Samuelson Dictum by Schiller

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SAMUELSONS DICTUM AND THE STOCK MARKET

BY

JEEMAN JUNG and ROBERT J. SHILLER

COWLES FOUNDATION PAPER NO. 1183

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

YALE UNIVERSITY

Box 208281

New Haven, Connecticut 06520-8281

2006

http://cowles.econ.yale.edu/


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SAMUELSONS DICTUM AND THE STOCK MARKET

JEEMAN JUNG and ROBERT J. SHILLER*

Samuelson has offered the dictum that the stock market is micro efficient butmacro inefficient. That is, the efficient markets hypothesis works much better forindividual stocks than it does for the aggregate stock market. In this article, we reviewa strand of evidence in recent literature that supports Samuelsons dictum and presentone simple test, based on a regression and a simple scatter diagram, that vividly illus-trates the truth in Samuelsons dictum for the U.S. stock market data since 1926.(JEL G14)

I. INTRODUCTION

Paul A. Samuelson has argued that onewould expect that the efficient markets hy-pothesis should work better for individual

stocks than for the stock market as a whole:Modern markets show considerable micro

efficiency (for the reason that the minority whospot aberrations from micro efficiency can makemoney from those occurrences and, in doing so,they tend to wipe out any persistent inefficiencies).In no contradiction to the previous sentence, I hadhypothesized considerable macro inefficiency, inthe sense of long waves in the time series of aggre-gate indexes of security prices below and abovevarious definitions of fundamental values.1

We shall see in this article that there is nowsubstantial evidence supporting Samuelsonsdictum where market inefficiency is definedas predictability of future (excess) returns.

We will also present a new test and scatter di-agram that clarifies the truth in this dictum.Samuelsons dictum is plausible if there is

much more information available to the mar-ket about future changes in fundamentals (the

dividends or earnings or cash flows) of indi-vidual firms than there is about future changesin the fundamentals of the aggregate stockmarket. Individual firms activities are highlydiverse: Some have breakthrough discoveriesor important new patents; others are in declin-ing industries or have fundamental structuralproblems. Hence some firms at some timesmay be well known to the market to havea highly positive expected growth of funda-mental value, whereas different firms or thesame firms at different times may be wellknown to the market to have a highly negativeexpected growth of fundamental value. Ifthere is enough variation in information thatthe market has about future fundamentalgrowth of individual firms, then these varia-tions might then be big enough to swampout the effect on price of time variation inother factors, such as speculative booms andbusts, making the simple efficient marketsmodel work fairly well as an approximationfor individual firms.

In contrast, there would seem not to be thesame kind of clarity in the market aboutchanges in the aggregate dividend or earningsflow for the stock market of a country.Changes in these flows for the aggregate stockare less dramatic than for individual firms,because the aggregate averages out the indi-vidual stories of the firms and the reasonsfor changes in the aggregate are more subtle

and harder for the investing public to under-stand, having to do with national economic

*We are indebted to John Y. Campbell, Erik Hjal-marsson, Paul A. Samuelson, and Tuomo Vuolteenahofor comments. Ana Fostel provided research assistance.

Jung: Associate Professor, Division of Economics andInternationalTrade,SangmyungUniversity,Seoul110-743 Korea. Phone 822-2287-5189, Fax 822-3965-702,E-mail [email protected]

Shiller: Professor, Cowles Foundation for Research inEconomics, International Center for Finance,Yale

University. 30 Hillhouse Ave., New Haven, CT06520. Phone 1-203-432-3708, Fax 1-203-432-6167,E-mail [email protected]

1. This quote is from a private letter from PaulSamuelson to John Campbell and Robert Shiller. Thequote appears and is discussed in Shiller (2001, p. 243).Samuelson has been making this point for many years;it is also made in Samuelson (1998).

