Do US Stock Prices Deviate From Their Fundamental Values - Some New Evidence

8/11/2019 Do US Stock Prices Deviate From Their Fundamental Values - Some New Evidence

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Do US stock prices deviate fromtheir fundamental values? Some new evidence q

Maosen Zhong a, Ali F. Darrat b,*, Dwight C. Anderson b

a Department of Finance, Kansas State University, Manhattan, KS 66506, USAb Department of Economics and Finance, Louisiana Tech University, College of Administration and

Business, P.O. Box 10318, Ruston, LA 71272-0046, USA

Received 20 October 2000; accepted 1 October 2001

Abstract

We propose a new methodology to test Famas [J. Finance 46 (1991) 1575] contention that

the present value model (PVM) should be augmented by time-varying expected inflation to

more adequately account for actual stock price behavior. Unlike other methods, our testingapproach can distinguish between the excess-price movement hypothesis of Shiller [Am. Econ.

Rev. 71 (1981) 421] and the dividend-smoothing hypothesis of Marsh and Merton [Am. Econ.

Rev. 76 (1986) 483]. We decompose the levels (as opposed to the variances) of stock prices into

their fundamental and non-fundamental elements in the context of a multivariate PVM co-

integrating framework and utilize the Gonzalo and Granger [J. Bus. Econ. Stat. 13 (1995)

27] procedure to formally test for the statistical significance of the non-fundamental com-

ponent. Our results from monthly data for the post-WWII period do not support the

inflation-augmented PVM since the non-fundamental component continues to achieve signif-

icance. This finding persists under alternative model specifications and data frequencies. The

apparent failure of the traditional, rational-expectation, PVM model to adequately account

for observed market behavior provides another piece of evidence supportive of Shillers

[Am. Econ. Rev. 71 (1981) 591] belief in some form of market irrationality.

2002 Elsevier Science B.V. All rights reserved.

q We thank, without implicating, two anonymous referees for helpful comments and suggestions. An

earlier version of this paper was presented to the Financial Management Association International s 30th

annual meeting, Seattle, Washington, October 2528, 2000, and to the SWFADs twenty-seventh annual

meeting, San Antonio, Texas, March 1518, 2000, and was awarded The American Association of

Individual Investors Best Paper in 2000 by the Southwestern Finance Association.* Corresponding author. Tel.: +1-318-257-3874; fax: +1-318-257-4253.

E-mail addresses: [email protected] (M. Zhong), [email protected] (A.F. Darrat), ander-

[email protected] (D.C. Anderson).

0378-4266/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0378-4266(01)00259-X

Journal of Banking & Finance 27 (2003) 673697

www.elsevier.com/locate/econbase
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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JEL classification: G12; G14Keywords: Present value model; Time-varying expected inflation; Fundamental and non-fundamental

values of stock prices; Market rationality

1. Introduction

One issue that occupies a great deal of time and attention in recent finance liter-

ature is whether the traditional present value model (PVM) can adequately explain

movements in stock prices. In an influential paper, Shiller (1981) argues that volatil-

ity in stock prices is too excessive to be justified by changes in market fundamentals

alone. He illustrates this phenomenon over the period 18711979 by plotting actual

S&P 500 stock prices with theirex postrational counterparts (as derived fromex post

forecasts of future discounted dividends). In Fig. 1, we extend Shillers plot with two

more decades of data. As Shiller suggests, actual stock prices over 18711997 are

indeed considerably more volatile than implied by market fundamentals. Hence,

Shiller (1981) and others (e.g., Campbell and Shiller, 1988) argue that some non-

fundamental factors may be behind the observed market behavior. As Lee (1998)

postulates, non-fundamental factors may include noise, feedback trading, or irra-

tional expectations.

Shillers argument is built on two critical assumptions: (a) real dividends are sta-

tionary around a historical trend; and (b) the discount factor (real interest rate) is

constant. Subsequent research rejects both assumptions and finds them largely re-sponsible for Shillers conclusion. In particular, Marsh and Merton (1986) challenge

the stationary assumption of stock prices and dividends. Responding to this criti-

cism, Campbell and Shiller (1987) incorporate the stationarity requirement, as well

Fig. 1. Real standard and Poors composite stock price index (solid line) andex post rational PVM price

(broken line) over the annual period 18711997. Both variables are detrended using Shillers (1981) meth-

odology.

674 M. Zhong et al. / Journal of Banking & Finance 27 (2003) 673697


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as any possible co-integration between stock prices and dividends, but still report

persistent deviations from the rational behavior implied by the PVM.

As to Shillers second assumption, several researchers (notably French et al., 1987;

Fama, 1991) suggest that a time-varying discount rate could also explain some of theexcessive volatility in stock prices. Reacting to this notion, Campbell and Shiller

(1988) derive a log-linear dividendprice ratio model that allows for the role of a

time-varying real discount rate. Yet, under alternative measures of the discount rate,

they continue to find that stock prices are too volatile to be compatible with market

fundamentals, concluding that there remain unexplained factors in the determina-

tion of the dividendprice ratio.

In the pursuit of such factors, two further possibilities receive much attention in

follow-up studies. The first line of thinking focuses either on memory in the duration

of dividend swings (like the recent work of Chow and Liu (1999)), or on linking dif-

ferent types of discount rates with macro state variables (for example, Abel (1993),MacDonald and Shawky (1995), Li (1998), and Campbell and Cochrane (1999)). The

second plausible explanation for excessive market volatility is advanced by those re-

searches who bring investorssentiment into the forefront of stock-price determina-

tion. Driven primarily by their psychology, investors become quasi-rational and

stock price may tend to deviate from market fundamentals. Examples of these be-

havioral finance models include the overconfidence and biased self-attribution model

of Daniel et al. (1998), and the interaction model of newswatchers and momentum

investors of Hong and Stein (1999).

The main purpose of this paper is to examine whether US stock prices significantly

deviate from their fundamental values predicted by the rational-expectation PVM.

Our approach differs from most previous studies in several respects. First, most ex-

isting research uses variancebound ratios to test whether the model does underesti-

mate volatility in the stock market (e.g. Shiller, 1981; West, 1988; Gilles and LeRoy,

1991; Mankiw et al., 1991). The logic behind these tests is that if stock prices are

indeed rational, they should not be more volatile than market fundamentals imply.

We adopt an alternativeand perhaps novellogic to test market rationality;

namely, if stock prices are indeed rational, then the levelsof prices should not deviate

excessively from their fundamental values. Marsh and Merton (1986) argue against

the use of variance-bound tests because of their inability to differentiate betweenexcess price volatility and inadequate dividend variability. Indeed, Marsh and

Merton provide an alternative managers dividend-smoothing hypothesis that is in-

consistent with the findings from variancebound tests. This criticism may not befoul

our approach since, regardless of any variability of dividends, the levels of rational

stock prices should not deviate from the levels dictated by market fundamentals.

