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Time series models are promising tools for forecasting commodity prices. However, their applications to guide producers in agricultural investments and marketing decisions are still limited. This article compares the ARIMA and Holt-Winters Exponential Smoothing models in terms of forecasting the monthly wholesale rice price in Tanzania. Even with very little difference, the Holt-Winters additive model showed the best results for forecasting rice prices compared to the ARIMA model. Thus, both models can be used to forecast the prices of agricultural products.

One of the main challenges facing smallholder farmers in developing countries is fluctuating prices. Agricultural prices tend to vary and fluctuate rapidly over time, as prices always adapt to various changes [

Tanzania is primarily an agricultural country, and rice being among the main crop produced; it’s associated with risks and uncertainties [

Given the economic importance of rice in Tanzania, which is the second most important crop after maize, the most traded crop than any other food crop, employs over 1.6 million people and is highly produced by small farmers (about 90 percent) [

The data used for empirical analysis are monthly rice prices (quoted in Tanzanian shillings per 100 kg) marketed at wholesale markets in the Mbeya region from January 2004 to September 2019, totaling 189 observations. The data set was obtained from the Ministry of Industry and Trade, Tanzania^{1}. The free software R was used to develop the ARIMA and Holt-Winters Exponential Smoothing models. For the choice of the best model, we reserved the last nine observations of the series for comparison with the predicted values. The descriptive results for the wholesale rice prices including mean, maximum, and standard deviations are presented in

Measure | Values |
---|---|

Mean | 116,371.23 |

Median | 108,541.57 |

Standard Deviation | 47,876.44 |

Coefficient of variation | 0.41 |

Minimum | 44,545.62 |

Maximum | 229,375.35 |

Observations | 189 |

Note: ^{a}Prices are in (1000) Tanzanian shillings.

The use of the ARIMA and Holt-Winters exponential smoothing models to forecast rice prices relied on their ability to predict time series data showing both trend and seasonal variation. The objective is to choose the best model for predicting the prices of agricultural products.

Among the statistical models developed to predict future values based on a historical database is the Autoregressive Integrated and Moving Averages (ARIMA) model. Proposed by Box & Jenkins [

The procedure for searching a stochastic model that represents a time series is an iterative process consisting of three phases: Identification phase, where the condition of stationarity, the behavioral structures such as trend, seasonality, periodicities, and the issues of data correlation (autocorrelation) are verified; Estimation phase through which suitability of the considered model is validated based on estimated parameters; and Diagnosis phase where the model adjustment through residual analysis, statistical tests, and model selection/adequacy criteria is verified [

When a time series has a time-dependent mean and variance, it is not stationary, and it will need to be transformed. The most common transformation is to take successive differences from the original series to a stationary series [

The first difference of Z t is defined by:

Δ Z t = Z t − Z t − 1 (1)

The second is given by:

Δ 2 Z t = Z t − 2 Z t − 1 + Z t − 2 (2)

In typical situations, according to the authors, it will be sufficient to take one or two differences for the series to become stationary. The number of differences needed to make the series stationary is called the order of integration d. The inclusion of the integration order term allows the use of the ARIMA models (p, d, q) given by the equation:

Δ d Z t = ϕ 1 Z t − 1 + ϕ 2 Z t − 2 + ⋯ + ϕ p Z t − p + ε t − θ 1 ε t − 1 − θ 2 ε t − 2 − ⋯ − θ q ε t − q (3)

where ϕ 1 , ⋯ , ϕ p , are the parameters for the autoregressive part and θ 1 , ⋯ , θ 2 , are the moving average, ε t is an error that cannot be estimated from the model, d represents the number of differences applied in the series.

As the main objective of the ARIMA model is to generate future forecast values, the paper uses the Akaike Information Criterion (AIC) proposed by Akaike [

AIC = ln ( σ ^ e 2 ) + 2 ( p + q ) n (4)

where σ ^ e 2 is estimated error variance; n is the sample size, and p, q are the parameter values.

From the above criteria values, it is possible to choose the most suitable model to make predictions h steps ahead; the lower the criteria values, the more suitable the model for making predictions. In addition to the selection criteria, we analyzed the white noise condition of the waste through the Ljung-Box test [

Holt-Winters exponential smoothing model can be divided into a multiplicative model and additive model. The choice of the model depends on the seasonal component of the series.

The model equations are given by:

Z t = ( β 0 + β 1 t ) + S N t + I R t → Additive (5)

Z t = ( β 0 + β 1 t ) × S N t × I R t → Multiplicative (6)

With three smoothing equations (Equations (7) to (9)):

l t = α ( y t / s n t − l ) + ( 1 − α ) ( l t − 1 + b t − 1 ) → Level (7)

b t = γ ( l t − l t − 1 ) + ( 1 − γ ) b t − 1 → Trend (8)

s n t = δ ( y t / l t ) + ( 1 − δ ) s n t − l → Seasonal (9)

where (γ, and δ) are smoothing constants, l is number of seasons in a year, S N t is a seasonal pattern and I R t refers to irregular components.

Before starting a time series analysis, it is necessary to identify the stationarity properties of the data series. The nature of the series (

The autocorrelation (ACF) and partial autocorrelation (PACF) graphs (

In order to identify the best ARIMA model, we tested several competing models.