ABBREVIATIONS

CRSP: Center for Research on Security Prices

Economic Inquiry

(ISSN 0095-2583) doi:10.1093/ei/cbi015

Vol. 43, No. 2, April 2005, 221228 Western Economic Association International

221


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growth, stabilizing monetary policy, and thelike. If changes in aggregate dividends areharder to predict, we might then expect thatfactors other than information about funda-mentals, factors such as stock market boomsand busts, would swamp out the effect of in-formation about future dividends in determin-ing price and make the simple efficientmarkets model a bad approximation for theaggregate stock market.

II. EVIDENCE IN THE LITERATURE FOR

SAMUELSONS DICTUM

There is now substantial evidence in thepublished literature for Samuelsons dictum.One of us (Shiller 1981) presented evidencethat was interpreted as finding evidence ofexcess volatility in the stock market relativeto the efficient markets model using U.S. data18711979 (see also LeRoy and Porter 1981;Campbell 1991). The same methods did notfind much evidence of inefficiency in otherprincipal components of industry stock mar-ket indexes over the same time interval (Shiller1989, ch. 11). In this sense, the aggregate mar-ket was found to be inefficient and the indus-try deviations from the aggregate market werenot found to be inefficient.

To deal with criticisms that these earlyefforts to detect excess volatility made inap-propriate assumptions about stationarityaround trend, Campbell and Shiller (1988a,b) derived a cointegrated model for divi-dend, price, and earnings, a model that wasthen recast as a vector autoregression in thelog dividend-price ratio, the change in log div-idends, and the log earnings-price ratio. Ap-plying this model to the aggregate U.S. stockmarket 18711987, they concluded that onlyabout 7% of the variance of annual stock mar-ket returns can be justified in terms of new in-formation about future dividends.

Vuolteenaho(2002) extended the CampbellShiller framework in such a way that it could beused to produce a decomposition of unex-

pected excess returns of a firm into a compo-nent due to information about future cashflows of the firm (what we will call the efficientmarket component) and a component due toinformation about future returns (what we willcall the inefficient market component ofreturns). Vuolteenaho studied 36,791 firm-years of data for the United States 195496.He found, based on a vector-autoregressive

analysis, that about 75% of thevariance of (un-expected) annual firm stock excess returns canbe justified in terms of the efficient marketcomponent.

Vuolteenaho also estimated the standarddeviation of the atypical discount (the dif-ference between the log stock price and thelog value that is justified by fundamentals ashe measures them) to be about 25%. Vuoltee-naho concludes that even though he foundthat most of the variance of returns can be jus-tified in terms of fundamentals, the inefficientcomponent of stock price variation that gener-ates predictable movements in future returnsstill has an economically significant impacton firm-level stock prices (p. 246). Samuel-sons dictum asserts that individual-firm stockprice variations are dominated by genuine in-formation about future cash flows of the firm,but they are not perfect indicators of thesecash flows either.

Cohen et al. (2001) used a method similarto that in Vuolteenaho (2002) to derive a de-composition of the cross-portfolio varianceof the log ratio of book value to market valueinto three components: a component thatcould be justified in terms of informationabout future cash flow, a component relatedto the persistence of the value spread, andwhat we might call an inefficiency compo-nent that generates predictable futurereturns.2 They also used a longer sample pe-riod, an international data set, and someimprovements in method. Their conclusionswith U.S. data 193797 and 192,661 firm-years were that 80% of the cross-sectional var-iance in the log ratio of book value to marketvalue can be justified by the first few compo-nents, only 20% by the inefficiency componentthat explains 15-year returns. Their conclu-sion with international data on 22 countries198298 and 27,913 firm-years was that 82%of the cross-sectional variance of book-to-market values was explained by the firsttwo components, and only 18% by the ineffi-ciency component that explains five-year

returns.

2. Cohen et al. (2001) assembled firms into portfoliosand looked at cross-portfolio variance rather than cross-firm variance. They did this because their (logarithmic)model does not allow zero or negative book values, butindividual firms sometimes have these. In this article,we would have a similar problem with zero dividends,for which a log isnot defined, but we do not use a logarith-mic model.