Comparing levels (rather than variances) of prices and market fundamentals can

avoid the above criticism.

If the levels of stock prices do in fact deviate from those of the PVM, then the non-

fundamental component of stock prices should prove significant. We propose a new

paradigm to test the hypothesis that stock prices possess an insignificant non-fundamental component. We model stock price as the sum of fundamental and

non-fundamental components. The fundamental component is derived, following

M. Zhong et al. / Journal of Banking & Finance 27 (2003) 673697 675


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Campbell and Shillers (1988) dividendprice ratio model, from the log-linear version

of the PVM. The non-fundamental component of stock prices, being the difference

between actual prices and their fundamental values, is any component unaccounted

for by price fundamentals. We show that the fundamental and non-fundamentalcomponents correspond respectively to the permanent and temporary components

in the sense of Gonzalo and Granger (1995). We then propose a likelihood ratio sta-

tistic to formally test the statistical significance of the non-fundamental component

of stock prices.

Second, our methodology differs from Campbell and Shillers (1988) in several

other respects. First, Campbell and Shiller focus on the dividendprice model (in-

stead of the price series themselves), since the dividendprice ratio model is indepen-

dent of the deflator used to generate real variables, and also because the log

dividendprice ratio is assumed stationary with an (1,1) co-integrating relation-ship. While incorporating both features, our tests directly focus on the prices them-selves (rather than on dividendprice ratios). Secondly, similar to the traditional

variancebound test approach, Campbell and Shiller compare the volatility of the

actual dividendprice ratio with the volatility of the forecast present value of the real

dividend growth rates. In contrast, we decompose the levels of stock prices into fun-

damental and non-fundamental components and explicitly test the statistical signifi-

cance of the non-fundamental component. Finally, unlike Campbell and Shiller, we

use the nominal (as opposed to the real) version of the PVM to analyze price funda-

mentals. We discuss the virtues of this approach in the next section.

Finally, Lee (1998) examines the response of US stock prices to various types of

shocks. Our study also differs from Lees in several respects. Unlike Lees real version

of the PVM, we use the nominal version of the model to test Fama s (1991) propo-

sition that expected inflation may be an important, but missing, fundamental factor

in the PVM. Moreover, while Lee focuses on variance decompositions of stocks

prices attributable to various shocks, we decompose the levels of stock price to avoid

Marsh and Mertons (1986) criticism against variancebound tests. Additionally, we

employ a different methodology (see footnote 1) to identify fundamental and non-

fundamental components of stock prices and then formally test the significance of

the non-fundamental component.

The rest of the paper is organized as follows. Section 2 extends the standard PVMand formulates the testing procedures. Section 3 outlines the data and discusses the

empirical results. Section 4 offers some implications. Section 5 concludes the paper.

2. An extended model and the testing methodology

2.1. Deriving fundamental and non-fundamental components of stock prices

In this section, we extend Campbell and Shillers dividendprice ratio model to a

stochastic one-period PVM, and then propose an econometric approach to opera-tionalize the model. We study the fundamental and the non-fundamental compo-

nents of stock prices in the context of a log-linear nominal PVM. Using the



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nominal version of the model seems more appropriate for investigating the validity

of the model. The real version of the PVM necessitates the use of unobservable real

interest rates that could introduce biases that are common with extracted data. We

also attempt to control for other potential determinants of stock prices that havebeen ignored in previous studies. It is conceivable that expected inflation is a poten-

tial candidate. Finance literature reveals that stock returns respond to changes in ex-

pected inflation (e.g., Bodie, 1976; Fama and Schwert, 1977; Stulz, 1986; Fama,

1991). As Fama (1991, p. 1586) notes, By the end of 1970s, however, evidence that

expected stock and bond returns vary with expected inflation was becoming com-

monplace. The nominal version of the PVM allows for testing Famas proposition,

whereas the real version does not.

We decompose the logarithm of nominal stock prices (pt) into a fundamental com-

ponent (pft) and a non-fundamental component (pnft ). Before deriving the two price

components, we first offer their formal definitions. These definitions are synthesizedfrom the literature on market efficiency and variancebound tests.

Definition 1. Let the log of stock prices (pt) be represented by the sum of the fun-

damental component (pft) and non-fundamental component (pnft ). That is

ptpftp

nft : 1

(a) The fundamental component of log stock price is the discounted value of all

future dividends using the log-linear PVM. It represents the perfectly rational expec-tations and information-efficient component of log stock price. Thus, it is not pre-

dictable and should follow a random walk process.

(b) The non-fundamental component of log stock price is the deviation from the

fundamental component of stock price. Thus, it is the difference between the log

actual price and its fundamental component.

By virtue of the above definition, stock prices can only consist of fundamental and

non-fundamental components. We first derive these two components theoretically,

and then identify them respectively as permanent and temporary components in a

co-integrated vector autoregressive framework. 1 Proposition 1 below describesour theoretical derivation of the fundamental and non-fundamental components

of stock prices. 2

Proposition 1.The fundamental component of log stock prices at the end of the period

(pft1) is given as follows:

1 Our decomposition framework differs from the Blanchard and Quah (1989) procedure used by Lee

(1998). Due to the restrictive nature of Blanchard and Quahs procedure (see Enders (1995)), Lee

decomposes stock prices into several different components depending upon the number of variables in his

model. Such a framework is incapable of identifying a unique fundamental or non-fundamental

component as stated in Definition 1.2 The Proof of proposition 1 and all theorems in the paper are relegated to the appendix.



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Etpft1 qpt1 1qdt1ert1 p

et1 k: 2

The non-fundamental component of log stock prices at the end of the period (pnft1) is

given as follows:

Etpnft1 1qpt1 1qdt1ert1p

et1 k; 3

whereq and k are linear parameters, p is the log of stock price, d is the log of dividends,

er denotes excess returns defined as the difference between gross stock returns and

riskless rates, andpe is expected inflation.

Proposition 1 provides testable economic models of the fundamental and non-

fundamental components of stock prices. 3 These models state that both of these

components are linear combinations of that periods log of stock prices, log of div-

idends, excess returns, and expected inflation. The value of q is equal to 1=1 expd p, which is very close to unity.

Unlike Campbell and Shillers dividendprice ratio model, our Proposition 1 pro-

vides a stochastic expression (rather than an accounting identity) for both funda-

mental and non-fundamental components of stock prices, and it also incorporates

time-varying expected inflation as another fundamental variable of stock prices.

The interpretation of this equation is that the fundamental value of a common stock

is expected to be the weighted average of the end-of-period ex post log stock price

and log dividend, minus excess returns and expected inflation during the period.

