Communications Economic models bases Prof.dr. Ion IVAN Adrian VIŞOIU Economic informatics department – A. S. E. 1. The economic models diversity The modern economy uses numerous analysis techniques and methods, out of which those orientated to the quantitative side of the process evolution are the most important. The building of economic models has as objective the analysis of the correlations between the factors of influence and the resultative variables out of which result analytical forms of the identified dependences. There is a large models diversity, each of the economic sciences branches saving very large presentation spaces for the results obtained on the basis of modeling. Direct models are the result of the immediate perception of the variables associated to the factors of influence. If it is considered a process to whom the resultative variable y is associated and who is determined in his evolution by numerous factors to which it associates, respectively, the variables , a direct model has the form: , where is a contribution coefficient, defined over the set . Therefore, the evolution of the stocks of the materials is modeled through the structure: where - the final stock of the material ; - the initial stock of the material ; - the entries through provisioning of the material ; - the exits for consumption of the material ; The contribution coefficients have the levels . In bookkeeping, for the accounts , the expenditures are entered in the debit side and the expenditures are entered in the credit side. For the calculation of the debit account balance, noted , it is estimated the expression: , where the contribution coefficients have the values: , . In the classic economic analysis, for the reproduction modeling in which two sectors are defined, the model is used S1 + V1 > F2 F1 + S1 + V1 > F1 + F2 S1 + V1 + S2 + V2 > F2 + S2 + V2 where: S1 – the product required in the first sector S2 – the product required in the second sector V1 – the surplus product from the first sector V2 – the surplus product from the second sector F1 – the substitution fund from the first sector F2 – the substitution fund from the second sector The econometric models represent a very important models category used especially for the study of the phenomenon and processes at macroeconomic level. It is considered a set of resultative variables , and a series of factors , to whom the variables are respectively associated. Both for the resultative variables and for the exogenous variables measures for moments are carried out, resulting a number of MN time series, where MN=M+N with: M - the number of resultative variables; N - the number of exogenous variables. These data series are entered in table 1. Table number 1. The data series registering Moment … … ... … … … … … … … … … … … … … .. … .. … It is important to establish: • the length of the time series; • the way in which the moment is selected; • the standardization of the data gathering periods; • the data gathering procedures The economic analysis causes the econometric models structures determination, resulting the dependences matrix , presented in table 2. Table number 2. Connections between the variables … … … … … … … … .. … where In the Klein-Goldberger econometric model, the following structural equations system is presented: The used variables have the following meaning: - the consumption in the year ; - the investments; - the private wages; - the demand at equilibrium; - the private profits; - the capital mass; - governmental expenditures others than the wages; - indirect taxes on profit and net exports; - governmental wages; - the yearly trend; The parameters are error variables, being called structural distortions. The parameters are structural parameters (regression coefficients), which connect the endogenous variables to the exogenous ones. In similar way, the parameters are structural parameters which connect the endogenous variables one to another. The dependences matrix is: 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 It results that if the econometric models equations coefficients are estimated for the resultative variable , it passes to the estimated levels calculation of these variables through the relation: , , equations number. - contribution coefficients ; - the coefficients of the model, estimated through the smallest squares method; - the estimated level of the resultative variable , from the Klein-Goldberger model Linear optimization models include in their structure functions with the form where - products number; - contribution coefficients defined over the set ; - multiplication coefficients; - the level of the variables associated to the factors, which influence the maximization or minimization processes. The linear models restrictions have the form: , where - contribution coefficients for the equation ; - limitary level for the defining of a limit of the variables interval for the used resources - unitary consumption of resource of type to achieve a unit. - restrictions number. The nonlinear models represent a large category used in the studying of the interdependences between factors. The great nonlinear models variety creates various effects in their systematized collecting and development, because at the description of this type of models rules must be defined, which conduct to correct descriptions and to implementations with a very high covering degree. There are situations in which the nonlinear models, through appropiate transformations are converted into linear models. For example, the nonlinear model: , through logarithmic transformation conducts to the structure . Performing the substitutions , , , , the linear model is obtained which can be rewritten in the form , where , and . All this conducts to the idea of finding some techniques and methods for stocking and administrating the economic models diversity, by using adequate software. 