3.1. Benchmark model

The most current and widely used futures-based forecasting model for cotton was developed by Hoffman and Meyer (Reference Hoffman and Meyer2018).Footnote ⁵ This model uses a 7-year Olympic moving average (by excluding the maximum and minimum for computing the average) of nearby basis to overcome nonstationarity and structural break. Thus, a relationship in Equation 1 is simplified to:

(2) $$\widehat{{\cal S}}_{k,t}=\overline{{\cal B}}_{k,t}+{\cal F}_{h,t,i} \quad t=2000,\ldots, 2023\quad i=1,\ldots, 12\quad k={\rm Jan},\ldots, {\rm Dec}$$

$$\overline{{\cal B}}_{k,t}={1 \over 5}\left[\sum _{j=1}^{7} {\cal B}_{k,(t-12\cdot j)}-\max \left(\sum _{j=1}^{7} {\cal B}_{k,(t-12\cdot j)}\right)-\min \left(\sum _{j=1}^{7} {\cal B}_{k,(t-12\cdot j)}\right)\right]$$

where $\widehat{{\cal S}}$ is the predicted spot price, $\overline{{\cal B}}_{k,t}$ is the predicted basis, calculated as a 7-year Olympic average of nearby basis, $({\cal{B}}_{k,t} = {\cal{S}}_{k,t} - {\cal{F}}_{h,t,0})$ , observed during spot month. This model assumes that β ₀ from Equation 1 can be approximated by a short-term historical basis, and β ₁ = 1. The benefit of this model is that it is simple, easy to use, and relies on readily available information. This model tends to perform well when its assumptions in Equation 1 are not violated, that is, β ₀ (represented by $\overline{{\cal B}}_{k,t}$ ) is stable, and β ₁ is close to 1. However, when the basis deviates from its historical average (as shown in Figure 2) and/or β ₁ diverges from 1, this method becomes less accurate.

3.2. Alternative model 1: Moving average model with basis deviation term

Several previous studies (Etienne et al., Reference Etienne, Farhangdoost, Hoffman and Adam2023; Liu et al., Reference Liu, Wade, Oellermann and Farris1994; Taylor et al., Reference Taylor, Dhuyvetter and Kastens2006; Tomek, Reference Tomek1980; Tonsor et al., Reference Tonsor, Dhuyvetter and Mintert2004, among others) showed that spot price projection can be enhanced by incorporating current market information into the forecasts. For example, Taylor et al. (Reference Taylor, Dhuyvetter and Kastens2006) demonstrated that the basis deviation from its 5-year moving average may be helpful to account for variability left in the residuals from the benchmark model. The basis deviation term is calculated as a difference between the current month’s nearby basis and its historical moving average:

(3) $$BDEV_{k,t}={\cal B}_{k,t}-\overline{{\cal B}}_{k,t}\quad t=2000,\ldots, 2023\quad k={\rm Jan},\ldots, {\rm Dec}$$

where ${\cal{B}}_{k,t}$ is the nearby basis, and $\overline{{\cal B}}_{k,t}$ is the predicted basis, calculated as a 7-year Olympic average of nearby basis observed during spot month (refer to Equation 2). For example, in December 2000, the current basis includes the December 2000 farm price and the futures price of the nearest-to-maturity March 2001 contract, and a 7-year Olympic moving average is calculated as an average of these differences over the last seven observations as per Equation 2.

Our first modification is to include the basis deviation term directly into the benchmark model as follows:

(4) $$\widehat{{\cal S}}_{k,t}={\cal F}_{h,t,i}+\overline{{\cal B}}_{k-2,t}+BDEV_{k-2,t}\quad t=2000,\ldots, 2023\quad i=1,\ldots, 12\quad k={\rm Jan},\ldots, {\rm Dec}$$

To incorporate the most recent market data into the model, in Equation 4, the $\overline{{\cal B}}$ and BDEV terms are lagged for two months. This is done because the USDA reports farm prices with a 2-month lag (i.e., June prices are published in August).

