外文资料--WHAT DRIVES FIRM-LEVEL STOCK RETURNS

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1、First draft: September 10, 1999This draft: October 16, 2000WHAT DRIVES FIRM-LEVEL STOCK RETURNS?Tuomo Vuolteenaho*Harvard UniversityDepartment of EconomicsLittauer Center 3121875 Cambridge StreetCambridge, MA 02138, USA * Correspondence: t_vuolteenahoharvard.edu. I would like to thank Tom Berglund,

2、Jonathan Berk,John Campbell, John Cochrane, Randy Cohen, Tim Doede, Eugene Fama, J. B. Heaton, John Heaton, PeterHecht, Anita Kelly, Matti Keloharju, Robert Kimmel, Owen Lamont, Aaron Lebovitz, Robert C. Merton,Tobias Moskowitz, Christopher Polk, David Robinson, Katherine Schipper, Erik Stafford, Pe

3、r Strmberg,Richard Thaler, and Pietro Veronesi for helpful discussions. I also received useful comments from theseminar participants at the Anderson School, Columbia Business School, EFA2000 Meeting, Fuqua Schoolof Business, Haas School of Business, Harvard Business School, Harvard University Depart

4、ment ofEconomics, Johnson Graduate School of Management, Kellogg Graduate School of Management, LondonBusiness School, MIT Sloan School of Management, Simon Graduate School of Business Administration,and Wharton School. I am grateful for the financial support of the Foundation for Economic Education

5、, theEmil Aaltonen Foundation, the Oscar Mayer Foundation, and the John Leusner Fund at the University ofChicago Graduate School of Business.WHAT DRIVES FIRM-LEVEL STOCK RETURNS?AbstractI use a simple vector autoregressive (VAR) model to decompose a typical firms stock return into twocomponents: cha

6、nges in cash-flow expectations (i.e., cash-flow news) and changes in discount rates (i.e.,expected-return news). The VAR model yields three main results. First, firm-level stock returns are mainlydriven by cash-flow news. For a typical stock, the variance of cash-flow news is more than twice that of

7、expected-return news. Second, expected-return-news series are highly correlated across firms, while cash-flow news can largely be diversified away in aggregate portfolios. Third, shocks to expected returns andcash flows are, perhaps surprisingly, positively correlated for a typical small stock.Keywo

8、rds: present value, volatility, capital markets, expected-return variation, return on equityJEL classification codes: G120, G140Astec Industries common stock, carrying a ticker symbol ASTE and trading on NASDAQ, returnedan enthusiastic 203% from February 1998 to February 1999. On February 22, 1999,

9、the Tennessee-basedmanufacturer of road building equipment announced record revenues of $363.9 million and net income of$24.4 million. The firms return on equity (ROE) improved from 7.2% for fiscal year 1997 to 18.4% for1998. Shortly after the earnings announcement, the stock traded at a price-earni

10、ngs ratio of 21 and price-to-book ratio of 3.9. The analysts were positively surprised by the operating results: Astecs quarterly earningsannouncements beat the analysts consensus estimates every time during 1998. The four analysts coveringthe stock forecasted continuing earnings growth for fiscal y

11、ear 1999 and recommended the stock as either astrong or moderate buy. Dr. J. Don Brock, Chairman and CEO, commented on the results: “1998 was anexcellent year for us in many respects.” 1By definition, a firms stock returns are driven by shocks to expected cash flows (i.e., cash-flownews) and/or shoc

12、ks to discount rates (i.e., expected-return news). In the above case, Astecs cash-flownews undoubtedly made the firm more valuable. However, it is unclear whether the magnitude of cash-flownews equaled the magnitude of Astecs stock return. It is possible that expected-return news wasresponsible for

13、a fraction of the 203% return. In this paper, I estimate how important these two sources ofstock-return variation are for a typical firm. In addition, I measure whether positive cash-flow news istypically associated with an increase or decrease in expected returns.While there is a substantial body o

14、f research measuring the relative importance of cash-flow andexpected-return news for aggregate portfolio returns (e.g., Campbell 1991 and Campbell and Ammer1993), there is virtually no evidence on the relative importance of these components at the firm level. Theabsence of firm-level empirical work

15、 is surprising, considering that theoretical research on the present-valueformula (e.g., LeRoy and Porter 1981) is often motivated at the firm level. Moreover, interpreting realizedfirm-level stock-price movements is difficult without such evidence.I estimate a simple, homogeneous vector autoregress

16、ion (VAR) from a large panel of firm-level data(1954-1996 CRSP-COMPUSTAT intersection). The VAR model and Campbells (1991) return-decomposition framework enable me to decompose the firm-level stock return into cash-flow and expected-return news and to see what the predictability results in the previ

17、ous literature imply about realized stockreturns. Specifically, my VAR model is designed to capture the following major empirically establishedreturn-predictability results. Historically, small firms have earned higher average stock returns than large2firms (Banzs 1981 “size effect”). Past long-term

18、 losers have outperformed past long-term winners (“long-term reversal,” DeBondt and Thaler 1985), while past short-term winners have outperformed past short-term losers (“momentum,” Jegadeesh and Titman 1993). High book-to-market-equity firms have earnedhigher average stock returns than low book-to-

19、market-equity firms (“book-to-market anomaly,” Rosenberg,Reid, and Lanstein 1985). Controlling for other characteristics, firms with higher profitability have earnedhigher average stock returns (Haugen and Baker 1996). Also, high-leverage firms have historicallyoutperformed low-leverage firms (Bhand

20、aris 1988 “leverage effect”). I avoid explicit and implicitmodeling of corporate dividend policy by using an accounting-based present-value model (Vuolteenaho2000) and excluding any dividend-based variables from the VAR model.2My first objective is to measure the importance of cash-flow and expected

21、-return news as drivers offirm-level stock returns. For excess returns (log stock return less log risk-free return), the variance ofexpected-return news (0.0645 or standard deviation 22%) is approximately one half of the variance of cash-flow news (0.1002 or standard deviation 32%). For market-adjus

22、ted returns (log return less cross-sectionalaverage log return), expected-return news is somewhat less important: The variance of expected-return news(0.0161 or standard deviation 13%) is one fifth that of cash-flow-news (0.0801 or standard deviation 28%).Thus, information about future cash flows is

23、 the main factor driving firm-level stock returns, but changes indiscount rates are also important.It is also interesting to partition the variance-decomposition results by firm size, i.e., the marketcapitalization of equity. Typically, small-stock returns are perceived to be more volatile than larg

24、e-stockreturns. However, it is not known whether this higher volatility stems from more volatile cash-flow orexpected-return news. For firms in the largest size decile, market-adjusted returns are driven by cash-flownews (variance 0.0319 or standard deviation 18%), while expected-return news (varian

25、ce 0.0040 or standarddeviation 6%) is less significant. For firms in the smallest decile, both cash-flow and expected-return-newsseries are much more variable than for large firms. The small-firm cash-flow-news variance (0.1673 orstandard deviation 41%) is four times the expected-return-news varianc

26、e (0.0410 or standard deviation20%). These cross-sectional differences in the news variances may have a natural explanation: It is possiblethat large firms are merely portfolios of small firms and, in such portfolios, a large part of the idiosyncraticnews variance is diversified away.3A second impor

27、tant objective is to reconcile the above firm-level-return-variance decomposition withthe aggregate-return-variance decompositions in the existing literature. Fama and Schwert (1977), Famaand French (1989), Jegadeesh (1990), Hodrick (1992), Campbell and Ammer (1993), Kothari and Shanken(1997), Ponti

28、ff and Schall (1998), Campbell and Shiller (1998), Lamont (1998), and others have shown thatthe expected returns on aggregate portfolios are time-varying. Campbell and Shiller (1988a, b), Campbell(1991), and Campbell and Ammer (1993) use the aggregate-return-predictability results to decompose stock

29、prices and the aggregate-stock-return variance. Generally, these studies find that, in the post-war aggregatedata, expected-return news dominates cash-flow news. Considering these aggregate results, the suggestionthat firm-level stock returns are mainly driven by cash-flow news may seem surprising a

