5.1. Background on cointegration

Achieving our analytical purpose would first require estimated price impacts of the event using a highly periodic (here monthly) time-series econometric model of the four non-durum U.S. wheat class prices (see Figure 1), because they are substitutes in production and consumption. For reasons discussed in the following three paragraphs and in Sections 5.2 and 5.3, we chose the modeling approach known as the cointegrated VAR or VEC model approach to model the four U.S. wheat class prices in order to capture the long-run forces (cointegrating relationships or cointegrating vectors [CVs]) that drive the individually nonstationary prices to move in a general tandem pattern.

It is well known that economic variables often fail to achieve conditions of weak stationarity required of valid inference.Footnote ¹ Prior to the work by Engle and Granger (Reference Engle and Granger1987) on cointegration, econometric analysts often first-differenced nonstationary variables that were used in modeled estimations.Footnote ²

In cases such as ours, with four individually nonstationary prices, Johansen and Juselius (Reference Johansen and Juselius1990), following Engle and Granger (Reference Engle and Granger1987), demonstrated that individually nonstationary series may form a group of interrelated or cointegrated variables that behave as a group in a stationary manner over time. As a result, the traditional remedy of differencing such individually nonstationary variables would jettison important long-run components (i.e., cointegrating relationships or CVs) that are crucial in explaining long-run behavior. So although differencing such cointegrated variables would achieve stationarity of the modeled time series, estimates from such differenced data would likely incur misspecification bias from omitted relevant information, the CVs. It is well-known that the cointegrated VAR approach permits the retention of levels-based CVs in a reduced-rank “error correction” space, so as to achieve stationarity requirements (and noncompromised inference) while avoiding difference-induced misspecification bias (Engle and Granger, Reference Engle and Granger1987; Johansen and Juselius, Reference Johansen and Juselius1990).

We first achieve a statistically adequate underlying (unrestricted) model. Second, we test for and impose cointegration restrictions on the four U.S. wheat class prices. We then focus on and interpret the cointegration parameter estimates necessary to formulate a methodology that estimates GE event-induced declines in prices and receipts at the national and plaintiff-specific levels for an illustrative class of U.S. wheat—namely, spring red hard wheat.

5.2. Statistically adequate underlying model and Haavelmo’s ceteris paribus conditions

It is well-known that an unrestricted VAR model of the noted wheat prices in (here logged) levels posits each U.S. wheat class price as a function of k (here k = 6) lags of itself and of k = 6 lags of the three remaining prices.Footnote ³ Johansen and Juselius (Reference Johansen and Juselius1990) demonstrated that such a levels VAR model may be rewritten into an algebraically equivalent unrestricted VEC model:

$$\Delta x\left( t \right){\rm{ }} = {\rm{ }}\Gamma \left( {\rm{1}} \right){\rm{ }} \times {\rm{ }}\Delta x\left( {t - {\rm{ 1}}} \right){\rm{ }} + {\rm{ }} \ldots {\rm{ }} + {\rm{ }}\Gamma \left( {k-{\rm{ 1}}} \right){\rm{ }} \times {\rm{ }}\Delta x\left( {t - k + {\rm{ 1}}} \right){\rm{ }} + {\rm{ }}\Pi {\rm{ }} \times x\left( {t - {\rm{ 1}}} \right){\rm{ }} + {\rm{ }}\Phi D\left( t \right){\rm{ }} + \varepsilon \left( t \right).$$ (1)

Supported by diagnostic evidence reported in the remainder of this section, we specified the model with linear trends in the variables and the cointegrated relationships. The number of endogenous variables, denoted as p, is 4. The ϵ(t) terms are white noise residuals, the delta is the difference operator, and x(t) and x(t − 1) are p by 1 vectors of the endogenous variables in current and lagged levels. The Γ(1), …, Γ(k − 1) terms are p by p matrices of short-run regression coefficients, and Π is a p by p long-run error correction term to account for endogenous levels. The other terms are collectively referred to as the model’s short-run/deterministic or “short-run” component. The ϕD(t) corresponds to a set of deterministic variables, including an array of binary variables that will be added to address stationarity and specification issues discussed in the remainder of this section. From Engle and Granger (Reference Engle and Granger1987) and Johansen and Juselius (Reference Johansen and Juselius1990), the error correction term is decomposed as follows:

$$\Pi {\rm{ }} = \alpha \times \beta\hskip0.75pt' .$$ (2)

The α is a p by r matrix of adjustment coefficients or “alphas” where r is the number of cointegrating relationships or the rank of Π discussed in Section 5.3. The β is a p by r vector of cointegrating parameters or “betas.” Π captures level-based long-run information: stationary linear combinations of the nonstationary levels data under cointegration, permanent shift binaries to capture long-run impacts of policies and/or market events, and a linear trend.

