Quarterly Accounting Earnings Forecasting: A Grey Group Model Approach

Transcription

1 Quarerly Accounng Earnngs Forecasng: A Grey Group Model Approach Zheng-Ln Chen Deparmen of Accounng Zhongnan Unversy of Economcs and Law # Souh Nanhu Road, Wuhan Cy, Hube People's Republc of Chna chzhln38@63.com Cha-Ju La Deparmen of Fnance and Bankng Dahan Insue of Technology # Shuren S., Sncheng Townshp, Hualen Couny 97, Tawan Republc of Chna la@ms0.dahan.edu.w Absrac: - Forecasng quarerly accounng earnngs s an mporan ask n prospecve analyss and seasonaly s an mporan phenomenon n hose daa s behavor. In hs paper, we nvesgae he effcency of applyng grey group model o forecas he earnng per share. Unlke radonal sascal model such as Foser Model, grey group model nheres no only easy o calculae bu also few observaons needed han sascs mehods when model buldng. Furhermore, he model lberaes he users from he sample daa assumpons from sascs analyss such as dencally ndependen dsrbuon. A daa sample wh ffy frms radng on Tawan Sock Exchange s employed here and he forecasng performances are compared wh hose obaned by Foser Model. The resuls demonsrae ha he grey group model s a compeve and compeen one n prospecve analyss. Key-Words: - Quarerly Accounng Earnngs, Forecasng, Grey Group Model. Inroducon Forecasng sales, earnngs, and earnngs per share are mporan asks n prospecve analyss. Quarerly accounng earnng forecass are of sgnfcan neres o nvesors, managers, fnancal analyss, and capal markes researchers [3]. However, for hose who are unfamlar wh he behavor of earnng daa o undersand how ends o evolve hrough me, some sascal mehods, for example, Box and Jenkns ARIMA models [2], Foser models [8] and Grffn models [9], are usually employed for he emprcal sudy of forecasng quarerly earnng per share. However, he classcal echnque for predcng a connuous arge varable s he sascal ordnary leas squares (OLS) regresson model. The sascal regresson model s assumpon-full such as he arge varable s normally dsrbued. When he laer s no me, he predcons are quesonable. Wh rapd developmen n he sudy of forecasng, several comparave mehodologes have been evolved n he pas decades. One specal echnque, grey model (denoed as GM) [5, 6], has been appled successfully n forecasng he behavor of a sysem or a seres [4, 0,, 2, 3, 5]. The seres s consdered as parly known, parly unknown, parly ceran and parly unceran n grey model. There are several advanages employng GM model o me seres, such as easy o calculae and few observaons needed han sascs mehods when model buldng [5]. Noce ha seasonal s an mporan phenomenon n hose quarerly earnng daa s behavor. Alhough [7] and proposed a hybrd mehod ha combnes he GM(,) model and he rao-average deseasonalzaon mehod o forecas seasonal me seres, no applcaon of he grey forecasng model was repored on forecasng quarerly earnng daa. Thus, he purpose of hs research s o nvesgae he effcency and ably of applyng GM(,) group model o forecas he earnng per share. Furhermore, he emprcal evdence o dae ndcaes ha he Foser or Grffn models domnae he forecasng ables of oher general Box-Jenkns models []. Therefore, he sudy compares he relave forecasng ably of he parsmonous Foser model wh he forecasng ably of hybrd grey model n quarerly accounng earnngs for a sample of frms radng on Tawan Sock Exchange (TWSE). The organzaon of hs paper s as follow. In secon 2, we make a bref descrbe of daa, he lnear models and grey group model. Secon 3 compares he forecas accuracy of grey model o ha of he Foser model. In secon 4, some concludng remarks are made. ISSN: ISBN:

2 2 Quarerly earnngs forecas models The model used n hs sudy consss of hree sages: daa preparaon, Grey Group model and foser model esablshed, and valdaon. Daa preparaon ncludes ask such as daa cleanng, compleeness and rearrangng. Then he second sage of models consrucon can be conduced by ulzed hose observaons. The valdaon sep confrms wh valdaons daa se. 2. Daa Quarerly earnngs per share daa were colleced from he value lne daa base for frms acvely radng on he Tawan Sock Exchange (TWSE). There are wo crera o choose hese frms. Frs, he frms mus be lsed n he TWSE Tawan Index Seres Compled by FTSE group, such as TSEC Tawan 50 Index, TSEC Tawan Md-Cap 00 Index, TSEC Tawan Technology Index and TSEC Tawan Dvdend Index. FTSE s a world-leader n he creaon and managemen of over 20,000 equy, bond and alernave asse class ndces, hus he frms lsed n hese compled ndexes are of clear rules ha are publcly avalable and whch govern ndex managemen n all foreseeable evenuales. The second creron dcaed ha each frms have a suffcenly lenghy me seres of quarerly earnngs per share daa for esmaon and valuaon purposes. Furhermore, he daa had o nclude he enre perod from he frs quarer of 996 unl he fourh quarer of Accordng o he crera descrbed above, 50 frms lsed n TWSE were seleced wh a oal of 48 observaons for each frm. In hs research, all daa are colleced from he daabase of CMoney Invesmen Decson Suppor Sysem. 2.2 The Foser model In hs subsecon, we frs descrbe he Foser model, whch s usually employed for he emprcal sudy of forecasng quarerly earnng per share. Le Q o denoe earnngs per share (or EPS) for quarer, hen he Foser models for EPS forecasng ake he followng form: ( Q Q ) = α + β ( Q Q ) + ε 4 5, where α and β are parameers o be esmaed and ε s random nose on he quarer. Noe ha equaon s a smple regresson model wh he dependen varable ( Q Q 4 ) and he ndependen varable Q Q ). Thus, o ( 5 esmae hose parameers, pror earnngs daa mus frs be expressed n erm of year-o-year changes. Then usng he processed daa, he coeffcens can be esmaed by a leas square approach or sandard maxmum lkelhood approach. Research has produced models ha forecas earnng per share based only on pror quarers observaons. Alhough such lnear regresson models are no used by many analyss, who have access o much more nformaon han such smple models conan. However, he models are useful for helpng hose unfamlar wh he behavor of earnngs daa o undersand how ends o evolve hrough me [4]. In hs sudy, he sample daa descrbed n subsecon 2. wll be used o compare he forecass obaned from grey group model. 2.3 Grey group model Gven he seres y = { x ( k), k =,2,... N}. A GM(,) s he soluon of he psudo dfference equaon formed as dy + az = b (2) d,where z ( ) = ( y ( ) + y ( )) for 2 n, 2 and y ( ) s accumulaed generang sequence by usng he accumulaed generang operaon (AGO). The parameers a and b n (2.) can be obaned by leas square mehod as a T T = ( B B) B y b (3), where z (2) z (3) B = M z ( N) and y (2) y (3) y =. M y ( N) Then x ˆ ( 0) ( k) s obaned by b a( k ) b xˆ ( k) = ( x ) e +, k = 2,3,..., N a a (4) and he esmae of x ( k), denoed by x ˆ ( 0) ( k), s x ˆ ( k) = xˆ ( k) ( k ), k = 2,3,..., N (5), where ISSN: ISBN:

3 ˆ ) (0 x = x = x. Assume y s seasonal wh perod p. To analyze he seres, s helpful o arrange he seres n a wo-dmensonal able accordng o he perod and season. If N = n p, hen a GM(,) group model for modelng y ncludes he followng seps:. Arrange y no a wo-dmensonal able accordng s seasonal perod p as shown n Table. 2. Defne he sub-sequence ss as follow: ss = { x ( k), k = ( m ) p +, m =,2,..., n }, p. (6) 3. Re-ndex subsequence ss as ss = { x ( k), x ( p + k), x (2 p + k),..., x (( n ) p + k)} = { xss, x (2), x (3),..., x ( n)} k ssk ssk ssk, where p. (7) 4. Ulzng n observaons n each subsequence ss o consruc GM(,). Then p GM(,)s are consruced and named as GM(,) group model. The forecass of x ( ), N < N + p hus can be obaned by he followng formula: xˆ ˆ ss ( ) x ( ), f ( ) = ss REM p ˆ ˆ ˆ x ( ) x ( ), f REM ( ) = 2 x ( ) = ss2 ss2 p... xˆ ( ) ss p ss p ( ), f REM ( ) = 0 p, where bss ass k b ( ) ss x ˆ ss ( ) = ( x ss ) e + (8) a a. a, b ss ss ss ss s he parameers vecor derved from he GM(,) model wh subsequence ss. And he symbol REM ( ) denoes he remander of P.Noce ha m = 4 s he smalles number for P GM consrucon and p = 4 s he perod of he Quarerly Accounng Earnngs seres. Table Arrange he seasonal me seres no a wo-dmenson able wh perod Comparng quarerly earnngs forecass In order o compare he forecasng ables and forecasng performance, each of he lnear me seres Foser model and he grey group model are esablshed by usng he n-sample daa from he frs quarer of 996 unl he fourh quarer of The n-sample daa are used o esmae he parameers of Foser model and grey group model. Afer he parameers are obaned, he seres from he frs quarer of 2007 unl he fourh quarer of 2007 are reserved as a pos-sample, whch s used o evaluae and compare he one sep ahead (one year) forecasng ables of boh models a each company. And he followng sascs are used o compare he forecasng performance:. Mean square error n 2 MSE = ( x ( ) ( )) (9) n = 2. Mean absolue error n MAE = x ( ) ( ) (0) n = To show he avalably of grey group model, he n-sample seres from he frs quarer of 996 o he fourh quarer of 2006 and hose predced wh grey group model and radonal Foser model are compared n he lef hand sde of Table 2. Noe ha he n-sample seres are used o parameers esmaed ISSN: ISBN:

4 and he n-sample MSE and MAE ndcae he performance of model fng. The MSE and MAE of one-year-ahead forecass are also showed on he rgh hand sde of Table 2. Table 2 shows he average n-sample square error and absolue error of Foser model are and respecvely. On he oher hand, he resuls predced by grey group model have he n-sample mean square error and mean absoulue error 0.208, whch are sgnfcanly smaller han hose obaned by Foser model. The resuls clearly show ha he lnear Foser model does no f he seres well. For he pos-sample forecass, he Mean square error and he Mean absolue error of Foser model are smaller han hose of grey group model. However, he pos-sample errors dfferences beween grey group model and Foser model are no sgnfcan and could be roughly acceped by he users who are unfamlar n quarerly earnngs forecass. Noe ha he observaons we used o esablsh grey group model and Foser model are 44 ( years) respecvely for each frms. As we know, here are several advanages employng grey model o me seres forecasng, such as easy o calculae and few observaons needed han sascs mehods when model buldng. Four observaons are he smalles requremen for GM(,) model consrucon[5]. Too many observaons way ncrease he model buldng cos. Thus he chosen of a proper amoun of observaons whle consrucng he grey model s an mporan ssue. To nvesgae he proper amoun of observaons and he forecass performance whle grey model buldng, we dscard he longes four observaons (one year observaons) sep by sep o esablsh he grey group models. The mean square error and mean absolue error of he forecasng performances are calculaed accordng o he dfferen daa lengh. Fgure 2 and Fgure 3 show he n-sample and pos-sample mean square error and mean absolue error wh dfferen daa lenghs respecvely. In Fgure, one may fnd ou ha he mean square error and mean absolue error look lke reversed V. We may conjecure ha ha he mproper amoun of daa lenghs n grey group quarerly earnngs forecasng models consrucon le n beween 6 o 8 years. On he oher hand, Fgure 2 ndcaes ha he pos-sample mean square error and mean absolue error are lowes wh 9 years daa lenghs. In hs research, we may conclude ha he proper amoun of daa lenghs n grey group quarerly earnngs forecasng model are nne years. Table 2, Fgure and Fgure 2 demonsrae ha he grey group model s a compable one n prospecve analyss. Table2 Forecasng Comparsons of Grey Group Model and Foser Model In-sample Pos-sample Grey group Model Foser Model Grey group Model Foser Model MSE MAE Fg. Grey group model n-sample MSE (sold lne), MAE (dashed lne) vs. Dfferen daa lengh ISSN: ISBN:

5 Fg. 2 Grey group model pos-sample MSE (sold lne), MAE (dashed lne) vs. Dfferen daa lengh 4 Concluson Forecasng represens he frs sep of prospecve analyss, and serves o summarze he forward-lookng vew ha emanaes from busness sraegy analyss, accounng analyss, and fnancal analyss. In hs sudy, we nvesgae he effcency of GM(,) group model on forecasng he earnngs-per-share. Emprcal sudy ndcaes mproper amoun of observaons s no helpful o reduce he forecasng errors bu ncreasng he cos of model consrucon. Furhermore, he n-sample and pos-sample forecas performances also show he grey group model wh he proper amoun of observaons s a compeve and compeen mehod for he non-saonary seasonal me seres analyss, forecas and conrol. No only do he advanages of he grey group model nhere he ably of forecasng bu also he characerscs of easy calculang and few observaons necessary. The grey group model has also lberaed he users from he procedures of he model-based selecon and he sample daa assumpons n sascs analyss. Analyzng he seasonal me seres, we can conclude ha he grey group model suggess a compeve mehod for he sysem analyss, forecas and conrol. Despe of he effcency of he grey group model, here reman some problems o be solved. For example: () Wha are he approprae mean facors n grey group model? () How many observaons are approprae for model buldng n dfferen applcaon cases? However, n order o ge an approprae seasonaly for quarerly accounng earnngs me seres, we hope he grey group model wll be a worhwhle approach and wll smulae more fuure emprcal work n me seres analyss. [] Bahke, A. W. and Lorek, K. S., The relaonshp beween me-seres models and he secury marke s expecaon of quarerly earnngs, The Accounng Revew, Vol. 59, 984, pp [2] Box, G. E. P. and Jenkns, G. W., Tme Seres Analyss: Forecasng and Conrol, San Francsco: Holden-Day, 970. [3] Brown, L., Earnngs forecasng research: s mplcaons for capal marke research, Inernaonal Journal of Forecasng, Vol.9, 993, pp [4] Chang, S.C., La, H. C. and Yu, H. C., A varable P value rollng Grey forecasng model for Tawan semconducor ndusry producon, Technologcal Forecasng and Socal Change, Vol.72, 2005, pp [5] Deng, G., Conrol Problems of Grey Sysem, Sysem Conrol Leer, Vol.5, 982, pp [6] Deng, G., Inroducon o Grey Sysem Theory, Journal of Grey Sysem, Vol., 989, pp.-24. [7] Fan, X.H., Grey forecasng Shor-erm Load usng GM(,)s, Journal of Grey Sysem, Vol.4, 2002, pp [8] Foser, G. The me seres behavor of quarerly earnngs: Prelmnary evdence, Journal of Accounng Revew, Vol.52, 977, pp [9] Grffn, P., The me seres behavor of quarerly earnngs: Prelmnary evdence, Journal of Accounng Research, Vol. 5, 977, pp [0] La, C. J., Improve GM(,) group model for non-saonary seasonal me seres forecasng, Journal of Grey Sysem, Vol.5, 2003, pp. pp [] La, C. J., Modfed Rollng Grey Model for Nonlnear Tme Seres Forecasng. Journal of Grey Sysem, Vol.4, 2002, pp References: ISSN: ISBN:

6 [2] L, G. D., Yamaguch, D. and Naga, M., A GM(,)-Markov chan combned model wh an applcaon o predc he number of Chnese nernaonal arlnes, Technologcal Forecasng and Socal Change, Vol.74, 2007, pp [3] Mao, M. and Chrwa, E. C., Applcaon of grey model GM(,) o vehcle faaly rsk esmaon, Technologcal Forecasng and Socal Change, Vol.73, 2006, pp [4] Palepu, K. G., Bernard, V. L. and Healy, P. M., Inroducon o Busness Analyss & Valuaon, Souh-Wesern Publshng Co., 997. [5] Y, D. S. Grey Model Forecas of Talened Persons, Sysem Engneerng, Vol., 987, pp ISSN: ISBN: