Price-earnings regressions in the presence of prices leading earnings

Journal

of Accounting

and Economics

15 (1992) 173-202.

Price-earnings regressions of prices leading earnings

North-Holland

in the presence

Earnings level versus change specifications alternative deflators

and

S.P. Kothari” UniversitJ’

of Rochester, Rochester, NY 14627. USA

Received October

1991, final version

received April 1992

The paper analytically evaluates alternative specifications of price-earnings regressions when prices lead earnings, i.e., reflect information about future earnings that is not reflected in the past time series of earnings. Because prices lead earnings, the specification using the earnings-level-deflated-by-price variable in a price-earnings regression is ‘better’, in terms of bias in the estimated earnings response coefficient and explanatory power, than specifications using earnings-change-deflated-by-price and earnings-deflated-by-lagged-earnings variables. An accurate proxy for unexpected earnings, however, outperforms the earnings-leveland earnings-change-deflated-by-price specifications.

1. Introduction Recently two topics in financial accounting research have attracted considerable attention: (i) price-earnings regressions specified in ‘levels’ versus ‘changes’ and (ii) strength of the price-earnings association. A price-earnings relation in the ‘levels’ is typically a regression of annual return on the contemporaneous year’s earnings deflated by beginning-of-the-year price [e.g., Easton and Harris (1991), Biddle and Seow (1991a), and Kothari and Sloan (1992)l.l In the *I thank Andrew Christie, Bob Lipe, Richard Sloan, Ross Watts (the editor), Jerry Zimmerman, and Douglas Skinner (the referee) for useful comments. I acknowledge financial support from the Bradley Policy Research Center at the Simon School, University of Rochester, and from the John M. Olin Foundation. ‘The term levels here is a misnomer because earnings deflated by price, being the accounting rate of return on the market value of equity, can be considered a change or return variable. I, however, use the term levels for consistency with the recent accounting literature and to differentiate it from a change-in-earnings-based explanatory variable.

0165-4101/92/$05.00

0

1992-Elsevier

Science Publishers

B.V. All rights reserved

174

S. P. Korhnri,

Price-earnings

regressions

‘changes’ specification, the independent variable typically is annual earnings change deflated by the previous year’s earnings or the beginning-of-the-year price [e.g., Beaver, Lambert, and Morse (1980) and Easton and Harris (1991)]. Ohlson (1989, 1991) and Biddle and Seow (199la,b) initiated interest in the levels and changes specifications of the price-earnings relation.2 Assuming earnings follow a random walk and current earnings are sufficient to determine the stock price, Ohlson (1991) and Biddle and Seow (1991a, b) show that earnings level and change deflated by price correlate equally with stock returns. Ohlson (1991) and, more recently, Ohlson and Shroff (1991)3 extend the analysis of the price-earnings relation in the levels and changes by formally incorporating the intuition that prices reflect a richer information set than that in the past time series of earnings or cash flows [see, for example, Beaver, Lambert, and Morse (1980), Fama (198 1, 1990) Kleidon (1983), Collins, Kothari, and Rayburn (1987), Freeman (1987), and Kothari and Shanken (1992)]. Assuming that prices contain information about one-period-ahead earnings change, they show that the levels specification can yield an unbiased slope coefficient of the price-earnings regression, whereas the change specification results in a biased slope coefficient. While they argue that specific assumptions will determine which of the two specifications is better, the thrust of their analysis appears to be that, because prices contain information about future (one-period-ahead) earnings changes, the earnings-level variable is ‘better’ than the earnings-change variable in a priceearnings regression. Many express concern that the strength of the priceearnings association is weak [e.g., Lev (1989), Bernard (1989), Lev and Thiagarajan (1991), and Klein and Todd (1991)]. The low correlation between returns and earnings changes is often attributed to deficiencies in the accounting measurement system [e.g., Lev (1989)]. This perhaps is a premature conclusion since it fails to recognize that the degree of contemporaneous correlation between stock returns and earnings changes hinges critically on how well an earnings change proxies for the unanticipated component of the contemporaneous period’s earnings change and serves as a proxy for the revision in the market’s expectations of future periods’ earnings changes. This paper makes an empirically plausible set of assumptions to examine the effect of prices containing information about future earnings changes (hereafter, prices lead earnings) on price-earnings relations specified in the levels and changes, with price and earnings as deflators.4 Information in prices is assumed lOthers who analytically or empirically examine the levels and changes specifications include Easton and Harris (1990, 1991) Ramesh and Thiagarajan (1991) Shroff (1991) Warfield and Wild (1991) and Lyon and Schroeder (1991). 3Analyses dently.

in this study and Ohlson

and Shroff (1991) are conducted

concurrently

and indepen-

‘%r comparison, analyses in Ohlson (1991) and Ohlson and Shroff (1991) reveal that, since they make very genera1 assumptions, conclusions will depend on the assumptions one is willing to make. See footnotes 6 and 8 below.

