The article deals with the method of calculating the fractal analysis, the time series of economic sustainability of the industrial enterprise on the trend-resistant sustainability were investigated by estimating the depth of the long-term memory of the time series and constructing a phase portrait. According to the approach used, the “depth of the long memory” is estimated in terms of fuzzy sets. The approach to the estimation of the index of economic stability is developed, based on the methods of forming an integrated indicator consisting of an assessment of such subsystems as the industrial and technical, financial-economic and subsystem of main parameters of the market environment. These helps to estimate the economic stability of the enterprise in the conditions of incomplete information from purpose of making effective management decisions. Combination of techniques for the formation of an integral index and a fractal analysis of the assessment of its trend stability showed an effective result, which was confirmed by the experiments.

The article deals with the analysis of existing approaches to exchange rate forecasting. It also includes the review of Ukrainian and foreign scientists on this topic. The authors of this article have considered the main disadvantages and benefits of existing forecasting dimensions, as well as individual methods and models. They indicated ways to facilitate the implementation of currency exchange rate forecasting using neural networks with software libraries for various programming languages and individual software applications, as well. As a result, the authors have systematized knowledge about existing approaches used in the process of currency exchange rate forecasting. There are two dimensions of currency exchange rate forecasting, in particular, intuitive and formalized ones. The intuitive dimension is peculiar to short-term forecasting and is often used in trading. Its main advantages include the ability to consider structural changes in the economy that can significantly affect the exchange rate formation itself and the speed of forecasting. However, the disadvantage of intuitive methods is the inability to prove formally the quality of the obtained forecasts. The advantages of the formalized dimension of forecasting include the ability to prove the quality. Businesses and government agencies use it the most often. Extrapolation methods and machine learning methods are mainly used to predict the exchange rate using formalized methods. Moreover, the reviewed studies indicate that among the well-known extrapolation methods for predicting the exchange rate, autoregressive models (VAR, AR, ARMA, ARIMA, SARIMA, ARCH, GARCH, ARDL) and smoothing methods (floating averages, adaptive methods and models) are used the most frequently. Machine learning methods include neural networks. Trend models have proved to be ineffective for currency exchange rate forecasting. The reason for this appeared to be using large amounts of data for currency exchange rate forecasting, and each fluctuation there directly affects the whole phenomenon.

The article is devoted to a comparative analysis of the use of adaptive methods and models, autoregressive models and neural networks in forecasting the exchange rate of the main reserve currencies: the euro, the Swiss franc, the Japanese yen and the British pound against the US dollar. In the course of the research, the works of Ukrainian and foreign scientists on this topic were reviewed and it was determined that the most used methods and models in forecasting the exchange rate based on time series are autoregression models (represented by ARIMA and SARIMA models), neural networks (represented by MLP and ELM architectures) and exponential smoothing methods. In the process of building the models, time series were examined for stationarity based on the Dickey-Fuller test and additive decomposition of the studied time series was performed to determine their main components (trend, seasonality, random component). Construction of forecast models was carried out, on the basis of which their comparative analysis took place. The main shortcomings and problems of using the selected methods are demonstrated and the best predictive models are determined. It is determined that the main drawback of all time series forecasting methods is their "adaptability" to the input data, and the desire to improve the estimation characteristics of the models as a result can lead to the fact that the forecasts differ significantly from the actual values. It was also determined that for forecasting the exchange rate of selected currency pairs, neural networks are best suited, which have both high evaluation characteristics and correspondence of the forecast to real values, and the MLP network shows better results compared to the ELM network. High evaluation characteristics are also demonstrated by adaptive models. However, the linear nature of the forecast does not allow adaptive models to make an accurate forecast in the long term. Although autoregressive models show worse estimation characteristics, they outperform neural networks in terms of matching real values for individual currency pairs.

The article reveals the essence of the concept of "free trade" and identifies the main tools of this policy. The method of calculating the index of free trade is studied, the dynamics of the index of free trade of Ukraine for 2011-2020 and the dynamics of the components of this index are analyzed. Determining the impact of key factors on the level of GDP with the help of economic and mathematical tools allows you to form and propose ways to increase the level of economic development for Ukraine. Purpose. The aim of the study is to assess the impact of free trade on the development of the country's economy using methods of economic and mathematical modeling. Method (methodology). To study the impact of free trade on the economic development of the country, several models based on panel data were built. Gross domestic product was chosen as the result variable, with independent factors such as exports and imports, inflation, net foreign direct investment, labor, and the free trade index. In the study of panel data, the normal regression model (combined), the fixed effects model, and the random effects model were constructed. The constructed models were tested using the Hausman, Broysch-Pagan and Wald test. The comparative analysis of modeling results is carried out. Results. The results showed the high accuracy of the constructed models and show that free trade, expressed by the index of free trade, has a negative impact on the economic development of the country, expressed in gross domestic product.

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