Socioeconomic Development: Search For Optimal Models For Forecasting Inflationary Processes

Abstract: The article is devoted to the analysis of the forecasting models for inflationary processes. The monetarist approach to inflation has been discussed in the article, and the inflation forecasting tools have been classified. Various inflation forecasting models have been studied, namely, one-factor models, Phillips-curve-motivated forecasts, vector autoregressive models, and dynamic models of general equilibrium. Comparative characteristics of the quality of the inflation forecasting models have been provided.


INTRODUCTION
Inflation is a fairly common process, especially in the modern world. In simple terms, inflation is the depreciation of money. A more precise definition is the increase in prices for various goods and services over a long period. Inflation also influences real purchasing power. A different amount of goods can be purchased for the same amount of money in different years (Kolmykova 2012;Nasakina 2011;Savitskaya 2014).
Inflation in the market economy is manifested in a direct increase in the price of goods. It can take a slightly different form during the administrative intervention in the economy: there is no price increase, but there is a shortage of goods.
Inflation can have both positive and negative impacts on the economy. It has a stimulating effect on commodity circulation, serves as a factor in the natural selection of economic evolution, and contributes to the growth of the competitiveness of domestic goods. The state with a progressive tax system can obtain additional income from inflation (Volkov, 2012;Nikitin 1995;Illarionov 1995). Inflation accelerated to 3.1 % in April 2020 compared to April 2019 and 0.8 % compared to March 2020, according to the Federal State Statistics Service. Annual inflation amounted to 2.5 % in March, 2.3 % in February, and 2.4 % in January. Consumer prices rose by 2.6 % in January -April compared to the same period in the previous year. Prices for food products rose by 1.7 % in April compared to March 2020, where prices for food products net of fruits and vegetables *Address correspondence to this author at the Stremyanny lane, 36, Moscow, 117997, Russia; Tel: + 7 (495) 958-21-56; E-mail: mikhail_gasparian@mail.ru rose by 0.9 %. Prices for non-food products rose by 0.4 % in April compared to the previous month, and prices for services rose by 0.1 % (Starostina, 2020).
Reasons for the appearance of inflation are as follows: • Continuous currency emission to cover various expenses (mainly government spending); • Monopolies. This allows monopolists to set their prices for goods and services, thereby covering their production costs; • Reduction in the real volume of national output, which leads to an increase in prices at a stable level of money supply, because a smaller amount of goods and services corresponds to the previous amount of money; and • Increase and introduction of new state taxes, duties, excise taxes, etc., at a stable level of money supply (Ilyashenko, and Kuklina 2017;Lambin 1996;Foss 2007).
The observed increase in inflation expectations is determined by temporary inflationary factors, including a drop in oil prices. However, their change will be shortterm in conditions of decreasing demand. According to the forecast of the Bank of Russia, annual inflation will amount to 3.8 -4.8 % following the results of 2020 and will stabilize near 4 % in the future, taking the current monetary policy into account (CBR 2020).

LITERATURE REVIEW
The founder of monetarism is the representative of the Chicago school Milton Friedman (2002), a 1976 Nobel Prize winner in Economics. K. Brunner, Ph.
In monetarism, it is believed that the appearance of inflation is caused mainly by the financial policy of the state. The monetarist approach is that the markets are sufficiently competitive and the market competition system secures a high degree of macroeconomic stability. As such, if not subjected to government intervention in the economy functioning, the market system secures significant macroeconomic stability (Snowdon, and Vane 2002).
Monetarists adhere to the neoclassical position that the economy always functions in a state of full employment in the long run. Therefore, a change in the money supply does not affect the real economy in the long run, i.e., does not affect the real interest rate or level of investment and employment, and leads to an adequate change in prices.
Monetarists believe that economic growth can be determined exogenously, it does not completely depend on the growth rate of the money supply, while the money velocity is relatively stable. Therefore, when taking into account the equation of exchange MV = PV (where М is the nominal money supply, V is the money velocity, P is the price level, and Q is the volume of output), inflation is equal to the growth rate of the money supply.
Monetarists usually propose the so-called "tight monetary policy" to fight inflation. Its main task is to reduce the amount of money in circulation or to slow down the money velocity. Several events can lead to this result:

Level of aggregation
Inflation is measured using various indices: • consumer price index (CPI), • producer price index, • cost-of-living index, • asset price index, • GDP deflator, • purchasing power parity, • online store price index, and • average check indices.

