Econometrics

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Econometrics (from the Greek οἰκονόμος oikonómos 'rule for household management' and μετρία metrics, 'relative to measure') is the branch of economics that makes extensive use of mathematical and statistical models as well as linear programming and game theory to analyze, interpret, and make predictions about economic systems, predicting variables such as the price of goods and services, interest rates, exchange rates, market reactions, the cost of production, the trend of business and the consequences of economic policy.

Introduction

The economy, belonging to the social sciences, tries to explain the functioning of the economic system in its different aspects, such as production, consumption, money, income distribution, etc. The tool most used by economists is the construction of theoretical and mathematical economic models that describe the behavior of economic agents. However, these models must be contrasted with the available data to find out if they have explanatory and predictive capacity, and ultimately to be able to choose between one or the other options. The construction of such models is the goal of econometrics.

Econometricians, econometricians or econometricians (quantitative economists) have tried to emulate the natural sciences (physics, chemistry) with better or worse results over time. It must be considered that they deal with one of the most complex phenomena we know, the behavior of people and their interaction.

In the elaboration of econometrics, mathematics, statistics, social research and economic theory come together. The biggest problem that econometricians face in their research is the paucity of data, the biases that existing data may present, the researcher's own biases, and the absence or insufficiency of adequate economic theory. Even so, econometrics is the only scientific approach to understanding economic phenomena.

Econometrics Definitions

Among the definitions of econometrics that relevant economists have formulated throughout history, we can highlight the following:

Ragnar Frisch (1930): 'Experience has shown that each of these three points of view, that of statistics, economic theory and mathematics, is necessary, but by itself not enough for a real understanding of the quantitative relations of modern economic life. It's the union of the three aspects which constitutes a powerful analysis tool. It is the union that constitutes econometry."

Paul Samuelson, Tjalling Koopmans and Richard Stone (1954): '... the quantitative analysis of current economic phenomena, based on the congruent development of theory and observations, and related by appropriate methods of inference'.

Valavanis (1959): 'The objective of econometry is to express economic theories in a mathematical way in order to verify them by statistical methods and to measure the impact of one variable on another, as well as to predict future events and give economic policy advice to desirable results'.

A.G. Barbancho (1962): 'Echonometry is the most operative branch of economic science, tries to numerically represent economic relations through an adequate combination of mathematical economic theory and statistics. Thus, mathematics, such as language and form of symbolic expression and effective instrument in the deductive process, represent the unifying medium; and economic theory, mathematical economy or economic statistics would be partial considerations of its content.'

Lawrence Klein (1962): 'The main objective of econometry is to give empirical content to the a priori reasoning of the economy.'

Malinvaud (1966): '... application of mathematics and statistical method to study economic phenomena.'

Christ (1966): 'Producing quantitative economy statements that explain the behavior of variables already observed, or predict the behavior of variables not yet observed'.

Intriligator (1978): 'Rama of the economy that deals with the empirical estimation of economic relations'.

G.C. Chow (1983): 'Art and science of using methods for the measure of economic relations'.

Carlos Sabino (1991): 'Name with which the application of mathematical techniques and statistics is designated to the resolution of economic problems. Econometry is usually based on the construction of formal models with which it is possible to verify hypothesis, measure statistical variables and perform simulation tests'.

A brief overview of econometrics

Econometrics deals with obtaining, from the real values of economic variables and through statistical and mathematical analysis (but not from economic theory, as if it is used in the natural sciences, such as physics), the values that the parameters would have (in the specific case of parametric estimation) of the models in which these economic variables appear, as well as checking the degree of validity of these models, and see to what extent these models can be used to explain the economy of an economic agent (such as a company or a consumer), or that of an aggregate of economic agents, such as a market sector, or a zone of a country, or a whole country, or any other economic zone; their evolution over time (for example, to say whether or not there has been structural change), to be able to predict future values of the variables, and to suggest economic policy measures in accordance with desired objectives (for example, to be able to apply mathematical optimization techniques to rationalize the use of resources within a company, or to decide what values a government's fiscal policy should adopt to achieve certain levels of tax collection).

Concept of econometric model

Econometrics, like economics, aims to explain one variable in terms of others. This implies that the starting point for the econometric analysis is the economic model and this will become an econometric model when the necessary specifications for its empirical application have been added. That is, when the variables (endogenous, exogenous) that explain and determine the model have been defined, the structural parameters that accompany the variables, the equations and their formulation in mathematical form, the random disturbance that explains the non-systematic part of the model, and statistical data.

