Application of Simultaneous Equation in Finance Research

Transcription

1 Applicaion of Simulaneou Equaion in Finance Reearch By Carl R. Chen an Cheng Few Lee A. Inroucion Empirical finance reearch ofen employ a ingle equaion for eimaion an eing. However, ingle equaion rarely happen in he economic or financial heory. Uing OLS meho o eimae equaion() which houl oherwie be reae a a imulaneou equaion yem i likely o prouce biae an inconien parameer eimaor. To illurae, le ar wih a imple Keyneian conumpion funcion pecifie a he follow: C = α β Y µ (.) Y = C I (.) I = I (.3) 0 Where C i he conumpion expeniure a ime, Y i he naional income a ime, I i he invemen expeniure a ime, which i aume fixe a I 0, an µ i he ochaic iurbance erm a ime. Equaion (.) i he conumpion funcion; Equaion (.) i he equilibrium coniion (naional income accouning ieniy); an equaion (.3) i he invemen funcion. Some of he variable in he moel are enogenou, oher are exogenou. For example, C an Y are enogenou, meaning hey are eermine wihin he moel. On he oher han, I i he exogenou variable, which i no eermine in he moel, hence no correlae wih µ. A imulaneou equaion bia arie when OLS i applie o eimae he conumpion funcion becaue Y i correlae wih he iurbance erm µ, which violae he OLS aumpion ha inepenen variable are orhogonal o he iurbance erm. To ee hi, hrough a erie of ubiuion, we can obain he reuce form equaion from he rucural equaion (.) hrough (.3) a: C Y α β µ = I0 β β β α µ = I0 β β β (.) (.)

2 Bae upon Equaion (.), clearly Y i correlae wih he iurbance erm, µ, which violae he OLS aumpion ha inepenen variable an he iurbance erm are uncorrelae. Thi bia i commonly referre o in he lieraure a he imulaneou equaion bia. Furhermore, he OLS eimae of β in equaion (.) i no only bia, bu alo inconien, meaning β eimae oe no converge o he rue β when he ample ize increae o vary large. B. Two-age an hree-age lea quare meho To reolve he imulaneou equaion bia problem a illurae in Secion A, in hi ecion we icu wo popular imulaneou equaion eimaion meho. Differen meho are available o hanle he eimaion of a imulaneou equaion moel: Inirec lea quare, inrumenal variable proceure, wo-age lea quare, hree-age lea quare, limie informaion likelihoo meho, an full informaion maximum likelihoo meho, ju o name a few. In hi ecion, we will focu on wo meho ha popular aiical an/or economeric ofware are reaily available.. Ienificaion problem Before geing ino he eimaion meho, i i neceary o icu he ienificaion problem. Ienificaion problem arie when we can no ienify he ifference beween, ay, wo funcion. Conier he eman an upply moel of gol. The rucure equaion can be wrien a: Q = α α P ε (3.) 0 Q = β β P e (3.) 0 Q = Q = Q (3.3) Equaion (3.) i he eman for gol funcion, where he eman Q i eermine by he price of gol, P; Equaion (3.) i he upply of gol, an i i a funcion of gol price; Equaion (3.) i an ieniy aing he marke equilibrium. Can we apply he OLS meho o Equaion (3.) an (3.) o obain parameer eimae? anwer hi queion, we fir obain he reuce form equaion for P an Q hrough ubiuion. To e α β α β 0 0 P = β α ε (4.)

3 = α β α β 0 0 Q α β α β α ε β e (4.) Obviouly i i impoible o eimae Equaion (4.) an (4.) uing OLS meho becaue here are four parameer ( α0, α, β0,anβ ) o be eimae, bu here are only wo equaion. Therefore, we canno eimae he parameer in he rucure equaion. Thi i he iuaion calle uner-ienificaion. To iffereniae eman equaion from upply equaion, now uppoe we aume ha eman curve for gol may hif ue o he change in economic uncerainy, which can be proxie by, ay, ock marke volailiy, V, which i aume o be exogenou. Hence Equaion (3.) an (3.) can be moifie a: Q = α α P α V ε (5.) Q = β β P e (5.) 0 The reuce form become P β α α ε e α β α β α β 0 0 = V = γ γ V π (6.) 0 Q ( 0 0) V α ε = β β γ β γ β e α β = λ λv π (6.) Becaue V i aume exogenou an uncorrelae wih reiual π an π, OLS can be applie o he reuce form equaion (6.) an (6.), an obain eimaor of γ 0, γ, λ 0, an λ. Examine Equaion (6.), we fin ha λ0 = β0 βγ 0, an λ = βγ. Since γ 0, γ, λ 0, an λ are all obaine from he OLS eimae, β 0 an β can be olve. Therefore, he upply funcion (Equaion (6.)) i ai o be ienifie. However, from he eman funcion, we fin ha β0 α0 α γ 0 =, an γ =. α β α β 3

