Predicting the future price of a commodity using the OWMA operator: An approximation of the interest rate and inflation in the brown pastusa potato price

Abstract

This main aim of this paper is to propose a methodology for the prediction of the future price of the brown pastusa potato in Colombia, taking into consideration the variables of interest rate, as measured by fixed term deposits (FTDs), and inflation rate, as measured by the consumer price index (CPI). The methodology conducts linear regression analysis and assesses the results using the significance test, the Durbin-Watson statistic, analysis of the variance inflation factor (VIF) and the coefficient of determination. After that, the forecast of the independent variables has been conducted with the ordered weighted moving average (OWMA) operator and new proposed OWA operators using probabilities that are presented in the paper. Using these new methods and the proposed econometric model, it is possible to establish future prices. The results show a greater impact of the interest rate than inflation, as well as the need to include supply and demand variables that have not been included due to the absence of systematic information.

Keywords

OWMA operator predicting commodity future price FTD CPI

1 Introduction

The destabilization of the spot market for commodities has been responsible for volatility and price increases [1, 2] showed that volatility has led to turbulent prices, production and inventories in two scenarios: first, in the short term with respect to marginal performance, the accumulation of inventories and price increases; and second, in the exhaustible resources over the total marginal cost of production. Particularly, “the prices for agricultural commodities might react heavily to adverse weather conditions, political announcements, or surprises in inventory announcements” [3], as well as economic cycles and monetary policy [4]. Hence, commodities play an influential role in modern financial markets by encouraging producers and consumers to cover their risks in the face of exposure to price fluctuations [5].

A particular case is that agricultural participation has great relevance in the Colombian economy. It is a significant sector in the generation of employment and income for the economically active population. However, its potential has been affected by economic, political, social, and environmental factors; armed conflict and the consequences of climate change.

Hence, this paper proposes a methodology to forecast the future price of a commodity (brown pastusa potato). This forecast includes monetary policy variables, such as the interest rate and inflation [6 –8]. Commodity prices are an indication of the real state of the economy due to their relationships with macroeconomic conditions. Likewise, price increases reflect accelerated economic growth and it is evident that increased inflation would cause more restrictive monetary policy. Hence, commodity prices are important measures of the economic situation and should be considered in monetary policy decisions.

In this sense, it is important for the producers and the consumers to know the future price of the potato in order to the government to make policies. For this reason, this research develops statistical (econometric model) and nonstatistical (OWA operator [9]) approaches that can be used to forecast the future price of the potato as a proposition to generate a commodity in the financial market in Colombia.

1.1 Commodity pricing

Commodities are resources that are classified by their various characteristics. They are real assets that generate intrinsic value and contribute to industry and consumers. However, the supply of some commodities is limited, depending on the availability in specific periods as a result of a strong seasonal component. In addition, they are heterogeneous assets since they have their own qualities. Hence, they can be classified as hard commodities (energy and metals) and soft commodities (livestock and agriculture). Finally, availability (or renewability) and storage are explanatory conditions for pricing [10].

Soft commodities, particularly agricultural ones, are related to social and economic development since they provide well-being and can satisfy human needs. However, the accelerated commodification due to financial forces has affected prices. Abrupt decreases in market prices are favorable for low income groups while high prices are better for the government, businesses and land owners [11]. Thus, commodity prices have been characterized by their volatility and continuous fluctuations, which are a result of the incidence of mainly economic variables, which have caused uncertainty in the markets. Regarding this case, [12] has determined that, based on the established prices of storable commodities, efficient information markets can be explained with the incorporation of information and the expectations of all participants.

Nevertheless, “the mainstream theory in commodity pricing, namely, the theory of storage, explains the behavior of commodity prices based on economic fundamentals. Furthermore, it has major implications for the volatility of commodity prices” [13]. According to this theory, [14] and others have determined that the circumstances of supply and demand cause differences between spot and future prices due to convenience yields, storage costs and inventory levels. Based on the above, seasonal variables and monetary policy have been included in the evolution of commodity price models to explain their behavior [15].

Thus, [16] has established that commodity price indices can be a decisive element of monetary policy (if prices increase, they can in turn increase inflation and decrease the money supply). Instead, [17] found that commodity price indices generate valuable data for monetary policy due to the information related to future price changes and production.

