price and return models

EI.SEVIER Journal of Accounting and Economics 20 (1995) 155-192

Price and return models

S.P. Kothari*, Jerold L. Zimmerman William E. Simon Graduate School of Business Administration, UniL'ersity of Rochester, Rochester,

NY 14627, USA

(Received March 1993; final version received January 1995)

Abstract

Return models (returns regressed on scaled earnings variables) are commonly ~eferred to price models (stock price regressed on earnings per share). We provide a framework for choosing between these models. An econom/cally intuitive rationale suggests that models are better specified in that the estimated slope coefficients from price models, but not return models, are unb/ased. Our empirical results confirm that price models" earnings response coefficients are less biased. However, return models have less serious econometric problems than price models. In some research contexts the comh;~ed use of both price and return models may be useful.

Key words: Capital markets; Price-earnings regressions; Earnings response coeffic/ents

JEL classification: MI4; C20

1. Introduction

Researchers in accounting must often choose between return models, in which returns are regressed on a scaled earnings variable, and pr/ce models, in which

* Corresponding author.

We thank Bill Beaver, Andrew Christ/c, N/ck Gonedes, Bob Holthausen, Dave Larcker, John Long, R/chard Sloan, Charles Trzincka, Mike Rozeff, G. William Schwert, Ross Watts, J~nice Wiltett, and especially Ray Ball {the editor) for ~.|pfu~ suggestions; Roger Edelen for excellent research a.~st- ance; and participants at Baruch College CUNY, the Stanford Summer Camp, University of Glasgow, University of Manchester, Michigan State University, University of Pennsylvania, Univer- sity of Rochester, and SUNY at Buffalo for useful comments. Financ/al support from the Bradk:y Policy Research Center at the Simon School University of Rochester and from the John M. O~n Foundation is gratefully acknowledged.

0165-410U95/$09.50 © 1995 Elsev/cr Sc/cnce B.V. All r/ghts reserved SSD/Ol 6541019500399 4

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stock prices are regressed on earnings per share. This paper provides an economic and econometric framework for assessing whether to use a price model, a return model, or both. Within the context of a typical valuation model, we provide an economically intuitive analysis which suggests that the estimated slope coefficient from the price model, but not the return model, is unbiased. We then provide empirical evidence on price and return model specifications. Finally, we draw on previous research to illustrate the contexts in which price and return models are helpful.

Previous research. Several papers discuss the conceptual advantages and disadvantages of price and return models. Gonedes and Dopuch (1974) argue that return models are theoretically superior to price models in the absence of well-developed theories of valuation. Lev and Orison (1982) describe the two approaches as complementary, whereas Landsman and Magliolo (1988) argue that price models dominate return models for certain applications. Christie (1987) concludes that, while return and price models are economically equivalent, return models are econometrically less problematic. Despite the criticism, price r, lodels persist (e.g., Bowen, 1981; Olsen, 1985; Landsman, 1986; Barth, Beaver, and Wolfson, 1990; Barth, 1991; Barth, Beaver, and Landsman, 1992; Harris, Lang, and Moiler, 1994).

l:~ onomic intuition for the return-earnings specification. Both price and return models begin with a standard valuation model in which price is the discounted present value of expected net cash flows. Both models also rely on the hypothesis that current earnings contain information about expected future net cash flows (e.g., Beaver, 1989, Ch. 4; Watts and Zimmerman, 1986, Ch. 2; Kormendi and Lipe, 1987; Ohlson, 1991). Since the market's expectations of future cash flows are unobservable, empirical specifications of the price-earnings relation often use current earnings as a proxy for the market's expectation.

Current earnings, however, reflect both a surprise to the market and a "staid component that the market had anticipated in an earlier period. In the return model, the stale component is irrelevant in explaining current return and thus constitutes an error in the independent variable, biasing the slope coefficient on earnings toward zero (e.g., Brown, Griffin, Hagerman, and Zmijewski, 1987). By contrast, the current stock price in the price model reflects the cumulative effect of earnings information, and thus varies due to bosh the surprise and stale components. Therefore, there is no errors-~in-variables bias in price-model regressions. Intuitively, current earnings are uncorrelated with the information about future earnings contained in the current stock price, the dependent variable. Econometrically, the price model thus has an uncorrelated omitted variable, which reduces explanatory power, but the estimated slope coefficient is unbiased (Maddala, 1990).

Criteria for evaluating alternative models. In evaluating price and return models, we measure the extent to which the estimated slopes and intercepts approximate their predicted values. In particular, assuming that earnings follow

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a random walk, the intercept in price- and return-earnings regressions should be zero. The slope, commonly referred to as the earnings response coe~%nt , should be the reciprocal of the firm's expected rate of return, l/r. Its magnitude should be about 10-12 for the sample of U.S. firms examined over the past 38 years. The second criterion we use to evaluate price and return models is the extent of misspecification and/or heteroscedasticity as indicated by the White (1980) statistic. The criteria that we employ to evaluate price and return models are neither unique nor the only ones that can be used. Depending upon one's research design and loss function, different criteria will apply.

To keep the analytics tractable and to more confidently assess the degree of bias in the estimated slopes, we restrict our analysis to a regression with earnings as the only explanatory variable. Balance sheet and income statement variables could certainly be examined (e.g., Barth, Beaver, and Landsman, 1992; Barth and Kallapur, 1994), but there is less agreement among researchers about both the information content of and the coefficient magnitudes on such variables. Evaluation of price and return models would, therefore, be more tenuous.

Empirical evidence. Our results indicate that the slope or earnings response coefficients are substantially less biased in price models than in return models. Coefficients from the price model, but not the return model, imply cost of capital estimates that are more in line with those observed in the market. Also, the time series of implied cost of capital estimates from cross-sectional price models more closely approximates long-term interest rates plus a risk premium than does the corresponding time series from return models. Nevertheless, price models do not unambiguously dominate return models. Price models more frequently reject tests of heteroscedasticity and/or model misspecification than return models. Therefore, researchers are confronted with two flawed functional forms: one that gives more economically sensible earnings response coefficients (price models) and another with less severe White (1980) specification problems (return models) but more biased slope coefficients. An obvious implication is that researchers using price models must exercise more care in drawing statistical inferences, e.g., by using White's (1980) heteroscedasticity-consistent standard errors. Since each functional form has its weakness, researchers should be aware of the econometric limitations in designing their experiments. When possible, using both functional forms will help ensure that a study's inferences are not sensitive to functional form.

Limitations. One limitation of our analysis is that we do not explicitly incorporate the implications of departures from the random walk property of firms" annual earnings. A coefficient of 1/r is implied by our simple valuation model that assumes a random walk. However, Be~l and Watts (1979, p. 205) explain that management's errors in accrual forecasts (e.g., bad debt expense) reverse themselves, and revenues and costs contain tra~g[tory components (e.g., special items, restructuring costs, and other one-time revenues, gains, and losses=: included in earnings). Consequently, earnings changes will be negatively serially

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correlated. If the earnings variable is not adjusted for this predictable component, then the estimated earnings response coefficient is biased towards zero for both price and return models (see Ali and Zarowin, 1992, and Sections 2 and 5 below).

Another limitation of our analysis is that we do not examine in detail the economic reasons for and the econometric consequences of nonlinearities in the price-earnings relation. These issues are beyond the scope of this paper. Basu (1995) explains how conservatism in accounting induces an asymmetric and nonlinear price-earnings relation. Hayn (1995) explores the consequences of losses. Freeman and Tse (1992), Cheng, McKeown, and Hopwood (1992), Das and Lev (1995), and Beneish and Harvey (1992) are other examples of research examining the linearity of the price-earnings relation.

Section 2 presents the intuition underlying the argument that the earnings response coefficient estimates from the price models are unbiased, whereas return models yield biased estimates. Section 3 describes the data and provides descriptive statistics, and Section 4 presents the empirical results. Section 5 explores the sensitivity of the results to several specification tests. Section 6 dis- cusses implications for other research.

2. Price and return specifications

This section examines alternative price-earnings specifications when the "nformation set in prices is richer than that in the current and past time series of earnings, ke., prices lead earnings. We formalize the prices-lead-earnings assumption when earnings follow a random walk.~ Thus~ the market's expectation of future earnings, conditional on all the information that it has, differs from the time-series {random-walk) expectation. We further assume that expected rates of return are constant through time and state a simple valuation model. Condi- tional on the valuation model, we show that, when prices lead earnings, the price model yields an unbiased earnings response coefficient, but the return specification yields biased estimates. The latter has also been analyzed in Ohlson and Shroff (1992) and Kothari (1992), among others. Finally, we examine implications of earnings containing value-irrelevant noise, i.e., there is a component of earnings that is unrelated to current, past, and future stock returns. The presence

~The analytics are simplified by assuming a random walk time-series property for annual earnings. However, it is well-known that time-series properties of earnings deviate mildly from a random walk (e.g., Ball and Watts, 1972; Brooks and Buckmaster, 1976) and that expected equity rates of return vary cross-sectionally and change through time. Implications of these violations of the assumptions underlying the analysis in this section for the various price-earnings specifications are discussed in Section 5. That analysis suggests that the tenor of the analytical and empirical results in the paper is unaffected.

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of value-irrelevant noise results in biased slope coefficients for all price-earnings specifications (e.g., Landsman and Magliolo, 1988; Ryan and Zarowin, 1995). All the assumptions and derivations underlying the analysis in this section are provided in the Appendix, whereas only the important assumptions and the intuition behind the results are discussed in this section.

2.1. Al ternat ive price-earninf fs specif icat ions under a s ty l i zed valuation model

Alternative price-earnings specifications commonly estimated in the accounting literature are:

Price model: P~ = ~ + f iX , + et, (1)

Re turn model: P , / P , - i = • + f l X j P , _ 1 + e,, (2)

Differenced-price ,nodel: alP, = ~ + f A X , + e,, (3)

where P, is the ex-dividend price at time t, X, is earnings for period t, 0t and fl are the intercept and slope coefficients, and e, is an error tenn. The differenced-price model is included because differencing often yields a stationary," series. Some of the econometric problems in using the price model can thus be overcome by using first differences (Christie, 1987).