The log stock price pt1 has a weight q close to one, while the log dividend gets aweight of (1 q) that is close to zero since the dividend is, on average, much smallerthan the price. A higher risk premium (excess return) and a higher inflation premium

should raise the discount rate of the stock and thus lower its fundamental value.

Shocks to cash flows (dividends) generate much smaller impacts on the fundamental

value than do shocks to shocks prices, risk premia, or expected inflation.

2.2. Empirical identification of fundamental and non-fundamental components of stock

prices

After theoretically deriving the fundamental and non-fundamental components ofstock prices, we utilize a co-integrated vector autoregressive model to empirically

identify the two components of stock prices. We first demonstrate that the funda-

mental and non-fundamental components constitute a permanent/temporary decom-

3 Solving forward Eq. (2) and imposing the transversality condition that limj!1qjptj 0 yields

pft EtX1j0

qj1

" qdt1j ert1j p

et1j

#k=1 q;

which is implied by Campbell and Shiller (1988) dividend-ratio model. However, this expression of the

fundamental value eliminates the possibility for rational bubbles. Acknowledging that the rational bubbles

may not hold in practice (as documented by Donaldson and Kamstra (1996)), we allow for the possibility

of a bubble term and use Eq. (2) to denote the fundamental value of stock prices.



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position of stock prices. We then advance several hypotheses to identity the non-fun-

damental component as a temporary component in the Gonzalo and Granger (1995)

sense. Finally, we propose a testing procedure to determine the statistical significance

of the non-fundamental component. Rejecting the null of a zero non-fundamentalcomponent in stock prices suggests that stock prices significantly deviate from their

fundamental values.

Hypothesis 1 (H1). The fundamental component of stock prices (pft) is difference

stationary [I1], and the non-fundamental component of stock prices (pnft ) is co-variance stationary [I0].

If the fundamental component derived in Proposition 1 indeed follows a random

walk process as defined in Definition 1, this fundamental component cannot be an

I0 process.

Hypothesis 2 (H2). In the vector autoregressive representation of the fundamental

and non-fundamental components of stock prices,

B11L B12LB21L B22L

Dpftpnft

u1tu2t

; we have B121 0 4

where u1t and u2t are white-noise residuals, BL are lag polynomials such thatB

L

I

b

1L

b

2L2 . . .

bnLn, and L is the lag operator.

Theorem 1 below shows that the fundamental and non-fundamental components

as stated in Proposition 1 represent the permanent and temporary processes of stock

prices.

Theorem 1. Under Hypotheses H1 andH2, the fundamental component of log stock

prices (pft) and the non-fundamental component of log stock prices (pnft ) constitute,

respectively, the permanent and temporary components of log stock prices (pt).

If both H1 and H2 are not rejected, then a shock to the fundamental componenthas a persistent (permanent) effect on the level of stock prices, while a shock to the

non-fundamental component has only temporary (transitory) effects on the level of

stock prices (see Proof of Theorem 1 in the Appendix A). Theorem 1 is consistent

with the common conjecture that the fundamental component is a permanent mean,

whereas the non-fundamental component has a temporary, but mean-reverting,

component of the price (see Fama and French, 1988; Lee, 1998).

As shown by Johansen and Juselius (1994), the empirical identification of a given

economic model involves testing the hypothesis that the coefficients estimated by an

econometric representation equal the parameters suggested by the model. Below, we

present a time-series econometric model of stock prices, dividends, excess returns,and expected inflation. We then propose a number of testable hypothesis to empir-

ically identify the fundamental and non-fundamental components of stock prices.



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We now outline an econometric model to test for significant deviations of stock

prices from their fundamental values. Let Zt be an n-dimensional vector of stock

prices (pt) and other fundamental variables such as dividends (dt), excess stock re-

turns (ert), and expected inflation (pet). That is, Zt ptdtertpet0. If the elements inZtare all difference-stationary and the rank of the co-integration is r, then Zthas a

vector error-correction model (VECM) representation of the form

DZtab0Zt1

X1h1

ChDZth nt; 5

whereD 1L; b is an (nr) co-efficient matrix of co-integrating vectors; a is an(nr) adjustment co-efficient matrix;Ch is an (nn) matrix of parameters reflectingthe short-run dynamics; and nt is a white-noise error vector IIDn0;K.

As Gonzalo and Granger (1995) demonstrate, vector Ztcan be explained in termsof a linear combination of (nr) numberI1common factors denoted as ft, plusa linear combination ofr number of co-integrating vectors. That is,

Ztn1

A1nk

ftk1

A2nr

b0

rnZtn1

; 6

where knr, A1 and A2 are loading matrices, ft are linear combinations of theelements Zt, and A1ft and A2b

0Zt represent a permanent/transitory decomposition.

The common factors ftare estimated as

fta0?Zt; 7

wherea0? is a (kn) vector and a0?a 0.

4 The factor loadings areA1b?a0?b?

1

and A2 ab0a1. We then obtain the permanent and transitory components of

stock prices as follows:

ptA11ft A

12b

0Zt; 8

where A1j is the first row of co-efficient matrix Aj, j 1; 2.We show below that the permanent and transitory components of stock prices in

Eq. (8) have linear relationships with the fundamental and non-fundamental compo-

nents of stock prices derived in Proposition 1. We propose several pertinent hypoth-

eses to investigate whether the estimated coefficients in model (8) correspond to the

parameters predicted by the economic model as stated in Proposition 1.

Hypothesis 3(H3). Stock prices, dividends, excess returns, and expected inflation are

co-integrated.

4 Under the null of co-integration, the maximum likelihood estimator ofa? can be found by solving the

equation

jkS00 S0kS1kkSk0j 0;

where S00, S0k, and Skkare the residual moment matrices and cross-product moment matrix of the least

square regressions ofDZtand Ztkon DZt1; . . . ;DZtk1, respectively. The estimate ofa?is the eigenvectorassociated with the (nr) smallest eigenvalues of the above eigenvalue problem.



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Hypothesis 4 (H4). b Hu, where the restriction matrix H is

H 1 1 0 0

0 0 1 1 0

;

and u is a (2 r) matrix of restricted long-run estimated coefficients.

Theorem 2. Under Hypotheses H3 andH4, the non-fundamental component is em-

pirically identified as a proportion of the temporary component indicated in Eq. (8).

That is,

pnft cpTt cA

12b

0Zt; 9

where c is a scalar parameter of proportionality.

Intuitively, Theorem 2 means that the non-fundamental component is propor-tional to the temporary component of stock prices under Hypotheses H3 and H4.

The proportionality factorc picks up the temporary component of stock prices from

the vector A02b0Zt.

