Contemporaneous causal orderings of US corn cash prices through directed acyclic graphs

Abstract

This study investigates dynamic relationships among US corn cash prices for the years 2006–2011. Using daily data from 182 markets in seven Midwestern states, an error correction model is estimated and directed acyclic graphs characterize the contemporaneous causal relationships among prices from different states. Empirical results based on the PC algorithm show that three states, Iowa, Ohio, and Minnesota, dominate corn cash prices. Four potential causal paths among the three states also are identified. Given that physical flows of grain are different at different times of the year, the data are divided into storage periods and harvest periods, and a VAR in differences is adopted to model the price relationships. While Iowa and Ohio continue to dominate corn prices during the storage period, the causal flows are mixed during the harvest period. An application of the LiNGAM algorithm refines the results relative to those derived from the PC algorithm and reveals that Iowa, the leading corn-producing state, is the only state that dominates pricing over the crop year.

Appendices

Appendix 1: Three possible cases of Equation (1)

Three possible cases for Eq. (1) of interest are: (a) if $\Pi$ has full rank p, $X_{t}$ is stationary in levels and a VAR in levels should be adopted, i.e., $X_{t}=\mu +\sum _{i=1}^{k}\Pi _{i}X_{t-i}+e_{t}$, where $\Pi _{1}=\Gamma _{1}+\Pi +I_{p}$, $\Pi _{i}=\Gamma _{i}-\Gamma _{i-1}$ for $i=2,\ldots ,k-1$, and $\Pi _{k}=-\Gamma _{k-1}$; (b) if $\Pi$ has zero rank ($\Pi =\mathbf {0}$), it does not contain long-term information and a VAR in differences should be adopted, i.e., $\Delta X_{t}=\mu +\sum _{i=1}^{k-1} \Gamma _{i}\Delta X_{t-i}+e_{t}$, for $X_{t}$ being I(1) and not cointegrated; (c) if $\Pi$ has rank $r\in [1,p-1]$ for $X_{t}$ being I(1), i.e., $X_{t}$ has r linearly independent cointegrating vector(s) and $p-r$ common stochastic trend(s), it can be written as $\Pi =\alpha \beta ^{\prime }$, where $\alpha$ and $\beta$ are $p\times r$ matrices with rank r, and $\beta ^{\prime }X_{t}\thicksim I(0)$ is stationary. $\Pi$ is the long-run impact matrix, and $\Gamma _{i}$ for $i=1,\ldots ,k-1$ are the short-run impact matrices. The rows of $\beta ^{\prime }$ constitute a basis for the r cointegrating vectors and the elements of $\alpha$ apportion the effect of the cointegrating vectors to the evolution of $\Delta X_{t}$. Equation (1) thus can be written as $\Delta X_{t}=\mu +\alpha \beta ^{\prime }X_{t-1} +\sum _{i=1}^{k-1}\Gamma _{i}\Delta X_{t-i}+e_{t}$. Because for any $r\times r$ non-singular matrix M, we have $\alpha \beta ^{\prime }=(\alpha M)(M^{-1} \beta ^{\prime })=(\alpha M)[\beta (M^{-1})^{\prime }]^{\prime }=\alpha ^{\Delta }\beta ^{\Delta }$, the factorization $\Pi =\alpha \beta ^{\prime }$ is not unique and only identifies the space spanned by the cointegrating relations. Further restrictions on the model are needed to establish the uniqueness of $\alpha$ and $\beta$.

Appendix 2: Cholesky decomposition

We can consider a p-variable structural VAR model which is expressed as:

$$\begin{aligned} AX_{t}=A_{1}X_{t-1}+A_{2}X_{t-2}+\cdots +A_{k}X_{t-k}+v_{t},\quad \text {for} \; t=1,\ldots ,T, \end{aligned}$$

(8)

where k is the lag order and $A_{i}$’s are $p\times p$ matrices of coefficient parameters. We can normalize the variance-covariance matrix of the structural shocks such that $E(v_{t}v_{t}^{{\prime }})\equiv \Sigma _{v}=I_{p}$ without a loss of generality if the diagonal elements of matrix A are unrestricted. The corresponding reduced-form of Eq. (8) is represented as:

$$\begin{aligned} X_{t}= & {} A^{-1}A_{1}X_{t-1}+A^{-1}A_{2}X_{t-2}+\cdots +A^{-1}A_{k}X_{t-k} +A^{-1}v_{t},\; \nonumber \\&\text {where}\; \; t=1,\ldots ,T ,\; \text {which is:} \end{aligned}$$

(9)

$$\begin{aligned} X_{t}= & {} B_{1}X_{t-1}+B_{2}X_{t-2}+\cdots +B_{k}X_{t-k}+e_{t},\;\;\text {where}\;\; t=1,\ldots ,T, \end{aligned}$$

(10)

if we let $B_{i}=A^{-1}A_{i}$ for $i=1,2,\ldots ,k$. As a weighted average of the structural shocks, the reduced-form shocks can be written as:

$$\begin{aligned} v_{t}\equiv \left[ \begin{array}{c} v_{t}^{1}\\ v_{t}^{2}\\ \vdots \\ v_{t}^{p} \end{array} \right] =\left[ \begin{array}{cccc} a_{11} &{} a_{12} &{} \cdots &{} a_{1p}\\ a_{21} &{} a_{22} &{} \cdots &{} a_{2p}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{p1} &{} a_{p2} &{} \cdots &{} a_{pp} \end{array} \right] \left[ \begin{array}{c} e_{t}^{1}\\ e_{t}^{2}\\ \vdots \\ e_{t}^{p} \end{array} \right] \equiv Ae_{t}\text {.} \end{aligned}$$

(11)

Equation (10) has been estimated by writing it as an ECM. To reveal the relative effect of each variable, we need the response of $X_{t}$ to the structural innovation $v_{t}$ and hence require the recovery of the elements of matrix $A^{-1}$ from reduced-form parameters, which can be accomplished by solving $\frac{p(p+1)}{2}$ equations using the information from the variance and covariance matrix of the reduced-form residuals $e_{t}$’s, i.e., $\Sigma _{e}=A^{-1}A^{-1^{\prime }}$. Due to symmetry of $\Sigma _{e}$, it is obvious that at most $\frac{p(p+1)}{2}$ unknowns in matrix $A^{-1}$ can be uniquely identified. If we use a Cholesky decomposition, $A^{-1}$ is a lower triangular matrix. However, the problem with this method is that it makes sense only if the causal chain used, which may be unrealistic, is accepted by the underlying model.