Electronic Library of Scientific Literature - © Academic Electronic Press


Volume 49 / No. 5 / 2001


A Quarterly Econometric Model of the Slovak Economy QEM-ECM-1.0


The quarterly econometric model QEM-ECM-1.0 represents an experimental version of a new model of the Slovak economy which is based on the application of the ECM methodology. It was developed during 2000 and, from its economic content point of view, it follows the econometric models of the Slovak economy being developed in INFOSTAT, Bratislava during the previous time period.

In line with the principles of the market economy, the demand character is the dominating feature of this model which is expressed by a demand specification of regression equations presenting prevailingly the development of the main macroeconomic indicators of the real economy. The limiting factor in this sense is the short history of market relations in the Slovak economy which has influenced also the specification of the final forms of model’s regression equations.

The presented experimental version of the econometric model of the Slovak economy is formed by a simultaneous system of 80 dynamic, linear and non-linear equations and identities which express the relationships between 135 variables. The core of the econometric model QEM-ECM-1.0 is formed by 26 regression equations of which 25 are estimated based on the ECM methodology and one has the classical econometric form.

It is well known that the ECM methodology, being created by R. Engle and C. Granger, is based on the combination of statistical and econometric methods. Its attractiveness lies in the fact that, on the contrary to the classical econometric approach, it allows to express separately the short-term and long-term influences of explanatory variables on the development of the explained variable within the estimation of the regression equation.

The development of the main macroeconomic aggregates of the Slovak economy from the area of its real, financial and banking sectors is presented by the means of particular regression equations of the model. The following variables belong to the main endogenous variables of the model: GDP and the components of its use, trade balance and the balance of services, employment and unemployment, deflators of GDP and its components, nominal and real wages in the Slovak economy, state budget revenues and expenditures, money supply, amount of credits, interest rates etc.

Among the key exogenous variables expressing the main tools of the macroeconomic policy belong the following ones: exchange rate and the state budget balance. Among the so called truly exogenous variables one can put the indicators of the import to the EU and the Czech Republic and the price index of imports to the EU member countries.

The set of regression equations and identities of the model, from their economic content point of view, can be divided into the following blocks:

The working data base of the econometric model of the Slovak economy QEM-ECM-1.0 contains more than 150 quarterly time series of real, derived and auxiliary variables. The time horizon of the data base covers the period from the 1st quarter 1993 up to 2nd quarter 2000, i. e. 30 quarterly observations. The real values of time series of particular variables in combination with the seasonal filters were used for the estimation of the parameters of regression equations of the model. All regression equations of the model were estimated by the means of the OLS method where the starting year of the estimation was, as a rule, the year 1994. The system Eviews was used for the creation of data base as well as for the estimation of regression equations.

The article consists of two parts. The first part briefly describes the methodological approach which has been applied when constructing and estimating the regression equations with the error correction term. These equations form the core of the new experimental version of the econometric model of the Slovak economy QEM-ECM-1.0. The general characteristics of this model, results of the estimation of regression equations and their brief interpretation are the subject of the second part of the article. The notation of the model in the form of a simultaneous set of equations is, together with the list of its variables, presented in the Annex.



Viliam PÁLENÍK – Vladimír KVETAN – Katarína KRIVANSKÁ – Jaroslav VOKOUN

The paper presents medium – term prediction of the macro – economic indicators in the Slovak economy. The prediction originated in the Institute of Slovak and World Economic of the Slovak Academy of Sciences (ISWE SAS) by the application of an econometric model ISWE01q4. To understand better the economic development in Slovakia, development since 1998 is briefly characterised. Starting points and assumptions concerning future development of the internal and external economic environment define limited space for the future development estimates. As for the institutional environment one respected the intentions of the Parliament, the Government and the Slovak National Bank.

The concentration of singular influences contributes to the future development indeterminacy. The political cycle conclusion and the implementation delay of the Government coalition programme concerning the reforms and privatising belong to the strongest internal influences. The changes in external environment are the product of the dynamic development in the international political scene where the estimation of relevant impacts is difficult. The prediction is focussed at the estimate of price development, labour market, wages, monetary aggregates, and GDP demand components.