Hypotheses H3 and H4 above can be tested in a co-integrated framework using the

Johansen and Juselius (1992) procedure. To test H3, we compute the maximal eigen-

value and trace statistics. 5 As to testing H4, the test is derived from a likelihood ratio

statistic which is distributed as v2 with degrees of freedom equal to the number of

restrictions on b (2r in our cases). 6

Once we empirically identify the fundamental and non-fundamental components

of stock prices, we test the significance of each fundamental variable in both compo-nents. We do that in the context of Hypothesis H5 below:

Hypothesis 5 (H5). None of the four variables pt; dt; ert; pet in Zt can be excluded

from the permanent and temporary components ofZt.

We should give attention to testing whether expected inflation (pet) can be excluded

from vector Zt. As discussed earlier, if Famas (1991) conjecture that the PVM must

allow for the role of expected inflation were to be true, then this variable should not

be excluded from either the fundamental (permanent) component or the non-fun-

damental (transitory) component of stock prices. Thus, we perform an exclusion

test on the fundamental component in Hypothesis H5 using the Gonzalo and

5 The two statistics are: kmax Tlog1 kkr1), and ktrace TPn

ir1log1 kki), where T is the

sample size,n is the number of variables in the system ( 4 in our case), and kkis are the solutions to theeigenvalue problem:j kSkk Sk0 S

100S0k j.

6 The test statistic is

TXri1

ln1ekiki1 kki

v2rns;

where ~kki is the solution to the following restricted eigenvalue problem:

j ~kkiH0SkkH H

0Sk0S100S0kHj0:



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Granger (1995) test of the common factors (ft). The null hypothesis is that a partic-

ular series can be excluded from the fundamental component. That is,

a?nnr Gnm hmnr withnr6m6 n; 10

where G is a (nm) restriction matrix on the common factor loadings, a? and h isthe estimated parameter matrix in the common factors. The test statistic is derived by

comparing the eigenvalues from the restricted and the unrestricted VECMs and then

formulating a likelihood ratio statistic. This statistic follows a v2-distribution with

degrees of freedom equal to nr nm. 7 The exclusion test on the tempo-rary component can be implemented using the Johansen and Juselius (1992) pro-

cedure outlined above.

2.3. Testing the significance of non-fundamental component of stock prices

After empirically identifying the non-fundamental component of stock prices, we

now test whether this component is significantly different from zero. Theorem 3 for-

mulates this testing procedure.

Theorem 3.We re-write the VECM system (5) as follows:

Dp1

Dd1Der1Dpet

2664 3775a1

a2a3a4

2664 3775b0Zt1 X1h1

ChDZth nt: 11

Under Hypotheses H3H5, testing the notion that the non-fundamental component of

stock prices is zero is equivalent to testing the following hypothesis:

Hypothesis 6 (H6). a10 in the VECM (11).

If stock prices do deviate from their fundamental values that are predicted by the

PVM, market rationality becomes in doubt, regardless of the origin of such deviation(be it excess price volatility or dividend smoothing). By virtue of the analysis in Engle

et al. (1983), testing the hypothesis that a10 may also be called a weak exogene-ity test of stock prices with respect to b (see also Johansen and Juselius, 1990). As

such, inspecting this hypothesis becomes a straightforward test in the context of the

vector-error-correction equation (11).

7 The test statistic takes the form

TXn

ir1

ln1 ~kkimn

1 ^

kki

;

in which T is the sample size, ~kkimn represents the solutions to the restricted eigenvalue problems in

Gonzalo and Granger (1995); namely,j ~kkG0S00G0 G0S0kS

1kkSk0Gj0.



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3. Data and empirical results

We use monthly data over the post-World War II period (1947:011997:12). The

data sources are as follows. The value-weighted NYSE stock index is obtained fromthe CRSP stock index file. Following Lees (1995) method of compiling dividend se-

ries, we first denote nominal stock prices and dividend series as Pt and Dt, respec-

tively. The value-weighted index returns including dividends are RDt fPt Dt=Pt1 1g, where dividends, D, is also obtained from the CRSP data file. Thevalue-weighted index returns excluding dividends areRt Pt=Pt1 1. Therefore,the dividend seriesDtis defined as Dt RDt Rt Pt1. Following Lee (1992), ex-pected inflation, pe, is modeled rationally as pet Et1ptjpt1; MBt1; it1;IPt1,where MB is the monetary base, i is the three-month treasury bill rate, and IP is

the industrial production average. As to excess stock returns, er, they are measured

by the differences between gross stock returns and nominal risk-free rates (three-month treasury bill rates). 8

We test for the presence of unit roots (non-stationarity) in the four variables as

well as in the fundamental and non-fundamental components which we construct

using Proposition 1. We report in Table 1 the results from the augmented Dickey

Fuller (ADF) and the weighted symmetric (WS) tests. 9 We choose proper lags based

on the Akaike Information Criterion (AIC) provided that the residuals are also

white-noise processes. As seen in Table 1, the null of non-stationarity is not rejected

for all four variables in levels, but the hypothesis is rejected if the variables are ex-

pressed in first-differences. Therefore, each variable in the vector Zt is I(1), and

it is possible that the levels of these variables are co-integrated. Results in Table 1

also show that the fundamental and non-fundamental components appear differ-

enced-stationary and leveled-stationary, respectively, consistent with Hypothesis H1.

To test Hypothesis H2, we select a high lag-order to eliminate residualsautocor-

relation (15 lags in our case). The t-statistic for the sum of coefficients on the non-

fundamental component, B12, is not significant (1.03), suggesting that HypothesisH2 cannot be rejected. This finding proves robust to using different lag orders in

the VAR. Therefore, consistent with Theorem 1, the fundamental and non-funda-

mental components as stated in Proposition 1 constitute the permanent and tempo-

rary components of stock prices.Next, we empirically identify the fundamental and non-fundamental components

of stock prices. We employ the Johansen and Juselius (1992) approach to test for co-

integration among stock price, dividends, excess returns and expected inflation (as

8 The fundamental component of stock price thus generated indeed follows a random walk process

according to Lo and MacKinlay (1998) variance ratio test (results are available upon request). A Ljung

BoxQ-test indicates that the non-fundamental component follows an AR(1) process. A preliminary Box

Jenkins identification procedure also reveals that the level of log stock prices follows an ARIMA(1,1,1).

This suggests that the sum of a random walk component and a stationary AR(1) component characterize

stock prices in the post-World War II period quite well.9 Pantula et al. (1994) argue that the WS procedure is more powerful than several alternatives, including

the ADF test. We employ both tests to ensure robustness of our inferences.



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depicted in Hypothesis H3). Table 2 displays the test results. The lag order of the un-

derlying VAR is 15 as jointly required by the AIC procedure and white-noise resid-

uals (Cheung and Lai, 1993). As Table 2 shows, there is sufficient evidence for two

non-zero co-integrating vectors in the model. 10;11

We now test whether the parameter restrictions implied in Hypothesis H4 can em-

pirically identify the fundamental and non-fundamental components of stock prices.