It is easy to show that all three specifications are equivalent in the sense that all three models yield a slope coefficient of I/r, where r is the (constant) expected rate of return (Christie, 1987). Two critical assumptions are that earnings follow a random walk and that only the information in the current and past time series of earnings is used by the market in setting prices, that is, prices do not lead earnings [see Eqs. (A.1)-(A.6) in the Appendix for details]. Thus, in the context of a stylized constant price-earnings ratio model, there is no economic difference between the price, return, and differenced-price specifications. The choice among the three alternatives must therefore be guided by econometric issues (Christie, 1987} or because violation of one or more of the underlying assumptions has a differential effect across these specifications. 2

2In addition to the return model in Eq. (2}, recent research on the price-earnings relation examines two other return model specifications. While the dependent variable is always stock return, alternative earnings variables are earnings change deflated by price, dX,/P,- 1 • and earnings change deflated by last period's earnings, dX,/X,_ x. If the 'price is a constant multiple of earnings" valuation model is assumed, regressions employing all of these earnings variables yield results identical to those using Eqs. {1)--{3}. We focus only on the earnings-deflated-by-price variable [i.e., the return model (2)] because, once prices lead earnings, previous research indicates that return model (2) outperforms the other two earnings variables in terms of bias in the estimated earnings response coefficient {e.g., Ohlson, 1991; Ohlson and Shroff, 1992; Kothari, 1992; Easton and Harris, 1991).

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2.2. Alternative specifications when prices lead earnings

Prices-lead-earnings assumption. Considerable research since Beaver, Lam- bert, and l~,~orse (1980)demonstrates that prices lead earnings, i.e., the information set in stock prices is richer than that in the past time series of accounting earnings (recent examples include Kothari and Sloan, 1992; Easton, Harris, and Ohlson, 1992; Wa=field and Wild, 1992; Collins, Kothari, Shanken, and Sloan, 1994). Therefore, even though the time-series properties of annual earnings are well-approximated by a random walk (e.g., Ball and Watts, 1972; Albrecht, Lookabill, and McKeown, 1977), the market anticipates a portion of the time-series earnings surprise or change in earnings. The forecasting power of prices with respect to future earnings changes arises because historical cost accounting, with its emphasis on conservatism, objectivity, and revenue-recog- nition conventions, has a limited ability to reflect the market's expectations of future earnings.

To formalize the prices-lead-earnings assumption, we begin with a random walk in earnings:

X, = X,-1 + aX, , (4)

where, conditioning only on the past observations, AX, has a zero mean, a constant va[iance, and is serially uncorrelated. However, when prices lead earnings, only a portion of d X is a surprise to the market; the rest is anticipated during periods t - 1, t - 2, and so on. If we assume that the market anticipates earnings only two periods ahead, Eq. (4) becomes

X f -= X , - t + S t + at.t_ 1 + at . t_2, (5)

where s, + a~.,_ 1 + a,.,-2 = 3X,, st is the component of AX, that is a surprise to the market in period t, and a,.,_ 1 and a,.,_ 2 are the components of 3X, that the market anticipates in periods t - 1 and t - 2 (s, a,.,_ 1, and a,.t-2 are assumed uncorrelated)) The first subscript, t, of a,.,_ t refers to the year of the earnings X,, and the second subscript, t - 1, refers to the pe,~od in which the market anticipates a component of the earnings X,. The following numerical example illustrates the notation. Last year's earnings, X,_ t, were $3 and current earnings, Xt, are $4. The surprise, s,, is $0.30. Last year the market anticipated that this year's earnings would be up $0.90, and two years ago the market anticipated a decrease in earnings of $0.20. We decompose the earnings change into:

$4 = $3 + $0.30 + $0.90 - 0.20,

Xt = X t - 1 4" st 4" at . t - t 4" at.t-2.

3The formulation in Eq. (5) is identical to that in Kothari (1992). For related, but conceptually different approaches, see Lipe (1990) and Ohlson (!991, App. B).

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Because price reflects anticipated future earnings, the price at time t, P~. will not be a constant multiple (1 it) of current earnings, X,. Pt will exceed X,/r if the market expects positive ee.rnings changes over the next two years (e.g., a growth firm), whereas P, will be lower than X,/r if the market expects negative future earnings changes. Eqs. (A.7)--(A.9) in the Appendix formalize this intuition. The earnings response coefficient, defined as the coefficient that maps a unit surprise into stock pric~, will be 1/r even though prices lead earnings because st is assumed permanent.

Price specification. When prices lead earnings, the current price, in addition to all the information in current and past earnings, contains information about future years" earnings that is absent from current earnings. This information (i.e., a,+ 1., and a,+2.,) generates variation in price, Pt, but is uncorrelated with X,. Therefore, price model (1) is missing an independent variable that would explain the variation in P, due to the anticipated components of future periods" earnings changes. Econometrically, this is an uncorrelated-omitted-variable problem that reduces the price model's explanatory power, but the estimated coefficient on X, is unbiased (see, for example, Maddala, 1990). Eq. (A.10) in the Appendix shows the derivation.

Return specification. Previous research shows that when prices contain information about future years' earnings changes (e.g., Brown, Foster, and Noreen, 1985; Collins, Kothari, and Rayburn 1987; Freeman, 1987), the estimated earnings response coefficient from the return model (2) is biased toward zero. While the Appendix contains the derivation [Eqs. (A.11) and (A.12)], the intuition is as follows. The dependent variable, Rt, in the return model reflects information about current and future earnings arriving over the current period. However, the independent variable, Xt, contains information arriving over both current and past periods. That is, X, contains both the surprise component, s , and stale components, at.t- 1 and at.,- 2- The stale components are irrelevant in explaining current returns (which are generated by s,), and newly anticipated components of future earnings changes (i.e., at+ 1.t and a,+ 2.,). Since Xt's stale components cannot explain Rt, the independent variable in the return model measures the variable of interest with error. This errors-in-variables problem biases the estimate of the return-model earnings response coefficient toward zero.

While we have analyzed the return model using a simple setting of prices leading earnings, the nature of historical-cost accounting suggests a more com- plicated structure foi- prices leading ~rnings. Basu (1995) shows that conservatism in accounting leads to an asymmetric nonlinear return-earn/rigs relation because earnings are more timely in capturing bad news than good news. The explanatory power and slope of a linear return model, therefore, are expected to be less than those from a welbspc:.'.':.~ n,~nlinear me:lel. The nonlinearity analysis within the context of issues discussed in Basu (1995) are beyond the scope of this study. The main point, however, is that whether tae prices-lead-earnings phenomenon is symmetric or not, the return model yields a biased slope.

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Differenced-price specification. The differenced-price specification, Eq. (3), also yields a biased earnings response coefficient when prices lead earnings. The intuition behind the result is similar to that described for the return model and, therefore, is not repeated here.

2.3. Alternative specifications when earnings contain value-irrelevant noise

Considerable research examines the price-earnings relation under the assumption that earnings consist of a value-relevant and a value-irrelevant component. The former is typically assumed to be perfectly correlated with stock prices, whereas the latter is uncorrelated with stock prices. For example, Beaver, Lambert, and Morse (1980) propose a model in which accounting earnings, Xt, contain a 'garbling" component, zit, that is uncorrelated with stock returns in all periods; they model the predictive ability of prices with respect to future earnings by assuming that the ungarbled component of earnings, x, ( = X, - ut), follows a first-order integrated moving average process [IMA(1,1)]. Should x, be observable, then, within the framework of the Beaver, Lambert, and Morse model, the information set in the time series of x, is identical to that in stock prices. Choi and Salamon (1990), Landsman and Magliolo (1988, model 3, pp. 598-599), and Ryan and Zarowin (1995) introduce models in which accounting earnings consist of value-relevant and value-irrelevant components. Finally, Ramakrishnan and Thomas (1994) also assume that earnings contain a price- irrelevant component, although they impose a negative serial correlation structure on it to capture accrual reversals.

Following some of the previous modeling in this area, we consider a simple formulation:

X~ = x, + u,, (6)

where xt is the value-relevant component and u, is the value- irrelevant component of accounting earnings, X~. x~ is assumed to follow a random walk and ut

2 is a zero mean, serially uncorrelated, white noise term with variance o-,. xt is perfectly correlated with price and ut is uncorrelated with xt and price.

In the pre~nce of value-irrelevant noise in earnings, all three specifications, Eqs. (1) to (3), yield downward-biased earnings response coefficient estimates (see Landsman and Magliolo, 1988; Ryan and Zarowin, 1995; and the Appen- dix). The intuition behind the result is that the independent variable in all three specifications measures the "true" variable of interest with error, the value- irrelevant noise, biasing the estimated slope towards zero? For all three models,

4Obviously, this result hinges on the assumption that the noise is uncorrelated with both price and earnings variables. If opportunistic accruals by management are the primary soure of noise and if this noise is assumed to be (perfectly) negatively correlated with the surprise component of value-relevant earnings, then such noise biases upward the slope coefficient in all three models. To keep the analysis simple and to focus on the forward-looking nature of prices to discriminate between alternative models, we ignore this and other formula,ions of noise.

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the degree of bias increases in the ratio of the variances of ut and zlx,. An indication of some researchers" assessment of the relative magnitudes of the variances of u, and Ax, is given by Ryan and Zarowin (1995). They conclude that the ratio is about 13, which would imply that, in the price model, the estimated earnings response coefficient is aboat 7% of the 'true" earnings response coefficient.

2.4. Summary

All three specifications yield unbiased coefficients (earnings response coefficients are expected to be 1/r) when prices do not lead earnings and earnings do not contain noise. The price specification yields an unbiased earnings response coefficient when prices lead earnings, but it yields a biased coefficient estimate in the presence of value-irrelevant noise in earnings. The return and differenced- price specifications, on the other hand, yield biased coefficients when prices lead earnings and also when earnings contain value-irrelevant noise.

While we provide predictions about the slope coefficient estimate from various price-earnings specifications, our analysis primarily highlights the differential implications of prices leading earnings for the price, return, and differenced-price specifications. Obviously, the predictions critically depend on the descriptive validity of the simplifying assumptions. In particular, the slope will be smaller than 1/r if there is negative serial correlation in earnings changes (e.g., Ball and Watts, 1972; Brooks and Buckmaster, 1976; All and Zarowin, 1992).

3. Data and descril~ive statistics

We use 1952-89 earnings and return data from the Compustat Annual Industrial tape and the Annual Research tape and the Center for Research in Security Prices (CRSP) monthly tape. Since we perform time-series, cross- sectional, and pooled analyses, we construct two samples with different data availability requirements. For the cross-sectional and pooled analyses, we in- dude any firm that has at least two consecutive annual earnings and return obse~ rations. For the time-series analysis we require a minimum of 20 consecutive annual earnings and return observations. The time-series analysis sample is thus a subset of the pooled and cross-sectional analyses sample. Annual earnings excluding those from discontinued operations and extraordinary items are used. Only firms with a December fiscal year-end are included to facilitate inferences from a cross-sectional analysis (e.g., to test whether the sample mean of slope coefficient estimates from successive cross-sectional regressions is reliably positive). This restriction induces a bias in favor of including larger stocks (Smith and Pourciau, 1988).