3.3. Alternative model 2: OLS model with nearby contract prices

The next alternative specification utilizes rolling regression analysis to develop a forecast based on Equation 1. The main difference between this specification and the benchmark approach is the use of regression analysis instead of moving averages (similar to Etienne et al., Reference Etienne, Farhangdoost, Hoffman and Adam2023). This relaxes the potentially restrictive assumptions related to the model coefficients, as discussed in Section 3.1.

Another notable difference between this specification and the benchmark model lies in the data used for basis prediction. In the benchmark approach, the measure of basis expectation does not change over various forecasting horizons as it is always calculated as a moving average of nearby basis during the spot month. In our approach, the estimated coefficient of the intercept, β ₀, represents the difference between the farm price and its nearby futures price at a selected horizon. For instance, a forecast of August 2016 spot prices (k = 8, t = 2016) generated in January 2016 uses the October 2016 futures contract prices observed in January 2016 (see Table 1). Here, the estimated coefficient of the intercept, β ₀, will represent the difference between the farm price and its nearby futures price observed 7 months prior (January 2016), when the forecast is made.

To develop spot price forecasts, we begin by estimating Equation 1 with a 5-year rolling window approach using the data starting in January 2000 to December 2004 (n = 60) to compute out-of-sample forecasts for January 2005. This method is similar to Hoffman’s benchmark model technique for addressing the issues of non-stationarity and structural breaks by focusing on the short-time period (discussed in Section 3.1). Hence, our approach of relaxing these assumptions and allowing the coefficients to vary by moving to regression analysis may help improve these forecasts.

3.4. Alternative model 3: OLS model with basis deviation term

Another alternative specification combines the new information given by the basis deviation term and the regression approach in Equation 1, as discussed in the previous section, to capture situations when the basis becomes unstable. This leads to estimating the following regression:

(5) $$\small{{\cal S}_{k,t}=\beta _{0}+\beta _{1}{\cal F}_{h,t,i}+\beta _{2}BDEV_{k-2,t}+\nu _{t,i}\quad t=2000,\ldots, 2023\quad i=1,\ldots, 12\quad k={\rm Jan},\ldots, {\rm Dec}}$$

This model is used to generate out-of-sample forecasts of farm prices using a 5-year rolling window procedure described in Section 3.3.

3.5. SAP forecast

After forecasting monthly cotton prices using the alternative model specifications, we combine those forecasts with actual farm prices received (when available) to generate projections for the SAP. We follow the procedure outlined in Hoffman and Meyer (Reference Hoffman and Meyer2018) to use the monthly forecast of farm prices, the actual realized spot prices, and marketing year weights assigned to each month within the marketing year to generate SAP forecasts (see Table 3).

Table 3.

Season-average price forecasts by forecast months and marketing year months

This table, adapted from Hoffman and Meyer (Reference Hoffman and Meyer2018), presents the forecasting procedure we follow to generate SAP. As discussed in Section 3.5, the SAP forecast for August and September months fully relies on futures prices since the USDA publishes the first observed monthly farm prices with a 2-month lag. Once the actual August 2016 prices become available in October 2016, we generate a composite forecast by combining the observed cash prices with futures-based forecasts for the remaining ten months of the marketing year. From October 2016 onward, as each subsequent month’s actual price is published, we replace the corresponding futures-based forecast with the realized data point, thereby enhancing the accuracy of our SAP projection.

To illustrate this process, consider a scenario where we need to generate the SAP forecast for the 2016/17 marketing year. Our forecasting process begins in August 2016, before any actual prices for that marketing year are available. The USDA publishes the first observed monthly farm price with a 2-month lag. Thus, the August 2016 price will not be released until October 2016. As a result, the initial SAP forecast for 2016/17, generated in August 2016, must rely entirely on futures prices using either the Hoffman or an alternative model specification. The September 2016 forecast will also depend solely on futures prices.

Once the actual August 2016 farm price becomes available in October 2016, we generate a composite forecast by combining this observed farm price with futures-based forecasts for the remaining ten months of the marketing year. From October 2016 onward, as each subsequent month’s actual price is published, we replace the corresponding futures-based forecast with the realized data point, thereby enhancing the accuracy of our SAP projection. This procedure is schematically presented in Table 3.

