Methodological Aspects in Problem Solution of the Determined Factor Analysis

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1 OPEN ACCESS IEJME MATHEMATICS EDCATION 06, VOL., NO. 7, Methodological Aspects in Poblem Solution o the Detemined Facto Analsis Olga F. Chistik a, Vladimi A. Pokoev b, Aleksand G. Mokonosov c and Alina F. Kosilova d a Samaa State nivesit o Economics, RSSIA; b Saatov Social and Economic Institute (Plekhanov Russian nivesit o Economics), RSSIA; c Russian State Vocational Pedagogical nivesit, RSSIA; d Samaa State Technical nivesit, RSSIA ABSTRACT The ugenc o the analzed poblem is due to the act that the need to stud the elationship o economic phenomena and the compaative analsis o the acto atings not onl in time but also in space equies the development o detemined acto analsis. The pupose o the aticle is to develop a genealized method o detemined acto analsis, which inceases the accuac o its esults, and it will boaden and deepen its scope. The leading appoach to solve the poblems o detemined acto analsis is a sstematic appoach that allows establishing a moe pecise assessment o multiplicative component. Results: The poposed genealized integal method o detemined acto analsis has a high pecision o actoial assessment, and the gound constuction allows caing out a compaative analsis o the impact o actos on a poductive indicato o the suveed economic sstem o an level. The data o this aticle ma be useul o the analsis o economic activit o oganizations, as well as o the development o the economic development stateg. KEYWORDS Detemined acto analsis; Index method; Integal method; The ates o gowth ARTICLE HISTORY Received 06 Ma 06 Revised 9 June 06 Accepted 9 June 06 Intoduction Establishing a context At pesent the detemined acto analsis (DFA) to deine inteelations o economic development indicatos o an obects and teitoies is the most popula and widel spead on the inal stage o statistic eseach and analsis o the entepise economic activities. The methods ae the basis o text-books on statistics and analsis o economic activities and ae oten used in scientiic eseach. The esults o DFA ae used b administative bodies at all levels to develop decisions in tactic and stategic planning, contol and coodination o CORRESPONDENCE Olga F. Chistik uichistik@andex.u 06 Chistik et al. Open Access tems o the Ceative Commons Attibution 4.0 Intenational License ( appl. The license pemits unesticted use, distibution, and epoduction in an medium, on the condition that uses give exact cedit to the oiginal autho(s) and the souce, povide a link to the Ceative Commons license, and indicate i the made an changes.

2 76 O. F. CHISTIK ET AL. poduction development. The need to stud the elationship o economic phenomena and the compaative analsis o the acto atings not onl in time but also in space, which allows ensuing the accuac o its esults, equies the development o detemined acto analsis. Reviewing the liteatue The taditional method o DFA, a amilia method to students, eseaches and pactitiones, is the index method that has in its basis the ode o changing actos om quantitative to qualitative o an othe conditional pemise o the ode o changes, not continuous, smooth but intemittent which is hadl eve obseved in ealit (Lebedev, 0; Saonov, 04). Instead o the taditional appoach thee wee epeatedl oeed dieent appoaches to the possible tpes o changes in the actos and the extent o thei paticipation in esults assessment (Atemenko & Anisimova, 0; Lubushin, 00), but the wee eected b dieent opponents because o the identiied shotcomings o thei theoetical ustiication except the integal methods o DFA oiented opposed to the index (taditional) method towads simultaneous changes o acto indicatos (Bagautdinova, Makov & Jashin, 03). Howeve, integal methods wee not ull ecognized b the taditional apologists due to the inabilit to speci (estimate) the level o dnamic seies at an point in time (date) o the analzed inteval, compaable with the levels o calenda peiods, although this epoach could be attibuted to all othe DFA methods. Establishing a eseach gap All the above-mentioned data deines the poblems o DFA which obstacle to its development and impeection:. The absence o the univesal and scientiicall-gounded DFA method.. Simpliication o DFA b wa o chain substitutions in man cases leads to distotion o the ealit and causes unpedictable eos and contadictions up to opposite intepetation o the actos inluence - one - in the diection o inceasing the esult, not the decline in eal lie, and the othe - on the conta - in the diection o educing athe than inceasing the esult. 3. The absence o the possible level intepetation o dnamics seies at an date o the analzed inteval. 4. The discepanc between the aggegation assessment esults o the acto poductive indicato assessment o sub-peiods with its acto assessment o the peiod as a whole. Stating the pupose The pupose o the aticle is to develop a genealized method o detemined acto analsis, which inceases the accuac o its esults, and it will boaden and deepen the scope o the DFA. Mateials and methods Methods o eseach In the couse o the eseach, the ollowing methods wee used: theoetical (analsis, snthesis, dialectical method, scientiic genealizations); statistics (method o aveages (aithmetic and geometic), the index method, integal

