International Journal of Advance Study and Research Work (2581-5997)/ Volume 2/Issue 7/July 2019

A Mathematical Model for Estimating Retail Price Movements of Basic Fruit and Vegetable Commodities Using Time Series Analysis Villaren M. Vibas &Avelina R. Raqueño Graduate School, Centro Escolar University, Manila, Philippines villarenvibas@yahoo.com DOI: 10.5281/zenodo.3333529 Abstract Prices of basic agricultural commodities in the market truly concern the entire populace in a region or country. They directly affect the consumers, farmers, traders, entrepreneurs, and even the government and policymakers. Developing a mathematical model in relation to the retail price movements of these basic agricultural commodities could possibly help every concerned individual with regard to economic matters as well as in planning the future. Specifically, the study included basic commodities such as fruits (banana and mango) and vegetables (tomato, cabbage, and pechay) in the National Capital Region (NCR) of the Philippines. The data were obtained from the Philippine Statistics Authority (PSA) in coverage of ten (10) years, from 2009 – 2018, while the time series modeling techniques used, were the ARIMA, SARIMA, and ARIMAx. After undertaking proper procedures and processes in developing the model, it was found that each of the commodities investigated in the study showed an increasing trend of monthly prices for a ten-year period (2009-2018). In terms of estimating the monthly retail prices of fruit commodities, ARIMAx (5, 2, 2, x=mango) emerged as the finest model for banana and ARIMAX (2, 2, 1, x=banana) for mango. For vegetable commodities, the best model to use for estimating monthly prices of cabbage was ARIMAX (3,2,1,x=pechay), SARIMA (1,1,1)(1,1,1) 12 for pechay and SARIMA (2,1,1)(2,1,1) 12 for tomatoes. Keywords: Price Movements, Forecasting, Time Series Analysis, Mathematical model, Agricultural commodities.

Introduction Agricultural commodities are important sources of export earnings and an overall macroeconomics performance is greatly affected by commodity price movements. Commodity price forecasts are vital in economic policy and formulation. Among those who played great roles are the individual farmers, the consumers as well as public agencies. An individual farmer needs output prices to determine the volume of sales and proper pacing to have an optimal income from his farm production. Formulation of his investment plans and decisions on his enterprises will be widened through his knowledge of price trends. The price information is also beneficial to common people as consumers. It affects directly the income of consumers because a lot of it influence their expenditures as well as access to food as net buyers. Likewise, public agencies utilize price data in planning agricultural programs and ensuring that the allocation of available resources to different uses is consistent with the price system (Nagaraja, G.N., & Shruthi, 2015). In the Philippine setting, the Department of Agriculture (DA) emphasized the role of the commodity prices especially in the economy as they are primarily determined by local market and policies. The issue of unexpected price rise of agricultural commodities such as in the case of fruits and vegetables has become a major concern that led to several actions and initiatives undertaken by the Philippine government to counter abrupt price changes in these basic agricultural commodities. However, a strong indication of unexpected price changes in fruit and vegetable commodities in the local markets are still evident (The Philippine Agricultural Economy, 2017). It was, therefore, an attempt of this investigation to develop the forecasting model on basic fruits and vegetable commodity retail price movements. It was hoped further by the researcher that an appropriate mathematical forecasting model would help the farmers, consumers, suppliers, traders as well as the policymakers and the government in taking appropriate future decisions in connection with the retail prices of these basic agricultural commodities. For the purpose of the study, the researcher selected the National Capital Region (NCR) of the Philippines as its setting. The basic agricultural commodities include fruits (banana and mango) and vegetables (tomato, cabbage, and pechay). Particularly, the data on the retail prices of these agricultural commodities were obtained from the Philippine Statistics Authority (PSA) in a ten-(10) year coverage - from 2009 – 2018.