To obtain a weighted forecast for each month, we use the 7-year Olympic average of marketing weights for that particular month within the marketing year.Footnote ⁶ Then, we multiply the forecasted prices by these assigned weights and compute the SAP by summing the weighted price projections.

4.1. Forecast accuracy and bias

We evaluate the accuracy of out-of-sample forecasts using traditional evaluation criteria described in this section. First, following MacDonald and Isengildina-Massa (Reference MacDonald and Isengildina-Massa2012) and Isengildina-Massa et al. (Reference Isengildina‐Massa, Karali, Kuethe and Katchova2021), we define the prediction error in percentage terms, denoted by pe _{k, t} :

(6) $$pe_{k,t}=(\ln {\cal S}_{k,t}-\ln \widehat{{\cal S}}_{k,t})\,{\rm *}\,100\quad t=2005,\ldots, 2023\quad k={\rm Jan},\ldots, {\rm Dec}$$

where $\ln{\mathcal{S}}$ and $\ln \widehat{{\cal S}}$ are the logarithmic transformations of the actual and predicted spot prices, respectively.

We use the two most common measures of forecast error size – MAPE and root mean squared percentage error (RMSPE) defined as:

(7) $$MAPE={1 \over n}\sum _{t=1}^{n} |pe_{k,t}|\quad t=2005,\ldots, 2023\quad k={\rm Jan},\ldots, {\rm Dec}$$

(8) $$\textit{RMSPE}=\sqrt{{1 \over n}\sum _{t=1}^{n} pe_{k,t}^{2}}\quad t=2005,\ldots, 2023\quad k={\rm Jan},\ldots, {\rm Dec}$$

A significant advantage of using MAPE measure is that it is relatively unaffected by occasional large prediction errors. In contrast, RMSPE is scale-independent, which ensures consistent and reliable performance evaluation across different measurement scales (Hyndman et al., Reference Hyndman2006). When comparing the performance of the alternative model with that of the benchmark model, smaller values of both MAPE and RMSPE measures indicate more accurate and precise projections.

While the MAPE and RMSPE measures describe the magnitude of forecast errors, they do not say much about the presence of bias or systematic errors in the generated price projections. Assuming that forecasters try to minimize a symmetric loss function, we assess the presence and direction of bias, using the MPE criterion:

(9) $$MPE={1 \over n}\sum _{t=1}^{n} pe_{k,t}\quad t=2005,\ldots, 2023\quad k={\rm Jan},\ldots, {\rm Dec}$$

We apply a two-tailed t-test to the MPE to assess the statistical significance of potential forecast biases. The null hypothesis of the t-test is that the average percentage error is not significantly different from 0. Rejection of the null hypothesis will indicate whether the forecasts under- or over-predict the observed spot prices. Negative and significant MPE values will indicate that the model over-predicts the actual price. In contrast, positive and significant MPE measures indicate an under-prediction of spot prices.Footnote ⁷

4.2. Test for differences between benchmark and alternative forecasts

Following previous studies (see Manfredo and Sanders, Reference Manfredo and Sanders2004; Colino et al., Reference Colino, Irwin and Garcia2011, for instance), we test for statistically significant differences in the forecast accuracy of the benchmark and alternative forecasts across different months using the MDM test developed by Harvey et al. (Reference Harvey, Leybourne and Newbold1997):

(10) $${\rm MDM}_{k}={\overline{d} \over \sqrt{{1 \over n}\left(\delta _{0}+2\sum _{q=1}^{t-1} \delta _{q}\right)}}\cdot \sqrt{{n+1-2h+{{h(h-1)}\over{n}} \over n}}$$

$$\delta _{0}={1 \over n}\sum _{t=1}^{n} (d_{t}-\overline{d})^{2};\quad \delta _{q}={1 \over n}\sum _{t=q+1}^{n} (d_{t}-\overline{d})(d_{t-q}-\overline{d})\quad t=2005,\ldots, 2023$$

where $h$ is the forecast horizon, $d_t = |pe_{1,k,t}| - |pe_{2,k,t}|$ , $pe_{1,k,t}$ is the percentage error from the benchmark model, $pe_{2,k,t}$ is the percentage error from the competing/alternative model specification. $\bar{d}$ is the average difference across the forecasts for each forecast period, ${\delta_0}$ is the variance of ${d_t}$ , while ${\delta_q}$ is the ${q}$ -th order auto-covariance term.

