South America and Australia and coal from North America, Australia, and South Africa. Panamax vessels (around 70,000 dwt) are used primarily to carry grain from North America, Argentina, and Australia and coal from North America, Australia, and South Africa. Finally, handysize vessels (around 35.000 dwt) transport grain, mainly from North America, Argentina, and Australia, and minor bulk products, such as sugar, fertilizers, steel and scrap, forest products, non-ferrous metals, and salt, virtually from all over the world.

(3) Spot-charters and time-charters are the two major vessel employment contracts in the shipping industry. In a spot-charter, a shipowner undertakes the responsibility to transport a cargo from the loading port to the destination port. The freight paid by the charterer (cargo owners) to the shipowner is expressed as USD ($) per ton of cargo and covers all of the shipowner's expenses in performing that voyage. A voyage charter may be thought of, therefore, as the equivalent of hiring a taxi to take you from A to B. In a time-charter, the shipowner agrees to hire out his vessel to a charterer for a specified time period. The freight rate paid by the charterer in this case is calculated as $ per day of hire. The charterer is directly responsible for all the voyage expenses, such as bunkers, port charges, canal dues, and so forth, but has much more flexibility, compared to a voyage charter, as to where he trades the ship. A time-charter is therefore much more akin to hiring a car. The relationship between spot- and time-charter rates has always been a pivotal issue in modeling shipping freight markets. Several studies in the literature are devoted to examining this relationship utilising different theories, methodologies, and various data sets (see Alizadeh and Nomikos [2002] for a review). The strong presumption is that spot- and time-charter rates are related in a systematic way, in a similar way that short and long bonds are related via a "term structure" relationship. Kavussanos and Alizadeh (2002), for instance, test the validity of this hypothesis using a series of tests, proposed initially by Campbell and Shiller (1987) for the bond market. They find evidence against this hypothesis, which they attribute to the existence of risk premia in the time-charter rates. More specifically, they find that there are negative risk premia in the market, which implies that shipowners are prepared to offer a discount in order to fix charter contracts with longer maturity, compared to the spot market. This discount reflects the additional risks that shipowners operating in the spot market face, compared to the time-charter market. We would like to thank an anonymous referee for pointing this out to us.

(4) For instance, during the period from February 1996 to June 2000, the average trading volume in the market was only 146 contracts. The monetary value of these contracts roughly corresponds to the average freight cost of transporting 108,000 tons of grain from the U.S. Gulf to Japan (that is, two voyages in Route 2 of the BP[); market sources estimate that this level of futures trading activity corresponds to only 10% of the total physical activity in the dry-bulk shipping market. It is also worth noting that the average trading volume after the introduction of the BPI, in November 1999, fell to only 17 contracts a day.

(5) In 1995, trading began in over-the-counter (OTC) forward contracts--the Forward Freight Agreements (FFAs)--on the routes that constitute the BPI. The market has seen remarkable growth since that date and indeed, many practitioners suggest that part of the reason that the BIFFEX futures market has met its demise is because of the development and subsequent growth in the FFAs.

(6) Orcutt (1952), Simon (1953), Reichenbach (1956), and Papineau (1985) offer similar expressions of asymmetries in causal relations. For a description of various causal asymmetries, see Hausman (1998).

(7) As we can see in Table 1, data for the routes that compose the BPI are available since February 5, 1993, when R4, then known as R9, was introduced. However, on February 2, 1996, the vessel size for R1A, R2A, R3A, and R4 increased from 64,000 tonnes to 70,000 tonnes causing a jump in the level of freight rates of approximately $1000 a day. Consequently, to avoid this structural break, we employ data for the BPI going back only to February 2, 1996.

(8) We also allow for the existence of a constant ([mu]) in the co-integrating relationship.

(9) Results from these tests indicate that routes R1A and R4 are individually weakly exogenous at the 5% level of significance. However, jointly testing that these shipping routes are weakly exogenous results in a Z2 value of 13.59 (with an associated p-value of 0.03). Such a finding does not provide conclusive evidence whether the markets are exogenous at stringent statistical levels. Hence, no restrictions on weak exogeneity are imposed in the estimated model. These results, like all other excluded to conserve space, are available upon request.

(10) Consider the innovation vector (et) from the ECM as: A[[epsilon].sub.t] = [v.sub.t], where A is a 7 x 7 matrix of coefficients and [v.sub.t] is a 7 X 1 vector of orthogonal shocks. As documented by Doan (1992), a factorization is identified if there is no combination of i and j (i [not equal to] j) for which both {a,j} and {[a.sub.ji]} are non-zero (where {[a.sub.ij]} is element ij of the matrix A).

(11) In subsequent innovation accounting analysis, we direct these edges to imply acyclic rather than cyclic graphs. For a discussion of problems arising from cyclic graphs, the reader is directed to Spirtes et al. (1999). The same analysis was conducted at the 1% level of significance. Similar (undirected linkages) are found connecting the markets with the exception of the links between R4 and R1A, R2 and R1A, and R2A and R1.

(12) Though DAGs alone seem to verify the importance of R2 in the price discovery process, data provided by the United States Federal Grain Inspection Service (FGIS)--which provides data on volume of trade and number of ships leaving the U.S. ports--also point out the significance of R2 in the world sea-borne trade. To illustrate, between February 2, 1996, and May 7, 2001 (the period of time studied here), a total of 86.5 million tonnes of grain was transported from the U.S. Gulf to South Japan (R2) on 4556 different vessels. This compares to a total of 14.5 million tonnes of grain being transported from the U.S. Gulf to the Amsterdam-Rotterdam-Antwerp (ARA) region of Europe (R1) on 414 vessels, and 37 million tonnes were shipped from the U.S. North Pacific to South Japan (R3) on a total of 1908 vessels. Using R2 as the base route, we can see that in terms of tonnage shipped, volume does have some influence on price leadership. In particular, volume on R1 is only about 17% of that shipped to Japan via R2, and volume on R3 is about 43% of the total volume that is shipped via R2.

(13) In addition to the forecast error decompositions, we also estimate the impulse response functions based on the DAGs. Results from the impulse response analysis are qualitatively similar to those presented for the forecast error decomposition. Consequently, in order to save space, they are not presented here and are available from the authors.

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Michael S. Haigh,* Nikos K. Nomikos, ([dagger]) and David A. Bessler ([double dagger])

* U.S. Commodity Futures Trading Commission, 8010 Three Lafayette Center, 1155 21st Street, NW, Washington DC, 20581; and the University of Maryland; E-mail [e-mail]; corresponding author.

([dagger]) Faculty of Finance, Cass Business School, London EC1Y 8TZ, UK; E-mail [e-mail].

([double dagger]) Department of Agricultural Economics, Texas A&M University, 349A Blocker Building, College Station, TX 77840; E-mail [e-mail].

Received February 2002; accepted October 2003.