Regression tests of the present value model and speculative bubbles

Faculty Working Paper 91-0150

330 STX

B385

1991:150 COPY 2

Regression Tests of the Present Value
Model and Speculative Bubbles

1 2 1991

Yoon Dokko

Department of Finance

Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 91-0150

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

June 1991

Regression Tests of the Present Value
Model and Speculative Bubbles

Yoon Dokko, Assistant Professor
Department of Finance

1 would like to thank Robert Shiller for the data base and James Davis and Louis Scott
for comments. I am responsible for any remaining errors.

Abstract

This paper develops a regression test of the present value model, which
holds regardless of whether rational bubbles are present in stock prices. We
test whether the forecast error of the stock price with respect to the ex post
rational price is consistent with the present value model. The statistical
results cannot reject the null hypothesis.

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Introduction

The objective of this paper is two-fold. First, this paper demonstrates that
the failure of some regression tests to accept the present value model cannot
be attributed to the existence of rational speculative bubbles. Second, the
paper proposes an alternative regression test of the present value model
that is valid regardless of whether rational bubbles are present in stock
prices.

One may suppose that the stock price can be decomposed into a fun-
damental value and a rational bubble and that the fundamental value is
determined by the present value model:

Pt = Ft + Bt

where Pt is the stock price at the beginning of period t + 1 (or at time tf),
Ft is the fundamental value at time t, Bt is the bubble at time t, Dt+i is
the dividend paid during period t + i (or from time t + i : — 1 through time
t + i), k is the discount rate, which is assumed to be constant, and Et is the
investor's expectations operator conditional upon information available at
time t. Flood and Garber (1980) show that when bubbles satisfy the Euler
equation,1 i.e., when bubbles are rational, we have

Bt = YT~kEtBt+l' ^

Following LeRoy and Porter (1981) and Shiller (1981), we define the

lrThe Euler equation is the first-order condition for a representative consumer's lifetime
expected utility maximization.

"ei post rational" price, P*, as

p<- = SoW (3)

Since EtP* = Fu we have

p; = pt-jft + ^ (4)

where r)t is a rational forecast error, in that it is uncorrelated with Pt and
Bt. In the absence of bubbles (i.e., Bt — 0), we have

p; = Pt + m (5)

and

var(P;) > var(Pt). (6)

The failure of the statistical results of various studies to accept inequal-
ity condition (6) has ignited heated debates among researchers. Some re-
searchers suggest that the failure of variance bounds tests to accept the
present value model can be attributed to the existence of speculative bub-
bles. However, Flood and Hodrick (1986) demonstrate that most variance
bounds tests preclude bubbles as an explanation. Others doubt, for various
reasons, the usefulness of variance bounds tests. For example, Marsh and
Merton (1986) suggest that since stock prices are non-stationary, imposing
an upper bound on stock price volatility is not meaningful for testing the
present value model.

From these debates, an alternative approach for testing the present value
model has emerged. Scott (1985) suggests that ordinary least squares (OLS)

regression tests of equation (7) are more powerful than variance bounds
tests:2

P; = a + bPt + r,t (7)

where trends are removed from the data, and a = 0 and 6 = 1 under the
null hypothesis.

Scott suggests that the existence of bubbles could cause his regression
test to reject the present value model. As will be shown later, Scott's fail-
ure to accept the null hypothesis cannot be attributed to the existence of
rational bubbles. Also, application of OLS to equation (7) yields ineffi-
cient coefficient estimates because the rational forecast error rjt is serially
correlated. To show this, we utilize the definition of P* such that

p* _ Pjt+i + A+i ,Qs

Pt ~ T+i ' (8)

From equations (1) and (2), we describe the stock price as

_ £7,(Pt+1 + A+i) ,Q,

Let

Pm + A+i = £*CP*+i + A+i) + ^+i (10)

where \xt is, by definition, a rational forecast error, which is serially uncor-
related. Equation (9) becomes

Pt - m • (11)

Subtracting equation (11) from equation (8) yields

*-* = TTk^ ~ Pl+') + TTk"^ (12)

2See alsoShiller (1990).

In the absence of bubbles, rjt = P* — Pt, and rjt is the present value of future
forecast errors (i.e., HSi <i+k)< )• OLS estimation of equation (7) thus yields
inefficient coefficient estimates under the null hypothesis.

Applying generalized least squares (GLS) or maximum likelihood es-
timation (MLE) to equation (5) is equivalent to estimating (note that
Dt = (1 + k)P;_x - P;) equation (13):

Pt = (l + A:)Pt-i-A + /it. (13)

Chow (1989) finds that the data are not consistent with equation (13).
However, Dokko (1991) suggests that without a theory of why and how
dividends are paid, one may not test the present value model in the form of
equation (13) using the observed stock price-dividend relation. For exam-
ple, as discussed in Dokko, if we incorporate the informational effect of the
dividend and dividend smoothing behavior into the present value model, we
see that the relationships of the current stock price with the lagged stock
price and the current dividend are determined by several forces, such as the
capitalization of an unexpected dividend, the dividend smoothing policy,
and the discount rate.

We may summarize the problems with testing the present value model
in the following ways: First, the variance bounds test may not be useful if
the stock price is non-stationary. Second, OLS estimation of equation (7)
is inefficient since the rational forecast error rft is serially correlated under
the null hypothesis. This inefficiency will be exacerbated by measurement
errors in the estimated P*. Finally, without a theory of dividends, it would
be difficult to be sure that the observed stock price-dividend relation is
capable of testing the present value model, as opposed to inadvertently

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estimating another relation.

The rest of the paper is organized as follows: Section I proves that the
Scott-type regression tests of the present value model cannot detect the
existence of rational bubbles. Section II presents an alternative regression
test. Our approach does not require a theory of dividends, and the statisti-
cal results are valid regardless of whether speculative bubbles are present.
We find that the data appear to be consistent with the present value model.
Section III contains a brief conclusion.

I. On Testing for Rational Bubbles

Since P* is not observed, researchers usually assume that the observed
market price at the end of the sample period, Pj, is the same as the ex post
perfect foresight price at that time, Pf. The estimated ex post rational
price P* is obtained in the following way:

P* = PT = p* + BT- rjT
Pt-i = Y^(^t + DT) = Pr_, + ^(Br - riT)

Using P* and Pt, the least squares estimate b in regression (7), with the
null hypothesis of b = 1, is

b = cov(p;,Pt)/var(Pt)

= C°V (^ + (l+\)T-t(BT - *»•)' Pt) /Var^^

= cov

U -Bt+ + \ Br, Pt) /varfP,)- (15)

The last equality holds because cov(r)t, Pt) = cov(tjt, Pt) = 0. Since bubbles
evolve over time, the bubble at time T (Bt) is correlated with the bubble
at time t (Bt, t < T) and thus with Pt. In other words, we can express Bt
as

BT = (1 + fc)#r-i + eT

= (l + A:)2BT_2 + (l + A:)eT_1 + eT

= (\ + k)T-tBt + (l + k)T-t-let+l + -- + eT (16)

where et is, by definition, white noise and uncorrelated with information
available at time t. Since cov(et+t, Pt) = 0 for all i > 1, it follows that

b = 1 (17)

This proves that failure to accept the null hypothesis of 6 = 1 in regression
(7), using P* and Pt, cannot be attributed to the presence of rational
bubbles.

II. An Alternative Regression Test of the Present Value Model

A. Methodology

The empirical analysis will be directed to examining whether the present
value relation in equation (12) holds. The test of the present value model in
the form of equation (12) has several advantages over the test of the present
value model in the form of P* = Pt + r]t. First, as Flood and Hodrick (1986)

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warned, model specification for testing rational bubbles, using ex post data,
is not an easy task. It is desirable to develop a model, such as equation (12),
which holds irrespective of the presence of rational bubbles. Second, since
fit is serially uncorrelated under the null hypothesis, OLS estimation of
equation (12) is efficient. Third, even though P* is unobserved, estimation
of equation (12) is robust with respect to measurement errors in P* . This
can be seen as follows:

= rb(i7+» - p^ + IT***1 + (i +fcr-'(i?r " ^

(18-a)

P;+1-Pt+1 = P;+1 - Pt+i + (1 + klTHt+1) (BT - iyr). (18-b)

Substituting Ptm+l - Pt+1 - {1+k)}-it+x)(BT - r]T) for Pt*+l - Pt+l in equation
(18-a) yields

p'-p' = lhiP^-p^) + TTk^- (19)

We use equation (19) to test the present value model.

