Includes bibliographical references (p. 10.

Dokko, Yoon

English

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Regression tests of the present value model and speculative bubbles

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Faculty Working Paper 91-0150330 STXB3851991:150 COPY 2Regression Tests of the Present ValueModel and Speculative Bubbles1 2 1991Yoon DokkoDepartment of FinanceBureau of Economic and Business ResearchCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignBEBRFACULTY WORKING PAPER NO. 91-0150College of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignJune 1991Regression Tests of the Present ValueModel and Speculative BubblesYoon Dokko, Assistant ProfessorDepartment of Finance1 would like to thank Robert Shiller for the data base and James Davis and Louis Scottfor comments. I am responsible for any remaining errors.AbstractThis paper develops a regression test of the present value model, whichholds regardless of whether rational bubbles are present in stock prices. Wetest whether the forecast error of the stock price with respect to the ex postrational price is consistent with the present value model. The statisticalresults cannot reject the null hypothesis.Digitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/regressiontestso91150dokkIntroductionThe objective of this paper is two-fold. First, this paper demonstrates thatthe failure of some regression tests to accept the present value model cannotbe attributed to the existence of rational speculative bubbles. Second, thepaper proposes an alternative regression test of the present value modelthat is valid regardless of whether rational bubbles are present in stockprices.One may suppose that the stock price can be decomposed into a fun-damental value and a rational bubble and that the fundamental value isdetermined by the present value model:Pt = Ft + Btwhere 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, D t+i isthe dividend paid during period t + i (or from time t + i: — 1 through timet + i), k is the discount rate, which is assumed to be constant, and Et is theinvestor's expectations operator conditional upon information available attime t. Flood and Garber (1980) show that when bubbles satisfy the Eulerequation, 1 i.e., when bubbles are rational, we haveBt = YT~k EtBt+l ' ^Following LeRoy and Porter (1981) and Shiller (1981), we define thelrThe Euler equation is the first-order condition for a representative consumer's lifetimeexpected utility maximization."ei post rational" price, P*, asp<- = SoW (3)Since EtP* = Fu we havep; = pt -jft + ^ (4)where r)t is a rational forecast error, in that it is uncorrelated with Pt andBt . In the absence of bubbles (i.e., Bt — 0), we havep; = Pt + m (5)andvar(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 thepresent value model can be attributed to the existence of speculative bub-bles. However, Flood and Hodrick (1986) demonstrate that most variancebounds tests preclude bubbles as an explanation. Others doubt, for variousreasons, the usefulness of variance bounds tests. For example, Marsh andMerton (1986) suggest that since stock prices are non-stationary, imposingan upper bound on stock price volatility is not meaningful for testing thepresent value model.From these debates, an alternative approach for testing the present valuemodel has emerged. Scott (1985) suggests that ordinary least squares (OLS)regression tests of equation (7) are more powerful than variance boundstests:2P; = a + bPt + r,t (7)where trends are removed from the data, and a = and 6 = 1 under thenull hypothesis.Scott suggests that the existence of bubbles could cause his regressiontest 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 ofrational bubbles. Also, application of OLS to equation (7) yields ineffi-cient coefficient estimates because the rational forecast error rj t is seriallycorrelated. To show this, we utilize the definition of P* such thatp* _ Pjt+i + A+i , Q sPt~ T+i ' (8)From equations (1) and (2), we describe the stock price as_£7,(Pt+1 + A+i) ,Q ,LetPm + A+i = £*CP*+i + A+i) + ^+i (10)where \x t is, by definition, a rational forecast error, which is serially uncor-related. Equation (9) becomesPt- m • (11)Subtracting equation (11) from equation (8) yields*-* = TTk^ ~ Pl+ ') + TTk"^ (12)2 See alsoShiller (1990).In the absence of bubbles, rjt = P* — Pt , and rjt is the present value of futureforecast errors (i.e., HSi <i+k)< )• OLS estimation of equation (7) thus yieldsinefficient 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 thatDt = (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 howdividends are paid, one may not test the present value model in the form ofequation (13) using the observed stock price-dividend relation. For exam-ple, as discussed in Dokko, if we incorporate the informational effect of thedividend and dividend smoothing behavior into the present value model, wesee that the relationships of the current stock price with the lagged stockprice and the current dividend are determined by several forces, such as thecapitalization of an unexpected dividend, the dividend smoothing policy,and the discount rate.We may summarize the problems with testing the present value modelin the following ways: First, the variance bounds test may not be useful ifthe stock price is non-stationary. Second, OLS estimation of equation (7)is inefficient since the rational forecast error rf t is serially correlated underthe null hypothesis. This inefficiency will be exacerbated by measurementerrors in the estimated P*. Finally, without a theory of dividends, it wouldbe difficult to be sure that the observed stock price-dividend relation iscapable of testing the present value model, as opposed to inadvertently4estimating another relation.The rest of the paper is organized as follows: Section I proves that theScott-type regression tests of the present value model cannot detect theexistence of rational bubbles. Section II presents an alternative regressiontest. 