We use per-share values of prices and earnings to reduce the presence of heteroscedastic disturbances (e.g., Barth, Beaver, and Landsman, 1992). The earnings variable is either earnings per share, X, (price model), ?arnings per share deflated by price at the beginning of the year, X,/P,_ ~ (return model), or change in earnings per share, AX, (differenced-price model). The corresponding price or return variable is price, P,, return, P,/P,_ ~, or change in price, ziP,. Annual calendar-year (i.e., fiscal-year) buy-and-hold returns, exclusive of dividends, are used. Earnings and price data are adjusted for stock splits, stock dividends, and stock issues. To avoid any undue influence of extreme observations, we exclude the largest and smallest 1% of observations for each variable from the sample (which also affects the time-series analysis sample because it is a subset of the pooled and cross-sectional analyses sample). While arbitrary, exclusion of extreme observations is consistent with a similar practice in previous research, e.g., Easton and Harris (1991) delete observations of X, deflated by Pt-I that exceed 1.5 in absolute value. The resulting sample sizes are 38,890 firm-years for the cross-sectional and pooled analyses and 27,127 firm-years representing 1,017 different firms for the time-series analysis.

Table 1 reports descriptive statistics for the various price and earnings variables separately for the two subsamples. Panel A contains descriptive statistics for the cross-sectional and pooled analysis sample. The minimum and maximum values of the various variables indicate that degpite exclusion of extreme 1% of observations, the range of values is substantial. For example, the minimum value of X,/P,_ ~ is more than 50 standard deviations smaller than its average value of 0.09.

Comparison of the descriptive statistics for the time-series sample in panel B with those in panel A suggests that the time-series sample consists of more successful, surviving firms. The average earnings per share and the average change in earnings per share are somewhat larger for the time-series analysis sample compared to the cross-sectional and pooled analyses sample. However, average returns for the time-series analysis sample are slightly lower than for the pooled sample.

4. Empirical evidence

This section presents results of estimating the various price--earnings models. Across all estimations, the results indicate that the price specification provides estimates of the earnings response coefficient that are closer in magnitude to those implied by expected rates of return observed in the marketplace. Nonetheless, as predicted by Christie (1987), tile price specification suffers more from heteroscedasticity/misspecification problems than the return model.


Table I Descriptive statistics for price and earnings variables used in valuation, return, and differenc~-prk~ specifications of the price-earnings relation; cross-sectional and pooled regression analysis sample and time-series regression analysis sample; annual data from 1952-1989

Variable Mean Std. dev. Min. Med. Max.

Panel A: Pooled and cross-sectional analyses sample, N = 38,890 firm-years

Xr 1.97 1.81 - 4.61 1.79 9.33 P, 24.41 18.23 0.22 20.25 1~.00 XJP,_ ~ 0.09 0.20 - 10.48 0.09 3.09 P,/P,- t 1.14 0.60 0.06 1.06 48.93 AXt 0.13 1.10 - 11.74 0.15 10.87 dPt 1.77 8.88 - 61.25 0.88 74.92

Panel B: Time-series analysis sample, N = 27,127 firm-years

Xr 2.27 1.79 - 4.61 2.09 9.33 P, 27.77 I8.76 0.25 24.25 106.00 X,/P,_ 1 0.09 0.15 - 8.72 0.09 3.09 P,/P,- 1 1.13 0.56 0.11 1.06 48.93 AX~ 0.15 1.12 -- ! 1.74 0.16 9.66 APt 1.99 9.24 -- 61.25 1.12 74.92

Sample: The cross-sectional and pooled analyses sample includes any firm that has at least two consecutive annual earnings and return observations. For the time-series analysis sample, a minimum of 20 consecutive annual earnings and return observations is required. X, is annual earnings per share excluding the extraordinary items and earnings from discontinued operations. Price relatives exclusive of dividends, P,/P,_ 1, are measured over calendar years. Only firms with a December fiscal year-end are included in both samples. The largest and smallest 1% of observations for each variable are excluded from both samples. Earnings, X,, and prices, Pt, are adjusted for stock splits and stock dividends when deflating by P,_ 1 and obtaining differenced variables.

4.1. Pooled time-series and cross-.'-ectional analysis

Results o f poo led t ime-series cross-sect ional es t imat ion of '~he var ious mode l s are repor ted in Tab l e 2. F o r each specification, the es t imated intercept, earnings response coefficient, and adjusted R 2 of the mode l are repor ted. O r d i n a r y least squares (OLS) s t anda rd e r rors and the Whi te (1980) heteroscedast ic i ty-consistent s t anda rd e r rors for each p a r a m e t e r es t imate and the Whi te (1980) chi- square test statistic are also repor ted. Both O L S and Whi te s t andard e r rors likely unders ta te the t rue s t anda rd e r ro r of the es t imated coefficients because we do not adjust for the posi t ive cross-corre la t ion a m o n g the regression residuals (Bernard, 1987). T h e repor ted s t anda rd er rors should therefore be viewed only as descript ive statistics.

Price mode l T h e price specification yields an earnings response coefficient es t imate of 6.55, with a Whi te s t andard e r ro r o f 0.049. T h e annua l expected ra te of re turn implied by the es t imated coefficient is 15.3% l- = 1/6.55]. The

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estimated intercept, 11.47, is more than 100 standard errors greater than zero, although again the standard error is understated because we ignore positive cross-correlation among the residuals. Recall that all specifications predict a zero intercept. Within the context of our simple valuation model, a nonzero intercept implies that the slope coefficient is biased. More generally, nonzero intercepts indicate either mr~el specification problems or an omitted- variables problem. As seen from Table 2, all specifications yield highly significant intercept estimates, as in previous research (e.g., Barth, Beaver, and Lands- man, 1992, Table 2; Easton and Harris, 1991, Table 1). Section 5 addresses the question of model misspecification, including the presence of transitory earnings, nonlinearities due to small-price deflators, and correlated omitted variables.

The White statistic for the price model, 1,025, indicates severe heteroscedasticity and/or specification problems. 5 The 42.2% adjusted R 2 from the price model is likely to overstate the information content of contemporaneous earnings because the model is estimated in levels. Scale differences across firms in the sample and autocorrelatcd errors, because of the use of levels, together can contribute to the model's explanatory power (Maddala, 1990, pp. 190-d99).

Re turn model. Compared to the estimated slope from the price model, the coefficient from the return model is only 0.45 (White standard error 0.112). The magnitude of the estimated earnings response coefficient from the return specification is comparable to the 0.84 (standard error 0.02) that Easton and Harris (1991, Table 1, first row) report using 19-year data from 1968-86. 6 The estimated coefficient implies an unreasonably high expected rate of return of over 200%. The White statistic, 14.2, rejects homoscedasticity of errors at an ~t level of 0.001. The 2.3% adusted R 2 indicates relatively low information content of current earnings. It is important to recall, however, that, if prices are forward-looking, both the slope and the explanatory power of the return model are biased toward zero. The adjusted R 2 therefore underestimates the information content of current earnings.

Differenced-price model. The earnings response coefficient estimate from the differenced-price model is 2.09 (White standard error 0.050), which is higher than that from the return model but considerably smaller than that from the price model. The implied expected rate of return is about 50%, which is still too

SThe significant White statistic can also occur if the theoretical relation between stock prices and earnings differs from the simple valuation model assumed here (White, 1980). Only if the assumed model is correct does the significant White statistic indicates the presence of beteroskedastic errors alone. ~T~ difference in the coeffment magnitude between Easton and Harris and this study is attribut- able to time-period differences and their use of 12-month returns ending in the third month after tbe fiscal year-end versus our use of returns measured over the fiscal year.


Table 2 Pooled time-series and cross-sectional estimation of the price, return, differenced-price, and deflated-price specifications of the price-earnings relation; sample of 38,890 firm-year observatiom from 1952-1989

Earnings response coefficient

Standard error Standard error White

Model Estimate OLS White Estimate OLS White R z star.

Price 11.47 0.104 0.100 6.55 0.039 0.049 42.2% 1025 Return I. 11 0.003 0.011 0.45 0.015 0.112 2.3 14.2 Diff. price 1.51 C.044 0.043 Z09 0.040 0050 6.7 141.3 Deft. price 4.69 0.037 0.087 9.03 0.062 0.066 37.2 432.6

Price model: Pr = • + ffXt + ~,

Return model: P,/Pt- ,. = ~ + [JX,/P,_ t + e,

Differenced-price model: APt = • + [JAXt + e,

Deflated-price model: P,/X, = a{l/X,) +/~ + ~,

Sample: The sample includes any firm that has at least two consecutive annual earnings and return observations. X, is annual earnings per share excluding the extraordinary items and earnings from discontinued operations. Price relatives exclusive of dividends, PdPr- i, are measured over calendar years. Only firms with a December fiscal year-end are included. The largest and smallest I% of observations for each variable are excluded from the sample. Earnings, X , and prices, P , are adjusted for stock splits and stock dividends when deflating by Pt-t and obtaining differenced variables.

h igh to be plausible. T h e W h i t e s tat is t ic indicates tha t , l ike the pr ice mode l , the differenced pr ice m o d e l suffers f r o m he te roscedas t i c i ty - re la ted misspecif icat ion. 7

D e f l a t e d - p r i c e m o d e l Whi le the pr ice specif ica t ion a p p e a r s t o yield a less b iased s lope coefficient es t imate , there a re at least two po ten t i a l p rob lems . First , e r ro r s f r o m the pr ice m o d e l are l ikely t o be he te roscedas t i c because o f scale

TOne interpretation of the relatively small coefficient estimate from the differenced-pri~ model compared to that from the price model is that the price model is misspecified. If regressions using first-differenced data yield different results, they frequently indicate misspecification of the undif- ferenced Ior levels) regression model tPiosser, Schwest, and White, 1982). However, since prices are forward-looking, the independent variable, X,, is expected to be correlated with lagged errors. Under these conditions. Plo~ser et aL point out that the slope coefficient from a differenced regression model is inconsistent and, therefore, results from a differenced regression cannot unambiguously indicate misspecification. Ira suitable set of instrumental variables is used to eliminate the correlation between X, and lagged errors, then the Plosscr et ai. (1982) differencing test can be applied. However, suitable instrumental variables are not easily obtained and, therefore, we do not perform the Plosser et al. differencing test of specification.