222 ECONOMIC INQUIRY


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The CampbellShiller results, the Vuoltee-naho results, and the Cohen et al. results to-gether make a case for Samuelsons dictum:Campbell and Shiller found that little of thevariability of aggregate stock market returnsare explained by fundamentals, whereasCohen and colleagues found that much ofthe variability of individual stock returns orratios of book value to market value areexplained by fundamentals.

III. A FURTHER DEMONSTRATION OF

SAMUELSONS DICTUM

We now show a simple demonstration ofevidence for Samuelsons dictum withoutany of the paraphernalia of vector autoregres-sions, which introduce assumptions about the

stochastic properties of the variables underanalysis, and we will do this by following indi-vidual firms for a long time, avoiding makingany assumptions that the outlook for one firmcan be inferred by data on other firms.3

We will show evidence that most of the var-iability of the individual-firm stock price rela-tive to one measure of fundamental value, thedividend, can be justified in terms of informa-tion about future changes in dividends. Wewill do this by a simple regression of futuredividend changes on the dividend price ratio,proving that this ratio substantially correctlyforecasts the future growth rate of dividends

and hence that variations of price relative todividends are largely justified in terms of mar-ket efficiency

Assuming a constantdiscount rate but vary-ing growth rate of real dividends, the dividend-price ratio Dt/Pt can be derived from thesimpleexpected present value relation with discountrater as

Dt=Pt rEtgDt ; whereg

Dt

XN

k1

DDtk=Pt=1 rk1:1

Ptis the real (inflation corrected) stock price atthe end of yeart,Dtis the real dividend duringthe year t, DDt DtDt1; r is the discount

factor used in the present value formula forstock prices, and Et denotes expectationconditional on information at timet.4

Note that in the equationgDt ; representinga dividend growth rate, is expressed as thesum of discounted amounts of future dividendchanges from a $1 investment at time t.5 Inother words, the growth rates are computedrelative to price Prather than D, and this isimportant because with individual firms thereare in fact some zero dividends, and so growthrates of dividends themselves could not be cal-culated. The equation can be viewed as a dy-namic counterpart of the Gordon model, D/P rg, wheregis the constant expected div-idend growth rate.

We could in theory evaluate this model, af-ter turning the efficient markets equationaround to Etg

D

t

rDt=Pt; by regressing,with time-series data,gDt onto a constant andthe dividend price ratioDt/Pt, and testing thenull hypothesis that the coefficient ofDt/Pt isminus one. Such a test of the efficient marketshypothesis would be recommended by its sim-plicity and immediacy. There is, however, thepractical difficulty that the summationextends to infinity and so the right-hand sidecan never be computed with finite data. Mostof the studies cited solved this problem by in-ferring the summation using an autoregressivemodel. A simpler and more direct way, with-out adding the additional assumptions im-plicit in the vector autoregressive model, isto approximate the right-hand side and runa regression of the approximated right-handside onto the dividend price ratio. This wasdone in Campbell and Shiller (1998, 2001)for aggregate stock market indexes. Campbelland Shiller (2001) regressed 10-year logdividend growth rates ln(Dt10/Pt) ontoln(Dt/Pt) with annual Standard & Poor com-posite stock price data using the long time-se-ries data of 1871 to 2000. The coefficient of

3. Cohen et al. (2001) also use an approach that doesnot rely on vector autoregressive returns, based on trun-cated log-linearized present values extending as far intothe future as 15 years. The lumping of firms into portfoliosmeans that Cohen and collegues were not following indi-vidual firms through time as we do.

4. Note that efficient markets theoryimplies(1) even iffirms repurchase shares in lieu of paying as much divi-

dends, the share repurchase has the effect of raising sub-sequent per-share dividends.5. Campbell and Shiller (1988a,b) used a log-

approximation of the dividend-price model as follows:

logDt=Pt Et logDt=Pt*where logDt=Pt*

XN

j1

qj1DlogDtj C:

The formula is closely analogous to equation (1) in thisarticle.

