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Cointegration and price discovery in US corn cash and futures markets

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Abstract

Using prices from 182 cash markets from seven states and the Chicago Board of Trade futures, we investigate cointegration and price discovery for corn. Analysis based on cash–futures pairs reveals that cointegration holds for 52 cash markets and failures tend to happen farther away from futures delivery locations. Cash generally are as important as futures prices as information sources in the long run and cash to futures information flow is most likely in the short run. Contributions to price discovery also are measured quantitatively for cointegrated cases. Analysis based on state-level cash prices indicates bidirectional information flow between cash and futures prices under a bivariate model, and futures to cash information flow under the octavariate model with all cash and the futures series. Comparisons of the two models show that including local cash markets in a price relationship model highlights cointegration and the futures’ price discovery role and could benefit cash price forecasting. Finally, evidence of nonlinear causality is found.

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Notes

  1. Approximately 39.5, 30.8, 9.2, 8.4, 4.3, 2.5, 2.1, 1.8, 1.2, and 0.2% of corn is used in the feed/residual, fuel/ethanol, DDGs (Distiller’s Dried Grains with Solubles), direct export, HFCS (High-Fructose Corn Syrup), sweeteners, starch, cereal, beverage/alcohol, and seed segment, respectively.

  2. As shown in the seminal work of Engle et al. (1983), weakly exogeneity and Granger causality are conditions required for forecasting.

  3. On days such as holidays where prices are missing in each market, we omit the observations and assume smooth data continuity (Goodwin and Piggott 2001).

  4. A possible periodic jump in this futures series caused by contract rolling biases against identifying cointegration for each cash market equally and should not affect empirical results substantially.

  5. Unless stated otherwise, we will refer to “log prices” as “prices” hereafter.

  6. Johansen’s trace test has been shown to be more robust as compared to the maximum eigenvalue test (see, e.g., Cheung and Lai 1993). For the current study, these two tests arrive at the same results about cointegrating ranks.

  7. As discussed in Sect. 3, unit root tests at the 5% significance level show that all of the price series are not stationary in levels but stationary in differences.

  8. Before the nonlinear Granger causality test, the BDS test (Broock et al. 1996) is applied to investigate nonlinearities of the residuals and thus determines the appropriateness of the nonlinear test.

  9. For our case, \(p=2\) and \(X_{t}=\left( \begin{array}{c} C_{\mathrm{{t}}} \\ F_{\mathrm{{t}}} \end{array} \right) \), where \(C_\mathrm{{t}}\) and \(F_\mathrm{{t}}\) stand for cash and futures prices, respectively. Equation (1) can be written as:

    $$\begin{aligned} H_{0}{:} \left( \begin{array}{c} \Delta C_\mathrm{{t}} \\ \Delta F_{\mathrm{{t}}} \end{array} \right) =\mu +\left( \begin{array}{cc} \Pi _{11} &{} \Pi _{12} \\ \Pi _{21} &{} \Pi _{22} \end{array} \right) \left( \begin{array}{c} C_{\mathrm{{t}}-1} \\ F_{\mathrm{{t}}-1} \end{array} \right) +\sum _{i=1}^{k-1}\left( \begin{array}{cc} \Gamma _{i,11} &{} \Gamma _{i,12} \\ \Gamma _{i,21} &{} \Gamma _{i,22} \end{array} \right) \left( \begin{array}{c} \Delta C_{\mathrm{{t}}-i} \\ \Delta F_{\mathrm{{t}}-i} \end{array} \right) +e_{t}. \end{aligned}$$

  10. For our \(p=2\) case, (a) corresponds to:

    $$\begin{aligned} \left( \begin{array}{c} \Delta C_\mathrm{{t}} \\ \Delta F_\mathrm{{t}} \end{array} \right) =\mu _{0}+\left( \begin{array}{c} \alpha _{1} \\ \alpha _{2} \end{array} \right) \left( \begin{array}{cc} \beta _{1}&\beta _{2} \end{array} \right) \left( \begin{array}{c} C_{\mathrm{{t}}-1} \\ F_{\mathrm{{t}}-1} \end{array} \right) +\sum _{i=1}^{k-1}\left( \begin{array}{cc} \Gamma _{i,11} &{} \Gamma _{i,12} \\ \Gamma _{i,21} &{} \Gamma _{i,22} \end{array} \right) \left( \begin{array}{c} \Delta C_{\mathrm{{t}}-i} \\ \Delta F_{\mathrm{{t}}-i} \end{array} \right) +e_{t}, \end{aligned}$$