4 Since here are only wo equaion, we can no poibly eimae hree unknown, α0, α,anα, hence he eman funcion i no ienifie. Bae upon he icuion of Equaion (3.) an (3.), we hu know ha in a wo-equaion moel, if one variable i omie from one equaion, hen hi equaion i ienifie. On he oher han, here i no omie variable in Equaion (3.), hence he eman funcion i no ienifie. Now le furher moify Equaion (3.) an (3.) a follow: Q = α α P α V ε (7.) Q = β β P β D e (7.) All variable are efine a before excep now we have ae a new variable D in he upply equaion. Le D be he governmen efici of Ruia, which i aume exogenou o he yem. When Ruia buge efici eeriorae, he governmen increae he gol proucion for cah. The reuce form of P an Q bae upon Equaion (5.) an (5.) i β α α β ε e = α β α β α β α β 0 0 P V D = γ γ V γ D π (8.) Q ( 0 0) ( ) V D α ε = α α γ α γ γ α γ β e α β = λ λv λ D π (8.) Bae upon he OLS eimae of Equaion (8.) an (8.), we can obain unique eimae for he rucure parameer α0, α, α, β0, β, an β, hence boh he eman an upply funcion are ienifie. In hi cae, we call he iuaion a exacly ienifie. In a cenario when here are muliple oluion o he rucure parameer, he equaion i ai o be over-ienifie. For example, in Equaion (9.) an (9.), we moify he upply equaion by aing anoher exogenou variable, q, repreening lagge quaniy of gol prouce (i.e., upply of gol in he la perio), which i preeermine. 4

5 Q = α α P α V ε (9.) Q = β β P β D β q e (9.) 3 The reuce form become β0 α0 α β β3q ε e P = V D α β α β α β α β α β = γ γ V γ D γ q π (0.) 3 αε βe Q = ( α0 αγ 0) ( αγ γ ) V αγ D αγ 3 α β = λ λv λ D λ q π (0.) 3 Bae upon Equaion (0.) an (0.), we fin αγ = λ, hence rucure equaion parameer α can be eimae a γ λ. However, we alo fin αγ 3 = λ3, hence α can alo ake anoher value, γ 3 λ 3. Therefore, α oe no have a unique oluion, an we ay he moel i over-ienifie. The coniion we employ in he above icuion for moel ienificaion i he o-calle orer coniion of ienificaion. To ummarize he orer coniion of moel ienificaion, a general rule i ha he number of variable exclue from an equaion mu be he number of rucural equaion. Alhough orer coniion i a popular way of moel ienificaion, i provie only a neceary coniion for moel ienificaion, no a ufficien coniion. Alernaively, rank coniion provie boh neceary an ufficien coniion for moel ienificaion. An equaion aifie he rank coniion if an only if a lea one eerminan of rank (M-) can be conruce from he column coefficien correponing o he variable ha have been exclue from he equaion, where M i he number of equaion in he yem. However, rank coniion i more complicae han orer coniion, an i i ifficul o eermine in a large imulaneou equaion moel. The following example bae upon Equaion (.) hrough (.3) provie ome baic iea abou rank coniion. Noe Equaion (.) hrough (.3) are imilar o Equaion (5.), (5.), an (3.3) wih erm rearrange. The icuion of he orer coniion raw heavily from Ramanahan (995). 5