2. The entry data series All the economic models use entry data registered according to strict procedures. In the financial-bookkeeping system there are typified documents, there are indicators registering and calculation rules - for example, for filling in an invoice there is a procedure that allows that every person who develops a process, obtains an identical content with the content of the documents filled in free by all other persons who reproduce also identically the respective process. Data series are built that include identification elements and the levels of the variables associated to the factors. All the data are systematized in tables with the form , where: X - number of rows; Y - number of columns; A complete cash voucher has the form , because it contains: • a number of n rows corresponding to the acquired products; • a row for the total value registration without tax on added value? • a row for the total tax on added value level? • a row for the sum to be paid. The two columns correspond to: column 1 - the product code/name; column 2 - the bought product price. The table with the participation of the persons which work in administration has the form , where: • number of rows corresponds to the employees number which work in administration; • number of columns corresponds to the 31 days that can be maximum in a month A matrix ,is built, where an element will have one of the values: 8 - if the worker a has worked in the day j full program; - if the worker has worked in the day j a number of hours from the set {1,2,3,4,5,6,7}; / - if the day isn't a labor-day - if the worker was absent in the day j; M – if the worker is on leave in the day j; The tables where the evolution of the economic phenomenon progress is registered have the form , where K– the number of the moments , for which the data gathering is effectuated; L – the number of the factors, for which associated variables levels are registered, by using very strict procedures. For example, for the study of the monthly labour productivity over the last 3 years, , ,where: column 1 - moment ; column 2 - achieved production value; column 3 - workers number; column 4 - products complexity coefficient. In case that: =constant the data series are uniform in comparison with the gathering moments. Because the economic models entry data appear as a great diversity, it is important to define even from the beginning the aimed objective. If the data form the object of the concatenation, table structures are designed and solid gathering procedures are made. The census data from the districts level are formed in such tables. The economic agents balance sheets, also, have printed forms associated and the procedures describe explicitly from where the data for each form row is taken. Also, the procedures show the way of the general totals calculation. For the calculation of the final result of an economic agent , for a year, it is registered: - expenditure with the materials and the energy in the month of the year; -expenditure with the amortization in the month of the year; - expenditures with the wages; - penalization payments; - earnings from the sold production; For the performance level evaluation, at global level, for the economic agents , we calculate the efficiency indicator , where: - the contribution coefficients level ( means loss, means profit). In case that the data come from different sources it is necessary to specify how the gathering procedures are represented. The following situations appear: • all the data sets were gathered after the same procedures, have the same number of series sets, the series have the same length, are equidistant, the data gathering starting moment is the same, for example, in the 40 districts (there are 40 data sets) are registered; the data are taken as such because they are: correct, complete, comparable; • the data sets were gathered after the same procedures, the series have the same number of terms, the terms are equidistant and the starting moments are different, as in table 3. Table number 3. Data series with different starting moments T - - - - - - - - - - - - - In case that the result of the intersection of all the series terms has sufficient length, it is worked with the resulted series. In case that this desideratum isn't achieved, it is proceeded to extrapolation. Very complex extrapolation algorithms are used to include the processes natural trends. 3. The models complexity The complexity in Halstead sense for a model is given by the relation: , where - number of operands (variables and coefficients); - number of operators. The linear model defined through the equation: , contains: - - estimated model coefficients, in number of , ; - - exogenous model variables and the resultative variable, in total number of , ; - * - the operators of the multiplication of the coefficients with exogenous variables, in number of , ; - + - the operators which connect the exogenous variables forming the right member of the linear model equation, in number of , ; - = - the assigning operator. . Performing the substitutions, there are obtained: , . In case of the linear model given through the equation: , there are obtained: . It results that the complexity level in Halstead sense of the linear regression economic model is: . After the elementary calculations effectuation it is obtained: , . For the equation , results and the complexity in Halstead sense of the model is For the model , with , the complexity in Halstead sense of this model is: In case of the nonlinear model . - the operands are: , resulting ; - the operators are: , resulting . The raisings to a power are noted with , and the multiplication operator has been written explicitly twice because he appears twice in the equation. The complexity in Halstead sense of this nonlinear model is: . An unitary mode of complexity calculation must be imposed in order to avoid confusions. For example in case of the model , before the complexity is to be calculated, the extended transformed form must be obtained through the effectuation of the algebraic calculations: The appearance frequencies of the elements which appear in the model equation in the extended form are given in table 4: Table number 4. Appearance frequencies from the extended model Component Frequency A 3 B 3 2 2 2 1 2 2 3 2 4 2 Total coefficients 19 1 2 2 2 3 3 Total variables 13 Operator Frequency * 18 + 6 6 = 1 Total 31 The complexity in Halstead sense of the model is: The economic model refinement conducts to the diminution of the complexity in Halstead sense. If a linear model has in the initial form exogenous variables and respectively the complexity , through the diminution