Likewise, [18] found that commodities experience increased real prices due to lower interest rate shocks; both oil prices and industrial inputs may behave excessively; and the prices of other commodities, such as food and metals, behave gradually. [19] considered that the shocks recorded in real interest rates significantly influence commodity prices based on the economic cycles and extreme factors that cause changes in economic variables over time.

On the other hand, [20] explained that “generalized commodity price increases lead consumer price inflation as a manifestation of the differing rates of adjustment of the prices of both types of goods to monetary developments and not necessarily the result of exogenous.” For its part, [8] has established the importance of the relationship between commodity prices and inflation. Price is used as a hedge element generating investments on these resources and this can be used in monetary policy decisions when they precede general inflation.

1.2 The OWA operator: method for the financial forecast

The ordered weighted average (OWA) operator is a tool that allows capturing decision-makers’ attitude. The attitude emphasizes the degree of optimism or pessimism in the selection of an alternative in the rational decision-making process, which can be considered as the attitudinal character of the decision maker under different decision criteria. This aggregation operator is proposed by [9], and it has the property of being located between the “and” (t-norm) and the “or” (t-conorm), i.e., all criteria must be satisfied or at least one of the criteria must be satisfied. Likewise, this operator allows one to appropriately adjust the degree of “and-ing” or “or-ing” within a multicriteria decision making process [9]. Similarly, the OWA operator allows one to represent the different decision criteria, such as the Optimistic criterion, Pessimistic or Wald criterion, Hurwicz criterion and Laplace criterion, in a simple formula [9]. Thus, under this method, numerous extensions have been made that focus on the aggregation of new variables and hybrid methods, such as induced variables [21, 22]; a Heavy weighting vector [23]; prioritized [24]; power [25]; linguistic [26]; distance measures [27]; measures of central tendency [28, 29]; probabilistic [30]; Bonferroni means [31 –33]; logarithms [34 –36]; Pythagoreans [37] and forecasting [38, 39], among others. They allow the development of multiple applications in different areas of research.

Based on the above, the versatility of the OWA operator allows it to be combined with other aggregation methods to obtain new operators with associated characteristics. These new operators provide solutions to specific decision making problems in complex and uncertain systems where the meaning of information is more important than their own measurement [40]. In this sense, applications have boomed in systems where human reasoning is predominant over the data. One of the fields where it has gained great strength is within business studies since it has been involved in areas such as strategy [41], human resources [42, 43]; marketing [44]; entrepreneurship [45, 46]; stakeholders [47, 48]; and risk management [49], among others, in which the human factor has a determining influence on the orientation of the decisions. This occurs since they are affected by the degree of preference and attitude towards a given situation. Thus, the possibility that a fact can happen depends on the capacity to aggregate subjective information with objective information in order to have a more accurate view of reality.

Under this premise, applications and formulations that combine both objective and subjective data have been developed to solve decision-making problems in finance. The contributions made in financial management include the analysis of perception in the management of financial risk [50, 51]; the selection of financial products [52, 53]; the design of an intelligent monitoring system for the financial market [54]; the evaluation of financial portfolios for the selection and location of assets [55]; the decision making process for selecting financial strategies [56]; the evaluation of the attributes in the evaluation of financial performance as a facilitator in the decision making of investment and management of stakeholders [57]; and the investment process and financial forecast [39, 58], among others. These approximations have introduced decision making attitude to different financial topics where the decision maker directly influences the final result. In this sense, the development and implementation of new hybrid extensions based on the OWA operator for financial forecasting problems contributes to the configuration of the methods applied within the aggregation operators.

Thus, the remainder of this article is organized as follows. Section 2 gives the data sources and estimation methods including showing the data set, the statistical methods, the econometric model, the OWA operator and a new extension for forecasting. Section 3 presents the methodology. Section 4 highlights the main results on the analysis of the econometric model and the prediction of the future price. Finally, the main conclusions and future works are presented.

2 Data sources and estimation methods

This section introduces the basic concepts that are used throughout the paper including the data set, the statistical methods, the econometric model and the OWA. Additionally, a new proposed OWA operator extension is presented.