S.P. Koihari, Price-earnings regressions

175

to be useful in forecasting more-than-one-period-ahead earnings changes in examining its effect on the estimated slope coefficient and explanatory power of alternative specifications of the price-earnings relation. Brown, Foster, and Noreen (1985), Freeman (1987), Kothari and Sloan (1992), and Collins, Kothari, Shanken, and Sloan (1992) collectively provide compelling evidence that prices are informative about more-than-one-period-ahead earnings changes. While much of the analysis assumes earnings follow a random walk, implications of the previously documented negative serial correlation in earnings [e.g., Ball and Watts (1972) and Brooks and Buckmaster (1976)] and accrual manipulation are also examined in the paper. The main theoretical conclusions are that, because prices reject information about future earnings changes: (i) compared to the change specification, the levels specification yields higher explanatory power and a less biased earnings response coefficient estimate, where the ‘true’ coefficient is the slope coefficient from a time-series regression of unexpected return on scaled unexpected earnings; (ii) the levels specification yields a biased earnings response coefficient when prices contain information about more-than-one-period-ahead earnings changes; (iii) if an accurate proxy for the market’s unexpected earnings is used, the earnings response coefficient estimate is unbiased and the explanatory power is greater than that using the levels and change specifications; (iv) beginningof-the-year price as the deflator, compared to the previous year’s earnings, yields a less biased earnings response coefficient estimate and higher explanatory power; and (v) the explanatory power of the typically estimated price-earnings regression is expected to be low, perhaps only about 15-20 percent. The intuition underlying the above results is as follows. If annual earnings are decomposed into anticipated and unanticipated components, only the latter is relevant in explaining the contemporaneous annual return. When the earnings-deflated-by-price variable is used, in efect, the anticipated component of earnings, at least in part, is eliminated from the explanatory variable. Since the anticipated component is irrelevant in explaining the contemporaneous return, its reduction/elimination from the earnings variable results in a less biased slope coefficient estimate and higher explanatory power. When the price-earnings regression is specified in changes, regardless of price or earnings as the deflator, the earnings-change variable is not as effective as the levels variable in reducing the irrelevant anticipated component. Consequently, the estimated slope coefficient is biased toward zero and the explanatory power is lower. While the levels specification ‘outperforms’ the change specification of the price-earnings regression, it is important to recognize that both earnings level and earnings change are noisy measures of the market unexpected earnings. If feasible, use of the latter is preferred over both the earnings-level and earnings-change variables. The paper extends the research on price-earnings relations in several dimensions. First, unlike Lipe (1990), Ohlson (1991) and Ohlson and Shroff (1991), who limit the information in prices to be helpful in forecasting only

176

S.P. Kothari. Price-earnings

regressions

one-period-ahead two-period-ahead

earnings changes, I allow prices to be informative about up to earnings changes.5 The two-period-ahead case is interesting because: (i) unlike the one-period-ahead case, analysis of the two-period-ahead case reveals that the estimated earnings response coefficient is biased even if the earnings-level-deflated-by-price variable is used, and (ii) it is empirically more plausible [see, e.g., Kothari and Sloan (1992) and Collins et al. (1992)]. Thus, the earnings-deflated-by-price specification yields an unbiased earnings response coefficient estimate only under the special assumption that prices lead earnings by only one period. Second, none of the previous studies, including Ohlson (1991) and Ohlson and Shroff (1991), formally analyzes the effect of price and earnings as alternative deflators on price-earnings regressions when prices are assumed to lead earnings. The analysis in this paper demonstrates that the errors-in-variables problem in a price-earnings regression caused by a noisy proxy for the market’s earnings expectation is exacerbated by earnings’ inability to accurately reflect the market’s expectation of future earnings. Since price incorporates the market’s earnings expectation, the errors-in-variables problem is mitigated. Finally, recent research questions the longstanding use of unexpected earnings as the explanatory variable in price-earnings regressions [e.g., Ohlson (1991) Easton, Harris, and Ohlson (1992), and Ohlson and Shroff (1991)]. I argue that unexpected earnings is the logical variable in association-study (long-window) as well as event-study (short-window) price-earnings regressions. In price-earnings association studies [e.g., Ball and Brown (1968), Beaver, Clarke, and Wright (1979)], previous research has often employed earnings changes to proxy for unexpected earnings because: (i) it implicitly assumes the market’s expectation is approximated by time-series expectation (i.e., prices do not lead earnings), and (ii) empirically, annual earnings follow a random walk. Thus, the inappropriateness of earnings change as a proxy for unexpected earnings stems from a lack of descriptive validity of the underlying assumptions (i.e., prices do not lead earnings and/or earnings follow a random walk), rather than a weakness in researchers’ logic. In event studies, time-series earnings expectation proxies are likely to be noisy, in part, because the market has access to information since the previous earnings announcement that is helpful in more accurately forecasting the current period’s earnings. Accordingly, researchers frequently employ analysts’ forecasts to obtain a more accurate proxy for the 5Easton and Harris (1991, p. 21) also recognize that prices lead earnings. They, however, do not derive/discuss the implications of prices leading earnings for alternative pricexarnings specifications. In comparing the levels and change specifications, they conjecture that levels could outperform the earnings-change-deflated-by-price specification. They argue that earnings change, deflated by price, could be a poorer proxy for unexpected earnings than earnings level, deflated by price, because of mean reversion in E/P ratios driven by the earnings variable (p. 32). The basis for expecting earnings level to outperform earnings changes is thus unrelated to prices leading earnings. Implications of earnings reversion for earnings response coefficient estimates is addressed in Ali and Zarowin (1992) and section 6 below.