From the specified econometric model, in a second stage we proceed to the estimation, a statistical phase that assigns numerical values to the parameters of the model equations. For this, statistical methods are used, such as: ordinary least squares, maximum likelihood, two-stage least squares, etc. Upon receiving the parameters the numerical value, they define the concept of structure that must have a stable value at the specified time.

The third stage in the elaboration of the model is the verification and contrast, where the parameters and the random variable are subjected to some statistical contrasts to quantify in probabilistic terms the validity of the estimated model.

The fourth stage consists of the application of the model according to its objective. In general, econometric models are useful for:

Structural analysis and understanding how the economy works.
Prediction of future values of economic variables.
Simulate for planning purposes different possibilities of exogenous variables.
Simulate for control the optimal values of instrumental variables of economic and business policy.

Econometrics Methods

The Method of Least Squares (OLS Estimation)

It is also known as linear regression theory, and will be more developed in the statistical part. However, a general summary of the application of the method of least squares will be given here.

The starting point is to represent the relationships between an endogenous economic variable and one or more exogenous variables in a linear fashion, as follows:

$Y=alpha +beta _{1}X_{1}+beta _{2}X_{2}+beta _{3}X_{3}+...+beta _{n}X_{n}$

or:

$alpha +sum _{{{i=1}}}^{{n}}beta _{{i}}X_{{i}}=Y$

"Y" is the endogenous variable, whose value is determined by the exogenous, $X_{{1}}$ until $X_{{n}}$ . What are the chosen variables depends on the economic theory in mind, and also on previous statistical and economic analysis. The aim sought would be to obtain the values of the parameters from $alpha$ until $beta _{{n}}$ . Often this model is usually completed by adding one more term to the sum, called an independent term, which is one more parameter to look for. So:

$Y=beta _{{0}}+beta _{{1}}X_{{1}}+beta _{{2}}X_{{2}}+beta _{{3}}X_{{3}}+...+beta _{{n}}X_{{n}}$ .

or:

$beta _{{0}}+sum _{{{i=1}}}^{{n}}beta _{{i}}X_{{i}}=Y$

In which $beta _{{0}}$ It's a constant, we have to find out. Sometimes it is useful, for statistical reasons, to assume that there is always a constant in the model, and to contrast the hypothesis of whether it is different, or not, from zero to rewrite it according to it.

In addition, it is assumed that this relationship is not entirely deterministic, that is, there will always be a certain degree of random error (actually, it is understood that it covers all those variables and factors that could not be included in the model) which is usually represented by adding a letter to the sum represents a random variable. So:

${displaystyle Y=mu +beta _{0}+beta _{1}X_{1}+beta _{2}X_{2}+beta _{3}X_{3}+...+beta _{n}X_{n}+/}$

or:

$beta _{{0}}+left(sum _{{{i=1}}}^{{n}}beta _{{i}}X_{{i}}right)+mu =Y$

It's usually supposed to be $mu$ is a normal random variable, with a mean zero and constant variation in all samples (although unknown), represented mathematically as $mu sim N(0,sigma ^{2})$

A statistical sample is taken, corresponding to observations of the values that these variables have taken at different moments in time (or, depending on the type of model, the values that they have taken in different areas or zones or economic agents to consider).

For example, in a certain model we may be interested in finding out how income has depended on the levels of prices, employment and interest rates over the years in certain country, while in another we may be interested in seeing how, throughout the same year, the income of different countries has depended on the same variables. So we would have to observe, in the first case, income, employment levels, prices and interest rates for year 1, the same, but for year 2, etc., to obtain the sample over several years, while that in the second case we would have to take into account the values of each one of the countries to obtain the sample. Each of those observations for each year, or country, would be called a sample observation. Note that a more ambitious analysis could still be made taking into account country and year.

Once the sample is taken, a method is applied, which has its mathematical and statistical justification, called the method of least squares. This basically consists of minimizing the sum of the errors (squared) that would be obtained, assuming different possible values for the parameters, when estimating the values of the endogenous variable from those of the exogenous variables in each of the sample observations, using the proposed model, and compare those values with those actually taken by the endogenous variable. The parameters that will achieve this minimum, that of the sum of the squared errors, are accepted to be the ones we are looking for, according to statistical criteria.