We followed the recommended methods of Juselius (Reference Juselius2006) to include several sorts of binary variables to address an array of stationarity and specification issues. A set of 11 binary variables was included to account for seasonality of U.S. wheat crop prices. In addition, we included the following “permanent shift” binary variables in levels within Π and in differenced form within the short-run component of equation (1):

• RECESS in order to account for the wheat market effects of the U.S. economic recession during December 2007 through June 2009.

• FB2008 in order to account for the U.S. wheat market impacts of the 2008 U.S. farm bill.

• GEEVENT in order to capture the immediate influences on the four endogenous wheat class prices of such an event as the GE event that occurred in May 2013. Valued at unity for June 2013 and July 2013, and at zero otherwise, this binary variable accounts for the event-induced market effects such as cancellation of foreign import orders for U.S. wheat, coincidental wheat futures market price declines, among other things, reported during the immediate months following the event’s May 2013 occurrence by Bjerga (Reference Bjerga2013), Boone (Reference Boone2013), Klein (Reference Klein2013), Jalonick (Reference Jalonick2013), and Thukral and Maeda (Reference Thukral and Maeda2013). Although the U.S. wheat contamination was discovered in April, we opted to not to include April in GEEVENT’s definition because we located no evidence of a news leak before the USDA’s official announcement on May 29, 2013. Because the USDA’s announcement of the GEEVENT event was on May 29, 2013, with most of the month (all but two full days) having already passed, and with little or no possible GE/RR event influence on the USDA’s estimates of the May 2013 wheat class prices, we chose to begin GEEVENT with June 2013 (see USDA-APHIS, 2018) and exclude May 2013 that some may have preferred to have been included. This GEEVENT binary is of particular use in our study because in addition to being consistent with the national contamination-induced effect estimates in tier 1, the binary’s definition is also consistent with tier-2 effect estimates because farmer plaintiffs who filed in the actual court case did so for marketings of spring red hard wheat sold over the June–July 2013 period.

Moreover, following Juselius (Reference Juselius2006), other “outlier” binary variables were included in the short-run component of equation (1). These binary variables were designed to account for “extraordinary” and exogenous wheat market influences attributed to short-run events not of direct relevance to the model being estimated.Footnote ⁴ Juselius (Reference Juselius2006) recommends this binary variable-based procedure, which focuses on achievement of Haavelmo’s (Reference Haavelmo1944) well-known conditions that use “statistical devices” that clear the data of extraneous exogenous effects not of direct theoretical relevance to the modeled market relationships. These binary variables are included only in the model’s short-run component and not in the long-run cointegration space. Consequently, we follow Clements, Tidwell, and Jin (Reference Clements, Tidwell and Jin2017) and, aside from indicating from which exogenous events the outlier binary variables may arise, we do not analyze or discuss these binary variables at length.

Following Juselius (Reference Juselius2006), when a binary variable was identified, the variable or variable group was included in the appropriate parts of equation (1), equation (1) was reestimated, and the binary variable was retained if a battery of diagnostics suggested enhanced specification. We do not focus on the econometric output that is not directly related to our analytical goal of developing a methodology to assess damages of such an event as the May 2013 GE/RR event. However, as exhibited in Table 1, we demonstrate that we achieved a statistically adequate underlying model.

Table 1. Misspecification tests for the unrestricted model before and after specification enhancement efforts

Notes: Kurtosis values are in parentheses. ARCH, autoregressive conditional heteroscedasticity; D-H, Doornik-Hansen; LM, Lagrangian multiplier; PSPRRDHD, price of spring red hard wheat; PWINRDHD, price of winter red hard wheat; PWINRDSF, price of winter red soft wheat; PWINWHSF, price of winter white soft wheat.