168 S.P. Kothari. J.L. Zimmerman / Journal o f Acco,.mting and Economics 20 (1995) 155--192

differences and cross-sectional variation in the slope coefficient, which the model assumes to be a constant. Easton (1985) and Christie (1987) recommend deflating the price model by a variable that is a function of the independent variable (earnings) to reduce heteroscedasticity. 8 Second, there is a danger of obtaining a significant coefficient on an (economically or theoretically) irrelevant included variable in a price regression (see Christie, 1987). 9 The intuition is that the significant coefficient on an irrelevant variable might merely be capturing scale differences across the sample of firms. Use of a suitable deflator in estimating price models is therefore recommended. We reestimate the price model using X, as the deflator: ~°

P,/X, = ~(I/X,) + )6 + ~,. (7)

The last row of Table 2 reports results of estimating Eq. (7), the deflated-price model. The i n t e rcep t , ~, is an estimate of the earnings response coefficient. The coefficient, 9.03 (White standard error 0.066), is greater than that from the price model, reducing the concern that the price models yield spuriously large coefficients. The White statistic, 432.6, remains significant, however. The coefficient on 1 i X , , which is an estimate of ~ in the price model, is 4.69 (White standard error 0.087), which is more than 50% smaller than the estimated ~ from the price model, indicating a better-specified model. The 37% adjusted R 2 of the deflated-price model is only marginally smaller than the 42.2% explanatory power of the price model. Since the regression errors of individual firms in the sample are likely to be positively autocorrelated, the explanatory power could be overstated (Maddala, 1990, p. 199) even with the deflated-price model.

The smaller degree of bias in the coefficients from the price and deflated-price models in Table 2 is consistent with the prices-lead-earnings phenomenon contributing to biased coefficients from the return- and differenced-price specifications. Recall that velue-irrelevant noise in earnings biases the estimated

SDeflating by X, also mitigates heteroscedastic errors in a price model that may be caused by an earnings process that follows a multiplicative random walk [i.e., X, = X,_ ,(1 + a,), a, is a zero- mean, constant-variance shock].

9Christie (1987) points out that time-series regressions in levels are an alternative to mitigate concern over obtaining a significant coefficient on an irrelevant variable.

~°The book value of assets is an alternative to earnings as the deflator for the price model. The book value of assets as the deflator has the advantage of always being positive and is less likely to generate outliers than when earnings are used as the deflator. The book-value-deflated price model is

PJBVA,_ , = I/BVA,_ l + [3X,/BVA,_ * + error,

where BVA,_ ~ is the book value of assets at the beginning of year t. The earnings response coefficient estimate from this model is 11.91 (standard error = 0.07) and the adjusted R 2 is 44.6%.

S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (199S) 155-192 169

coefficients of all specifications. Empirically, however, only coefficients from the return and differenced-price specifications appear to he severely biased. Obvi- ously this conclusion rests on the assumption that coefficients from the price model are not upward biased. It is possible, however, that this assumption is violated. If cross-sectional variation in the slope coefficient (which is inversely related to the expected rate of return) is correlated with the independent variable (earnings per share) then the estimated slope coefficient is biased (Christie, 1987). Since empirically both earnings per share and stock price are positively correlated with market capitalization, and market capitalization is well-known to be inversely related to the expected rate of return (Banz, 1981), a positive correlation between the firm-spedfic earnings response coefficient and earnings per share is expected. This conclusion will indeed result in upward-biased slope coefficient estimates. Similar analysis suggests that the coefficient from the return and differenced-price specifications would he biased downward. There- fore, we examine whether biases account for observed differences in the earnings response coefficients reported in Table 2. Howevei-, ~e demonstrate in Section 5 that including finn size and beta in the various models to mitigate the bias has little effect on the inferences from the price and deflated-price model estimations described in this section.

Comparing estimated earnings response coefficients to the market price-earnings ratios. While the estiraates of the earnings response coefficient from the price and deflated-price models are considerably greater than those from the return or differenced-price specifications, they are somewhat smaller than the price-earnings ratio of a typical stock in our sample. The ratio of average price to average earnings per share for the pooled sample (Table 1, panel A) is 12.4 (24.41/1.97). 11 The ratio of medians is 11.3. The coefficients from the various models range from 0.45 to 9.03 (see Table 2). One reason for the difference between the estimated coefficients and the average price-earnings ratio of the sample firms is the presence of transitory components of earnings, violating the random walk assumption underlying the prediction about the earnings response coefficient magnitude. Analysis described in Section 5 and previous research indicate that the presence of transitory components in earnings explains a large portion of the difl~rence between 1it and the estimated earnings response coefficient from the price or deflated-price model. The considerably smaller coefficients frcm the return model, however, are due largely to the effect of prices leading earnings.

~ The ratio of average price to average earnings would generally be smiler than the average of the firm-specific ratios of prices to earnings because of the Jensen inequality. The average of the price-earnings ratios would be unduly influenced by outliers generated by firms reporting earnings that are close to zero.

170 S.P. Kothari. J.L. Zimmerman / Journal of Accounting and Economics '0 (1995) 155-192

4.2. Cross-sectional analysis

Results of 38 annual cross-sectional estimations of the price, return, differenced-price, and deflated-price specifications of the price-earnings relation are reported in Table 3. For each model, we report sample statistics of the 38 intercept and slope coefficient estimates. One benchmark against which the estimated slope coefficients can be evaluated is the average or median price-earnings ratio over the 38 years. The time series of annual price-earnings ratios, defined as the ratio of average price in year t to average earnings in year t has an average of 13.1 and a median of 12.9.

Price model. In Table 3, panel A, the average estimated slope coefficient from the price model is 7.9, with a standard error of 0.41. All 38 coefficient estimates are positive and range from 3.9 to 14.2. The reported standard error is likely to understate the true standard error for at least two reasons. First, since the residuals from a levels regression in year t are likely to be positively correlated with the residuals in years t + 1 and beyond, the estimated coefficients are also expected to be positively correlated. The reported standard error ignores this dependence in the estimated coefficients through time, so it under- states the true standard error. Second, because the true slope coefficient in year t is a function of the expected rate of return in year t and expected rates of return exhibit high positive serial correlation, the estimated coefficients should also exhibit positive autocorrelation. This too will lead to understated standard e r r o r s .

To incorporate the effect of serial correlation in the estimated coefficients on the standard errors, we also calculate, but do not report, adjusted standard errors. The adjustment reflects the Newey and West (1987) correction, with six lags, for serial dependence in the estimated coefficients. Not surprisingly, these standard errors are larger (by a factor of about 2) than those reported in Table 3, but the average slope coefficients are always at least two standard errors away from zero. The iaferences, thus, are not altered.

The time-series average slope coefficient of 7.9 from the price model implies an expected rate of retul'n of 12.7%. This is smaller than the 15% expected rate of return implied by the coefficient from the pooled price model regression reported in Table 2, but still greater than the average expected rate of return of 7.6% implied by the sample firms" average price-earnings ratio of 13.1. Since the estimated coefficients range from 3.9 to 14.2, the implied expected rates of return range from 7.0% to 25.6%. While undoubtedly the implied expected rates of return exaggerate the dispersion in true expected rates of return over this period because of sampling error in the estimated coefficients, the range of values does not appear unreasonable.

Return and differenced-price models. The estimated average slope coefficient from the return model, 1.65, is considerably greater than that obtained from the pooled regression in Table 2. However, both return and differenced-price model

S.P. Kothari. J.L. Zimmerman / Journal o f Accounting and Economics 20 (1995) 155-192 171

Table 3 Annual cross-sectional estimation of the price, return, differenced-price, and deflated-price specifications of the price-earnings relation; 38 annual regressions using a sample of 38,890 firm-year observations from 1952-1989

Coeff. Mean Std. err. Min. QI Med. Q3 Max.

Pane~ A: Price model, P, = :~ +/1Xt + et

:c 10.2 0.58 3.9 7.5 10.0 12.6 7.9 0.41 3.9 5,9 7.3 9.7

Adj. R 2 53.1 40.2 47.0 51.1 59.8 White 23.1 35 significant at 15.4 22.2 33.3

stat. 0.05 p-value

19.1 14.2 67.5

Panel B: Return model, Pr/Pt- i = ~ + 3 X , / P t - 1 + er

1.01 0.04 0.52 0.85 1.01 1.20 1.65 0.23 - 1.50 0.40 1.45 2.90

Adj. R z 12.8 0.0 6.2 12.5 17.4 White 3.9 6 significant at 2.1 3.7 5.2

star. 0.05 p-value

1.46 4.17

33.9

Panel C: Differenced-price model, APt = • +/~AXt + et

1.79 0.78 -- 7.3 -- 0.1 2.14 3.8 3.24 0.27 0.8 1.8 2.98 4.6

Adj. R e 13.8 1.6 10.0 12.7 18.4 White 10.4 28 significant at 5.3 8.0 11.7

star. 0.05 p-value

15.2 7.8

27.6

Panel D: Deflated-price model, Pt /Xt = {1/Xt) 4- ~ 4- ~,t

5.0 0.30 1.8 3.7 4.9 6.2 9.6 0.51 4.2 7.1 8.6 12.7

Adj. R z 38.7 14.4 29.6 40.0 49.1 White 11.7 27 significant at 5.9 10.8 15.9

stat. 0.05 p-value

10.4 16.7 60.5

On each panel, average value of the estimated coefficients, adjusted RZs, and White (1980) chi-square statistics from 38 cross-sectional regressions are reported. Standard errors are for the time-series sample mean coefficients. Qt and Q3 are 25th and 75th fractiles of the distribution of estimated coefficients, adjusted RZs, or White statistics.

Sample. The sample includes any firm that has at least two consecutive annual earnings and return ob~rvations. X, is annual earnings per share excluding the extraordinary items and earnings from discontinued operations. Price relatives exclusive of dividends, P,/P,_ 1, are measured over calendar years. Only firms with a December fiscal year-end are included. The largest and smallest 1% of observations for each variable are excluded. Earnings, X,, and prices, P , are adjusted for stock splits and stock dividends when deflating by Pt- 1 and obtaining differenced variables.

r e g r e s s i o n s (see p a n e l s B a n d C) y ie ld s l o p e coef f ic ien ts t h a t a r e s u b s t a n t i a l l y

s m a l l e r t h a n t h o s e f r o m t h e p r i c e m o d e l r eg ress ion . T h i s r esu l t is c o n s i s t e n t wi th

t he i n tu i t i ve a n a l y s i s in S e c t i o n 2 a s s u m i n g p r ices c o n t a i n i n f o r m a t i o n a b o u t

f u t u r e ea rn ings .

172 s.P. Kothari. J.L. Zimmerman / Journal of Accounting and Economics 20 (! 995) ! 55- ! 92

Deflated-price model. Panel D reports results of estimating the deflated-price model. The estimated slope coefficient, 9.6, is greater than that from the price model. I~ The implied expected rate of return is 10.4%. The 38.7% average adjusted R 2 of the deflated-price model is considerably less than the 53.1% average explanatory power of the price model in panel A. The explanatory power of the return and differenced-price models is considerably lower, consistent with the earlier analysis indicating downward bias in the estimated slope coefficient and explanatory power of these models,~3

Evaluation of the various models using the results in Table 3 has so far been based on the closeness of the coefficient estimates to PIE ratios. A related criterion of interest could be to assess an estimate relative to its standard error (i.e., power). That is, is the return-model-based coefficient more likely to be statistically significant, even though it is biased? Using the standard errors reported in Table 3 as well as those adjusted for serial dependence in the coefficient estimates, the price and deflated-price models yield coefficient estimates that are about twice as many standard errors away from zero as those from the return model. The pooled regression analysis results also yield a similar inference.