30、t first.I show that the cash-flow-news component that is important at the firm level is largely diversifiedaway in aggregate portfolios. Using a bottom-up approach, I first calculate the two news series forindividual firms and then aggregate them into expected-return and cash-flow-news series for th

31、e equal-weight index portfolio. While the variance of cash-flow news is twice that of expected-return news forfirm-level excess returns, for an equal-weight portfolio the cash-flow-news variance is only three quarters ofthe expected-return-news variance. It appears that while cash-flow information i

32、s largely firm specific,expected-return information is predominantly driven by systematic, macroeconomic components. Hence,my firm-level results are consistent with the earlier aggregate results.My third objective is to measure the correlation between expected-return and cash-flow news for atypical

33、stock. Perhaps surprisingly, I find that cash-flow news is positively correlated with shocks toexpected returns for a typical stock. Although my main tests use annual returns, the correlation pointestimates obtained from longer-horizon (e.g., five-year) news series are also positive. The positive ne

34、wscorrelation appears to be largest for the smallest stocks, declining (nearly) monotonically in size.If cash-flow and expected-return news are positively correlated, why do many researchers (e.g.,DeBondt and Thaler 1985 and Chopra, Lakonishok, and Ritter 1992) find evidence that, at first, appearst

35、o be implying negative correlation between expected-return and cash-flow news? These studies examinethe univariate autocovariance function of returns. I argue that return autocovariances alone are insufficientto identify the contemporaneous correlation between the two news terms. Hence, the results

36、in above-cited“overreaction” studies should be interpreted as evidence showing that expected returns are time-varying, but4not necessarily as evidence implying that news about cash-flows coincides with an “excessive” pricereaction.The rest of the paper is organized as follows. Section I outlines the

37、 stock-return-decompositionmethodology. Section II describes the sample. Section III presents and discusses the results. Section IVconcludes.I. Decomposing the stock returnI use an approximate present-value formula to decompose the stock return into cash-flow andexpected-return news. The formula exp

38、resses the current stock price as a discounted sum of current bookequity, future stock returns, and future accounting returns. Unexpected stock returns must, therefore, beexplained by changes in expectations about future stock returns and/or future accounting returns.Typically, the previous literatu

39、re (e.g., studies by Campbell 1991 and Campbell and Ammer 1993)uses the dividend-growth model of Campbell and Shiller (1988a) to decompose aggregate returns. In thispaper, I use a slightly different, accounting-based present-value formula derived by Vuolteenaho (2000).Because this formula uses ROE (

40、earnings over book equity) instead of dividend growth as the basic cash-flow fundamental, it may be more intuitively appealing than the Campbell-Shiller dividend-growth formula,especially at the firm level. In addition, this version of the present value relation provides a natural basis forvariable

41、selection in my empirical firm-level analysis. Whether one chooses to think about infinite-horizoncash-flow fundamentals in terms of dividend growth or ROE is a matter of taste, however.Three main assumptions are made in order to derive the ROE-based version of the approximatepresent-value model. Fi

42、rst, book equity, Bt, dividend, Dt, and market equity, Mt, are assumed to be strictlypositive. Second, the difference of log book equity, bt, and log market equity, mt, and the difference of logdividend, td, and log book equity, bt, are assumed to be stationary, even though the series individually h

43、avean integrated component. Third, earnings (denoted by tX), dividends, and book equity must satisfy thefamiliar clean-surplus identity:ttttDXBB+=1(1) book equity this year equals book equity last year plus earnings less dividends. Armed with theseassumptions, Vuolteenaho (2000) derives a model for

44、the log book-to-market ratio (denoted by ):=+=+=0011)(jjtjtjjjtjttferk,(2)5where ROE is denoted by et )/1log(1+=ttBX and the excess log stock return by rt )1log(ttFR +=- ft.tR is the simple excess stock return, Ft is the interest rate, ft is log one plus the interest rate, and k is aconstant plus th

45、e approximation error. As long as some dividends are paid, the discount coefficientsatisfies1bin the following regression:ttcftwNbar+=,(15)Conversely, the underreaction case corresponds to 1b) and$115, it is reasonable to call this behavior overreaction.Substituting the definition of return as a sum

46、 of cash-flow news, expected-return news, and one-period expected return:)()1 () 1(1,1,1tttttcftrttcftrttttcftrtcfttwrEwNbaNwNbaNrEwNbaNNrE+=+=+=+(16)(Note that the one-period expected return is by definition independent of both the cash-flow and expected-return news.) The overreaction hypothesis ca

47、n, therefore, be restated in terms of the correlation between theexpected-return and cash-flow news: The stock market overreacts if expected-return news and cash-flownews are negatively correlated. Conversely, the market underreacts to cash-flow news if the two newsseries are positively correlated.T

48、able V and Figure 4 examine the regression coefficient as a function of the firm size. For a typicallarge stock in decile ten, the estimated correlation between the news terms is close to zero (2% correlationwith 20 %-point standard error), and the estimated regression coefficient of returns on cash

49、-flow news isclose to one (0.99 with 0.07 standard error). For a typical small stock in decile one, the news correlation isstrongly positive 58% with 9 %-point standard error, and the estimated regression coefficient is significantlybelow one (0.72 with 0.07 standard error). Furthermore, the estimat

50、ed correlation is (nearly) monotonicfunction of the size-decile assignment. Thus, under the above-stated restrictive assumptions, the datasuggest small stocks underreact and large stocks react correctly to cash-flow news on average.The above discussion raises a more general methodological point abou

51、t testing over- andunderreaction theories: How can my results be reconciled with the findings of DeBondt and Thaler (1985)and Chopra, Lakonishok, and Ritter (1992) who conclude that the stock market overreacts in general andthat the overreaction is particularly strong for the smallest stocks? A typi

52、cal overreaction study onlyexamines the univariate autocovariance function of returns. Unfortunately, as noted by Campbell (1991) ina different context, a univariate time-series approach cannot unambiguously estimate both the variance ofexpected-return news and covariance between expected-return and

53、 cash-flow news.For 0j,20),cov(),cov(),cov(),cov(),cov(1,1,11,1,1jtjttcfjtjttrjtjttjtcfjtrjtjttcftrttjttrENrENrErENNrENNrErr+=+=(17)As one can see from (17), it is possible that positive cash-flow news implies high future expected returns,yet returns are negatively autocorrelated. In other words, re

54、turns may be negatively autocorrelated even ifthe market underreacts to relevant cash-flow news. Even if the last term of (17) is positive (implyingunderreaction to cash-flow news), if expected returns are highly variable for reasons unrelated to over- orunderreaction, then the negative second term

55、can overwhelm the last term, yielding a negative sum.Concluding overreaction to relevant cash-flow news (as I define it) from negative long-horizonautocorrelations is erroneous, because the long-horizon negative autocorrelation may be induced by anytype of time-variation in expected returns, not onl

56、y by the type consistent with the overreaction hypothesis.E.2 Positive news correlation: an efficient-markets explanationIn an efficient market, uncertainty about the risks of projects can create an underreaction-like patternin cash-flow and expected-return news.8 Assume for simplicity that the prod

57、uct market is competitive andevery new investment projects net present value (NPV) is zero. Consider a firm that announces it hasstarted a new investment project. Because all projects are zero NPV, the announcement does not affect thestock price or cause an unexpected stock return. Although no value

58、 is created by the project, the expectedreturns and expected cash flows of the firm may be affected. If the firm unexpectedly announces a high-riskproject, expected returns on the firms stock increase. For this high-risk project to have zero NPV, it alsomust have high level of cash-flows. Via this m

59、echanism, news that a firm takes a zero-NPV, high-riskproject necessarily implies positive cash-flow news. Hence, firms taking zero-NPV projects with varyinglevels of risk can generate the positive news-correlation pattern observed in the data.Extensions of this simple rational story can also offer

60、an explanation for the size pattern in the firm-level news correlations. It is possible that a single project is a larger fraction of total assets for small firmsthan for large firms. Therefore, it is plausible that the project uncertainty induces more positive correlationto small-stock news series

61、than to large-stock news series. Furthermore, allowing for positive NPV projectsand assuming that recently started projects are riskier than mature projects allows one to match the price-momentum patterns in the data. (For a closely related formal model, see Berk, Green, and Naik 1999.)21F. Addition