The null hypothesis of the MDM test is $H_0: \boldsymbol{\xi} = 0$ , where ${\boldsymbol{\xi}} = {\boldsymbol{\mathbf{E}}}[{\boldsymbol{{\mathbf{V}}}}(\,p{e_{1,k,t}} )-{\boldsymbol{\mathbf{V}}}(\,pe_{2,k,t} )]$ , and $\mathbf{V}(pe_{\cdot,k,t} )={|pe_{\cdot,k,t}|}$ is a loss function when RMSPE is used as a measure of forecast accuracy (Franses, Reference Franses2016; Hyndman et al., Reference Hyndman2006). This tests whether the forecast performance of the benchmark and alternative model specifications are significantly different from each other. Rejection of the null hypothesis indicates that the difference in the size of errors between the two models is significantly different from zero. We also confirm our results with one-tailed tests.

4.3. Test for forecast encompassing

Sometimes forecasts with larger errors may contain incremental information that is missing in the alternative forecasts (Granger and Newbold, Reference Granger and Newbold1973). Harvey et al. (Reference Harvey, Leybourne and Newbold1998) developed the encompassing test to evaluate whether (i) a given forecast incorporates the information content present in another projection, or (ii) the information in the benchmark model outweighs that of the alternative model, or vice versa. In effect, this test operates on the premise that if one forecast fully encompasses the other– the information content of the superior forecast would be so comprehensive that the inferior forecast would add little to no valuable insight to a combined forecast. Essentially, the contribution of the inferior forecast to the combined forecast would be minimal, with its optimal weight approaching zero.

If our forecasts at time t conditional on the full information set ${\Omega_i}$ , $y^{(a)}_{t|\Omega_i}$ , encompasses the benchmark $y^{(b)}_{t|\Omega_i}$ , the difference between both forecasts ought not to be a significant predictor of our forecast errors. This notion yields the regression-based test for estimating the optimal weights for each component of the composite forecast as follows:

(11) $$\matrix{ {\underbrace {{y_t} - y_{t|{{\rm{\Omega }}_i}}^{\left( a \right)}}_{p{e_{2,k,t}}}} \hfill & = \hfill & {\phi (\underbrace {y_{t|{{\rm{\Omega }}_i}}^{\left( b \right)} - y_{t|{{\rm{\Omega }}_i}}^{\left( a \right)}}_{p{e_{2,k,t}} - p{e_{1,k,t}}}) + {\xi _t}} \hfill \cr {} \hfill & \Updownarrow \hfill & {} \hfill \cr \hskip 12pt {{y_t}} \hfill & = \hfill & {\phi y_{t|{{\rm{\Omega }}_i}}^{\left( b \right)} + \left( {1 - \phi } \right)y_{t|{{\rm{\Omega }}_i}}^{\left( a \right)} + {\xi _t}} \hfill \cr } $$

where $pe_{1,k,t}$ is the percentage error from the benchmark model, and ${pe_{2,k,t}}$ is the percentage error from our alternative model specifications.

We estimate the regression in Equation 11 and test the null hypothesis of ${H_0: \phi = 0}$ using the two-tailed t test. The null hypothesis assumes that the benchmark forecast encompasses the competing forecast. Consistent with the second equation in 11, rejection of the null suggests that a combination of the two forecasts (a composite forecast) would yield a more informative forecast and, simultaneously, lower prediction errors. Put simply, the alternative forecast contains incremental information not otherwise present in the benchmark model forecasts. Alternatively, failure to reject the null hypothesis implies that the benchmark model forecast fully encompasses the information content of the alternative model projections and further implies that the combined forecast would be composed entirely of the projection from the benchmark model.

