Short-term Forecasting of Reimbursement for Dalarna University

Transcription

1 Shor-erm Forecasing of Reimbursemen for Dalarna Universiy One year maser hesis in saisics 2008 Auhors: Jianfeng Wang &Xin Wang Supervisor: Kenneh Carling

2 Absrac Swedish universiies are reimbursed by he governmen according o a scheme relaed o he regisraion of sudens (HSTK) and he sudens performance (HPRK). On a disaggregaed level, such as a deparmen or a field, he reimbursemen is uncerain as he number and performance of sudens are flucuan. So he adminisraion faces he challenge o balance he reimbursemen and he expense. In his hesis, we ry o disinguish a beer forecasing model for he educaional fields or he deparmens of Dalarna Universiy. We analyze he ime series by wo mehods, namely Census II mehod and he ARIMA mehod. We apply he wo mehods o wo approaches, direc and indirec approaches. The firs one is o use he ime series of reimbursemen direcly; he second one is o forecas he number of sudens and heir performance separaely, and hen conver hese wo forecased values ino reimbursemen. To compare he resuls, we choose he indicaion of MAPE. We use monhly daa from Jan o Feb Finally, we find ha, on he deparmen level, he ARMA model of he indirec approach is he bes one for forecasing he reimbursemen. Keywords: Census II Mehod, ARMA Mehod, MAPE

3 Conens 1. Inroducion Daa Descripion Mehod Resuls for he Field of Naural Science Comparison beween he Census II Mehod and he ARMA Mehod Conclusion Reference Appendix

4 1. Inroducion In Sweden, he reimbursemen for he higher educaion secor is always increasing unil recenly. In 2004, he reimbursemen became o decline parly because he educaional insiuions enrolled less sudens han before 1 as he reimbursemen is highly relaed o he number and he performance of he college sudens. To some exen, he reimbursemen for a universiy will flucuae around a rend. Our hesis is o esablish an accurae model o forecas he reimbursemen for he fields or he deparmens in he universiy. The model may help hem o balance heir expense and reimbursemen beer. In his hesis, we ake he Dalarna Universiy as an example. Number Year Figure 1: The number of full ime equivalen (FTE) sudens in Sweden Source: 2005 Annual Repor of Swedish Naional Agency for Higher Educaion Figure 1 shows he number of sudens regisered in In 2005, he oal volume of higher educaion comprised 295,150 full-ime sudens. This is a reducion of 2.5 percen compared wih he previous year and his is he firs ime for weny years ha he number of full-ime sudens declined 2. Dalarna Universiy was esablished in Today here are abou sudens sudying 1 hp://hsv.se/saisics/heannualrepor.4.539a949110f3d5914ec hml 2 hp://

5 in his universiy. Disance educaion is also offered in various forms. There are more han 200 courses wihin he areas of welfare, business, infrasrucure media, culure and ourism and so on 3. All of he courses are classified ino 11 fields: he Humaniy, Spor, Law, Educaion, Media, Medicine, Naural Science, Social Science, Technique, Healh and Care, and he Ohers, he amoun of he reimbursemen depends on hese fields. The courses can also be classified ino 5 deparmens: Culure, Social, Healh, Humaniy, and Indusry. The fields and he deparmens a Dalarna Universiy are shown in Table 1. Table 1: The share of he fields and deparmens in erms of he average reimbursemen from 2000 o 2008 (%) Deparmen Culure Social Healh Humaniy Indusry Marginal size Field Oher Humaniy Spor Law Educaion Media Medicine Naural Science Social Science Technique Care Marginal size This able shows he size each field akes up in he deparmens and he marginal size of he deparmens and he fields in Dalarna Universiy. For example he field Humaniy akes up 5.39 % in he Culure deparmen, and abou 7.33 % in his school and he Culure deparmen akes up 17.1 %. From his able we can see he Spor field is he smalles one, whereas he 3 hp://du.se/templaes/sarpage 1622.aspx?epslanguage=SV - 2 -