Table 1 permits a comparative analysis of evidence from a battery of diagnostics generated by the first and last estimations of an unrestricted VEC (equation 1) in our quest for enhanced specification. Prior to specification enhancement efforts, evidence suggested issues with nonnormal residual behavior, particularly at the univariate level for all four endogenous prices. After a series of reestimations with statistically supported binary variables discussed previously, the underlying unrestricted model saw the trace correlation, a goodness-of-fit indicator, nearly double to 0.626; evidence suggested that serial correlation and heteroscedasticity were not issues; and indicators of skewness and kurtosis improved and ultimately fell within accepted levels. From this information, we conclude that we met literature-established evidential standards that permit us to conclude that we achieved a statistically adequate underlying unrestricted model for which cointegration properties may be tested and exploited. The diagnostic results exhibited in Table 1 not only render us confident that inclusion of the various binary variables noted previously was accomplished in a properly implemented procedure as described in Juselius (Reference Juselius2006) but also that we avoided well-known econometric consequences that potentially arise from excessive use and/or misspecification of such binary variables.

5.3. Testing for cointegration, imposing appropriate rank of Π, and testing for time invariance of parameter estimates

Equation (2) shows that Π is a p x p (here 4 by 4) matrix equal to the product of two p by r matrices, where r is the reduced rank of Π and represents the number of cointegrating relations or CVs that error correct the system of our four individually nonstationary prices. Testing for the appropriate r is a test for cointegration. Under cointegration, the rank of β’ × x(t − 1) is reduced from p = 4 (i.e., r < p), and r may range from 1 to 3. The literature has widely employed Johansen and Juselius’s (Reference Johansen and Juselius1990) nested trace tests to ascertain Π’s reduced rank or r. Table 2 provides nested trace test statistics and test results. Evidence at the 5% significance level was sufficient to reject the first null hypothesis that r ≤ 0 and was insufficient to reject the second (that r ≤ 1), as well as the third and fourth, null hypotheses. These nested results suggest that Π’s reduced rank is r = 1 and that a single CV corrects the system. A reduced rank of r = 1 was then imposed on Π.

Table 2. Nested trace test statistics and test results

Note: At the 0.05 level of significance.

Source: CATS2 routines developed by Dennis (Reference Dennis2006) in RATS (Version 8.2) of Estima.

As instructed by Juselius (Reference Juselius2006, chap. 9), one should test for the time invariance (or constancy) of estimated parameters, and she offers the “known-beta” test programmed in Estima (2012) to do so. The known-beta procedure (see Juselius Reference Juselius2006, chap. 9) formally tests the null hypothesis that parameter estimates of the cointegrated model (with the imposed restriction of Π having the rank of here r = 1), including GEEVENT defined for the 2013 U.S. wheat contamination, are constant or time invariant over the entire sample including the sample’s June–December 2013 subperiod after the time of the contamination’s occurrence. Following Juselius (Reference Juselius2006, chap. 9) and Babula et al.’s (Reference Babula, Newman and Rogowsky2006, p. 47) application of the test to a cointegrated VAR analysis of monthly prices of U.S. sugar–related products, test values (normalized by a 5% critical value and asymptotically distributed as a chi-squared distribution) were all less than unity. This finding in turn suggests that the parameter estimates of our model (including the GEEVENT estimate) are time invariant and, hence, are valid over the entire sample period. Evidence and analysis in the preceding paragraph and in Section 5.4 suggests that although the contamination event modeled by GEEVENT did have an effect, the contamination event did not elicit time variance of coefficients (did not elicit a structural break), and the estimates imputed from GEEVENT are valid. In other words, formal statistical evidence suggests that for the 2013 wheat contamination event, a structural break or parameter time variance was not an issue.

This statistical evidence that the 2013 wheat contamination of focus did not elicit a structural break is the principal difference between this study and the Carter and Smith (Reference Carter and Smith2007) and Li et al. (Reference Li, Wailes, McKenzie and Thomsen2010) inquiries on the U.S. price effects of the U.S. corn and rice contaminations, respectively. Ours is the only one of the three studies where formal statistical analysis (known-beta test results) suggested that there was not a structural break. So given the time invariance and valid interpretation of the shifter or the binary GEEVENT’s coefficient over the whole sample, one is not compelled to split the full sample, as did Carter and Smith (Reference Carter and Smith2007) and Li et al. (Reference Li, Wailes, McKenzie and Thomsen2010), into pre- and postcontamination subsamples so as to ultimately estimate contamination-induced price effects.

