Dividend Variability and Stock Market Swings

Martin D. D. Evans∗

Department of Economics

Georgetown University

Washington DC 20057

The Review of Economic Studies

Abstract

This paper examines the extent to which swings in stock prices can be related to variations

in the discounted value of expected future dividends when investors face uncertainty about

their future behavior. I develop an econometric model that accounts for the instability of U.S.

dividend growth and discount rates during the past 120 years. Estimates of the model reveal

that changing forecasts of future dividend growth account for more than 90% of the predictable

variations in dividend-prices. The estimates also imply that instability in the dividend and

discount rate processes contribute signiÞcantly to the predictability of long-horizon stock returns.

∗I would like thank the referees and The Foreign Editor, John Cochrane for many helpful suggestions. The paperhas also beneÞtted from the comments of seminar participants at The Universities of Bristol, Exeter, Liverpool, andYork, The London School of Economics, The Wharton School, The Federal Reserve Bank of Philadelphia, and the1995 meetings of the Econometric Society. The research was completed while visiting the The Financial MarketsGroup at The London School of Economics and The Bank of England. I also gratefully acknowledge the Þnancialsupport of the Houblon-Norman Fund at The Bank of England.

1

1 Introduction

The goal of relating stock price movements to fundamentals remains elusive. Following Shiller

(1981) and LeRoy and Porter (1981), a large literature has developed documenting the fact that

stock prices are excessively volatile compared to the prices implied by the discounted value of

expected and actual future dividends. In addition, Campbell (1991), Hodrick (1992), Bekaert and

Hodrick (1992) and others have found that stock returns appear predictable over long horizons. In

the light of these Þndings, recent research has focused on whether the variations in discount rates

necessary to account for both the volatility-test rejections and the predictability of stock returns

can be reconciled with the discount rate variations inferred from the economy [see, for example,

Abel (1993), Campbell and Cochrane (1994), Cecchetti, Lam and Mark (1990, 1993), Kandel and

Stambaugh (1990)].

This paper re-examines the extent to which swings in stock prices can be related to variations in

the discounted value of expected future dividends. The principle innovation in my analysis is that

I allow for the possibility that aggregate dividends and discount rates have not followed stable time

series processes over the past 120 years. Although this possibility has been noted previously by

Lehman (1992) and Campbell and Ammer (1993), the effects of instability have yet to be explicitly

incorporated into empirical models of stock prices.

I use the log-linear pricing framework developed by Campbell and Shiller (1989) to develop and

estimate models for the dividend-price ratio in which investors rationally account for instability

in the dividend and discount rate processes. In principle, instability may affect the behavior of

stock prices through two channels. First, stock prices may vary as investors learn about the current

process fundamentals are following. Barsky and DeLong (1993) investigate this possibility in a

model where investors used a simple learning rule to revise their estimates of long-term dividend

growth. Similarly, Timmermann (1994) studies the convergence properties of a model in which

investors learn about the long-run dynamics of dividends. Second, investors may rationally antic-

ipate a change in the future behavior of fundamentals when forming their forecasts. Under these

conditions, a peso problem may affect the dynamics of stock prices - a phenomena that has been

studied in foreign exchange models, [Evans and Lewis (1995)]. This paper develops models that

allow for both learning and peso problems resulting from the instability in the behavior of dividends

and discount rates.

The paper begins by presenting the dividend-ratio model of Campbell and Shiller (1989). This

model relates the behavior of the log dividend-price ratio to investors expectations of future fun-

damentals; the difference between dividend growth and the discount rate. Within this framework,

I examine the theoretical consequences of instability using a simple regime switching speciÞcation

for fundamentals. I show how fundamentals switching can induce predictability in excess returns,

1

measured as the difference between the return on stocks and the discount rate, even when investors

expect excess returns to be zero. This Þnding contradicts standard inferences based on rational

expectations. However, it is perfectly consistent with rational investor behavior in samples where

the distribution of regime switches differs from the underlying distribution used by investors. I also

show that under these conditions the variance of actual dividend-prices can exceed the variance

of warranted dividend-prices calculated from the ex post realizations of fundamentals. Such small

sample effects provide alternative interpretations for the return predictability and excess volatility

Þndings in the literature.

The importance of these small sample effects depends upon the degree of instability in funda-

mentals and the extent to which rational investors account for switches when forecasting. Section

3 presents an econometric model to examine these issues. The model takes the form of a Vec-

tor Autoregression (VAR) for dividend-prices and fundamentals with coefficients that vary across

regimes. This model is a multivariate generalization of the switching the models pioneered by

Hamilton (1988). Because the switching VAR allows the autocorrelation function for fundamentals

and dividend-prices to vary across regimes, it belongs to the class of nonlinear time series models.

My analysis provides an example of how the use of a nonlinear model can alter our perspective on

the relevance of certain economic models.

Section 4 analyses the model estimates based on 120 years of U.S. stock market data. There are

large cross-regime differences in the autocorelation structure of fundamentals and many changes in

regime during the sample period. As a result, the forecasts of fundamentals implied by the switching

VAR estimates are quite different from the forecasts implied by standard VARs. This difference is

important for understanding the origins of dividend-price movements. I Þnd that approximately

60% of the variance of dividend-prices can be attributed to changing forecasts of fundamentals

derived from switching VARs, compared to 35% when the forecasts are based on standard VARs.

Importantly, these statistics are derived from switching VARs that do not impose the restrictions

of any theoretical model for stock prices. As such, they speak quite generally to the origins of stock

price movements and contrast with the results in Cochrane (1991) and Campbell (1991).

I also use the switching VAR estimates to test particular versions of the dividend-ratio model.

Overall, I Þnd that the behavior of dividend-prices can be well characterized by the dividend-ratio

model when fundamentals are identiÞed by the difference between dividend growth and the commer-

cial paper rate. Although the tests of the cross-equation restrictions implied by the dividend-ratio

model can be formally rejected, the model performs well in a number of other economically meaning-

ful respects. Importantly, the model estimates indicate that changing forecasts of future dividend

growth account for more than 90% of the variations in dividend-prices. This result rehabilitates

the idea that swings in stock prices are primarily associated with news about dividends.

Section 5 examines the robustness of these Þndings to the assumed presence of only two regimes

2

and the absence of learning. There is little evidence that more than two regimes are necessary

to characterize the instability in fundamentals over the sample. At the same time I can strongly

reject the presence of a single regime. I also estimate models that allow investors to learn about the

current regime. The introduction of learning adds a good deal of complexity to these models. As a

result, I am only able to construct an approximate test of the cross-equation restrictions implied by

the dividend-ratio model. Subject to this caveat, the results from these learning models are similar

to those presented in Section 4.

Section 6 examines the implications of the switching VAR estimates for the behavior of stock

returns. Using my preferred set of estimates (with the restrictions of the dividend-ratio model

imposed), I Þnd that only 14% of the variance in dividend-prices is attributable to changing forecasts

of future returns. I also show that approximately 80% of the variance in unexpected stock returns

are due to revisions in the expected present value of fundamentals. These Þndings contrast with the

results in Campbell and Ammer (1993) based on standard VARs. The switching VAR estimates

are also used to study long-horizon returns. The model estimates imply that there is a good

deal of ex post predictability in unexpected returns. This small sample phenomena arises from

the insatiability of fundamentals and is quite consistent with rational investor behavior. The

instability in fundamentals also accounts from the observed degree of predictability in stock returns

at horizons beyond a year. These Þndings show that there is no inconsistency between the observed

predictability in stock returns and the idea that stock prices vary primarily with dividend news.

The paper ends with a summary of the main results.

2 Dividend-Price Variation with Fundamentals Switching

This section examines the theoretical consequences of instability in the time series behavior of

fundamentals for the behavior of stock prices. I begin by presenting the dividend-ratio model. This

model serves as the framework for both the theoretical and empirical analyses below.

2.1 The Dividend-Ratio Model

Let pt be the log real stock price at the beginning of year t and dt+1 the log of real dividends paid

over the year. The realized log return from the beginning of year t until the beginning of year t+1

is ht+1 ≡ log(exp(pt+1) + exp(dt+1))− pt. Campbell and Shiller (1989) show that this return maybe well approximated by

ht+1 = κ+ δt − ρδt+1 +∆dt+1, (1)

3

where δt ≡ dt − pt is the log dividend-price ratio at the beginning of year t, and ∆dt+1 is thedividend growth rate over the year. ρ is a parameter close to but smaller than 1 and κ is a positive

constant. Iterating (1) forward and imposing the terminal condition, limt→∞ρiδt+i = 0, gives anexpression for the log dividend-price ratio in terms of the discounted value of future returns and

dividend growth:

δt =∞Xi=1

ρi−1(ht+i −∆dt+i)− κ1− ρ . (2)

Equation (2) is not an economic model for dividend-prices because all the variables are measured

ex post. To derive a model, Campbell and Shiller restrict the behavior of stock returns with the

assumption that

E[ht+1|Ωt] = E[γt+1|Ωt], (3)

where γt+1 is the ex post discount rate and E[.|Ωt] denotes investors expectations given informationat the beginning of year t, Ωt. Throughout I assume that Ωt includes It, the information set

containing economic variables observable at the beginning of year t, that includes δt. To derive an

economic model, Þrst take expectations of the left and right-hand sides of (2) conditional on Ωt.

Next, use the law of iterated expectations with (3) to substitute for E[ht+j |Ωt]. Since δt is equal toits expectation, E[δt|Ωt], we can write the resulting expression as

δt = −∞Xi=1

ρi−1E[yt+i|Ωt]− k1− ρ , (4)

where yt+1 ≡ ∆dt+1 − γt+1.Campbell and Shiller refer to (4) as the dividend-ratio model. It says that the log dividend-

price ratio [hereafter simply dividend-prices] is equal to a constant minus the expected present

value of future fundamentals, yt+i. If the discount rate is constant, (4) implies that all variations in

dividend-prices are due to changes in the expected present value of future dividend growth. When

the expected future discount rate varies, dividend-prices will change with the expected present

value of dividend growth less the discount rate, ∆dt+i − γt+i.Before the empirical implications of (4) can be examined, investors expectations must be iden-

tiÞed. The common approach in the literature is to invoke the standard rational expectations

assumption that investors forecast errors are uncorrelated with prior information in Ωt. When

fundamentals follow a stable time series process that is understood by investors, their forecast

errors retain this property within a Þnite data sample. Under these circumstances, the empirical

implications of (4) can be derived with the techniques developed by Campbell and Shiller (1989).

4

The premise of this paper is different. My analysis is based on the idea that fundamentals may

have followed an unstable time series process that was poorly understood by investors. I begin by

examining the theoretical consequences of instability in the fundamentals process for the behavior

of dividend-prices and stock returns. My aim is to show how standard inference methods about

the origins of dividend-price movements can be unreliable when there is instability. To keep things

simple, I will not attempt to relate this instability to changes in the dividend policies of individual

Þrms or to developments in the economy that could affect the behavior of the discount rate. I will

also assume that the discount rate is observable so that data on fundamentals are available to the

researcher.

2.2 Fundamentals Switching

Suppose that dividend-prices are determined by the dividend-ratio model as shown in (4) and

fundamentals, yt, switch between two processes. Switches are determined by changes in a discrete-

valued variable, zt = {0, 1} which is known to investors at the beginning of year t. I further assumethat realizations of fundamentals during year t, yt+1, depend on the process being followed during

year t, determined by the value of zt. These realizations are written as yt+1(z).

To see how switches in fundamentals affect the behavior of dividend-prices, consider the variance

decomposition for dividend-prices implied by the dividend-ratio model. For this purpose, multiply

both sides of (2) by δt and take expectations. Substituting the identity ∆dt+i ≡ yt+i + γt+i intothe result gives,

V ar(δt) = − Covµ ∞Pi=1ρi−1E[yt+i|Ωt], δt

¶−Cov

µ ∞Pi=1ρi−1et+i, δt

¶+Cov

µ ∞Pi=1ρi−1(ht+i − γt+i), δt

¶. (5)

where V ar(.) and Cov(., .) denote the sample variance and covariance respectively, and et+i ≡yt+i − E[yt+i|Ωt]. Here the variance of dividend-prices is decomposed into the covariance betweendividend-prices and the expected present value of fundamentals, the present value of the forecast

errors, et+i, and the present value of excess returns, ht+i − γt+i.The dividend-ratio model places restrictions on this decomposition. Using (4) to substitute for

the expected present value of fundamentals, we see that the Þrst term on the R.H.S. is equal to

V ar(δt). Making this substitution, (5) implies that

Cov

µ∞Pi=1ρi−1(ht+i − γt+i), δt

¶= Cov

µ ∞Pi=1ρi−1et+i, δt

¶. (6)

Thus, the dividend-ratio model restricts the covariance between dividend-prices and the present

5

value of excess returns to equal the covariance between δt and the present value of the fundamen-

tals forecast errors. Under the standard rational expectations assumption that forecast errors are

uncorrelated with prior information, Ωt, that includes δt, the covariance on the right equals zero.

Thus, the dividend-ratio model implies that the present value of excess returns cannot be forecast

with dividend-prices.

To see how switching in the fundamentals process can affect this implication, I begin by writing

the realized value of fundamentals during period t+ i, for i > 0, as

yt+i = E[yt+i(0)|Ωt] +∇E[yt+i|Ωt]zt+i−1 +wt+i, (7)

where ∇E[yt+i|Ωt] ≡ E[yt+i(1)|Ωt] − E[yt+i(0)|Ωt]. Equation (7) decomposes future fundamentalsinto the conditional forecasts of yt+i under each process, E[yt+i(z)|Ωt] = E[yt+i|Ωt, zt+i−1 = z],and a residual, wt+i.

1 When investors hold rational expectations, their forecasts of yt+i(z) coincide

with the mathematical conditional expectation of yt+i. Taking expectations on both sides of (7)

conditioned on the Ωt for zt+i−1 = {0, 1} implies that E[wt+i|Ωt] = 0. Thus, wt+i inherits theproperties of conventional rational expectations forecast errors.

As investors are unaware of future regimes, their forecast errors will differ from the wt+i errors.

To see this, we Þrst take conditional expectations on both sides of (7):

E[yt+i|Ωt] = E[yt+i(0)|Ωt] +∇E[yt+i(0)|Ωt]E[zt+i−1|Ωt]. (8)

Taking the difference between (7) and this expression gives

et+i = ∇E[yt+i|Ωt](zt+i−1 −E[zt+i−1|Ωt]) +wt+i. (9)

Equation (9) shows investors forecast errors to be comprised of wt+i, and a term that depends upon

the error in forecasting the future regime, zt+i−1 −E[zt+i−1|Ωt]. In large samples, both terms willbe uncorrelated with elements of Ωt, including δt. Hence, the presence of switching does not affect

the implications of the dividend-ratio model for the forecastability of excess returns under these

circumstances. This result may not hold in small samples however. Here the empirical frequency

of regime switches is unrepresentative of the underlying distribution of regime changes used by

investors to forecast future fundamentals. Under these circumstances,the rational ex post errors in

forecasting zt+i−1 may be correlated with elements in Ωt, including δt.To illustrate, consider the extreme case where the sample only contains observations from regime

1Notice that it is always possible to write future fundamentals in this way irrespective of the switching processthey follow or the speciÞcation of investors information.

6

z = 0. Here the R.H.S. of (6) becomes

Cov

µ ∞Pi=1ρi−1wt+i, δt

¶−Cov

µ ∞Pi=1ρi−1∇E[yt+i|Ωt]E[zt+i−1|Ωt], δt

¶. (10)

During the sample, investors are likely to revise their expectations of future regimes, E[zt+i|Ωt],and/or their forecasts of how fundamentals will differ across regimes, ∇E[yt+i|Ωt]. This has theeffect of changing dividend-prices, δt, because it implies a change expected future fundamentals

[see (8)]. As a result, the last term in (10) will generally differ from zero even though there is no

change in regime over the sample. This means from (6) that excess returns will appear forecastable

within the sample.

Although this example is an extreme one, the basic point carries over to general cases where

the frequency of regimes changes within a sample differs signiÞcantly from the expected frequency

implied by the underlying distribution used by investors to forecast [Evans (1997)]. In a small

sample the covariance between (ht+i − γt+i) and δt may appear signiÞcantly different from zeroeven though the dividend-ratio model holds and investors have rational expectations. This means

that it is dangerous to judge the performance of the model from the apparent predictability of

excess returns, or equivalently, a model for expected excess returns.

Similar problems occur if we instead focus on the realizations of fundamentals during the sample.

To see why, consider the warranted value of δt implied by the dividend-ratio model:

Wt ≡ −∞Xi=1

ρi−1yt+i − k1− ρ = δt −

∞Xi=1

ρi−1et+i.

Multiplying both sides by Wt, and taking expectations, we obtain

V ar(Wt) = V ar(δt) + V arà ∞Xi=1

ρi−1et+i

!− 2Cov

à ∞Xi=1

ρi−1et+i, δt

!.

In large samples, the assumption of rational expectations implies that the covariance term is close

to zero so that V ar(Wt) ≥ V ar(δt). This variance bound on dividend-prices lies at the heart ofthe volatility tests pioneered by Shiller (1981) and LeRoy and Proter (1981). By contrast, when

switching induces a small sample covariance between forecast errors an dividend-prices, it is possible

for V ar(Wt) < V ar(δt). Instability in the fundamentals process can therefore lead to a violationof the variance bound in small samples. Hence, apparent excess volatility of actual dividend-prices

within a sample need not be interpreted as evidence against the dividend-ratio model.

This simple example demonstrates how standard inferences based on the forecastability of excess

returns and the (excess) volatility of dividend-prices can be misleading as to the factors governing

dividend-prices in small samples when there is instability in the fundamentals process.

7

3 The Econometric Model

This section presents an econometric model that allows us to examine the behavior of dividend-

prices in the presence of fundamentals switching. We will be able to examine the origins of

movements in dividend-prices quite generally with the model. It is also designed to test two

particular versions of the dividend-ratio model.

3.1 Dividend-Ratio Models

The versions of the dividend-ratio model I consider use the following discount rate speciÞcations:

E[γt+1|Ωt] = (1− ϕ)µ+ ϕE[γt|Ωt−1] + et.E[γt+1 − rt+1|Ωt] = (1− ϕ)µ+ ϕ(E[γt|Ωt−1]− rt) + et. (11)

In Model A, the expected discount rate follows an AR(1) process with innovations et. Since

E[ht+1|Ωt] = E[γt+1|Ωt] by assumption, this equation implies that expected stock returns fol-low an AR(1) process as in Campbell (1991). In Model B, the expected discount rate varies with

the real return on bonds, rt+1, again according to an AR(1) process. This model implies that the

expected excess return on stocks over bonds, or risk premium, follows an AR(1) process. Notice

that neither speciÞcation restricts the size of investors information, Ωt. Rather, they restrict the

way in which next periods forecast is related to last periods forecast.

We can now rewrite the equation for the dividend-ratio model in terms of an observable measure

of fundamentals, xt. Substituting for γt+1, using (A) or (B), we can rewrite (4) as

δt =µ− k1− ρ −

∞Xi=1

ρi−1E[xt+i|Ωt] + ξt (12)

with ξt = ϕξt−1 + εt,

where xt is equal dividend growth,∆dt, in Model A, and adjusted dividend growth, ∆dt − rt, inModel B. ξt is equal to the present value of expected returns in Model A and expected excess

returns in Model B, with εt = (1−ρϕ)−1et. Equation (12) represents two particular versions of thedividend-ratio model that form the basis for the empirical model. It differs from the dividend-ratio

model in (4) in that dividend-prices are now related to an observable measure of fundamentals, xt,

rather than the unobservable measure, yt ≡ ∆dt − γt.

8

3.2 The Switching VAR

The econometric model extends the VAR methodology developed by Campbell and Shiller (1989)

to allow for switches in the process for observed fundamentals. The model is based on (12) and the

following switching equation for fundamentals

xt+1 = a1(zt) + b11(zt)xt + b12(zt)δt + c11(zt)xt−1 + c12(zt)δt−1 + vt+1, (13)

with vt+1 ∼ N(0,σ2(zt)). Here realizations of fundamentals during year t depend upon the regime atthe beginning of the year, denoted by zt, through the coefficients a1(.), bij(.) and cij(.), and through

the variance of the innovations, σ2(.). As above, zt is assumed to follow an independent Þrst-order

Markov process with constant transition probabilities, λz ≡ Pr(zt = z|zt−1 = z), z = {0, 1}.If the dividend-ratio model in (12) holds and investors know the current regime as well as the

history of dividend-prices and fundamentals [i.e., Ωt = {It, zt−i,}i≥0], then the joint behavior ofdividend-prices and fundamentals can be described by a second-order switching VAR:"

xt+1

δt+1

#=

"a1(zt)

a2(zt+1, zt)

#+

"b11(zt) b12(zt)

b21(zt+1, zt) b22(zt+1, zt)

#"xt

δt

#(14)

+

"c11(zt) c12(zt)

c21(zt+1, zt) c22(zt+1, zt)

#"xt−1δt−1

#+

"v1,t+1

v2,t+1

#,

where v1,t+1 and v2,t+1 are serially uncorrelated innovations with a regime-dependent covariance

matrix. Notice that realizations of δt+1 can depend upon both zt+1 and zt. In this way, the model

allows investors to incorporate information about the current process for fundamentals, governed

by zt+1, when dividend-prices, δt+1, are determined. The VAR also allows dividend-prices at the

beginning of year t to have predictive power for fundamentals over the year via the b12(.) coefficient.

As (12) shows, δt depends on E[xt+1|Ωt]. Thus, b12(.) is likely to differ from zero when investorshave more information about future fundamentals than is contained in their past values alone. By

allowing b12(.) to vary with zt, the switching VAR can take account of cross-regime differences in

the information investors have about future fundamentals.

The dividend-ratio model places restrictions on switching VAR shown in (14). In particular,

using the method of undetermined coefficients, Appendix A shows that the coefficients in the

9

dividend-price equation must satisfy

a2(zt+1, zt) = π0(zt+1) + π1(zt+1)a1(zt)− ϕπ0(zt)π4(zt+1)/π4(zt),b21(zt+1, zt) = π3(zt+1) + π1(zt+1)b11(zt)− ϕπ1(zt)π4(zt+1)/π4(zt)b22(zt+1, zt) = π2(zt+1) + π1(zt+1)b12(zt),

c21(zt+1, zt) = π1(zt+1)c11(zt)− ϕπ3(zt)π4(zt+1)/π4(zt),c22(zt+1, zt) = π1(zt+1)c12(zt)− ϕπ2(zt)π4(zt+1)/π4(zt),

(15)

where

π0(i) =³ρπe0(i) + µ− κ− [1− ρπe1(i)]a1(i)

´φ(i), π2(i) =

³[ρπe1(i)− 1]c12(i)

´φ(i),

π1(i) =³ρπe2(i)− [1− ρπe1(i)]b11(i)

´φ(i), π3(i) =

³[ρπe1(i)− 1]c11(j)

´φ(i),

φ(i) =³1− ρπe2(i) + [1− ρπe1(i)]b12(i)]

´−1, π4(i) =

³1− ρϕ+ ρϕπe4(i)

´φ(i), (16)

with πe(i) ≡ π(i)λi + π(k)(1 − λi) for any π(.), i 6= k. These cross-equation restrictions hold forthe four possible combined values of zt+1 and zt. Thus the dividend-ratio model places a total of

20 restrictions on the coefficients of the switching VAR.

Below, I present estimates of the switching VAR in (14) with the coefficients of the dividend-

price equation satisfying (15). This reduces the number of parameters to be estimated but does

not constrain the estimates of πi(.) to conform with the restrictions of the dividend-ratio models

in (16). SpeciÞcally, I estimate the coefficients of the fundamentals process, {a1(.), b11(.), b12(.),c11(.), c12(.)}, the AR parameter, ϕ, the Markov transition probabilities, λz, the covariance matrixof innovations to the VAR, together with the π0s. The maximum likelihood estimates are derivedusing Hamiltons (1988) algorithm assuming that the innovations v1,t+1 and v2,t+1 are normally

distributed.

4 Results

4.1 Data and Stability Results

The empirical analysis uses the annual series on stock prices and dividends for the Standard and

Poors Composite Stock Price Index, extended back to 1871 by using the data in Cowles (1939).

Real stock prices are computed by deßating the January price of the stock index with the annual

average of the producer price index before 1990, and the January value of the index thereafter.

Real dividends are similarly calculated from the total dividend per share accruing to the index.

Real returns, rt+1, are calculated by subtracting the rate of inßation from the return on commercial

paper.

The upper panel of Table 1 reports the sample autocorrelations for δt and the two measures of

10

fundamentals; real dividend growth, ∆dt, and adjusted dividend growth, ∆dt− rt. In all cases, theautocorrelations die out quickly indicating that the processes are stationary - consistent with the

structure of the switching VAR. The lower panel examines the time series properties of the data

with a series of regressions. The right hand columns show the R2 of each regression, Q statistics for

serial correlation in the regression residuals, tests for structural stability. The latter are based on L

statistics [Hansen (1991)] that test for parameter stability against the alternative hypothesis that

the parameters follow a martingale process. The L1 statistic tests for stability in all the coefficients

and the L2 statistic tests for stability in the coefficients and the variance of the residuals.2

The top portion of the panel reports results for dividend growth regressions. Although there

is little evidence of parameter instability in the AR(1) model, the regressions R2 is only 0.023.

When lagged dividend-prices and dividend growth are included in the regression, the R2 statistics

are a good deal higher. This is consistent with the idea that investors have more information about

future dividend growth than is contained in the lagged values of ∆dt alone. Now both L statistics

indicate that we can reject the null hypothesis of structural stability at the 5% level.

The middle portion of the panel shows results for the adjusted dividend growth regressions. As

above, there is more evidence of instability in models that include dividend-prices as a regressor

than in the AR(1) model. With one lag of dividend-prices, both L statistics are signiÞcant at the

5% level. With two lags, the statistics are signiÞcant at the 10% level.

The lower portion of the panel examines simple regression models for dividend-prices. The

Þrst two rows consider regressions of δt on lagged fundamentals. While there is strong evidence of

parameter instability in these cases, both models do a very poor job of tracking the movements

in dividend-prices. As the last two rows of the table show, adding further regressors considerably

improves the predictive power of the regressions. The R2 statistics are now over 0.6 and there is

little evidence of residual serial correlation.

Further evidence on the instability of fundamentals is presented in Table 2. Here I report results

from estimating different versions of the fundamentals process in (13). The left hand columns show

estimates for speciÞcations where fundamentals are identiÞed by dividend growth. Although the

estimated transition probabilities do not differ greatly between the two speciÞcations, the estimates

of c11(z) and c12(z) indicate that xt−1 and δt−1 have signiÞcant predictive power for forecastingxt+1 particularly in regime z = 0. Moreover, as the Q-statistics reported at the bottom of the

table show, in this speciÞcation there is no signiÞcant evidence of residual serial correlation within

2The L1 statistic is calculated from the regression

yt = β1x1t + β2x2t + . . .βmxmt + et

with E(et) = 0, E(e2t ) = σ

2t and limT→∞

1T

PTt=1 σ

2t = σ

2, as 1T

PTt=1 S

0t(PT

t=1 ftf0t )−1St where ft ≡ [f1,t. . . . fm+1,t]0

and St ≡ [S1,t. . . . Sm+1,t]0 with Si,t =Ptj=1 fi,j , fi,t = xi,tet for i ≤ m and fm+1,t = e2t − σ2t . fm+1,t is omitted fromft and St when calculating the L2 statistic. Asymptotic critical values are tabulated in Hansen (1991).

11

either regime. In the case where xt = ∆dt− rt, there is signiÞcant serial correlation in the regime 0residuals when δt−1 and xt−1 are omitted from the speciÞcation. When these variables are included,the estimates of c12(z) are highly signiÞcant and most of the residual serial correlation disappears.

Overall, these results indicate that at least two lags of δt and xt should be included in switching

speciÞcations if they are to provide a good characterization of fundamentals over the sample. This

Þnding is consistent with the use of (13) as the basis for the switching VAR.

4.2 Model Estimates

Table 3 reports estimates of the switching VAR in (14) for both deÞnitions of fundamentals. The

reported estimates are based on models that impose the restrictions in (15) and ϕ = 0. The latter

restriction rules out serial correlation in expected returns and appears supported in the data. When

the restriction was dropped, the estimates of ϕ were imprecise and very close to zero. Moreover, I

could not reject the restriction of ϕ = 0 using likelihood ratio tests for either Model A or B at the

5% level. Despite these Þndings, I will consider the robustness of the results below to alternative

values for ϕ.

From the table we can see that the probabilities of remaining in either regime, λz, from one year

until the next are between 70% and 90%. These estimates imply that the unconditional probability

of being in regime z = 1 is 0.73 for Model A and 0.534 for Model B. The estimates of ai(z), bij(z)

and cij(z) show how the predictability of fundamentals varies across regimes. In Model A, the

parameters on lagged fundamentals and dividend-prices are statistically signiÞcant in the regime

1 process while only lagged dividend-prices appear signiÞcant in the regime 0 process. Similarly,

in Model B, the estimates indicate that lagged fundamentals are more signiÞcant predictors of

future adjusted dividend growth in regime 1 than regime 0. There are also differences in the

variability of fundamentals across regimes. In both models, the estimated standard deviation of the

innovations to fundamentals in regime 1, σ1(1), is more than twice the size of the estimated standard

deviation of the regime 0 innovations, σ1(0). By contrast, there is little evidence of regime-dependent

heteroskedasticity in the innovations to the dividend-prices. In both models, the estimates of σ2(z)

are similar across regimes.

Figure 1 shows implications of the model estimates. The upper two graphs plot the smoothed

probability of being in regime 1. These plots show that both estimated models imply numerous

switches in the process for fundamentals over the sample. The lower two graphs plot the auto-

correlation functions for fundamentals in each regime implied by Model A and B together with

the correlation function calculated from a standard 2nd order VAR for comparison. As the plots

show, the dynamics of fundamentals differ across regimes particularly over horizons of four years

or less. While dividend growth is positively correlated at all lags in regime one, the regime zero

correlations alternate in sign as do the standard VAR correlations. Thus the regime one dynamics

12

of dividend growth appear quite different from those identiÞed by the standard VAR. Similarly, the

standard VAR correlations for adjusted dividend growth differ from the within regime correlations.

These plots show positive correlations at all lags and indicate the adjusted dividend growth is more

persistent in regime zero.

These autocorrelation functions highlight two features of the switching VARs. In the early

switching models pioneered by Hamilton (1988), zt only entered linearly into the process for the

continuous variable with the result that the autocorrelation function implied by the model was

constant.3 The switching VAR generalizes this structure by allowing zt to enter the process lin-

early via the intercept coefficients and nonlinearly through the autoregressive coefficients. This

nonlinearity is the source of the regime-dependent autocorrelations shown in Figure 1. Given the

large cross-regime differences in the autoregressive parameters of the estimated xt processes re-

ported in Tables 2 and 3, this appears to be an important feature of the dynamics for fundamentals

in the sample.

The autocorrelations also make clear that the estimated models identify instability in the short-

term dynamics of fundamentals. As a consequence, the analysis based on these model estimates

will focus on how changes in these short-term dynamics for fundamentals affects dividend-prices

and stock returns. This contrasts with Barsky and DeLong (1993) and Timmermann (1994) who

focus on the implications of investor uncertainty about the long-term dynamics of dividends. Since

there is little evidence of instability in long-term dividend growth, or adjusted dividend growth in

the sample, the question of whether investors were heavily inßuenced by this form of uncertainty

is open.

4.3 Implications for Dividend-Prices

I begin the analysis of the switching VAR estimates by considering how large a fraction of the

variance in dividend-prices can be attributed to changing forecasts of future fundamentals. To

calculate this fraction, I substitute the identity ht+i − ∆dt+i ≡ ht+i − xt+i + xt+i − ∆dt+i intoequation (2), multiply the result by δt, and take expectations conditioned on the information set

Jt = {δt−i, xt−i}i≥0. This gives

V ar(δt) = Cov

µ−

∞Pi=1ρi−1E[xt+i|Jt], δt

¶+Cov

µ∞Pi=1ρi−1E[ηt+i|Jt], δt

¶(17)

3For example, Hamilton (1988) considers models of the form

xt+1 − θ(zt+1) = α(L)et+1,where α(L) is a polynomial in the lag operator. This process can be expressed as a linear combination of twoindependent AR processes, and consequently, has a constant autocorrelation structure.

13

where ηt ≡ ht + xt −∆dt.In Model A, ηt equals the return on stocks, ht. In this case, (17) decomposes variations in

dividend-prices into the covariance between δt and changing expectations of future dividend growth

and stock returns. In Model B, ηt is equal to the excess return on stocks, ht − rt. Here (17) allowsus to study how much of dividend-price variability can be attributed to variations in the risk

premium on stocks, as measured by E[ht+i − rt+i|Jt]. Appendix B shows how the switching VARestimates can be used to calculate both covariance terms. Notice this variance decomposition is

not derived from an economic model for dividend-prices. It is an implication of (2) for any time t

information set that contains δt. Here I have used an information set Jt that includes current and

past values of δt and xt. This choice allows me to calculate the forecasts in (17) from the switching

VARs. Since these forecasts are not constrained by any theoretical model, estimates of the variance

decomposition in (17) should be informative about the source of dividend-price variability quite

generally.

Panel I of Table 4 reports the estimate of Cov¡P∞

i=1 ρi−1E[xt+i|Jt], δt

¢as a fraction of V ar(δt).

The statistics in row 1 are based on the switching VAR estimates.4 As the table shows, the fraction

of dividend-price variability attributed to changing forecast of fundamentals are 62% and 58% for

Models A and B. The second row reports corresponding statistics of 37% and 32%calculated from

standard VARs. From the differences between the statistics in rows 1 and 2, changing forecasts of

fundamentals appear to account for a good deal more of the variation in dividend-prices once we

allow for the effects of instability in the fundamentals process.

The remaining rows of Panel I report the fractions calculated from switching VARs that impose

different values for ϕ. In rows 3 and 4 the fractions were calculated from model estimates where ϕ

equals 0.128 and 0.224. These values are implied by autoregressive parameters of 0.5 and 0.75 in

expected monthly returns. As the table shows, the introduction of serial correlation has a relatively

small impact. In all cases the statistics remain well above their counterparts based on the standard

VARs. The estimated fractions in row 5 are a good deal lower. These estimates are based on models

that impose ϕ = 0.48, the value implied by autocorrelation of 0.9 in expected monthly returns. As

the Table shows, the log likelihoods for these models are a good deal lower than the likelihoods

in row 1. Thus, while it is possible to restrict the switching VAR so that the fraction of returns

attributable to changing forecasts of fundamentals is lower than that implied by a standard VAR,

there is no statistical support for these restrictions. From this I conclude that the estimates in row

1 are robust to the presence of a reasonable degree of serial correlation in expected returns.

Panel II of Table 4 reports χ2 tests of the cross-equation restrictions in (16) implied by the

dividend-ratio model. The statistics in row 1 test for the presence of a regime-speciÞc risk premium

4As in Campbell and Shiller (1989), all the calculations set ρ equal to 0.937, the exponential of the differencebetween the sample average of ∆dt and ht in the data.

14

in expected stock returns. Here I combined the equations for π0(1) and π0(0) to eliminate µ − κand test the resulting restriction between π0(1), π0(0) and the other coefficients. As the table

shows, this restriction is not signiÞcant at the 5% level in either model. The statistics the next row

examine the remaining restrictions on π1(z),π2(z) and π3(z). These statistics are signiÞcant at the

5% level.

To assess the economic signiÞcance of these statistics, I compared the implications of the model

estimates in Table 3 against the predictions of a VAR where the cross-equation restrictions in

(16) are imposed. First I used the estimates of the π(.)0s and the cross-equation restrictions in(16) to Þnd values for the coefficients of the fundamentals switching process consistent with the

dividend-ratio model. With these values [denoted by * ], I then compared

E[x∗t+1|It] = a∗1(zt) + b∗11(zt)xt + b∗12(zt)δt + c∗11(zt)xt−1 + c∗12(zt)δt−1, (18)

against the unrestricted VAR forecasts:

E[xt+1|It] = a1(zt) + b11(zt)xt +b12(zt)δt + c11(zt)xt−1 + c12(zt)δt−1. (19)

This comparison allows us to see how different investors short-term forecasts of fundamentals would

have to be in order to make the observed behavior of dividend-prices consistent with the predictions

of the dividend-ratio model.

The upper panel of Figure 2 plots E[xt+1|It] and E[x∗t+1|It] implied by the estimates of ModelA where xt+1 ≡ ∆dt+1. As the Þgure shows, the restricted forecasts are much more variable thanthe unrestricted VAR forecasts; the sample variance of E[x∗t+1|It] is 25.03 compared to 1.17 forE[xt+1|It]. The lower panel plots the forecasts implied by the estimates of Model B where xt+1 ≡∆dt+1 − rt+1. Here the switching VAR forecasts appear similar to the forecasts for fundamentalsneeded to rationalize the observed swings in dividend-prices, the correlation between the forecasts

is 0.80.

How could the cross-equation restrictions be so strongly rejected for Model B while the forecasts

of future fundamentals look so similar? The test statistics in Table 3 examine whether the expected

present value of fundamentals based on restricted and unrestricted forecasts are equal rather than

just the short-term forecasts. The strong rejection of the cross-equation restrictions for Model B

must therefore be due to the differences between the restricted and unrestricted long-term forecasts

of fundamentals.

Long-term forecasts depend upon estimates of the transition probabilities,λz, that are closely

related to the number of times during the whole sample that fundamentals continued to follow the

regime z process between t and t + 1, measured as the fraction of the number of times zt = z.

Clearly these estimates are heavily inßuenced by the realized behavior of fundamentals and may

15

have differed from the probabilities rational investors used to form expectations at the time. For

example, investors views about the prospects for future fundamentals at the beginning of the Great

Depression might quite reasonably have been based on different probabilities than were consistent

with the incidence of regimes over the previous 50 years. Such differences are not allowed for in the

cross-equation tests reported above. It is therefore possible that the economic signiÞcance of these

tests could be reduce if the restricted and unrestricted long-term forecasts can be reconciled with

the choice of transition probabilities that are similar but not identical to those estimated from the

data.

To investigate this issue, I found the transition probabilities than minimized the sum of squared

differences between the present value of restricted and unrestricted forecasts of fundamentals based

on the coefficient estimates of Model B. This procedure yields transition probabilities of λ1 = 0.95

and λ0 = 0.73, and a correlation of over 0.99 between restricted and unrestricted present values.

While it hard to give an objective assessment of whether these probabilities are consistent with

views of rational investors during the sample period, they are relatively close to the estimated values

of λ1 = 0.78 and λ0 = 0.75. This Þnding reduces the economic signiÞcance of the test statistics in

Table 3.

Figure 3 compares the performance of Model B against the standard VAR. The Þgure plots

dividend-prices, δt, the unrestricted negative present value of future fundamentals calculated from

Model B using λ1 = 0.95 and λ0 = 0.73 to calculate the forecasts, δ∗t , and the negative present valueimplied by the standard VAR. As the Þgure shows, δt and δ

∗t move together except for a few years

between 1915 and 1945 where δt peaked indicating a market crash. Aside from these episodes, the

movements in dividend-prices appear quite closely related to changes in the expected present value

of future fundamentals consistent with the dividend-ratio model. By contrast,the estimates derived

from the standard VAR appear much less closely related to dividend-prices. These differences

provide quite striking evidence of how the presence of switching contributes to the variability of

fundamentals forecasts.

The analysis so far gives no indication of how changes in expected future dividend growth

and discount rates individually contribute to the movements in dividend-prices. To investigate this

issue, I used the distribution of regimes, zt, estimated from Model B to estimate a switching process

for the real rate:

rt+1 = a3(zt) + b31(zt)xt + b32(zt)δt + b33(zt)rt

+c31(zt)xt−1 + c32(zt)δt−1 + c33(zt)rt−1 + v3,t+1. (20)

Appendix C describes how the estimated forecasts of future real rates from (20) were combined

with (18) to obtain the present values of future dividend growth,P∞i=1 ρ

iE[∆d∗t+i|It], and real rates,

16

P∞i=1 ρ

iE[∆r∗t+i|It], consistent with the cross-equation restrictions of the dividend-ratio model.Panel III of Table 4 shows that movements in the expected present value of future dividend

growth contribute over 90% to the variability of dividend-prices in the sample. By contrast, the

expected present value of future real rates only contribute about 8%. These statistics suggest

that predictable discount rate variations are much less important than changing forecasts of future

dividend growth in explaining the behavior of dividend-prices. From the sample correlations we

also see that innovations in dividend-prices are almost always associated with news about expected

future dividend growth.

Overall, the results above indicate that variations in dividend-prices can be fairly well charac-

terized by the dividend-ratio model if fundamentals are identiÞed by adjusted dividend-growth and

allowed to switch between processes. Although we can formally reject the cross-equation restrictions

implied by the dividend-ratio model, it appears that rational investors forecasts of fundamentals

would not have to differ a great deal from the switching VAR forecasts over most of the sample

in order for these restrictions to hold. Furthermore, when investors forecasts are restricted to be

consistent with the dividend-ratio model, we Þnd that changing forecasts of future dividend growth

are by far the most important determinant of dividend-price movements.

5 Alternative Models

The switching VARs in Table 3 are based on the assumption that the instability in fundamentals

can be adequately represented by switches between two regimes. The models also assumed that

investors knew the current regime when forecasting future fundamentals. In this section I shall

consider models based on different assumptions.5

5.1 Three Regimes?

Making inferences about the appropriate number of regimes to include in a switching model raises

some thorny econometric issues.6 To circumvent these issues, I will focus on the implications of

alternative switching speciÞcations for the forecasts of fundamentals. In particular, I will examine

the conditions under which the behavior of the present value of fundamentals forecasts derived

from the switching VAR is robust to the number of regimes.

The approach is most easily understood if we focus on a speciÞc example. Suppose that fun-

damentals switch between three regimes. The VAR forecasts of future fundamentals can then be

5The reader may skip this section without loss of continuity.6As Engel and Hamilton (1990), Hansen (1992) and others have noted, standard tests cannot be used to compare

switching models with different numbers of regimes because the usual asymptotic distribution theory needed to drawinferences no longer applies. For a discussion see Evans (1997).

17

written as

∞Xi=1

ρiE [xt+i|Jt] =∞Xi=1

ρihE[xt+i(1)|Jt] Pr(zt+i−1 = 1 |Jt) +E[xt+i(2)|Jt] Pr(zt+i−1 6= 1 |Jt)

i+

∞Xi=1

ρiE[xt+i(2)− xt+i(3)|Jt] Pr(zt+i−1 = 3 |Jt). (21)

The probabilities in (21) are determined by the elements of the regime transition matrix which can

be written as

Pr(zt+1 = j|zt = i) =

ψ1 (1− ψ1)θ1 (1− ψ1)(1− θ1)1− ψ2 ψ2θ2 ψ2(1− θ2)1− ψ3 ψ3(1− θ3) ψ3θ3

, (22)where 1 > ψi, θi > 0.

We can now Þnd the conditions under which the present value in (21) is invariant to whether

xt switches between two or three regimes. In particular, suppose that

ψ3 = ψ2, θ1 = θ2, and θ2 = 1− θ3. (23)

With these restrictions, the three regime Markov chain can be represented by a two regime chain

with regimes st = {1, 0} where st = 1 when zt = 1, st = 2 when zt 6= 1 and Pr(st+1 = i|st = i) = ψi[see Hamilton (1994)]. If, in addition, E[xt+1(2)|Jt] = E[xt+1(3)|Jt], by iterated expectations,E[xt+i(2)− xt+i(3)|Jt] = 0 for i > 0. So, when both conditions hold, we can rewrite (21) as

∞Xi=1

ρiE [xt+i|Jt] =∞Xi=1

ρi2Ps=1

E[xt+i(s)|Jt] Pr(st+k−1 = s|Jt), (24)

which is the present value for a two regime model with regimes determined by st. Thus, the

behavior of the present value will be robust to the presence of two or three regimes when (i)

E[xt+1(2)− xt+1(3)|Jt] = 0 and (ii) the restrictions in (23) hold.Table 5 examines these conditions based on estimates of the fundamentals models in Table 2

allowing for three regimes. The table reports estimates of the transition matrices and test statistics

for the restrictions. Row 1 reports χ2 tests for the null hypothesis that ψ3 = ψ2, θ1 = θ2, and

θ2 = 1− θ3. This hypothesis cannot be rejected at the 5% level for either model. The χ2 statisticsin row 2 test the null hypothesis that the coefficients have the same value in regimes 2 and 3 so

that E[xt+1(2) − xt+1(3)|Jt] = 0. Here there is some evidence against the null in the case of thedividend growth model; the marginal signiÞcance level is 4.8%. Row 3 reports statistics for both

sets of restrictions. Neither test statistic is signiÞcant at the 5% level.

18

We can also use the estimates from Table 2 to examine robustness to the presence of 2 rather

than 1 regime. In particular, we can test whether the coefficients are equal across all regimes. A

rejection of this restriction implies that E[xt+1(i)|Jt] 6= E[xt+1(j)|Jt] for j 6= i so that the behaviorof the present value of fundamentals would not be invariant to the presence of one or two regimes.

As the χ2 statistics reported in rows 4 and 5 show, these restrictions are strongly rejected for both

fundamentals models.

Based on these results, there is little evidence to indicate that the expected present value of

fundamentals estimated from the switching VAR varies signiÞcantly according to whether funda-

mentals are modelled as switching between 2 or 3 regimes. At the same time, we can strongly

reject one of the necessary conditions for robustness to the presence of 2 rather than 1 regime.

This Þnding is consistent with the large differences between the variance ratios calculated from the

switching and standard VARs in Table 4.

5.2 Models with Learning

So far the analysis has proceeded under the assumption that investors knew the current regime

and faced uncertainty concerning future regimes. To examine the impact of this assumption, I

estimated modiÞed versions of the switching VAR that allow investors to be uncertain about the

current and future regimes. These models continue to assume that fundamentals switch between

two regimes as in (13). The key difference is that market participants do not know the current or

past regimes when forecasting fundamentals.

The estimated models have the same general form as (14) except that the π(.) coefficients are

functions of the estimated value of the regime, zet , rather than the actual regime. For the purpose

of estimation I assume that

πi(zet ) = πi(0) + [πi(1)− πi(0)] zet , (25)

where πi(1) and πi(0) are coefficients. I also assume that zet can be approximated by E[zt|Jt−1, xt].

According to this speciÞcation, investors estimate the regime at t is based on past data (contained

in Jt−1) and current fundamentals, xt, but not the current value of dividend-prices. This assumptiongreatly simpliÞes estimation because it eliminates the simultaneous dependency between δt and z

et

that would be present if zet were identiÞed by E[zet |Jt]. Appendix D describes how the estimates of

E[zt|Jt−1, xt] are calculated as part of Hamiltons (1988) algorithm and how the model estimateswere used to check the accuracy of the approximation for zet .

Table 6 reports the model estimates for both deÞnitions of fundamentals. The estimates appear

generally similar to those reported in Table 3. This impression is supported by the middle panel

that reports correlations between the two sets of estimates. Here we see that the forecasts of

19

fundamentals implied by the models are highly correlated, as too are the predicted movements in

dividend-prices. The greatest difference between the models shows up in the estimated regimes. In

the dividend-growth models, the correlations between the two sets of regime estimates is 0.68.

When investors are uncertain about the current as well as future regimes, the dividend-ratio

model takes the same form as (12) except that expectations are now conditioned on investors

information, Ωt, that excludes {zt−i}i≥0. This version of the dividend-ratio model imposes thefollowing restrictions on the π(.) functions:

π0(zet ) = µ(z

et ) [µ− k + ρπ0(Γ(zet ))− φ(zet )a1(zet )] π2(zet ) = −µ(zet )φ(zet )c12(zet )

π1(zet ) = µ(z

et ) [ρπ3(Γ(z

et ))− φ(Γ(zet ))b11(zet )] π3(zet ) = µ(zet )φ(zet )c11(zet )

µ(zet )−1 = (1− ρπ2(Γ(zet )− φ(zet )b12(zet )) φ(zet ) = [1− ρπ1(Γ(zet ))] (26)

where Γ(zet ) ≡ E[zet+1|Jt] = (1 − λ0) + (λ1 + λ0 − 1)zet . Since these conditions must hold for allvalues of zet , in general πi(.) must be a nonlinear function of z

et under the null hypothesis that

the dividend-ratio model holds true. Under this null, (25) must therefore be viewed as linear

approximations to the true πi(.) functions. Consequently, tests based on the estimates of πi(1) and

πi(0) can only provide an approximate test of the dividend-ratio model.7 In the cases where zet = 1

or 0, (25) and (26) imply that πi(1) and πi(0) satisfy the restrictions in (15). These equations are

the basis for the cross-equation tests reported in Panel III. Here we see that neither test statistic

is signiÞcant at the 5% level in the case of Model B.

Clearly, the introduction of learning adds a good deal of complexity to the switching VARs that

necessitates the use of approximations that were hitherto unnecessary. This complicates formal

comparisons of the test results in Tables 4 and 6. Nevertheless, the results from the learning

models do appear quite similar to those presented in Section 4. Subject to the caveats above, there

is little here to indicate that the switching VAR results are unduly sensitive to the assumption that

investors knew the current regime.

6 Stock Returns

I shall now examine how the presence of switching in the process for fundamentals affects the

behavior of stock returns. Recall that the variance of dividend-prices can be decomposed as

V ar(δt) = Cov

µ−

∞Pi=1ρi−1E[xt+i|Jt], δt

¶+Cov

µ ∞Pi=1ρi−1E[ηt+i|Jt], δt

¶. (16)

7An exact test would require the estimation of non-linear π() functions that could satisfy (26) for all values of zet ,a task beyond the scope of this paper. Note too that (26) does not include restrictions on π4(.) because, consistentwith the Þndings above, the learning models were estimated with ϕ = 0 [see Appendix C]

20

Panel I of Table 7 reports estimates of the second term in (16) as a fraction of estimated variance of

dividend-prices, V ar(δt), together with standard errors. The estimates in columns (1) and (2) are

derived from Model A where ηt equals the return on stocks. According to these estimates, changes

in expected returns account for less than 40% of the variability in dividend-prices if we allow for

switches and over 60% if we do not. The contribution of returns also appears much lower when we

allow for switching in the case of model Model B where ηt equals excess stock returns. Here the

ratios fall from 68% to 41%. Although these differences are quite large, it should be noted that

fractions calculated from the switching models are less precisely estimated than those based on the

standard VARs.

The estimates in columns (1) - (4) are based on unrestricted VAR estimates. Column (5) reports

the estimate derived from the restricted version of Model B with coefficients that are consistent

with the dividend-ratio model. As the table shows, only 14% of the variability in dividend-prices

is attributed to changes in expected excess returns. Recall from Figure 3 that dividend-prices and

the restricted present value of future fundamentals move closely together except in the few years

where there was severe market crash. The contribution of expected excess returns to the variability

of dividend-prices comes mainly from these periods.

We can also use the model estimates to study the factors affecting the behavior of unexpected

returns. Combing equations (1) and (2) with the deÞnition of returns, we can write

ηt+1 −E [ηt+1|Jt] =∞Xi=0

ρi (E[xt+i+1|Jt+1]−E[xt+i+1|Jt]) +∞Xi=1

ρi (E[ηt+i+1|Jt+1]−E[ηt+i+1|Jt]) .

Multiplying both sides by unexpected returns and taking expectations, gives

V ar(eηt+1) = Covµ ∞Pi=0ρi ^E[xt+i+1|Jt+1], eηt+1¶+Covµ ∞P

i=1ρi ^E[ηt+i+1|Jt+1], eηt+1¶ ,

(27)

were wt+1 denotes the innovation in a variable wt+1 between t and t + 1. Like (16), this variance

decomposition is not based on any model of dividend-prices. The Þrst term identiÞes the fraction

of the variance in unexpected stock returns that can be attributed to news about fundamentals.

As the equation shows, all other unexpected movements in returns must be attributable to news

about future expected returns.

Panel I of Table 7 shows estimates of the Þrst term in (27) as a fraction of V ar(eηt+1). In the caseof Model A, the estimates appear relatively insensitive to the presence of switching. News about

future dividend growth only accounts for approximately 15% of the variance in unexpected returns.

Columns (3) and (4) report the estimates based on the unrestricted versions on Model B. Again

the estimates appear quite robust to the presence of switching and are similar to those derived

21

from Model A. The table also shows how the variance decomposition changes when the restrictions

of the dividend-ratio model are imposed. Based on the restricted estimates of Model B, column

(5) shows that 80% of the return variance can be attributed to news about future fundamentals.

Clearly, this variance decomposition is sensitive to the presence of the dividend-ratio restrictions

on the forecasts of fundamentals. Recall from Section 4 that the economic evidence against these

restrictions is weaker than the statistical evidence. If we are willing to place more weight on the

economic evidence, the results in Table 7 suggest that news about fundamentals accounts for most

of the variance in returns. If not, then the results suggest the fundamentals news makes a much

smaller contribution, a Þnding consistent with Campbell (1991) and Campbell and Ammer (1993).

In section 3 we saw how switching in the process for fundamentals could affect the predictability

of returns in small samples. To examine the empirical signiÞcance of these small sample effects, I

regressed the k-period return realized at t+ k, ηkt+k ≡Pki=1 ηt+i, on dividend-prices:

ηkt+k = α+ α(k)δt + ut+k. (28)

Inferences about the signiÞcance of the estimates of α(k) in this regression are complicated by

the presence of serial correlation in the error term ut+k; under the null of no predictability, ut+k

will follow an MA(k−1) process. Appropriate asymptotic standard errors that allow for bothserial correlation and conditional heteroskedasticity are derived from Hansens (1982) Generalized

Method of Moments estimator.8

Panel II of Table 7 reports the estimated slope coefficients and standard errors from (28) in

columns (1) and (3) for ηt = ht and ηt = ht − rt respectively. All the estimates of α(k) aresigniÞcant at the 5% level. The conventional interpretation of these results is that the (excess)

returns expected by investors co-varied with dividend-prices.

Is there another interpretation of these Þndings based on regime switching? To investigate this

possibility, I also regressed estimates of unexpected returns on dividend-prices:

ηkt+k −E[ηkt+k|Jt] = β + β(k)δt +wt+k. (29)

Since δt ∈ Jt, under conventional rational expectations, the estimate of β(k) should be insigniÞ-cantly different from zero. However, as we saw in Section 3, this property of rational forecast errors

need no longer hold in small samples in the presence of switching. It is therefore possible that

predictability of returns implied by the estimates of α(k) mainly reßects this small sample effect.

Columns (2) and (4) report the estimates of β(k) using the estimates of Models A and B

to calculate expected returns, E[ηkt+k|Jt]. These estimates incorporate the effects of fundamentals

8I also considered an alternative regression suggested by Hodrick (1992) that does not contained overlapping errorsunder the null. Results from these regressions are similar to those reported.

22

switching but not the restrictions of the dividend-ratio model. As the table shows, the estimates

of β(k) indicate that unexpected returns are negatively correlated with dividend-prices during the

sample at all horizons. Moreover, in 6 of the 8 cases, the coefficients are signiÞcantly different from

zero at the 5% level. Since these Þndings are not based on any economic model for dividend-prices,

at the very least they should make us cautious about interpreting return predictability.

Column (5) reports a Þnal set of regression results that use the restricted version of Model B

to calculate expected returns. In contrast to the estimates in (2) and (4), here all the estimates of

β(k) are positive and signiÞcant at the 5% level for horizons over 1 year. This is the same pattern of

predictability in realized returns implied by the estimates of α(k) in (3). To investigate whether all

of the observed predictability in returns could be accounted for by switching in the restricted version

of Model B, I conducted a small Monte Carlo Experiment. Based on the restricted estimates, I

generated 1000 samples of dividend-prices and returns over 117 years. The regression in (28) was

then estimated in each of the 1000 experiments and the results recorded. The terms in brackets

shown in column (3) are the p-values for the t-statistics on α(m) estimated in the data calculated

from the Monte Carlo distributions. These p-values denote the probability of observing a t-statistic

as large as those found in the S&P data. As the p-values indicate, this version of the dividend-ratio

model is capable of producing the observed predictability in realized returns over 2 years or more

with probabilities ranging from 6% to 28%. These results support the idea that regime switching

can signiÞcantly affect the predictability of returns in typical samples.

7 Conclusion

I have examined how instability in the time series process for fundamentals can affect the behavior of

dividend-prices within the framework of Campbell and Shillers dividend-ratio model. Estimates of a

switching VAR showed that there has been a good deal of instability in the process for fundamentals

during the past 120 years. Based on these model estimates, changing forecasts of fundamentals

account for far more of the variation in dividend-prices than standard VARs that ignore instability

when forecasting fundamentals.

The switching VAR estimates also indicate that variations in dividend-prices can be fairly well

characterized by the dividend-ratio model if fundamentals are identiÞed by adjusted dividend-

growth. Although we can formally reject the cross-equation restrictions implied by the dividend-

ratio model, it appears that rational investors forecasts of fundamentals would not have to differ a

great deal from the switching VAR forecasts over most of the sample in order for these restrictions to

hold. Furthermore, when investors forecasts are restricted to be consistent with the dividend-ratio

model, we Þnd that changing forecasts of future dividend growth are by far the most important

determinant of dividend-price movements. These Þndings rehabilitate the idea that stock prices

23

primarily respond to dividend news. However, they are also consistent with observed predictability

of stock returns over long horizons. The switching VAR estimates imply that the predictability in

ex post returns can be attributed to the small sample effects of fundamentals switching.

These Þndings are subject to the caveat that the switching VAR does not completely capture

all the variations in the dividend-price ratio, particularly around major market crashes. Also, the

model takes no account of how switches in fundamentals may contribute to risk premia. Further

research into both these issues is clearly warranted.

24

References

Abel, A. B. (1993), Exact Solutions for Expected Rates of Returns under Markov Regime

Switching: Implications for the Equity Premium Puzzle, Journal of Money, Credit, and

Banking 26, 345-361.

Barsky, R. B. and J. B. De Long (1993), Why does the Stock Market Fluctuate? Quarterly

Journal of Economics CVIII, pp291-311.

Bekaert, G. and R. J. Hodrick (1992), Characterizing the Predictable Components in Equity and

Foreign Exchange Rates of Return, Journal of Finance 47, pp.467-509.

Campbell, J. Y. (1991), A Variance Decomposition of Stock Returns, Economic Journal 101,

pp.152-179.

Campbell, J. Y. and J. Ammer (1993), What Moves Stock and Bond Markets? A Variance

Decomposition for Long-Term Asset Returns, Journal of Finance 48 pp.3-37.

Campbell, J. Y. and H. J. Cochrane (1994), By Force of Habit: A Consumption-Based Ex-

planation of Aggregate Stock Market Behavior working paper, Federal Reserve Bank of

Philadelphia.

Campbell, J. Y. and R. J. Shiller (1989), The Dividend-Price Ratio and Expectations of Future

Dividends and Discount Factors, Review of Financial Studies 1, pp.195-228.

Cecchetti, S. J., P. Lam and N. C. Mark (1990), Mean Reversion in Equilibrium Asset Prices,

American Economic Review 80, pp. 398-418.

Cecchetti, S. J., P. Lam and N. C. Mark (1993), The Equity Premium and the Risk-Free Rate:

Matching the Moments, Journal of Monetary Economics 31, pp. 21-46.

Cochrane, J. H. (1991), Volatility Tests and Efficient Markets: A Review Essay, Journal of

Monetary Economics 27, pp. 463-485.

Cowles, A. (1939), Common Stock Indexes (2nd. Ed.) Principia Press, Blooomington, Ind.

Engel, C. and J. D. Hamilton (1990), Long Swings in the Dollar: Are They in the Data and Do

the Markets Know It? American Economic Review 80, 689-713.