    and (b) corresponds to:

    $$\begin{aligned} \left( \begin{array}{c} \Delta C_\mathrm{{t}} \\ \Delta F_\mathrm{{t}} \end{array} \right) =\left( \begin{array}{c} \alpha _{1} \\ \alpha _{2} \end{array} \right) \left( \left( \begin{array}{cc} \beta _{1}&\beta _{2} \end{array} \right) \left( \begin{array}{c} C_{\mathrm{{t}}-1} \\ F_{\mathrm{{t}}-1} \end{array} \right) +\delta ^{^{\prime }}\right) +\sum _{i=1}^{k-1}\left( \begin{array}{cc} \Gamma _{i,11} &{} \Gamma _{i,12} \\ \Gamma _{i,21} &{} \Gamma _{i,22} \end{array} \right) \left( \begin{array}{c} \Delta C_{\mathrm{{t}}-i} \\ \Delta F_{\mathrm{{t}}-i} \end{array} \right) +e_{t}. \end{aligned}$$

  11. Shu and Zhang (2012), Francis et al. (2010), Dergiades et al. (2013), Fujihara and Mougoué (1997), Silvapulle and Moosa (1999), Ajayi and Serletis (2009), and Dergiades (2012) gave a formal description of the nonlinear Granger causality test.

  12. Before the nonlinear Granger causality test, data nonlinearities are examined (Francis et al. 2010; Dergiades et al. 2013) by applying the BDS test suggested by Broock et al. (1996) to the residuals from an ECM or a VAR in differences (Dergiades et al. 2013; Fujihara and Mougoué 1997). The BDS test essentially inspects the validity of the identically and independently distributed (i.i.d) assumption on time series.

  13. All tests in Table 1, 2 are performed at the 5% significance level.

  14. We follow Bekiros and Diks (2008a, b) and discuss results for one lag. However, it is possible to perform the test for longer lags (see, e.g., Diks and Panchenko (2006).

  15. Johansen’s trace statistic tests the nested hypotheses: \(null:r=r_{0}\) versus \( alternative:r>r_{0}\) for \(r_{0}=0\), 1, 2, ..., \(p-1\), where p is the number of variables. Johansen (1992) proposed a sequential testing procedure to determine the number of cointegrating vectors. Hypotheses are tested in the following order: \(H_{1}^{*}(0)\), \(H_{1}(0)\), \(H_{1}^{*}(1)\), \( H_{1}(1)\), ..., \(H_{1}^{*}(p)\), \(H_{1}(p)\). For example, \(H_{1}^{*}(1)\) can only be rejected if also \(H_{1}^{*}(0)\) and \(H_{1}(0)\) are rejected, and \(H_{1}(1)\) can only be rejected if also \(H_{1}^{*}(0)\), \( H_{1}(0)\), and \(H_{1}^{*}(1)\) are rejected. Testing is terminated and the corresponding hypothesis is not rejected at the first failure to reject the null in the testing sequence. Johansen’s maximum eigenvalue statistic tests the hypotheses: \(null:r=r_{0}\) versus \(alternative:r=r_{0}+1\) for \( r_{0}=0 \), 1, 2, ..., \(p-1\).

    Readers may be interested in cointegration results based on unlogged prices. For 171 (94%) of the 182 cash markets, the results are qualitatively the same as those based on logged prices.

  16. For example, the null hypothesis of the ADF test without (with) trend at the 5% significance level is rejected for 85% (77%) of the basis of the 52 cash markets cointegrated with the futures and only 10% (3%) of that of the 130 cash markets not cointegrated with the futures.

  17. Weekly Midwest No 2 diesel retail prices are obtained from the US Energy Information Administration.

  18. Empirical results based on state-level cash price series in Sect. 6 reveal that adding more cash markets to a model could help identify cointegration.

  19. Indiana is an exception because it 8 cointegrating and 7 non-cointegrating cash markets of the futures.

  20. One exception exists for Iowa and Nebraska (see Table 3) because the ratio of cash markets cointegrated with the futures in Iowa is smaller than that in Nebraska.