The correspondingv2-test statistic proves insignificant (3.71 with 4 d.f.). Hence, the

Table 2

Johansen co-integration tests (testing Hypothesis H3) (monthly data: 1947:011997:12)

Trace statistics: LRtrace TPpir01ln1 kki)H0: r 0 H0: r6 1 H0: r62 H0: r6 3

Zt pt; dt; ert;pet; k 117.89

53.84 9.04 2.59

Notes: indicates rejection of the null of no co-integration at the 5% significance level. The proper (AIC/

white-noise) lags in the underlying VARs are 15. The corresponding critical values for the above four

hypotheses are 53.12, 34.91, 19.96, and 9.24, respectively.

Table 1

Unit root test results (monthly data: 1947:011997:12)

Levels

pt dt ert pet pft pnft

Augmented DickeyFuller test 1:57 1:89 1:99 1:87 1:63 9:01

Weighted symmetric test 1:61 2:78 2:23 2:01 1:68 9:09

First-Differences

Dpt Ddt Dert Dpet Dp

ft

Augmented DickeyFuller test 9:10 13:69 12:67 8:59 14:16 Weighted symmetric test 9:17 13:45 12:74 8:85 9:08

Notes: Variables are defined as follows: pt is the log of stock prices, dt is the log of dividends, ert is excess

stock returns defined as the difference between gross stock returns and riskless interest rates, and pet is

expected inflation.pft is the time-varying fundamental component of stock prices defined as the discounted

value of all future dividends using the dynamic log-linear PVM (see Eq. (2)), and pnft denotes the non-fundamental component of stock prices defined as the ex postprice deviation from the fundamental value

(see Eq. (3)). A time trend is included in all unit root test regressions. The testing equations use lags as

determined by the Akaike Information Criterion (AIC). An indicates rejection of the null of non-

stationarity at the 5% level. The corresponding critical values for the augmented DickeyFuller and

weighted symmetric tests are2:86 and3:18, respectively.

10 Following Cheung and Lai (1993), we rely on the trace statistic since it is relatively insensitive to

skewness and excess kurtosis in the residuals. However, results from the maximal eigenvalue version

provide qualitatively similar results.11 The Schwartz Basian Criterion (SBC) suggests the appropriateness of 3 lags instead, but without

altering the conclusion of two non-zero co-integrating vectors. With its heavier emphasis on conserving

degrees of freedom, the SBC usually chooses a simpler model to that selected by the AIC (Greene, 2000).



13/25

non-fundamental component is empirically identifiable as the temporary component

of stock prices as stated in Eq. (8).

Finally, Hypothesis H5 can be tested by examining the significance of each vari-

able in the fundamental and non-fundamental components of stock prices. Table

3 reports the outcomes of exclusion tests for the fundamental and non-fundamentalcomponents. As indicated in the table, none of the variables can be excluded from

either component. This suggests that all four variables (including expected inflation)

should be maintained in the model to avoid a serious loss of information about fun-

damentals. This finding buttresses Famas (1991) contention that expected inflation

is an important fundamental underlying stock prices.

After empirically identifying the two components of stock prices, we use Theorem

3 to test the validity of the PVM, that is, whether the non-fundamental component is

zero (H6). Table 4 displays the test results for the statistical significance of the non-

fundamental component under Hypothesis H4. Note that the non-fundamental com-

ponent is empirically identifiable under Hypothesis H4. One may argue that a generalstationary (temporary) component should be allowed in the model and then inspect

its statistical significance. 12 Table 4 also reports the outcomes from testing the null

of a zero temporary component with and without the restriction of Hypothesis H4.

As is clear from the table, the null is soundly rejected in both cases. 13

Table 3

Exclusion tests (testing Hypothesis H5) (monthly data: 1947:011997:12)

pt dt ert pet

Fundamental component (v2(2) statistics) 39.35 62.24 39.33 17.06

Non-fundamental component (v2(2) statistics) 7.73 7.45 17.46 46.88

Notes: indicates rejection of the null of exclusion at the 5% significance level. The corresponding critical

value is 5.99.

Table 4

Significance tests of the non-fundamental component (testing Hypothesis H6) (monthly data: 1947:01

1997:12)

Null hypotheses v2-statistics

Zero non-fundamental component (under Hypothesis H4) v26 21:96

Zero temporary component (without the restriction of Hypothesis H4) v22 16:97

Notes: indicates rejection of the null of zero non-fundamental (transitory) component of stock prices at

the 5% significance level. The corresponding critical values for v2(6) and v2(2) are 12.60 and 5.99,

respectively.

12 Without restriction of Hypothesis H4, the temporary component may not exactly conform to the

definition of non-fundamental component of stock prices.13 The evidence proves robust to using alternative data frequencies (quarterly or annual) and to using

different processes to generate inflation (e.g., static expectations or nominal interest rates as in James et al.

(1985)). Details of all sensitivity tests are relegated to an appendix available from the authors upon

request.



14/25

4. Implications for stock market volatility

Our test results suggest that the stochastic fundamental values of US common

stocks do not closely mimic the behavior of actual stock prices. There exists a signifi-cant non-random-walk, non-fundamental, component in the levels of these prices.

What implications do these results have for the observed phenomenon of excessive

volatility in stock prices, and how do they relate to previous volatility research? 14

Since the non-fundamental component is the predictable, mean-reverting compo-

nent of stock prices, we may compute the fraction of stock return variance that is

explained by this mean-reverting component. According to Definition 1, the natural

log of stock prices (pt) can be modeled as the sum of the fundamental component (pft)

and the non-fundamental component (pnft ). The pft is the discounted value of all

future dividends using the log-linear version of the PVM. It represents the informa-

tion-efficient component ofptand, as such, pftis not predictable and should follow arandom walk process.pnft , on the other hand, represents the difference betweenptand

pft. The two components of stock prices can be simply modeled as

pft bpft1 ut; 12

pnft apnft1 vt; 13

where b is a drift, 0< a < 1, and ut and vt are white-noise processes. This simplemodel is analogous to that in Fama and French (1988). However, their model de-

duces the existence and properties of the unobservable non-fundamental componentfrom the behavior of stock returns. In our model, however, the non-fundamental

component is derived from observable factors (see Proposition 1).

Eq. (13) assumes that the non-fundamental component follows an AR(1) process.

This is a common assumption in literature [see, for example, Summers (1986),

Poterba and Summers (1988), and Fama and French (1988)]. Given this assumption,

it can be shown that the non-fundamental component is also a stationary predictable

process.