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International Journal of Advance Study and Research Work (2581-5997)/ Volume 2/Issue 7/July 2019

Methods Data Source and Collection The study was conducted to identify and develop a mathematical model for the retail price movements of basic fruit and vegetable commodities in the National Capital Region (NCR) of the Philippines. Specifically, the retail prices of the basic fruit and vegetable commodities in the local markets of the said region were utilized, which include banana, mango, tomato, cabbage, and pechay). These are the basic local fruit and vegetable commodities where the residents of NCR, Philippines bought and consume on a daily basis. The ten–year records (2009 – 2018) of the retail prices of the said commodities were taken from the Philippine Statistics Authority (PSA) and the Philippine Department of Agriculture (DA). This includes secondary data from the agencies office as well as from their website database after proper coordination and permission were made. Time - Series Techniques The study made use of three-time series techniques such as Auto-Regressive Integrated Moving Average (ARIMA), Seasonal Auto-Regressive Integrated Moving Average (SARIMA) and Auto Regressive Integrated Moving Average-Exogenous Variable (ARIMAx). They were employed in developing mathematical models for the monthly prices of basic fruit and vegetable commodities in the National Capital Region of the Philippines from 2009-2018. ARIMA has the ability to represent several varieties of time series and is more useful to linear time series data as well as stationary and non-stationary types of data. On the other hand, Seasonal ARIMA (SARIMA) is a time series techniques that deal with seasonality. It is an extended version of ARIMA in which seasonal differencing of appropriate order is used to remove non-stationarity from the time series involved. Lastly, ARIMAX is another variation of ARIMA aside from SARIMA. It extends ARIMA models through the inclusion of exogenous variables. This modeling technique is usually used as a support to ARIMA or a comparison to other forecasting models (Hamzacebi, 2008; and Lombardo, 2000). Procedure in Model Development The main data analyzed for the development of the models were the monthly retail prices of fruits (banana and mango) and vegetables (cabbage, tomato, and pechay) in the markets of the National Capital Region of the Philippines from 2009 – 2018. It was divided into two main divisions in which half of it represented the training set (2009-2013) and the other half was utilized as a test set (2014-2018). The data from the training set will be used in the initial development of the mathematical models while the data from the test set will be employed for testing the accuracy of the models. In addition, the data from the training set was subjected to ADF test for stationarity while Ljung-Box test and Jarque-Bera test for the residuals independence and normality. Statistical software such as R-program and Python-program were used in the construction of the models in each commodity. The models developed using the training set were then subjected to forecasting accuracy tests (RMSE, MAPE, MPE, and MASE) using the data from the test set. After completing the appropriate procedures, a more appropriate mathematical model for estimating the retail price of each commodity was achieved. The step by step procedure is done in model development is shown in the research paradigm below.

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International Journal of Advance Study and Research Work (2581-5997)/ Volume 2/Issue 7/July 2019

Statistical Tools The study utilized the following statistical techniques and software in the development of the model in each commodity. A. R-Program Statistical Software and Python Statistical Software. These two main statistical software were used in the construction and development of the model in each commodity as well as their corresponding components and equations. B. Augmented Dickey-Fuller Test (ADF). This was utilized to accommodate more complex models with unknown orders. It uses statistic as a negative number, and the more negative it is, the stronger the rejection of the hypothesis that there is in the unit root. C. Ljung-Box Test. This was employed to determine whether the residuals are independent by comparing the p-value to the significance level for each chi-square statistic D. Jarque Bera Test. It was used to determine if the residuals are normally distributed. E. RMSE Accuracy Test. It pertains to the root measure of the average squared deviation of forecasted values. It was employed to give an overall idea of the error occurred during the forecasting F. MAPE Accuracy Test. It is a measure of accuracy used to compare forecast performance between different data series. It usually expresses accuracy as a percentage of average absolute error occurred G. MPE Accuracy Test. This measure was used to represent the percentage of average error occurred while forecasting. It also possessed the properties of MAPE except it shows the direction of error occurred H. MASE Accuracy Test. This is a scale-free error metric that was used to compare forecast methods on a single series and also to compare forecast accuracy between a given series Results and Discussion The results of the study revealed an increasing trend in the monthly prices of the basic fruit and vegetable commodities in the NCR, Philippine markets. Further, after the development and subjecting the different mathematical models to forecasting and accuracy tests, it was found that, when estimating the monthly prices of fruits, ARIMAx (5,2,2,x=mango) appeared as the best model for banana while the ARIMAx (2,2,1,x=banana) emerged as the finest model for mango. With regard to vegetable commodities, the best model to use for estimating monthly prices of cabbage is ARIMAx (3,2,1,x=pechay), SARIMA (1,1,1)(1,1,1)12 for pechay and SARIMA (2,1,1)(2,1,1) 12 for tomatoes. A summary of the best model identified in each commodity as well as their corresponding equations and interpretations are shown in Table 1. Table 1: Summary of Best Mathematical Models Identified for Each Commodity