of the number of exogenous variables to , it is obtained a complexity , the complexity difference . By performing the calculations it is obtained: .Because a refined model has between and variables, , what conducts to the inequality . Because , it results in case that a complexity reduction is obtained. In case that through refinement it is passed from a nonlinear model to a linear model, also, the model complexity decreases. It is the nonlinear model: : and through linearization it is obtained : . The appearance frequencies of the two models operands and operators are given in table 5. Table number 5. Structures basis model and refined model Component Frequency in the model Frequency in the model 1 1 1 1 1 1 1 - 4 3 1 1 2 1 2 1 5 3 10 6 + 3 2 = 1 1 * 4 2 2 - 10 5 . It is important to analyse the relative complexity of the model in comparison with a maximum possible complexity that is obtained through the operands and operators aggregation. The relative complexity , is calculated through the relation For example for the model , with 4 operands and 3 operators, , , it results . For the model it is noticed that he contains the operands , with and the operators =,*, , and has therefore the same complexity calculated with the previous indicator, although the models have not the same operators and the operands have different functions. If the priorities of the operators , with , are considered, pointing out that the operation associated to the priority is executed before the one associated to the weight , it is proceeded to the building of a new weight system that associates the importance coefficients in decreasing order in comparison with the priorities. If it is considered for example the priority table: Operator Priority 1 *,/ 2 +,- 3 = 4 in the evaluation of the complexity in Halstead sense, the appearance frequencies of the operators are replaced with the associated weights. Operator Weight 4 *,/ 3 +,- 2 = 1 Therefore, the model: , has the adequate appearance frequencies and weights given in table 6. Table number 6. Weighted components nonlinear model Component Frequency Weight Frequency X Weight 1 1 1 1 1 1 1 1 1 2 2 1 2 5 1 1 1 1 1 1 1 1 1 3 8 = 1 1 1 + 2 2 4 * 2 3 6 2 4 8 19 while the model has the data in table7. Table number 7. Weighted components linear model. Component Frequency Weight Frequency X Weight 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 3 6 = 1 1 1 + 2 2 4 * 2 3 6 11 For the models : : It is obtained , which shows that the model is sensitive at the operators differentiation. The general model of the complexity in Halstead sense is: m – number of operands; - appearance frequency of the operand ; - weight associated to the operand in construction; - number of operators; - appearance frequency of the operator ; - weight associated to the operator ; in the considered case the weigh was built starting from the operators priority. For the polynomial equation of degree n, through: , there are identified: - the coefficients that must be estimated in number of n+1; the power coefficients 1,2,...,n in number of n; on the whole ; - model variables; ; = - an assigning operator with the frequency ; + - adding operator with the frequency ; - the operator for raising the power with the frequency . It results: The complexity C in Halstead sense for the polynomial regression economic model is given by the relation Taking into account that the evaluation of a function has the highest priority, the weight associated to the evaluation of a function is , with . If it is maintained: = +,- *,/ Therefore, for the complexity evaluation of the model first, the model is written under the form . Table number 8. Weighted components model with radical Component Frequency Weight Frequency X Weight 1 1 1 1 1 1 1 1 1 1 1 1 4 = 1 1 1 + 2 2 4 1 5 5 10 For the model Table number 9. Weighted components trigonometric functions model Component Frequency Weight Frequency X Weight 2 1 2 2 1 2 2 1 2 2 1 2 1 1 1 2 1 2 11 = 1 1 1 + 5 2 10 * 4 3 12 2 5 10 33 The complexity in Halstead sense is: For the nonlinear models complexities are calculated using the same rules. 4. Models generators The economic models collections point out the existence of some models families. A model M is considered that it has in the structure a set of exogenous variables. The transition to model is achieved through the inclusion in the variables set of a subset of new variables. The transition from the model M, through the elimination of a variable, conducts to the acquisition of a model with a more simple structure . When variables are added to the model I, it is a development process and the operator is noted with : . When it is passed from a model M to another model , through the elimination of a variable, it is a simplification process and the operator is : . It is observed that in case that the added variable is at the model M, that already contains the variables . and, . In case that through simplification the information acquired from the model continues to be relevant, the process is called the model refinement. The linear models generators represent very important mechanisms for the acquisitions of representative economic models. The independent variables and the dependent variables Y are considered. The economic practice conducted, in general, to the elaboration of linear models because: • the studied phenomenons aim a linear dependence; • the parameters estimation methods are customary for this type of models; • the results interpretation is lightened if the linearity hypotheses are taken into account. Models with one variable are created, with the form: , and is an order number of the model. Models with two variables are created: , , . In the same way are created models with three, four variables and the most complete of the models includes the n independent variables having the form: , where on the whole, for the n independent variables, the linear models structures are in number of . With each model structure a vector B with n components is associated, if the variable belongs to the model, in rest. By using the smallest squares method it is proceeded to the estimation of the coefficients of the models . … : … where are the coefficients that have to be estimated for the model k. For the models structures, the sums are