62、al robustness checksF.1 Log returns vs. simple returnsThe variation in expected log returns does not necessarily imply variation in expected simple returns.For example, let log returns be conditionally normally distributed. Then, the conditional expected simplereturn can be expressed as)1(log(var)1(

63、log(exp)1 (12111ttttttRRERE+=+ (20)Hence, expected log returns may vary while expected simple returns are constant, if the increase inconditional expected log return is compensated by a decrease in conditional log-return variance.I estimate the VAR model with simple instead log returns, and set the

64、discount coefficient to 1. 9(Details of the results are available on request.) This substitution does not materially alter the results. Theratio of expected-return-news variance to total unexpected-return variance is practically unaltered: 0.2412for log returns and 0.2355 for simple returns. The cor

65、relation between the news terms is slightly reducedfrom 0.40 for log returns to 0.26 for simple returns, both more than two standard errors from zero. Itappears that my main results are not driven by the choice between log and simple returns.F.2 Approximation errorI investigate the size of the cumul

66、ative approximation error t in equation (3) with an additionalVAR specification. I add a fourth variable, market-adjusted clean-surplus ROE )(CSe to the state vector.Addition of this fourth variable enables me to calculate the cumulative approximation error. Table IXreports the covariance matrix of

67、expected-return news, cash-flow news using indirect method (i.e.,computing cash-flow news as a residual and thus including the approximation error in the cash-flow-newsterm), cash-flow news using direct method (i.e., directly calculating the change in the discounted sum ofclean-surplus ROEs and thus

68、 not including the approximation error), and approximation error.Three observations are apparent from Table IX. First, the approximation error is positively correlatedwith expected-return news and negatively correlated with both indirect and direct cash-flow news. Second,at first, it may seem that i

69、ndirect method of calculating cash-flow news as a residual inflates the cash-flow-news variance relative to the method of directly calculating the change in the discounted sum of clean-surplus ROEs does. This, however, is not the case. From Table IX it is clear that direct computation of thecash-flo

70、w news results in a larger cash-flow-news variance than indirect computation. The indirect method22(slightly) understates the variance of the cash-flow news and is thus the more conservative choice relative tomy finding that the cash-flow-news variance dominates the firm-level returns. Third and mos

71、t significantly,the magnitude of the approximation error is so low that the choice between indirect and direct methods isinconsequential to the results. The standard deviation of is 3% (variance 0.0009 with 0.0003 standarderror). Because the approximation error is so small, the decision to include t

72、he approximation error in cash-flows or expected-return news, or to omit it from both terms, is inconsequential to my results.F.3 Horizon issuesThe measurement horizon may be an important factor to the regression coefficient of return on cash-flow news. The existing literature10 suggests that while

73、the high-frequency expected-return and cash-flownews may be positively correlated, securities with long strings of good cash-flow news (typically three tofive years) receive atypically high valuations and, hence, have lower expected returns in the future. Becausethe VAR model completely specifies th

74、e state-variable dynamics, the VAR parameters enable me tocompute the variance decomposition for lower-frequency news.I decompose the q-period discounted return into q-period cash-flow and expected return news. I takethe change in expectations of equation (2) from 1t to 1+ qt and reorganize:1,1, 110

75、,10110+=+=+=+=qtrtqtqqjjtcfjqjjtjtqjjtjpENrEr,(18)where 1, 1 +tqtE denotes the change in expectations from 1t to 1+ qt. Defining the two componentsof q-period return as q-period cash-flow news (qcfN) and expected-return news (qrN ):1,1, 11,10,1,+=+qtrtqtqqqtrqjjtcfjqqtcfpENNN.(19)From equation (19)

76、one can see that q-period cash-flow news is a discounted sum of one-period cash-flownews. The expected-return news term is related to the change in expectations about the log price of returnsq-periods in the future. Appendix 3 contains a derivation, as well as formulas for computing the news termsan

77、d variance decomposition from the VAR-model parameters.Figure 5 shows the variance-decomposition results as a function of the return measurement horizon,calculated from the short VAR in Table II. The expected-return-news variance increases up to five-yearhorizon, and then begins a slow decay. The ca

78、sh-flow-news variance grows nearly linearly, as expectedbased on equation (19). The figure also plots the regression coefficient of q-year returns on q-year cash-23flow news. The regression coefficient begins from 0.8 at one-year horizon, rising to 0.95 at the four-yearhorizon and to the 1.00 at ten

79、-year horizon. Hence, the point estimates suggest positive, not negativecorrelation of 3-to-5-year news.F.4 1975-1996 subsampleKothari, Shanken, and Sloan (1995) argue that book-to-market related return predictability isspuriously induced to the COMPUSTAT database by the process of back-filling the

80、data for successfulfirms. In order to show that this back-filling bias is not driving my results, I re-estimate the model using1975-1996 subsample (22 years, 29123 firm-years). Because back-filling is not a serious problem inCOMPUSTAT data in the later period and because I require an extensive histo

81、ry of data, this subsampleshould be free of look-ahead biases.Using 1975-1996 subperiod and the short VAR specification to decompose market-adjusted returnsconfirms my earlier results. (Full results are available on request.) The expected-return-news standarddeviation is 20% (variance 0.0395 with 0.

82、0138 standard error), while the cash-flow-news standard deviationis 38% (variance 0.1412 with 0.0267 standard error). The ratio of expected-return-news variance to totalunexpected-return variance is approximately 0.47 (with 0.16 standard error). The subperiod estimatesindicate a positive 65% correla

83、tion, over seven standard errors from zero.First, the 1975-1996 subperiod news-correlation results are actually stronger than the 1954-1996results. Second, because the smaller firms are more important in later years of the sample, the subperiodresults are consistent with the conditional variance dec

84、ompositions in Table V and Figure 4.IV. ConclusionsThe present-value formula enables one to divide unexpected stock return into two components:changes in cash-flow expectations (i.e., cash-flow news) and changes in discount rates (i.e., expected-returnnews). As shown by Campbell and Shiller (1988a,

85、b), stock return volatility must originate from volatilecash-flow and/or expected-return news.The relative volatility of these two components is an empirical question. While an extensive body ofresearch investigates the sources of volatility for aggregate portfolios, there is virtually no evidence o

86、n thesubject at the firm level. My objective is to measure the importance of cash-flow and expected-return news24as drivers of firm-level stock returns. In addition, I test hypotheses about the correlation between the newsterms for a typical firm.I estimate a simple, homogeneous vector autoregressio

87、n (VAR), which provides the means fordecomposing firm-level stock returns and return variance. The VAR model yields essentially three results.First, firm-level stock returns are predominantly driven by cash-flow news. For excess log returns, thevariance of expected-return news is approximately one h

88、alf of the variance of cash-flow news. For market-adjusted log returns, the variance of expected-return news is one fifth of the cash-flow-news variance.Thus, expected cash flows are the primary drivers of firm-level stock returns, but discount-rate variation isalso important.Second, cash-flow news

89、which is dominant at the firm level is largely diversified away inaggregate portfolios. Although the variance of cash-flow news is twice that of expected-return news forfirm-level excess returns, for an equal-weight portfolio the cash-flow-news variance is only three quarters ofthe expected-return-n

90、ews variance. This finding suggests that cash-flow information is largely firm-specificand that expected-return information is predominantly driven by a systematic, market-wide component.Third, I find that cash-flow news is positively contemporaneously correlated with shocks to expectedreturns for a

91、 typical stock. Good news about fundamentals is typically accompanied by higher expectedreturns. Furthermore, this correlation appears to be more positive for the smallest stocks than for mid-capstocks, and more positive for mid-cap stocks than large stocks. While the estimated correlation is positi

92、vefor small and mid-cap stocks, it is about zero for the largest stocks.25Appendix 1: Approximate expression for the book-to-market ratioThe linearized expression of the book-to-market ratio is derived from three assumptions. First, bookequity, Bt, dividend, Dt, and market equity, Mt, are assumed to

93、 be strictly positive. Second, the difference oflog book equity, bt, and log market equity, mt, and the difference of log dividend, td , and log book equity,bt, are assumed to be stationary, even though the series individually have an integrated component. Third,earnings (denoted by tX ), dividends,