2.1 Data sources

The data used are the daily recorded prices for the Brown pastusa potato from January to October 11 of 2018, which have been taken from the website of the Mercantile Exchange of Colombia. Likewise, data on the interest rates of 90-day fixed-term deposit certificates (FTDs) from the Bank of the Republic of the Colombia website and the consumer price index (CPI) with a monthly frequency from the National Administrative Department of Statistics of Colombia (DANE) are used.

2.2 Economic model and statistical methods

[59] consider that “one needs simple models to characterize the types of interconnections just described between commodity prices and other macroeconomic developments”. Thus, a linear regression model is used to analyze the future price of the brown pastusa potato. The model takes into account two variables: the interest rate and inflation. The model is as follows. $Price = β interest + β inflation$ (1)

The Durbin-Watson statistic is also used to find the serial correlation, and it is defined as follows: $d = \frac{\sum_{i = 2}^{n} {(z_{i} - z_{i - 1})}^{2}}{\sum_{i = 1}^{n} z_{i}^{2}}$ (2)

Subsequently, Variance Inflation Factor (VIF) analysis is developed to determine the increase in the estimated variance of the ith regression coefficient if $R_{i}^{2}$ is equal to zero. [60] states that the “VIF provides a reasonable and intuitive indication of the effects of multicollinearity on the variance of the regression coefficient.” This is represented by the following equation: ${VIF}_{i} = \frac{1}{1 - R_{i}^{2}}, i = 1, \dots, p,$ (3) where $R_{i}^{2}$ is the multiple correlation coefficient of X_i regressed on the remaining explanatory variables, and p is the number of predictor variables.

2.3 Ordered weighted average (OWA)

Finally, mobile averages are used in the ordered weighted average (OWA) operator to consider several periods of time and use dynamic data for the generation of different potential results, as stated by [38]. The OWMA equation has taken the following form.

Definition 1. An ordered weighted moving average (OWMA) of dimension m is a mapping OWMA : R^m → R that has an associated weighting vector W of dimension m with $W = \sum_{j = 1 + t}^{m + t} w_{j} = 1$ and w_j ∈ [0, 1] such that $OWMA (a_{1 + t}, a_{2 + t}, \dots, a_{m + t}) = \sum_{j - 1 + t}^{m + t} w_{j} b_{j},$ (4) where b_j is the jth largest argument a_i, m is the total number of arguments in the sample and t is the change in the mean from the initial analysis.

Additionally, some extensions to the OWMA operator such as those using induced variables, a heavy weighting vector and probabilities are used to forecast the exogenous variables that are included among the econometric variables [21 , 61]. These formulations are presented as follows.

Definition 2. An induced ordered weighted moving average (IOWMA) operator of dimension m is a mapping IOWMA : R^M × R^M → R that has an associated weighting vector W of dimension m with $W = \sum_{j = 1 + t}^{m + t} w_{j} = 1$ and w_j ∈ [0, 1] such that

$\begin{matrix} IOWMA (\begin{matrix} 〈 u_{1 + t}, a_{1 + t} 〉, 〈 u_{2 + t}, a_{2 + t} 〉, \dots, \\ 〈 u_{m + t}, a_{m + t} 〉 \end{matrix}) \\ = \sum_{j = 1 + t}^{m + t} w_{j} b_{j}, \end{matrix}$ (5) where b_j is the a_ith value of the IOWMA pair 〈u_i, a_i〉 having the jth largest u_i, u_i is the order inducing variable, a_i is the argument variable, m is the total number of arguments considered from the whole sample and t indicates movement in the average from the initial analysis.

Definition 3. A heavy ordered weighted moving average (HOWMA) operator is defined as a given sequence ${a_{i}}_{i = 1}^{N}$ in which a new sequence ${s_{i}}_{i = 1}^{N - n + 1}$ is multiplied by a heavy weighting vector such that $HOWMA (s_{i}) = \sum_{j = 1 + t}^{m + t} w_{j} b_{j},$ (6) where b_j is the jth largest element of the collection a₁a₂, …, a_n, and W is an associated weighting vector of dimension m with $W : 1 \leq \sum_{i = 1 + t}^{m + t} w_{i} \leq n$ and w_i ∈ [0, 1]. Observe that we can also expand the weighting vector from -∞ to ∞. Thus, the weighting vector w becomes $- \infty \leq \sum_{j = 1}^{n} w_{j} \leq \infty$ .