calculated using the data arranged as in table 2 where represents the analytic expression of the model with the coefficients, R is the number of effectuated observations, the value of the dependent variable at the moment , and is the order number of the model. For the coefficients estimation of the linear models , a software product that functions under Windows operating systems has been elaborated. The product performs the following proceedings: • taking of data regarding the dependent variable at the moments , namely ; • taking of levels for the independent variables . The user can add, respectively he can delete independent variables. The free term generation is implicitly. • the data can be taken from files created previously by the program, which permits the taking again of the models defining; • the generation of all the models structures configurations and the calculation of the coefficients corresponding to each model; • the calculation, for each structure of model k of the sum , of the squares of the differences , where is the value at the moment of the dependent variables, estimated with the help of the previously calculated parameters; • the decreasing arranging of the models after S; • omission of those models k for which the ratio , where and it is a value chosen by the user and . The models quality is made obviously through the test data. It is considered that the dependent variable Y depends on the factors , after the rule with the form , with , being the observations number. (5.1) A number of observations regarding the levels of the independent variables and of another variable , which does not influence the level , are gathered by using the test data generator. The experimental data are found in table number 10. The level of the dependent variable Y at the moment i recorded in the table, is the one obtained through the substitution of the levels of the variables at the moment i in the upper formula. Table number 10. Experimental data Moment Y X U Z W 1 223 41 53 28 58 2 172 28 73 21 65 3 277 40 26 42 53 4 35 58 32 52 1 5 150 83 67 20 35 6 531 31 12 3 99 7 509 88 9 85 70 8 306 4 24 58 72 9 239 57 90 90 77 10 154 47 45 90 37 11 131 25 23 84 28 12 80 15 100 1 68 13 196 20 23 42 43 14 101 30 98 80 65 15 242 59 37 25 45 16 420 56 24 90 74 17 223 59 90 71 73 18 107 23 38 89 33 19 270 6 49 19 79 20 329 66 61 78 74 It is wanted the verification of the fact that the best model is the one in which only the variables influence . For the upper test data, used as entry data for the models generator and for , the following results have been obtained: • the first result returned by the generator is an analytical form for the resultative variable Y, identical with the one proposed for testing, , which confirms the generator capacity to choose the best suitable model; it is observed that all the factors that influence the level of Y are taken into account and that the unimportant factors are ignored indifferently from their number; the sum S of the squares of the differences between the effective levels of Y and the levels estimated with the help of the calculated coefficients is theoretically null, in actual fact having the size order because of the rounding of errors introduced by the intern representation of the numbers in the calculation system. • the second result returned by the generator is almost identical with the first, being introduced in the analytical form also the variable with the coefficient ; this insignificant coefficient correlated also with the size order of the values corresponding to this variable only underline the practice null influence of Z over Y; the size order of the sum S is also negligible. • for the value the generator does not offer other models structures, in exchange the obtained results are the most relevant. Another test to which the models structures generator is presented, is also through the insertion of deviations of different magnitudes, in a direction or the other, at the calculated levels of the variable Y through the formula (1) to verify if it is capable to identify correctly the influence factors. As a matter of fact, by keeping the values of the variables corresponding to the independent variables in the table, changes have been made to the column Y, this one becoming The measure of these changes is , . This corresponds to a square average deviation of units given the initial value. In this case, the results were the following: • the first result returned by the generator was , ; the analytical form approaches very much to (5.1); the presence of the variable Z in the model results from an accidental correlation between this and the studied variable Y, which is however so significant that it is included even in the optimal solution; the influences of the main factors are returned correctly, fact noticed through the coefficients, the influence from Z being smaller taking into account the size order of its values and the one of the coefficients • not very far from the optimal solution is an analytical form in which there are present variables considered from the beginning to be influence factors , having , ; the generator identifies the influence factors correctly, the model built with their help being considerable close to the best solution. The linear models generators with delayed arguments allow the elaboration of constructions which permit the modeling of the multiple stimulation effects which are found on short term in influences from all the sets. It is considered a set formed from equidistant moments , where and . The restriction regarding the equidistance does not affect the generality of the present approach. It is considered an element t from the set of the moments and the exogenous variables , for which there has been effectuated measurements in the moments . Also, for the endogenous variable Y, measurements have been effectuated in the same moments, resulting the data from table 11. Table number 11. Data set for set T Moment … … … … … … … … .. … where, - measured level of the dependent variable at the moment ; - measured level of the independent variable at the moment . The phenomenon evolution shows that the factors influence differently the resultative variable. More, the variation at a moment o a factor spread them with a delay abroad the