Also, this method will provide us with information (in the form of some additional statistical values, which are obtained in addition to the parameters) to see to what extent the parameter values we have obtained are reliable, for example, to make hypothesis testing, that is, to see if certain assumptions that had been made about the model are true or not. You can also use this additional information to check if some of those variables can be dispensed with, to see if the parameter values may have changed over time (or if the parameter values are different in a different economic area). those of another, for example), or to see to what degree predictions about the future value of the endogenous variable are valid if it is assumed that the exogenous variables will assume new values.

Least Squares Method Problems

The method of least squares has a whole series of problems, whose solution, often approximate, has been occupying the work of researchers in the field of econometrics.

From the outset, the method assumes that the relationship between the variables is linear and well specified. For cases of non-linearity, either methods are used to obtain a linear relationship that is equivalent, or linear approximations, or optimization methods that absorb the non-linear relationship to also obtain values of the parameters that minimize the error. quadratic.

Another assumption of the model is the normality of the model errors, which is important when testing hypotheses with small samples. However, in large samples the central limit theorem justifies assuming a normal distribution for the least squares estimator.

However, the problem is considerably complicated, especially when testing hypotheses, if it is believed that the variance of the model errors changes over time. It is the phenomenon known as heteroscedasticity (the opposite phenomenon is homoscedasticity). This phenomenon can be detected with certain statistical techniques. To solve it, it is necessary to use methods that try to estimate the changing value of the variance and use what is obtained to correct the sample values. This would lead us to the method known as generalized least squares. A more complicated version of this problem is when it is assumed that, in addition, not only the variance of the error changes, but also that the errors of different periods are correlated, which is called autocorrelation. There are also methods to detect this problem and to correct it to some extent by changing the sample values, which are also part of the generalized least squares method.

Another problem that occurs is that of multicollinearity, which generally occurs when one of the exogenous variables actually depends, also statistically, on another exogenous variable of the same model considered, which introduces a bias in the information provided to the endogenous variable and may cause the method of least squares to not be applied correctly. Usually the solution is to find out which variables are causing the multicollinearity and rewrite the model accordingly.

It must also be taken into account that in certain models there may be dynamic relationships, that is, that an exogenous variable also depends on the values that it itself and/or other variables took in previous times. To solve these problems, what is called time series models are studied.

Econometric software

Among the most used programs are SAS, Stata, RATS, TSP, SPSS, Limdep and WinBugs. For more details, the following references can be noted.

EViews
Gauss
Gretl
Matlab
Microfit
R
Limdep
SAS
SPSS
Stata
SHAZAM

R as such is a programming language at the same time that it is a tool to apply this econometrics in a very powerful way. On the other hand, econometrics with the help of programs or programming languages and in a strict sense, does not require specialization. The econometric analysis can be done in Java, J, C, C++, C#, Python, Perl, Scheme, K, S (the main base of R together with Scheme) and derivatives of these languages as well, among other important amount of dialects or programming languages.

As an example of the previous paragraph, SPSS is a software initially created for statistical analysis in social sciences (see Wikipedia article). R initially as a project derived from S and with a more statistical purpose. Another example in this regard, Stata is a statistical program, but it allows powerful analyzes in econometrics.

Gretl is focused on making the interface very friendly to the econometrician, in addition to serving time series efficiently. Eviews, which owes its name to Econometrical Views (Econometric Views), has a purely initial purpose, the econometrics; for this reason, it displays an appropriate amount, but not very customizable, of highly useful information for these analyses.

Even the most advanced scientific calculators may have some basic elements for the development and testing of econometric models. It is enough that you can graph and in regressions you can calculate, by any means, that $mu$ is a normal random variable ( $mu sim N(0,sigma ^{2})$ ). If not, further steps would be required in the calculator. Even without being calculators, econmetric analysis can be done, such as MATLAB, Maple, Scilab. Clearly, mathematical programs that have just been mentioned have limitations, such as the number of observations they can support (e.g., the Scilab 5.5.1 version barely supported a matrix that between columns and rows reached five thousand).

Notwithstanding the benefits of both software, it generally depends on the devices in which the tool is going to be used. If, for example, Windows is preferred as the operating system, a significant number of licensed and free programs can be used; not so in GNU Linux. In this latest distribution and operating system, not many license distributions will be able to be used, although other equally powerful forms will. In Mac OS there are also problems with some paid or Open Source, Free License or Free Software programs.

The purposes of the econometric analysis will also determine the use of a certain program. For example, if what you want is something completely personalized, with levels of professionalism that are very suitable for international publications, programming languages are appropriate. These allow information to be exposed in its own way more easily than in others with predetermined interfaces.

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