Monetary policy will act in an anti inflationary way. Price deregulation, the development of wages, and the growth of import prices will act in pro-inflationary way. Based on these influences predicted inflation in the years 2002–2006 will be in the interval from 6.7 % down to 6.2 % with the gradually decreasing tendency.

The growth of consumer prices exerts pressure from the employees on the increase of wages. The increase of unemployment on the other way enables the employers to revise the development of wages. Predicting wages development for the year 2002 one takes into account the election cycle. We expect, that to achieve labour productivity the enterprises will prefer wage increases to the increase of employee numbers. Nominal wage increase can thus reach about 8 % level.

In the model equation the level of nominal wages influences predominantly the labour demand development. The level of nominal wages acts negatively on the labour demand, while credits granted to the enterprises help to increase employment rate. Export is an important factor too; it promotes industrial production thus influencing positively employment rate. The employment rate development has a significantly inertial character. The labour demand development expected in the years 2002–2006 implies the increased significance of favourable influences concerning the employment rate.

One can explain the conclusions relating to the monetary policy in various ways, which are to a certain extent contradictory. Limited functioning of the transmission mechanisms can cause the questionability of monetary market reaction to the development of nominal or rather to real interest rates. Marked decrease of interest rates in the years 1999 and 2000 has had so far no impact on the development of interest emission and on the acceleration of monetary development. It is caused also by the development of the structure of monetary survey, above all by the increase of net foreign assets and credit for the general government. In spite of a certain inconsistency we regard the character of the monetary policy in the year 2001 and in the forthcoming years as moderately expansive and expansive.

The decelerated inflation development and the growth of domestic demand should become the pro – increase development of the Slovak economy. The continuance and slight increase of the GDP rate of growth from 3.2 up to 5.0 % will be the outcome of the revitalisation of the Slovak economy. The emphasis of the Slovak National Bank on the inflation goals and the implementation of proprietary changes in banks will create the environment with the lower preference for risk. Lower competitiveness will be the risk of the unbalance restoration in the economy.


On Foreign Capital Investment Bifurcations

Miloslav S. VOSVRDA

An economic model of cycles focused on the foreign capital investment phenomenon will be briefly introduced. We consider a system of the first order nonlinear differential equations where a feedback is controlled by a capital/output ratio parameter. The influence of the parameter changes on a macroeconomic stability is analyzed. A convergence of this economic system either to stable state, or limit cycle, or chaotic state is demonstrated.


Seasonality modelling of time and savings deposits of housEholds by state-space models


Lots of econometric models discussed in the literature are the models that contain trend and error components only. We now present the method seldom used for modelling trend (deterministic component), seasonal and error component using the dynamic models in state-space form. Model parameters are time varying Their development is estimeted adaptively by the Kalman recursions. In recent years quantitative systems based on the state-space representation have been used.

Based on the fact that economic and financial time series of quarterly and monthly or high frequency data contain seasonality, we do not take the seasonality into account, because the econometric model and forecasting method become only second best. A widely used method to incorporate the seasonality into the econometric models is the approach known as the decomposition method.

Unfortunately, the decomposition method has some disadvantage. The decomposition method is basically intuitive and there are a number of theoretical weaknesses in this approach and in estimating model parameters. State-space representation and the associated Kalman recursions have had a profound impact on time series modelling and many related areas.

In this paper we investigate the application of state-space models and the Kalman recursions to quarterly time and savings deposits of households forecasting and stress of the problems encountered with this modelling technique. The time and savings deposits of households is a model with a quarterly time step. Original data are quarterly time series providing a total of 24 observations (see [2]).

For this data, we can express the underlying model in the version with the local linear trend and constant seasonality by the following model equations system


where vt is the white noise variable with zero mean and s 2v variance. The first equation is a transition equation. By third equation is modeled seasonal component suggesting that the seasonal component is the some any one season each year. The first two equations are also the behavioral equations, the next equations are identities.