94、 and book equity must satisfy the familiar clean-surplus identity:ttttDXBB+=1(A1.1) book value equals lagged book value plus earnings less dividends.Define rt )1log(ttFR +- ft as excess log stock return, where tR is the simple excess stock return,Ft is the interest rate and ft is log one plus the in

95、terest rate. Market and accounting returns (i.e., ROE) can beexpressed as()ttttttttttFRMDMMDMfr+=+=+1log1loglog11(A1.2)()ttttttttEBDBBDBe+=+=+1log1loglog11(A1.3)Substituting the log dividend-growth rate, td, the log dividend-price ratio, t, and the logdividend-to-book-equity ratio, tttbd , to the re

96、turn definitions (A1.2) and (A1.3):1) 1)log(exp(+=+tttttdfr.(A1.4)1) 1)log(exp(+=ttttde.(A1.5)The nonlinear functions (A1.4) and (A1.5) can be approximated around and . Specifically, usesome convex combination of the unconditional means of the variables as an expansion point for bothfunctions. Subtr

97、acting rt + ft from et, the approximate expression is111)() 1)log(exp() 1)log(exp(+=tttttttttfre.(A1.6)Above, the log book-to-market ratio is denoted by t. Note that as dividend yields drop, instead of fallingapart the approximation becomes more accurate, while approaches unity.Finally, the one peri

98、od approximation is iterated forward to yield:NtNNjjtjNjjtjNjjtjNjjtjtefr+=+=+=+=+=100001(A1.7)26In equation (A1.7), t denotes the approximation error in equation (A1.6). Because 1, the limitN of equation (A1.7) converges to equation (2) in the text. (The approximation error of equation (2)in the te

99、xt is defined as =+01jjttk.)Which value to pick for is an empirical question. In my data, the three model variables(profitability, stock return, and the lagged book-to-market ratio) and the approximation explain 99.82% ofthe variation in the book-to-market ratios. The 99.82% R2 is achieved with 967.

100、=. Because isestimated accurately and my main results are not sensitive to the -choice, I use this -value in theanalysis and treat it as a constant. Table A1.I also shows similar regressions using US GAAP ROE insteadof clean-surplus ROE (row 2). Using US GAAP ROE will reduce the R2, but does not mat

101、erially affect .Appendix 2: Cross-correlation consistent standard errorsIn many finance applications, the available data set contains perhaps twenty cross-sections, each withthousands of data points. In such cases, incorrectly assuming that the errors are cross-sectionallyuncorrelated can yield stan

102、dard errors that are biased downwards by a factor of five (Fama and French2000b). This bias is due to the fact that error correlations are often systematically related to theexplanatory variables. Fortunately, the statistics literature has proposed two solutions for a similar problemfrequently arisi

103、ng with complex surveys: Rogers (1983, 1993) robust standard errors and the jackknifestandard errors of Shao and Rao (1993). Both methods yield asymptotically correct standard errors for theOLS and WLS estimators under a general cross-correlation structure. Compared to the popular Fama-MacBeth (1973

104、) procedure, these two methods have the practical advantage of giving the standard errors forpooled-OLS/WLS coefficients allowing for, among other things, the use of common time-series variablesin the regressions.A simple exposition of Rogers (1983, 1993) standard errors starts from the familiar for

105、mula for OLSstandard errors. Let X denote the panel of explanatory variables, the covariance matrix of the panel oferrors, and tX and t a single cross-section of explanatory variables and the corresponding errorcovariance matrix. Assuming that the errors are independent across cross-sections allows

106、writing11111)()()()(=XXXXXXXXXXXXTtttt.(A2.1)27Denote regression errors by , and notation for fitted values is modified with a hat. Since)(XXEXXtttttt=, Rogers substitutes estimated errors for true errors to get a variance estimator ofregression coefficients:111)()(=XXXXXXTttttt(A2.2)Under plausible

107、 assumptions, the standard errors are consistent in T. That is, they converge as the timedimension of the panel grows. In order to ensure that the effect of a single cross-section on the coefficientestimates vanishes as more and more cross-sections are included, Rogers assumptions include that the t

108、ime-series of ttttXX is well behaved. The above standard error formula can be interpreted as generalizedWhites standard errors; in the special case of only one observation per cross-section, the standard errors areequivalent to White (1980) heteroscedasticity consistent standard errors.Rogers (1993)

109、 performs a small-scale Monte Carlo study of the finite sample properties. Accordingto his findings, “as long as the largest cluster is 5 percent or less of the total sample, this bias in varianceformula (12) should be negligible.” Since I weigh cross-sections equally and have over forty cross-secti

110、ons, small sample biases are unlikely to affect my regression results.An alternative to Rogers robust standard errors is the jackknife method. Shao and Rao (1993) provethat a delete-cross-section jackknife method produces consistent standard errors for a linear regression evenif the errors are cross

111、-sectionally dependent. Shao and Rao further prove that their method producesconsistent standard errors for well-behaved functions of regression coefficients, too.The jackknife begins by computing the normal OLS or WLS regression coefficient estimate from theentire sample (denote this by “normal est

112、imate”). Then, T new samples of size 1T are recorded omittingone cross-section at a time. Next, the regression coefficient estimates are computed from each one of thenew samples (denote these estimates “delete-cross-section estimates”). Then, 1T pseudo-values arecalculated as T times the normal esti

113、mate less 1T times the delete-cross-section estimate. Finally, thevariance of pseudo-values is the estimated variance of the regression coefficient estimator. For a moredetailed description of the procedure and proofs, see Shao and Rao (1993).Appendix 3: q-period return decompositionI decompose the

114、q-period discounted return into q-period cash-flow and expected-return news. I takethe change in expectations of equation (2) from 1t to 1+ qt and ignore the approximation error:28=+=+011011)()()(0jjtjtqtjjtjtjtqtrEEfeEE(A3.1)=+=+=+=+=qjjtjtqtjjtjtjtqtqtqtqjjtjtqjjtjrEEfeEEEErEr)()()(110122110110K(A

115、3.2)Substituting the definitions of one-period cash-flow news (4) and the log-price of returns (10):1,1, 110,10110+=+=+=+=qtrtqtqqjjtcfjqjjtjtqjjtjpENrEr,(A3.3)where 1, 1 +tqtE denotes the change in expectations from 1t to 1+ qt.The VAR model presented in the text can be used to calculate the news t

116、erms and variancedecompositions. Equations (A3.4) and (A3.5) state the relevant expressions:=+=+qjjtjqqqqtrqjjtjqqtcfuNueN111,111,) 1(A3.4)=+=+qjjqjqqrqcfqjjqjqqqrqjjjqcfeNNNeeN1111) 1(2111) 1(,covvar,) 1() 1(var(A3.5) 1 The data are from PRNewswire, the firms 10-K filings, Zacks, and Market Guide.2

117、 Modeling dividend policy is difficult for two reasons. First, the time-series stability of a firms dividendpolicy is suspect (Fama and French 2000a). For example, changing tax laws and increasing use of (non-dividend-protected) executive stock options may have caused near-permanent shifts in a typi

118、cal firmsdividend policy (Jolls 1998). Second, because one can observe a wide range of dividend policies (froma high, stable pay-out ratio to not paying dividends at all), a homogeneous VAR would not adequatelycapture this persistent cross-sectional heterogeneity.3 Market-adjusted or benchmark-adjus

119、ted returns can be decomposed, as well. Apply equation (3) toindividual firm-level stock returns and market returns separately, and subtract the latter from the former.As a result, the (unexpected) market-adjusted stock return can be decomposed into components due toabove-market expected stock retur

120、ns and ROEs. When the discussion applies only to market-adjustedquantities, I modify the notation by a tilde. For example, tr denotes the market-adjusted log stock return.4 The delisting-return assumptions follow Shumways (1997) results. Shumway tracks a sample of firmswhose delisting returns are mi

121、ssing from CRSP and finds that performance related delistings areassociated with a significant negative return, on average approximately 30 %. This assumption isunimportant for my final results, however.5 At first, it may seem that the above indirect method of calculating cash-flow news as a residua