3 IEJME - MATHEMATICS EDCATION 763 methods o DFA - decomposition o the absolute gowth index and poductive indicato on actos; analsis methods o dnamics); methods o mathematical statistics and gaphic esults. Expeimental eseach base Expeimental eseach base is the kind o economic activit - "manuactuing entepises" in the Saatov egion in tems o thei aveage sala. Stages o eseach The eseach was conducted at thee stages: - at the ist stage was caied out a theoetical analsis o the existing methodological appoaches to the philosophical, mathematical, and economic liteatue, as well as the theo and methodolog o statistical eseach: wee emphasized poblems, the pupose and methods o eseach, and was made up the pilot plan o the stud. - at the second stage wee pesented methods o impoving the accuac o the esults o DFA; wee given mathematicall easonable classiication o categoies in absolute and elative ate changes o economic indicatos depending on the method o compaing the distance and the time o thei levels; was intoduced the concept o tend levels measued o "astonomical" peiods that allowed ecognizing the legitimate application o highe mathematics o continuous unctions in the development and impovement o methods o DFA, was conducted expeimental wok, wee analzed and eined the indings obtained in the couse o expeimental wok. - at the thid stage was concluded expeimental wok, wee claiied theoetical and pactical conclusions and sstematized the eceived esults. Results Methods to incease the esults accuac o detemined acto analsis While conducting a citical analsis o the vast maoit o statistical methods, techniques and methods o DFA thee was omed the most impotant peequisite o a clea intepetation o the oiginal continuit o changes in economic indicatos which ae given in pactice b discete levels, and ixed onl at the beginning and end o the espective peiods elated necessail to the calenda and documenta ixed length (month, quate, ea). In ode to educe the discete level o social and economic phenomena, in othe wods - its oncoming to continuit, it is necessa to enhance the levels o calenda peiods b the levels o astonomical peiods o the same duation, stating and ending at the same date o the elated calenda peiods. Figue shows the natue o points location in astonomical and calenda peiods and maks the linea tend levels o the conditional indicato: 0, 30, 40, 50 o the calenda eas: 0, 03, 04, 05 at the end o each o them and, as an example, thee ae onl two additional levels o astonomical eas: 37 - om Octobe, 0 to Octobe, 03 and 4 - om Mach, 03 to Mach, 04 (Fig.).

4 764 O. F. CHISTIK ET AL. Figee. The points (dates) location at the beginning and end o the ea peiods on the X- axis and the levels o the indicato in coesponding units on the Y-axis Such astonomical levels can be deined o the moment as well as o the inteval pima and deivative dnamics seies at an point o the epoted peiod (ive eas, ea, hal a ea, quate, month), i not exactl, then appoximatel, on the assumption that thei value coesponds to the tend o changes in the calenda levels. Thus, the poposed assessment methods (exact o appoximate) o astonomical peiod levels at time points (dates) o calenda peiods negate the thid poblem o the eal existence and detemination o indicato values in continuous economic pocesses. The mathematicall gounded inequalit intepetation o the gowth ate o poductive indicato o each acto o the peiod to multiplicative acto assessment o the esult o sub-peiods that is calculated b the method o chain substitutions allowed to develop a numbe o impotant povisions to obtain estimates o most adequate DFA esults to simultaneous and continuous development o inteelated economic phenomena, leveling the second and the outh poblems o impoving DFA. Recommendations to incease the DFA esults accuac The useul esult o the eseach is the ollowing conclusion: It is impossible to develop a univesal method o DFA to calculate the absolutel exact acto assessment o an sstem o inteelated economic indicatos o an consequent calenda peiod. That is wh, to obtain the desied acto assessment it is advisable to do the ollowing: ) ill up the oiginal data base o calenda levels o economic indicatos b astonomical ones, ) deine the tendencies o changing inteelated economic indicatos, 3) use the integal DFA methods oiented towads these tendencies,

