# Econometrics

A direction of applied mathematical methods in economic research, the aim of which is to describe quantitatively the regularity and correlations of economic objects and processes on the basis of theoretical representations of their most important determining factors by means of mathematical models and statistical methods of data processing. Characteristic to econometrics is the assumption concerning the implicit occurrence of this regularity in indices that are measurable and applicable in economic statistics and prediction, in the background of an action of secondary random factors and auxiliary phenomena and, as a consequence, the aim to detect theoretically-predictable relations with a simultaneous choice of a concrete form for expressing them.

Isolated attempts at a quantitative description of the correlations between economic variables were already made in the 19th century. Evolving within the framework of capitalist economic science, the econometric direction uses its theoretical foundations, which restricts the possibility of a scientific analysis and a prediction of social-economic processes by means of econometric models. At the same time, the practical need to study phenomena at all economic levels — from national to individual firms — prompted the development and multiple application of a collection of typical formulations of economic-mathematical models and mathematical methods, which permit the isolation and analysis of quantitative correlations occurring in statistical data, and their substantiation, under some hypotheses on the admissible data and the relations concerned.

The detection and investigation of quantitative regularity and correlations in the socialist-state economy is subject to the problems of the development of economic theory and the need for planning and control. Compared to the econometric trend in the capitalist economic science, the scientific direction followed in the USSR and the socialist countries is more general in terms of its problem and applicable methods, studying the quantitative regularity and correlations in economics on the basis of the marxist-leninist political economy and the theory of control of the state economy and its components. It is designated in the literature by the term "economic-mathematical methods" . It includes not only methods for constructing quantitatively-defined models in terms of statistical data, but also all problems connected with the preparation and foundation of optimal economic solutions, the analysis of necessary conditions for their effective realization, the modelling of processes of the functioning of complex social-economic systems in which material and informational interactions occur simultaneously, and the elaboration and fulfillment of plans and control solutions at various levels, including the creation of conditions for the active radical change of relations and tendencies that have developed fairly steadily in the past.

Econometrics uses concepts, formulations and methods of solution of problems from many domains of mathematics, including mathematical statistics; probability theory; mathematical programming, numerical methods solving problems in linear algebra and for solving systems of non-linear equations, and numerical methods for finding fixed points of mappings; in many cases it has to deal with inverse and ill-posed problems in stochastic formulations. At the same time, econometrics applies mathematical methods not only to determine relations in a form suitable for statistical investigation (for example, as a regression equation with non-random factor arguments that is linear with respect to the parameters that are being evaluated), but also to more complex mathematical models formalizing typical problems for economic studies. Such typical economic-mathematical models that can be constructed and studied by means of econometric methods are: generating functions, which express stable regular correlations between the input and the results of the productive activity of economic systems at various levels; factor models of productivity; systems of demand functions for groups of consumers and specific functions of consumer preference; statistical and dynamic interdisciplinary models of manufacture, distribution and consumption of production; special models of interdisciplinary and interregional distribution and redistribution of resources; models of general economic equilibrium; models of foreign trade exchange for groups of countries; etc.

Faced with the practical necessity of extending the collection of initial hypotheses concerning the random factors occurring in the statistical data, econometrics formulates problems that are non-traditional for applied statistical analysis and generates methods for their solution. Thus, a number of econometric models treat the independent variables and parameters not as determined, but as random variables, include time-distributed interdependencies of the variables, use latent variables that are not directly observable, take into account a priori restrictions on the relevant parameters, postulate the change of the relations under consideration in time or in the space of the factors, and try to find the time moment of such changes or the sets of values of the factors on which they occur. Besides generalizations of the statements of problems on the detection of relations, a characteristic of econometrics is the theoretical and empirical (for example, by the Monte-Carlo method) study of properties and the relative effectivity of various methods of solution of problems of one class, as well as the tendency to formulate a system of recommendations by means of which the hypotheses concerning the models of the phenomena and objects under consideration can subsequently be verified and refined.

The development of an economic-mathematical direction employing a rich arsenal of applied mathematical methods and the possibility of using the tools of computational technology for the collection, storage and processing of data lead to a more accurate definition of classes of models and of the methods of their elaboration and analysis, unified by the generality of admissible systematic aspects. In particular, in a wide-ranging econometric approach to the development of economic-mathematical models it is accepted to distinguish a class of proper econometric models. Their generalized formulation, which can be made concrete in many directions, is as follows:

Let $X _ {i}$, $i = 1 \dots n$, be a collection of variables — economic indices — and let $X _ {i} ^ {t}$ be their corresponding measurable values in an interval of time $t$. It is assumed that these variables satisfy the system of relations

$$\tag{1 } F _ {k} ( X ^ {t} \dots X ^ {t - \tau } ; \ t ; a ; \epsilon _ {k} ^ {t} ) = 0 ,\ k = 1 \dots m ,$$

where $X ^ {t}$ is the state vector of the system under consideration during the period $t$, $X _ {i} ^ {t}$, $i = 1 \dots n$, are its coordinates, $F _ {k}$ is a function prescribed up to the value of the parameters represented by the vector $a$, and $\epsilon _ {k} ^ {t}$ is the realization of a random variable $\xi _ {k} ^ {t}$, $k = 1 \dots m$, such that the conditional probability distribution for the random variables $\xi _ {k} ^ {t}$, $k = 1 \dots m$, for fixed values of the vectors $\epsilon ^ {t-} 1 = \{ \epsilon _ {k} ^ {t-} 1 \}$, $\epsilon ^ {t-} 2 = \{ \epsilon _ {k} ^ {t-} 2 \} \dots$ denoted by

$$\tag{2 } {\mathsf P} ( \epsilon ^ {t} \mid \epsilon ^ {t-} 1 ,\dots ; b ) ,$$

is assumed to be known up to the values of some parameter vector $b$.

For the econometric model (1)–(2) the following basic problems are considered:

To determine the values of the parameters $a$ and $b$ in terms of the known collection of data $X ^ {T-} Q \dots X ^ {T}$ so that these data satisfy the relations (1) in some specially determined sense.

To construct a conditional prediction of the values of $m$ variables $X _ {j} ^ {t}$, $j = i _ {1} \dots i _ {m}$, in terms of prescribed values of the remaining $n - m$ variables $X _ {i} ^ {t}$, $i = 1 \dots n$, $i \neq i _ {1} \dots i _ {m}$, for $t = T + 1 \dots T + \tau$.

To compare several alternative models of the form (1)–(2), which differ by the chosen functions $F _ {k}$ and the law ${\mathsf P} ( \epsilon ^ {t} \mid \epsilon ^ {t-} 1 ,\dots ; b )$, for definite values of their parameters, and to determine whether they include models that are substantially better (worse), corresponding to the available data and qualitative arguments about the nature of the detected relations.

These qualitative formulations of the problems of estimating the parameters of econometric models, the construction by means of them of predictions, and the comparison of such models are defined in econometrics so that for some classes of functions $F _ {k}$ and laws ${\mathsf P}$ it turns out that it is possible to propose methods of solution and, as a rule, to realize them in the form of appropriate computer programs.

In practical investigations econometric methods are applied not only to the development of proper econometric models, but also in the process of creating more general and versatile models that also use normative, optimized and imitated approaches to modelling and overcome in this way the descriptiveness inherent in the econometric approach.

#### References

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