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 BubblesSince P* is not observed, researchers usually assume that the observedmarket price at the end of the sample period, Pj, is the same as the ex postperfect foresight price at that time, Pf. The estimated ex post rationalprice P* is obtained in the following way:P* = PT = p* + BT - rjTPt-i = Y^(^t + DT ) = Pr_, + ^(Br - riT )Using P* and Pt , the least squares estimate b in regression (7), with thenull hypothesis of b = 1, isb = cov(p;,Pt)/var(Pt )= C°V (^ + (l+\)T-t (BT - *»•)' Pt) /Var^^= covU -Bt ++ \ Br,Pt) /varfP,)- (15)The last equality holds because cov(r)t , Pt ) = cov(tjt, Pt) = 0. Since bubblesevolve over time, the bubble at time T (Bt) is correlated with the bubbleat time t (Bt , t < T) and thus with Pt . In other words, we can express BtasBT = (1 + fc)#r-i + eT= (l + A:) 2BT_ 2 + (l + A:)eT_ 1 + eT= (\ + k)T- tBt + (l + k)T- t- le t+ l + -- + eT (16)where e t is, by definition, white noise and uncorrelated with informationavailable at time t. Since cov(e t+t , Pt ) = for all i > 1, it follows thatb = 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 rationalbubbles.II. An Alternative Regression Test of the Present Value ModelA. MethodologyThe empirical analysis will be directed to examining whether the presentvalue relation in equation (12) holds. The test of the present value model inthe form of equation (12) has several advantages over the test of the presentvalue model in the form of P* = Pt + r] t . First, as Flood and Hodrick (1986)6warned, 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, sincefit is serially uncorrelated under the null hypothesis, OLS estimation ofequation (12) is efficient. Third, even though P* is unobserved, estimationof equation (12) is robust with respect to measurement errors in P* . Thiscan 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) yieldsp'- p'= lhiP^- p^ ) + TTk^- (19)We use equation (19) to test the present value model.B. The Data BaseThe data base is the same as that used in Campbell and Shiller (1987). Ptis 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. D t is the dividend foreach year t from 1872 through 1986, deflated by the average price deflatorfor the corresponding year. 3 The assumed discount rate is 8.35%, which3If Pt is the stock price for January 1987, the corresponding D t is the dividend paidduring the year of 1986.is the average annual real rate of return on common stocks for the sampleperiod.C. Statistical ResultsIf P* — Pt is stationary, the testing equation isp;-pt = A + A(p;+1 -pt+1 ) (20)where /?'s are regression coefficients to be estimated, and the null hypothesisis 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 resultsP;-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-orderautocorrelation of the regression residual, and F is the ^-statistic with 24One may suggest that the absolute magnitude of //< grows over time as the stock priceand the dividend grow over time.8and 113 degrees of freedom for testing the joint hypothesis of (3 = andft = 0.923. (0.923 = 1/1.0835)The statistical results appear to be consistent with the present valuemodel with a constant discount rate. We reject none of the following: (i)fa = 0; (ii) ft = 0.923; (iii) the joint hypotheses of fa = and ft = 0.923.The regression residual, which is the estimate of the present value of theforecast error (f¥j), is not serially correlated. This is also consistent witha rational forecast error. The latter result should be interpreted cautiously.As argued by Summers (1986), tests for autocorrelations of single periodreturns could have extremely low power against the inefficient market hy-pothesis.III. Concluding RemarksThis paper has shown that the regression of P* onto Pt cannot detectspeculative bubbles and has proposed an alternative test of the presentvalue model. The paper has tested whether the present value relation holdsfor the rational forecast error rj t . This approach has several advantages overthe regression of P* onto Pt . The statistical results could not reject thepresent value model.ReferencesCampbell, John Y., and Robert J. Shiller, "Cointegration and Tests ofPresent 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 ValueModels," Review of Economics and Statistics vol. 71, no. 3 (August1989), 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 PriceLevel 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 (September1986), pp. 831-42.LeRoy, Stephen, and Richard Porter, "The Present Value Relation: TestsBased on Implied Variance Bounds," Econometrica vol. 49, no. 3 (May1981), pp. 555-74.Marsh, Terry A., and Robert C. Merton, "Dividend Variability and VarianceBounds Tests for the Rationality of Stock Market Prices," AmericanEconomic Review vol. 76, no. 3 (June 1986), pp. 483-98.Scott, Louis 0., "The Present Value Model of Stock Prices: Regression Testsand 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 bySubsequent 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 RatioModel," 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.591-601.10

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