6 Q α α P α V = ε (.) Q β β P = e (.) Q 0 Q = 0 (.3) In he following able, all rucural parameer are rippe from he equaion an place in a marix. Variable Equaion Inercep Q Equ. (.) α0 Q P V α α 0 Equ. (.) β0 0 β 0 Equ. (.3) Since variable Q an V are exclue from Equaion (.), he eerminan of remaining parameer in column Q an V i α 0 = α Becaue hi ha a rank of, which i equal o he number of equaion ubrac, Equaion i ienifie. On he oher han, a eerminan of rank canno be conruce for Equaion (.) becaue i ha only one zero coefficien, hence Equaion (.) i uner-ienifie.. Two-age lea quare Two-age lea quare (SLS) meho i eay o apply an can be applie o a moel ha i exacly- or over-ienifie. To illurae, le ue Equaion (7.) an (7.) for emonraion, an rewrie hem a he follow: For more eaile icuion of he rank coniion, ee economeric book uch a Greene (003), Juge e al. (985), Fiher (966), Blalock (969), an Fogler an Ganapahy (98). 6

7 Q = α α P α V ε (.) Q = β β P β D e (.) Bae upon he orer coniion, Equaion (.) an (.) each ha one variable exclue from he oher equaion, which i equal o he number of equaion minu one. Hence he moel i ienifie. Since enogenou variable P i correlae wih he iurbance erm, he fir age for he SLS call for he eimaion of preice P ( ˆP ) uing a reuce form conaining all exogenou variable. To o hi, we can apply he OLS o he following equaion: P = η η V η D τ (3) OLS will yiel unbiae an conien eimaion becaue boh V an D are exogenou, hence no correlae wih he iurbance erm τ. Wih he parameer in equaion (3) eimae, he preice P can be calculae a: P ˆ = ˆ η ˆ η V ˆ η D. Thi ˆP i he inrumenal variable o be ue in he econ age eimaion, an i no correce wih he rucure equaion iurbance erm. Subiuing ˆP ino Equaion (.) an (.), we have Q = α α Pˆ α V ε (4.) Q = β β Pˆ β D e (4.) Since ˆP i no correlae wih he iurbance erm, OLS meho can be applie o Equaion (4.) an (4.). 3. Three-age lea quare The SLS meho i a limie informaion meho. On he oher han, he hree-age lea quare (3SLS) meho i a full informaion meho. A full informaion meho ake ino accoun he informaion from he complee yem, hence i i more efficien han he limie informaion meho. Simply pu, 3SLS meho incorporae informaion obaine from he variance-covariance marix of he yem iurbance erm o eimae rucural equaion parameer. On he oherhan, 7

8 SLS meho aume ha ε an e in Equaion (4.) an (4.) are inepenen an eimae rucural equaion parameer eparaely, hu i migh loe ome informaion when in fac he iurbance erm are no inepenen. Thi ecion briefly explain a 3SLS eimaion meho. Le he rucural equaion, in marix, be: Yi = Zψ i i εi, where i=,,.m (5) In Equaion (5), Y i i a vecor of n obervaion on he lef-han ie enogenou variable; Z i a marix coniing of he righ-han ie enogenou an exogenou i variable, i.e., Z [ y x ] = ; an ψ i i a vecor of rucural equaion parameer uch i i i ha ψ = [ α β ]'. Le i i i Y Y = Y m Z 0 Z = 0 Z m ψ ψ = ψ m ε ε = ε m Then, Y = Zψ ε (6) If we muliply boh ie of Equaion (6) by a marix X, where x ' 0 X ' =, 0 x ' i.e., X ' Y = X ' Z X ' ε (7) Then he variance-covariance marix of he iurbance erm in Equaion (7), will be X ' ε 8