6 Technique field is he larges one. The key of his hesis is ha we classified he daa no only by he fields, bu also by he deparmens. To forecas he reimbursemen, we choose wo approaches: he direc approach and he indirec approach. The firs one is o use he ime series of he reimbursemen direcly; he second one is o forecas he number of sudens and heir performance separaely and hen conver hese forecased values ino he reimbursemen. We use wo mehods, namely he Census II mehod and he ARIMA mehod, o model hese wo approaches on he field level and he deparmen level. 2. Daa Descripion We use he monhly daa from Jan o Feb The Swedish universiies are reimbursed by he governmen according o a simple scheme relaed o he regisraion of he sudens and he suden s performance. The allocaion rule of he reimbursemen is: 60% of he reimbursemen is based on he regisraion and 40% is based on he performance 4. In his hesis, we use HSTK as he shor form for he regisraion of he sudens and HPRK for he performance. We ge he daa of HSTK and HPRK from all he courses. All he courses are classified ino 11 fields and how he reimbursemen depends on fields is shown in he able A and B of he Appendix C. The HSTK is calculaed by he number of regisered sudens muliplying wih he credis of he course and dividing by 60 credis which is he oal number of credi poins for one year full-ime sudy. The calculaing mehod of he HPRK is similar o he HSTK. The only difference is he number of regisered sudens is changed ino he number of sudens who have passed he examinaion. The wo formulaions are shown below: HSTK = number of regisered sudens credis of he course /60 HPRK = number of sudens who have passed he exam credis of he course /60 4 The Annual Repor of Swedish Naional Agency for Higher Educaion hp://hsv.se/saisics/heannualrepor.4.539a949110f3d5914ec hml - 3 -

7 We received he daa saved in 9 Excel files, which presen he 9 years from , and each file conains 12 spreadshees presening he 12 monhs of he year. In each row of he spreadshee, i lays ou he records of he HSTK and HPRK for each course offered by he universiy. Daa classified by fields To arrange he daa, we merged he records of he courses which belong o he same field in every monh by summing hese records and named hem as THSTK and THPRK. Because he records are accumulaive, o ge he value of he HSTK and he HPRK happened in his monh we should use he one monh laer daa subrac he daa of his monh. For example, when we calculae he HSTK in he field of Naural Science (NAT) in December 2007, we should use he formulaion: THSTK2007,12 THSTK2007,11 = = To calculae he HPRK in he same field in December 2007, we should use he formulaion: THPRK2007,12 THPRK2007,11 = = Wih he number of he HSTK and he HPRK of each monh in hand, we can conver hese values ino he monhly reimbursemen wih he following funcion: Reimbursemen = 19465*(THSTK THSTK 1) *(THPRK THPRK 1). = 19465* * = Where represens he monhs, he presens he reimbursemen for each uni of he HSTK in he field of Naural Science and he is for each uni of he HPRK, which is shown in Appendix C. The annual oal reimbursemen of he universiy is shown in Table 2. Table 2: The annual reimbursemen of Dalarna from (Million SEK) This informaion presens he rend of he reimbursemen, and we can see he flucuaion of he reimbursemen during he years. For example in year 2004 he reimbursemen is

