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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