122、l relies onheavier assumptions than directly calculating the change in the discounted sum of clean-surplus ROEsdoes. This, however, is not necessarily the case. If both the book-to-market ratio and stock return areincluded in the VAR as state variables, the VAR implies a process for clean-surplus RO

123、Es. This impliedmodel results from the identity; any time-series model that maps the lagged book-to-market ratio andlagged stock return to the current book-to-market ratio and current stock return also implies current clean-surplus ROE. Furthermore, if clean-surplus ROE is also included as a state v

124、ariable, the above indirectmethod and a direct calculation of cash-flow news within the VAR will yield the same result (up to theapproximation error, which is minimal and negatively correlated with the directly calculated cash-flow-news series.) Hence, whether the estimated cash-flow-news series is

125、reliable depends mainly on variableselection, not on the method of obtaining the cash-flow news as a residual. The robustness-check sectionF.2 discusses these issues in more detail. 6 Because I constrain the transition matrix to be equal across size groups, my results of higher expected-return-news

126、variance of small firms cannot be driven by variation in the predictive coefficients. Instead,all size-related heterogeneity in my results must by construction come from the error covariance matrix.In principle, the size-related heterogeneity in the variance-decomposition results might be an artifac

127、t ofthe incorrect restriction of equal transition matrices across size groups. To investigate this possibility, Iassume that each firms assignment to a size group is permanent and estimate separate matrices foreach size group. The results obtained under these assumptions actually strengthen my resul

128、ts: The small-stock cash-flow-news and expected-return-news variances, as well as the news-correlation, increaserelative to corresponding large-firm statistics.7 I thank the anonymous referee for suggesting this interpretation.8 I thank Jonathan Berk and Robert C. Merton for clarifying discussions o

129、n this topic.9 Unreported regressions indicate that the approximation is most accurate for simple returns when discountcoefficient is set close to one.10 See the summary of empirical evidence in Daniel, Hirshleifer, and Subrahmanyam (1998).ReferencesBall, R., Brown, P., 1968. An empirical evaluation

130、 of accounting income numbers. Journal of AccountingResearch 6, 159-178.Banz, R. W., 1981. The relationship between return and market value of common stocks. Journal ofFinancial Economics 9, 3-18.Berk, J. B., Green, R. C., Naik, V., 1999. Optimal investment, growth options, and security returns. Jou

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134、-228.Campbell, J. Y., Shiller R. J., 1988b. Stock prices, earnings, and expected dividends. Journal of Finance 43,661-676.Campbell, J. Y., Shiller R. J., 1998. Valuation ratios and long-run stock market outlook. Journal ofPortfolio Management, 11-26.Chopra, N., Lakonishok, J., Ritter, J. R., 1992. M

135、easuring abnormal performance: Do stocks overreact?Journal of Financial Economics 31, 235-268.Cochrane, J. H., 1991. Volatility tests and efficient markets: review essay. Journal of Monetary Economics27, 463-85.Cochrane, J. H., 1992. Explaining the variance of price-dividend ratios. Review of Financ

136、ial Studies 5, 243-280.Daniel, K., Hirshleifer, D., Subrahmanyam, A., 1998. Investor psychology and securities market under- andoverreactions. Journal of Finance 53, 1839-1885.DeBondt, W. F. M., Thaler, R. H., 1985. Does the stock market overreact? Journal of Finance 40, 557-581.Fama, E. F., French,

137、 K. R., 1989. Business conditions and expected returns on stocks and bonds. Journal ofFinancial Economics 25, 23-49.Fama, E. F., French, K. R., 2000a. Disappearing dividends: Changing firm characteristics or increasedreluctance to pay? Working paper, University of Chicago, Graduate School of Busines

138、s.Fama, E. F., French, K. R., 2000b. Forecasting profitability and earnings. Journal of Business 73, 161-75.Fama, E. F., MacBeth, J., 1973. Risk, return, and equilibrium: empirical tests. Journal of PoliticalEconomy 81, 607-636.Fama, E. F., Schwert W. G., 1977. Asset returns and inflation. Journal o

139、f Financial Economics 5, 115-146.Ferson, W. E., Harvey, C. R., 1991. The variation of economic risk premiums. Journal of PoliticalEconomy 99, 385-415.Haugen, R. A., Baker, N. L., 1996. Commonality in the determinants of expected stock returns. Journal ofFinancial Economics 41, 401-439.Hodrick, R. J.

140、, 1992. Dividend yields and expected stock returns: alternative procedures for inference andmeasurement. Review of Financial Studies 5, 357-386.Ikenberry, D., Lakonishok, J., Vermaelen, T., 1995. Market underreaction to open market sharerepurchases. Journal of Financial Economics 39, 181-208.Jegadee

141、sh, N., 1990. Evidence of predictable behavior of security returns. Journal of Finance 45, 881-898.Jegadeesh, N., Titman, S., 1993. Returns to buying winners and selling losers: implications for stock marketefficiency. Journal of Finance 48, 65-91.Jegadeesh, N., Titman, S., 1999. Profitability of mo

142、mentum strategies: An evaluation of alternativeexplanations. Journal of Finance, forthcoming.Jolls, C., 1998. Stock repurchases and incentive compensation. NBER Working Paper 6467.Kothari, S. P., Shanken, J., 1997. Book-to-market, dividend yield, and expected market returns: a time-series analysis.

143、Journal of Financial Economics 44, 169-203.Kothari, S. P., Shanken, J., Sloan, R. G., 1995. Another look at the cross-section of expected stock returns.Journal of Finance 50, 185-224.Lamont, O., 1998. Earnings and expected returns. Journal of Finance 53, 1563-1587.Lakonishok, J., Shleifer, A., Vishn

144、y, R. W., 1994. Contrarian investment, extrapolation, and risk. Journalof Finance 49, 1541-1578.LeRoy, S. F., Porter, R. D., 1981. Stock price volatility: tests based on implied variance bounds.Econometrica 49, 97-113.Loughran, T., Ritter, J. R., 1995. The new issues puzzle. Journal of Finance 50, 2

145、3-51.Michaely, R., Thaler, R., Womack, K., 1995. Price reactions to dividend initiations and omissions. Journalof Finance 50, 573-608.Pontiff, J., Schall, L., 1998. Book-to-market ratios as predictors of market returns. Journal of FinancialEconomics 49, 141-160.Rogers, W. H., 1983. Analyzing complex

146、 survey data. Rand Corporation memorandum. Santa Monica,CA.Rogers, W. H., 1993. Regression standard errors in clustered samples. Stata Technical Bulletin ReprintsSTB-13 STB-18, 88-94.Rosenberg, B., Reid K., Lanstein, R., 1985. Persuasive evidence of market inefficiency. Journal ofPortfolio Managemen

147、t 11, 9-17.Shao, J., Rao, J. N .K., 1993. Jackknife inference for heteroscedastic linear regression models. CanadianJournal of Statistics 21, 377-385.Shumway, T. G., 1997. The delisting bias in CRSP data. Journal of Finance 52, 327-340.Vuolteenaho, T., 2000. Understanding the aggregate book-to-marke

148、t ratio. Working paper, University ofChicago, Graduate School of Business.White, H., 1980. A heteroscedasticity-consistent covariance matrix estimator and a direct test ofheteroscedasticity. Econometrica 48, 817-838. Table I: Descriptive statisticsThe table presents selected descriptive statistics.