Definition 4. A induced heavy ordered weighted moving average (IHOWMA) operator is defined as a given sequence ${a_{i}}_{i = 1}^{N}$ in which a new sequence ${s_{i}}_{i = 1}^{N - m + 1}$ is multiplied by a heavy weighting vector such that

$\begin{matrix} IHOWMA (\begin{matrix} 〈 u_{1 + t}, a_{1 + t} 〉, 〈 u_{2 + t}, a_{2 + t} 〉, \dots, \\ 〈 u_{n + t}, a_{m + t} 〉 \end{matrix}) \\ = \sum_{j = 1 + t}^{m + t} w_{j} b_{j}, \end{matrix}$ (7) where b_j is the jth element that has the largest value of u_i, u_i is the order inducing variable, and W is an associated weighting vector of dimension m with $W : 1 \leq \sum_{i = 1 + t}^{m + t} w_{i} \leq n$ and w_i ∈ [0, 1]. Observe that it can also expand the weighting vector from -∞ to ∞. Thus, the weighting vector w becomes $- \infty \leq \sum_{j = 1}^{n} w_{j} \leq \infty$ .

In financial forecasting, the probabilities that the same scenarios happened are important in order to understand if outcomes can be repeated or approximated. In this sense, the use of the Probabilistic OWA (POWA) operator [62] can complement the forecasting and provide an interesting aggregation operator to consider in this kind of problem. The traditional POWA formulation is as follows.

Definition 5. A POWA operator of dimension n is a mapping POWA : Rⁿ → R having an associated weighting vector P, where p_i ∈ [0, 1] and $\sum_{i = 1}^{n} p_{j} = 1$ , that is expressed as follows:

$POWA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} \hat{p_{j}} b_{j},$ (8) where b_j is the jth element, which is the largest of the collection a₁a₂, …, a_n. In the collection, each argument a_i is associated with a probability p_i where $\sum_{i = 1}^{n} p_{i} = 1$ and p_i ∈ [0, 1]; ${\hat{p}}_{j} = β w_{j} + (1 - β) p_{j}$ with β ∈ [0, 1]; and p_j is the probability of p_i ordered according to b_j, the jth largest element of a_i. Additionally, if β = 0, we obtain the PA operator; and if β = 1, we obtain the OWA operator.

2.4 New propositions using probabilities with OWMA operators

The contributions developed from the probabilistic ordered weighted average (POWA) proposed by Merigó [62] have opened a range of possibilities for extensions. Within this extension, a new method is presented that combines the characteristics of four operators, including the OWMA [38], IOWA [22], HOWA [23] and POWA [62], which can help to solve the problems of transaction price and financial forecasting. Hence, the most complex aggregation operator that can be formed is the probabilistic induced heavy ordered weighted moving average (PIHOWMA) operator and some other special cases can be defined in specific situations. These operators are defined as follows.

Proposition 1. A PIHOWMA operator of dimension n is a mapping PIHOWA : Rⁿ → R with a given sequence ${a_{i}}_{i = 1}^{N}$ in which a new sequence ${s_{i}}_{i = 1}^{N - m + 1}$ is multiplied by a heavy weighting vector. It is expressed as follows:

$\begin{matrix} PIHOWMA (\begin{matrix} 〈 u_{1 + t}, a_{1 + t} 〉, 〈 u_{2 + t}, a_{2 + t} 〉, \dots, \\ 〈 u_{n + t}, a_{n + t} 〉 \end{matrix}) \\ = \sum_{j = 1 + t}^{m + t} {\hat{p}}_{j} b_{j}, \end{matrix}$ (9) where b_j is the jth element that has the largest value of u_i; u_i is the induced order of variables; ${\hat{v}}_{ij}$ is the corresponding weight of the jth criteria in the ith category; i = 1, …, q ; j = 1, …, i_i; each argument a_i is associated with a probability p_i where $\sum_{i = 1}^{n} p_{i} = 1$ and p_i ∈ [0, 1]; ${\hat{p}}_{j} = β w_{j} + (1 - β) p_{j}$ with β ∈ [0, 1]; and p_j is the probability of p_i ordered according to b_j, the jth largest element of a_i. W is an associated weighting vector of dimension m with $W : 1 \leq \sum_{i = 1 + t}^{m + t} w_{i} \leq n$ and w_i ∈ [0, 1]. Observe that we can also expand the weighting vector from -∞ to ∞. It is important to note that sometimes the weighting vector can be measured as $- \infty \leq \sum_{j = 1}^{n} w_{j} \leq \infty$ .

Among the properties of the PIHOWMA operator are the following. a) Commutativity: Assume f is the PIHOWMA operator. Then, f (〈u_i+1, a_i+1〉, …, 〈u_n+t, a_n+t〉) = f (〈u_i+1, b_i+1〉, …, 〈u_n+t, b_n+t〉). b) Monotonicity: Assume f is the PIHOWMA operator. If |u_i+t, a_i+t| ≥ |u_i+t, b_i+t| for all i_i+t, then f (〈u_i+1, a_i+1〉, …, 〈u_n+t, a_n+t〉) ≥ f (〈u_i+1, b_i+1〉, …, 〈u_n+t, b_n+t〉). c) Idempotency: Assume f is the PIHOWMA operator. If |u_i+t, a_i+t| = a for all i + t, then f (〈u_i+1, a_i+1〉, …, 〈u_n+t, a_n+t〉) = a . d) Unbounded: Assume f is the PIHOWMA operator. Then, $1 \leq \sum_{i = 1 + t}^{m + t} w_{i} \leq n$ .

Additionally, if the weighting vector is bounded, then the PIHOWMA operator becomes the probabilistic induced ordered weighted moving average (PIOWMA) operator and it is defined as follows.

Proposition 2. A PIOWMA operator of dimension n is a mapping PIOWMA : Rⁿ → R with a given sequence ${a_{i}}_{i = 1}^{N}$ in which a new sequence ${s_{i}}_{i = 1}^{N - m + 1}$ has an associated weighting vector P, where p_i ∈ [0, 1] and $\sum_{i = 1}^{n} p_{j} = 1$ . It is expressed as follows:

$\begin{matrix} PIOWMA (\begin{matrix} 〈 u_{1 + t}, a_{1 + t} 〉, 〈 u_{2 + t}, a_{2 + t} 〉, \dots, \\ 〈 u_{n + t}, a_{n + t} 〉 \end{matrix}) \\ = \sum_{j = 1 + t}^{m + t} {\hat{p}}_{j} b_{j}, \end{matrix}$ (10) where b_j is the jth element that has the largest value of u_i; u_i is the induced order of variables; ${\hat{v}}_{ij}$ is the corresponding weight of the jth criteria in the ith category; i = 1, …, q ; j = 1, …, i_i; each argument a_i is associated with a probability p_i where $\sum_{i = 1}^{n} p_{i} = 1$ and p_i ∈ [0, 1]; ${\hat{p}}_{j} = β w_{j} + (1 - β) p_{j}$ with β ∈ [0, 1]; and p_j is the probability of p_i ordered according to b_j, the jth largest element of a_i.

Finally, the PIOWMA operator becomes the prioritized heavy ordered weighted moving average (PHOWMA) operator when there is not an order induced association between arguments and weights. The definition is the following.

Proposition 3. A PHOWMA operator of dimension n is a mapping PHOWMA : Rⁿ → R with a given sequence ${a_{i}}_{i = 1}^{N}$ in which a new sequence ${s_{i}}_{i = 1}^{N - m + 1}$ is multiplied by a heavy weighting vector. It is expressed as follows:

$\begin{matrix} PHOWMA (a_{1 + t}, a_{2 + t}, \dots, a_{n + t}) = \sum_{i = 1 + t}^{m + t} \\ (\sum_{h = 1 + t}^{n_{i}, m + t} w_{ij} C_{ij} (x)), \end{matrix}$ (11) where b_j is the jth element, which is the largest of the collection a_1+t, a_2+t, …, a_n+t. In the collection, each argument a_i+t is associated with a probability p_i where $\sum_{i = 1}^{n} p_{i} = 1$ and p_i ∈ [0, 1]; ${\hat{p}}_{j} = β w_{j} + (1 - β) p_{j}$ with β ∈ [0, 1]; and p_j is the probability of p_i ordered according to b_j, the jth largest element of a_i+t. C_ij (x) measures the satisfaction of the jth criterion in the ith group by alternative x ∈ X for each i = 1, …, q and, j = 1_t, …, i_i+t. It is important to note that sometimes the weighting vector can be measured as $- \infty \leq \sum_{i = 1 + t}^{m + t} w_{i} \leq \infty$ .

As explained above, these new operators are important when there is information about the probabilities that the same economic or financial scenarios repeat. Additionally, these are useful when information about the expectations and knowledge of the expert should be included in the results. Furthermore, when there is high uncertainty in the market, the use of the traditional historical data is not enough to make a decision. Hence, by using these aggregation operators, it is possible to combine different sorts of information that support the decision-making process and take into account the decision maker’s criteria.

3 Methodology

To observe the usability and viability of the proposed method, two methodological approaches are developed. First, the econometric model that is made up of the monthly FTD and the Consumer Price Index is presented. Second, the PIHOWMA operator is used to forecast the future price. This allows aggregating the criteria and subjectivity of the decision-maker. The details of both procedures are shown below.

To analyze if the economic model is statistically correct, the following tests is used.

Significance test. This test is useful to analyze if the information provided by the independent variable is important. To do this, the P value is analyzed under the next assumptions:

H₀ is accepted if the P value is greater than 0.05; and

H₀ is rejected if the P value is less than 0.05.

Autocorrelation. This test is used to find patterns in the error that the model presents. Thus, in order to detect it, the Durbin-Watson statistic is used.

Multicollinearity. To find if there is correlation between the predictors in the model, an analysis of the Variance Inflation Factor (VIF) is done under the following assumptions:

If VIF less than 5, there is low multicollinearity;

If VIF is more than 5 but less than 10, there is medium multicollinearity; and

If VIF is more than 10, there is high multicollinearity.

Coefficient of determination. This indicator is important to detect the proportion of the variance in the dependent variable that is predictable from the independent variables.

With these tests of the economic model, it can be established whether it is statistically correct to predict the price of the brown pastusa potato with the interest rate and inflation.

Then, to forecast the future price of the brown pastusa potato in Colombia, it is necessary to forecast the values of the independent variables. In this case, a forecast for the first half of 2019 is done using the PIHOWMA operator and its extensions using the induced, heavy and probabilistic operators. The main advantages of this operator are the following:

the induced operator lets the decision maker arrange the weighting vector so that it is not based on the maximum or minimum criterion, but instead arranged according to their ideas;

the heavy operator is a weighting vector that is unbounded, which lets us under or overestimate the results according to the decision maker’s expectations and knowledge of the future market; and

a probabilistic vector that indicates the probability that the results of a specific month can happen again.

4 Results

In this section, the econometric model analysis and the application of the forecast of the future price using the PIHOWMA operator are presented.

4.1 Econometric model analysis

The information used to generate the econometric model is the monthly information from January to October 2018 and the results are the following.

The econometric model is $Price = β interest + β inflation$ $Price = - 905 + 37050 interest + 242 inflation,$ (12) where interest is the monthly FTD and inflation is measured using the Consumer Price Index.

Table 1 shows the results of the statistical tests used to determine if the model is statistically correct.

Table 1

Statistical tests

Test used	Result
Significance test	Interest 0.046
	Inflation 0.145
	Model 0.001
Autocorrelation	1.03600
Multicollinearity	Interest 3.454
	Inflation 3.454
Coefficient of determination	R-squared 90.3%
	R-squared (adjusted) 87.1%

Source: Own elaboration.