Selection of Inflation Forecasting Tools
The following groups of forecasting methods can be used to predict the dynamics of prices during inflation: • Surveys

Mathematical methods and
• Modeling based on expert judgment.
Mathematical models are a very large class of methods used to build models and forecast, consisting of several subgroups: • One-factor models, • Use of trends (GDP deflators, CPI), • Models of autoregression, moving average, ARIMA, • Multifactor models, • Appeared in the 1960s, and • Modeling the inflation process based on expertly selected variables.
Many models have appeared over the entire existence of mathematical modeling, which is constantly being modified and refined (Ol'derogge et al. 2014;Volkov 2015;Gromov et al. 2007;Kiseleva et al. 2019). For example, the following models are in the greatest demand among the 20 types: • Phillips-curve-motivated forecast, • Vector autoregressive models (VAR), • Dynamic stochastic general equilibrium models (DSGE), and • Models using neural networks.
All classes of mathematical models for forecasting inflationary processes are based on the use of mathematical formulas and properties to increase the accuracy of the model, and therefore the accuracy of the prediction. The inflation gap indicator is one of the most striking examples. This indicator shows the gap or difference between actual prices and equilibrium prices for a given situation.
Another, no less popular way to increase the accuracy of modeling is to test the model on sufficiently large data samples. This allows not only to select the model coefficients with greater accuracy but also to check the significance of each of the factors included in the original model.

ONE-FACTOR INFLATION FORECASTING MODELS
It makes sense to use one-factor models for small periods or when the accuracy or details of the model are not critical. The main advantages of one-factor models are the following:

•
They are easy to build, • They can be easily adjusted, • These models are easy to understand, as are the results obtained after building and forecasting, and

•
The result can easily be interpreted.

The Most Common One-Factor models
"Random walk" models. Random walk models allow analyzing processes devoid of trends, seasonality, and other patterns. A variation of Atkeson and Ohanian is used in the modeling: where πt is the inflation rate in period t; h is the planning horizon and ɛt+h is the random error.
Direct autoregression models. DAR models are as follows: where πt is the inflation rate in period t; α, b are the coefficients for variables; h is the planning horizon, and ɛt+h is the random error.

Recursive autoregression models (RAR):
where πt is the inflation rate in period t; α, b are the coefficients for variables; and ɛt+h is the random error.
ARIMA models: where p is the order of autoregression; q is the order of the moving average; d is the order of difference; πt is the inflation rate in period t; α, b are the coefficients for variables; and ɛt+h is the random error.

Phillips-Curve-Motivated Forecast
The idea that inflation and unemployment were related turned out to be unstable during long periods of modeling. This is because all economic agents have their expectations, which greatly influence their behavior.
Due to this remark, a "triangular" model appeared, where the inflation rate depends on three variables: • Unemployment rate, • The shock of costs, and

• Previous inflation rate
where: • πt is the inflation rate in period t, • α, b, c, d are the coefficients for variables, • ut is the unemployment rate, • it is the variable describing the supply shock, • L is the lag value, and • ɛt is the random error.
This model was improved after successful tests, during which all variables were clearly expressed. Namely, dummy variables that described the state policy in the field of wage regulation and real purchasing opportunities were added.

Vector Autoregressive Models
The VAR models have been actively used since 1992. The general view of the VAR models is as follows: where: • Y1t, Y2t are the endogenous variables, • Х1t, Х2t are the exogenous variables, • α, b, c, d, e are the coefficients for variables, and • ɛ1t, ɛ2t is the random errors.
It was concluded through the study and application of the VAR that prices increased along with an increase in the interest rate. This was the so-called "price paradox" that arose due to the existence of a missing factor in the VAR model. It later turned out that this factor was a change in the interest rate.

B. Bernanke's Inflation Model
Another well-known inflation model was proposed by B. Bernanke. It is a factor-expanded VAR model that allows using about 120 factors, which can be combined into the following groups: • output and income, • employment, • consumption, • permits for the construction and sale of real estate, • stocks and orders, • stock indices, • exchange rates, • interest rates, • monetary and credit aggregates, • price indices, and • average salary.
This model also allowed to get rid of the "price paradox" due to the peculiarities of the interaction between the exogenous and endogenous variables.
One of the biggest advantages of the VAR models is the lack of restrictions on a wide range of specifications: • Economics, • Factors, • Interaction between the factors, and • Economic meaning.
However, these advantages can turn into big disadvantages if the situation and the interaction between the factors are not studied deeply and in sufficient detail. In practice, the high accuracy of the forecast is required in most cases, rather than the observance of all conditions and the precise adjustment of all factors in terms of meaning and economic concepts. The VAR models are popular due to this flexibility.

RESULTS
All models are usually selected by the criterion of higher accuracy of modeling and forecasting compared to previous models ( Table 2).
The use of different selection criteria on different models is a common occurrence. For example, many experts compare several models, which include both their development and third-party models, on their data to study the resulting model. This leads to the emergence of mathematical models that greatly exceed the accuracy of many existing ones (Bogoviz et al. 2019;Thompson 2009;Shapkin and Shapkin 2013).
However, there are minor reservations even in such methods, which can sometimes make it impossible to use the results in the future or other samples.
The process of creating such models usually involves the perfection of the existing modeling methods and techniques. This is achieved by expanding the number of variables, adding new functionality, etc.
This entails a significant complication of preparing data for processing. As such, all the initial data must be modified in a complex manner, which ceases to be observed over time, according to experience. Such an integrated approach leads to the supply of poor-quality data to the model, which either reduces the accuracy of forecasting or gives completely incorrect results (Morrow et al. 2007;Eliferov, and Repin 2013;Keller 2008).

DISCUSSION
The synergy of various models, opinions, and forecasts is an equally important way. Using equalweighted averaging and given the weight of the forecasts, or using the Bayesian model averaging method, even more, accurate forecasts can be obtained that will take into account all aspects that influence the inflation rate.
For example, large banks, consulting agencies, and other companies and corporations have long been using a peculiar synthesis of a fairly large number of models. They can take anything into account: from prices for goods and services to the global news ( Table  2).
The experience of the world powers testifies to the efficiency of the inflation targeting policy. However, it is still in its infancy and developing in Russia, unlike in developed countries. Therefore, it is too early to talk about its results. As such, inflation remains one of the most acute economic problems of modern Russia, which requires careful study and analysis. The further prosperity of Russia and its competitiveness at the global level depends on how fast, high-quality, and Inflation in the economy of the Russian Federation was caused by a decrease in the efficiency of social production, had a hidden nature, and was manifested in a shortage of goods and a gap in prices: low for final products and high for raw materials. Russia has seen a decrease in inflation over the past 13 years (Kondrus, and Kozyrkova 2017;Nazirov, Mehdiyev, and Dovletmurzaeva 2016).
The synergy of various models, opinions, and forecasts is the main way. More accurate forecasts can be obtained using equal-weighted averaging, given the weight of forecasts, or using the Bayesian model averaging method, which will account for all the aspects that influence the inflation rate. They can take anything into account: from prices for goods and services to global news.

CONCLUSION
Many specialists make calculations based on the obtained data to track the level of inflation. The mathematical models for forecasting inflationary processes are based on the use of mathematical formulas and properties to increase the accuracy of the model, and therefore the accuracy of the forecasting.