JUNG & SHILLER: SAMUELSONS DICTUM 223


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ln(Dt/Pt) turned out to be positive, to have thewrong sign. The result was interpreted as indi-cating that in the entire history of the U.S.stock market, the dividend-price ratio hasnever predicted dividend growth in accor-dance with the simple efficient markets theory.

IV. RUNNING THE REGRESSION WITH

INDIVIDUAL STOCK DATA

A fundamental problem with testing thismodel with individual stock data is, as wehave noted, that although the model concernsgrowth rates of dividends over a time rangeextending from decade to decade, there arenot many firms that survive for many decades.In fact, when we did a search on the Center forResearch on Security Prices (CRSP) tape, we

found that there were only 49 firms that ap-pear on the tape continuously without missinginformation during the period of 1926 to2001.6 Because the number of surviving firmsis so small, there is a risk that they are atypi-cal, not representative of all firms. This riskmust be borne in mind in evaluating ourresults, but we believe that looking at thisthe universe of surviving U.S. firms on theCRSP tape still offers some substantialinsights, at least as a case study. Note thatthe mere fact of survival would be expectedif anything to put an upward bias on the aver-age return on the stocks. It would have no ob-

vious implication for either the time-series orcross-sectional ability of the dividend-price ra-tio to predict future changes in dividends.

Using monthly data from the CRSP tape,we create the series of annual dividends, Dt,by summing up 12 monthly dividends fromJanuary to December of the year; the price

Ptis for the end of the year.7 We exclude from

the series nonordinary dividends due to liqui-dation, acquisition, reorganization, rights of-fering, and stock splits. All the dividends andstock prices are adjusted by the proper priceadjustment factors obtained from the CRSPtape and then are expressed in real terms usingthe Consumer Price Index. As a proxy for thefuture dividend growthgDt we useg

Dt ;the sum-

mation truncated afterKyears:

^gDt XK

k1

DDtk=Pt=1 rk1;2

and we set r equal to 0.064, which is the an-nual average return over all firms and datesin the sample.8

To confirm statistical significance, we re-

gress ^gDt onto a constant and Dt/Pt with the49 individual firm data in three different ways:(A) separately for each of the 49 firms (49regressions each with 76-Kobservations), (B)pooled over all firms with a dummy for eachfirm (one stacked regression with 49 [76-K]observations), and (C) for the equallyweighted portfolio composed of the 49 firms(one regression with 76-K observations).Table 1 shows the three results for K 10,15, 20, and 25, while for the pooled regression,K 75 is also shown. When appropriate, t-statistics were computed using a HansenHodrick (1980) procedure to correct these sta-

tistics for the effects of serial correlation in theerror term due to the overlapping 10-, 15-, 20-,or 25-year intervals with annual data. For thestacked regressions (B) forK 10, 15, 20, and25, the HansenHodrick procedure was mod-ified to take account as well of contem-poraneous correlation of errors across firms.9

If there were no problem of survivorshipbias and if the truncation of our infinite sumfor ^gDt were not a problem, then we would ex-pect, assuming the simple efficient markets

6. When Poterba and Summers (1988) did a similarsearch of the CRSP tape, they found 82 survival firms dur-ing the 192685 period. The smaller number here appar-ently reflects the continuing disappearance of firmsthrough time. Though the number of firms is small, we ob-serve that they span a wide variety of industries. Amongthe 49 firms, there are 31 manufacturing firms, 5 utility

companies, 5 wholesale and retails, 3 financial firms, 4mines and oil companies and 1 telecommunicationcompany.

7. The results are invariant to the starting month forthe calculation of annual dividends. We also work on thesame estimation using the data of survival firms afterWorld War II.There are125 firms that have existed duringthe 19492001 period without any missing information onstock prices and dividends. The results of the regressionson these samples are basically similar to those reported inthe article.

8. We avoid the common practice of using the termi-nal price,PtKto infer dividend changes beyondt K(as

Cohen et al. 2001 essentially did)because that would bringus back to using a sort of return variable as the dependentvariable in our regressions; we want our method to havea simple interpretation, here just whether the dividend-price ratio predicts future dividend growth.

9. The variance matrix X of the error term in thestacked regression, for computation of the variancematrixof the coefficients (X#X)

1(X#XX)(XX)1 consists of 49

49 blocks, one for each firm pair. Each block has the usualHansenHodrick form, but we allow for cross-covariancein the off-diagonal blocks.



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model, that the slope in the regressions shouldbe minus one and the intercept be the averagereturn on the market. In fact, the truncationof the infinite sum means that the coefficientmight be something other than minus one.Hence, we merely test here for the negativityof the coefficient of the dividend-price ratio,looking only to see if it is significant in predict-ing future dividend changes in the right direc-tion. Because of survivorship bias, the factthat we are looking only at surviving firmswould appear to put a possible upward biason the intercept, and therefore we do not focuson the intercept here.

Table 1 part A reportsthe summary result ofthe 49 individual regressions. ForK 10, theaverage coefficient and the average t-statisticon Dt/Pt are 0.440 and 2.11, respectively.We find that forK 10, 42 out of the 49 firmshad negative coefficients as predicted by thetheory, thatis, thedividend price ratio doespre-dict future changes in dividends in the right di-rection for the simple efficient markets model,

and 20 of the coefficients are statistically signif-icant at 5% significance level.10 As K is in-creased, the averaget-statistic andr2 decrease.The coefficient ofD/Palways has the negativecoefficient predicted by the simple efficientmarkets model, though far from1.00. Thus,

D/P does seem to forecast future dividendgrowth, although the coefficient is shrunkenfrom minus one toward zero, as one might ex-pect if there is some extraneous noise D/P(caused, say, by investor fads), causing anerrors-in-variables bias in the coefficient.

Table 1 Part B shows the results when theregressions were pooled, so that there are (ex-cept where K 75) many more observations inthe regression than in Part A and hence morepower to the test. In theK 75 case, the lim-iting case with our 76 annual observations, theregression reduces to a simple cross-section re-gression of the 49 firms for t 1926. Sincethere are only 49 observations in theK 75case, the test is not powerful here, and we re-port it only for completeness. For K 10,15, 20, and 25 the t-statistic is highly signifi-cant and negative. AsKis increased, the coef-ficient of the dividend-price ratio decreases,and atK 75, the coefficient is very close toits theoretical value of1.00 (though poorlymeasured because only 1926 D/P are used).

These results provide impressive evidence forthe simple efficient markets model as appliedto individual firm data in the sense that the es-timated coefficients are significantly negative,though usually above minus one.

Table 1 Part C shows the results when theregressions were put together into one regres-sion (by using an equally weighted portfolio)so that we can test the simple efficient markets

TABLE 1

Results for Regressions of Future Dividend Growth on Current Dividend-Price Ratio:

^gDt a bDt=Pt et

Coefficient ofDt/Pt t-Statistic r2

A. Average of 49 separate regressions

i. K 10, n 66 each regression 0.440 2.11 0.182

ii. K 15, n 61 each regression 0.498 1.85 0.167

iii. K 20, n 56 each regression 0.490 1.67 0.173

iv. K 25, n 51 each regression 0.499 1.55 0.162

B. Pooled over all firms

i. K10,n 3,234 0.589 5.91 0.174

ii. K15,n 2,989 0.648 5.69 0.217

iii. K20,n 2,744 0.666 4.82 0.216

iv. K25,n 2,499 0.711 4.84 0.149

v. K75,n 49 1.087 1.41 0.041

C. Using the portfolio of the 49 firms

i. K 10, n 66 0.336 1.79 0.084

ii. K

15, n

61 0.322 1.52 0.063

iii. K 20, n 56 0.463 1.84 0.101

iv. K 25, n 51 0.697 2.40 0.175

10. Those results, not reported in the table to conservethe space, are available on request.



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for the aggregate stock market over the sam-ple period. The sharply reduced range of thedividend-price ratio for the aggregate stockmarket would itself be a strong reason to sus-pect that the simple efficient markers modelwill perform less well for it when comparedwith individual firms.

Over this entire range, there is a distinctnegative slope to the curve as the efficient mar-kets theory would predict: Firms with lowerdividend-price ratios did indeed have highersubsequent dividend growth, offering someevidence for micro-efficiency. Plots for K10, 15, and 20 look very similar to Figure 1.

One should be cautious in interpreting thisdiagram, however. Note that by constructionall points lie on or above a line from (0,0) witha slope of minus one, reflecting the simple factthat dividends cannot go below zero. The effi-cient markets model and our assumption thatdividends beyondKyears into the future can-not be forecasted instead says that the scattershould cluster around a line from (0,rc) witha slope of minus one, a line that lies above theother line and is parallel with it, wherec is themean of the truncated portion of the presentvalue formula, as well as any possible survi-vorship bias. But our results are not guaran-teed by construction. Indeed when thescatter of points for the aggregated firms (cor-responding to the third regressions, Part C inTable 1) is plotted, it lies above this line butdoes not have a negative slope.

This line from (0,0) with a slope of minusone is easily spotted visually as the lower enve-lope of the scatter of points. Any observationofDt/Ptthat is followed by a dramatic drop individends (to approximately zero forKyears)will lie approximately on this line. Some of themost visible points on the scatter representsuch firms. For example, the extreme rightoutlier on the scatter, representing Schlum-berger in 1931, represents nothing more thana situation in which the firm attempted tomaintain its dividend level in spite of rapidlydeclining fortunes. Its stock price fell precipi-

tously after the 1929 crash, convertinga roughly 8% dividend into a 40% dividend,which was cut to zero in 1932 and held at zerofor many years. This extreme case may beregarded as a victory for the efficient marketsmodel, in that it does show that the dividend-price ratio predicts future dividend growth,though not the usual case we think of whenwe consider market efficiency. It is plain from

the fact that the points are so dense around thelower envelope line that much of the fit derivesfrom firms whose dividends dropped sharply.

Another simple story is that of firms thatpay zero dividends. Note that all firm-yearpairs with zero dividends can be seen arrayednext to the vertical axis and that the dividendgrowth for these firms tends to be higher thanfor the firms with nonzero dividends, as thesimple efficient markets model would predict.Firms with zero dividends showed higher divi-dend growth as measured bygDt :the mean g

Dt

for the zero-dividend observations is 0.149,which is greater thanr 0.064, possibly reflect-ing the selection bias for surviving firms noted.The fact that these points along the vertical axiscluster above 0.064 might also be considereda sort of approximate victory for market effi-ciency. Also note that even if we deleted thesefirms, there still is a pronounced negative slopeto the scatter. The predictive ability of the sim-ple efficientmarketsmodel is notjustdue to thephenomenon of zero dividends.

Even if we delete all observations of zero div-idends, andlook at dividend price ratiosless thanthe discount rater, that is, less than 0.064, thentheslope of theregression line forK 25 changesto 0.479, not much closer to zero. This meansthat there are also observations of a low but nonzero dividend-price ratio successfully predictingabove normal dividend growth.

Regression diagnostics following Belsleyet al. (1980) revealed that no particularly influ-ential observations were responsible for theresults in the pooled regressions.

VI. SUMMARY

There is now substantial evidence in theliterature, based on vector autoregressivemethodology, in favor of Samuelsons dictum.In this article, we augmented this evidence us-ing data on long-surviving firms and a very di-rect approach to examining market efficiency.With these data on the universe of U.S. indi-vidual firms on the CRSP tape with continu-

ous data since 1926, Samuelsons dictumappears to have had some validity. Over thein-terval of U.S. history since 1926, individual-firm dividend-price ratios have had somesignificant predictive power for subsequentgrowth rates in real dividends; this is evidenceof micro-efficiency. A look at a scatter plot ofthe data confirms that this result is not exclu-sively due to firms paying zero dividends.