evolution of the resultative variable. Further on, there are considered models of the form: , . For example, labor productivity W at a given moment t is influenced by the level of investments in the higher education at the moment , where h is the duration of the studies of a series of graduates integration. The linear model is: But the models with delayed argument can be also of the form: When the model structure is not known, it is necessary that the effectuation of a proper qualitative analysis and of a proper quantitative analysis. With that purpose, algorithms must be defined for the generation of the models with delayed argument which conduct to operational models structures. For the generation of linear models with a single delayed argument, there are considered the independent variable X and the dependent variable Y for which measurements corresponding to the moments , as in table 12. Table number 12. Data for the linear model with a single delayed argument Moment By using the smallest squares method, the coefficients of the model are estimated and is calculated. Where is the value of the dependent variables estimated at the moment and (1) is the order number of the model. At the next step of the algorithm, the initial data table is transformed in a modified table, through the sliding with a position of the terms corresponding to the independent variable, obtaining table 13. Table number 13. The argument is delayed by a period Moment - By ignoring the first row and by using the smallest squares method, it results the model and he sum of the square differences . For a delay of the argument by two periods, it results the model , the data being represented in the table number 14. Table number 14. The argument is delayed by two periods Moment - - The method is repeated, obtaining models of the form and the sums where s indicates the model order, and k, the delay taken into account. These models are arranged ascendingly depending on the calculated sums and the first l models with delayed argument which we consider good are chosen. Nonlinear models generators target the utilization of some structures of analytical forms. It is considered a set of analytical forms given in table 15 [6]. Table number 15. Analytical forms of the nonlinear models Number criteria Name of the model Analytical form 1. parabolic function 1 2. parabolic function 2 3. parabolic function 3 4. parabolic function 4 5. parabola of Neile 6. power function 7. exponential function 1 8. exponential function 2 9. exponential function 3 10. exponential function 4 11. logarithmic function 1 12. logarithmic function 2 13. semi logarithmic function 14. inverse log function 15. inverse log-log function 16. inverse function 17. Prais function 18. Function of order 3 19. hyperbolic function 1 20. hyperbolic function 2 21. Tornqvist function 1 22. Tornqvist function 2 23. Tornqvist function 3 24. Johnson function 25. parabolic log function 26. logistic function 27. square logistic function 28. Cobb-Douglas function 1 29. Cobb-Douglas function 2 30. CES function 1 31. CES function 2 32. CES function 3 33. CES function 4 34. Allen function 35. Sato function For a set of endogenous variables formed of and a set of exogenous variables formed of , a nonlinear model with one unknown factor is generated the variants being , obtaining models structures. So as it results from table 11, R analytical forms are defined before with a single exogenous variable, this involving the fact that the generator will conduct to the acquisition of a number of G models structures given by the relation . For example, for , . If the analytical forms exist , then the generated models structures are: It is very important that in addition to the analytical forms defined in table 10, the user has at his disposal facilities for defining personal analytical forms, to which he applies already existing structures generation mechanisms. For example, for models with the form the generation is effectuated both inside the structures for imposed values of the variables number and for transformations of the variables with a set of functions . For example, for the variables , for and for the functions and ,the models generator conducts to the acquisition of the following nonlinear models structures : The models generators have the mission to provide complete lists of different analytical expressions which cover the whole combinations diversity. With the developments towards the artificial intelligence of an informatics application, the models generator will include also self training elements extracted from the analysis of the models collection stored in the models bases and the premise of obtaining new models classes will be created, what corresponds to the achievement of a qualitative leap in the economic models structures. 5. Procedures Until the utilization of an economic model, the covering of numerous stops is necessary. The procedures meant for the coefficients estimation have as entry data: • the length of he data series associated to the influence factors and to the resultative variables; • the proper data series; • the structure of the model for which the coefficients are estimated, the code associated to the used estimation method. The results offered by these procedures are: • the estimated coefficients string; • the level of the sums of the squares of the differences between the resultative variables real levels and the estimated ones; • values associated to the result of the testing of some statistical hypotheses regarding the estimators quality; • information regarding the estimation development. In case that the estimation methods are used, which impose some restrictions on the model entry data series, before the development of the respective estimation algorithm, the hypotheses regarding those restrictions are verified. Otherwise it is passed to the utilization of other estimation algorithms free from these restrictions. After the utilization of these procedures, the information returned by the procedures has to be analyzed to validate the model in which the estimation process has been developed and to use the estimators in a consistent way, in comparison to the ensemble of the designing activities, analysis, realization and current utilization of the economic models. The