8 million SEK. Daa classified by deparmens In he original daa, we do no have he level of he reimbursemen for deparmens. To calculae he reimbursemen of each deparmen, firsly we merged he records of he courses which belong o he same deparmen in every monh, and hen in each deparmen we merged he records of he courses which belong o he same field. Afer hese procedures, wih he same calculaing mehod for he field jus as we saed above, we can ge he reimbursemen of he deparmen. Then we go he rue value happened in every monh. Then we had monhly ime series daa for 9 years for each of he deparmens. 3. Mehod To forecas he reimbursemen for one field or one deparmen, we consider wo approaches, namely indirec and direc approaches. The reimbursemen is deermined in a simple way. Le y be he reimbursemen of one field or one deparmen, z denoes he number of he HSTK, and w denoes he number of he HPRK. Then we have he formulaion: y = c 1*z + c 2*w (1). Where he consans c 1 and c2 are he reimbursemen depending on he field and shown in Appendix C. The direc approach is o model ŷ direcly, where ŷ is a funcion of pas value of y say y, y 1..., and ime, For he indirec mehod, firs we forecas he z and w separaely, which means we build one model for ẑ, a funcion of z,z 1...,, 1... and one for he ŵ, a funcion of w,w 1...,, 1... hen conver he wo forecased values ino reimbursemen calculaed by equaion (2). yˆ = c *zˆ + c *wˆ (2). 1 2 Where he" Λ" means he prediced value. To ge he model we choose he Census II - 5 -

9 mehod and he ARIMA mehod. Firsly, we inroduce he knowledge of he Census II mehod. Any ime series X is composed of rend (T), season (S), cycle (C), and random influences (E). This mehod is o derive he rend, season, and oher componens from X. There are wo common models: Muliplicaive: X = (T C) S E Addiive: X = (T + C)+ S + E In his hesis we choose he addiive mehod, because he muliplicaive model is commonly used for he growh curve raher han saionary curve. Anoher reason why we choose he addiive model is ha he original daa conains he 0 value, which can no be ransaced by he muliplicaive model. The main sep of he Census II is compuing T+C by moving average and hen ge S+E=X-(T+C). Then ge S by moving average and calculae X-S = T+C+E, he seasonal adjused value. In his process, we use he moving average many imes. Finally we can calculae he value of ˆX : Xˆ = Tˆ + Cˆ + Sˆ. All hese seps are performed wih saisic sofware Eviews 3. We also ried he ARIMA model (Auoregressive Inegraed Moving Average model). The main procedure of his model is explained as follow: when we ge he observed series, he firs hing is o decide if he series is saionary. We use he es of uni roo o es if he series is saionary. We found ha he daa is saionary and hence we need no o difference he series. Then we can consruc an ARMA model, and check wheher he residuals of he ARMA model are whie noise. If no, we repea he procedure above, if so, he procedure is over. 4. Resuls for he Field of Naural Science There are 11 fields and 5 deparmens in Darlarna Universiy and i would be oo many o show all he resuls. Insead we will describe he modeling process of he field of Naural Science as an example. The proceeding is similar for he oher fields and he deparmens. Summary resuls are given in secions 5 and

10 Afer removing he seasonal facor, we ge he seasonal adjused series of he HSTK in he field Naural Science. HSTK Year Figure 2: Seasonal adjused series of he regisraion in he field Naural Science Le z be he seasonal adjused series. We ry several funcions o model z, such as he * * error funcion, logisic ransform, linear funcion of and ^n and log funcion and so on. In Table 3 we show he AIC for some compeing models. Table 3: The Values of AIC for compeing models Funcion AIC Error Funcion 9.55 Logisic Transform 9.58 Linear Funcion of and ^ n 9.89 Log Funcion 9.85 According o AIC, we would like o choose he Error Funcion: 2 * 2 z ẑ = b * e dz MA(1) π + 0 * This is a funcion of ẑ and ; b is a parameer o be esimaed. The residuals of he model pass he whie noise es. So he model is reasonable. Then we show he graph of seasonal - 7 -

11 facor in Figure 3: Figure 3: Seasonal facor series of he regisraion in he field Naural Science Le z be he seasonal facor series. According o he formulaion: z ˆ = * ˆ z + z ˆ (3) So we can calculae he value of he z ˆ. Figure 4 shows linear graph of he forecased daa and he original daa. HSTK HSTK Year Year Figure 4: Comparison beween original value and forecased value of he regisraion in he field Naural Science The correlaion of z ˆ and z is We also check he index of mean absolue - 8 -