5 IEJME - MATHEMATICS EDCATION 765 4) implement the second and the thid tems in adacent sub-peiods o the analzed peiod and then to aggegate acto decomposition assessment o the gowth ate and (o) absolute gowth o the esult obtained o sub-peiods in genealized o the whole peiod. Economic and mathematical gounds o categoies in ate changes o economic indicatos and genealized DFA method The implementation o the thid and the outh conditions ecommended in Section should base on the clea intepetation o categoies in ate changes o the continuous economic pocess which does not contadict to the povisions o highe mathematics on the one hand and to its analogues in the theo o statistics (absolute gowth and the ate o incease) on the othe hand. The authos suggest the ollowing explanations to the intepetation o categoies in ate changes o economic indicatos. Absolute ate deinition As ou know, the numeato o the aveage speed is identical to the basic absolute gowth duing the inteval spanning the ull dnamic seies equal to the sum o chain gowth o adacent sub-intevals, and the denominato is equal to the duation o the whole inteval measued b the unit o time b which the aveage speed is calculated. Fo example, let us calculate the aveage ate change o emploee s annual wage who is engaged in manuactuing in the Saatov egion: ; ubles at the end o Decembe, 05 and 04: 34-0=4 (thousand ubles) in 05; quatel 4/4 = 6 (thousand ubles); monthl 6/3= (thousand ubles), dail 000/30=66,7 (ubles). Deining a linea tend o two eeence levels: t whee t 0,,,...,365, we calculated the aveage dail ate change o the ea levels: duing andoml selected ist thee das o Febua ( t 3, t 34 ): H (ubles); K K H h (ubles). t t 34 3 (ubles) и 66 K H Deceasing the length o peiod t K t H om 3 das to da, to0,; to 0,0;, shaes o the da, we obtained the same values o the aveage dail ate change o the annual level o wage: 66 ubles equal to the paamete a o the tend a a t whee t a is its ist deivative. 0 In geneal case it is convenient to call moment aveage absolute ate change o the tend level, to call d dt dieential absolute ate and to call absolute gowth o the level duing the inteval ( a, b ) equal to the integal sum o dieentials o the tend inteval absolute ate and to estimate it b the b value o the deinite integal dt a H K

6 766 O. F. CHISTIK ET AL. Relative ate deinition In conditions o the given example the ate change o emploees annual wage o the ea 04 was 34, 4 (.4 %). Then the aveage ate 0 changes o the annual levels o the ea 04 will be equal: quatel,4 4, 074, monthl,4,074 3, 0089, , 0003, dail. It means that in 04 the annual size o the emploee s aveage wage inceased eve quate.0003 times o b 0.03%. The aveage dail elative ate change o the annual levels o wages o small peiods o time in Febua T t K th K H is: o the ist thee das o Febua 34 3, 0003 ; o 0.o a da; o 0.0 o a da,0003. In the limit tansition in geneal case, i 0 The value level and will be maked as T h we have T e. e will be called the moment ate change o the tend T In ou example T e, Decembe 30 the aveage dail ate o sala incease was 00.08%., i.e. at the end B analog with the dieential absolute velocit the authos intoduce the concept o dieential ate change o the tend (unction) o an ininitesimal peiod o time dt which is maked as t dt ; t dt T dt. Dieential ate change is equal to the moment ate change in the degee dt and it is a necessa link o multiplicative integation o the pivate ate change o the ininitesimal adacent intevals in the inteval ate change: b b b T dt T t t dtt. a a a () Genealized method DFA o the elative ate change o economic indicatos