149、The first panel (A) shows the descriptivestatistics for basic data and the second panel (B) shows the descriptive statistics for market-adjusted data.The descriptive statistics are estimated from pooled data.Panel A reports means, standard deviations, and percentiles (minimum, 25%, 50%, 75%, andmaxi

150、mum) of excess log return, r , log US GAAP return on equity in excess of risk-free rate, feGAAP,log leverage, lev , and log book-to-market, .Panel B shows the descriptive statistics for market-adjusted log data and standard deviations andpercentiles (minimum, 25%, 50%, 75%, and maximum) of market-ad

151、justed log return, r, market-adjusted log US GAAP return on equity, GAAPe, market-adjusted log leverage, ve l, and market-adjustedlog book-to-market, . The variables are market-adjusted by subtracting the cross-sectional averageeach year.The sample period is 1954-1996 (43 years), consisting of 36791

152、 firm-years.Panel A: Descriptive statistics, basic dataVariableMeanSt. Dev.Min25%-pctMedian75%-pctMaxr0.03270.3223-2.1497-0.12830.04190.21572.0014feGAAP0.00750.2664-2.5755-0.00270.04280.00833.8693lev-0.57940.4987-5.5215-0.7831-0.4684-0.23950.0000-0.20690.6131-3.7796-0.5783-0.16730.20163.9827Panel B:

153、 Descriptive statistics, market-adjusted dataVariableMeanSt. Dev.Min25%-pctMedian75%-pctMaxr00.2867-2.1718-0.14660.00630.16021.9011GAAPe00.2640-2.5210-0.01860.02800.07933.8861ve l00.4944-4.8967-0.19980.10340.32960.645300.5606-3.3295-0.10340.04480.34913.8282 Table II: Short VAR model for market-adjus

154、ted returnsThe table reports the parameter estimates for the short VAR model. The model variables includemarket-adjusted log stock return, r, (the first element of the state vector z ), the market-adjusted logbook-to-market ratio, , (the second element), and market-adjusted log profitability, e, (th

155、e thirdelement).Parameters in the table correspond to the following system:)(,1,tititititiuuEuzz=+=For each parameter, I report three numbers. The first number (bold) is a weighted least squares estimateof the parameter, where observations are weighted such that each cross-section receives an equal

156、weight.The second number (in parentheses) is a robust standard error computed using the Rogers (1983, 1993)method. The third number (in brackets) is a robust jackknife standard error computed using a jackknifemethod outlined by Shao and Rao (1993). Both standard errors are consistent even if the VAR

157、 errors arecorrelated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in total 36791 firm-years.For a more detailed definition of the variables and data, see the data section of the text.Coefficient estimates for the first order market-adjusted VAR(estimate), (s.e.), j.s.

158、e. tr0.1182(0.0224)0.02290.0477(0.0131)0.01360.1464(0.0308)0.03150.0668(0.0040)0.0040-0.0544(0.0027)0.00280.0130(0.0019)0.0020t0.0554(0.0327)0.03360.8953(0.0174)0.01800.0570(0.0494)0.0512-0.0544(0.0027)0.00280.0967(0.0151) 0.01520.0114(0.0021)0.0021te0.1042(0.0110)0.0112-0.0264(0.0061)0.00630.4939(0

159、.0595)0.06200.0130(0.0019)0.00200.0114(0.0021)0.00210.0344(0.0053) 0.0052Table III: Variance decomposition of market-adjusted returnsThe table reports a variance decomposition of market-adjusted returns and other derived statisticscalculated from the two VAR specifications. Both VAR specifications h

160、ave the structure)(,1,tititititiuuEuzz=+=The short VAR model (Panel A) state vector includes market-adjusted log stock return, r, themarket-adjusted log book-to-market ratio, , and market-adjusted log profitability, e.The long VAR model in Panel B includes four lags of market-adjusted log stock retu

161、rn, r, themarket-adjusted log book-to-market ratio, , two lags of market-adjusted log profitability, e, two lagsof market-adjusted leverage, ve l, and the size variable, ezsi. Because of the high dimensionality, thecoefficient matrices are not reported.The table shows the covariance matrix of expect

162、ed-return and cash-flow news. Unexpected stockreturn can be divided into two components, one due to shocks to expected returns and the other due toshocks to expected cash flows. The reported covariance matrix is computed from the particular VARmodel using equations (8) and (9) in the text. For clari

163、ty, this panel also reports the correlationcoefficient between the news terms and the ratio of expected-return-news variance to total unexpected-return variance.For each derived statistic, I report two numbers. The first number (bold) is a derived statisticcomputed using the weighted least-squares e

164、stimates of the parameters. The second number (in brackets)is a robust jackknife standard error computed using a jackknife method outlined by Shao and Rao (1993).The standard errors are consistent even if the VAR errors are correlated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as t

165、he sample, in total 36791 firm-years.For a more detailed definition of the variables and data, see the data section of the text.Panel A: Distribution of the expected-return and cash-flow news from the short VAR(estimate), j.s.e.Cov. matrixNrNcfExpected-returnnews (Nr)0.01610.00690.01470.0074Correlat

166、ion between expected-returnand cash flow news:0.40920.1200Cash-flownews (Ncf)0.01470.00740.08010.0130Ratio of expected-return-news varianceto total unexpected-return variance:0.24120.0984Panel B: Distribution of the expected-return and cash-flow news from the long VAR(estimate), j.s.e.Cov. matrixNrN

167、cfExpected-returnnews (Nr)0.02030.01070.01950.0176Correlation between expected-returnand cash flow news:0.46790.2392Cash-flownews (Ncf)0.01950.01760.08560.0285Ratio of expected-return-news varianceto total unexpected-return variance:0.30290.1501Table IV: Distribution of atypical discounts implied by

168、 the short VAR modelThis table reports the unconditional variance of market-adjusted atypical discounts calculated fromthe short VAR model in Table II. The short VAR model variables include market-adjusted log stockreturn, r, the market-adjusted log book-to-market ratio, , and market-adjusted log pr

169、ofitability, e.The log stock price can be decomposed into a permanent and temporary component, where the temporarycomponent is called atypical discount. The market-adjusted atypical discount is defined as:=+011,jjttjtrrEpThe panel also reports the fraction of fitted market-adjusted atypical discount

170、s that reside outsidelog(1/2), log(2) range.The first number (bold) is a derived statistic computed using the weighted least-squares estimatesof the VAR parameters. The second number (in brackets) is a robust jackknife standard error computedusing a jackknife method outlined by Shao and Rao (1993).

171、The standard errors are consistent even if theVAR errors are correlated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in total 36791 firm-years.For a more detailed definition of the variables and data, see the data section of the text.Distribution of market-adjusted aty

172、pical discounts(estimate), j.s.e.Unconditional variance of market-adjusted atypical discounts(computed from the VAR parameter estimates)0.06730.0278Unconditional standard deviation of market-adjusted atypicaldiscounts (computed from the VAR parameter estimates)0.25950.0518Fraction of the fitted valu

173、es of market-adjusted atypicaldiscounts that are less than log(1/2)0.0077N/AFraction of the fitted values of market-adjusted atypicaldiscounts that are greater than log(2)0.0011N/AFraction of the fitted values of market-adjusted atypicaldiscounts that are outside +/- log(2) interval0.0088N/ATable V:

174、 Local variance decompositionsThe table reports a local variance decomposition of market-adjusted returns for small and largestocks. The VAR specification has the structure)sticscharacteri()sticscharacteri(,1,tititititiuuEuzz=+=The state vector includes market-adjusted log stock return, r, the marke

175、t-adjusted log book-to-market ratio, , and market-adjusted log profitability, e. The characteristic affecting the errorcovariance matrix is firm size (i.e., market value of equity)The table estimates a separate variance decomposition for each of the size deciles. For every cell,I report five statist

176、ics: the variance of expected-return news (var(Nr), the variance of cash-flow news(var(Ncf), the covariance between the news terms (N-Cov), the correlation between the news terms (N-Corr), and the regression coefficient of return on cash-flow news (b).For each statistic, I report two numbers. The fi

177、rst number (bold) is a derived statistic computedusing the weighted least-squares estimates of the parameters. The second number (in brackets) is arobust jackknife standard error computed using a jackknife method outlined by Shao and Rao (1993).The standard errors are consistent even if the VAR erro

178、rs are correlated across firms.Variance decomposition as a function of the firm sizeVar(Nr)Var(Ncf)N-CovN-CorrbSmall0.04100.01660.16730.03030.04760.01910.57510.08760.71540.068320.03260.01330.13940.02460.03590.01530.53200.10390.74260.072930.02100.00920.10130.01790.01970.01010.42640.12510.80590.072440

179、.0160 0.00690.08340.01400.01270.00720.34770.13220.84790.066050.01490.00670.08310.01470.01440.00820.41050.14200.82630.073860.01050.00440.06360.00990.00740.00450.28660.13890.88340.058570.00880.00440.05160.00770.00480.00410.22620.15620.90630.071280.00710.00360.04230.00640.00310.00310.17770.15980.92730.