4.2.1. Estimated earnings response coefficients and interest rates We next discuss correlating implied expected rates of return from the annual

estimated slope coefficients with long-term government bond yields as proxies for the expected rates of return through time. Previous research (e.g., Collins and Kothari, 1989) indicates that the return-model-based earnings response coefficient estimates through time exhibit variation that is correlated with the expected rate of return proxies. Our focus here is on the extent to which the implied expected rates of return from various models are correlated with long-term government bond rates.

We estimate the following regression separately for each price-earnings specification:

Rt = 70 + 7t It + errort, (8)

where Rt is the implied expected rate of return for year t and equals 1/bt, bt is the estimated earnings response coefficient from one of the four price-earnings specifications, t covers 1952 to 1989 (38 years), and L is the long-term government bond rate for year t, taken from Ibbotson and Sinquefield (1989).

Z ZThe average coefficient estimate using the book value of assets as the deflator is 12.9. 13We also estimated all four models using the smaller time-series analysis sample which includes only those firms with a minimum of 20 observations per firm. The results are virtually indistinguishable from those reported here for the cross-sectional sample.

S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (I995) 155--I92 173

Assuming that the long-term government bond rates capture variation in the expected rates of return over time, ?t will be 1 if the implied expected rates of return from a particular price-earnings specification track the true expected rates of return over the years. Under these assumptions, ~'o is an estimate of the market risk premium. 14

OLS parameter estimates of model (8) are reported in columns 2-5 of Table 4. The estimated 3'o and 71 using the implied expected rates of returns from the price model are 0.06 (standard error = 0.01) and 1.10 (standard error = 0.16). There is roughly a one-to-one correspondence between variation in the implied expected rates of return and the bond yields, and the expected rates of return estimates are about 6% greater than the bond yields. This fielding is consistent with the economic intuition that the 6% intercept approximates the market risk premium and the expected market return equals the sum of the bond yield and the risk premium.

Since the expected rate of return estimates are autocorrelated, regression errors of model (8) will be autocorrelated, as confirmed by the Durbin-Watson statistic. Since under t he~ conditions the OLS estimates are unbiased but less efficient than generalized least squares estimates (Maddala, 1990), we also report results of estimating the regressions using the Cochrane-Orcutt (1949) procedure assuming that the OLS residuals follow a first-order autoregressive process. The results are reported in the last two columns of Table 4. The estimated 71, 0.98 (standard error = 0.23), is indistinguishable from the OLS estimate of "/1. In addition, results for the deflated-price model in the last row are similar to those for the price model. This finding is not surprising given the similarity in earlier results on earnings response coefficient estimates.

The estimated 71s using the Cochrane-Orcutt procedure for the return and differenced-price models are 5.73 (standard error = 1.94) and 4.18 (standard error = 1.60). Is They indicate a positive association between the bond yields and implied expected rates of return through time. However, since government

14By correlating the implied expected rates of return with the government bond yields, we implicitly assume that variation in the pr/ce-earnings ratios is related to nominal yields. However, if inflation has no reaJ effect on valuation, then price-earnings ratios reflect the real discount rates. In the presence of real effects of inflation (e.g., Fama, 1981), we expect an association between the implied expected rates of return and nominal yields. The reason for this association is that in periods of high (unexpected) inflation, because of the adverse effects of inflation on the economy, earnings (or dividend) growth is expected to be low. Consequently, price-earnings ratios will be low and nominal yields will be high, leading to a positive association between implied expected rates of return and nominal yields. t s In estimating model (8) for the return and differenced specifications, we set Rt equal to 100% if the implied expected rate of return exceeded 100% or was negative. The implied expected rate of return was set equal to 100% 17 (2) times for the return (differenced) model. If these implied expected rates of return had not been set equal to 100%, the estimated "/i and R z for the return model would have been 30 and 23%, respectively.

174 S.P. Kothari. J.L. Zimmerman / Journal o f Accounting and Economics 20 (1995) 155-192

Table 4 Regressions of expected rates of return thro,lgh time implied by the estimated slope coefficients from the price, return, differenced-price, and deflated-price models on the long-term government bond yields: Rt = 7o + ",'l i~ + errort

OLS estimation ~ Cochrane-OrcutP

Standard error Standard error

Durbin- Model 7o 7~ Adj. R" Watson 70 7~

Price 0.06 1.10 54.9% 0.99 b 0.07 0.98 0.01 0.16 0.02 0.23

Return 0.16 7.66 48.2 0.88 b 0.30 5.73 0.10 1.29 0.15 1.94

Diff. pr/ce 0.09 4.62 33.4 0.97 b 0.13 4.18 0.08 1.04 0.12 1.60

Deft. price 0.06 0.81 35.0 0.76 b 0.07 0.71 0.01 0.18 0.02 0.28

Rt = I/(bt) where Rt is the expected rate of return estimate for year t, b, is the estimated earnings response coefficient from one of the four price-earnings specifications, and 1, is the long-term government bond rate. If b, is less than 1. then the Rt is set equal to 1 to avoid extreme Rt observations in a regression. R, is set equal to 1 seventeen times for the return specification, two times for the differenced-price specification, and never for the price and deflated-price specifications. There are 38 annual expected rate of return estimates from 1952 to 1989. The long-term government b~,~d y/elds are taken from Ibbotson and Sinquefield (1989).

The price-earnings specifications are:

Price model: Pt = • + f iX, + er

Return model: Pt/Pr- ~ = • + f l X , / P t - ~ + e,

Differenced-price model: 3Pt = • + flzJX, + e,

Deflated-price model: P~/Xt = ~t(l/Xt) + fl + et

Results of ordinary least squares estimation and those from estimating the model after performing the Cochrane-Orcutt AR~ 1) transformation of the data.

bThe Durbin-Watson statistic is significant at 5%.

b o n d y i e lds o v e r t h e s a m p l e p e r i o d r a n g e d f r o m 3 % to 14%, w i th a n a v e r a g e o f

6 . 8 % , t h e l a r g e coef f ic ien t e s t i m a t e s in T a b l e 5 for t he r e t u r n a n d d i f f e renced- p r i c e m o d e ! s ~,-~nh, , ,~n~r~lh, h igh l~',,p!s o f e x p e c t e d "~'"~ , ,r , , , , , , , , , ~:,,,. . . . . r-.,. ~, . . . . . . . . .., . . . . ,,,at.~,o ~,J J~t.,.alai. • ,L.,i e x a m p l e , if t h e b o n d y ie ld is 6 % , t hen the r e t u r n - m o d e l - b a s e d e x p e c t e d r a t e o f

r e t u r n o n a t y p i c a l s t o c k in o u r s a m p l e will be 3 4 % . T h e l a rge 71 e s t i m a t e s a l s o

i m p l y tha~ the e x p e c t e d r a t e s o f r e t u r n b a s e d o n the p r i c e a n d d i f f e r e n c e d - p r i c e

m o d e l s e x h i b i t a ve ry h igh d e g r e e o f s ens i t i v i t y to c h a n g e s in b o n d y ie lds .

S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (1995) 155-192 |75

Overall, the results from cross-sectional estimation of the various specifications of the price-earnings relation indicate that the price-model-based parameter estimates assume economically sensible values. The results are consistent with the analysis in Section 2 under the assumption that prices contain information about future ,.., nings changes.

4.3. Time-series analysis

Results of 1,017 firm-specific time-series estimation of the various price- earnings specifications are reported in Table 5. In panel A, the average estimated slope coefficient from the price model is 4.6, with a standard error of 0.13. However, since we ignore cross-correlation among the coefficients' estimation errors, the repo:~ed ~andard error undL,~.tates the true standard error. The average explanatory power, 21.8%, is consia,., ably lower than that observed in cross-sectional price model estimation. White's test is rejected at the 0.05 level for 95 of the 1,017 firms.

The time-series estimation suffers from at least two problems. First, since price and earnings levels are serially correlated through time, the regression errors are autocorrelated. We therefore reestimate the regressions using the Cochrane-Orcutt procedure (results are not reported). The average estimated slope coefficient increases marginally to 4.9 (standard error = 0.12). Second, because prices contain information about future earnings, the earnings variable for period t is likely to be positively correlated with the regression disturbance terms for periods t - 1 and earlier. [Keim and Stambaugh (1986), Stambaugh (1986), and Fama and French (1988) discuss this problem, which is geNaane to many studies.] The positive correlation between the independent variable and lagged regression errors induces a finite-sample downward bias in the estimated coefficient. The bias decreases in the number of time- series observations and increases in the first-order autoregressive coefficient on earnings, the independent variable (Stambaugh, 1986). Since the sample size here is often 20-30 annual observations, and because time-series properties of annual earnings are we!l-approximated by a random walk, the bias could be serious.

The average estimated slope coefficient from time-series price-model regressions, 4.6, is considerably smaller than that from cross-sectional estimation, 7.9 (see Table 3). It is not obvious that the difference is due entirely to the downward bias in the time-series estimate of the average slope coefficient. Differences in the samples employed in the time-series and cross-sectional analyses are unlikely to explain the smaller average slope coefficient estimate from the time-series estimation (see Footnote 10L Moreover, since the time-series sample comprises r¢iatively large surviving filans, and because previous research indicates that the expected rate of return on stocks is negatively related to firm size (Banz, 1981), the average slope coefficient for the time-series sample should be larger

176 S.P. Kothari, J.L. Zimmerman / Journal o f Accounting and Economics 20 (1995) 155-192

Table 5 Firm-specific time-series estimation of the price, return, differenced-price, and deflated-price specifications of the price-earnings relation; 1,017 firm-specific regressions using a sample of 27,127 firm-year observations from 1952-1989

Coeff. Mean Std. err. Min. QI Med. Q3 Max.

