Antoniuk O.P.

Associate Professor of National Mining University, Dnepropetrovsk

Using simultaneous equations for Modeling of the

interrelationship of basic macroeconomic indicators of Ukraine

A simplified model of macroeconomics was proposed in [1], which was based on the following assumptions: consumption was an increasing function of the available income; investments were an increasing function of national income; national income was the sum of consumer spending, investments and government procurement of goods and services. A mathematical model that related the above mentioned macroeconomic indicators was built on the basis of these provisions.

Based on this, a simplified model of the main macroeconomic indicators of Ukraine has been proposed, which in contrast to [1] has not considered state regulation of the economy; has accounted external borrowings and payments on foreign loans. The model represents the following system of three equations:

, (1)

, (2)

, (3)

In the above equations is consumption, is investment, is national income, is government purchases of goods and services, – is income tax in the year; and – are random variables with respect to which the standard assumption is given about their centredness (expectations are zero) and noncorrelatedness for different time periods , .

Estimation of structural coefficients has occurred in two stages. At the first stage, due to lack of information about the error variance in the regressions (1) – (3) it was assumed that they were the same, which led to the problem of estimating

, (4)

where the minimization was performed by , , and ,

, (5)

, (6)

. (7)

In (5) – (7) is the duration of the observation period, in our case years.

As the result of solving the problem of estimating (5) – (7), estimates of structural coefficients =0,02; =0,68; =5,55; =0,21 have been obtained. They satisfy a priori restrictions on the coefficients of the model in structural form . Moreover, the relative errors for the three regression equations are e₁=5%; e₂=8%; e₃=3%. These errors have been defined as the ratio of the average values , , for the range to the standard deviation of the residuals in the corresponding regressions (4) – (7).

The residues have been determined by the formulas

, , , . (8)

In the paper, the estimates parameters of the model for determining the national income as the sum of consumer spending, investment and government procurement of goods and services were determined.

The direction of future research is to test the quality of the resulting model, to make the statistical analysis of the accuracy of the constructed model and to identify ways to improve it.

References.

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2. Statistical Yearbook of Ukraine for 2012 / Ed. O.H. Osaulenko. / State Statistics Service. – K.: 2013. – 552 p. (in Ukraine).

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5. Christopher D. Introduction to Econometrics – 4h ed. – Oxford: Oxford University Press, 2011. – 560 p.

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8. Knopov P.S., Korkhin A.S. Regression Analysis Under A Priori Parameter Restrictions. – Springer, 2012. – 244 p.