S.P. Korhari, Price-earnings regressions

177

market’s expectation of earnings at the time of the earnings announcement [e.g., Collins and DeAngelo (1990)]. I show analytically that, under a more descriptive prices-lead-earnings assumption, an accurate proxy for unexpected earnings yields an unbiased earnings response coefficient estimate and correlates more highly with stock returns than the earnings-deflated-by-price variable. In practice, however, an accurate proxy for the market-unexpected earnings may be difficult to obain and the researcher may therefore use earnings-deflated-by-price instead. The choice thus hinges on which proxy for unexpected earnings is less noisy, rather than which should be the explanatory variable of interest. This conclusion is not unique to the research on the price-earnings relation. Brown and Warner (1980, 1985) find that simpler, less rigorous methods (e.g., mean-adjusted returns) are at least as powerful as the more theoretical approaches (e.g., market- and risk-adjusted returns) to detect event-period abnormal performance. Results of analyzing price-earnings data for over 2,700 firms with at least five years of data available over the years 1951 to 1988 are generally consistent with the theoretical analysis in this study. Median explanatory power and median earnings response coefficient estimates of firm-specific time-series price-earnings regressions are higher for the levels specification using price as the deflator than for the change specification using price or earnings as the detlator. The average explanatory power for the levels specification using price as the deflator is 15 percent, which is comparable to the degree of price-earnings association that we expect given prices lead earnings. Section 2 of the paper examines the price-earnings relation under a stylized valuation model that assumes prices do not lead earnings. Implicitly or explicitly, the valuation model that assumes prices do not lead earnings underlies much of the price-earnings relation literature. However, this model is introduced only as a benchmark against which results assuming prices lead earnings can be evaluated. Section 2 establishes that, when prices do not lead earnings, the levels and change specifications are identical on the basis of bias in the estimated slope coefficient and explanatory power. Section 3 examines the price-earnings relation when prices lead earnings. This is the only assumption that is different from those underlying the stylized model in section 2. The effect of the prices-lead-earnings assumption is that the level specification outperforms the change specification on both dimensions. Section 3 also discusses the appropriateness of estimating price-earnings regressions using unexpected earnings versus earnings level, both deflated by price. Section 4 focuses on price versus earnings as the deflator. The analysis is carried out initially assuming prices do not lead earnings, and then assuming prices lead earnings. The empirical evidence is presented in section 5. Sensitivity of the results in this study to some of the underlying assumptions is examined in section 6. In particular, effect of

178

S.P. Korhari, Price-earnings

the observed small changes is analyzed. 2. Price-earnings

regressions

degree of negative serial correlation in annual earnings A summary and conclusions are provided in section 7.

relation under a stylized model

Assumptions: (i) earnings for a period contemporaneously reflect all the information that is reflected in return over the period, i.e., prices do not lead earnings; (ii) only infromation about earnings (more precisely, expected future ca& flows) affects stock prices (thus, discount rate changes, for example, are ignored); (iii) earnings follow a random walk; and (iv) the dividend payout ratio is 100 percent. All the assumptions are made to highlight the effect of earnings on stock prices, ignoring other factors [see, for example, Beaver, Lambert, and Morse (1980) and Fama (1990) for a similar approach]. In particular, the lOO-percent-payout-ratio assumption is made to simplify the econometric analysis of the price-earnings relation. Economic intuition is, however, not sacrificed by this assumption and implications from the analysis below hold under the more realistic assumption of a less-than-loo-percent dividend payout. While not descriptive, the assumptions are useful because, once assumption (i) is relaxed, the resulting analysis predicts differential effects of alternative specifications of the earnings variable. Under assumptions (i) through (iv), price is given by