7 IEJME - MATHEMATICS EDCATION 767 Given b the omula () the inteelation o the moment, dieential and inteval ate changes o the unction t allowed establishing the impotant thing: the inteval ate change in the sum o the end numbe o unctions is equal to the end poduct o multiplications o dieential ate changes in the degee o speciic values weight o each unction in the total numbe o items. Thus, the attaction o highe mathematics methods in the theo o DFA assues that when assessing the degee o inluence o each additive acto on the elative ate change o thei oveall esult we should conside the ate change value o this acto as well as its contibution (shae) in the total amount, and it is sensible as om two actos with simila ate changes the geate impact o the sum on the dnamics will be made b the acto that has a bigge shae in this sum. The theoetical eseaches accompanied b numeical examples give gounds to asset that divesiied expansion was o the ate change o the poductive indicato o vaious cases o changing actos can be appoximated b the omula o the integal DFA method, and the accuac o the esults will steadil incease with the degee o agmentation o the analzed peiod in equal time peiods and involvement o additional astonomical levels (Pokoev, Nosov & Salomatina, 04): I whee m m n m ()... s,,..., 0 n l m is the numbe o totalit pats (o example, i l is the numbe o entepises in the complex, -is the numbe o complexes in the copoation, etc.); n is the numbe o time segments that the analzed peiod is divided in; s l -additive actos; l l - multiplicative actos l,s ; s - the numbe o multiplicative actos making coesponding additive acto ; m i - the individual index o the acto l l duing the peiod [, ]; m - aveage shaes o the intevals [, ] o the coesponding stuctual subdivision in the total value o this acto in the poduction omation (o example, is the aveage shae o the acto o

8 768 O. F. CHISTIK ET AL. the entepise in the total value o this acto in the complex; is the aveage shae o the acto o the entepise in the summa value o this acto in the copoation;, m is the aveage shae o the acto o the stuctual subdivision o the level m o the hieach in the total value o this acto in the poduction omation, o example, the aveage shae o the acto in the goup o copoations in thei totalit in the egion. To calculate the indicatos o the degee, indexes l i we can onl use thee dieent methods descibed in scientiic woks and text-books on statistics to deine the aveage chonological value o the moment seies o dnamics in the ollowing situations: piece-linea changes ; in case o small changes hal-sum o its levels at the beginning and at the end o the peiod; assuming exponential changes. sing a simple numeical example the authos illustate the assessment behavio o the actos impact (T - labo costs, W - poductivit) on the dnamics o the poductive indicato (Q - volume o output) o two entepises deined b dieent methods and was o DFA on the assumption o the ollowing altenatives: ) simultaneous change o single-stage sequence o actos duing the whole peiod; ) quatel eined dnamics o simultaneous o step-b-step actos development (Table ). Table. The indicatos o the entepise activities o the calenda epoted ea and astonomic eas at the beginning and at the end o eve quate o the epoted ea Entepise Yea levels at the beginning and at the end o the Indicatos quates o the epoted ea А B Q the volume o output, thousand ubles the shae in 0,6770 0,5485 0, ,500 0,4795 association T the numbe o wokes W output/woke, 47,87 43,4 40,83 4,5 40, thousand ubles Q - the volume o output, thousand ubles - the shae in 0,330 0,4575 0, , ,5075 association T - the numbe o wokes W - output/woke, thousand ubles 46,98 57,8 59,3 55,74 58,0

9 IEJME - MATHEMATICS EDCATION 769 Totall in association Q - the volume o output, thousand ubles T - the numbe o wokes W - output/woke, thousand ubles ,54 48,73 48,30 47,3 47,80 With the data om Table the omula will be: Q4 4 Q I Q Q Q 0 (3) whee is the numbe o the entepise; 4 Q Q - the aveage shae o the entepise output Q Q, ]; in the association in -quate [ i T i W, i, i W o i T, i W ae individual indexes o the numbe o the T wokes and the eicienc o labou o the dnamics o the esult Q o the entepise o the quate [, ] o the whole epoted ea; i T и i W - assessment o the changes inluence o labou costs and poductivit o two entepises on the value o the output index o the complex o the quate o a ea [, ]; i T и 4 4 i W - assessment o the changes inluence o each acto T and W on the value o the output index o the complex o a ea. You can see the single and total values o the changes inluence o the actos on the dnamics o the poductive indicato in Table. Table. The values o the inluence o the actos T and W entepises on the dnamics o the output obtained b integal method Entepise Duing the At the end o quates o the epoted ea s Indicatos whole peiod I II III IV (.0-3.)