  21. The logit model estimated is \(logit(cointegration)=2.5978-0.7259\times \textit{distance}\). The marginal effect of distance on cointegration is \(\frac{{\partial p(cointegration)}}{{\partial {\textit{distance}}}} = \frac{{ - 0.7259{e^{ - 2.5978 + 0.7259 \times {\textit{distance}}}}}}{{{{(1 + {e^{ - 2.5978 + 0.7259 \times {\textit{distance}}}})}^2}}}\), revealing that distance has a negative effect on the probability toward cointegration.

  22. Empirical results based on state-level cash price series in Sect. 6 reveal that adding more cash markets to a model could highlight the price discovery role of the futures market. Future research comparing cointegration and price discovery results for a cash and the futures market with different numbers of other cash markets incorporated into a model is thus of interest.

  23. Coefficients \(\Gamma _{i,12}\)’s and \(\Gamma _{i,21}\)’s associated with cash markets 48, 50, 56, 103, 129, and 130 are not plotted in Fig. 6 . The optimal numbers of lags corresponding to these markets are 3, 4, 3, 4, 4, and 2, respectively, under the VAR in levels representation. \(\Gamma _{1,12}=0.467\), \(\Gamma _{2,12}=0.154\), \(\Gamma _{1,21}=0.0595\), and \(\Gamma _{2,21}=0.0201\) for market 48; \(\Gamma _{1,12}=0.544\), \(\Gamma _{2,12}=0.305\) , \(\Gamma _{3,12}=0.148\), \(\Gamma _{1,21}=0.0941\), \(\Gamma _{2,21}=0.0346\), and \(\Gamma _{3,21}=0.0253\) for market 50; \(\Gamma _{1,12}=-0.00555\), \( \Gamma _{2,12}=0.0426\), \(\Gamma _{1,21}=0.192\), and \(\Gamma _{2,21}=0.0813\) for market 56; \(\Gamma _{1,12}=-0.0175\), \(\Gamma _{2,12}=0.0327\), \(\Gamma _{3,12}=-0.0973\), \(\Gamma _{1,21}=0.337\), \(\Gamma _{2,21}=0.0738\), and \( \Gamma _{3,21}=0.192\) for market 103; \(\Gamma _{1,12}=-0.0992\), \(\Gamma _{2,12}=0.0729\), \(\Gamma _{3,12}=0.211\), \(\Gamma _{1,21}=0.0544\), \(\Gamma _{2,21}=-0.0109\), and \(\Gamma _{3,21}=-0.000210\) for market 129; \(\Gamma _{1,12}=0.165\), and \(\Gamma _{1,21}=0.0937\) for market 130.

  24. See Footnote 10.

  25. Coefficients \(\Gamma _{i,12}\)’s and \(\Gamma _{i,21}\)’s associated with cash markets 133, 144, 145, 147, 148, 158, 168, 176, 179, and 182 are not plotted in Fig. 6. The optimal numbers of lags corresponding to these markets are 4, 2, 2, 3, 3, 2, 2, 3, 2, and 2, respectively, under the VAR in levels representation. \(\Gamma _{1,12}=-0.0637\), \(\Gamma _{2,12}=0.0310\), \( \Gamma _{3,12}=0.0916\), \(\Gamma _{1,21}=0.00901\), \(\Gamma _{2,21}=-0.0665\), and \(\Gamma _{3,21}=0.0794\) for market 133; \(\Gamma _{1,12}=-0.120\), and \( \Gamma _{1,21}=0.0889\) for market 144; \(\Gamma _{1,12}=0.0972\), and \(\Gamma _{1,21}=0.0895\) for market 145; \(\Gamma _{1,12}=0.321\), \(\Gamma _{2,12}=0.149 \), \(\Gamma _{1,21}=0.169\), and \(\Gamma _{2,21}=0.0109\) for market 147; \(\Gamma _{1,12}=0.350\), \(\Gamma _{2,12}=0.145\), \(\Gamma _{1,21}=0.121\), and \(\Gamma _{2,21}=-0.0212\) for market 148; \(\Gamma _{1,12}=0.0313\), and \(\Gamma _{1,21}=0.121\) for market 158; \(\Gamma _{1,12}=-0.247\), and \(\Gamma _{1,21}=0.306\) for market 168; \(\Gamma _{1,12}=0.507\), \(\Gamma _{2,12}=0.209\), \(\Gamma _{1,21}=0.0637\), and \(\Gamma _{2,21}=0.00820\) for market 176; \(\Gamma _{1,12}=-0.103\), and \(\Gamma _{1,21}=0.128\) for market 179; \(\Gamma _{1,12}=-0.0571\), and \(\Gamma _{1,21}=0.167\) for market 182.

