# Forecasting Cost Risks of Corn and Soybean Crops through Monte Carlo Simulation
## 2 Materials and Methods
### Monte Carlo Simulation
#### 2.5.2 Autoregressive Integrated Moving Average (ARIMA)  
ARIMA is used for forecasting time series data that may have a trend but no seasonal component. It combines autoregressive (AR) terms, differencing (I) to make the series stationary, and moving average (MA) terms to model the time series--Equation (3). \[\phi(B)(1-B)^{d}y_{t}=\theta(B)\in^{t} \tag{3}\]  
where \(\Phi(B)\) are polynomial operators for the AR and MA components, respectively, d is the differencing order, and \(e^{t}\) is the white noise error term. 2.5.3. Damped Trend Non-Seasonal (DTN-S)  
DTN-S is used for time series data with a trend but no seasonal pattern, where the trend is expected to diminish over time. It extends the [PERSON]'s linear trend model by introducing a damping parameter to reduce the influence of the trend over time--Equation (4). \[\hat{y}_{t+k}=l_{t}+\phi t\text{-}b_{t} \tag{4}\]  
where \(l_{t}\) is the level, \(b_{t}\) is the trend, \(\phi\) is the damping parameter, and \(t\) is the time period. 2.5.4. Double Moving Average (DMA)  
DMA is used to smooth out time series data by averaging data points over a specified period, and then applying another moving average to the result. It involves applying a simple moving average twice, first to the raw data and then to the smoothed data--Equations (5) and (6). \[SMA_{1}(t)=\frac{1}{m}\sum_{i=t-m+1}^{t}y_{i} \tag{5}\]  
\[SMA_{2}(t)=\frac{1}{m}\sum_{i=t-m+1}^{t}SMA_{1}(i) \tag{6}\]  
where SMA\({}_{1}\)(t) is the first moving average, SMA\({}_{2}\)(t) is the second moving average, and \(m\) is the period length. 2.5.5. Non-Seasonal Smoothed Trend (TANS)  
TANS is used for time series data with a smoothed trend but no seasonal component. It involves smoothing the time series to account for trends while ignoring seasonality. This can vary, but typically involves smoothing techniques that adjust for trend while ignoring seasonal fluctuations, such as a moving average or smoothing spline. 2.5.6. Double Exponential Smoothing (DES)  
DES is used for time series data with a trend but no seasonality. It extends SES by adding a component to model the trend in the data. It provides forecasts by combining the level and trend components--Equations (7)-(9). \[\hat{y}_{t+1}=l_{t}+b_{t} \tag{7}\]  
\[l_{t}=\alpha y_{t}+(1-\alpha)(l_{t-1}+b_{t-1}) \tag{8}\]  
\[b_{t}=B(l_{t}-l_{t-1})+(1-\beta)b_{t-1} \tag{9}\]  
where \(\alpha\) is the smoothing parameter for the level, \(\beta\) is the smoothing parameter for the trend, \(l_{t}\) is the level at time \(t\), and \(b_{t}\) is the trend at time \(t\). 3. Analysis and Discussion of Results  
3.1. Descriptive Analysis  
The results of the descriptive analysis (Table 1) revealed a significant variability in the total range of all the variables chosen for this study. This fact can be explained by the increase in prices of inputs used in the production of corn and soybeans from 2018 to 2022. The 59% appreciation of the dollar is related to the rise in prices of fertilizers, soil amendments, and agricultural pesticides during the studied period [34; 35; 36; 37; 38], as the dollar is the currency used in international transactions, including the trade of agricultural inputs imported by farmers. In addition to these factors, uncertainties caused by the COVID-19 pandemic have led to imbalances between supply and demand of inputs in the global market, which have influenced significant fluctuations in prices of fertilizers, herbicides, fuel, and other inputs analyzed in this study [39]. The uncertainties stemming from the pandemic have affected all sectors of agribusiness, from production to distribution and marketing, resulting in negative economic impacts across all segments, particularly in input costs [4; 40]. In contrast, the increase in the value of the dollar also stimulated the exports of corn and soybeans, thereby pushing the domestic supply and further raising prices.