10 770 O. F. CHISTIK ET AL. А B Totall in associati on T 0,995 0,9798 0,988,009 0,96 W 0,9407 0,976,0039 0,9876 0,906 Q 0,936 0,959 0,9866 0,9905 0,87 T,0037,0373,0,06,07 W,0893,007 0,970,005,09 Q,0933,0484 0,988,037,68 T 0,9989,063 0,9948,09,09 W,047 0,980 0,9748,0079 0,988 Q,034 0,9980 0,9697,073,07 Final assessments o the inluence o actos on the dnamics o the poductive indicato ae located in the intesection o the column 5 with the last thee o its lines Checking the index equalit o the poductive indicato, to the poduct o the multiplicative components o its acto decomposition conims the coectness o calculations made b all ou methods speciied in the columns o Table 3: the ist and the second,0,006,07 ; the thid and the outh,0 0,996,07 ; the ith and the sixth,07 0,990,07,09 0,988,07. ; the seventh and the eighth As Table 3 contains inomation on quatel simultaneous (paallel) changes o the ea levels o the acto indicatos but not on step-b-step changes, the esults obtained with the help o the integal method in the gaphs 3, 4 o Table 3 show the esults o the multidiectional impact o the actos: T towads the incease and W towads the decease o the poductive indicato Q moe exactl than the acto assessments o the index method in the gaphs, povoking the alse impession o the unidiectional impact o the actos T and W towads the incease o Q. The esults obtained ate implementing astonomical quatel annual levels in DFA using the index method - in columns 5, 6, and the integal one in columns 7, 8, ae signiicantl dieent om the pevious ones, while maintaining the pioit opinion that the inluence o the actos is multidiectional: T - towads the incease o Q and W towads its decease. Both methods even with a small split o the epoted peiod (in quates) athe than the moe signiicant (in months, decades) gave ve simila and moe accuate multiplicative components assessment esults o the acto decomposition in the index o the poductive indicato. Discussions The stud o liteatue allows the authos to conclude that the analsis methods o the elationship o socio-economic phenomena and pocesses ae widel applied to an obect in the inal stage o the stud (Aomand, Emami & Salimi, 06; Pocházka & Čemáková, 05; Savić & Zubović, 05; Mahlbeg et al., 03; Moskowitz, Yao & Pedesen, 0).

11 IEJME - MATHEMATICS EDCATION 77 In the taditional method o detemined acto analsis is the pemise o the conditional ode o inluence actos (Atemenko & Anisimova, 0; Lebedev, 0; Pokoev, Nosov & Salomatina, 04). K.N. Lebedev (0) and N.P. Lubushin (00) oeed a dieent appoach, chaacteizing the degee o paticipation o the consideed actos in the assessment o the esult. Howeve, integal methods, ocused on the simultaneous change o actoial indicatos, along with the taditional have a theoetical disadvantage - take into account the intemittent natue o the change actos. The pape poposes a genealized method o detemined acto analsis (DPA) in the stud o the elationship o economic phenomena. The authos illustate was to impove the accuac o the esults o DFA, and a mathematicall easonable classiication o categoies in absolute and elative ate changes o economic indicatos depending on the method o compaing the distance and the time o thei levels. The concept o tend levels measued o "astonomical" peiods allows ecognizing the legitimate application o highe mathematics o continuous unctions in the development and impovement o methods o DFA. The authos also highlight useul eatues o the genealized method o DFA to boaden and deepen the scope o its application The aticle epesents methodological aspects in solving development poblems o detemined acto analsis, which have not been consideed in pevious studies. Conclusion The eseach showed that the genealized integal DFA method is chaacteized not onl b high accuac o acto values but also b othe positive qualities, which make it possible to develop and widen the sphee o its use. Because o its constuction containing the poduct o acto values o the peiods o time o the epoted peiod it is possible to make compaative analsis o the actos inluence on the poductive indicato o an level o the studied economic sstem o sub-peiods o the same and dieent duation. The next peculiait is the availabilit o the detemining actos inluence assessment o stuctual unit at an level o the hieach in complex poduction omation on a poductive indicato. At the same time acto assessment oms the sstem o inteelated vaiables which allows obtaining assessment o the industial omations o highe level (entepises, complexes, deinite economic activities) om assessment o the actos o the stuctual divisions o lowe level. This ensues the possibilit o deepening the compaative analsis o actoial assessment not onl in time but also in space, exploing the eect o simila actos o dieent depatments within the same poductive omation. The pincipal assumptions and the genealized DFA method ae ee o consevatism and subectivit inheent in the taditional method, and theeoe it has a high degee o impovement, modenization and adaptation in elation to detemined modeling o the elationship o economic indicatos having an complexit. Acknowledgement