Lemma 1. The non-fundamental component of stock prices (pnft ) obtained from The-

orem 1 follows a stationary predictable process if it is captured by an AR(1) process.

Lemma 1 supports Lees (1998) finding the behavior of stock prices is better cap-

tured by a fads than by a bubble process since fads component is a stationary

(mean-reverting) component, whereas the bubble is not.

Based on the simple model of Eqs. (12) and (13), the fraction of the variance of

stock returns explained by expected changes in the non-fundamental component is

approximated by the negative slope (beta(Dt)) in the regression of the return from

t to t Dt, Rt;tDton the return from time t Dt to t, RtDt;t. (See Appendix A for

the derivation of this result). Thus,

14 We thank an anonymous referee for guiding us through the following discussion.



15/25

Variance fractionVarEtp

nftDt p

nft

VarRtDt;t betaDt

CovR

t;tDt;R

tDt;t

VarRtDt;t : 14

As shown in the Appendix A, the first-order autocorrelation ofDt-period changes

in the non-fundamental component (UDt) is the negative of the proportion of thevariance of the changes in the non-fundamental component (pnftDt p

nft ) due to ex-

pected changes (EtpnftDt p

nft ). That is,

VarEtpnftDt p

nft

VarpnftDt pnft

UDt: 15

We note that the fraction of the variance of stock returns explained by expectedchanges in the non-fundamental component should form an inverse U-shaped pat-

tern. Further, as Fama and French (1988) suggest, the ratio of the variance of

expected changes to the variance of actual changes in the non-fundamental compo-

nent should fluctuate around one half.

Fig. 2 displays the empirical pattern of stock returns and their non-fundamental

components. The beta starts at around 0.0 and then displays positive values for

the first five months of return horizons. This is consistent with positive autocorrela-

tion in short-term returns (the short-term momentum behavior) discussed in Lo and

MacKinlay (1998). Thereafter, stock returns exhibit negative autocorrelation of

about0:38 at the 23rd month, and then gradually recede toward zero. Clearly, thisfinding accords well with the long-term mean reversion of stock returns (the con-

trarian behavior) of Fama and French (1988). As predicted by the proposed model,

UDt) fluctuates around 0:5, implying that the ratio of the variance of expectedchanges to the variance of actual changes in the non-fundamental component is ap-

proximately one half.

Two implications bear emphasis here. First, the behavior of the non-fundamental

component implied by our Proposition 1 is consistent with the prediction from such

Fig. 2. The Pattern of first-order autocorrelations of stock returns and changes in the non-fundamental

component over increasing return horizons. The solid line (

) is the plot of first-order autocorrelations

of stock returns over the Dtperiod (beta(Dt)). The broken line (- - -) is the plot of first-order autocorrela-

tions of changes in the non-fundamental component (UDt).



16/25

conventional models of stock price as those in Fama and French (1988). Secondly,

the fraction of return volatility explained by expected changes in the non-fundamen-

tal component increases with return horizons of up to two years. Thereafter, this

fraction diminishes, and is completely eliminated by the mean reversion behaviorof stock returns after an elapse of almost three years. This suggests that stock prices

do deviate from their fundamental values in the short term, and then begin to revert

back in the long term.

5. Conclusions

In their seminal papers, Shiller (1981) and Campbell and Shiller (1988) contend

that the behavior of US stock prices appears incompatible with market rationality

as represented by the PVM. Fama (1991) points out that time-varying expected in-flation may partially account for the observed behavior of stock prices. We incorpo-

rate Famas suggestion and propose a new methodology to test the statistical

significance of the non-fundamental component of stock prices. To control for the

role of expected inflation and, at the same time, perform reasonable empirical tests,

we use a nominal (rather than a real) interpretation of the PVM. In contrast to Lee

(1998) and to the variancebound literature, we focus on the levels (not the vari-

ances) of stock prices and their market fundamentals in order to avoid Marsh and

Mertons (1986) criticism that traditional tests of markets volatility cannot distin-

guish between excessive price volatility and dividend smoothing.

We first show that the fundamental and non-fundamental components of stock

prices correspond, respectively, to the permanent and temporary components. We

then empirically decompose the levels of stock prices into their fundamental and

non-fundamental components in a multivariate co-integrated context and test the

null hypothesis that the non-fundamental component is not significantly different

from zero. Rejecting this null implies that stock prices do significantly swing away

from their fundamental values of the rational-expectation PVM.

Our results for the US post-WWII monthly data (1947:011997:12) confirms the

inability of the PVM, even after augmentation by time-varying expected inflation,

to adequately explain actual market behavior, whereby the non-fundamental compo-nent of stock prices is significantly different from zero. It appears, therefore, that the

conventional, rational-based PVM is inadequate to account for observed market be-

havior even after controlling for Famas (1991) time-varying expected inflation.

Clearly, this finding appears congruous with Shillers belief in some form of irratio-

nality in the US stock market.

As with previous studies, our analysis faces the difficulty arising from the joint hy-

pothesis problem. That is, we test the hypothesis of market rationality jointly with

the hypothesis that the augmented-PVM captures all rational variations in stock

prices. Clearly, there may be other plausible, but missing, fundamental factors be-

sides expected inflation. For example, Darrat (1990) reports that, besides inflation(monetary policy), budget deficits (fiscal policy) also possess explanatory power

for US asset prices. Campbell and Cochrane (1999) also argue that time-varying con-



17/25

sumption in excess of habit could forecast expected stock returns. Therefore, enlarg-

ing the co-integrated space of the modified PVM with more theoretically relevant

fundamentals, and then retesting, appears a fruitful research topic.

Appendix A

Proof of Proposition 1. Following Campbell and Shillers (1988) derivation ap-

proach, we define the continuously compounded nominal stock returnsrt1for timet

to t 1 as

rt1 logPt1 Dt1 logPt: A:1

Applying Taylors approximation theorem and putting log (Pt) on the left-hand side

of (A.1) yields the following log-linear expression for stock prices:

pt logq 1q log1=q 1 qpt1 1qdt1 rt1; A:2

where lower-case letters indicate the logarithms of variables;q 1=1expd p.Ifpt is non-stationary, the right-hand side of (A.2) is also non-stationary.

Eq. (A.2) is measuredex post, and the discount rates rt1 are unobservable. To as-

sign an economic meaning to expression (A.2), and following Campbell and Shiller

(1988), we impose some restrictions on the behavior of nominal discount rates. In

particular,

Etrt1 r0 Etert1 Etp

e

t1; A:3whereEtdenotes a rational expectation operator formed by using the information set

Itthat is available to market participants at the end of period t;r0 is a constant real

riskless rate; ert1is the risk premium (excess stock returns) measured by the nominal

gross return on a given stock during timet to t1 relative to the nominal return onshort-term debt it1; and p

et1 is inflation expected at time t to prevail at time t1.