9 σx ' x σmx ' x E( X ' εε ' X ) = Ω x ' x = σ mx ' x σ mmx ' x (8) The 3SLS rucural equaion parameer can hu be eimae a ˆ ˆ ψˆ = { Z ' X[ Ω ( x ' x) ] X ' Z} Z ' x[ Ω ( x ' x) ] X ' Y (9) A queion arie in he eimaion proce becaue he σ ' are unknown, hence he marix Ω i alo unknown. Thi problem can be reolve by uing he reiual from he rucural equaion eimae by he SLS o form he mean um of reiual quare an ue hem o eimae Ω. Sanar economeric ofware uch a SAS can be eaily ue o eimae Equaion (9). In um, 3SLS ake hree age o eimae he rucural parameer. The fir age i o eimae he reuce form yem; he econ age ue SLS o eimae he ˆΩ marix; an he hir age complee he eimaion uing Equaion (9). Since Ω conain informaion perinen o he correlaion beween iurbance erm in he rucural equaion, 3SLS i calle a full informaion meho. 3 Since 3SLS i a full informaion eimaion meho, he parameer eimae are aympoically more efficien han he SLS eimae. However, hi aemen i correc only if he moel i correcly pecifie. In effec, 3SLS i quie vulnerable o moel mipecificaion. Thi i becaue moel mipecificaion in a ingle equaion coul eaily propagae ielf ino he enire yem. C. Applicaion of imulaneou equaion in finance reearch In hi ecion, we ue an example employing he imulaneou equaion moel o illurae how he yem can be applie o finance reearch. Corporae governance lieraure ha long ebae wheher corporae execuive inere houl be aligne wih ha of he hareholer. Agency heory argue ha unle here i an incenive o align he manager an hareholer inere, facing he agency problem, manager are likely o exploi for peronal inere a he expene of hareholer. One way o align he inere i o make he execuive compenaion 3 For more eaile icuion, ee Ghoh (99), Juge e al. (985), an Greene (003). 9

10 incenive-bae. Chen, Seiner, an Whye (006) uy he effec of bank execuive incenive compenaion on he firm rik-aking. A ingle equaion moel of he effec of execuive compenaion on firm rik-aking woul look like: m Rik α α ( Comp) α ( LTA) α ( Capial) α ( NI ) α ( Dgeo) = i= 5 n α i ( Dyear) µ (0) i= m i Where Rik i meauremen of firm rik; Comp i he execuive incenive compenaion (i.e., opion-bae compenaion); LTA i he oal ae in log form; Capial i he bank capial raio; NI i he non-inere income, in percenage; Dgeo i a binary variable meauring bank geographic iverificaion; an Dyear i a yearly ummy variable. Chen e al. (006) argue ha OLS eimae of Equaion (0) will prouce imulaneiy bia becaue execuive compenaion i enogenou o he moel, an i likely o be correlae wih he iurbance erm µ. Therefore, Chen e al. (006) inrouce anoher equaion o meaure execuive compenaion. m Comp = β0 β( Rik) β( LTA) β3( SP) βi ( Drae) υ () Where SP i he unerlying ock price an Drae i a erie of ummy variable meauring annual inere rae. For example, Rae9 i efine a he T-bill rae of 99 if he aa i from year 99; oherwie, a value of 0 i aigne o Rae9. Inere rae ummie conrol for he impac of inere rae on opion value. i= 4 Taking Equaion (0) an () ogeher, we fin ha applying OLS o hee wo rucural equaion will no yiel unbiae eimae becaue he righ-han-ie variable inclue enogenou variable Comp an Rik. Therefore, Chen e al. (006) apply SLS o hee wo equaion, an Table repor ome of heir reul. Chen e al. fir repor OLS eimae of he rik equaion an fin ha execuive compenaion rucure oe no impac firm rik-aking. SLS reul repore in Table, however, reveal ha once he imulaneiy of firm rik eciion an execuive compenaion are aken ino accoun, execuive compenaion oe affec firm rik-aking. The incorrec inference erive from he OLS eimae i hu ue o imulaneiy bia. 0