12 77 O. F. CHISTIK ET AL. The authos ae gateul o the suppot in ecognizing thei economic and mathematical intepetation o the poblems o DFA development and peection - the poblems solving and implementation in DFA b the amous scientists in the ield o statistics, Doctos o Economics - L.S. Kazinets, I.I. Eliseeva, L.M. Kaplan, M.M. Yuzbashev, G.S. Kildishev, V.M. Rabtsev, E.V. Zaova, V.N. Aanasev and man othe scientists. Scientiic, educational and methodological publications (that have been done b emploees o Statistics Depatment o SSSE) on DFA integal methods ae embedded in the educational pocess o Saatov State Social and Economic nivesit (SSSE) and late in Samaa State nivesit o Economics (SSE). Notes on contibutos Olga F.Chistik is Poesso o Samaa State nivesit o Economics, Samaa, Russia. Vladimi.A. Pokoev is Poesso o Saatov Social and Economic Institute, Russia. Aleksand G. Mokonosov is PhD, poesso o Russian State Vocational Pedagogical nivesit, Ekateinbug, Russia. Alina F. Kosilova- is an applicant o the degee o Candidate o Econom o Samaa State Technical nivesit, Samaa, Russia. Reeences Aomand, M., Emami, K. & Salimi, F. (06) Gowth and Poductivit: The Role o Budget Deicit in the MENA. Selected Counties Pocedia Economics and Finance, 36, Atemenko, V. G. & Anisimova, N. V. (0) Economic Analsis. Мoscow: Knous. 88p. Bagautdinova, N. G., Makov, V. A. & Jashin, N. S. (03) Impovement o Instuments o the State Cluste-Based Polic in the Contexts o Economic Entities Inteelation Asmmet. Wold Applied Sciences Jounal, 7, Lebedev, K. N. (0) Poblems o Facto Analsis Based on the Methods o Deteminate Facto Analsis. Economic Analsis: Theo and Pactice, 3, 4-3. Lubushin, N. P. (00) Economic Analsis. Мoscow: nit. 575p. Mahlbeg, B., Feund, I., Cuaesma, J. C. & Pskawetz, A. (03) Ageing, poductivit And wages in Austia. Labou Economics,, 5-5. Moskowitz, T. J., Yao H. O. & Pedesen, L. H. (0) Time seies momentum. Jounal o Financial Economics, 04(), Pocházka, P. & Čemáková, K. (05) Inluence o Selected Institutional Factos on the Economic Gowth: Case Open Makets. Pocedia Economics and Finance, 30, Pokoev, V. A., Nosov, V. V. & Salomatina, T. V. (04) Peconditions and Conditions o Development o Deteminate Facto Analsis (The Poblems o Economic Analsis ). Economic Analsis: Theo and Pactice, 4, Saonov, А. А. (04) The Theo o Economic Analsis: textbook. Vladivostok: VS. 64p. Savić, M. & Zubović, J. (05) Compaative Analsis o Labou Makets in South East Euope. Pocedia Economics and Finance,,