1032 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 Proof.By Assumption 2 Y.ELEtDo (9) (10) aF-Fa□ =T+r-diF (11) To obtain the market equilibrium,we need to describe the investment oppor- tunities first.Let I7i denote the undiscounted cumulative cash flow from a zero-wealth portfolio long one share of stock i financed by selling the risk-free bond.We have Ii+1(Pis+1+Dit+1)-(1 +r)Pir (12) ein vin, (13) where Pi is the price of stock i,ein =E,[i+1]is the one-period-ahead expecta- tion of excess return and vin=Ii-ein is the corresponding expectation error. 3.2.The equilibrium According to Assumption 7,a representative economic agent's optimization problem can be written as maxE[-exp(-Wt+i门， (14) subject to W:+1 =(1 r)W:Qen.Qien.+1, (15) where W is the agent's wealth and o is the vector of his or her stock holdings. Let us conjecture that Un is Gaussian.2 With the conjecture,we immediately have that Q=En'en EnE[P+1+D+1-(1+r)P ] (16) where En is the variance-covariance matrix of the innovations Un. 2We will see shortly that the conjecture is true since Eq.(18)implies that P,and therefore I,are linear functions of D.F:and N
2We will see shortly that the conjecture is true since Eq. (18) implies that Pt and therefore Pt are linear functions of Dt , Ft and Nt . Proof. By Assumption 2 »it,EtC = + s/1 1 (1#r)s Di,t`sD (9) " = + s/1 A a iF 1#rB s Fit (10) " a iF 1#r!a iF Fit h (11) To obtain the market equilibrium, we need to describe the investment opportunities first. Let Pit denote the undiscounted cumulative cash flow from a zero-wealth portfolio long one share of stock i financed by selling the risk-free bond. We have Pi,t`1 ,(Pi,t`1 #Di,t`1 )!(1#r)Pit (12) "e i P#v i P, (13) where Pi is the price of stock i, e i P"Et [Pi,t`1 ] is the one-period-ahead expectation of excess return and v i P"Pi !e i P is the corresponding expectation error. 3.2. The equilibrium According to Assumption 7, a representative economic agent’s optimization problem can be written as max Qt Et [!exp(!¼t`1 )], (14) subject to ¼t`1 "(1#r)¼t #Q@ t eP,t #Q@ t P,t`1 , (15) where ¼ is the agent’s wealth and Q is the vector of his or her stock holdings. Let us conjecture that P is Gaussian.2 With the conjecture, we immediately have that Q"R~1 P eP"RP~1Et [Pt`1 #Dt`1 !(1#r)Pt ], (16) where RP is the variance—covariance matrix of the innovations P. 1032 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1033 The total supply of stocks to rational economic agents is 1+N,where 1 is a two-dimensional vector of ones.Market clearing condition =1 +N then implies P=(1+r)-1EP+1+D+)-(1+)-1E1+N) (17) which may be solved forward to yield P,=V,-(1/r)En1-ΣnΦW, (18) whereΦisa2×2 diagonal matrix 1 0 1+r-aiN Φ= (19) 0 1+r-a2N」 Theorem 2.The equilibrium conditions of the model imply that en=En(1 +N) (20) and that En satisfies the following matrix equation: En-∑nΦEvΦ'En=ED+P∑FP', (21) where里=I+⊙*isa2×2 natrix. Proof.The first part of the theorem about en is pretty straightforward since market clearing implies o=1 N. Note that D+1=F+1+p.+1 as specified in Section 2.From price equa- tion,Eq.(18),we have P+1+D+1=V+1+D+1-(1/r)En1-EnΦN+1 =0*F+1+F+1+D.+1-(1/r)En1-EnN+1 =ΨF+1+p.+1-(1/r)Enl-∑nΦN,+1, (22) where平=I+⊙*. On the other hand,the definition of I implies that Σn=Var(Pr+i+D+i (23) As a result,we have Σn=∑D+平ErΨ'+EnΦEwΦΣm口 (24) Eq.(21)has a real-valued solution if and only if the matrix is not too large in magnitude.Therefore,if the market is too noisy and/or the noise is too