Panel A: Price model, P, = ~t + f iX, + e,

17.3 0.41 - 28.5 8.0 15.0 24.0 fl 4.6 0.13 - 10.0 2.0 4.1 6.4 Adj. R 2 21.8 - 32.5 4.0 17.4 34.5 White 3.03 95 significant at 1.5 2.7 4. I

star. 0.05 p-value

85.2 30.0 95.3

Panel B: Return model, P,/P~_ i = ~ + [3X,/P,_ i + e, :c 0.90 0.01 - 0.03 0.80 0.90 0.10 fl 2.3 0.06 -- 14.1 1.3 2.0 3.1 Adj. R z 15.5 - 50.0 3.7 12.4 23.5 White 2.19 26 significant at 1.2 1.9 2.9

star. 0.05 p-value

2.40 :3.8 91.0

Panel C: Differenced-price model, ziP, = ~ + f l dX t + e,

1.3 0.05 - 5.8 0.4 1.0 2.1 fl 3.5 0.13 -- 10.6 1.1 2.5 4.7 Adj. R 2 9.1 - 11.1 - 1.0 4.9 14.7 White 2.02 6 significant at 1.1 1.9 2.8

star. 0.05 p-value

10.9 34.4 82.3

Panel D: Deflated-price model, P J X , = ~t( l /X,) + fl + e,

15.6 0.43 - 75.5 6.1 12.3 21.8 fl 5 .0 0 .14 - 11.6 2.3 4.7 7.2 Adj. R z 47.3 - 29.5 23.5 48.6 72.4 White 3.07 93 significant at 1.6 2.6 4.0

star. 0.05 p-value

78.2 39.0 98.6

In each panel, average value of the estimated coefficients, adjusted R2s, and White (1980) chi-square statistics from 1,017 time~ries regressions are reported. Standard errors are for the cross-sectional sample mean coefficients. Qt and Q3 are 25th and 75th fractiles of the distribution of estimated coefficients, adjusted RZs, or White statistics.

Sample: The sample includes any firm that has at least 20 consecutive annual earnings and return observations. X, is annual earnings per share excluding the extraordinary items and earnings from diseontinucd operations. Price relatives exclusive of dividends, P,/P,_ ~, are measured over calendar years. Only firms with a December fiscal year-end are included. The largest and smallest 1% of observations for each variable are excluded. Earnings, X , and prices, P,, are adjusted for stock splits and stock dividends when deflating by P,_ i and obtaining differenced variables.

t h a n t h a t for the c ross - sec t iona l sample . T ime-se r i e s e s t i m a t i o n m a y thus yield

c o n s i d e r a b l y d o w n w a r d - b i a s e d s lope coefficient e s t ima tes a n d / o r cross-sec-

t i ona l e s t i m a t i o n m a y resul t in u p w a r d - b i a s e d s lope coefficient es t imates , b u t

a c o m p l e t e r e s o l u t i o n of this issue is b e y o n d the scope of this paper .

S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (1995) 155-192 [77

Results of the return and differenced-price model estimations are provided in panels B and C of Table 5. The return model yields an average c ~ t ~timate of 2.3 and the differenced-price model yields 3.5. The average c ~ n t from the time-series estimation of the return model is larger than that from cross-sectional estimation. In contrast, both time-series and cross-sectional estimations of the differenced-price model yield average slope coefficient estimates of similar niagnitudes. Thus, comparison of the cross-sectional and time-series estimation of the price, return, and differenced-price models does not reveal a consistent pattern in the estimated slope coefficient magnitudes. There- fore, one cannot conclude that, because the average coefficient estimate from time-series estimation of the price model is smaller than cross-sectional estimation, the latter is biased upward.

The deflated-price model results are presented in Table 5, panel D. The average slope coefficient estimate, 5.0, is close to that obtained from the price model estimated using the Cochrane-Orcutt procedure (not reported in Table., 5). ~6 Deflation by Xt, however, is not particularly helpful in reducing the frequency of White test rejection.

In summary, all specifications yield cost of capital estimates that exceed the estimate implied by the average price-earnings ratio of 12-13. The cost of capital estimates from the price model are close to economically sensible values and they track time-series variation in interest rates. The cost of capital estimates from the price model arc also relatively invariant to using cross-sectional, time-series, or pooled estimation methods. All of the models have serious problems of hetcroscedasticity and/or other misspecification, with the price model being particularly severely affected. The deflated-price model reduces these problems.

5. Sensitivity of results to ,arious assumptions

This section examines whether the inferences in previous sections are sensitive to the assumptions underlying the analysis. In particular, we examine whether: i) transitory components in earnings affect the various models differently and thus explain the observed differences across models; ii) omitted variables explain the nonzero intercepts and whether the estimated slope from the price model is upward biased because of a positive cross-correlation between the earnings variable and the firm-specific coefficient; and iii) instrumental-variable regressions yield 'better" coefficient estimates.

trAverage coeff-ic/ent from the book-va|ue-deflated model, 8.91, is comparable to that from pooled and cross-sectional estimations.


5.1. Transitory components in earnings and linear regression

5. !. 1. Transi tory earnings components" effect on coefficient magnitudes To assess the degree of bias in the estimated coefficients from the various

models, we use I /r as the benchmark, based on 'earnings following a random walk" as one of the assumptions. However, earnings changes exhibit mild negative serial correlation. If this violation of the underlying assumption affects the various models differently, then we cannot unambiguously conclude that earnings response coefficient estimates from the price model are less biased. We briefly discuss the effect of transitory components in earnings on the coefficient estimates from the various models.

If earnings are a mixture of a random walk and a zero-mean transitory process, then the coefficient on earnings (i.e., the sum of these two components) will be a weighted average of the coefficients on the random walk and transitory components. Since the coefficient on the random walk component is fl = 1/r and that on the transitory component is 1, the coefficient on earnings in a price model will be smaller than fl (assuming r < 100%). Specifically, the coefficient will be k(fl - 1) + 1, where k is the ratio of the variance of the random walk component of earnings to the sum of variances of the random walk and transitory components of earnings, i.e., 0 < k _< 1 (proof available on request). If earnings are entirely permanent, then k = 1 and the coefficient will be ft. Alternatively, if earnings are entirely transitory, then k = 0 and the coefficient is 1. Note, howe'-:er, that the average-price-to-average-earnings ratio for a sample of firms will be close to fl precisely because earnings averaged over time and across firms, by definition, are expected to have a zero transitory component.

Our results in Table 2 and from cross-sectional regressions in Section 4.2 are consistent with the presence of transitory components: coefficients from the deflated-price model are 9.0 to 9.6, compared to an average price-earnings ratio of 12-13. The difference between the estimated slope coefficient and the average price-earnings ratios indicates that transitory components account for about 25% of the variation in earnings. We also find (but do not report) that, as expected, including special items and earnings without special items as two distinct independent variables in the price and deflated-price models yields a larger earnings response coefficient on earnings without special items.

Transitory earnings components affect the return-model analysis in much the same way as the price-model analysis. Expectational error in the earnings variable due to the forward-looking nature of price, however, complicates the returr~Imodel analysis. In the absence of prices leading earnings, however, transi,~ory components" effect on the return-model-based earnings response coefficient estimate will be identical to that for the price model. When prices lead earnings, to the extent that a portion of the transitory component is also anticipated, the estimated earnings response coefficient from the return model will be biased downward.

S.P. Kolhari. J.L. Zimmerman / Journal of Accounting and Economics 20 (1995) 155-192 179

5.1.2. Nonlinearity due to transitory components Freeman and Tse (1992) argue that the return-earnings change relation is

nonlinear due to transitory earnings components. They argue further that an earnings change is more likely to be transitory if it is relatively large (e.g., when current earnings are negative and the earnings change is large and negative). Thus, the return-earnings change relation will be relatively flat when earnings changes are negative or large and positive, giving rise to the S-shaped return- earnings change relation first documented by Beaver, Clarke, and Wright (1979). A similar intuition underlies the nonlinear return-earnings changes specifications examined by Cheng, Hopwood, and McKeown (1992), Beneish and Har- vey (1992), and Das and Lev {1992). Our discussion of transitory components is applicable even if the importance of transitory components is not uniform across different magnitudes of earnings changes. The main point of our paper is that when prices lead earnings, return and differenced-price models yield biased coefficient estimates, but the price model does not. To keep the analytics simple in demonstrating this point, ¢¢e do not address nonlinear price-earnings specifications. Nevertheless, to demonstrate nonlinearity in the price-earnings relation, we estimate various models excluding observations that are dominated by transitory earnings, namely earnings loss observations. The average earnings response coefficient from the return model increases from 1.65 in Table 3 to 3.18, and that from the deflated-price model increases from 9.6 to 11.0.17 Thus, controlling for the effect of tcansitorj earnings components does not fully explain the differences in the magnitudes of coefficients from price and return models.

5.1.3. Nonlinearity due to small-price deflator Another source of nonlinearity is the use of price as a deflator in the return

model, particularly when the stock price approaches zero. Extremely low-priced stocks behave more like at-the-money options. For these stocks, even modest good news in earnings can produce a dramatically high rate of return, thereby inducing a nonlinear price-earnings relation, is To assess whether the low coefficient magnitudes from the return model are due to deflating by low prices, we reestimate all the models using only observations with Pt- i > $3.00 and using the book value of assets zs the deflator in the return model. There are relatively small changes in the coefficient estimates relative to those reported

tTHayn (1993) reports similar results for the return model.

tSThe high rate of return is likely to be a consequence of the market revising its cash flow expectations upward and a decline in the stock's expected rate of return. Very low-pr/ced stocks typically have an extremely high market-valued leverage ratio, which will likely drop substantially if good earnings news is reported. This unanticipated change in leverage will be associated with a decline in the stock's expected rate of rate and therefore a positive stock return will be realized.

180 S.P Kothari. JL. Zimmerman / Journal of Accounting and Economics 20 (1995) 155-192

earlier. For example, the average coefficient from annual cross-sectional price- model regressions ci~anges from 7.9 in Table 3 to 7.8. The corresponding numbers for the r~turn model are 1.65 and 2.0.

5.2. Omitted-variables bias

This section investigates whether the nonzero intercepts reported in Section 4 are explained by firm size as a summary proxy for omitted variables from the various models. We first explain the choice of firm size as a proxy for omitted variables and then report results. This section also examines whether a correlated-omitted-variables bias explains the observed differences in the earnings response coefficient magnitudes from the various models. In particular, the correlated-omitted-variables bias arises because earnings response coefficients are an inverse function of systematic risk and there is an empirical correlation between risk and the earnings variables in the various models. Since risk is not included in the price-earnings models as a separate explanatory variable, a correlated-omitted-variable bias arises. Our tests suggest that the resulting bias is too small to explain the large difference between the price- and return- model earnings response coefficient estimates.

Nonzero intercept. Since the intercept is predicted to be zero, the significant nonzero estimates reported in the previous section suggest omitted variables from the model. In the absence of theoretical guidance on possible omitted variables, we add the natural logarithm of the market value of equity at the beginning of period t as a second independent variable. Firm size is known to be correlated with a wide range of variables including expected stock returns, systematic risk, variance of raw and abnormal returns, stock price, and dividend yield. Given that firm size is a catch-all proxy, we conjecture that it would be correlated with the omitted variable(s) as well. In the price and deflated-price models we also expect size to be useful because it captures some of the forward-looking information in price (the dependent variable) that is missing from current earnings. The explanatory power should therefore increase~

The average estimated intercepts of the price and deflated-price models are insignificantly different from zero when firm size is included. The earnings response coefficient estimates remain highly significant, but they decline to 6 for the price model and 7.7 for the deflated-price model. The coefficients on firm size are also reliably positive. The average explanatory powers increase by 12% (i.e., from 53% to 65%) and 15% ~i.e., from 39% to 54%), consistent with firm size prox)ing for omitted variabJes.