The above equations system can be represented in linear state-space form, i. e., the discrete series {yt} satisfies the equations of the form


where Xt is the v x 1 vector of inputs (state) or explonatory variable (in our case ), v is number of inputs, Yt is the w x 1 vector of observed (measured) or explained variables (in our case Yt = {yt}), w is the number of outputs. Wt, Vt are the v x1 or w x 1 vectors of independent Gaussian white noise variables with zero mean and covariance matrices by E () = Qt and E () = Rt respectively.

The first state space equation defines a sequence of observations. The second one is interpreted as describing the evolution of the state Xt of a system at time t in terms of a known sequence of v x v matrix Ft. F, G are known matrices or vector at time t. Assuming that the coefficients of the matrices F, G and the initial estimates of the unobservable state vector X are known, one may to obtain an estimate of the state vector Xt| t based on the information of Yt available at time t, i. e., based on the information of Y up to Yt. Denote the current period by s. We wish to estimate the state vector X at period based on the information at time s. Let t| s represents the estimate of Xt based on the information up to and including period s and Pt| s represents its mean square error matrix. Kalman recursions find out the best linear estimate of the state vector Xt based on the observations Y1, Y2, ..., Ys. Then the appropriate set of Kalman recursions for the estimate Xt| s in terms of:

As mentioned earlier the first v values of data were used to calculate the starting values of X and P. To estimate the variance of the random component vt in the behavioral equations, the log likelihood function of the state-space representation was used in the following form

Now assuming that the seasonal component of the state variable xt,2 in equation (1) is multiplicative – the magnitude of the seasonal swing is proportional (a ratio) to the trend xt,1. We can define the version of the structural model with multiplicative seasonal movements over time with respect to the local linear trend and constant seasonality model by


Similarly these equations can be represented in linear state-space form with

The Kalman recursions give the some estimates of the state values (trend and seasonal component) as in the case of the model with additive seasonal movements over time. This is because the trend xt,1, seasonal component xt,2 and the estimations of Xt,1, Xt,2 given y1, y2, ..., yN is given by the Kalman filter. These estimations obtaining by Kalman filtering are dependent only on the initial values of Xt, Pt and the known matrices F, Q, G, R only and in this case both the information set y1, y2, ..., yN and the matrices F, Q, G, R are the some.

For comparison purposes, we have generated forecast and computed the coefficient of the determination R2 for the time and savings deposits of households using the decomposition method, assuming an additive model. The values of R2 of the state-space models and the Kalman recursions are slight better.


The Model Structure for Forecasting, Optimization and Analysis of Tax Revenues in the Slovak Economy

Michal FENDEK – Michal HATRÁK – Vladimír MLYNAROVIČ

The paper combines an econometric and optimization approach to the analysis, simulation, optimization and forecasting the tax revenues in Slovakia. The macroeconomic background is represented by the relatively small macroeconomic model which is the base for the macroeconomic optimization. Macroeconomic optimized forecasts enter the tax submodel and forecasts of taxes are computed.

The approach that has been developing and gradually implementing at the Ministry of Finance of the Slovak Republic since 1997 links the operations research techniques and econometric modeling with economic theory to analyze, forecast and optimize tax revenues in the Slovak economy. The resulting model structures are made computable through their implementation in Excel using its solver and developed VBA user functions and procedures. In the current version on the base of data for the period from 1993 to 2000 the results for the period from 2001 to 2004 are provided and analyzed.

Planning of macroeconomic policy making plays an essential role in the development process of countries in the period of their transition. The experience of planning, however, witnesses failure in the use of appropriate quantitative tools. The project integrates econometric model of the Slovak economy and econometric sub-model of the tax system with multi-criteria optimization methods into a decision support system with its own set of scenarios that are fulfilled and directed by user for forming and evaluating corresponding macroeconomic policy.

On the top level of the system there is the econometric model of the Slovak economy that is implemented in the form of multiple criteria and multiple stage optimization problem where:

From the technical point of view the resulting problem is formulated through VBA user functions in the form of the Excel sheet and the corresponding optimized macroeconomic forecast is computed by the application of the Excel Solver.

