switching equilibria: the present value model for stock prices revisited

Journal of Economic Dynamics & Control 28 (2004) 2297–2325www.elsevier.com/locate/econbase

Switching equilibria: the present value model forstock prices revisited

Mar-.a-Jos-e Guti-errez, Jes-us V-azquez∗

Departamento de Fundamentos del An�alisis Econ�omico II, Universidad del Pa��s Vasco,Av. Lehendakari Aguirre 83, 48015 Bilbao, Spain

Received 16 July 2002; accepted 22 October 2003

Abstract

This paper analyzes the dynamic features displayed by alternative rational expectations equi-libria in the context of the present value model for stock prices with feedback. In particular,it shows that there exists a unique equilibrium implying cointegration and that equilibrium ischaracterized by either the fundamental or, alternatively, the backward solution depending on thesize of the feedback parameter. It is shown analytically that the existence of switching equilib-ria induces large stock market swings. Using US data and structural estimation, the hypothesesof feedback and switching equilibria are tested. The empirical results provide evidence of bothswitching equilibria and feedback.? 2004 Elsevier B.V. All rights reserved.

JEL classi*cation: G12; C62

Keywords: Multiple equilibria; Feedback; Cointegration and stock price volatility

1. Introduction

The present value (PV) model of stock prices assuming rational expectations (RE)was extensively tested during the 1980s (Campbell and Shiller, 1987; Chow, 1989;West, 1988, among others). Many of these studies Bnd US stock prices to be morevolatile than implied by the PV model. These studies share two common assumptions.First, they assume a unique RE equilibrium for stock prices. Second, they considerthat the dividend process has remained unchanged over the whole sample period. The

∗ Corresponding author. Tel.: +34-94 601 3779; fax: +34-94 601 3774.E-mail address: [email protected] (J. V-azquez).

0165-1889/$ - see front matter ? 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jedc.2003.10.006

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2298 M.-J. Guti�errez, J. V�azquez / Journal of Economic Dynamics & Control 28 (2004) 2297–2325

relaxation of either of these assumptions provides a potentially good approach to ex-plaining the excess volatility found in the literature. 1

Following the Brst approach, Froot and Obstfeld (1991) propose the Brst parsimo-nious PV model of stock prices that has found empirical support. They characterizestock prices using a rational intrinsic bubble which depends exclusively on dividends.Recently, Ackert and Hunter (1999) have shown that Froot and Obstfeld’s modelis observationally equivalent to a PV model of stock prices that in addition explic-itly includes control over dividends by managers. Also following the Brst approach,Timmermann (1994) shows that the existence of feedback in a PV model generatesmultiple (bubble-free) RE solutions. Moreover, Timmermann provides evidence thatstock prices appear to Granger-cause dividends, which he interprets as evidence offeedback from stock prices to dividends. 2 Furthermore, he suggests that the excessvolatility observed in stock prices may be explained by switches among the set of REequilibria. However, Timmermann neither explains what eIects the switching betweenRE equilibria may induce nor empirically studies the existence of switching equilibria.

The second approach is followed by DriJll and Sola (1998) and Evans (1998).Using the PV model of stock prices, the two articles provide evidence that a regime-switching model describing the evolution of dividends accounts for much of the vari-ation of US stock prices. However, neither of them considers the eIects of switchingequilibria.

This paper builds upon these two approaches. We analyze a simpler version ofTimmermann’s (1994) model in order to fully characterize how changes in the divi-dend process alter the dynamic features displayed by the alternative (bubble-free) REequilibria. We Brst show that the presence of the feedback mechanism produces threealternative RE equilibria.

Secondly, it is shown that there exists a unique (bubble-free) RE equilibrium im-plying cointegration between stock prices and dividends. This equilibrium is character-ized by a diIerent equilibrium solution depending on the dividend process parameters.Finally, this paper illustrates that the existence of switching equilibria induces largechanges in the variance of the spread (i.e., the stationary linear combination betweenstock prices and dividends as deBned by Campbell and Shiller, 1987) and large changesin the response of stock price variation to the spread. These results imply that largestock market swings are feasible, even with a constant discount factor, when switchingequilibria occur.

We assume cointegration between stock prices and dividends in the empirical anal-ysis. We believe it is the relevant case based on the empirical evidence reported byCampbell and Shiller (1987). Moreover, the error-correction structure implied by the

1 As reviewed by Cochrane (1991), numerous attempts at improving the PV model of stock prices havebeen made. Thus, the implications of a time-varying discount rate have been investigated at length. However,as pointed out by Froot and Obstfeld (1991), there is little positive empirical evidence that discount factorvariation alone can explain the excess volatility of stock prices.

2 Timmermann (1994, p. 1109) recognizes that Granger-causality does not necessarily constitute proofof a feedback relation. However, in the light of a model with imperfectly and heterogenously informedagents, Timmermann argues that the evidence of stock prices Granger-causing dividends can be interpretedas evidence of feedback from stock prices to dividends.

M.-J. Guti�errez, J. V�azquez / Journal of Economic Dynamics & Control 28 (2004) 2297–2325 2299

PV model for stock prices under cointegration suggests that the model can be estimatedin two steps. Thus, we Brst estimate the error-correction representation of the cointe-grated system in order to summarize the dynamic features exhibited by stock pricesand dividends. Second, we apply the simulated moments estimator (SME) suggestedby Lee and Ingram (1991) to estimate the parameters of the stock prices model usingannual data for the US. We estimate the model for the full sample and the resultingsub-samples detected by DriJll and Sola (1998) using the same data set.

The empirical results show a good Bt of the PV model for the sub-samples1910–1955 and 1955–1975. They also provide evidence supporting the hypothesisof switching equilibria that explains the diIerent stock price–dividend relationship ob-served in the sub-samples 1910–1955 and 1955–1975. Furthermore, we Bnd evidenceof a small but very signiBcant presence of feedback from stock prices to dividends ineach sub-sample considered.

The rest of the paper is organized as follows. Section 2 introduces the PV modelfor stock prices and obtains the alternative RE solutions. Section 3 characterizes thedynamic properties displayed by the alternative RE equilibria in the space of the param-eters describing the dividend process. Section 4 shows that switches between alternativeRE equilibria lead to regime-switching in the ARMA representation of dividend pro-cess. Section 5 describes the estimation procedure and the empirical evidence found.Section 6 concludes.

2. The present value model for stock prices

Under the RE hypothesis, the PV model for stock prices assuming a constant discountfactor states the following relationship between stock prices and dividends:

pt = �Et(pt+1 + dt) + ut ; (1)

where pt is the real stock price at the beginning of time t, dt is the real dividendobtained from the stock during period t, 0¡�¡ 1 is the constant discount factor andEt denotes the conditional expectation operator given the information set, It , availableto the economic agents at the beginning of time t. It includes current and past valuesof all random variables of the model but the current value of dividends, dt , whoserealization occurs during the period. 3 ut denotes random shocks to the stock pricedue, for instance, to noise traders’ activities.

The PV model is completely characterized by specifying the process followed bydividends. We assume a rather general process for dividends,

dt = 0 + 1pt + 2dt−1 + 3pt−1 + 4dt−2 + vt ; (2)

where vt denotes the dividend process innovations. vt is not included in It because dt

is not included either.

3 In some papers (e.g. West, 1988), dividends are dated at t + 1 instead of t in (1). It must be clear thatthis distinction is meaningless whenever it is assumed that dt (or dt+1 using West notation) does not belongto the information set at the start of period t, but does belong to the information set at the start of periodt + 1.


Eq. (2) allows for the presence of a positive feedback from stock prices to dividends.As discussed at length by Timmermann (1994, p. 1094), there are many ways ofrationalizing this feedback. 4 One possibility is that the feedback may reMect the eIectof stock prices on dividends via the cost of capital restriction faced by the Brm. Anotherpossibility is that stock prices may summarize private information in a context ofasymmetric information. In particular, lagged stock prices may capture the inMuence ofother relevant variables such as expected future interest rates and output gaps. In thiscontext, the dividend policy followed by a Brm may be conditional on stock prices.

The feedback parameters, 1 and 3, may vary over time because the cost of capitalrestriction faced by a Brm changes and/or because of the manner in which dividenddecisions are conditional on stock price changes. The possibility of these variationscan be easily rationalized if the dividend process (2) is understood as a reduced formwhere the dividend process parameters depend upon deeper fundamental parametersthat characterize, for instance, Bscal and monetary policy. Thus, policy changes arelikely to cause changes in the dividend process parameters.

With no loss of generality, we consider the following simpliBed version of thedividend process (2) in order to simplify the exposition of the theoretical results

dt = 0 + 1pt + 2dt−1 + vt ; (3)

where 1 and 2 are both included in the interval [0; 1]. We further assume that ut andvt are i.i.d. random variables.

Eqs. (1) and (3) form a bivariate system of diIerence equations. Using the unde-termined coeJcient method (Muth, 1961b; McCallum, 1983, among others), we beginby writing pt as a linear function of ut and the predetermined state variable dt−1, plusa constant,

pt = �0 + �1dt−1 + �2ut : (4)

For appropriate real values of �0, �1 and �2, the expectational variable Etpt+1 willthen be given by

Etpt+1 = �0 + �1Etdt = �0(1 + �11) + �10 + �1(�11 + 2)dt−1 + �1�21ut :(5)

To evaluate the �s, we substitute (3), (4) and (5) into (1), which gives

�0 + �1dt−1 + �2ut = �[�0(1 + 1 + �11) + 0(1 + �1)]

+ �[�1(�11 +2 +1)+2]dt−1 +[1+�1�2(1+�1)]ut :

This equation implies identities in the constant term, dt−1 and ut as follows:

�0 = �[�0(1 + 1 + �11) + 0(1 + �1)];

�1 = �[�1(�11 + 2 + 1) + 2];

�2 = 1 + �1�2(1 + �1): (6)

4 Timmermann (1994, pp. 1103–1106) provides two examples of optimizing models of the stock marketwhich exhibit feedback.


After some algebra, we can show that there are two solutions to the system ofEqs. (6),

�1 = (�10; �

11; �

12) = [ 1; �1; ’1]; (7)

�2 = (�20; �

21; �

22) = [ 2; �2; ’2]; (8)

where

�1 =1

2�1(1 − �(1 + 2) + ([1 − �(1 + 2)]2 − 4�212)1=2);

�2 =1

2�1(1 − �(1 + 2) − ([1 − �(1 + 2)]2 − 4�212)1=2);

i =�0(1 + �i)

1 − �− �1(1 + �i); i = 1; 2;

’i =1

1 − �1(1 + �i); i = 1; 2:

Notice that when dividends are endogenous (i.e., 1 ¿ 0) the two fundamental solutionsonly exist when 5

[1 − �(1 + 2)]2 − 4�212¿ 0; (9)

that is, when �1 and �2 are both real numbers. Moreover, in the particular case wheredividends are exogenous (i.e., 1 =0), solution (7) does not exist and solution (8) canbe expressed as

pt =�0

(1 − �2)(1 − �)+

�2

1 − �2dt−1 + ut :

This is the solution studied in the traditional bubble-free RE literature when dividendsare considered exogenous.

In addition to RE solutions (7) and (8), the PV model for stock prices, Eq. (1),exhibits other RE equilibrium solutions. These alternative solutions are obtained byfollowing the backward approach for solving linear RE models (see Broze and Szafarz,1991, Chapter 2). This approach starts by using the premise that the RE of stock pricesand dividends at period t are given by

Etpt+1 = pt+1 − �t+1; (10)

Etdt = dt − vt ; (11)

5 Following McCallum (1983, p. 146, footnote 9), we implicitly believe that if some of the equilibriumsolutions are real and the others are complex, then the latter are irrelevant because they are not economicallysensible solutions.


respectively; where �t+1 (the rational prediction error) is an arbitrary martingale dif-ference with respect to the agents’ information set at period t, It . By using (10), (11)and rearranging, the PV model (1) can be written as

pt = �−1pt−1 − dt−1 − �−1ut−1 + vt−1 + �t : (12)

We refer to (12) as the set of backward solutions to the PV model. Among the set ofbackward equilibrium solutions (12) we pay attention to the unique backward solutioncharacterized by �t = 0 for all t, that is,

pt = �−1pt−1 − dt−1 − �−1ut−1 + vt−1; (13)

since we want to focus our attention on bubble-free RE equilibrium solutions.Solutions (7) and (8) are called fundamental solutions in the sense that the coeJ-

cients associated with the state variables in these two solutions depend on the dividendprocess parameters. This dependence is known as the hallmark of RE models. 6 Fromnow on we refer to solutions (7) and (8) as the �1-fundamental and �2-fundamentalsolutions, respectively. Notice that all three solutions are RE equilibria and satisfy (12)with diIerent �t . 7

3. A characterization of the dynamic properties of alternative equilibria

This section fully characterizes how the dynamic properties displayed by alternativeequilibria may change for small perturbations of the dividend process parameters.

3.1. More on the existence of alternative equilibria

The following proposition establishes the region in which only the backward solution(13) exists because the other bubble-free equilibrium solutions are complex. From nowon we refer to RE equilibrium solution (13) as the backward solution.

Proposition 1. When inequality (9) holds all three solutions considered exist, other-wise only the backward solution exists. Moreover, the set of values of 1 and 2 forwhich only the backward solution exists is empty for �6 1=4. The set of values of1 and 2 for which all three solutions exist is non-empty for any 0¡�¡ 1.

Proof. See Appendix A.

6 Notice that this dependency does not show up in any backward solution.7 It can be shown that the fundamental solutions, (7) and (8), satisfy the linear diIerence equation (12)

with the diIerence martingale given by

�t =1

1 − �1(1 + �i)ut

for i = 1; 2, respectively.


δ1

1

1

δ41

δ41

1ρ

2ρ

ρρ21=

( )[ ] 04121

2221 =−+− ρρδρρδ

δ1

Fig. 1. Existence of alternative equilibria.

Fig. 1 summarizes the results stated in Proposition 1. For a given 0¡�¡ 1, theshaded region displays the combinations of values for 1 and 2 for which the REequilibrium is characterized only by the backward solution (13) (that is, when inequality(9) is not satisBed).

3.2. The stationary case

The following proposition states the conditions in the region where all three alterna-tive solutions exist under which they are stationary in a statistical sense. The varianceof stock prices under each solution is also characterized.

Proposition 2. Denote �i = 2 + �i1 for i = 1; 2. If �i ¡ 1 for i = 1; 2, then(i) fundamental solutions (7) and (8) are stationary, respectively. The variance of

stock prices under the fundamental solutions, (7) and (8), is given by

�i0 =

�2i

1 − �2i�2

v +1 − �2

i + �2i

21

1 − �2i

’2i �

2u; (14)

for i = 1; 2, respectively.(ii) the backward solution, (13), is stationary. The variance of stock prices char-

acterized by the backward solution, �b0, is given by

�b0 =

(1 + �1�2)22

(1 − �21)(1 − �2

2)(1 − �1�2)�2

v

+(1 + �1�2)(1 + 2

2) − 22(�1 + �2)(1 − �2

1)(1 − �22)(1 − �1�2)�2

�2u: (15)



1ρ

2ρ

δ

1

1( )δδ 21−

δδ−1

12 =µ

B

CA

D

X

Y

W Z

21 µµ =

11 =µ

Fig. 2. Features of alternative equilibria.

From part (i) of this proposition it is straightforward to show that the variance instock prices for the �2-fundamental solution, (8), is always lower than the variance instock prices for the �1-fundamental solution, (7).

Fig. 2 illustrates the regions in which the �1-fundamental and �2-fundamental so-lutions are stationary. Curve �1 = �2 displays combinations of 1 and 2 such that[1 − �(1 + 2)]2 − 4�12 = 0. Therefore, region B is the area where only the back-ward solution exists. The level curve �1=1 is fully described by the segment connectingthe points (1; 2) = ((1 − �)=�; 0) and (1; 2) = ((1 − �)2=�; �). Points located to theright of this segment imply combinations of 1 and 2 such that �1 ¡ 1 for which the�1-fundamental solution is stationary. Otherwise the �1-fundamental solution is not sta-tionary. The level curve �2=1 exactly corresponds to the segment from (1; 2)=(0; 1)to (1; 2) = ((1 − �)2=�; �). Points above this segment result in combinations of 1

and 2 such that �2 ¿ 1 for which the �2-fundamental solution is not stationary; other-wise, the �2-fundamental solution is stationary. Since the backward solution is station-ary when �1 ¡ 1 and �2 ¡ 1, then the backward solution is only stationary in regionC. 8 All these statements are proved in Appendix B. 9

Parameters �i deBned in the above propositions can be interpreted by looking at thediIerence equation system formed by (1) and (3). This system can be expressed inmatrix form as

A0 + A1Yt + A2Yt+1 + A3Ut+1 = 0;

8 The backward solution is not stationary in region D because �i ¿ 1, for i = 1; 2 and it is not stationaryin region A since �1 ¿ 1.

9 Notice that Fig. 2 represents a case where �¿ 1=2, otherwise point ((1 − �)=�; 0) would lie to the rightof (1; 0).


where

Yt =

(pt

dt−1

); Ut+1 =

ut

pt+1 − Et(pt+1)

vt

; A0 =

(0

−0

);

A1 =

(1 0

−1 −2

); A2 =

(−� −�

0 1

); A3 =

(−1 � �

0 0 −1

):

Operating and rearranging, this matrix system can be written as

Yt+1 = BYt + CUt+1 + D; (16)

where B = −A−12 A1, C = −A−1

2 A3 and D = −A−12 A0.

It is easy to show that �i for i = 1; 2 are the eigenvalues of transition matrix B. Inthis context the areas illustrated in Fig. 2 can be interpreted in terms of the stationaryproperties of the diIerence equation system (16): (i) If no equilibrium solution isstationary (that is, in area D), diIerence equation system (16) is a source; (ii) whenonly the �2-fundamental solution is stationary (that is, the parameters of 1 and 2 liein area A), diIerence equation system (16) is a saddle path; (iii) if all three equilibriumsolutions are stationary (that is, in area C), diIerence equation system (16) is a sink.Moreover if either �1 or �2 is equal to one, then stock prices and dividends arecointegrated because in this case |B− I |=0. It is easy to show that if �i=2+�i1=1,then 2=1−�1=(1−�), and the cointegrated vector is (1;−�=(1−�)). The latter resultsimply says that the spread between stock prices and dividends, pt − (�=(1 − �))dt−1,is stationary when pt and dt−1 are cointegrated.

3.3. The cointegration case

The following proposition states the properties of the solutions when prices anddividends are cointegrated.

Proposition 3. If �i = 1 for some i = 1; 2 then the �1-fundamental solution does notexist and the �2-fundamental solution takes the form

�2 = (�20; �

21; �

22) =

( −�0

(1 − �)(� − 2);

�1 − �

;12

):

Moreover, among all solutions there exists a unique equilibrium implying cointegra-tion between stock prices and dividends with two possible scenarios:

1. If �1 = 1¿�2, the backward equilibrium solution implies that stock prices anddividends are cointegrated whereas the �2-fundamental solution is stationary.

2. If �1 ¿�2 =1, the �2-fundamental solution implies that stock prices and dividendsare cointegrated whereas the backward equilibrium solution is explosive.



This proposition points out that there is a unique equilibrium solution implyingcointegration between stock prices and dividends. As shown in Fig. 2, cointegrationappears along the line connecting points (0; 1) and ((1 − �)=�; 0).

We can illustrate a simple mechanism that may lead to switches between the back-ward and the �2-fundamental equilibria triggered by changes in the feedback parameter.First, consider that the economy is initially located at a point such as X in Fig. 2. Atthis point, as previously shown, the only equilibrium implying cointegration betweenstock prices and dividends is characterized by the backward equilibrium solution. 10

Now assume that there is a change in economic policy that induces a suJcient de-crease in the value of the feedback parameter, 11 1, such that the economy is nowlocated at a point such as Y in Fig. 2 where the only equilibrium implying cointegra-tion is characterized by the �2-fundamental solution. A second switching equilibriummechanism is just the opposite. 12

The existence of switching equilibria provides an additional source of variation instock prices. In addition to Muctuations in stock prices caused by innovations in thedividend process, stock price movements can be induced by switches between REequilibria.

In order to understand how switching equilibria may imply large stock market swingsunder the hypothesis of cointegration it is useful to write down the expressions of theBrst-diIerences of stock prices under the �2-fundamental and the backward equilibriumsolutions, denoted by Qpf

t and Qpbt , respectively,

Qpft =

�0(� − 2 − 1)(1 − �)(� − 2)

− 2

(pt−1 − �

(1 − �)dt−2

)+

12

ut +�

(1 − �)vt−1;

Qpbt = −0 + 2

(1 − �)�

(pt−1 − �

(1 − �)dt−2

)− 1

�ut + vt−1:

A comparison between these two expressions shows that the response of Qpt tochanges in the spread varies widely depending on which equilibrium solution char-acterizes the unique (bubble-free) equilibrium under the hypothesis of cointegration.Thus, an increase in the spread decreases Qpt under the �2-fundamental solution,whereas it increases Qpt under the backward solution.

Switching equilibria also imply changes in the variance of the spread as shown inthe following proposition. Denote the spread at period t by spt .

10 Recall that the �2-fundamental solution at a point such as X does not imply cointegration between stockprices and dividends.

11 As stated above, recall that the dividend process can be viewed as a reduced form where its parametersare likely to change with variations in the parameters characterizing economic policy.

12 We have focused only on switches between equilibria characterized by the cointegration restriction.However, when the cointegration restriction does not hold, we can illustrate other examples. For instance,consider that the economy is located at a point such as W in Fig. 2 where the equilibrium is characterizedby the �2-fundamental solution. If there is a change in the value(s) of 1 or/and 2 such that condition (9)does not hold (for instance, a switch from W to Z in Fig. 2), then the economy jumps to an equilibriumcharacterized by the backward solution.


Proposition 4. If stock prices and dividends are cointegrated, then

(i) the unconditional mean and variance of the spread under the �2-fundamentalequilibrium, spf

t , are given by

E(spft ) =

−�0

(1 − �)(� − 2);

var(spft ) =

122�2

u;

respectively.(ii) the unconditional mean and variance of the spread under the backward equilib-

rium, spbt , are given by

E(spbt ) =

−�0

(1 − �)(� − 2);

var(spbt ) =

1(�2 − 2

2)�2

u +�4

(1 − �)2(�2 − 22)

�2v ;

respectively.


Proposition 4 shows that switching equilibria induce only changes in the functionalform of the variance of the spread since the mean of the spread is the same underthe two alternative solutions. The variance of the spread under the backward solutiongoes to inBnity when 2 goes to � from below, that is, when X approaches the point((1 − �)2=�; �) in Fig. 2. 13 Therefore, a switch from Y to X due to small increasein the feedback parameter on the one hand dramatically increases the variance of thespread and on the other hand results in a large increase in the way stock price variationsrespond to a change in the spread as shown above. These results point to the conclusionthat large stock market swings are feasible, even with a constant discount factor, whena switch between alternative equilibria occurs.

4. Regime-switching in the dividend process

This section shows how switches between alternative RE equilibria for stock pricesmay also lead to regime-switches in the ARMA representation characterizing the divi-dend process.

13 Notice that the variance of the spread under the backward solution is only well deBned when �¿2(that is, along the line connecting the points ((1 − �)=�; 0) and ((1 − �)2=�; �) in Fig. 2.


By plugging fundamental solutions (7) and (8) into Eq. (3), we obtain ARMArepresentations for the dividend process under the two alternative fundamental solutions

dt =0(1 − �)

(1 − �) − �1(1 + �i)+ �idt−1 +

1

1 − �1(1 + �i)ut + vt ;

where �i =2 +�i1, for i=1; 2. Notice that the dividend process associated with eachstock price fundamental solution follows a Brst-order autoregressive process. Moreover,any such process is stationary if and only if the associated stock price equilibriumsolution is itself stationary (that is, �i ¡ 1). After some algebra, we can show fromthis expression that the mean of the dividend process is an increasing function of�i. Thus, the mean dividend is higher for the �1-fundamental solution than for the�2-fundamental solution.

In order to obtain the dividend process under the backward solution, we Brst writethe backward solution (13) as an ARMA(2; 1) process (see part ii) of the Proof ofProposition (2) in Appendix A):

q(L)pt = −0 + m(L)vt + n(L)ut ; (17)

where L is the lag operator, q(L) = 1 − (�−1 − 1 + 2)L + 2�−1L2, m(L) = −2L2

and n(L)=−�−1(1−2L)L. By substituting equation (17) into Eq. (3) and after somealgebra, we obtain that the dividend process under the backward equilibrium solutionis characterized by the following ARMA(3; 2):

(1 − 2L)(1 − �1L)(1 − �2L)dt = 0(1 − �1)(1 − �2)

+ (1 − (�1 + �2)L + (�1�2 − 2)L2)vt

− 1

�(1 − 2L)ut−1:

If 2 ¡ 1, the dividend process characterized by the stock price backward solution isstationary if and only if the backward solution is itself stationary (that is, �2 ¡�1 ¡ 1).Extending these results to the cointegration case is straightforward.

Using US stock market data, DriJll and Sola (1998) have recently found evidence ofregime-switching in the ARMA representation of the dividend process without explain-ing what forces are causing this regime switching. Our results point out the theoreticalpossibility that the regime-switching in the ARMA representation found by DriJll andSola may be explained by switches between alternative RE equilibria. In short, the re-sults in Sections 3 and 4 have shown that changes in the dividend process parametersmay lead to switching equilibria and vice-versa.

5. Empirical evidence

We use historical US data on stock prices and dividends. More precisely, thesedata come from the Standard and Poor’s stock price and dividend indexes taken fromthe Securities Price Index Record. We study the period 1871–1989. Stock prices areJanuary values whereas dividends are annual averages for the calendar year. Nominal


stock prices and dividends are deMated by the producer price index (1982 = 100) forJanuary of each year in order to get real Bgures. 14 ; 15

5.1. Preliminary evidence

Assuming that the dividend process is exogenous, DriJll and Sola (1998, p. 369)identify three regime switches located around the years 1910, 1955 and 1975. We ex-tend this analysis by allowing the presence of feedback from stock prices to dividends.To that end, we estimate a Markov switching regression of the Brst-diIerences of divi-dends on the spread (that is, a Markov switching estimation of the dividend process (3)imposing the cointegration restriction) following the procedure described by Hamilton(1994, Chapter 22). Formally, the dividend process is described by

dt − dt−1 = 0(st) + 1(st)[pt − �

1 − �dt−1

]+ vt ; (18)

where the regime variable st is either 1 or 2 and parameters 0(st), 1(st) and �v(st) areassumed to follow a Brst-order two-state Markov process with prob(st =1|st−1=1)=pand prob(st = 2|st−1 = 2) = q. Table 1 shows the estimation results for (18). 16 Thepoint estimate of the standard deviation of the innovations is more than four timeslarger in the Brst regime (state 1) than in the second (state 2). Moreover, the pointestimate of the feedback parameter is signiBcantly diIerent from zero in the secondregime and this estimated value is more than four times larger than the estimated valuefound for the Brst.

Fig. 3 shows the allocation of time periods to the two states. In particular, it showsthe Bltered probabilities of being in state 2 conditional on the information availableat time t. From 1880 to 1910 the periods are allocated mostly to state 2. The period

14 The producer price index is taken from Table 26.2, series 5, in Shiller (1989, Chapter 26).15 We restricted the data sample to the period 1871–1989 for two reasons. First, this is the sample period

usually considered in the relevant literature, which makes comparison of the empirical evidence found inthis paper with that found in other studies easier. Second, empirical results (not reported in this paper) showthat the model is rejected by recent data involving the 1980s and 1990s. In particular, real dividends followa linear upward trend during the period 1871–2000. Similarly, real stock prices follow a linear upward trendduring the period 1871–1975. However, real stock prices behave in a highly non-linear manner during thelast 25 years of 20th century which it is hard to reconcile with the PV model for stock prices. A possiblereason for this failure is that, as reported by Fama and French (2001), US Brms have become gradually lesslikely to pay dividends regardless of their characteristics since 1978. Then, dividends seem to have becomeless and less important in explaining the evolution of stock prices since the end of the 1970s. However, byno means should we consider that the PV model for stock prices is meaningless nowadays. The essence ofthis model is that stock prices equal the expected future stream of Brm net earnings. Fama and French’sBnding only implies that dividends have ceased to be as good a proxy for Brm net earnings as they were inthe past in Brm share pricing. The question is then what could be a good proxy for Brm net earnings since1978. The answer to this question is beyond the scope of this paper and is left for future research.

16 We have also estimated the dividend process (18) assuming a three-state Markov regime switching modeland including more lags as in (2). The estimation results are quantitatively similar to those found in thesimple two-state model. These results are available from the authors upon request.


Table 1Two state Markov switching regression of the dividend process; period 1871–1989

Estimated value Standard error

0(1) −0.0005 0.00170(2) −0.0030 0.00071(1) 0.0026 0.00151(2) 0.0116 0.0016�v(1) 0.0067 0.0009�v(2) 0.0015 0.0002p 0.9015 0.0269q 0.9794 0.0172Mean log-likelihood 4.1314

Note: The numbers in parentheses denote the state (1 or 2).

TIME

1.0

0.1

0.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1860 1880 1900 1920 1940 1960 1980 2000

Stat

e 2

prob

abili

ty

Fig. 3. Filtered probabilities associated with (18).

from 1910 to 1955 is characterized by frequents jumps from one state to another. Theperiod from 1955 to 1975 is entirely attributed to state 1. These estimation resultssuggest that the feedback parameter corresponding to the period 1955–1975 (state 1)must be smaller than the one associated with the period 1910–1955 (characterizedby a combination of states 1 and 2). Our results basically conBrm the timing of theregime switches suggested by DriJll and Sola (1998). However, given that the period1910–1955 is characterized by frequent jumps from one regime to another, we proceed


Table 2Phillips–Perron Z tests

Period Variable With trend Without trend

Sample pt −13.32 −3.591871–1989 dt −17.36 −1.17

Qpt −107.09 −107.11Qdt −92.66 −92.71pt − (�=(1 − �))dt−1 −40.72 −40.28

Subsample pt 10.02 9.331871–1910 dt 10.15 8.81

Qpt −34.72 −34.72Qdt −31.96 −32.59pt − (�=(1 − �))dt−1 −24.47 −20.91

Subsample pt 9.91 9.311910–1955 dt 9.99 8.77

Qpt −36.33 −36.30Qdt −36.17 −36.27pt − (�=(1 − �))dt−1 −23.64 −23.79

Subsample pt 10.10 9.511955–1975 dt 10.28 8.99

Qpt −15.29 −14.78Qdt −4.41 −2.75pt − (�=(1 − �))dt−1 −4.87 −7.70

Note: The Phillips–Perron Z statistics are corrected for fourth-order serial correlation. A table displayingthe critical values for the Z test is reported in Fuller (1976, p. 371).

below to estimate alternative sub-samples for 1871–1910, 1910–1955 and 1955–1975separately. As shown in Table 2, standard unit root and cointegration tests suggestthat stock price and dividend processes are I(1) and the two variables are cointegratedprocesses. 17

17 Augmented Dickey–Fuller tests and Phillips–Perron tests (Phillips and Perron, 1988) were carried outfor the whole sample and the alternative sub-samples. Table 2 shows the Phillips–Perron Z statistics. Theresults are qualitatively similar when considering Phillips–Perron Zt and augmented Dickey–Fuller tests andwhen considering alternative orders of the serial correlation in computing Phillips–Perron statistics. Forthe full sample the test results are qualitatively similar to those provided by Campbell and Shiller (1987).Moreover, the cointegration tests carried out in the paper do not reject the hypothesis of cointegration for thealternative sub-samples with the exception of the sub-sample 1955–1975. At this point of the analysis, we seetwo possibilities. One is to consider that results suggest that there is no cointegration for this sub-sample. Butthis conclusion would already imply the existence of switching equilibria from a cointegrated RE equilibriumto one where cointegration is not present. Alternatively, we could consider that the latter sub-sample is soshort that any test of cointegration lacks power. Thus, we could decide to maintain the hypothesis of astable cointegration relationship to detect the presence of switching equilibria because, as our theoreticalresults show (see Proposition 3), switching equilibria do not necessarily imply a switch in the cointegrationrelationship. The latter approach is the one followed in this paper.


5.2. The error-correction model representation

We estimate the PV model given by (1)–(2) under the cointegration hypothesis. Thethree solutions associated with this model are derived in Appendix C.

The error-correction model of the fundamental solutions is given by(Qpt

Qdt−1

)=

(�0 + �10

0

)+

(�113 + �2 − 1 �124 + �3

13 24 − 1

)(pt−1

dt−2

)

+

(−�13 −�14

−3 −4

)(Qpt−1

Qdt−2

)+

(�4 �1

0 1

)(ut

vt−1

); (19)

where Qpt = pt − pt−1, Qdt−1 = dt−1 − dt−2, 13 = 1 + 3, 24 = 2 + 4, and the�s correspond to the solutions derived in Appendix C.

As pointed out above, the backward solution (13) is an RE equilibrium solution nomatter what process is followed by dividends. Thus, Eqs. (13) and (2) can be writtenafter some algebra in the following error-correction form:(

Qpt

Qdt−1

)=

(−0

0

)+

(�−1 − 13 − 1 −24

13 24 − 1

)(pt−1

dt−2

)

+

(3 4

−3 −4

)(Qpt−1

Qdt−2

)+

(�−1ut−1

vt−1

): (20)

It is easy to show that the cointegration restriction is given by 24=1−�13=(1−�). 18

5.3. Estimation procedure

In order to estimate the structural parameters of the model we follow the indirectinference principle proposed by Gouri-eroux and Monfort (1996) where the error-correction representation model is considered as the auxiliary model. 19 Thus, we Brstestimate the error-correction representation of the cointegrated system in order to sum-marize the dynamics characterizing stock prices and dividends. Second, we apply thesimulated moments estimator (SME) suggested by Lee and Ingram (1991) to estimatethe underlying structural parameters.

The use of this estimation strategy is especially appropriate in this context since thealternative RE equilibrium solutions follow an error-correction model. Then, we caneasily test the restrictions imposed by the diIerent solutions on the error-correctionmodel.

The SME makes use of a set of statistics computed from the data set used and froma number of diIerent simulated data sets generated by the model being estimated. A

18 Notice that this cointegration restriction is similar to the cointegration restriction associated with (3) inthe sense that the sum of the parameters associated with the dividend lags, 24, is linked to the sum ofparameters associated with stock price variables, 13, that appears in Eq. (2).

19 Related papers using a parametric model to deBne the auxiliary model are Smith (1993) and Gouri-erouxet al. (1993).


suJcient condition for the SME to be consistent and asymptotically normal is that thetime series be covariance stationary. Therefore, we work with Brst-diIerences of stockprices and dividends and with the spread at time t deBned by pt − (�=(1 − �))dt−1.More speciBcally, the statistics used to carry out the SME are the coeJcients fromthe error-correction representation of the two-variable cointegrated system formed byBrst-diIerences of pt and dt−1 with eight-lags. To Bnd the appropriate lag of theerror-correction representation, the likelihood ratio test was used. The null hypothesistested was s lags versus s + 1 lags. The lowest number of lags s associated with thenon-rejection of the null was chosen. For the whole sample we Bnd s=8 whereas forthe sub-samples 1871–1910, 1910–1955 and 1955–1975, we Bnd s equal to 3, 1 and 1,respectively. 20

To implement the method, we construct a p × 1 vector with the coeJcients of theerror-correction representation obtained from real data, denoted by HT (,0), where p inthis application is 37, 21 T denotes the length of the time series data, and , is a k × 1vector whose components are the structural parameters. The true parameter values aredenoted by ,0. In our model, the parameters are , = (1; �z; �s; �; 0; 1; 3; 4). 22

Given that the real data are by assumption a realization of a stochastic process, wedecrease the randomness in the estimator by simulating the model m times. Time seriesof simulated data are obtained recursively using the matrix system (19) (alternatively,the matrix system (20)), imposing the cointegration restriction. For each simulation ap × 1 vector of error-correction representation coeJcients, denoted by HNi(,), is ob-tained from the simulated time series of pt and dt−1 generated from the model, whereN=nT is the length of the simulated data. For the sake of simplicity, we set n=m. Av-eraging the m realizations of the simulated coeJcients, i.e., HN (,)=(1=m)

∑mi=1 HNi(,),

we obtain a measure of the expected value of these coeJcients, E(HNi(,)). Since weestimate the model many times (we analyze three alternative RE solutions and fourdata sets: the whole sample and three sub-samples), and after checking the robustnessof the results we make n = m = 5 in this application. To generate simulated valuesof Qpt and Qdt−1 we need the starting values of the Brst-diIerences of stock pricesand dividends. In the estimation, we have arbitrarily computed these starting valuesby using the Brst observed values of stock prices and dividends in our sample. Forthe SME to be consistent, the initial values must have been drawn from a stationarydistribution. In practice, to avoid the inMuence of the starting values we follow Leeand Ingram’s suggestion of generating a realization from the stochastic processes ofQpt and Qdt−1 of length 2N , discard the Brst N -simulated observations, and use only

20 As shown by Hosking (1981), a portmanteau test for checking whether residuals from the error-correctionmodel are white noise may be viewed as a sequence of Lagrange ratio tests for zero restrictions on thecoeJcients of the error-correction representation such as the one carried out in this paper.

21 We have 32 coeJcients from an eight-lag, two variable system, two coeJcients for the error correctionterm, since the cointegration vector is known, and three more coeJcients from the non-redundant elementsof the covariance matrix of the residuals. Notice that p becomes 17, 9 and 9 for the Brst, second and thirdsubsamples, respectively.

22 We consider that ut and vt follow AR(1) processes given by ut = 0ut−1 + zt , and vt = 1vt−1 + st ,respectively; where 0 and 1 are both included in the interval [−1; 1], and zt and st are i.i.d. random errorswith mean zero and variance �2

z and �2s .


the remaining N observations to carry out the estimation. After N observations havebeen simulated, the inMuence of the initial conditions must have disappeared.

The SME of ,0 is obtained from the minimization of a distance function of error-correction representation coeJcients from real and simulated data. Formally,

min,

JT = [HT (,0) − HN (,)]′W [HT (,0) − HN (,)];

where the weighting matrix W−1 is the covariance matrix of HT (,0).Denoting the solution of the minimization problem by ,̂, Lee and Ingram (1991)

prove the following results:

√T (,̂ − ,0) → N

[0;(1 +

1n

)(B′WB)−1

];

(1 +

1n

)TJT → 22(p − k);

where B is a full rank matrix given by B = E(9HNi(,)=9,). 23

5.4. Empirical results

First of all we show the estimation results for the full sample in Table 3. The resultsshow that the �+

4 -fundamental solution obtained in Appendix C provides the best Bt.Moreover, the estimated value of 1 is small but statistically signiBcant, supportingthe hypothesis of a feedback relationship from stock prices to dividends. The value ofthe goodness-of-Bt statistic for the fundamental solution (1 + 1=n)TJT = 51:311, whichis distributed as a 22(29) for the whole sample, clearly shows that the cross-equationrestrictions imposed by the RE in this equilibrium solution are not supported by thedata.

In order to detect the presence of switching equilibria, we estimate the alternativesolutions of the model for the three sub-samples previously detected.

Table 4 displays the estimation results for the period 1871–1910. The best Bt isobtained with the �−

4 -fundamental solution. The estimation results show that feedbackparameters (1 and 3) are small but statistically signiBcant. For this fundamentalsolution the value of the goodness-of-Bt test (1+1=n)TJT =38:06, which is distributedas a 22(9) for this sub-sample, implies that the cross-equation restrictions imposed bythis solution are also rejected at standard critical values.

The estimation results for the period 1910–1955 are shown in Table 5. For thissub-sample, the backward solution Bts the data better than any other solution.

23 The objective function JT was minimized using the optimization package OPTMUM programmed inGAUSS language. The Broyden–Fletcher–Glodfard–Shanno algorithm was applied. To compute the covari-ance matrix we need to obtain B. Computation of B requires two steps: Brst, obtaining the numerical Brstderivatives of the coeJcients of the error-correction model with respect to the estimates of the structuralparameters , for each of the n simulations; second, averaging the n-numerical Brst derivatives to get B.


Table 3Empirical results for the US stock market; period 1871–1989

Solution Fundamental(+) Backward

(1 + 1n )TJT 51.311 125.816

1 0.01518 0.01302(0.00201) (0.00202)

�z 0.20260 0.18990(0.00238) (0.00973)

�s 0.00391 0.00385(0.00004) (0.00004)

� 0.76390 0.94632(0.03741) (0.04727)

0 0.95941 0.02108(0.02029) (0.11217)

1 −0.14479 0.20249(0.15851) (0.22110)

3 −0.01494 −0.00859(0.00204) (0.00252)

4 −0.11909 0.15031(0.12545) (0.17438)

Note: Standard errors are in parentheses in this and all subsequent tables.


Solution Fundamental(−) Backward

(1 + 1n )TJT 38.056 36.842

1 0.00618 0.00607(0.00121) (0.00123)

�z 0.00236 0.06320(0.00501) (0.00377)

�s 0.00211 0.00211(0.00005) (0.00005)

� 0.95215 0.95182(0.06646) (0.04808)

0 −0.21099 −0.21725(0.18490) (0.15308)

1 0.17955 0.46373(0.28047) (0.19497)

3 0.01479 0.02369(0.00586) (0.00565)

4 −0.02173 0.12209(0.04935) (0.26174)

Moreover, it can be established that the backward solution is supported by the databecause the estimate of the discount factor � when estimating the backward solutionis quite reasonable (� = 0:9792), but the equivalent estimate is quite unrealistic when



Solution Fundamental(+) Backward

(1 + 1n )TJT 1.985 1.630

1 0.02984 0.02643(0.02194) (0.00525)

�z 0.16515 0.17342(0.02113) (0.00444)

�s 0.00233 0.00402(0.10117) (0.00235)

� 0.58352 0.97924(0.50701) (0.01032)

0 0.79451 0.08320(0.17384) (0.18960)

1 −0.95258 −0.76106(4.92710) (0.43867)

3 −0.03220 −0.02045(0.04595) (0.00557)

4 −0.44003 −0.42536(3.71654) (0.22793)

estimating the fundamental solution (� = 0:5835). 24 Focusing on the estimation re-sults from the backward solution, we observe that the signiBcant values of 1 and3 also show evidence of feedback for this sub-sample. The goodness-of-Bt statistic(1 + 1=n)TJT = 1:630, which is distributed as a 22(1) implies that the cross-equationrestrictions imposed by this equilibrium are not rejected at any standard critical value.

Table 6 shows the estimation results for the sub-sample 1955–1975. The �4-fundamental solution Bts the data better than the backward solution. 25 The signiBcantvalues of 1 and 3 again show evidence of feedback. Moreover, the goodness-of-Btstatistic for this solution (1 + 1=n)TJT = 3:09, which is distributed as a 22(2), clearlyshows that the data do not reject the cross-equation restrictions imposed by this equi-librium at any standard critical value.

Comparing the estimation results we observe that each sub-sample is characterized bya diIerent RE equilibrium. These estimation results provide some evidence in support ofthe hypothesis of switching equilibria. Moreover, the parameter values of the dividendprocess change when alternative sub-samples are considered. 26

24 As an alternative, we have estimated the fundamental solution imposing a reasonable value for �. Inthis case, the Bt of the fundamental solution is signiBcantly worse than the one obtained for the backwardsolution.

25 We say that the �4-fundamental solution Bts the data because the data support the restriction that theterm inside the square root deBning �+

4 and �−4 must be equal to zero. This additional restriction can be

written in such way that 4 is no longer a free parameter.26 When the dividend process is (2) and stock prices and dividends are cointegrated it is the size of the

sum of the feedback parameters, 13, that determines which solution characterizes the equilibrium. Therefore,the results for the cointegration case can still be represented by the line connecting the points [(1 − �)=�; 0]and (0,1) in Fig. 2 with 1 and 2 being replaced by 13 and 24, respectively.



Solution Fundamental Backward

(1 + 1n )TJT 3.086 106.237

1 0.00864 0.00951(0.00169) (0.00229)

�z 0.14378 0.25288(0.04350) (0.02805)

�s 0.00170 0.00034(0.00012) (0.00739)

� 0.96737 0.96604(0.01656) (0.06237)

0 0.47985 −0.11582(0.25039) (0.30601)

1 0.55585 0.11210(0.44430) (40.50342)

3 −0.00313 0.00352(0.00187) (0.00293)

4 −0.15867(0.48295)

We observe that the relative volatility of the innovations entering the stock priceand dividend equations (that is, �z=�s) during the sub-sample 1955–1975 is twice ashigh as that for the sub-sample 1910–1955. In addition, we can observe an inverserelationship between the estimated values for the sum of the feedback parameters, 13,and the relative volatility. This empirical evidence supports the Muth–Lucas hypothesisthat the higher the relative volatility of the innovations is, the lower the informationalcontent given to stock prices (measured by the size of 13) must be when dividenddecisions are made. 27

By assuming a reasonable value for the discount rate (say � = 0:93) we have that(1− �)2=�=0:0052688 in Fig. 2. The point estimates of 13 for the sub-samples 1910–1955 and 1955–1975 are 0:00598 and 0:00510, respectively. 28 These point estimatessuggest, on the one hand, that the equilibrium for the sub-sample 1910–1955 is de-scribed by a point such as X in Fig. 2 where the backward solution characterized theunique equilibrium implying cointegration. On the other hand, the point estimates sug-gest that the equilibrium for the sub-sample 1955–1975 is described by a point such

27 Muth (1961a) posited this hypothesis in the context of the permanent income hypothesis of consumption.Lucas (1973) postulated this hypothesis to analyze the output-price relationship. Lucas noted that whenmonetary Muctuations become more volatile, agents pay less attention to the price signal when making theirdecisions.

28 Obviously, the conclusions are based on point estimates that are subject to sample variability. However,Bnding more robust evidence on whether 13 changes signiBcantly between sub-samples is rather problematic.The reason is that for reasonable values of �, the threshold value of (1 − �)2=� has to be quite small, so tohave evidence of switching equilibria caused by small changes in the feedback parameters the size and thechanges in the estimates of 13 have to be very small, as our estimates show.


as Y where the fundamental solution characterizes the unique equilibrium implyingcointegration between stock prices and dividends. 29

Our estimation results can be summarized as follows. First, our empirical results,using structural estimation, provide additional evidence supporting the hypothesis ofa feedback mechanism from stock prices to dividends found by Timmermann (1994)using (OLS) reduced form estimation. Second, our empirical evidence supports thehypothesis of switching equilibria and thus provides an explanation of why the PVmodel Bts the data poorly in terms of the goodness-of-Bt statistic when analyzing thewhole sample, but the Bt is much better when considering the alternative sub-samples.Finally, our estimation results suggest a negative relation between the size of feedbackparameters and the volatility of stock price innovations relative to the volatility ofdividend innovations which supports the Muth–Lucas hypothesis.

6. Conclusions

Many studies have found that US stock prices are more volatile than is impliedby the PV model. This paper shows theory and evidence that the observed excessvolatility can be attributed in principle to switches between alternative (bubble-free)RE equilibrium solutions of the PV model. On the one hand, this paper shows ana-lytically that exogenous small changes in the dividend process parameters may induceswitching equilibria and switching equilibria result in both large changes in the vari-ance of the spread and large changes in the response of stock price variation to thespread.

On the other hand, we use historical annual US data and an indirect inference methodto estimate the PV model for stock prices under the assumption that stock prices anddividends are cointegrated. Full-sample estimation results show signiBcant evidenceof feedback, although all cointegrating RE equilibria are rejected by the data. Whenanalyzing diIerent sub-samples, the empirical results provide evidence supporting thehypothesis of switching equilibria and a good Bt of the PV model for the sub-samples1910–1955 and 1955–1975. Moreover, we Bnd evidence of a small but very signiBcantpresence of feedback from stock prices to dividends in each sub-sample considered.This feedback is smaller in those periods in which the volatility of stock price inno-vations relative to the volatility of dividend innovations is higher. The latter empiricalresult supports the Muth–Lucas hypothesis which, in this context, postulates that theinformational content given to stock prices when deciding on dividends is inverselyrelated to the volatility of stock price innovations relative to the volatility of dividendinnovations.

Acknowledgements

We are grateful for comments and suggestions from Antonio D-.ez de los R-.os,Ricardo Dom-.nguez, Eva Ferreira, Javier Gardeazabal, Alfonso Novales, Pedro Pereira,

29 Notice that purely for the sake of illustration we have assumed a relatively large change in 1 whendescribing the switch from X to Y in Fig. 2.


Luis Puch, Marta Reg-ulez, Gonzalo Rubio, anonymous referees and participants atseminars at 2002 Econometric Society European Meetings, XXV Simposio de An-alisisEcon-omico, X Foro de Finanzas, the Universidad Complutense de Madrid and Universi-dad del Pa-.s Vasco. This research project started when the authors were visiting the De-partment of Economics at the University of California, San Diego. They are grateful forthe hospitality received there. Financial support from Ministerio de Educaci-on, GobiernoVasco and Universidad del Pa-.s Vasco through projects BEC2000-1393, PI-1999-131and 9/UPV00035.321-13511/2001, respectively, is gratefully acknowledged.

Appendix A.

Proof of Proposition 1. We consider that an equilibrium solution exists when the so-lution is real rather than complex. When inequality (9) holds, all three solutions con-sidered exist. However when inequality (9) does not hold, �i is a complex numberfor i = 1; 2. Therefore, only the backward solution exists because the two fundamentalsolutions are complex.

Let us consider the zero-level curve of the term inside the square-root in the deBnitionof the �i:

(1 − �(1 + 2))2 − 4�212 = 0: (A.1)

By diIerentiating this curve, we obtain

92

91= −1 − �(1 − 2)

1 + �(1 − 2);

which is negative for all 0¡1 ¡ 1 and 0¡2 ¡ 1. Therefore, by the implicit functiontheorem we know that there exists a unique continuously diIerentiable function 2 =3(1), which has negative slope, characterizing the zero-level curve, (A.1). It is easyto see that 3 is a convex function whose intercepts are (1; 2)=(0; �−1) and (1; 2)=(�−1; 0). Furthermore, pairs of (1; 2) in the lower (upper) contour set of (A.1) do(not) satisfy inequality (9) (see Fig. 1).

It is easy to prove that along the zero-level curve, (A.1)

91

9�

∣∣∣∣2 constant

= −21

�(1 − 2�2) + 1(2 − �)

1 + �(1 − 2)¡ 0:

This means that the smaller the discount factor is the farther from the origin the levelcurve (A.1) is. Furthermore, for � = 1=4, (A.1) crosses the pair (1; 2) = (1; 1); thismeans that the upper contour set of (A.1) does not intersect the region in which0¡1 ¡ 1 and 0¡2 ¡ 1. Therefore, for �6 1=4, the region in which the backwardsolution is the unique real equilibrium solution is an empty set.

On the other hand, the point (1; 2)=(1=4�; 1=4�) is in level curve (A.1). Therefore,at least any combination (1; 2) = (1=4� − �; 1=4� − �) where 0¡�¡ 1=4� always


belongs to the lower contour set of (A.1) for any 0¡�¡ 1. This means that theregion in which the three solutions considered are real is always a non empty set.

Proof of Proposition 2. Proof of (i): Using (3), after recursive substitutions and rear-ranging, fundamental solutions (7) and (8) can be written as follows

a(L)pt = k + b(L)vt−1 + c(L)ut ; (A.2)

where L denotes the lag operator and a(L)=1−�iL, b(L)=�i, c(L)=’i and k=�i0+’i

for i = 1; 2, respectively.As is well known, the process characterized by (A.2) is stationary if |�i|¡ 1. When

�i is real, �i = 2 + �i1¿ 0 for i = 1; 2, and therefore the condition for fundamentalsolutions to be stationary is simply �i ¡ 1.

Using standard results (for instance, see Granger and Newbold, 1977, pp. 26–27), theautocovariance generating function for the stock price process (A.2), when the processis stationary, can be written as follows

�(L)i =b(L)b(L−1)a(L)a(L−1)

�2v +

c(L)c(L−1)a(L)a(L−1)

�2u:

The variance of the stock price processes characterized by fundamental solutions (7)and (8) (�i

0, for i = 1; 2; respectively) is equal to the coeJcient associated with L0

in the power series expansion of the autocovariance generating function �(L)i, which,after some algebra, can be written as expression (14)

�i0 =

�2i

1 − �2i�2

v +1 − �2

i + �2i

21

1 − �2i

’2i �

2u:

Notice that �i=2+�i1 and ’i=1=(1−�1(1+�i)). Therefore, to show that the varianceof the �2-fundamental solution, (8), is lower than the variance of the �1-fundamentalsolution, (7), it is suJcient to show that �i

0 is increasing in �i since �1¿ �2. Aftersome calculation this is easily proved. Let us denote the Brst and second terms in (14)by A and B, respectively. Then

9�i0

9�i=

2�2v�i(1 − �i2)(1 − �2

i )2+

2�i21(1 − �i2)1 − �2

i+ 2’2

i 1�(1 − �2i + �2

12i ):

Notice that this partial derivative is positive under the stationary condition, �i ¡ 1. Thisimplies that the variance in stock prices is increasing in �i. This completes the proof.Proof of (ii): The bubble-free backward equilibrium solution, (13), can be written

as an ARMA process as follows. By taking into account the dividend process, Eq. (3),the backward solution (13) can be written as

pt = −0 + (�−1 − 1)pt−1 − 2dt−2 − �−1ut−1:

Adding and subtracting 2pt−1 from this and using (13), we have that

q(L)pt = −0 + m(L)vt + n(L)ut ; (A.3)

where q(L)=1−(�−1−1+2)L+2�−1L2, m(L)=−2L2 and n(L)=−�−1(1−2L)L.


Since the backward solution, (13), can be expressed as the ARMA(2; 1) process(A.3), this equilibrium solution is stationary whenever all roots of q(L)= 0 lie outsidethe unit circle. Let us denote by q1 and q2 the roots of q(L) = 0. Thus,

q(L) =2

�(L − q1)(L − q2) = 0;

where

q1 =(�−1 − 1 + 2) +

√(�−1 − 1 + 2)2 − 42�−1

22�−1 ;

q2 =(�−1 − 1 + 2) −

√(�−1 − 1 + 2)2 − 42�−1

22�−1 :

After some algebra, it is easy to show that the square root in the deBnition of the qs isthe same as that in the deBnition of the �s. Therefore, we can establish the followingrelationship between these qs and the �s that deBne the fundamental solutions:

qi =�2

(2 + �i1);

for i = 1; 2. Moreover, it is easy to see that q1q2 = �=2. Therefore, for i; j = 1; 2 andi �= j,

qi =1

2 + �j1≡ 1

�j;

for i �= j. Therefore, q1 and q2 are both outside the unit circle if and only if �i ¡ 1,for i = 1; 2.

In order to obtain the variance of the backward solution when the process is station-ary, let us rewrite q(L) in (A.3) as follows:

q(L) = (1 − �1L)(1 − �2L);

where 0¡�i = q−1i ¡ 1. The autocovariance generating function for the stock price

process under the backward solution, �(L)b, can be written as follows

�(L)b = �2vm(L)m(L−1)q(L)q(L−1)

+ �2un(L)n(L−1)q(L)q(L−1)

=�2

v22 + �2

u�−2(1 − 2L)(1 − 2L−1)

(1 − �1L)(1 − �2L)(1 − �1L−1)(1 − �2L−1):

The variance in the stock price process characterized by the backward solution isderived from the coeJcient associated with L0 in the power series expansion of theautocovariance generating function �(L)b. After simple but tedious algebra, one canshow that the variance of stock prices process is given by Eq. (15). This completesthe proof.


Proof of Proposition 3. In Appendix B we show that when �i =2 +�i1 =1 for i=1or i=2, we are along the line 2 =1− (�=(1− �))1. After some calculation it is easyto see that along this line

�i =

1 − ��1

− 1 for i = 1;

�1 − �

for i = 2:

Taking this result into account we can see that the �1-fundamental solution does notexist because �1

0 is not deBned and the �2-fundamental solution is given by

�2 = (�20; �

21; �

22) =

( −�0

(1 − �)(� − 2);

�1 − �

;12

):

On the other hand, taking into account that the �2-fundamental solution can be ex-pressed as the ARMA process (A.2), it is immediate to see that when �1 = 1¿�2,the �2-fundamental solution is stationary because �2 is inside the unit circle. Howeverif �1 ¿�2 = 1, the �2-fundamental solution implies that stock prices and dividends arecointegrated.

In the same way the backward solution can be written as the ARMA process (A.3).Therefore when �1 = 1¿�2, the backward solution implies cointegration between thestock prices and dividends. However if �1 ¿�2=1, the backward solution is explosivebecause one of the two roots is outside the unit circle.

Proof of Proposition 4. From Proposition 3 and the backward equilibrium solution(13), it is easy to show that the spread under the �2-fundamental and the backwardsolutions are given, respectively, by

spft =

−�0

(1 − �)(� − 2)+

12

ut ;

spbt = − 0

1 − �+

2

�spb

t−1 − �−1ut−1 − �1 − �

vt−1:

Given that ut and vt follow i.i.d. processes with mean zero and variances �2u and �2

v ,respectively, the expressions in the proposition are straightforwardly derived.

Appendix B.

In this appendix we prove the statements about Fig. 2 made in Section 3.Let us consider the level curve

�1 ≡ 2 + �11 = 1: (B.1)


By diIerentiating this curve, we obtain

92

91= −

1 − �(1 − 2) +√

[1 − �(1 + 2)]2 − 412

1 + �(1 − 2) −√

[1 − �(1 + 2)]2 − 412

: (B.2)

Taking into account that throughout (B.1),

2�(1 − 2) − [1 − �(1 + 2)] =√

[1 − �(1 + 2)]2 − 412; (B.3)

we can rewrite (B.2), as92

91= − �

1 − �:

Therefore, the implicit function associated with (B.1) is a linear function with negativeslope. On the other hand, (1; 2) = ((1 − �)=�; 0) and (1; 2) = ((1 − �)2=�; �) lieson (B.1). Notice that the latter pair also satisfy the zero-level curve (A.1) deBned inProposition 1

[1 − �(1 + 2)]2 − 4�212 = 0; ⇔ �1 = �2:

Moreover, since the term on the right-hand side of (B.3) is non negative, all pairs of(1; 2) in the level curve (B.1) must satisfy

262� − 1

�+ 1:

Notice that the pair (1; 2) = ((1 − �)2=�; �) satisfy the latter equation with equality.All these results mean that the level curve (B.1) is equivalent to the set of all pairs

(1; 2) on the line 2 = 1 − �=(1 − �)1, such that 26 �.Furthermore, it is easy to see that,

9�1

91¡ 0: (B.4)

This means that all pairs satisfying inequality (9) and located to the right (left) of thelevel curve (B.1) satisfy �1 ¡ 1 (�1 ¿ 1).

On the other hand, for any combination satisfying inequality (9),

�1 = 2 + �11¿1 + �2

2�:

Since all pairs satisfying inequality (9) and located northwest of combination((1 − �)2=�; �) are such that 2 ¿�,

�1¿1 + �2

2�¿

1 + �2

2�¿ 1:

This result and (B.4) imply that, only for those pairs satisfying inequality (9) andlocated to the right of the level curve (B.1), the �1-fundamental solution is stationary.

Following the same steps, we can see that the level curve �2 ≡ 2 + �21 = 1 isequivalent to the set of all pairs (1; 2) on the line 2 = 1 − �=(1 − �)1, such that26 �. In other words, in Fig. 2 the level curve �2 = 1 is equivalent to the segmentconnecting the points (1; 2) = ((1 − �)2=�; �) and (1; 2) = (0; 1).


Moreover, it is easy to prove that the �2-fundamental solution is not stationary onlyfor those pairs satisfying inequality (9) and located on and above that level curve.

The backward solution is stationary only in region C since in this region �1 ¡ 1 and�2 ¡ 1.

Appendix C.

In this appendix we obtain the fundamental solutions when the dividend process isdescribed by (2). We begin by writing pt as a linear function of ut and the predeter-mined state variables dt−1, pt−1, dt−2 plus a constant,

pt = �0 + �1dt−1 + �2pt−1 + �3dt−2 + �4ut : (C.1)

For appropriate real values of �0, �1, �2, �3 and �4, the expectational variable Etpt+1

will then be given by

Etpt+1 = �0(1 + �11 + �2) + �10 + (�1(�11 + �2 + 2) + �3)dt−1

+ (�2(�11 + �2) + �13)pt−1 + (�3(�11 + �2) + �14)dt−2

+ �4(�11 + �2)ut : (C.2)

To evaluate the �s, we substitute (2), (C.1) and (C.2) into (1). That equation impliesidentities in the constant term, dt−1, pt−1, dt−2 and ut as follows:

�0 = �(�0(1 + �11 + �2 + 1) + 0(1 + �1));

�1 = �(�1(�11 + �2 + 1 + 2) + 2 + �3);

�2 = �(�2(�11 + �2 + 1) + 3(1 + �1));

�3 = �(�3(�11 + �2 + 1) + 4(1 + �1));

�4 = 1 + ��4(�11 + �2 + 1):

Taking into account the cointegration restriction, we solve this Bve equation systemwith Bve unknowns. One can show that �4 has two solutions given by

�4 =�(�3 + 1) − (1 − �)(1 − �4)

2(1 − �)�4

± ([(�(�3 + 1) − (1 − �)(1 − �4)]2 + 4(1 − �)2�4)1=2

2(1 − �)�4:

Notice that 2 does not appear in this expression because the cointegration restrictionhas been already imposed. We denote the two alternative solutions of �4 by �+

4 and �−4 ,

respectively. Once a solution of �4 is obtained, the associated solution for �0, �1, �2

and �3 can be derived recursively. We do not show the expressions of the �s becausethey are (cumbersome) non-linear functions of �4 and the structural parameters of themodel. These expressions are available upon request.


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