  26. Two exceptions are cash markets 148 and 173 for which the information share average finds that the cash market contributes more to price discovery while the common factor weight reveals an opposite result. However, the numerical results provided by the two approaches are almost identical for each of these two markets (see Table 2).

  27. As pointed out by Baillie et al. (2002), this does not indicate that one method is better than another because they differ in the focuses of price discovery measurements and have their own merits (de Jong 2002).

  28. Detailed numerical results are available upon request.

  29. The ADF, PP, and KPSS test with or without trend reveal that the basis is not stationary for Iowa, Indiana, Ohio, Minnesota, Nebraska, and Kansas. For Illinois, the KPSS test without trend shows that its basis is not stationary. The plot of the basis is available upon request.

  30. In the long run, the null hypotheses that \(\alpha _{Futures}=0\), \(\alpha _{IA}=0\), \(\alpha _{IL}=0\), \(\alpha _{IN}=0\), \(\alpha _{OH}=0\), \(\alpha _{MN}=0\), \(\alpha _{NE}=0\), and \(\alpha _{KS}=0\), where \(\alpha _{Futures}\), \(\alpha _{IA}\), \(\alpha _{IL}\), \(\alpha _{IN}\), \(\alpha _{OH}\), \(\alpha _{MN} \), \(\alpha _{NE}\), and \(\alpha _{KS}\) are elements of the loading matrix of the octavariate ECM, are rejected at the 5% significance level, indicating that futures and seven state-level cash series all respond to disturbances in the long-run equilibrium. We also test the null hypotheses that \({\left| \alpha _{F}\right| =\left| \alpha _{IA}\right| }\), \({\left| \alpha _{F}\right| =\left| \alpha _{IL}\right| }\), \({\left| \alpha _{F}\right| =\left| \alpha _{IN}\right| }\), \({\left| \alpha _{F}\right| =\left| \alpha _{OH}\right| }\), \( {\left| \alpha _{F}\right| =\left| \alpha _{MN}\right| }\), \({\left| \alpha _{F}\right| =\left| \alpha _{NE}\right| }\), and \({\left| \alpha _{F}\right| =\left| \alpha _{KS}\right| }\), and they are all rejected at the 5% significance level, indicating that futures and each state-level cash series respond to disturbances at different magnitudes.

  31. As we can see in Table 4, Nebraska and Kansas never have nonlinear causal relationships with the futures market because their observed individual markets are far away from futures delivery locations. However, while observed markets in Iowa and Minnesota also are farther away from futures delivery points than those in Illinois, Indiana, and Ohio, they are much closer to these points as compared to markets in Nebraska and Kansas. Thus, nonlinear relationships can still be detected. This again reveals that the price connection between a cash and the futures market tends to weaken as the distance between the cash and futures delivery point increases.

  32. Log prices are used in the models. For our case, the log transformation stabilizes the variance of the underlying unlogged series and could be beneficial for forecasting (Lükepohl and Xu 2012). Readers are referred to Lükepohl and Xu (2012) for a detailed investigation into conditions under which taking logs is beneficial for forecasting. It is also worth comparing the bivariate with a trivariate, quadvariate, quinvariate, hexavariate, and heptavariate model. However, we need to consider 6, 15, 20, 15, and 6 versions of the trivariate, quadvariate, quinvariate, hexavariate, and heptavariate model, respectively, for each state. For simplicity, we focus on the comparison between the bivariate and octavariate model.

    Xu and Thurman (2015) provided evidence that including more cash series in a VAR model can improve price forecasts of individual cash markets.

  33. 10-, 20-, 40-, 60-, and 90-day correspond to 2-, 4-, 8-, 12- and 18-week ahead forecasts, respectively.

  34. Ohio is not considered because the bivariate model forecasts better for it at h3 and h5. h1 is not considered because the bivariate model forecasts better for all states at this horizon.

  35. Separate from one-quarter ahead forecasts, Colino and Irwin (2010) also provided results for two- and three-quarter ahead forecasts. One-quarter ahead forecasts are selected to illustrate the average RMSE reduction because they represent a horizon that is closest to those considered in the current study.

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  1. Department of Economics, North Carolina State University, Raleigh, NC, 27695, USA

    Xiaojie Xu

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Xu, X. Cointegration and price discovery in US corn cash and futures markets. Empir Econ 55, 1889–1923 (2018). https://doi.org/10.1007/s00181-017-1322-6

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