On the second level of the system there is the econometric model which describe in the detailed structure the tax revenues. The optimized macroeconomic forecast provides inputs of the second level model. From the technical point of view model of the tax revenues is again realized in the form of Excel sheets through VBA user procedures.

The system of optimization and econometric model is completed with a set of scenarios that provides possibilities:

  1. to analyze effects of an assumed macroeconomic development on the tax system;

  2. to compare predicted values with expected effects of changes in the legislative of the tax system in Slovak Republic.




One intensely applies Data Envelopment Analysis (DEA) on the enterprise and bank levels. The author contrariwise tries to apply DEA analysis on the macro-level, specifically to evaluate predictions.

When evaluating the prediction efficiency one has to fulfil two basic prerequisites:

The transformation efficiency in the DEA analysis is expressed by a number within the interval (0,1). For the relevant number one uses the indication score. Efficient option is of value 1, to an inefficient one 0 is assigned. Less efficient option lies inside this interval. The relevant number expresses the value of the ratio of evaluated outputs divided by evaluated inputs. Evaluations are reached algorithmically by the solution of task (2).

Using DEA analysis the author evaluates the predictions of the Slovak economy for the years 1999–2000. Options indicated as a and b were prepared by INFOSTAT, Bratislava and the option indicated as c was designed by the Institute of Slovak and World Economics of the Slovak Academy of Sciences (ISWW SAS), Bratislava. The results of prognoses were dissected into four separate parts:

One presents the input and output data for each part.

To solve the problem, one uses the code DEA-SOLVER, which forms the annex of the publication Data Envelopment Analysis [3] in the compact disc form. The code DEA-SOLVER operates within the software Microsoft Excel 97/2000.

On the Tables 1 and 2 of the paper the author presents the data on inputs and outputs by individual institutions and years. In Table 3 there is the score for the individual parts of the system (real, taxes, money and payment balance); its values lie within the interval 0, 1. Score for the years 1999–2005 is synthesised in the form of sums or products. Relevant options can be mutually compared within individual years and for the predicted period as a whole. One produces the sums for the whole period by individual options a, b, c and after the value of the sum one considers the highest value of efficiency. In this paper one uses the calculations of the most efficient option c (the option of variant ISWE SAS).

The quality of the results of the efficiency evaluations depends on how we manage to design for the individual parts of the reproduction process the input-output macroindicators, which in turn depend on how well one prepared the predictions.

Relevant methodology can evaluate and compare individual predictions for a certain period, and find out, which of them delivers better results and which of them presents worse results, or enables to analyse deeper these predictions. This methodology, however, cannot evaluate assumptions, and cannot eliminate extreme or less realistic assumptions that support predictions; it can, however, contribute to discover them.

The application of these methods on the macro-level will enable their improvement or development.


Data envelopment analysis

František SUDZINA

The goal of this article is to extend data envelopment analysis by reverse envelope, worst coefficients and identification of outliers.

Reverse data envelopment analysis solves the problem whether there are weights which, if applied to all units, would make the selected unit the worst efficient. In other words, whether it is possible to choose common weights, so no other units would be less efficient than the selected one.

It is possible using reverse data envelopment analysis to identify the worst units in some sense. But it does not say anything about their real relative efficiency under the worst conditions. Measurement of the worst and the best possible efficiency gives a better idea about efficiency of decision-making units.

Outliers are the units, which lie in extreme positions. These are the units that use the fewest amount in at least one input component and/or produce the highest amount in at least one output component (depending on the model orientation).

Very often is a small increase in one output compensated by a significant drop in other output or a small decrease of an input is compensated by a big increase of another one. They are efficient because they lie on a production frontier, so there is no other unit, which managed to produce better results. But in the same time, they do not fulfill the idea of being efficient because their marginal rate of substitution of inputs or outputs is the worst one.

Outliers can be identified as the ones, which lie on the reverse envelope of efficient units. So, they can be easily identified in two steps by current DEA programs.