According to the Akaike information criterion (AIC) [

ARIMA | AIC |
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(0, 1, 3) (2, 1, 0) [ | 1279.418 |

(3, 1, 0) (2, 1, 0) [ | 1281.175 |

(1, 1, 2) (2, 1, 0) [ | 1282.015 |

(0, 1, 0) (2, 1, 0) [ | 1284.557 |

(2, 1, 1) (2, 1, 0) [ | 1284.995 |

(0, 1, 2) (2, 1, 0) [ | 1285.58 |

(1, 1, 0) (2, 1, 0) [ | 1286.215 |

(0, 1, 1) (2, 1, 0) [ | 1286.297 |

(2, 1, 0) (2, 1, 0) [ | 1286.623 |

(1, 1, 1) (2, 1, 0) [ | 1288.122 |

Note: A good model is the one that has minimum AIC among all the other models (Akaike, 1974).

and PACF plots (

After identifying the best fit model, the next step was to forecast rice prices for the next 24 months. The results of the fitted values (180 months) (

To further justify the results, we compared the forecast results of the ARIMA model with nine steps ahead and the actual rice prices for the same period. The results in

The Holt-Winters model falls into two categories, additive and multiplicative. In order to choose the best model, we first find the fitted values for the additive and multiplicative models. In the second step, we compared the fitted values of the models to the original series. The model that best corresponds to the price series studied was then chosen to compare with the ARIMA model. According to

Holt-Winters additive model.

In order to compare the two models, we used the MAPE (Mean Absolute Percent Error) selection criteria. As proposed in the methodology, the last nine observations of the nominal rice price series were reserved for the comparison of the proposed models. The results in

Period | Predicted Price Values | Actual Price values | ||||
---|---|---|---|---|---|---|

Point Forecast | Lo 80 | Hi 80 | Lo 95 | Hi 95 | ||

Jan 2019 | 137.676 | 124.102 | 151.249 | 116.917 | 158.435 | 140.120 |

Feb 2019 | 149.031 | 128.863 | 169.199 | 118.187 | 179.876 | 143.630 |

Mar 2019 | 159.297 | 132.958 | 185.636 | 119.015 | 199.579 | 150.450 |

Apr 2019 | 158.986 | 129.149 | 188.822 | 113.356 | 204.616 | 153.675 |

May 2019 | 153.901 | 120.937 | 186.865 | 103.487 | 204.315 | 145.678 |

Jun 2019 | 140.985 | 105.165 | 176.804 | 86.203 | 195.766 | 133.750 |

Jul 2019 | 142.618 | 104.154 | 181.082 | 83.792 | 201.443 | 130.950 |

Aug 2019 | 138.452 | 97.514 | 179.389 | 75.842 | 201.061 | 140.120 |

Sep 2019 | 144.117 | 100.846 | 187.388 | 77.939 | 210.294 | 143.630 |

Note: Prices are in (1000) Tanzanian shillings.

Period | Holt-Winters (MAPE = 5.3314) | ARIMA (0, 1, 3) (2, 1, 0) [ | Actual price values |
---|---|---|---|

Jan 2019 | 140.393 | 137.675 | 140.120 |

Feb 2019 | 141.033 | 149.031 | 143.630 |

Mar 2019 | 147.561 | 159.297 | 150.450 |

Apr 2019 | 148.129 | 158.986 | 153.675 |

May 2019 | 141.136 | 153.901 | 145.678 |

Jun 2019 | 128.449 | 140.985 | 133.750 |

Jul 2019 | 128.574 | 142.618 | 130.950 |

Aug 2019 | 123.266 | 138.452 | 125.500 |

Sep 2019 | 123.330 | 144.117 | 127.496 |

Note: Prices are in (1,000) Tanzanian shillings.

used to forecast rice prices studied. However, according to the MAP selection criteria, the price values predicted by the Holt-Winters additive model are closer to the original rice prices than those predicted by the ARIMA model (see

In this study, we looked at forecasting agricultural commodity prices using the time series model. The data used are the wholesale rice prices in Tanzania for the period January 2004 to September 2019. The study found that both of the studied models (Holt-Winters and ARIMA) performed well in predicting rice price values. However, in the model comparison, although by very little difference, the Holt-Winters additive model was closer to the actual data. Thus, it is clear from this article that time series can be used to forecast the prices of agricultural products, not only rice but also other crops, thus helping producers, policymakers and other actors to make informed production and marketing decisions.

Furthermore, this study is limited to rice prices in Tanzania. Therefore, similar studies on other agricultural products are also recommended.

YJM designed and analyzed the statistical data for the study. ST contributed to the data collection and analysis. YY supervised the study. All authors have read and approved the final and revised version of the manuscript.

The authors are grateful to everyone who facilitated the study.

The authors declare no conflicts of interest regarding the publication of this paper.

Mgale, Y.J., Yan, Y.X. and Timothy, S. (2021) A Comparative Study of ARIMA and Holt-Winters Exponential Smoothing Models for Rice Price Forecasting in Tanzania. Open Access Library Journal, 8: e7381. https://doi.org/10.4236/oalib.1107381