The total supply of stocks to rational economic agents is 1#N, where 1 is a two-dimensional vector of ones. Market clearing condition Q"1#N then implies Pt "(1#r)~1Et (Pt`1 #Dt`1 )!(1#r)~1RP(1#Nt ) (17) which may be solved forward to yield Pt "Vt !(1/r)RP1!RPUNt , (18) where U is a 2]2 diagonal matrix U" C 1 1#r!a 1N 0 0 1 1#r!a 2N D . (19) ¹heorem 2. ¹he equilibrium conditions of the model imply that eP"RP(1#N) (20) and that RP satisfies the following matrix equation: RP!RPURN U@RP"RD #WRF W@, (21) where W"I#H* is a 2]2 matrix. Proof. The first part of the theorem about eP is pretty straightforward since market clearing implies Q"1#N. Note that Dt`1 "Ft`1 #D,t`1 as specified in Section 2. From price equation, Eq. (18), we have Pt`1 #Dt`1 "Vt`1 #Dt`1 !(1/r)RP1!RPUNt`1 "H*Ft`1 #Ft`1 #D,t`1 !(1/r)RP1!RPUNt`1 "WFt`1 #D,t`1 !(1/r)RP1!RPUNt`1 , (22) where W"I#H*. On the other hand, the definition of P implies that RP"Vart (Pt`1 #Dt`1 ). (23) As a result, we have RP"RD #WRF W@#RPURN U@RP. h (24) Eq. (21) has a real-valued solution if and only if the matrix URN U@ is not too large in magnitude. Therefore, if the market is too noisy and/or the noise is too C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1033
1034 C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 persistent,no stable equilibrium can be established.We will exclude this possi- bility in the subsequent analysis. 4.Information filtration and perceived investment opportunities 4.1.Perceived laws of motion Now we consider the differential information model.To solve for an equilib- rium with non-completely-ranked information sets,we need a tractable method to deal with the information extraction problem. According to Assumption 5,agents in class a observe a record of current and past values Sat [Pit P2n Die D2e FitT, (25) where Pi=Pir-p1 and P2=P2-P2 are 'demeaned'stock prices.Pi and p2 reflect the unconditional expected risk premia,which will be discussed later. Define Xar =Sar-E[S]as the period-ahead conditional expectation error in Sr.Following Sargent(1991),we assume that the filtration rule of agent a,or equivalently,the agent's perceived law of motion for Sat,is a first-order ARMA process of the form Sat AaSa.t-1 BaXa.t-1 Xat (26) We will solve for matrices A and B and show that this assumption is appropri- ate to establish a rational expectations equilibrium. The above perceived law of motion can also be written as []-]+] (27) or Yat gaYa.t-1 +UYa. (28) where (29) Se Xor Yor and go can be defined and analyzed symmetrically for agents in class b.For example,Spr is defined as Sot [Pi P2e Di D20 F2t] (30) and then Xor =Sor-E[Sh1]is defined straightforwardly
persistent, no stable equilibrium can be established. We will exclude this possibility in the subsequent analysis. 4. Information filtration and perceived investment opportunities 4.1. Perceived laws of motion Now we consider the differential information model. To solve for an equilibrium with non-completely-ranked information sets, we need a tractable method to deal with the information extraction problem. According to Assumption 5, agents in class a observe a record of current and past values S at"[PI 1t , PI 2t , D1t , D2t , F1t ]@, (25) where PI 1t "P1t !p 1 and PI 2t "P2t !p 2 are ‘demeaned’ stock prices. p 1 and p 2 reflect the unconditional expected risk premia, which will be discussed later. Define Xat"S at!E[S atDFa t~1] as the period-ahead conditional expectation error in S at. Following Sargent (1991), we assume that the filtration rule of agent a, or equivalently, the agent’s perceived law of motion for S at, is a first-order ARMA process of the form S at"Aa S a,t~1#Ba Xa,t~1#Xat. (26) We will solve for matrices Aa and Ba and show that this assumption is appropriate to establish a rational expectations equilibrium. The above perceived law of motion can also be written as C S at XatD "C Aa Ba 0 0 DCSa,t~1 Xa,t~1D #C Xat XatD, (27) or Yat"ua Ya,t~1#Ya,t , (28) where Yat"C S at XatD, Ya,t "C Xat XatD, and u a "C Aa Ba 0 0 D . (29) S bt, Xbt, Ybt and u b can be defined and analyzed symmetrically for agents in class b. For example, S bt is defined as S bt"[PI 1t , PI 2t , D1t , D2t , F2t ] (30) and then Xbt"S bt!E[S btDFb t~1] is defined straightforwardly. 1034 C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027–1051
C.Zhou Journal of Economic Dynamics and Control 22 (1998)1027-1051 1035 Given the perceptions outlined above,agents form period-ahead forecasts according to E[Yatl Ya.t-1]=gaYa.t-1, (310 E[Yhal Yb.-1]=goYb.t-1. (32) The actual law of motion for prices results from the dynamic market equilib- rium that equates asset supplies and asset demands arising from these expecta- tions.The rational expectations assumption requires that agents'perceptions be consistent with the actual law of motion. Let denote the state vector of the economy which contains Yt,Yor and some other state variables.With a proper choice of elements,will evolve according to =T(g-1+H(g)we (33) where w,is a vector of innovations.For a given set of perceptionsg=(ga g),the actual law of motion,Eq.(33),can be used to obtain the projections of Yir on Yja-1 for j=a,b. E[Yal Ya.t-1]=Ia(g)Ya.t-1, (34) E[YbrlYb.t-1]=Th(g)Yb.t-1, (35) where rj(g)(j=a,b)are obtained using the linear least squares projection formula. For the current asset pricing model,state vector z can be expressed as =(Pie P20 Die D2e Fi F2e Nie N2 Xat Xit' (36) and the vector of innovations in w,can be written as W:=[U1D U2D,UiF,U2F,UIN,U2N]' (37) The equilibrium of the market can be formally defined as: Definition.A(limited-information)rational expectations equilibrium(REE)with heterogeneous information is the fixed point (gg)=(r(g),T(g))such that the market clears in equilibrium. This kind of equilibrium concept was previously used by Sargent(1991)in investigating optimal investment in a production economy,and was then used by Hussman(1992)in an asset pricing model similar to ours.3 Both authors have 3 Hussman(1992)assumes that there is a single risky asset in the market.As we mentioned in the introduction,this single-asset setup is not appropriate to address the effects of private information on portfolio choices and related issues
3 Hussman (1992) assumes that there is a single risky asset in the market. As we mentioned in the introduction, this single-asset setup is not appropriate to address the effects of private information on portfolio choices and related issues. Given the perceptions outlined above, agents form period-ahead forecasts according to E[YatDYa,t~1]"ua Ya,t~1, (31) E[YbtDYb,t~1]"ub Yb,t~1. (32) The actual law of motion for prices results from the dynamic market equilibrium that equates asset supplies and asset demands arising from these expectations. The rational expectations assumption requires that agents’ perceptions be consistent with the actual law of motion. Let z t denote the state vector of the economy which contains Yat, Ybt and some other state variables. With a proper choice of elements, z t will evolve according to z t "T (u)z t~1#H (u)wt , (33) where wt is a vector of innovations. For a given set of perceptions u"(u a , u b ), the actual law of motion, Eq. (33), can be used to obtain the projections of Yjt on Yj,t~1 for j"a,b. E[YatDYa,t~1]"Ca (u)Ya,t~1, (34) E[YbtDYb,t~1]"Cb (u)Yb,t~1, (35) where Cj (u)( j"a, b) are obtained using the linear least squares projection formula. For the current asset pricing model, state vector z t can be expressed as z t "MPI 1t , PI 2t , D1t , D2t , F1t , F2t , N1t , N2t , Xat, XbtN@ (36) and the vector of innovations in z t , wt , can be written as wt "[v 1D , v 2D , v 1F , v 2F , v 1N , v 2N ]@ (37) The equilibrium of the market can be formally defined as: Definition. A (limited-information) rational expectations equilibrium (REE) with heterogeneous information is the fixed point (u a , u b )"(Ca (u), Cb (u)) such that the market clears in equilibrium. This kind of equilibrium concept was previously used by Sargent (1991) in investigating optimal investment in a production economy, and was then used by Hussman (1992) in an asset pricing model similar to ours.3 Both authors have C. Zhou / Journal of Economic Dynamics and Control 22 (1998) 1027—1051 1035