Ph. D. in Physics and Maths Mezhyeva T.I.

Amur State University (Birobidzhan Branch), Russia

The models of long-term forecasting of economic processes and complex systems

The science-prognostics in recent decades, uses a great number of methods, procedures, techniques of forecasting, unequal in its application.

According to the assessments of foreign and domestic scientists of prognostics, there are over a hundred methods of forecasting, so professionals need to use methods that will be adequate for the studied economic phenomena, processes or systems.

In the literature there are a large number of classificatory schemes of forecasting techniques. Most of these techniques are not acceptable or have insufficient cognitive value. The basic inaccuracy of existing classificatory schemes is a violation of the principles of classification.

If a researcher is not an expert in applied mathematics, statistics, econometrics, difficulties will appear in using most methods of forecasting while implementing them in order to obtain high qualitative and exact forecasts. The most simple and widespread methods of forecasting are classic adaptive models forecasting in MS Excel. More complex methods are classical non-linear multifactorial models and neuron network methods of forecasting.

Complex non-linear multifactorial models cannot be calculated manually, so it is recommended to use the package Statistica.

To understand what benefits give the offered methods of data analysis and forecasting, it is important to note three fundamental problems occurring while forecasting.

1. Defining of the necessary and sufficient parameters to assess the condition of the researched subject area.

2. The large dimension of the model. The desire to take into account indicators and evaluation criteria in the model as much as possible leads to the fact that the model cannot be calculated.

3. The interaction of systems. Interacting systems form the system of higher level which has its own properties and this fact makes it absolutely impossible to analyze systems, included in the “abovesystem”.

To overcome these problems, attempts are made to apply such sections of modern fundamental and calculus mathematics as neuron computers, the theory of stochastic modeling (chaos theory), risk theory, catastrophe theory, synergetics theory and theory of self-organizing systems (including genetic algorithms).

These methods will increase the depth of forecast by identifying hidden regularities and relationships among macroeconomic, political and global financial indicators badly formalized by the usual methods.

According to the degree of formalization all the predictive methods are divided into intuitive and formalized. The intuitive prediction is applied when an object of prediction either too simple or complex so it is nearly impossible to consider the influence of many factors analytically. In these cases, the expert analysis is applied. The given individual and collective expert estimations are used as final predictions or as raw data in complex systems of forecasting.

The following models are used:

1.                     The integrated matrixes system of financial flows according to the estimations’ outcomes of the previous year.

2.                     The system of inter-branch balances in comparable and current prices.

3.                     The models of medium-term forecasting.

The models of medium-term forecasting are:

-              the model of time series analysis and making of inertia forecasts,  which allows preprocessing of data and determines the trajectory of the inertial development;

-              the medium-term econometric model of macroeconomic parameters of socio-economic development in the structure of the system of national accounts.

-              the balanced-econometric multibranch dynamic model, intended to build scenery forecasts for 15 years.

The model includes the following interrelated contours: production of goods, the income turnover, the payment balance, the monetary sphere of inflation, the workforce [10].

The long-term model lets:

-              to describe the possibilities of development and basic risks connected with microeconomic development, technological development;

-              to reveal arising in the forecasting period (15 years) risks and the threats to progress connected both with changes in the world markets, and with internal structural problems;

-              to estimate possible consequences of accepted administrative decisions as in the sphere of macroregulation (for example, the policy of the exchange rate), and in the sphere of branch progress (dynamics of the prices for production of natural monopolies, the results of realization of progress strategy, etc.);

-              to provide the coordination of macroeconomic and structural (branch) indices of the macroeconomic forecast.

The model is based on the developed information bases of various spheres of economics (microeconomics, industry and its branches, state finance, monetary bank credit investment external economic spheres, domestic sectors). This base includes about three thousand dynamic numbers of monthly, quarterly and annual statistics indices.

The simplest task of optimal management. One of the methods, applied at solving extreme problems, is identifying of some problem supposing rather simple decision. We shall consider the elementary problem of management. It looks like:

 

(1)

The general idea of the problem’s solution is the function of management quality reduces to its "splitting" on subtasks for each separately taken period of time, and the assumption, that they can be successfully solved. [8]. We shall construct Lagrange function for the task:

(2)

Where  vectors of Lagrange multiplies  Contingencies which bring a common character are not included in the target function in this case.

Write down the task in another form:

(3)

Necessary conditions of function extremum on a set of vectors  in the system are carried out by the recurrent way in the reverse order. The extremum’s conditions on a set of vectors  are as the results of task solution:

(4)

Thus, the task of searching for optimum management comes down to searching for managements, suspicious at optimality, specifically to finding such  satisfied the system of conditions which are called a discrete principle of Pontryagin maximum. [7].

Finding a maximum profit of a company. As an example, consider the problem of optimal production firm, the function of profits can be modeled on the relation:

(5)

1. Find the derivative of this function:

(6)

2. Equate the derivative to zero:

(7)

Is the volume of outputs equal to four optimus for the company? To answer this question, it is necessary to analyze the nature of the change the sign of the derivative in passing through the point of the extremum

3. Analyze the nature of the change of the sign of the derivative.

At  and profit decreases,

At  and profit increases.

(8)

Therefore, at the point of the extremum  profit takes a minimum value, and, thus, the production volume is not optimal for the company

4. Taking a decision.

The goals of economic mathematical models are varied: they are built for an analysis of those or other conditions and regulations of the economic theory, logic grounds of economic laws, processing and actuation to the system of empirical data. In practical terms, the economic mathematical models are used as a tool for forecasting, planning and management of the national economy and other economic activities of the society.

Literature:

1.                     Vyazgin V.A. Mathematic methods of atomized projection: train. manual for stud. higher educ. institutions / V.A. Vyazgin, V.V. Fedorov. – M.: High school, 1989. – 184 p.

2.                     Gluhov V.V. Mathematic methods and models for management / V.V. Gluhov, M.D. Mednikov, S.B. Korobko. – StP.: Publishing house “Lan’”, 2005. – 528 p.

3.                     Dubov U.A. Multicriterial methods of forming and choice systems’ variants / U.A. Dubov, S.I. Travkin, V.N. Yakimetch. – M.: Science, 1986. – 287p.

4.                     Konuhovskey P.V. Mathematic methods of operations’ research in economy / P.V. Konuhovskey / - StP.: Peter, 2002. 208 p.

5.                     Krichevetch A.N. Maths for psychologists / A.N. Krichevetch, E.V. Shishkin, A.G. D’ychkov. – M.: Flinta. 2003.

6.                      Rozen V.V. Aim – optimality – decision (mathematic models of taking optimal decisions) / V.V. Rozen. – M.: Radio and link, 1982. – 168 p.

7.                     Watshem T.J. Quantitative methods in finance / T. J. Watshem, K.K. Parramow. – M.: UNITY, 1999. – 528 p.

8.                     Chernorudskey I.G. Methods of optimization and taking decisions / I.G. Chernorudskey. – StP.: Lan’, 2005. – 384 p.

9.                     Shikin E.V. Mathematic methods and models in management / E.V. Shikin, A.G. Chhartishvilly. – M.: DELO, 2004. – 440 p.

10.                 Economic mathematic methods and fined models / Under editorship of V.V. Fedoseeva. – M.:Youright, 2012. – 328 p.