The time-invariant GEEVENT coefficient is valid over the full sample (before and after the 2013 contamination event), and being defined for the contamination event per se, the GEEVENT coefficient estimate may be used as discussed in Section 5.4 to discern empirically the contamination-induced price effects (Juselius Reference Juselius2006; Patterson Reference Patterson2000, p. 195). In our study’s case without a contamination-induced structural break, and under the important caveat that our four modeled U.S. wheat price variables provide a statistically adequate reduced-form cointegration model as our specification discussion in Section 5.2 shows that we have done, one should not split the sample into pre- and postcontamination subsamples. In addition, the importance of finding a nonwheat or non–U.S. wheat substitute price as a noncontaminated “control” as Carter and Smith (Reference Carter and Smith2007) found for corn in sorghum price is not as crucial or necessary. Given a statistically adequate cointegrated VAR model of the four substitute wheat prices, and given our finding (Table 2) of one cointegrated relationship normalized on spring red hard wheat price, then the GEEVENT estimate likely ferrets out the contamination-induced effects on U.S. spring red hard wheat price without needing to throw in a control variable from an outside market. Finally, Carter and Smith’s (Reference Carter and Smith2007) recommended modeling strategy of using a cointegrated VAR or VEC model with a shifter must have been for cases such as ours without structural breaks over the entire sample; otherwise the recommended shifter’s coefficient estimate would not have been constant (i.e., would have been time variant) and not valid for discerning contamination-induced effects.

5.4. An estimate of GE event–induced price declines on spring red hard wheat

Following Johansen and Juselius (Reference Johansen and Juselius1990) and Juselius (Reference Juselius2006), we normalized the cointegrating relationship on the price of spring red hard wheat (PSPRRDHD) and subsequently conducted a series of hypothesis tests concerning model specification. We imposed the restrictions that were accepted by the hypothesis tests and then reestimated the single CV normalized on PSPRRDHD.Footnote ⁵ The coefficient on the GEEVENT binary variable was the part of the CV of direct relevance to our previously discussed analytical purpose.

When the CV is normalized on PSPRRDHD, and because the estimations were done in double-log format, then one may invoke Halvorsen and Palmquist’s (Reference Halvorsen and Palmquist1980) well-known convention of interpreting the GEEVENT coefficient estimate of −0.03909159. Following this convention, an estimate arises of −3.83% for the GE event–induced reduction in the price of spring red hard wheat. More specifically, we raised e, the base of the natural logarithm, to the power of the econometric GE-event binary coefficient estimate (–0.03909159) and subtracted 1 to get −0.03833787. The coefficient suggests that on average, the price of spring red hard wheat was about 3.83% lower during the 2 months following occurrence of the May 2013 GE event (the GEEVENT binary’s period of definition) than during other parts of the sample period. This reduction in price is consistent with findings of a contamination-elicited price decline for the commodities of focus in Carter and Smith (Reference Carter and Smith2007) for corn and in Li et al. (Reference Li, Wailes, McKenzie and Thomsen2010) for rice.

5.5. Estimated national effects on price and receipts from the May 2013 GE event

Table 3 provides a method to calculate the national effects attributed to the May 2013 GE event on price and receipts of U.S. spring red hard wheat farmers. These effects demonstrate the calculation procedures employed in late 2013/early 2014 when actual court cases were filed in response to the noted event. As such, the procedures use the most reasonable assumptions and the most accurate projections of price reductions, spring red hard wheat production, and crop percentages sold on the spot market as were available by economists and attorneys who participated in the noted class action cases.

Table 3. Estimates of national damages to U.S. spring red hard wheat farmers

Notes: Columns A and B are based on U.S. Department of Agriculture projections that were available during the noted litigation in late 2013 (U.S. Department of Agriculture, Economic Research Service, 2013). Column C is based on assumptions made by the authors. Column D is based on calculations using columns A, B, and C and the price reduction estimate. The estimated coefficient on the GEEVENT binary is −0.03909159. When the base of the natural logarithm is raised to the power of this coefficient estimate, and 1.0 is subtracted from it, and this result is multiplied by 100, one obtains the econometric price reduction estimate of −3.8337374% or −0.0383 reported in the table. So consider the −$98.28 million loss of receipts in column D under the assumption that 75% of the U.S. spring red hard wheat was marketed on the spot market. This estimate of lost receipts is calculated as the $6.97 price times −0.038337374 times 490.39 million bushels times 0.75. Column F is calculated as $6.97 times −0.038337374. PSPRRDHD, price of spring red hard wheat.