180、067290.00530.00290.03730.00540.00120.00260.08440.18650.96820.0688Big0.00400.00220.03190.00460.00030.00210.02570.19610.99090.0681 Table VI: Short VAR model for excess returns with aggregate variablesThis table reports the parameter estimates for the short VAR model for excess returns. The modelinclud

181、es both firm specific and aggregate variables. The firm specific variables included are excess logstock return, r , (the first element of the state vector z ), the log book-to-market ratio, , (the secondelement), and excess log profitability, feGAAP, (the third element). The aggregate variables (vec

182、torx ), are cross-sectional median excess log return, cross-sectional median log book-to-market, and cross-sectional median excess log profitability. For a more detailed definition of the variables, see the datasection of the text.Parameters in the table correspond to the following system, where the

183、 left-lower block of isconstrained to a zero matrix:)(,11,tititittittiuuEuxzAxz=+=For each parameter, I report three numbers. The first number (bold) is a weighed least squares estimateof the parameter. The second number (in parentheses) is a robust standard error computed using theRogers (1983, 199

184、3) method. The third number (in brackets) is a robust jackknife standard errorcomputed using a jackknife method outlined by Shao and Rao (1993). Both standard errors areconsistent even if the VAR errors are correlated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in tot

185、al 36791 firm-years.For a more detailed definition of the variables and data, see the data section of the text.Coefficient estimates for the first order market-adjusted VAR(estimate), (s.e.), j.s.e.A tr-0.0527(0.0611)0.07240.1238(0.0231)0.02400.0531(0.0140)0.01500.1333(0.0299)0.0309-0.4311(0.1187)0.

186、13810.2563(0.1111)0.13583.0362(1.1974)1.5273t-0.0068(0.0841)0.09450.0585(0.0287)0.02980.8891(0.0185)0.01940.0501(0.0399)0.04140.2240(0.1478)0.1694-0.1664(0.1161)0.1398-1.6814(1.3478)1.6535ttfe-0.0261(0.0057)0.00650.1051(0.0111)0.0113-0.0252(0.0061)0.00630.5007(0.0590)0.0615-0.1017(0.0214)0.02450.025

187、5(0.0123)0.01400.7328(0.1028)0.1183trmedian0.0093(0.0532)0.0636000-0.2428(0.1547)0.15930.1718(0.1030)0.09921.3922(1.0209)1.2156tmedian-0.0229(0.0572)0.06230000.2019(0.1664)0.17220.8554(0.1108)0.1014-0.3454(1.0981)1.1201ttfemedian0.0060(0.0038)0.00470000.0120(0.0110)0.0113-0.0070(0.0073)0.00930.8207(

188、0.0727)0.0750Coefficient estimates for the first order market-adjusted VAR (continued) trtttfetrmediantmedianttfemediantr0.0883(0.0074)0.0082-0.0772(0.0074)0.00820.0130(0.0019)0.00200.0191(0.0050)0.0056-0.0199(0.0056)0.00620.0006(0.0002)0.0003t-0.0772(0.0074)0.00820.1256(0.0213)0.02160.0114(0.0022)0

189、.0021-0.0205(0.0055)0.00610.0234(0.0067)0.0073-0.0005(0.0003)0.0003ttfe0.0130(0.0019)0.00200.0114(0.0022)0.00210.0346(0.0053)0.00520.0000(0.0003)0.00020.0000(0.0003)0.00020.0001(0.0000)0.0000trmedian0.0191(0.0050)0.0056-0.0205(0.0055)0.00610.0000(0.0003)0.00020.0183(0.0046)0.0051-0.0190(0.0051)0.005

190、70.0006(0.0002)0.0002tmedian-0.0199(0.0056)0.00620.0234(0.0067)0.00730.0000(0.0003)0.0002-0.0190(0.0051)0.00570.0212(0.0059)0.0065-0.0006(0.0002)0.0003ttfemedian0.0006(0.0002)0.0003-0.0005(0.0003)0.00030.0001(0.0000)0.00000.0006(0.0002)0.0002-0.0006(0.0002)0.00030.0001(0.0000)0.0000Table VII: Varian

191、ce decomposition of excess returnsThis table reports the variance decomposition of excess returns and other derived statisticscalculated from two VAR specifications. Both models include both firm specific and aggregate variables.The VAR specifications have the structure shown below, with the left-lo

192、wer block of is constrainedto a zero matrix:)(,11,tititittittiuuEuxzAxz=+=The short VAR model (used to compute the variance decomposition in Panel A) contains thefollowing variables. Firm specific variables included are excess log stock return, r , the log book-to-market ratio, , and excess log prof

193、itability, feGAAP. The aggregate variables (vector x ), are cross-sectional median excess log return, cross-sectional median log book-to-market, and cross-sectionalmedian excess log profitability. Coefficient estimates are reported in Table VI.The long VAR model (used to compute the variance decompo

194、sition in Panel B) contains thefollowing variables. Firm specific variables include three lags of excess log stock return, the log book-to-market ratio, two lags of excess log profitability, two lags of leverage, lev , and my size variable, size .The aggregate variables are cross-sectional median ex

195、cess log return, cross-sectional median log book-to-market, and cross-sectional median excess log profitability. Because of the high dimensionality, thecoefficient matrices are not reported.The table shows the covariance matrix of expected-return and cash-flow news. Unexpected stockreturn can be div

196、ided into two components, one due to shocks to expected returns and the other due toshocks to expected cash flows. The reported covariance matrix is computed from the short VAR modelusing equations (8) and (9) in the text. For clarity, this panel reports the correlation coefficient betweenthe news t

197、erms and the ratio of expected-return-news variance to total-unexpected-return variance.For each statistic, I report two numbers. The first number (bold) is a statistic computed using theweighted least-squares estimates of the parameters. The second number (in brackets) is a robustjackknife standard

198、 error computed using a jackknife method outlined by Shao and Rao (1993). Thestandard errors are consistent even if the VAR errors are correlated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in total 36791 firm-years.For a more detailed definition of the variables and

199、data, see the data section of the text.Panel A: Distribution of the expected-return and cash-flow news from the short VAR(estimate), j.s.e.Cov. matrixNrNcfExpected-returnnews (Nr)0.04650.03110.02920.0213Correlation between expected-returnand cash flow news:0.42790.1643Cash-flownews (Ncf)0.02920.0213

200、0.10020.0247Ratio of expected-return-news varianceto total-unexpected-return variance:0.52640.3416Panel B: Distribution of the expected-return and cash-flow news from the long VAR(estimate), j.s.e.Cov. matrixNrNcfExpected-returnnews (Nr)0.04540.02970.02920.0271Correlation between expected-returnand

201、cash flow news:0.42920.2239Cash-flownews (Ncf)0.02920.02710.10210.0378Ratio of expected-return-news varianceto total-unexpected-return variance:0.50970.3206Table VIII: Implied variance decomposition of the equal-weight index returnsThis table reports variance decompositions for the equal-weight inde

202、x returns implied by two VARspecifications. The VAR specifications have the structure shown below, with the left-lower block of is constrained to a zero matrix:)(,11,tititittittiuuEuxzAxz=+=The short VAR model (used to compute the implied variance decomposition in Panel A) containsthe following vari

203、ables. Firm specific variables included are excess log stock return, r , the log book-to-market ratio, , and excess log profitability, feGAAP. The aggregate variables (vector x ), are cross-sectional median excess log return, cross-sectional median log book-to-market, and cross-sectionalmedian exces

204、s log profitability. Coefficient estimates are reported in Table VI.The long VAR model (Panel B) contains the following variables. Firm specific variables includethree lags of excess log stock return, the log book-to-market ratio, two lags of excess log profitability,two lags of leverage, lev , and

205、my size variable, size . The aggregate variables (vector x ), are cross-sectional median excess log return, cross-sectional median log book-to-market, and cross-sectionalmedian excess log profitability. Because of the high dimensionality, the coefficient matrices are notreported.The table reports th

206、e variance decomposition for equal-weight portfolio of firms. I first computethe news series for each firm using the particular VAR model. I then form two new series by taking thecross-sectional average of expected-return and cash-flow news. The formulas for these two series aregiven in equation (12