Unlike the price- and deflated-price model results, the average estimated intercept from the return model, however, remains reliably positive. In fact, it increases from 1.01 to 1.14 upon the inclusion of firm size in cross-sectional regressions. The average estimated coefficient on firm size in the return model is

S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (1995) 155--192 18|

reliably negative, consis:e.nt with a negative relation between firm size and expected rates of return (e.g., Banz, 1981; Reinganum, 1981),

While firm size is helpful in rendering the average estimated intercept insigni- ficant in the price model, White's (1980) test continues to indicate heteroscedasticity and/or specification problems for the price, deflated-price, and differenced-price models. More importantly, the White test statistic for the return model is significant in 33 of the 38 cross-sectional regressions. Without firm s/ze, it was significant in only six years. Since firm size is negatively correlated with the variance of returns, the residual variance is correlated with firm size as an independent variable, causing heteroscedasticity. Consequently, the White test statistic is frequently significant.

Correlated-omitted-variable bias. Since the earnings response coefficient in all the models is assumed to be a cross-sectional constant, regression errors include (fli - fl)Xi,, where fli is firm fs coefficient, (fl~ - r ) is the deviation of f irm/ 's coeffic/ent from the cross-sectional average, r , and X~, is earnings per share. In addition to contributing to heteroscedastic errors, constraining the coefficient to be a constant creates an upward bias in the coefficient estimate in the price model because of a positive correlation between (fl~ -- r ) and X~t. To see this, first note that, ceteris paribus, low-risk stocks have higher eacnings respr, nse coefficients, i.e., fl~ - ,6 > 0; conversely, fl~ - / / < 0 for high-risk stocks. Thus, f l i - / / and risk are negatively correlated in the cross-section. Next, as an empirical matter, earnings per share and stock price are both decreasing func- tions of risk, because both correlate positively with firm size, which is well- known to be inversely related to the expected rate of return. 19 The net result is that f l i - fl and earnings per share are positively correlated. The resulting upward bias can be viewed as arising from a firm's risk being a positively- correlated-omitted variable from the regression. A similar analysis indicates that the coefficient from the return model is likely to be downward-biased.

To assess the degree of bias, we reestimate the price and deflated-price models with the capital asset pricing model (CAPM) beta included in the regressions. The CAPM beta is estimated using 60 monthly returns prior to year t. e° The average coefficient on earnings from 38 cross-sectional price-model

ZgThe analysis is more compl/cated when one accounts for the positive relation between earnings changes and risk changes (Ball, Kothafi, and Watts, 1992). 2°Betas estimated using monthly returns might not adequately capture cross-sectional variation in expected returns ~e.g., Handa, Kothari, and Wasley, I989: Kothari, Shanken, and Sloan, 1995), which might inhibit finding the correlated-omitted-variable bias in the price-earnings regressions. We use betas estimated using monthly returns because for individual firms the use of longer-than- monthly return observations to estimate betas sacrifices statistical p~'ecision potentially due to nonstationarity. Another reason is that the effect of inclnding betas estimated from monthly returns in the price-earnings regressions [s so small that it seems unlikely that better estimates of beta would alter the tenor of the resu,ts h~ the paper.

182 S.P. Kothari, J.L. Zimmerman / Journal of Accounting and Economics 20 (1995) 155-192

regressions is 7.6 (standard error = 0.40) compared to 7.9 reported in Table 3. The corresponding numbers for the deflated-price model are 9.1 (standard error = 0.53) and 9.6. As expected, the coefficient on beta is negative. Inclusion of beta has a negligible effect on the average coefficient on earnings from the return model: it increases from 1.65 in Table 3 to 1.68.

As a second proxy for risk in the price model to mitigate the upward bias in the estimated earnings response coefficient, we employ the natural logarithm of firm size. These results were briefly summarized earlier in this section in the context of using firm size as a proxy for omitted variables from the price-earnings models. Overall, results using both beta and firm size to mitigate the upward bias in the price- and deflated-price models" coefficient estimates indicate that the bias is small and, perhaps more importantly, the main result of the paper that price and deflated-price models yield substantially less biased coefficient estimates is unaffected.

5.3. Instrumental-variable estimation

Since the estimated slope coefficients from the price model are smaller than the sample firms" average price-earnings ratio, the price-model estimated slopes are potentially biased downward. Instrumental-variable estimation is a common approach to obtain less biased slope coefficient estimates. We use average earnings of all the sample firms belonging to a two-digit SIC industry as the instrumental variable for earnings on all the firms in that industry. We obtain an average estimated instrumental-variable earnings response coefficient of 7.9, which is the same as that obtained from the OLS estimation of the price model. Alternative interpretations of the results are that two-digit SIC code membership is not a very good instrumental variable, and/or the value-irrelevant noise is cross-correlated such that industry-level regressions are not belpful in mitigating the bias they cause, or there are omitted variables.

6. lmlflieations for other research

We show that, if earnings follow a random walk and if prices reflect a richer information set than in the current and past time series of earnings, then price models (in which stock prices are regressed on earnings per share) yield unbiased slope or earnings response coefficients. By contrast, return models (in which stock returns are regressed on earnings per share deflated by beginning-of-year stock price) and differenced-price models (in which stock price changes are regressed on changes in earnings per share) yield slope coefficients which are biased downward. However, price, return, and differenced-price models all yield significant nonzero intercept coefficient estimates, inconsistent with the theory.

S.P. Kothari. J.L. Zimmerman / Journal of Accounting and Economics 20 ~! 995) 155- i 92 | 83

The return model exhibits less serious heteroscedasticity and/or other spec/fication problems compared to the price and differenced-price models.

The tests indicate that transitory components in earnings explain the fact that the price model yields coet~cieats that are smaller than average price-earnings ratios. Transitory components, however, do not explain the observed differences between the estimated coefficients from the price and return or differenced-price models. Including size in the price and deflated-price models yields intercepts that are indistinguishable from zero. Bias due to a correlation between earnings per share and the expected rate of return appears small. The instrumental- variable regressions do not yield earnings response coefficient estimates that are any closer to the price-earnings ratios than those reported earlier. Researchers should be aware of the econometric limitations of the various models in designing their tests. Future research should address the issue of nonlinearit/es in the price-earnings relation.

Our findings have implications for capital market research in accounting. Currently, much of the research uses the return model as the functional form. The findings in this paper do not suggest using either price or return models exclusively, because both have serious econometric problems and both have impo~_ant deviations from the underlying theoretical model. Future studies can be enriched by testing for sensitivity to the functional form and by incorporating the relative strengths and weaknesses of alternative specifications. Using the price model, perhaps in addition to the return model, could permit more definitive inferences.

For example, many studies hypothesize that the earnings response coeffcient will change around or during an information event because of accrual manipulation (e.g., Collins and DeAngelo, ! 990, Collins and Salatka, 1991 ). Alternatively, the coefficient also changes if the information content of prices or earnings changes such that the unexpected earnings proxy used by ~ researcher becomes more or less noisy over time, e.g., Skinner (1990) studies option-listing and Rao (1989) studies firms after their initial public offerings. Use of price models, in addition to return models, in the above research contexts could be useful in drawing inferences about managemeat's accrual manipulation (via transitory earnings) or the timeliness of earnings. If accrual manipulation introduces random noise or transitory earnings, then both price and return regressions should yield smaller earnings response coefficient estimates. If accrual manipulation biases the earnings per share of all finns by a constant fraction, then, depending on the upward or downward bias, both price and return models should yield lower or higher slope coefficient estimates in the event period compared to coefficients in the nonevent period. Finally, if the earnings response coefficient is expected to change because prices are forward-looking, then, unlike the return model, the price model does not predict a change in the earnings response coefficient (because the forward-looking nature of prices does not bias the earnings response coefficient estimate from the price model). The


use of both return and price models has the potential to yield more convincing evidence.

T --ts of information content of accounting earnings and its components are also" .mmon in the accounting literature. These tests either assess the signifi- cah~ of estimated slope coefficients or test incremental explanatory power of a set of variables (e.g., Beaver, Griffin, and Landsman, 1982; Beaver and Landsman, 1983; Barth, Beaver, and Landsman, 1992). While we analyze only a simple regression of prices on earnings, the advantage of price studies is that even in a regression of prices on various revenue and expense items they yield unbiased (or less biased) slope estimates compared to return studies. One must, however, be careful in interpreting the coefficients on various revenue and expense variables from a price regression. The coefficient magnitudes will depend on the time-series properties of these items and the riskiness of the various items. In addition, the empirical correlation among the variables can be important. Therefore, coefficients on various revenue and expense items are not expected to be equal (Jennings, 1990).

Finally, while price mcdels likely yield less biased slope estimates in information content studies, it is important to recognize that price models do not measure information arrival over a period. The dependent variable, price, is not a measure of the impact of information arriving in the current period. In an efficient market, the impact of information over a period is measured by stock returns (i.e., the deflated change in the price level). However, the explanatory power of return models provides only a lower bound on the information content of an accounting variable because of the errors-in- variables problem discussed herein. Stated differently, unless the market's earnings expectations are proxied accurately, the return model R2s understate the extent to which current period's accounting numbers reflect the information affecting security prices. While a significant association between returns and accounting numbers indicates information content, the low R2s of the return studies might potentially lead researchers to draw incorrect inferences, e.g., Lev's (1989) inference that earnings contain "noise' or Shiller's (1989) conclusion that much of the stock market's volatility reflects investor irrationality.

Appendix

This appendix describes a stylized valuation model that assumes that prices are set in the market using only the information in the current and past time series of earnings. Under such a valuation model, and when earnings follow a random walk, the price, return, and differenced-price specifications of the price-earnings relation are equivalent. We then examine alternative models by allowing prices to reflect a richer information set than the current and past time


series of earnings. Finally, implications of value-irrelevant noise on alternative specifications are briefly examined.

Alternative specifications under the stylized valuation model. We make the following assumptions: (i) earnings for a period con temporaneous ly reflect all the information that is in the return over the period, i.e., prices do not lead earnings; (ii) the expected rate of return is constant through time; (iii) earnings follow a random walk; and (iv) the dividend payout ratio is 100% (i.e., earnings equal dividends). Assuming dividends equal earnings (assumption iv) simplifies the analytics. We also assume that shareholder wealth is unaffected by the d iv idend-payout policy, because the firm's dividend policy per se is assumed not to signal anything abou t the profitability of future investments. Therefore, implications of the analysis below hold under the more realistic assumpt ion of a less-than-100% dividend payout . Under the assumptions (i) through (iv), prk:e is given by Pt = E,(Xt+ t ) /r = X J r .