Note that there is no inflation premium in excess stock returns since the inflationary

effect is canceled out after subtraction. Eq. (A.3) implies that the ex ante return over

periodt1 equals a constant real riskless rate plus a risk premium and an inflationpremium.

Eqs. (A.2) and (A.3) yield the following model of the fundamental value of stockprices:

pft Etqpt1 1qdt1ert1pet1logq 1q log1=q 1 r0:

A:4

By Definition 1, the fundamental component follows a random walk process (with a

drift b). That is,

pft1 bpftet1; where et1 is white-noise; A:5

orEtp

ft1 bp

ft: A:6



18/25

Eqs. (A.4) and (A.6) yield the expression of the fundamental component in Propo-

sition 1.

Etpft

1 qpt1 1qdt1ert1 pet

1 k;

where k logq 1q log1=q 1 r0 b: A:7

To derive the non-fundamental component, we get from Definition 1 of the text:

Etpnft1 pt1Etp

ft1

pt1 qpt1 1qdt1ert1pet1 k; A:8

we obtain Eq. (3) in the text by rearranging (A.8).

Proof of Theorem 1.We begin with the definition of permanent and temporary (P

T) decomposition that is described in Gonzalo and Granger (1995).

Definition.Let Xtbe a difference-stationary sequence. A PT decomposition for Xt is

a pair of stochastic processes Pt, Ttsuch that:

(i) Pt is difference stationary and Ttis covariance stationary,

(ii) Var(DPt) and VarTt> 0,(iii) XtPt Tt,

(iv) letBL DPtTt

uPtuTt

be the autoregressive (AR) representation of (DPt; Tt;

with uPt and uTtuncorrelated, then

(a) limh!1oEtXth=ouPt60 and(b) limh!1oEtXth=ouTt0,

whereEtis the conditional expectation with respect to the past information.

Condition (i) of the above Definition is satisfied under Hypothesis H1. Condition

(ii) is satisfied as long as pft and pnft are not constants. Condition (iii) is satisfied ac-

cording to Eq. (1) of Definition 1 in the text. We demonstrate below the Condition

(iv) is also satisfied under H2.

We re-write the first-order bivariate VAR in Eq. (4) into a vector moving-average

representation:

Dpftpnft

A11L A12LA21L A22L

u1tu2t

: A:9

Thus, following Gonzalo and Granger (1995), we obtain the moving average rep-

resentation ofDpft.

Dpft A111u1t A121u2t I Lf~AA11Lu1t ~AA12Lu2tg; A:10

where

A1jL A1j1 IL ~AA1jL; j 1; 2; A:11

and ut u1t; u2t) is a vector of white noise with covariance matrix

R R11 R12R21 R22

:



19/25

Assuming that u1t and u2tare not perfectly correlated, they can be decomposed as

u1tuPt and u2tR111R21u1t uTt: A:12

From (A.10) and (A.12), we obtain

limh!1

oEtpth

ouPtA111 A121R

111R12 A:13

and

limh!1

oEtpth

ouTtA121: A:14

Since

limh!1

oEtpth

ouPt lim

h!1

oEtpfth

ouPt; A:15

(pft;pnft) will be the PT decomposition according to the above Definition if and only

ifA121 0 in (A.14). Note that

A121 B1111B121B211B111

1B121 B221

1: A:16

Therefore, Condition (iv) is satisfied iffB121 0, as dictated by H2.

Proof of Theorem 2. Proposition 1 predicts that the non-fundamental component

of stock prices is a linear combination of pt; dt; ert; it with parameters 1q;1 q; 1; 1plus an intercept k(qand kcan be estimated). Hence, two conditionscharacterized the non-fundamental component:

(i) pt and dthave the (1; 1) relationship, and(ii) ert and ithave the 1; 1 relationship.

Under H4, these two conditions are satisfied. Note that the co-integrating vector

b

0

Ztcan be normalized on any element in Zt. Thus, the non-fundamental componentwould be a linear relation to the co-integrating vector (hence the temporary compo-

nent) as indicated in Eq. (9).

Proof of Theorem 3. This proof contains two steps. First, we demonstrate that

testing the hypothesis of zero non-fundamental component of stock prices amounts

to testing H0: A120 in Eq. (9).

Under Hypothesis H3, there is one co-integrating vector among the four variables

pt1; dt1; ert1; it1, i.e., r 1.Under Hypotheses H4H5, the temporary component p

Tt A

12b

0Zt is a linear rela-

tion topnft . That is, p

nft cp

Tt, the linear parameters c 60. From Eq. (8), we obtain:

pnft cA12b

0Zt: A:17



20/25

A12 is a (1 1) scalar parameter (since r 1), and b0Zt is the co-integrating vector.

Therefore, testing pnft 0 amounts to testing H0: A120.

Next, we demonstrate that testing H0:A12 0 in Eq. (9) is equivalent to testing H6:

a10 in the VECM (11).Intuitively, Hypothesis H0: A

120 means that the temporary component of stock

price has no impact on the stock price. Note that the temporary component can only

have short-run effects on pt. Ifa10, then the temporary component ofZthas noimpact on the changes in stock prices (therefore no short-run effect on levels of stock

prices). Thus, we may conclude that stock prices do not have a significant temporary

(or, equivalently, non-fundamental) component.

Technically, to test H6: a10 in the VECM (11), we formulate an equivalenthypothesis:

H0 : anr Wns Wsr; A:18

whereWis the restriction matrix andWis the (sr) matrix of estimated co-efficientsof speed of adjustment under hypothesis (A.18), the temporary component becomes

A2b0Ztab

0a1b0Zt

WWb0WW1b0Zt:A:19

When r 1, (b0WW) is a non-zero scalar. Let Q b0WW), then

A2b0

Zt Wns WsrQ1

11 b0

1n Zt

n1: A:20

To test whether a particular row in A2b0Ztis zero, we specify the Wrestriction matrix

as a (n1) vector composed of a zero element and ones for the rest of the elements.For example, to test whether stock prices pt in Zt contains a zero temporary com-

ponent, we specify Was follows:

Wn1

0

1

1

1

2

664

3

775; and W becomes a 11 scalar:

Then, we can rearrange (A.20) into the following equation:

A2b0

n1Zt W

11Q1

11

Wn1

b0Zt: A:21

The first element of the Wvector (i.e., 0) picks up the first element of the temporary

component A2b0Zt. Therefore, if hypothesis (A.18) is valid, Eq. (A.17) characterizes

the temporary component A2b0Zt. Hypothesis (A.18) can be formally tested in the

VECM (9) by specifying H0: a10, that is, the error-correction term is not signif-

icant in the VECM. Johansen and Juselius (1990) provide a likelihood ratio statisticto test this hypothesis (also called weak exogeneity test with respect to a and b). This

weak exogeneity test is briefly described below.