Evans, M. D. D. Peso Problems: Their Theoretical and Empirical Implications, Handbook of

Statistics: Statistical Methods in Finance. G. S. Maddala and C. R. Rao, eds, North Holland.

25

Evans, M. D. D. and K. K. Lewis, (1995), Do Long-Term Swings in the Dollar Affect Estimates

of the Risk Premia? Review of Financial Studies.

Hamilton, J. D. (1988), Rational Expectations Analysis of Changes in Regime: An Investigation

of the Term Structure of Interest Rates, Journal of Economics, Dynamics and Control 12,

pp. 385-423.

Hamilton, J. D. (1994), Time Series Analysis, Princeton, N.J.

Hansen, B. E. (1991), Testing for Parameter Instability in Linear Models, working paper, Uni-

versity of Rochester.

Hansen, B. E. (1992), The Likelihood Ratio Test under Nonstandard Conditions: Testing the

Markov Switching Model of GNP, Journal of Applied Econometrics 7, S61-S82.

Hansen, L. P. (1982) Large Sample Properties of Generalized Method of Moments Estimators,

Econometrica 50, 1029-54.

Hodrick, R. J. (1992), Dividend Yields and Expected Stock Returns: Alternative Procedures for

Inference and Measurement, Review of Financial Studies 5, 357-386.

Kandel, S. and R. Stambaugh, (1990), Expectations and Volatility of Consumption and Asset

Returns, The Review of Financial Studies 3, pp. 207-232.

Lehman, B. N. (1991), Earnings, Dividend Policy, and Present Value Relations: Building Blocks

of Dividend Policy Invariant Cash Flows, NBER working paper No 3676.

LeRoy, S. and R. Porter (1981), The Present-Value Relation: Tests Based on Implied Variance

Bounds, Econometrica 49, pp. 555-574.

Shiller, R. J. (1981), Do Stock Prices Move too Much to be JustiÞed by Subsequent Changes in

Dividends? American Economic Review 71, pp. 421-436.

Timmermann, A., (!994), Can Agents Learn to Form Rational Expectations? Some Results on

Convergence and Stability of Learning in the UK Stock Market, Economic Journal 104, pp.

777-797.

26

Table 1: Sample Statistics and Stability Tests

Sample StatisticsVariable Mean Autocorrelations

δt −304.549 0.713 0.510 0.457 0.362 0.315 0.307∆dt 1.343 0.152 -0.149 -0.110 -0.121 0.013 -0.040∆dt − rt -1.725 0.267 -0.084 -0.083 -0.197 -0.155 0.021

Regression EstimatesEquation Regressors R2 Q3 Q6 L1 L2

∆dt ∆dt−1 0.023 3.617 5.733 0.090 0.457∆dt ∆dt−1, δt 0.227 3.264 7.401 1.599∗∗ 1.786∗∗∆dt ∆dt−1, δt,∆dt−2, δt−1 0.416 1.394 1.827 1.352∗∗ 1.410∗∗

∆dt − rt ∆dt−1 − rt−1, 0.071 2.727 7.362 0.207 0.548∆dt − rt ∆dt−1 − rt−1, δt 0.240 5.035 7.518 1.452∗∗ 1.839∗∗∆dt − rt ∆dt−1 − rt−1, δt,∆dt−2 − rt−2, δt−1 0.434 0.352 1.596 1.098∗ 1.204∗

δt ∆dt−1 0.001 104.552∗∗ 145.879∗∗ 1.532∗∗ 2.220∗∗

δt ∆dt−1 − rt 0.001 111.311∗∗ 149.405∗∗ 1.522∗∗ 2.223∗∗δt ∆dt−1,∆dt−2, δt−1 0.604 3.639 7.883 0.592 0.791δt ∆dt−1 − rt−1,∆dt−2 − rt−2, δt−1 0.616 1.830 3.854 0.526 0.764

Notes: δt is the log dividend-price ratio multiplied by 100, ∆dt is the Þrst difference of log dividends, and

rt is the real return on commercial paper. Qi denotes Q-statistics for serial correlation in the residuals of

each equation up to order i. The L statistics test the null hypothesis of constant parameters against the

alternative that the parameters follow a Martingale process. L1 tests for stability in all the coefficients; L2tests stability of all the coefficients and the variance of the residuals. ∗∗ and ∗ indicate signiÞcance at the5% and 10% levels.

Table 2: Alternative Switching Models

Parameter Estimates (Standard Errors)

Dividend Growth: xt ≡ ∆dt Adjusted Dividend Growth : xt ≡ ∆dt − rt

Model 1 Model 2 Model 1 Model 2

a1(1) -11.919 (3.392) -7.149 (0.409) -1.179 (3.323) -5.551 (1.691)a1(0) -149.567 (13.031) -54.969 (11.307) -138.169 (16.094) -62.931 (14.674)

b11(1) 0.345 (0.040) 0.433 (0.043) 0.593 (0.051) 0.672 (0.018)b11(0) -0.519 (0.087) 0.474 (0.081) 0.157 (0.080) 0.555 (0.068)

c11(1) -0.013 (0.024) 0.006 (0.012)c11(0) -0.135 (0.098) -0.009 (0.058)

b12(1) -0.039 (0.011) -0.110 (0.018) -0.003 (0.010) -0.189 (0.010)b12(0) -0.519 (0.045) -0.768 (0.052) -0.461 (0.055) -0.656 (0.044)

c12(1) 0.085 (0.019) 0.176 (0.009)c12(0) 0.571 (0.062) 0.440 (0.057)

λ1 0.807 (0.055) 0.783 (0.062) 0.697 (0.066) 0.638 (0.071)λ0 0.662 (0.091) 0.642 (0.097) 0.657 (0.834) 0.631 (0.073)

σ(1) 2.851 (0.173) 2.677 (0.168) 1.982 (0.169) 1.268 (0.092)σ(0) 9.113 (0.498) 7.149 (0.409) 9.807 (0.702) 8.864 (0.485)

Residual Diagnostics

Regime Q6 M.S.L. Q6 M.S.L. Q6 M.S.L. Q6 M.S.L.

1 7.757 0.256 8.986 0.174 12.647 0.049 11.579 0.0720 148.143

Table 3: Switching VAR Models

Parameter Estimates (Std. Errors)I

Model A: xt ≡ ∆dt Model B: xt ≡ ∆dt − rrRegime z = 1 Regime z = 0 Regime z = 1 Regime z = 0

λz 0.893 (0.054) 0.716 (0.129) 0.779 (0.085) 0.746 (0.095)

a1(z) -0.424 (1.149) -15.877 (5.137) 0.336 (0.882) -7.990 (4.111)

b11(z) 0.280 (0.077) -0.009 (0.236) 0.252 (0.064) 0.400 (0.162)b12(z) -0.169 (0.043) -0.679 (0.186) -0.104 (0.051) -0.584 (0.154)

c11(z) 0.162 (0.067) -0.059 (0.305) 0.045 (0.061) 0.286 (0.217)c12(z) 0.197 (0.055) 0.219 (0.350) 0.150 (0.050) 0.443 (0.235)

σ1(z) 1.362 (0.151) 0.603 (0.053) 1.152 (0.152) 0.372 (0.054)σ12(z) -0.313 (0.104) -2.959 (1.589) -0.137 (0.073) -2.063 (0.520)σ2(z) 2.088 (0.455) 1.846 (0.392) 1.880 (0.220) 1.726 (0.174)

π0(z) -7.248 (2.625) -13.028 (9.195) -7.370 (2.564) -8.371 (5.380)π1(z) 0.593 (0.238) 0.363 (0.421) 0.403 (0.210) 0.626 (0.289)π2(z) 0.781 (0.084) 0.529 (0.315) 0.780 (0.081) 0.696 (0.182)π3(z) 0.059 (0.225) 0.353 (0.591) 0.004 (0.167) 0.251 (0.413)

Notes: The table reports estimates of the switching VAR for fundamentals and

dividend-prices shown in (13) and (14). The estimates are derived under the re-

striction that ϕ = 0,implying no serial correlation in expected returns.

Table 4: Model Implications

I Variance RatiosModel A Model B

ϕ Cov(∞Pi=1ρiE[∆dt+i|Jt], δt) Log L Cov(

∞Pi=1ρiE[∆dt+i − rr+i|Jt], δt) Log L

1 0 0.63 -145.46 0.59 -142.042 - 0.36 - - 0.32 -3 0.128 0.61 -198.00 0.47 -209.874 0.224 0.57 -199.30 0.59 -245.095 0.480 0.22 -220.16 0.22 -227.34

II Cross Equation Restrictions

Statistic M.S.L. Statistic M.S.L

1 1.732 0.188 0.957 0.3282 15.533 0.016 21.496 0.001

III Variance Decomposition

1 V ar(P∞i=1 ρ

iE[∆d∗t+i|Jt])/V ar(δ∗t ) 0.9012 V ar(

P∞i=1 ρ

iE[∆r∗t+i|Jt])/V ar(δ∗t ) 0.0823 Corr(

P∞i=1 ρ

iE[∆d∗t+i|Jt], δ∗t ) 0.9584 Corr(

P∞i=1

P∞i=1 ρ

iE[∆r∗t+i|Jt], δ∗t ) 0.303

Notes: Panel I reports the covariance as a fraction of V ar(δt) based on switching VAR estimates in rows1, 3, 4 and 5 that impose the values of ϕ shown. The statistics in row 2 are based on a standard 2nd-order

VAR with no switching. Panel II reports tests of the cross-equation restrictions implied by the dividend-

ratio model using the estimates from Table 3. The χ2 statistic in row 1 tests the restrictions on π0. The

statistic in row 2 tests for the restrictions on π1,π2 and π3. The ratios in Panel III are calculated from

Model B augmented by an equation for the real rate process. The present values are calculated from the

model estimates with the cross-equations restrictions of the dividend-ratio model imposed [see Appendix C

for details].