5.1. Bias

The tests of bias, conducted based on the MPE statistic (see Table 4 and Figure 3), show that WASDE forecasts had positive forecast errors indicating a tendency to under-predict cotton prices during November and December (harvest period). On the other hand, Hoffman’s model had negative forecast errors, over-predicting the SAPs in each forecast month. The alternative models proposed in this study provide a significant improvement in terms of bias over Hoffman’s model as Models 1 and 2 produce unbiased forecasts at all forecast horizons, and Model 3 generates unbiased forecasts through March (with under-prediction in April–July, postharvest). Moreover, the magnitude of any bias estimates throughout Table 4 is extremely small, indicating that while there are detectable biases in some forecasts, they are not substantial in absolute terms. This suggests that the improvements achieved by the alternative models, while statistically significant, result in only minor adjustments to the forecast values. Consequently, even the biased forecasts, such as those from Hoffman model, are relatively close to the actual SAPs.

Figure 3.

Mean percentage error (MPE) across SAP forecast months.

Notes: This figure presents the accuracy statistics of the MPE criterion (detailed in Equation 9) across models and horizons. For each forecast month, a two-tailed t-test is conducted to assess bias in MPEs and determine if they significantly differ from zero. Model details are provided in Section 3: Model 1 denotes a moving average approach with a basis deviation term, Model 2 presents a regression model with nearby futures prices, and Model 3 represents a regression model with a basis deviation term.

Table 4.

Mean percentage error (MPE) for season-average price forecast

Note 1: Model 1 represents the moving average approach with basis deviation term.

Note 2: Model 2 represents the regression model with nearby futures prices.

Note 3: Model 3 represents the regression model with basis deviation term.

Note 4: The null hypothesis states that the MPEs are not significantly different from 0.

Note 5: Statistical significance of a two-tailed test is indicated with the asterisk: *p < 0.10, **p < 0.05, ***p < 0.01.

5.2. Accuracy

To evaluate the accuracy of our models with respect to WASDE projections, we rely on the most commonly used accuracy measures of MAPE and RMSPE. Table 5 presents MAPE statistics for each forecast month. Our findings demonstrate that both Hoffman’s and Models 2 and 3 offer accuracy improvements to WASDE forecasts in August and September during growing period, but only Model 2 offers additional accuracy improvements in May–July after harvest. Furthermore, Model 2 outperforms the Hoffman model in 10 out of the 12 forecast month, while the other models provide more modest accuracy improvements. Figure 4 shows the differences in MAPEs between WASDE and alternative forecasts and indicates accuracy improvements where these differences are negative. This figure demonstrates that accuracy improvements in August and September were fairly substantial, while the ones later in the forecasting cycle (May–July) rather minor. This figure may also be interpreted relative to the size of Hoffman’s model forecasts and indicates very substantial improvements in accuracy for all of the proposed models from December through May.

Figure 4.

Mean absolute percentage error (MAPE) differences across SAP forecast months.

Notes: This figure illustrates the differences in MAPE statistics between forecasts from alternative models and WASDE projections across forecast months. Negative (positive) MAPE differences for each forecast month indicate that the alternative model produces lower (higher) MAPE values, indicating more (less) accurate forecasts compared to the WASDE model. Model details from Section 3 are as follows: Model 1 denotes a moving average approach with a basis deviation term, Model 2 presents a regression model with nearby futures prices, and Model 3 represents a regression model with a basis deviation term.

Table 5.

Mean absolute percentage error (MAPE) for season-average price forecast

Values marked with ‡ indicate that a specific model performs better in terms of lower forecast errors compared to the WASDE projections. Values marked with † denote that a specific model shows lower forecast errors compared to the Hoffman model projections. Model 1 represents a moving average approach with a basis deviation term. Model 2 represents a regression model using nearby futures prices. Model 3 denotes a regression model with a basis deviation term.