P, = E,(X,+ I l/r = X,/r,

(1)

where P, is the ex-dividend price at time t; E,( .) is the expectation operator conditional on all public information at time t; d, is dividends in period t; X, is earnings for period f; d, = X,, which means no distinction is made between earnings, net cash flows, and dividends; and r is the appropriate risk-adjusted discount rate that is assumed constant through time. If price inclusive of the dividend over a period is used, then (1) becomes

(f’, + 4) = (1 + l/W,,

(1’)

which is similar to (1). In (1’) P, + d, reflects the current period’s earnings and therefore the coefficient on earnings in (1’) is greater by one compared to that in (1) [see, e.g., Collins and Kothari (1989, table l)]. If the payout ratio is less than 100 percent, then earnings are expected to follow a random walk with drift. A positive drift is expected because of earnings reinvestment. Shareholder wealth, however, is unaffected because the dividend policy per se is assumed not to signal anything about the profitability of future investments [Miller and Modigliani (1961), Fama and Miller (1972, ch. 2)]. The valuation relation using prices inclusive of dividends under a less-than-loopercent-payout policy is therefore the same as (1’).

S.P.. Kothari, Price-earnings regressions

2.1.

Earnings-dejated-by-price

179

specijication

Using eq. (1) the earnings-deflated-by-price

specification

where l/r is the earnings response coefficient. The empirical analog of (2) using X, as the time-series proxy for E,(X,+ ), is

is

earnings-expectation

I

P,/P,_l

The earnings

= a + /3X,/P,-,

response

+ E,.

coefficient

(3)

estimate

is

b = Ccov(X,l~,-l,~,I~,-l)llvar(X,l~,-l)

(4)

and, since (1) is assumed, E(b) = fl = l/r. The intercept, (Y,is expected to be zero. Since only earnings information is assumed to affect stock prices and the market’s expectation of future earnings is based only on the information in the past time series of earnings, the price-earnings relation is deterministic. Certainty, however, is not one of the assumptions necessary to predict a deterministic price-earnings relation. The latter is a consequence of the assumptions that earnings are the only source of information and prices do not lead earnings. 2.2. Earnings-change-dejiated-by-price

specijication

This subsection establishes that the earnings-changeand earnings-leveldeflated-by-price specifications yield identical earnings response coefficient estimates. This analysis serves as a prelude to that in the following sections where prices are assumed to lead earnings. Ohlson (199 1) and Biddle and Seow ( 199 1a) also derive this result. The earnings-change-deflated-by-price specification is AP,IP,_,

= c( + PAX,JP,-,

+ E,,

(5)

where AX, = X, - X,_ 1. Eq. (5) can be rewritten P,/P,_I

-

1 = CI+ P(X, - X,_1)/P,_I

as

+ E,.

(6)

Since the assumptions underlying the valuation eq. (1) imply that X,_ 1/Pl_ r = r is a constant, (6) reduces to the earnings-deflated-by-price specification in (3). The only difference is that the intercept will be a’ = (u - fir) = (c( - 1) = - 1, because LI= 0 and fir = 1. Thus, under the assumptions of a random walk in

J.A.E.

C

S.P. Kothari. Price-earnings

180

regressions

earnings, constant expected return through time, and prices not leading earnings, the earnings-leveland earnings-change-deflated-by-price specifications have identical slope coefficients. The

Price-earnings

relation when stock prices lead earnings

This section examines the effect of relaxing the prices-do-not-lead-earnings assumption underlying the stylized model. Prices-lead-earnings is a more realistic assumption. The nature of the historical cost accounting measurement process (e.g., conservatism, objectivity, and revenue-recognition conventions) is such that it has a limited ability to reflect the market’s expectations of future earnings. Therefore, prices are expected to lead earnings. As noted earlier, previous research provides considerable evidence that prices lead earnings. This section shows that if prices lead earnings, the earnings response coefficient estimates from contemporaneous price-earnings regression models (3) and (5) are downward biased (i.e., less than l/r) and the explanatory power decreases. These effects are smaller when the explanatory variable and deflator are X,/P,_ 1 and P,_l than when AX,/P,_, and X,-l are used instead. This section also demonstrates that the use of an accurate proxy for the market’s unexpected earnings yields an unbiased earnings response coefficient estimate and the explanatory power is greater than that using X,/P,_, as the independent variable. 3.1. Prices-lead-earnings