procedures for operating on data sets target the realization: • the taking of the data sets; • the data sets homogenization through interpolation, extrapolation and elementary transformation; • the sets concatenation obtaining data sets with an increased number of data series; • the extension of the data sets for obtaining series with a greater number of terms; • the providing of the data comprehensibility through the using of terms transformation coefficients; • the data sets aggregation for obtaining new indicators The entry data contain: • the data series number • the data series length • processing typologies The obtained results are materialized in: • the new data series • the lengths of the new data series • the storage destinations of the series • information regarding the quality of the operating process. The procedures for the generation of test data have the mission to prepare entry data that will be used for the model research. There are numerous situations in which the existing data structures are incomplete, although the evolution trends of the phenomenon and variation limits of the associated variables levels are known. Also, the other existing data series lengths at one time are insufficient for producing qualitative estimations. For preparing an economic model, data sets have to be generated and the model properties have to be studied by using the generated data sets. In this way the behavior of the model is simulated. Procedures for the pseudoaleatory numbers generation are used, which follow different distributions rules. Entry data of the procedures for the data sets generations are: • number o data series which have to be generated; • lengths of the data series which have to be generated; • distribution rules which the generated pseudoaleatory numbers have to follow, the directions which have to be followed (any, ascending, descending); • the limits of the interval on which the generation is effectuated. The obtained results target: • the generated data series; • the data storage support; • the way in which the process has developed. For example, if the generation of a data series S is wanted, formed of n terms , with and , a procedure that realizes such a demand includes: • the mechanism for the selection of the instructions sequence for the generation of pseudoaleatory numbers which follow the distribution rule for the series S; • determination of the ratio r, with which the set of the intervals is built, in which the pseudoaleatory numbers series is generated, where …………………….. it results , , • referring to the pseudoaleatory numbers generation sequence of the interval for obtaining the value ; • repeating of the generation process with the obtaining of the string ; • repeating of the process for obtaining data sets. It exists the possibility to generate data sets, in which: • all the data sets have ascending tendency; • all the data sets have descending tendency; • some data sets have ascending tendency and others have descending tendency; • the ascending or descending tendencies are expressed on subintervals for all the data sets. The data sets are instituted in simulated expressions of some work hypotheses regarding the evolution of an economic process. The model will be tested on the generated data sets and the behavior in the different hypotheses will be seen. The procedures for the values calculation take models, estimated coefficients and data sets of the exogenous variables and calculate the simulations terms , for the models . The procedures for the models selection are used on conditions that there exist: • the data series containing the registered values of a resultative variable; • the data series with estimated data of the resultative variable obtained through the utilization of the models from the models base; • processes of securing the compatibility of all the data sets. An algorithm for selection from the multitude of models of those which belong to a homogeneity subinterval of specified length is defined. The procedures for refining the model have the mission to reduce the number of exogenous variables from the equation structure. In case of the models with more equations, besides the reduction of the exogenous variables number, the reduction of the equations number also takes place. An economic model with smaller dimensions is obtained. The role of the refinement is to reduce the administration effort of the model without losing the estimation quality of the resultative variables. For example it is considered the model M given by the equation: . After the effectuation of the coefficients estimation it is obtained: . By using the variables levels, the estimated values of the resultative variable are obtained. It is calculated: Through the factorial analysis the weights of the independent variables are obtained Variable Weight (%) Cumulated weights 87 87 9 96 4 100 TOTAL 100 - In the hypothesis that variables are included in the model so that the total weights are greater than , it results that the refined model has the structure : For the estimated values of the refined model it is obtained the sum of the differences squares The analysis model , where represents the acceptance limit of the cumulated weight of the factors, is given by the relation: here the models and his refined model were taken into account. If , the refined model isn’t representative and loses information. If , the refined model is good, the information loss is acceptable. If , the refined model is very good and it is taken in the economic analysis without any risks. Through the calculation of the indicator it results that the refined model must be accepted. In the same way it is handled also for the models with simultaneous equations. The procedure for the identification of the connections between the variables to set if between the two variables X and Y there are relations of the form (see the adjacent table): Also in case of three or more variables it is verified if one of them is the linear combination of two or more variables. For example for the variables Y, X, Z , it is verified if: the situations in which direct connections appear are pointed out and some variables are eliminated, resulting list of essential variables for the models. The procedures are based on the fact that, in