12 n ẑ z /z = 1 percenage error (known as MAPE = *100% ). When n=6, i equals o %. n The analysis of he HPRK is similar as he HSTK. Le * w be he seasonal adjused series, w be he seasonal facor series. The graph of seasonal adjused is shown below: Figure 5: Seasonal adjused series of performance in he field Naural Science The resul is quie similar as he HSTK. According o AIC, we sill choose he error funcion. Then we show he graph of seasonal facor as follow: HPRK HPRK Year Year Figure 6: Seasonal facor series of he performance in he field Naural Science - 9 -

13 According o he formulaion: wˆ = wˆ + w ˆ (4) * We can ge he value of he ŵ. Figure 7 shows he linear graph of he forecased daa and he original daa. HPRK Year Figure 7: Comparison beween he original value and he forecased value of he performance in he field Naural Science The mean absolue percenage error is %. Then oally reimbursemen of he field Naural Science can be calculaed by he funcion: yˆ = c 1 * wˆ + c 2 * zˆ ( c1 = , c2 = ). Figure 8 shows he forecased series of he reimbursemen

14 Reimbursemen (Million SEK) Year Figure 8: Forecased series of he reimbursemen in he field Naural Science Wih he similar analysis for he oher fields, we can ge he oal reimbursemen of Dalarna Universiy by summing he reimbursemen of he 11 fields. The resul has been shown in he nex secion. The analysis wih he direc approach is quie similar as he indirec approach. We jus model ime series ŷ direcly. So we will show he comparison of he forecased resuls in he nex secion. The analysis of he deparmens is also similar. Sill he resuls will be shown in he secion Comparison beween he Census II Mehod and he ARMA Mehod To compare which mehod is beer on hese wo approaches and wo levels, we choose he MAPE as an indicaion. Obviously he smaller he MAPE is, he beer he model should be. To find ou if here are some differences in shor-erm forecasing beween he mehods and he approaches on he wo levels, we esimaed hree differen forecasing lenghs of 3 monhs, 6 monhs and 12 monhs. In order o make he resuls comparable, we give he predicions from Mar o Feb Take 3 monhs forecasing as an example; firs we deem he daa inerval from Jan Feb as a se {A}. For he firs 3monh forecasing of Mar May 2007, we use he se {A}. Then we use {A} + he original daa of Mar May

15 2007 o forecas he daa of Jun Aug We use {A} + he original daa of Mar Aug o forecas Sep Nov Finally, we forecas Dec Feb by he se {A} + he original daa of Sep Nov Afer he seps above, we calculae he MAPE of he forecased daa from Mar Feb The resuls from Census II mehod are shown in he able below. We compare he resuls on he field level. Then we compare resuls on he deparmen level in he similar way. Table 4: Comparison in erms of MAPE beween he direc and indirec approaches wih he daa classified by fields where he Census II is used for forecasing (%) Direc Approach Indirec Approach Period 3 monhs 6 monhs 12 monhs 3 monhs 6 monhs 12 monhs Field Oher Humaniy Spor Law Educaion Media Medicine Naural Science Social Science Technique Healh Care Average MAPE Because he Spor field is a quie small field and he daa are oo few o forecas, so we can ignore he effec i made. The Average MAPE is he average of all he fields excep Spor

16 Table 5: Comparison in erms of MAPE beween he direc and indirec approaches wih daa classified by deparmens where he Census II is used for forecasing (%) Direc Approach Indirec Approach Period 3 monhs 6 monhs 12 monhs 3 monhs 6 monhs 12 monhs Deparmen Culure Social Healh Humaniy Indusry Average MAPE From Table 4 and Table 5, on he average level, we can conclude he direc approach is beer han he indirec approach. While he daa classified by deparmens do some help o ge a more accurae predicion han he daa classified by fields. As menioned above, in his hesis, we also ried he ARMA model. The resuls are shown in ables 6 and 7:

17 Table 6: Comparison in erms of MAPE beween he direc and indirec approaches wih daa classified by fields where he ARMA is used for forecasing (%) Direc Approach Indirec Approach Period 3 monhs 6 monhs 12monhs 3 monhs 6 monhs 12 monhs Fields Oher Humaniy Spor Law Educaion Media Medicine Naural Science Social Science Technique Healh Care Average MAPE Table 7: Comparison in erms of MAPE beween he direc and indirec approaches wih daa classified by deparmens where he ARMA is used for forecasing (%) Direc Approach Indirec Approach Period 3 monhs 6 monhs 12 monhs 3 monhs 6 monhs 12 monhs Deparmen Culure Social Healh Humaniy Indusry Average MAPE

18 Compared wih he able 6 and able 7, again we ge he conclusion ha he forecasing resul from he deparmen level is beer han he resul from he field level. From he able 6, on average, he direc approach is also beer han he indirec approach as menioned in analysis of Census II. Bu i shows an opposie siuaion in able 7, he indirec approach achieves a beer resul. Compared wih Census II mehod, he ARMA mehod seems o achieve a beer resul, no maer which approaches are applied o. 6. Conclusion Judging from he rule ha he smaller he MAPE is he beer he forecasing resul should be, we ge he conclusion ha on he average level, he indirec approach wih he ARMA mehod and classify daa on he deparmen level achieves a beer resul. Tha is o say, o give a more accurae forecasing reimbursemen of Dalarna Universiy we had beer ry his pah raher han he oher pahs. We analyze he reason why he ARMA model is beer han he Census II mehod. As we have menioned, Census II mehod pus emphasis on showing rend, season, circle and error. I decomposes he ime series ino hese four componens, bu i is hard o give an accurae decomposiion for he four pars. Wha s more, dealing wih he daa wih he moving average mehod for oo many imes may lead o he reducion of he lengh of he ime series. Compared wih he Census II mehod, he ARMA mehod can make up hese deficiencies. In addiion, ARMA mehod pays more weighs o he recen daa, which means he forecased resul could be closer o he realiy

19 Reference [1] Seven C. Wheelwrigh & Spyros Makridakis (1985), Forecasing Mehods for Managemen, 4TH Ed. New York: John Wiley & Sons. [2] Waler Vandaele (1983), Applied Time Series and Box-Jenkins Models, Unied Saes of America: Academic Press. [3] Brockwell P. J. & Davis, R. A. (2001), Time Series: Theory and Mehods, 2Ed.Beijing Universiy Press. [4] James Douglas. Hamilon (1994), Time Series Analysis, Princeon Universiy Press. [5] Jingshui Sun (2005), Economeric, Tsinghua Universiy Press. [6] Shiskin J. A. H. Young & J. C. Musgrave, The X-II Varian of he Census II Mehod Seasonal Adjusmen Program, Bureau of he Census. [7] Peer Dalgaard. (2002), Inroducory Saisics wih R, New York: Springer. [8] Swedish Naional Agency For Higher Educaion: hp://

20 Appendix Appendix A: Example of Original daa of HSTK in Naural Science Field Period observaion Period observaion Period observaion

21 Appendix B: Example of Original daa of HPRK in Naural Science Field Period observaion Period observaion Period observaion

22 Appendix C: Differen Reimbursemen Level for Differen Science Field Table A (HSTK) regisraion (SEK) Field Amoun Humaniy HU Spor ID Law JU Educaion LU Medicine MD Media ME Naural Science NA Social Science SA Technique TE Healh Care VÅ Oher ÖV Table B: (HPRK) - performance (SEK) Field Amoun Humaniy HU Spor ID Law JU Educaion LU Medicine MD Media ME Naural Science NA Social Science SA Technique TE Healh Care VÅ Oher ÖV