207、) in the main text. The reported variance decomposition is based on the samplecovariance matrix of these two series. I also report a diversification factor, i.e., the ratio of firm levelnews variance to aggregate news variance. The formulas (13) and (14) in the main text describe thediversification

208、factor. The first number (bold) is the point estimate. The second number (in brackets) isa robust jackknife standard error computed using a jackknife method outlined by Shao and Rao (1993).The standard errors are consistent even if the VAR errors are correlated across firms.I use the CRSP-COMPUSTAT

209、intersection 1954-1996 as the sample, in total 36791 firm-years.For a more detailed definition of the variables and data, see the data section of the text.Panel A : Variance decomposition for equal-weight portfolio, the short VAR(estimate), j.s.e.Cov. matrixNrNcfExpected-returnnews (Nr)0.02960.02970

210、.01540.0197Diversification factor forexpected-return news:0.63730.2582Cash-flownews (Ncf)0.01540.01970.02320.0201Diversification factor forcash-flow news:0.23090.1475Panel B : Variance decomposition for equal-weight portfolio, the long VAR(estimate), j.s.e.Cov. matrixNrNcfExpected-returnnews (Nr)0.0

211、2780.02730.01440.0178Diversification factor forexpected-return news:0.61320.2869Cash-flownews (Ncf)0.01440.01780.02220.0192Diversification factor forcash-flow news:0.21720.1445Table IX: The size of the cumulative approximation error This table reports the covariance matrix of expected-return news, i

212、ndirectly computed cash-flownews, directly computed cash-flow news, and the approximation error. The covariance matrix iscomputed from a VAR-model parameters:)(,1,tititititiuuEuzz=+=The VAR-model state vector includes (in order) market-adjusted log stock return, r, the market-adjustedlog book-to-mar

213、ket ratio, , market-adjusted log GAAP profitability, GAAPe, and market-adjusted logclean-surplus profitability ,CSe. Define 001 1Le, 1004Le, and 1)(1Ie. Expected-return news can then be conveniently expressed as tiu, indirectcash-flow news as ()tiue,1+, direct cash-flow news as i,tu)(Ie14, and the a

214、pproximation erroras the difference between indirect and direct cash-flow news. The covariance matrix of these four termsis computed using the VAR-error covariance matrix .For each element of the covariance matrix, I report two numbers. The first number (bold) is thepoint estimate. The second number

215、 (in brackets) is a robust jackknife standard error computed using ajackknife method outlined by Shao and Rao (1993). The standard errors are consistent even if the VARerrors are correlated across firms.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in total 36791 firm-years.For a mo

216、re detailed definition of the variables and data, see the data section of the text.Covariance matrix of news terms and the approximation error.(estimate), j.s.e.Cov. matrixNrNcf (indirect)Ncf (direct)Expected-return news(Nr)0.01610.00690.01440.00750.01320.00740.00120.0004Indirectcash-flownews (Ncf)0

217、.01440.00750.07950.01320.08090.0133-0.00140.0004Directcash-flownews (Ncf)0.01320.00740.08090.01330.08320.0134-0.00230.0007Cumulativeapproximationerror ()0.00120.0004-0.00140.0004-0.00230.00070.00090.0003Table A1.I: Return on equity (ROE) and the approximate identityThe table estimates an expansion p

218、oint for the approximate identity relating the book-to-marketratio, stock return, and return on equity. The table shows regression estimates of profitability less stockreturn plus lagged book-to-market on future book-to-market.The table contains two rows. The first row regresses the excess log clean

219、-surplus ROE, tCStfe,less excess log stock return, tr , plus lagged log book-to-market, 1t, on log book-to-market, t. Thesecond row regresses the excess log US GAAP ROE, tGAAPtfe, less excess log stock return, tr , pluslagged log book-to-market, 1t, on log book-to-market, t. tf is log one plus the i

220、nterest rate. Clean-surplus return on equity is calculated using the formula()+=1111logtttttttttcstBDBBMDMFRewhere M denotes market and B book equity, D dividends, and F the interest rate.I use the CRSP-COMPUSTAT intersection 1954-1996 as the sample, in total 36791 firm-years.For a more detailed def

221、inition of the variables and data, see the data section of the text.Accuracy of the approximationY-variableInterceptDiscount coef. X-variableR21)(+tttCStrfe-0.0000.967t99.82%1)(+tttGAAPtrfe-0.0170.987t88.25%Figure 1: Time-series evolution of variablesThe figure graphs the time series of cross-sectio

222、nal percentiles (5%, 25%, 50%, 75%, 95%) for thefollowing variables: excess log return, r , excess log US GAAP return on equity, feGAAP, logleverage, lev , and log book-to-market, . The figure is calculated from the CRSP-COMPUSTATintersection 1954-1996, in total 36791 firm-years. For a more detailed

223、 definition of the variables anddata, see the data section of the text.Figure 2: Cumulative response of returns to shocksThe figure contains two graphs. The first graph shows the cumulative response of returns to atypical 25% unexpected return. The impulse response function is calculated from the sh

224、ort VAR modelin Table II. The typical 25% return is induced by setting the first element of the VAR error vector to0.25. The other elements of the VAR error vector are set to their conditional expectations, conditional onthe first element being equal to 0.25.The second graph shows the cumulative res

225、ponse of returns to a typical 25% cash-flow shock.The typical 25% cash-flow shock is induced by setting the VAR error vector to a constrained maximumlikelihood value, imposing a constraint cash-flow news equals 0.25. The solid horizontal line is set to 25% on both graphs. Dashed lines denote +/- sta

226、ndard-errorbounds, calculated with the jackknife.Figure 3: Histogram of atypical discountsThe figure graphs a histogram of fitted values of market-adjusted atypical discounts, 36791observations. The market-adjusted atypical discount is defined as:=+011,jjttjtrrEpThe fitted values of atypical discoun

227、ts are computed from the short VAR model in Table II.Figure 4: Local variance decompositions for size decilesThe figure graphs the local variance decomposition for the size deciles. The upper-left plot showsthe variance of expected-return news, the upper-right plot the variance of cash-flow news, th

228、e lower-leftplot the regression coefficient of return on cash-flow news, and the lower-right plot the news correlationas a function of the firm size. Dashed lines denote +/- standard-error bounds, calculated with thejackknife. For details, see Table V.Figure 5: Variance decompositions for different

229、return measurement horizonsThe figure graphs variance decompositions for different return measurement horizons. Figures arecomputed from the short VAR model shown in Table II, using the formulas in Appendix 3. For allgraphs, the x-axis signifies the return measurement interval in years. The upper-le

230、ft plot shows thevariance of expected-return news, the upper-right plot the variance of cash-flow news, the lower-left plotthe regression coefficient of return on cash-flow news, and the lower-right plot the news correlation.Dashed lines denote +/- standard-error bounds, calculated with the jackknif

231、e.19501960197019801990200010.500.51Excess log stock returnYear1950196019701980199020000.60.40.200.20.4Excess log profitabilityYear19501960197019801990200021.510.50Log leverageYear195019601970198019902000321012Log booktomarketYear05101520250.20.220.240.260.280.30.32Years from the shockCumulative retu

232、rnsResponse to a typical 25% unexpected return05101520250.180.20.220.240.260.280.3Years from the shockCumulative returnsResponse to a typical 25% cashflow shock1.510.500.511.50100020003000400050006000700080009000FREQUENCY024681000.010.020.030.040.050.06Expectedreturn news variance (var(Nr)024681000.

233、050.10.150.2Cashflow news variance (var(Ncf)02468100.60.70.80.911.1b in r(t) = a + b*Ncf(t) + e(t)02468100.200.20.40.60.8Correlation between Nr and Ncf024681000.010.020.030.040.050.06The return measurement period in yearsExpectedreturn news variance (var(Nr)024681000.20.40.60.8The return measurement period in yearsCashflow news variance (var(Ncf)02468100.70.80.911.1The return measurement period in yearsb in r(t) = a + b*Ncf(t) + e(t)02468100.200.20.40.6The return measurement period in yearsCorrelation between Nr and Ncf