11 Table Simulaneou Equaion Moel Showing he Relaion beween Toal Rik an Opion-Bae Compenaion Eimae Uing Two-Sage Lea Square (SLS) Moel () σ j & OPTION/TOTAL_COMP () σ j & ACCUMULATED_OPTION Equaion σ j OPTION/TOTAL_COMP σ j ACCUMULATED_OPTION Variable Equaion Equaion Equaion Equaion OPTION/TOTAL_COMP (.7)** ACCUMULATED_OPTION (4.86)*** σ j (3.3)*** (5.93)*** LN(TA) (-3.73)*** (3.3)*** (-6.)*** (3.93)*** CAPITAL_RATIO (-.83)** (-6.)** NON_INT_INCOME% (0.7) (-0.83) GEO_DUMMY (-0.99) (0.85) STOCK_PRICE (.69)*** (5.99)*** D9 /DRae (3.8)*** (-.88)* (5.54)*** (-0.37) D93 / DRae (3.5)*** (-0.89) (4.98)*** (0.55) D94 / DRae (.4) (0.88) (.37)*** (0.5) D95 / DRae (0.05) (-.77)* (0.36) (-0.57) D97 /D Rae (.66)*** (0.57) (3.)*** (.58) D98 /D Rae (7.68)*** (.3)* (0.5)*** (.67)* D99 / DRae (5.47)*** (4.7)*** (7.55)*** (.38) D00 / DRae (0.53)*** (.0)** (4.65)*** (.47)*** R 30 % 7 % 45% 9% σ j i a meaure of oal rik; OPTION/TOTAL_COMP i he percenage of oal compenaion in he form of ock opion; ACCUMULATED_OPTION i he accumulae opion value meauring he execuive wealh; LN(TA) i he naural log of oal ae; CAPITAL_RATIO i he capial-o-ae raio; NON_INT_INCOME% i he percenage of income ha i from non-inere ource; GEO_DUMMY i a binary variable meauring geographic iverificaion; an Dum9 - Dum00 are ummy variable coe a or 0 for each year from i he exclue year. ***, **, * inicae ignificance a he percen, 5 percen an 0 percen level repecively.

12 D. Summary an concluing remark Rarely a ingle equaion arie in economic heory. In a muli-equaion yem, OLS fail o yiel unbiae an conien eimaor for he rucural equaion. Therefore, appropriae eimaion meho mu be applie o he eimaion of rucural equaion parameer. Thi paper fir icue iuaion where a imulaneou equaion yem may arie. We hen explain why OLS eimaion i no appropriae. Secion B inrouce wo mo frequenly ue meho o eimae rucural parameer in a yem of equaion. Before SLS an 3SLS meho are ynheize, we explain he orer coniion an he rank coniion of moel ienificaion. SLS an 3SLS are hen inrouce, an he ifference beween hee wo meho are icue. Secion C give example from he lieraure where applicaion of he imulaneou equaion moel in finance are hown. In he finance applicaion, Chen e al. (006) employ a wo-equaion moel o examine he relaionhip beween execuive incenive compenaion an firm rik-aking. Becaue boh execuive compenaion an firm rik are enogenou, SLS i more appropriae han OLS meho. When eciing on he appropriae meho o eimae rucural equaion, one mu be cauiou ha in he real worl he iincion beween enogenou an exogenou variable are ofen no a clear-cu a one woul like o have. Economic heory, herefore, mu play an imporan role in he moel conrucion. Furhermore, a explaine in Secion B, alhough full informaion meho prouce more efficien eimaion, hey are no alway beer han he limie informaion meho. Thi i becaue, for example, 3SLS i vulnerable o moel pecificaion error. If an equaion i mi-pecifie, he error will propagae ino he enire yem of equaion. E. Reference Blalock, H.M., (969), Theory Conrucion: From Verbal o Mahemaical Formulaion, Prenice-Hall, Englewoo Cliff, New Jerey. Chen, C.R., T. Seiner, an A. Whye, (006), Doe Sock Opion-Bae Execuive Compenaion Inuce Rik-Taking? An Analyi of he Banking Inury. Journal of Banking an Finance (30), pp Fiher, F.M., (966), he Ienificaion Problem in Economeric, McGraw-Hill, New York, NY. Fogler, H.R., an S. Ganapahy, (98), Financial Economeric, Prenice-Hall, Englewoo

13 Cliff, New Jerey. Ghoh, S.K., (99), Economeric: Theory an Applicaion, Prenice Hall, Englewoo Cliff, New Jerey. Greene, W.H., (003), Economeric Analyi, Prenice Hall, Upper Sale River, New Jerey. Juge, G.G., W.E. Griffih, R.C. Hill, H. Lükepohl, an T. Lee, (985), The Theory an Pracice of Economeric, John Wiley & Son, New York, New York. Ramanahan, R. (995), Inroucory Economeric wih Applicaion, he Dryen Pre, For Worh, Texa. 3