Table 6: Alternate Switching VAR Models

I Parameter Estimates (Std. Errors)

Model A: xt ≡ ∆dt Model B: xt ≡ ∆dt − rrRegime z = 1 Regime z = 0 Regime z = 1 Regime z = 0

λz 0.812 (0.074) 0.807 (0.082) 0.832 (0.079) 0.556 (0.223)a1(z) -0.373 (0.770) -9.057 (2.942) -1.997 (0.982) -12.443 (7.317)b11(z) 0.206 (0.099) 0.076 (0.143) 0.258 (0.091) 0.534 (0.230)b12(z) -0.133 (0.039) -0.496 (0.088) -0.218 (0.040) -0.767 (0.187)c11(z) 0.132 (0.097) 0.003 (0.241) 0.058 (0.080) 0.277 (0.390)c12(z) 0.135 (0.044) 0.292 (0.167) 0.216 (0.051) 0.396 (0.260)σ1(z) 1.247 (0.165) 0.387 (0.057) 1.380 (0.153) 0.564 (0.053)σ12(z) -0.207 (0.084) -2.135 (0.639) -0.415 (0.109) -4.577 (2.082)σ2(z) 2.075 (0.269) 1.584 (0.181) 2.443 (0.417) 2.314 (0.523)π0(z) -4.137 (3.439) -10.981 (4.267) -8.140 (2.693) -6.162 (6.635)π1(z) 0.278 (0.582) 0.421 (0.261) 0.721 (0.334) 0.475 (0.275)π2(z) 0.887 (0.111) 0.625 (0.141) 0.745 (0.085) 0.781 (0.229)π3(z) 0.844 (0.475) -0.001 (0.325) 0.071 (0.227) 0.123 (0.298)

II Cross-Model Correlations

Variable Correlation Variable Correlationzt 0.682 zt 0.799

E[xt+1|Jt] 0.833 E[xt+1|Jt] 0.737E[δt+1|Jt] 0.951 E[δt+1|Jt] 0.959

III HypothesisTests

Statistic M.S.L. Statistic . M.S.L1 0.945 0.331 0.381 0.5372 12.248 0.057 7.428 0.283

Notes: Panel I reports estimates of the 2nd-order switching VAR for fundamentals and dividend-prices that

allows for uncertainty about the current regime [see Appendix D]. Panel II reports correlations between

the estimates implied by these models and those in Table 3. Panel III reports tests of the (approximate)

cross-equation restrictions implied by the dividend-ratio model using the estimates from Panel I. The χ2

statistic in row 1 test the restrictions on π0. The statistic in row 2 test for the restrictions on π1,π2 and π3.

Table 7: The Predictability of Returns

I Model A Model BSwitching no yes no yes yesRestrictions no no no no yes

(1) (2) (3) (4) (5)

Cov(∞Pi=1ρiE[ηt+i|Jt], δt) 0.634 0.374 0.681 0.415 0.145

(0.050) (0.116) (0.059) (0.154) (0.133)

Cov(ρi∞Pi=1ρi gE[xt+i|Jt+1], eηt+1) 0.144 0.154 0.152 0.181 0.800

(0.147) (0.056) (0.169) (0.086) (0.020)

II

Return α(k) β(k) α(k) β(k) β(k)Regressions

k = 1 0.263 -0.316 0.247 -0.143 0.045(0.085) (0.072) (0.080) (0.066) (0.061)

[

Table 5: Tests for the Number of Regimes

Parameter Estimates (Standard Errors)

xt ≡ ∆dt xt ≡ ∆dt − rtψ1 0.270 (0.045) 0.428 (0.110)ψ2 0.302 (0.103) 0.392 (0.124)ψ3 0.486 (0.101) 0.475 (0.129)θ1 0.678 (0.092) 0.423 (0.096)θ2 0.631 (0.116) 0.616 (0.144)θ3 0.524 (0.088) 0.355 (0.135)

Tests

Statistic M.S.L. Statistic M.S.L.

1 1.400 (0.706) 2.560 (0.465)2 11.187 (0.048) 2.960 (0.706)3 12.988 (0.112) 5.491 (0.704)

4 1116.792 (

Appendices

A Cross-Equation Restrictions

To derive the cross-equation restrictions implied by the dividend-ratio model in (11), I proceed in

three steps. First, I iterate (11) one period forward to obtain

δt = µ− κ+ ρE[δt+1|Ωt]−E[xt+1|Ωt] + (1− ρϕ)ξt. (A1)

Next, I Þnd the rational expectations solution to (A1) consistent with the switching process for

fundamentals in the VAR. For this purpose, I posit and verify that the solution satisÞes

δt = π0(zt) + π1(zt)xt + π2(zt)δt−1 + π3(zt)xt−1 + π4(zt)ξt. (A2)

Using (A2) and (12) to substitute for δt+1 and xt+1 in (A1) gives

δt = µ(zt)£µ− k + ρπ0(zet+1)− [1− ρπ1(zet+1)]a1(zt)

¤(A3)

+µ(zt)£ρπ3(z

et+1)− [1− ρπ1(zet+1)]b11(zt)

¤xt

−µ(zt)[1− ρπ1(zet+1)]c12(zt)δt−1+µ(zt)[1− ρπ1(zet+1)]c11(zt)xt−1 + µ(zt) [1− ρϕ+ ρϕπ4(zet )] ξt.

(A3) has the same form as the posited solution in (A2). We can therefore equate the coefficients

in these equations to Þnd the set of restrictions that implicitly identify the π(.)0s in terms of theparameters of the fundamentals process. This gives the restrictions shown in (15).

Next, I lead (A2) one period, subtract ϕπ4(zt+1)δt/π4(zt) from the result, and substitute for δt

and xt+1 using (A2) and (12). After some rearrangement, this gives

δt+1 =

·π0(i) + π1(i)a1(j)− ϕπ4(i)

π4(j)π0(j)

¸+

·π1(i)b11(j) + π3(i)− ϕπ4(i)

π4(j)π1(i)

¸xt

+hπ1(i)b12(j) + π2(i)

iδt +

·π1(i)c11(j)− ϕπ4(i)

π4(j)π3(i)

¸xt−1 (A4)

+

·π1(i)c12(j)− ϕπ4(i)

π4(j)π2(j)

¸δt−1 + π4(i)ξt+1 + π1(i)vt+1.

(A4) is the same form as the equation for dividend-prices in the VAR. Equating coefficients on the

right hand side of (A4) with the terms in the VAR gives the restrictions in (14).

Finally, note that solution for dividend-prices in (A2) will satisfy limi→∞ ρiE[δt+i|Ωt] if δt followsa stationary I(0) process. Since xt+1 and zt are I(0) by assumption, δt will be I(0) when |π2(z)| < 1.

Table 3 shows this condition is met by both sets of model estimates. This condition insures that

the restrictions derived in the second step are consistent with the present value relation in (11)

rather than just the difference equation in (A1).

B Present Value Ratios

Tables 4 and 7 report variance ratios calculated from both the restricted and unrestricted versions

of the switching VAR. To facilitate the calculations, Þrst write the switching VAR shown in (13)

in companion form:

Yt+1 = A(zt+1, zt)Yt +Wt+1, (A5)

with Y 0t = [xt δt xt−1 δt−1 1]0. Notice that E[xt+1|Jt] = HE[A(zt+1, zt)|Jt]Yt with H = [1 0 0 0 0].The unrestricted present value estimates are derived from (A5) where the elements of A(.) arecalculated from the maximum likelihood estimates of a1, b11, b12, c11, c12, and the π0s. In the case ofthe restricted estimates, I use (15) to solve for b11, c11, c12 given estimates of b12 and the π

0s withρ = 0.937. The restricted A(.) matrix is based on these parameter values.

Consider the variance ratios involving the expected present value of observed fundamentals,

PVxt ≡∞Pi=1ρi−1E[xt+i|Jt]. (A6)

I posit and verify that

PVxt = E[G(zt)|Jt]Yt (A7)

for some vector of regime-dependent coefficients G(.). For this purpose, iterate (A6) one periodforward to get

PVxt = ρE[PVxt+1|Jt] +E[xt+1|Jt], (A8)

and substitute for E[xt+1|Jt] and PVxt using (A5) and (A7). The resulting equation must hold forall values of Yt, hence,

E [G(zt)|Jt] = ρE [G(zt+1)A(zt+1, zt)|Jt] +HE [A(zt+1, zt)|Jt] . (A9)

By the law of iterated expectations, if G(zt) solves

E [G(zt)|Jt, zt] = ρE [G(zt+1)A(zt+1, zt)|Jt, zt] +HE [A(zt+1, zt)|Jt, zt] ,

it must also solve (A9). The solution to the latter equation is

G(1)0G(0)0

= I − ρλ1A(1, 1)0 −ρ(1− λ1)A(0, 1)0−ρ(1− λ0)A(1, 0)0 I − ρλ0A(0, 0)0

−1 A(z, 1)0H0A(z, 0)0H0

. (A10)(A10) gives us the set of regime dependent coefficients in (A7). [NB the last vector are constants be-

cause HA(1, z) = HA(0, z).] Finally, note that under the posited solution limi→∞ ρiE£PVxt+i|Jt¤ =

0 so (A7) solves (A6) and not just the difference equation, (A8).

The unrestricted estimates of the present value in (A5) can now be calculated by Þrst Þnding

the values of G(1) and G(0) from (A10) using A(i, j) based on the unrestricted coefficient estimates.Next, Hamiltons algorithm is used to calculate the Þltered estimates of the state, i.e., E[zt|Jt] fort = 1, 2, . . . T. Finally, PVxt is estimated as (G(1)E[zt|Jt] + G(0)(1−E[zt|Jt]))Yt. The restrictedpresent values are found in a similar manner except that the A(.) matrix is based on restrictedcoefficient estimates. Table 4 also reports variance ratios derived from standard VAR. In these

cases the companion matrix A(.) is a constant A and the solution to (A10) is G = HA [I − ρA]−1 .The present value is therefore given by the standard formula PVxt = HA [I − ρA]−1 Yt. The tablereports Cov(PVxt , δt)/V ar(δt) where V ar(.) and Cov(.) denote the sample variance and covariance.

We can also use (A7) to estimate innovations in the present value

gPVxt+1 ≡ PVxt+1 −E £PVxt+1|Jt¤ = ∞Pi=0ρi (E[xt+i+1|Jt+1]−E[xt+i+1|Jt]) .

Combing this e