These findings are largely confirmed in RMSPE analysis which is more sensitive to large errors, as shown in Table 6 and Figure 5). The main difference is that there is no longer evidence of Hoffman’s forecasts being more accurate than WASDE, while the accuracy enhancements from Model 2 are confirmed for August–September and May–June. Using this metric, Model 2 consistently outperforms Hoffman’s model’s accuracy and Models 1 and 3 also offer consistent accuracy improvements. Figure 5 shows the differences in RMSPEs between WASDE and alternative forecasts, indicating an accuracy improvement during the forecast months where these differences are negative. The results are consistent with those from MAPEs. Among the three proposed models, Model 2 stands out by demonstrating the most accuracy improvements compared to WASDE, particularly in August and September, with relatively minor improvements from May to July months.

Table 6.

Root mean squared percentage error (RMSPE) for season-average price forecast

Values marked with ‡ indicate that a specific model performs better in terms of lower forecast errors compared to the WASDE projections. Values marked with † denote that a specific model shows lower forecast errors compared to the Hoffman model projections. To determine whether the improvements are statistically significant, we use a Modified Diebold–Mariano test (refer to section 5.3). Model 1 denotes a moving average approach with a basis deviation term. Model 2 represents a regression model using nearby futures prices. Model 3 denotes a regression model with a basis deviation term.

Figure 5.

Root mean squared percentage error (RMSPE) difference across SAP forecast months.

Notes: This figure illustrates the differences in RMSPE statistics between forecasts from alternative models and WASDE projections across forecast months. Negative (positive) RMSPE differences for each forecast month indicate that the alternative model produces lower (higher) RMSPE values, indicating more (less) accurate forecasts compared to the WASDE model. Detailed in Section 3, Model 1 denotes a moving average approach with a basis deviation term, Model 2 presents a regression model with nearby futures prices, and Model 3 represents a regression model with a basis deviation term.

5.3. MDM test

The MDM test results in Table 7 indicate that WASDE forecasts had significantly lower errors compared to Hoffman’s and our proposed models, particularly during the harvest and postharvest months (starting from November forward). In addition, for the months where our regression approaches outperformed WASDE in terms of lower MAPE and RMSPE (Tables 5 and 6), the MDM tests indicate that these improvements were not statistically significant. However, compared to the Hoffman model, Models 1 and 2 produce significantly lower forecast errors during March–May months, while Model 3 generates lower forecast errors only for the January and February forecast months. Overall, these findings indicate that our proposed model specifications offer significant improvements in forecast accuracy over the Hoffman’s model, demonstrating their potential for more accurate forecasting during certain periods.

Table 7.

Modified Diebold–Mariano test

Note 1: Model 1 represents the moving average approach with basis deviation term.

Note 2: Model 2 represents the regression model with nearby futures prices.

Note 3: Model 3 represents the regression model with basis deviation term.

Note 4: The null hypothesis for the two-tailed Modified Diebold–Mariano test states that the two forecasts have similar forecast accuracy.

Note 5: Statistical significance of a two-tailed test is indicated with the asterisk: *p < 0.10, **p < 0.05, ***p < 0.01.

5.4. Encompassing

The encompassing tests indicate that our proposed models incorporate additional information absent from the WASDE projections and the Hoffman model. Although the MDM test did not show a significant enhancement in forecast accuracy compared to the WASDE benchmark, the encompassing test revealed that our models contained incremental information missing from the WASDE forecasts during months when our models exhibited lower forecast errors. Furthermore, the encompassing test results (see Table 8) demonstrate that Models 2 and 3 encompass additional information absent from the Hoffman model across all forecast months, while Model 1 incorporates incremental information over the Hoffman model primarily during the harvest and postharvest periods, starting from November onward. These findings suggest that our proposed models offer additional information over the Hoffman approach and may result in more accurate combination forecasts.

Overall, our results highlight the superiority of Model 2 over the Hoffman model, demonstrating its ability to significantly enhance forecast accuracy throughout the marketing year and producing more informative price projections. While Model 2 does not outperform the WASDE benchmark, on average, it offers notable improvements in accuracy during certain months.