In the significance test, the P value for interest is 0.046 (H₀ is rejected) and that for inflation is 0.145 (H₀ is accepted). It can be observed that the information provided by inflation is not that relevant. Additionally, the P value for the complete model is 0.001 (H₀ is rejected), indicating that the model passes the significance test. In the case of autocorrelation, the result is 1.03600, indicating that the test is inconclusive. In this sense, it is possible to assume that there is no autocorrelation in the model. The multicollinearity is measured by the variance inflation factor (VIF). The value for inflation is 3.454 (low) and that for interest is 3.454 (low). In addition, the coefficients of determination, the R-squared and adjusted R-squared, for this model are 90.3% and 87.1%, respectively.

With the results obtained from all the tests performed, it is possible to assume that it is statistically correct to predict the brown pastusa potato price using the interest rate and inflation.

4.2 Forecast of the future price using the PIHOWMA operator

The following assumptions provide the necessary information to use the operators.

A moving average n = 3 (Quarterly)

A weighting vector W = (0.20, 0.30, 0.50) =1

An induced vector U = (5, 10, 15)

A heavy weighting vector H₁=(0.25, 0.30, 0.50)=1.05. This vector will be used to calculate interest because it is assumed that it will be higher than that in 2018.

A heavy weighting vector H₂=(0.20, 0.30, 0.45)=0.95. This vector will be used to calculate inflation because it is assumed that it will be lower than that in 2018.

A probabilistic vector p = (0.20, 0.20, 0.60)

The importance of the weighting vector is 60% and that of the probability vector 40%.

The results using the 4 types of operators are presented in Tables 2 and 3.

Table 2
2019 values for interest

Month OWMA IOWMA HOWMA IHOWMA

January 0.045436 0.04535 0.051607 0.05154

February 0.045427 0.045346 0.053019 0.052956

March 0.045431 0.045347 0.054505 0.05444

Month POWMA PIOWMA PHOWMA PIHOWMA

January 0.045426 0.045375 0.049129 0.049088

February 0.045402 0.045353 0.049956 0.049919

March 0.045405 0.045355 0.050849 0.050810

Month	OWMA	IOWMA	HOWMA	IHOWMA
January	0.045436	0.04535	0.051607	0.05154
February	0.045427	0.045346	0.053019	0.052956
March	0.045431	0.045347	0.054505	0.05444
Month	POWMA	PIOWMA	PHOWMA	PIHOWMA
January	0.045426	0.045375	0.049129	0.049088
February	0.045402	0.045353	0.049956	0.049919
March	0.045405	0.045355	0.050849	0.050810

Source: Own elaboration.

Table 3

2019 values for inflation

Month	OWMA	IOWMA	HOWMA	IHOWMA
January	0.127300	0.111850	0.110294	0.096592
February	0.131910	0.114575	0.112228	0.096928
March	0.131785	0.114343	0.108735	0.093716
Month	POWMA	PIOWMA	PHOWMA	PIHOWMA
January	0.128048	0.118778	0.117844	0.109623
February	0.133730	0.123329	0.121921	0.112740
March	0.133495	0.123029	0.119665	0.110653

Source: Own elaboration.

With this information and the econometric model, it is possible to generate future prices for the brown pastusa potato in 2019. The results are shown in Table 4. It is noteworthy that the results have 8 different scenarios about the future price of the brown pastusa potato in Colombia and the price ranges from 802.29 to 1140.73 in the first quarter of 2019. In addition, the scenario for the price of the brown potato is characterized by a high degree of uncertainty. Given the characteristics of production and trading that exist, this commodity can have losses. Thus, it is important to propose and develop models in order to avoid losses and to be able to make forecasts with an acceptable error, presenting the possibility to create derivative contracts.

Table 4

Brown Pastusa Potato price for 2019

Month	OWMA	IOWMA	HOWMA	IHOWMA
January	809.21	802.29	1,033.74	1,027.92
February	810.01	802.81	1,086.51	1,080.48
March	810.1	802.79	1,140.73	1,134.68
Month	POWMA	PIOWMA	PHOWMA	PIHOWMA
January	809.02	804.87	943.74	940.25
February	809.49	805.17	975.39	971.77
March	809.55	805.16	1,007.93	1,004.29

Source: Own elaboration.

