Development of the cultivated mushroom market from the standpoint of time series sustainability

. The article deals with the issues of sustainable development of the world market of cultivated mushrooms. The authors of the article have studied the time series of the world gross value, world volumes of imports and exports, the world price index for cultivated mushrooms and truffles in order to analyze their growth and dynamics. The results of calculations carried out using the statistical database of the Food and Agriculture Organization of the United Nations from 1961 to 2022 showed that the indicated time series are non-stationary. The gross value of mushroom production increased 70 times during the study period, the volume of imports increased 98 times, the volume of exports increased 93 times, and the producer price index doubled. The price index for cultivated mushrooms and truffles is stable without sharp fluctuations. The authors reveled shifts in a series of world gross value of mushroom production in 2000, 2009, 2011. Additive emissions, level shifts and temporary changes were found in the dynamics of world imports and exports.


Introduction
Issues of sustainable development of various branches of agriculture need to be considered due to global climate change, food, water and energy security, reduction of biological diversity, problems of access and sharing of natural resources [1][2][3][4][5][6].
The interpretation of the category "sustainable development" currently varies widely in different fields of science [7][8][9][10][11][12]. Generally it characterizes the property of various systems in the presence of a number of exogenous influences to maintain an equilibrium position for a long period of time.
The sustainable development of the market of cultivated mushrooms can be defined as its ability to restore and, possibly, increase the characteristics of the product offer and consumer demand for cultivated mushrooms, which were inherent to it before the period of external influence, in a fairly short period of time.
The market of cultivated mushrooms is a unique market, which combines the medicinal benefits of products, food culture, exotic flavors and an environment that is alternative to many traditional food products. However, according to researchers, there is a risk of consumers rejecting dangerous, harmful, in their opinion, food that does not bring benefits [13][14][15]. Cultivated mushrooms are a rich source of protein food, dietary fiber and minerals [16][17][18]. At the same time, the production of cultivated mushrooms is a high-tech, scienceintensive production that requires a constant search for scientific research in the integration of biology, mycology, mechanical engineering, automation, which makes it possible to predict the volume of production with a high probability, manage the crop and justify the financial and economic parameters of production [19][20][21][22][23].

Materials and methods
The article assessed the stability of the development of the cultivated mushroom market from the standpoint of the stability of time series -the presence of the necessary trend of the studied statistical indicator, which is under the influence of unfavorable conditions.
The study used data from the following time series of the Food and Agriculture Organization of the United Nations (FAOSTAT) statistical database: world gross value of cultivated mushrooms and truffles (at constant prices 2014-2016, thousand US dollars), world imports and exports mushrooms and truffles (in tons) from 1961 to 2021, mushroom and truffle price index (data price index for 2014-2016 is taken as 100) from 1991 to 2022.
To test the stationarity of these time series, the extended Dickey-Fuller test implemented in the R statistical package was used, and the autocorrelation coefficients were calculated.
The time series data was used in a growth model to measure the trend of these variables. According to the study [24], the use of linear growth rates (LGR) has limitations when comparing growth rates between separate periods. Therefore, it is more appropriate to use the compound annual growth rate (CGR) to describe the dependencies. To determine the trend function and the subsequent assessment of growth rates, a semi-logarithmic function was chosen [25,26]: x y 1 0 ln (1) where yan investigated dependent variable, хan independent variable, β0, β1model parameters, εa mistake. The parameter β1 shows a proportional change in у for a given absolute change in the factor, with an increase in the factor by one, the dependent variable will increase by 1 100  %. Wherein where r is a growth rate.
The authors used Chow's test [27] to check the stability of the trend time series model. The test statistic has a Fisher distribution and is calculated by the formula where ka number of model parameters, 1a sample size for the period 1, 2a sample size for the period 2, 1a period residual sum of squares 1, 2a period residual sum of squares 2, 0a sample residual sum of both squares. To determine the outliers in the time series, the authors used the approach of Chung Chen and Long-Mu Liu [28]. An iterative procedure for detecting and adjusting for additive emissions (AO), level shift (LS), temporal changes (TC), and innovation emissions (IO) was implemented to obtain estimates of model parameters and emissions effects.

Research results
The first step of the study was to test the studied series for stationarity using the Dickey-Fuller test. The results of the calculations (Table 1) show that the time series of the world gross production value, the world volume of imports and exports, and the price index for mushrooms and truffles are not stationary. The nonstationarity of the considered time series is also confirmed by the calculations of the autocorrelation coefficients ( fig. 1-4).  The study has shown that the world gross value of cultivated mushrooms and truffles, the world volume of imports and exports of mushrooms and truffles from 1961 to 2021, the price index for mushrooms and truffles from 1991 to 2022 increased significantly ( Table 2-  The results of the calculations show that the semi-logarithmic model of the gross value of cultivated mushrooms and truffles is significant at the level 5% (Fcalc=4,76, The stability of the growth parameters of the gross value of the production of cultivated mushrooms and truffles was tested using the Chow test. In 2000, there was a significant increase in this indicator by 85% compared to 1999. As a result of applying the test for structural changes at the point of 2000, the value of FЧоу = 65.471, р-value = 5.499e-09, therefore, the hypothesis of the absence of structural changes is rejected. A similar study for the series of world imports and exports of cultivated mushrooms and truffles also shows that there is enough evidence to say that there is a structural break point in the data. (Fimport = 25.877, p-valueimport = 5.074e-06; Fexport = 18.516, p-valueexport = 4.285e-05).
Chung Chen and Long-Mu Liu's method was used to determine other emissions in the series of gross production value of cultivated mushrooms and truffles. The authors have identified three level shifts (in 2000, 2009, 2011). This confirms the conclusion about the structural shift obtained using the Chow test.

Results
The analysis carried out has shown that the time series of the world gross value of production, the world volume of imports and exports, and the price index for mushrooms and truffles are non-stationary. The price index for cultivated mushrooms and truffles is stable, without sharp fluctuations. Three level shifts were identified in the gross production value series of cultivated mushrooms and truffles, several additive emissions, level shifts and time changes were found in the time series of world imports and exports.