Efficient unit identified by data envelopment analysis can be used as input for statistical (regression) analysis. By elimination of outliers my be regression analysis able to find out statistically significant information even in data where it would not be possible before. This way could the best practice be modeled. The other possibility is to use reverse data envelopment analysis and model the worst practices. Elimination of outliers may play an important role also here.



František PELLER – Katarína SAKÁLOVÁ

In this paper we will describe the application of mathematic methods in the conversion of extra risk into the special insurance conditions in the life insurance.

The main fields of risks and uncertainties in the life insurance company are: mortality and sickness, investment gain, taxation and inflation, and new products of life insurance company. We will deal mainly with the risks connected with mortality. Life Insurance Company is exposed mainly to these risks that are for it the main and typical risks. The size and importance of the mortality risk depends mainly on the product type. From this point of view we can divide the life insurance products into the two types. The so-called risk insurance (for instance the temporary insurance for the case of death), which are sensitive on the mortality premises and the so-called reserve-making insurances (for instance the combined insurance – endowment insurance), which are not sensitive on the mortality premises but on the financial premises, above all on the interest rate (investment gain).

One of the most important tools used by the insurance company to reduce the mortality risk is the careful subscription. The insurance company can reduce the mortality risk also by blocking relatively large part of each insurance. Also in this case, however, it is important to subscribe carefully, as covering insurance company does not take over from the insurance company the part of whichever risk. The main demand of this covering insurance company is the subscription process quality.

We will deal further above all to the non-standard mortality risks (extra risks) and their conversion into the price of insurance (premium) of such a risk. The insurance companies identify extra risk in the process of subscription. The subscription and the life insurance are defined as the creation and sale of insurance. It consists of the definition of the risk and conditions under which the insurance company takes over this risk as well as of the definition of guarantees and persons, who will be given such guarantees by the insurance company. Thus, before the insurance company insures a certain person, it tries during the subscription process to find out if the standard insurance is adequate to the health state of the relevant person. Most insurance companies accept by 90 to 95 % insured standard rates. This percentage varies depending on whether the insurance company prefers the competition values of premium or large-hearted subscription policy. The size of the group of standard insured persons depends however not only on the competition, but also on the sort of product and on the progress in medicine.

Extra risks are the risks where the insurance company applies the use of special conditions. Extra risk related to mortality can be quantitatively expressed in three forms: percentage of normal mortality, addition to age, increased mortality intensity. The insurance company must convert these forms of extra risk quantification into special insurance conditions. Several methods exist. The most important are: extra premium, deduction, exclusion clause, postponed decision, rejection, offer of an alternative contract or the change of original contract. Two methods – extra premium and deduction are exact, e. g. they enable conversion of the extra risk by the quantifiable form. We deal with them in more detail.

Extra premium is the supplementary premium by which standard premium is increased. It reflects the character of extra risk and of the contract type. The method of its definition depends on the method of the insurance company in quantifying the extra risk. We can demonstrate how one can model the increase of mortality intensity up to the value by the increase of interest rate. To describe the situation we can use the linear methods of insurance mathematics. We will use the above-mentioned modelling for the calculation of the extra premium for basic products of life insurance (pure endowment insurance, temporary anticipated pension, endowment insurance, temporary insurance for the case of death and combined insurance) by means of the standard mortality tables at the increased interest rate.

Deduction is the sum the insurance company will deduct from the insurance settlement in case of death. In case of endowment term fulfilment the insurance settlement is unchanged Interested person with an extra risk pays the same premium as the one with standard risk, in case of death, however, the insurance settlement is lower. Deduction can be set by two methods. The first one is simple – deduction is constant throughout defined fixed time. At the second method deduction at the insurance start is quite high and decreases to nil as the insurance time progresses. We will evolve the formula defining deduction Z (t) related to the unit of insurance sum in time t in the case of decreasing deduction. At that the first method of risk quantification will be used. We will present also the advantages and disadvantages of the deduction and extra premium as methods of risk conversion into the insurance price.


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