According to the results, the prediction of the price of the pastusa brown potato that has a greater adjustment to the results corresponds to the PIHOWMA. Regarding the average real prices in the year 2019, that for the month of January is adjusted to 94,02%, that for February is 97.17% and that for March is 100.42%. In the Colombian case, it is essential to point out that the studies on commodity price prediction that have been carried out so far are not conclusive. Likewise, with the inclusion of the OWA operators, which allow the values of the independent variables to be forecasted, the price prediction model has a greater adjustment to reality because there is restricted access to the information used to improve the Econometric model, such as hectares planted, tons produced and others relevant to the supply and demand of the product. This type of information provides a better understanding and enables better models to be generated. In addition, it is necessary to calculate the daily price through a systematic calculation of each of the variables in the model to include the product in the price index of basic products in Colombia.

5 Conclusions

There are multiple statistical models of commodity prices including monetary policy elements, such as the following: cointegration tests [16], the Durbin-Watson test and the Dickey-Fuller test [63], univariate time series models and standardized residuals [64], VAR models [18], cross-correlations using the multifractal detrended cross-correlation analysis method [19], and a cointegrating VAR framework [20], among others. The prediction model for the future price of the brown pastusa potato has used a linear regression of two variables, the OWMA operator and its extensions. The FTD interest rate variable corresponds to the weighted average of the effective interest rates of the fixed-term deposit certificates of financial institutions [65] and the consumer price index (CPI) variable is the main reference for calculating inflation for Colombia. According to the obtained results, the statistical model is correct, and it is mainly found that the variance inflation factor has low multicollinearity. However, the autocorrelation with the Durbin-Watson statistic is not conclusive. Likewise, it has established that there is no autocorrelation in the model. Thus, one of the limitations is the absence of a significant relationship between the brown pastusa potato price and inflation [8].

It is noteworthy that the results show four different scenarios depending on the use of the OWA operator and some of its extensions. These results are an important step to including the brown pastusa potato in the commodity price index, improving farmer’s income and providing the consumer with a more stable price. With the application of the OWMA operators and their extensions for the prediction of the independent variables of the model, it has been found that there are methodological and information limitations. First, as a consequence of using n = 3 for all moving averages, the price prediction is not applicable over the long term because there will be a loop in the results such that it tends to be the same over the time. That is why the systematic calculation of the formula and updating the information of the exogenous variables for the real results is very important. Second, according to the results of the econometric model, the interest rate variable has greater importance to the value of the brown pastusa potato versus inflation. In this sense, an important change in the interest rate would make the results drastically change. Third, there are restrictions due to the noninclusion of information on supply and demand because of the absence of official data by the Colombian government on these variables. Consequently, it is relevant that the agricultural sector and the government start to work together to generate more complete information that allows one to know the number of hectares sown and tons produced. Thus, the interest rate variable is shown to have a significant impact on commodity prices, which makes it convenient for those responsible for the design of monetary policy to take into account the side effects of the interest rate in the prices of different products (including agricultural prices) and visualize the impact through the elaboration of econometric models.

It is noteworthy that Colombia trades financial assets and commodities on the Colombian Stock Exchange (BVC) and the Mercantile Exchange of Colombia (BMC). First, the BVC has introduced a derivatives market, USD futures, government bonds futures (TES), representative stock futures (Grupo Sura, Éxito, Preferencial Bancolombia, Cemargos and Ecopetrol), stock index futures (colcap) and options on the exchange rate (TRM). Second, the BMC is the main administrator of the commodity market. In partnership with the Ministry of Agriculture and Rural Development, it promotes programs, incentives and compensation for rice, cotton, milk (milk powder and ripened cheeses), cattle, panela, yucca, yellow corn and white corn. Likewise, policies are carried out in the face of falling international prices for white and yellow corn and forwards in the public procurement market (MCP forwards). This scenario shows that “in Colombia there is not futures market on agricultural commodities and the BMC only focuses on registration operations” [66]. Hence, it is pertinent to develop a model that provides an approach to estimate the future price of a commodity. Finally, based on the above, future works can extend the approach to other commodity products since there are limited products in the commodity market in Columbia and more products can be included in it. A specific methodology that can include the main characteristics of the Colombian markets is needed.

Footnotes

Acknowledgment

Author Leon-Castro acknowledges support from Chilean Government trough FONDECYT initiation grant No. 11190056.

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