case of the dependences from the presented types, the correlation coefficient is equal to 1. The procedure consists of: • the calculation of the correlation matrix • the selection of the variables for which the correlation coefficient is 1 • the reconstruction of the reduced list of the variables between which there are not direct connections. There are considered the data from table 16 Table number 16. Number criteria X2 X3 X4 X5 X6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Adequately a correlation matrix is created and the obtained levels conduct to the elimination of the variables between which there are linear dependences. The procedures for the data regrouping depending on the orthogonality target the reorganization of the data sets used in models as there were provided by the user. As the tables with data are inserted, a unique reference number is assigned to them and the module activation leads automatically to the utilization of the data set after the reference number. The procedures for the determination of the models orthogonality effectuate the quantitative analysis for the models description, pointing out: • the number of equations • the number of distinct variables • the number of distinct operands • the appearance frequencies of the operators • the operands positions and the operators positions. The obtained values are compared and the degree of similitude between the analyzed models results. If the orthogonality indicator , it results that the models are identical. If the same indicator has the value 1, it results that the models are totally different. The models are considered: : The data from table 17 result through the effectuation of the structural analysis of the four models. Table number 17: The quantitative characteristics of the models Characteristic Number of equations 1 1 1 1 3 Number of variables 2 4 3 2 3 Number of coefficients 2 4 3 2 6 Number of constants 0 0 0 ?1 3?? Number of distinct operators 3 =,*,+ 7 =,*,+ 5 =,*,raising to power 5 15 Operations 1 3 0 1 4 * 1 3 2 1 7 = 1 1 1 1 0 < 0 0 0 0 2 Raising to power 0 0 2 0 Min 0 0 0 0 1 / 0 0 0 1 0 The orthogonality degree between the models and is deduced in this way: • the columns of the models and are compared • if the elements of the row k are identical, the value 1 is assigned to the variable , otherwise the value 0 is assigned to the variable . • the number of the n characteristics, for which the orthogonality is analyzed, is determined; in table 17, : • the symmetric matrix of the orthogonality degrees given in table 18 results. Table number 18. The orthogonality matrix of the models 0 0 0 0 0 in case of the models with null orthogonality degree, it is proceeded to the correspondence table normalization. The correspondence table is in fact a homogenous structure which contains M components, where M indicates the number of different models stored in the models base. The quadruples included into a component indicate: • the orthogonal name of the model • the reference number of the model from the base • the reference number of the variables list • the reference number of the data set For example in the models base BM, the models are given: For the model the data from the table 19 are used. Table number 19: The data used in model Number criteria 1 10 1 2 15 17 3 17 1 4 31 2 5 42 3 6 61 3 7 72 1 8 55 6 9 110 2 10 130 8 For the model the data are systematized in the same way. A synthetic form regarding the initial structure contains data grouped in table 20. Table number 20. The table structure of the models existing in the models base Model name Model reference number Reference number data lists and variables Reference number data sets Columns used in variables tables 1 1 1 1,2 2 2 2 1.3 3 3 3 1.23 4 4 4 1.2.4 After the models analysis the orthogonality degrees result: which leads to the idea of the restructuration of the models table contents. Also it is observed that the data contain numerous identical elements: 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 It will result a new models structure, a new models table will be created and it will be proceeded to the reorganization of the reference lists of the data sets. 6. Administration system of the models base From the presented facts results that the models base is a complex construction in which lists are included with: - dependent variables and independent variables; - linear and nonlinear models structures; - registered data sets and generated data sets; - implementation of the coefficients estimation algorithms; - procedures for the hypothesis verification; - procedures for the estimated values calculation; - models hierarchizing procedures. These very important components are elements with which the models base is populated. For the models base to become operational, an administration system has to exist. First of all, the administration system has to operate distinctly with data sets, with the procedures and with the models structures. Second of all, the administration system functions have to effectuate the fast finding of the data sets, of the models structures and of the procedures in order to secure the processes development in concordance with the demands of the analyst economist. Third of all, the administration system has to be equipped with functions which permit the data sets adding, the models adding and the procedures adding. The perspective has to be changed, through which the models base bringing up to date implied data / models / procedures deleting or changing of some parts of these with new sequences. The acceptance of the bringing up to date function exclusive through adding comes to bring a concordance between the natural way of understanding the evolution with the corresponding reflection of it on informatics level. The models base administration system operates with non homogenous entities, important aspect in securing the flows consistency. Fourth of all, an open character is secured, the users having at their disposal the possibility of defining personal algorithms for interpolation and extrapolation, for pseudoaleatory numbers generation, for the coefficients estimation, for implementing personal models selection criteria. Fifth of all the, defining of the specific concepts regarding the finding, selection, extraction, targets