Commodity/Mathematical Model A. Fruits 1. Banana ARIMAx(5,2,2,x=mango)

2. Mango ARIMAx(2,2,1,x=banana

B. Vegetables 1. Cabbage ARIMAx(3,2,1,x=pechay)

Interpretations This model revealed that the differences in the prices between the two succeeding months for the past six months had an effect on the current month’s price. Likewise, the random shocks in the previous two months exhibited an effect on the current price. Finally, the recent price of the mango, as affected by the other terms, had an effect on the current price of a banana. It means the current price of banana was affected by its previous price in the past six months. In addition, its present price also was affected by the current price of mango. This model shows that the differences in the prices between the two consecutive months for the past three months had an effect on the current month’s price. Likewise, the random shock in the previous month denoted a consequence with regard to the current price. Lastly, the present price of the banana, as influenced by the other terms, had an impact on the price of mango at present. This means the present price of mango was influenced by its price in the previous three months and is also affected by the current price of a banana. This model proves that the difference in the prices between the most recent four months had an effect on the current month’s price. Likewise, the random shock in the previous month elucidated an effect on the current price. Evidently, the price of the pechay at present, as affected by the other terms, had an effect on the current cabbage price. It means that the price of cabbage in

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International Journal of Advance Study and Research Work (2581-5997)/ Volume 2/Issue 7/July 2019

2. Pechay SARIMA(1,1,1)(1,1,1)12

3. Tomato SARIMA(2,1,1)(2,1,1)12

the previous four months has an effect on its present price. Also, the current price of pechay has an impact on the present price of cabbage. This shows that the price depends on the difference between the prices of the previous two months for the past 3 months as well as the prices for two months before the previous year. The shocks that transpired on the prices of the succeeding month of the current year and the shocks that ensued during the season of the foregoing year had an effect on the current price. This means the price of pechay was influenced by its price in the previous two months and also its price two months before the previous year. It further indicates seasonality in the prices of pechay as its model came up with SARIMA(1,1,1)(1,1,1)12. The model explains that the price depends on the difference of the prices of the preceding two succeeding months for the past 3 months along with the difference of the prices of the previous two consecutive months for the past 3 months of the previous year. Further, the shocks that occurred on the prices of the previous month of the current year and the shocks that transpired during the season of the earlier year had an effect on the price at present. It means the current price of tomato was affected by its price in the previous three months in the current year and also influenced by its price in the last season in the same month. This further shows seasonality in the prices of tomato by the model SARIMA(2,1,1)(2,1,1)12.

Conclusions Based on the results of the study, the following conclusions were drawn: 1. 2. 3.

4. 5.

The monthly retail prices of agricultural commodities in the National Capital Region of the Philippines from 20092018 has an increasing trend and is expected to continue for the next years to come. Time series models like ARIMA, SARIMA and ARIMAx could be used for estimating the prices of fruit and vegetable commodities depending on their accuracy and performance in forecasting. For fruit commodities, it was concluded that their monthly retail prices do not just depend on their previous prices but also affected by the prices of other fruits. In the study, both fruit commodities (banana and mango) came up with ARIMAx models. With regard to vegetables, seasonality is a factor (e.g. pechay and tomato). Their prices are influenced by their foregoing prices in the previous season of the same month. Forecasts of the best models identified for each commodity signify a continuous increase in the prices in the future years. Though this will provide more gains and profit for the producers, it will also lead to more cost and expenses for the consumers. Considering that the Philippines is a developing country, the increase in prices of basic commodities must be compensated by an income increase to compensate for the inflation rate.

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# A Mathematical Model for Estimating Retail Price Movements of Basic Fruit and Vegetable Commodities

Prices of basic agricultural commodities in the market truly concern the entire populace in a region or country. They directly affect the co...

# A Mathematical Model for Estimating Retail Price Movements of Basic Fruit and Vegetable Commodities

Prices of basic agricultural commodities in the market truly concern the entire populace in a region or country. They directly affect the co...