21/25

Consider the following notations: S00 and Skk are the residual moment matrices

from the least-square regressions ofDZtand Ztkon DZt1; . . . ;DZtk1, respectively;and S0k is the cross-product moment matrix of the residuals, and

Skk;bSkk SkbS1bbSbk;

Ska;bSka SkbS1bbSba;

Saa;bSaa SabS1bbSba:

Let B W? such that B0W 0; SkbS0kB, SbkB

0S0k; SbbB0S00B; and Sab

W0S00B. According to Johansen and Juselius (1990), under hypothesis a WW,the maximum likelihood estimator ofW can be solved as the eigenvector associated

with the following:

jkH0Skk;bHH0Ska;bS

1aa;bSka;bHj 0 A:22

for ~kk1> ~kk2> > ~kks1 ~kkn, and V vv1. . .vvn) normalized by vv0Skk:bvv I.

Now take bb vv1. . .vvrr) that yields the estimates W W0W1Sak:bbb, anda Www

WW0W1W0 S0k S00BB0S00B

1B0S0kbb. The maximized-likelihood function is

L2=Tmax H S00j jYri1

1 kki: A:23

The likelihood-ratio statistic of Hypothesis (A.23) is

T

Xr

i1

lnf1 ~kki=1 kkig; A:24

whereTis the sample size, ~kki is theith largest eigenvalue of Eq. (A.22), and kki is the

ith largest eigenvalue of jkSkk Sk0S1

00S0kj 0. Asymptotically, this statistic is v2

distributed withr pms degrees of freedom.

Proof of Lemma 1. According to Definition 1 in the text, the non-fundamental

component of stock prices can be expressed as follows:

pnft apnft1 vt; where a60; and vt is white-noise: A:25

Rearranging (A.25), we get

pnft1 1a

pnft vt: A:26

Thus,

Dpnft pnft p

nft1

pnft 1

a

pnft vt

a1

a pnft

vt

a:

Given Dpt Dpf

t

Dpnft

and Dpft

pft

pft1

et (see Eq. (A.6)), we obtain

Dptet Dpnft

a1

a pnft et

vt

a

; A:27



22/25

whereet vt=a are also white noise. Since Dptproves stationary, it follows thatpnft is also stationary. This indicates that the first-order autoregression co-efficienta

in Eq. (A.25) is less than 1.0.

Derivation of Eqs. (14) and (15). With ptas the log of stock prices, the continuously

compounded stock returns from t to t Dtcan be written as

Rt;tDtptDt pt

pftDt pft p

nftDt p

nft :

A:28

The random-walk price component yields white noise in returns, and the presence of

the non-fundamental component causes the observed mean-reversion behavior of

stock returns (Fama and French, 1988).

The first-order autocorrelation ofDt-period changes in pnf

t is the slope in the re-gression of (pnftDt p

nft ) on (p

nft p

nftDt), and is written as

UDt CovpnftDt p

nft ;p

nft p

nftDt

VarpnftDt pnft

; A:29

where the numerator is

CovpnftDt pnft ;p

nft p

nftDt Varp

nft 2Covp

nft ;p

nftDt

Covpnft ;pnft2Dt: A:30

The stationarity ofpnf

t implies that Cov[pnf

t ;pnf

tDt] and Cov[pnf

t ;pnf

tD2t] approach zeroas Dt increases, so Cov[pnftDt p

nft ;p

nft p

nftDt] approachesVar(p

nft ). Similarly,

VarpnftDt pnft 2Varp

nft 2Covp

nftDt;p

nft ; A:31

which approaches 2Var(pnft). The first-order autocorrelation ofDt-period changes in

pnft approaches 0:5 for large return horizons Dt.Note that the slope UDt) can be interpreted as (the negative of) the ratio of

the variance of the expected change in the non-fundamental component,

VarEtpnftDt p

nft , to the variance of the actual change, Varp

nftDt p

nft . To see this,

ifpnft follows an AR(1) process as specified in Eq. (13), the expected change fromt to

t Dt is obtained by solving forward Eq. (13):

EtpnftDt p

nft a

Dt 1pnft A:32

and the covariance in the numerator ofUDt) of Eq. (A.29) is

CovpnftDt pnft ;p

nft p

nftDt 12a

Dt a2DtVarpnft

1aDt2Varpnft : A:33

Hence, the covariance in (A.29) is (the negative of) the variance of expected change

in the non-fundamental component, or VarEtpnftDt p

nft .

The first-order autocorrelation ofDt-period changes in pnft can also be related tothe first-order autocorrelation of stock returns over the period from t to t Dt.The slope in the regression of the return Rt;tDt on RtDt, is



23/25

betaDt CovRt;tDt;RtDt;t

VarRtDt;t : A:34

Assuming that the two fundamental and non-fundamental components are uncor-related (a reasonable assumption given that the correlation coefficient between these

two components in the data is less than 0.004), the autocovariance of returns in

beta(Dt) is same as autocovariance of the changes in the non-fundamental compo-

nents pnft . Thus

betaDt CovpnftDt p

nft ;p

nft p

nftDt

VarRtDt;t A:35

UDtVarpnftDt pnft

VarRtDt;t A:36

VarEtp

nftDt p

nft

VarRtDt;t : A:37

Therefore, (the negative of) the slope in the regression of the return fromt to t Dt,Rt, t Dt, on the return from time t Dtto t;RtDt;tcan be used to approximate thefraction of the variance in stock returns explained by expected changes in the non-

fundamental component.

If the price consists only of a fundamental component (has a zero non-fundamen-tal component), then the betas should vanish for all Dt. On the other hand, if the

price only consists of a non-fundamental component, the beta(Dt) approaches

0:50 as Dt increases. However, if the price has both components, then the funda-mental component should push the beta toward zero,while the non-fundamental

component pushes the beta toward 0:50 for long horizons. As the return horizonDtincreases, the variance of the non-fundamental component approaches 2Var(pnft ),

and the fundamental component variance eventually dominates. Consequently, if the

non-fundamental component is non-zero, then the slope of regression ofRt;tDt on

RtDt;tshould form a U-shaped pattern, starting around zero for short horizons, be-

coming more negative as Dtincreases, and then moving back toward zero as the fun-damental component begins to dominate at long horizons.

Since the fraction of the variance of stock returns explained by expected changes

in the non-fundamental component is approximately the negative of beta(Dt), the

variance fraction should form an inverse U-shaped pattern. Moreover, the ratio of

the variance of expected change to the variance of actual changes in the non-funda-

mental component should fluctuate around 0.50.

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Do US Stock Prices Deviate From Their Fundamental Values - Some New Evidence

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