the triplets (data, models structures, procedures), which group complex proceedings. Sixth of all, a growth of the generalization degree for the transaction concept is produced, which in case of the models base implies the traversing of some flows in which it is operated simultaneously with data sets, with data structures and with procedures. The new conglomerate, more complex than the object structure that includes the operands and operators, develops a new projection on the philosophy of designing an administration system, the administration system of the models base, in which new typologies specific to the implemented processes in the field of artificial intelligence are included besides the already usual proceedings. Utilizations of the models base imply activations of some sequences of procedures from the models base. The administration system of the models base is a construction with a very high complexity degree. The users must have the possibility of starting a small diversity of economic analysis projects. For example, the coefficients estimation of an economic model on the basis of a data set consists of: • the specification of the exogenous variables number and of the endogenous variables number; • the specification of the data series terms number; • the insertion of the data table • the delimitation of the variables list corresponding to the data series position Also, some results regarding the estimations quality are displayed. This procedure is specific to the situation in which the user has a clear image on the phenomenon and this one’s analysis is already a routine activity. 7. Conclusions The new instruments have to be first presented, after that the activated functions for the most often encountered problems types have to be explained and only after that the comparative analysis of the given results is effectuated. The models base is a complex construction, which has the mission to reunite in a single whole, like a human body, components with separate functions, however cohesive, which through the created interdependences solve a great diversity of problems types. The user defines his own problems which he administrates. It is important to have a correct image on the problems which he holds or he considers new problems. The reconstruction is the result of the reproducibility of some sequences of steps, of data sets and with models structures with the same starting contents, which the abandoned problem had or the new problem obtained from an old problem through the insertion of major changes when the basis problem from which the development was obtained was not saved. The open character of this instrument is given by the functions through which the user defines data sets and models, which, if they respect the orthogonality demands, are included in the models base in the form of permanent entities, free to be referred by any other users. The problem of the inclusion by users of the procedures, of the processing options and of the menus in the interface comes up against the resistance of the restrictions connected with the quality of the procedures as software, to which the elaborator has to secure correctness, viability and portability at least at the level which the administration system of the models base has in his ensemble. Through the inclusion of the users procedures, a significant heterogeneousness degree is obtained, which is administrated through the securing of the transparence concerning the new included procedures origin or through the securing of the temporary character or through the maintaining of the open character for source texts so that other users test and intervene for the errors correction. The models base offers elements which integrate in other informatics applications. Also it takes data sets which have results in the users informatics applications. The opening to the Internet shows the new valences of the economic modeling, as well as the tendency to obtain models structures generalizations in order to extend their utilization. The models base complexity has the tendency to increase because: - through the increase of the data sets number, through the increase of their diversity and through the increase of the non homogeneousness degree, procedures which allow the unitary referring and procedures for the securing of the assimilation from qualitative point of view of these new data sets have to be elaborated; - through the adding of new models structures or of new models typologies, algorithms have to be implemented which secure the coefficients estimation, which conduct to the obtaining of final results and which integrate them into the existing models multitude as a simple process of diversity increase; - when new types of problems are proposed to the users, visible options must be in the accessed menus and the necessary connections between the components already existing in the models base and the new components which are included to implement the new problems types must be defined. The problem of the models base has tendencies of favorable evolution, the new researches conducting to approaches orientated to the natural language utilization and to the newest results in the field of artificial intelligence. The development however has to be approached gradually, in steps to secure the operational character of the construction and not the distortions specific to a product found in an endless elaboration process. Bibliography [1] M. H. HALSTEAD – Elements of Software Science, Elsevier – North Holland, Amsterdam, 1977 [2] Ion IVAN, Romulus ARHIRE, Marian MACESANU – Program Complexity Analysis, Hierarchy, Classification, SIGPLAN NOTICES, vol 22, nr. 4, 1984, pp. 94-102 [3] Marian MACESANU, Romulus ARHIRE, Ion IVAN – Clase de complexitate pentru produse-program, Buletinul Roman de Informatica, nr. 1, 1985, pg. 63-68 [4] Ion IVAN, L. TOVISSI, E. MOSCOVICI, Utilizarea analizei entropice in studiul complexităţii programelor cu aplicaţii la normarea activităţii de programare, Buletinul Român de Informatică, nr. 4, 1982, pg. 39-44 [5] Ion IVAN, Adrian VIŞOIU – Generator de modele cu argument intarzaiat, Revista de Comert, vol.5, nr. 1, 2004, pg. 47 - 50 [6] Doru DUŢĂ, Csaba FABIAN – Manual de utilizare a pachetului de programe VERONICA, Lito ASE, Bucureşti, 1972.