Column A provides the projected price of spring red hard wheat, PSPRRDHD, the average of the USDA’s monthly PSPRRDHD values for the June–December 2013 period immediately following the May 2013 GE event (USDA-NASS, 2018).Footnote ⁶ Column B provides the USDA’s projected June 2013–May 2014 spring red hard wheat production (490.39 million bushels) as reported in the Wheat Outlook report that had most recently been published in the immediate aftermath of the May 2013 event and available to litigators in late 2013 (see USDA-ERS, 2013). The impact estimates on national spring red hard wheat receipts are therefore conditional estimates and hinge on assumed proportions of the postevent June 2013/May 2014 market year crop that would have been sold on the spot market immediately after the event. Generally unknown, we provide four alternative assumptions for these proportions in column C.

Column D provides projected declines in U.S. receipts for spring red hard wheat attributed to the May 2013 GE event. The econometrically estimated event-induced decline in PSPRRDHD, −3.83% is in column E. As a result, column D provides the projected damages, in terms of decreased receipts to U.S. spring red hard wheat under four alternative assumptions of percentages of the crop sold onto the spot market in late 2013/early 2014.Footnote ⁷ Column F is the estimated 3.83% GE event–induced price reduction estimate applied to $6.97 projected price in dollars per bushel.

In summary, using our methodology as exhibited in Table 3 would generate estimated damages from the May 2013 GE event to U.S. spring red hard wheat farmers of lost receipts that ranged from $32.76 million to $131.04 million from a GE event–induced drop in PSPRRDHD of 3.83% or $0.27 per bushel. Because of space considerations, and given our goal of methodology development, we provide such estimates in Table 3 only for spring red hard wheat and remind readers that analogous tables may be calculated for the other three noted U.S. wheat classes.

5.6. Estimated effects on a hypothetical plaintiff farmer’s price and receipts from the May 2013 GE event

Table 4 provides how the econometrically estimated drop in price, −3.83%, may be used to estimate plaintiff-farm-specific declines in spring red hard wheat price and in receipts from the May 2013 GE event. Given the confidentiality of farm-specific financial records that may not be published herein, the data exhibited in Table 4 are based on records of a hypothetical plaintiff farmer’s wheat transactions during June 2013, soon after the event. In conferring with extension economists who possess actual records, these hypothetical records were deemed reasonable. The methodology suggests that occurrence of the May 2013 GE event resulted in a reduction of $0.31 per bushel price received from $8.04 to $7.73. The actual price received of $7.73 per bushel (column 3) is adjusted into a higher (hypothetical) price that would have occurred had the GE event of May 2013 not taken place—$8.04 per bushel (column 4). This latter hypothetical price of $8.04 per bushel was obtained by dividing $7.73 by the econometrically estimated price decline factor of 0.961662626 that in turn is the base of the natural logarithm, e, raised to the power of the previously noted GEEVENT coefficient estimate (−0.03909159).Footnote ⁸

Table 4. Estimation of hypothetical damages to a spring red hard wheat farmer

Further, this article provides a contribution by taking a national price effect estimate from modeling strategies similar to Carter and Smith’s (Reference Carter and Smith2007) and Li et al.’s (Reference Li, Wailes, McKenzie and Thomsen2010) general modeling strategies and then translating them, for the first time in the literature, to farm-specific plaintiffs for use in litigation. Actual and projected receipts are the volume sold multiplied by the actual and projected prices received in columns 5 and 6 (Table 4), respectively. In turn, the negative difference in the actual and projected receipts represents the damages in terms of lost farm receipts attributed to the May 2013 GE event in column 7 (Table 4). In this hypothetical plaintiff’s case, our methodology estimates that $4,807 in lost receipts was attributed to the May 2013 GE event.

As insightfully noted by an anonymous reviewer, there are other factors not accounted for that could increase or decrease the adverse economic effects of the 2013 GE/RR contamination of U.S. wheat supplies. These other factors include various transaction costs associated with “cleaning” the wheat in the U.S. supply chain, costs of testing for the presence of a GM contaminant, and restructuring of export shipments from countries that imposed the ban and those countries that did not.