Price specification. The price regression model is

P, = a + / 3 X , + et, (A.I)

where a and/3 are the intercept and slope coefficients, and the error term, e,, is included because empirically the assumptions underlying the valuat ion model may be violated. The estimate of/3 is 21

b = cov(Pt , X,) /var(X, ) . (A.2)

Eq. (A.2) is simplified by substi tut ing P, = X , / r from the price model (1) to obtain

b =" cov(Xt /r , X , ) /var(X,} = 1iv =/3. (A.3)

Using (1), {A.2), and (A.3), the est imated intercept can be shown to be zero. Return specification. T o derive the return specification, we divide Eq. (1) by

price at the beginning of period t:

Pt/Pt- l = (1/r}Et(X,+ l ) /P , - I. (A.4)

zt Since price and earnings follow a random walk, their (time-series) variances are undefined. [We define b in Eq. (A.2) merely for notational comparability with the expressions for b using other price-earnings specifications.] However, b is well-defined in the sen~ that it can he estimated as a projection of P, on X, using the sample observations. If the price and earnings vectors are cointegrated {i.e.. even though the two variables follow a random walk, the errors have a zero mean and constant variance because movements in the two variables are governed by common factors; see Engle and Granger, 1987; Maddala. 1990). then there is no econometric difficulty in estimating the price model (1). To mitigate potential econometric problems in estimating the valuation models, we also estimate model (I) using a suitable deflator (see estimation of deflated valuation models in Section 4).

186 S.P. Kothari, J.L. Zimmerman / Journal o f Accounting and Economics 20 (1995) 155-192

The empirical analog of (A.4), using X, as the time-series expectation proxy for E , ( X , + l), is

P , / P , _ ~ = • + X , / P , _ t + e.,. (A.5)

Analogous to Eq. (A.3), E(b) =/~ = 1/r, and • is expected to be zero. Therefore, the price and return models give equivalent estimation of the slope coefficient.

F i r s t - d i f f e r e n c e d - p r i c e spec i f i ca t ion is

P, - E , _ , ( P , ) = A P t = ( 1 / r ) [ X , - E,_ ~(X,)] = ( 1 / r ) A X t . (A.6)

By substituting the pricing model (1), the earnings response coefficient estimate from an empirical analog of (A.6) can be shown to have Elb) =/~ = l / r .

A l t e r n a t i v e spec i f i ca t i ons w h e n pr i ce s lead earn ings . Section 2.2 in the text describes the 'prices-lead-earnings' assumption. The analysis below examines the effect of prices containing information about two future periods' earnings changes on the alternative price-earnings specifications. Using Eq. (5) from the text, the market's expectation of future earnings is

E~(X,+ l) = X , + at+ , . , + at+ l.t- I (A.7)

and

E~{X,+k) = X , + a , + l . t + a , ~ l . , - t t - at+2.t for k >_ 2. (A.8)

Thus, the market expects a perpetuity of X ~ + a t + t . t + a t + t . t - i +a t+za , except that in period t + 1 earnings are expected to be lesser than the perpetuity by a,+2. , . Therefore, P, is given by the present value of the perpetuity minus a, + 2., discounted over one period:

P, = [(Xt + a , + t . , + a , + t . , - , + a ,+2 . , ) / r ] - [a,+2.,/{1 + r)]

~, ( X , + a,~ t . , + at+ l . r - 1 + at+ 2 . t ) / r , (A.9)

where the approximation is obtained because a,+ 2 . t / ( l + r) represents a discounted one-period cash flow from the anticipated earnings a,+2.t and thus makes only a small contribution to P,.

P r i c e spec i f i ca t ion . The earnings response coefficient estimate from a price regression is

b = cov{P,, X,)/var(X,)

cov[{P,-2 + A P , - I + 3P t ) ,Xz -2 + St--I + a t - t . , -2 + a t - l . t - 3 + st + at . t- l + ar.t-2]

var[X,-2 + s,-t + a , - t . , - z + a t - t . t -3 + st + ar,t-t + at.t-2]

coy [P,_ z.{X,-- z + a,_ t.,- 2 + a,_ l.,- 3 + a,.,_ 2 )] + coy [APt ._ 1 .{s, _ t + a .... t )] + cov[ AP,, s,]

var[X,_z + a,_ l . , -z + a,_ 1.,-3 + at . t -2] + var[s, t + a,.~ i] + var[s,]

I l / r )var[X,_ 2 + at- l . , - 2 + a,_ x.z- 3 + at.,- z] + { l/r)var[st _ 1 + a,.,_ t J + { l/r)var[st] =. var[X,_ 2 + a,_ t., - z + a,_ i . , - 3 + a,.,_ z] + var [s,_ t + a,.,_ t] + vat[s,]

= l/r. {A.10)


The price specification thus yields an unbiased estimate of the earnings response coefficiet,,. The intuition, once again, is that the econometric consequence of prices containing information about future periods' earnings changes is that there is an uncorrelated-omitted-variables problem that leaves the estimated slope coefficient unbiased.

Return specb'ication. The earnings response coefficient estimate from a return model is

b = coy [{Xt- 1 + st + at . t- l + a t . t - 2 ) / P t - l , P t / P t - l]

var[(Xt_ l + st + at . , - t + at . t - 2 ) lP t - 1]

cov(s t / e t - ~ , Pt / P t - a)

= var[{Xt_ ~ + at. t- ~ + a t . t - z ) / P t - l] + var [ s t /P ,_ 1]" (A.11)

To simplify the expression for b, first focus on the var[(X,_ ~ + at. t - 1 + at . , -2)~

Pt-1] term in the denominator of Eq. (A.I 1). Since (X,- l + a:.,_ i + at.t-,_ + at+ l . t - l ) / P t - i ] ~ r is a constant [see Eq. (A.9)], ( X t - 1 + at . t - l + a t . , - 2 ) / P t - 1 and a,+ L , - ~ / P t - ~ are (almost) perfectly negatively correlated and, therefore, var[(Xt_ 1 + at. t- 1 + a t . , - 2 ) / P t - 1] = var(at+ Lt- 1/Pt - 1). To derive the degree of bias in the earnings response coefficient estimate, we must make assumptions with respect to the relative magnitudes of the variances of st, a,.t- ~ and a, . t -2- If the variances of s,, at . t -~, and a,.t-2 are assumed equal, v a r l a t + L , - t / P t - l ) = var l s , /P t_ t). Substituting this result in Eq. (A.11),

b = c o v ( s t / P , - l , P t / P t - 1 )/2 * var ( s t /P t - l ), (A.12)

which implies E(b) = 0.5 ,(I/r). The return model yields biased slope coefficient estimates because earnings changes are anticipated. The greater the degree of earnings anticipation (i.e., larger variances of a,.,_ ~ and at.,-2) relative to the variance of the surprise component of X,, i.e., s,, the greater the degree of bias in the earnings response coefficient estimate. Also, the bias will be greater if prices anticipate earnings changes more than two periods ahead (e.g., Kothari and Sloan, 1992) because st will then be a relatively smaller component of A X t . Note, however, that our objective is not to determine the exact degree of bias, but merely to demonstrate that the return specification produces a biased coefficient when prices lead earnings.

Differenced-price specification. We derive the bias in the estimated earnings response coefficient from the differenced-price specification when prices lead earnings by one period. We do not explore the bias under the assumption that prices lead earnings by two periods because the estimated coefficient is biased even when pi~ce.~ anticipate one-period-ahead earnings changes. Barth, Beaver, and Landsman (1992), among others, provide an intuitive discussion of bias in the estimated slope coefficients from a differenced-price model that could arise if changes in earnings do not accurately proxy the surprise in earnings to the market.

188 S.P. Kothari. J.L. Zimmerman / Journal of Accounting .nd Economics 20 (1995) 155-192

The slope coefficient estimate from the differenced-price regression model is given by

b = coy [AP t, AXt] /var(Xz)

= coy [(l/r) (st + at + 1.t), (st + at.t - t )]/var [st + at.t - t ]

= (1/r)var{st){var(st) + var(at.t-1)}. (A.13)

Eq. (A.13) indicates that the earnings response coefficient estimate will be biased because of the vat(at.t- 1) term in the denominator. Once again, the degree of bias will be an increasing function of the extent to which earnings changes are anticipated by the market versus they are a surprise.

Alternative specifications when earnings contain value-irrelevant noise. The notion of a value-irrelevant component in earnings is as formalized in Eq. (6) of the text. The earnings component without noise, xt, is assumed to follow a random walk:

xt = xt-1 + ~h, (A14)

where rh has a zero mean and variance of tr 2 , and it is serially uncorrelated, u, is also a zero mean, serially uncorrelated, white noise term with variance a , z. x, is perfectly correlated with price and u, is uncorrelated with xt as well as price. The pricing equation is

Pt = (1/r) EtU(xt + l) = ( l /r)xt . ~A.15)

Price specification. The slope coefficient estimate from a price regression model is (see Landsman and Magliolo, 1988)

b = cov(Xt, Pt)/var(Xt)

= coy [(xt + u,), Pt] /var[x t + ut]

= coy(x,, Pt)/{var(x,) + var(ut)}

= (1/r)[1/{ 1 + var(ut)/var(xt)}], (A.16)

where, given the price equation (A.15), we substitute 1/r for cov(xt, Pt)var(xt). Eq. (A.16) indicates that, unless var(ut) = 0, b is biased toward zero. The bias increases in the ratio of the variances of ut and x,.

Return spec!t~cation. The slope coefficient from estimating the return regression model is

b = coy ( X J P t - a, Pt /Pt - t ) / var (X t /P t - 1)

= coy [-(xt + ut)/Pt-1, PdPt-1]/var[ ' (x t + ut)/P~-t]

= c o v ( x J P t - l, P J P t - 1 ) /{var(xJPt_ i) + var{uJPt- 1 )}

= {1/r)[l /{ 1 + var(ut/Pt- 1)/var{x,/P,- 1)}], (A.17)

S.P. Kothari, 3.L. Zimmerman / Journal 6f ,~.ccounting and Economics 20 (1995) 155-192 189

where we substitute 1/r for cov(xJP,_ 1, Pt/Pt- t) /var(x,/P,- 1 ). As in case of the price specification, the return specification also yields a biased slope c ~ t estimate because earnings contain valuation-irrelevant noise. The degree of bias is determined by the ratio var(u~/P,_ ~)/var(x,/P,_ ~).

Differenced-price specification. The analysis here is similar to that for the price specification. Specifically, the estimated slope coefficient is given by

b = (Â/r)[l/{1 + var(Aur)/var(Axt)}]. (A.18)

The degree of bias in Eq. (A.18) depends on the ratio of var(Au,) to var(Axr).

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