B. The Data Base

The data base is the same as that used in Campbell and Shiller (1987). Pt
is the stock price for January of each year t -f- 1 from 1872 through 1987,
deflated by the price deflator for January of that year. Dt is the dividend for
each year t from 1872 through 1986, deflated by the average price deflator
for the corresponding year.3 The assumed discount rate is 8.35%, which

3If Pt is the stock price for January 1987, the corresponding Dt is the dividend paid
during the year of 1986.

is the average annual real rate of return on common stocks for the sample
period.

C. Statistical Results

If P* — Pt is stationary, the testing equation is

p;-pt = A + A(p;+1-pt+1) (20)

where /?'s are regression coefficients to be estimated, and the null hypothesis
is that ft, = 6 and ft = 1/(1 + fc).

If P* — P* is non-stationary,4 the empirical model analog of equation
(19) is (following Scott)

For the 1872 through 1986 period, we obtain OLS results

P;-Pt = -0.007 + 0.908 (Pt+1-Pt+i)
(0.007) (0.039)

Adj. R2 = 0.82, p = 0.12, F = 0.55 (22)

(0.09)

5l_^ = _o.624 + 0.869 (P*+1 Pt+1)
Dt (0.404) (0.046) V A /

Adj. R2 = 0.76, /? = 0.06, F = 1.26 (23)

(0.09)

where the numbers in parentheses are standard errors, p is the first-order
autocorrelation of the regression residual, and F is the ^-statistic with 2

4One may suggest that the absolute magnitude of //< grows over time as the stock price
and the dividend grow over time.

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and 113 degrees of freedom for testing the joint hypothesis of (30 = 0 and
ft = 0.923. (0.923 = 1/1.0835)

The statistical results appear to be consistent with the present value
model with a constant discount rate. We reject none of the following: (i)
fa = 0; (ii) ft = 0.923; (iii) the joint hypotheses of fa = 0 and ft = 0.923.
The regression residual, which is the estimate of the present value of the
forecast error (f¥j), is not serially correlated. This is also consistent with
a rational forecast error. The latter result should be interpreted cautiously.
As argued by Summers (1986), tests for autocorrelations of single period
returns could have extremely low power against the inefficient market hy-
pothesis.

III. Concluding Remarks

This paper has shown that the regression of P* onto Pt cannot detect
speculative bubbles and has proposed an alternative test of the present
value model. The paper has tested whether the present value relation holds
for the rational forecast error rjt. This approach has several advantages over
the regression of P* onto Pt. The statistical results could not reject the
present value model.

References

Campbell, John Y., and Robert J. Shiller, "Cointegration and Tests of
Present Value Models," Journal of Political Economy vol. 95, no. 5 (Oc-
tober 1987), pp. 1062-88.

Chow, Gregory C, "Rational versus Adaptive Expectations in Present Value
Models," Review of Economics and Statistics vol. 71, no. 3 (August
1989), pp. 376-84.

Dokko, Yoon, "The Present Value Model and the Market Discount Rate,"
working paper, University of Illinois, 1991.

Flood, Robert P., and Peter M. Garber, "Market Fundamentals Versus Price
Level Bubbles: The First Test," Journal of Political Economy vol. 88 no.
3 (August 1980), pp. 745-70.

Flood, Robert P., and Robert J. Hodrick, "Asset Price Volatility, Bubbles,
and Process Switching," Journal of Finance vol. 41, no. 4 (September
1986), pp. 831-42.

LeRoy, Stephen, and Richard Porter, "The Present Value Relation: Tests
Based on Implied Variance Bounds," Econometrica vol. 49, no. 3 (May
1981), pp. 555-74.

Marsh, Terry A., and Robert C. Merton, "Dividend Variability and Variance
Bounds Tests for the Rationality of Stock Market Prices," American
Economic Review vol. 76, no. 3 (June 1986), pp. 483-98.

Scott, Louis 0., "The Present Value Model of Stock Prices: Regression Tests
and Monte Carlo Results," Review of Economics and Statistics vol. 67,
no. 4 (November 1985), pp. 599-605.

Shiller, Robert J., "Do Stock Prices Move Too Much to be Justified by
Subsequent Changes in Dividends?" American Economic Review vol.
71, no. 3 (June 1981), pp. 421-35.

Shiller, Robert J., "A Scott-Type Regression Test of the Dividend Ratio
Model," Review of Economics and Statistics vol. 72, no. 2 (May 1990),
pp. 356-61.

Summers, Lawrence H., "Does the Stock Market Rationally Reflect Fun-
damental Values?" Journal of Finance vol. 41, no. 3 (July 1986), pp.
591-601.

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