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When the firms are aggregated into an in-dex, the dividend-price ratio gets the wrongsign in the regressions and is usually insignifi-cant. If anything, high aggregate dividend-price ratios predict high aggregate dividendgrowth, and so there is no evidence ofmacro-efficiency.11

One might interpret these results as sayingthat the faith that has in the past beenexpressed for the simple efficient marketsmodel for the aggregate stock market is the re-sult of a faulty extrapolation to the aggregateof a model that did indeed have some value forindividual firms.

REFERENCES

Belsley, D. A.,E. Kuh, and R. E. Welsch. Regression Diag-nostics: Identifying Influential Data and Sources ofCollinearity.New York: Wiley, 1980.

Campbell, J. Y. A Variance Decomposition for StockReturns. Economic Journal, 101, 1991, 15779.

Campbell, J. Y., and R. J. Shiller. The Dividend-PriceRatio and Expectations of Future Dividends andDiscount Factors. Review of Financial Studies, 1,1988a, 195228.

. Stock Prices, Earnings, and Expected Divi-dends. Journal of Finance, 43, 1988b, 66176.

. Valuation Ratios and the Long-Run Stock Mar-ket Outlook. Journal of Portfolio Management,Winter 1998, 1126.

. Valuation Ratios and the Long-Run Stock Mar-ket Outlook: An Update. NBER Working Paper

No. 8221, 2001. (Forthcoming inAdvances in Behav-ioral Finance II, edited by Richard Thaler. NewYork: Sage Foundation, 2003.)

Campbell, J. Y., and M. Yogo. Efficient Tests of StockReturn Predictability. Unpublished paper, HarvardUniversity, 2002.

Cohen, R., C. Polk, and T. Vuolteenaho. The ValueSpread. National Bureau of Economic ResearchWorking Paper No. W8242. 2001.

. DoesRisk or Mispricing Explain the Cross Sec-tion of Stock Prices? Unpublished paper, 2002.

. The Price Is (Almost) Right. National Bureauof Economic Research Working Paper No.10131,2003.

Elliott, G., and J. H. Stock. Inference in Time Series Re-gression When the Order of Integration of a Regres-sor Is Unknown. Econometric Theory, 10, 1994,672700.

Hansen, L. P., and R. J. Hodrick. Forward ExchangeRates as Optimal Predictors of Future Spot Rates:An Econometric Analysis. Journal of PoliticalEconomy, 88, 1980, 82953.

Jung, J. Efficiency and Volatility of Stock Markets:Mean Reversion Detected by Covariance Ratios.Unpublished manuscript, 2002.

LeRoy, S., and R. Porter. The Present Value Relation:Tests Based on Variance Bounds. Econometrica,49, 1981, 55574.

Poterba, J. M., and L. H. Summers. Mean Reversion inStock Prices: Evidence and Implications.Journal ofFinancial Economics, 22,1988, 2659.

Samuelson, P. A. Summing upon Business Cycles: Open-ing Address, inBeyond Shocks: What Causes Busi-ness Cycles, edited by Jeffrey C. Fuhrer and ScottSchuh. Boston: Federal Reserve Bank of Boston,1998.

Shiller, R. J. Do Stock Prices Move Too Much to Be Jus-tified by Subsequent Changes in Dividends? Amer-ican Economic Review, 71, 1981, 42136.

. Market Volatility. Cambridge, MA: MIT Press,1989.

.Irrational Exuberance,2nd ed. New York: Broad-

way Books, 2001.Vuolteenaho, T. What Drives Firm-Level Stock

Returns? Journal of Finance, 57, 2002, 23364.

11. Jung (2002) finds using variance and covarianceratio tests that individual stock returns show quite differ-ent mean reversion characteristics than do the returns onthe portfolio of them.


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Samuelson Dictum by Schiller