assumption

If prices lead earnings, then the market’s and time-series expectations of earnings differ. That is, the market anticipates, and reflects in price, a portion of the time-series-earnings surprise. Since time-series properties of annual earnings are well approximated by a random walk [e.g., Ball and Watts (1972) and Albrecht, Lookabill, and McKeown (1977)], the time-series-earnings surprise is the change in annual earnings. In the presence of prices leading earnings however, only a portion of the earnings change, AX,, is a surprise to the market and the rest is anticipated in earlier periods, i.e., during t - 1, t - 2, and so on. Initially I assume that information about AX, reaches the market during

S.P. Korhari. Price-earnings regressions

181

period t - 1 and t (which will be referred to as prices lead earnings by one period). This is formalized as a special case of the Lipe (1990) and Ohlson (1991, app. B) models by defining X, as x, = x,-i

+ x, + Lx,,

(7)

where x, + Lx, = AX,, x, is the component of AX, that is a surprise to the market in period t (contemporaneously) whereas Lx, is the component of AX, that the market anticipates by the beginning of period t, and x, and Lx, are uncorrelated.6 Unlike Ohlson (1991), the information reaching the market in period t that forecasts next period’s earnings, i.e., Lx,+ 1, is assumed uncorrelated with x,, the information in period t’s earnings. This is a natural assumption since the intuition underlying eq. (7) is that the market has information that is incremental to that in the past time series of earnings. Lipe (1990) makes a similar assumption. Since information about Lx, reaches the market in period t - 1, but (accounting) earnings do not incorporate this until period t, the market-unexpected earnings are x,. The time-series-earnings surprise, on the other hand, is AX, ( = x, + Lx,) because earnings follow a random walk. The market’s expectation of future periods’ earnings at time t, under the above assumptions, is E,(X,+k) = X, + Lx,+~ The expectation p, =

of a perpetuity

for

k 2 1.

(8)

of X, + Lx,+ 1 implies the stock price will be

w, + LX,.lW.

(9)

It is important to note that the earnings behavior and valuation as modeled in eqs. (7)-(g) differ from the stylized model in section 2 in only one respect: prices are assumed to lead earnings here whereas in section 2 the information set reflected in prices was no better than that reflected in the past time series of earnings. 3.2. Price-earnings

regression

when prices lead earnings

by one period

This subsection compares the levels, X,/P, _ 1, and changes, AX,/P, _ I, specifications of the price-earnings regressions when the prices-lead-earnings assumption is formalized according to eq. (7) and the pricing equation is (9).

61 am grateful to Bob Lipe and Jim Ohlson for pointing out that eq. (7) is a special case of their respective models in Lipe (1990) and Ohlson (1991). Note, however, that Lipe (1990), Ohlson (1991), or Ohlson and Shroff (1991) do not entertain prices leading earnings by more than one period, which is considered in section 3.3 below.

182

3.2.1.

S. P. Korhari, Price-earnings regressions

Earnings-dejated-bqj-price

speciJication

The contemporaneous price-earnings regression of P,/ P, _ 1 on X,/P, _ 1 when prices lead earnings by one period yields the following:

b = CcoW,/P,=

lr

P,/f’- l)l/WX,/Pt-

ccov(w-1 + Lx, +

xt)/p,-l,ptIp,-l)1

+ Lx, + x,)/P,- 1 ).

/var((X,-l

1)

(10)

Eq. (10) is simplified by substituting (X,_ 1 + Lx,)/P,_ 1 = r [see eq. (9)], which is a constant. Thus, while X, contains X,_ 1 and Lx,, both of which are irrelevant in explaining contemporaneous return, P,/P, _ 1, deflation by P, _ 1 decomposes the earnings variable into a constant, r, and x,/P,_ 1, the component that is relevant to explaining P,/P, _ I. Therefore, eq. (10) reduces to

b = cov(x,Ip,-

1,

P,lP,-

1

)/var(x,/P,- 1 ).

(11)

The expected value of b in (11) is l/r, which is obtained immediately by substituting eqs. (9) and (7) for P,. The intuition underlying the result is that, while x, and Lx,+ I together generate the period t return, only the component of P,/P,_ 1 that relates to the information in x, covaries with x,/P,_ 1. Since x,, the included variable, and Lx, + 1, the omitted variable, are mutually uncorrelated, the estimated coefficient is not biased. Thus, (11) is similar to (4): the contemporaneous priceeearnings regression in the absence of prices leading earnings. The conclusion is that the earnings response coefficient estimate from a contemporaneous priceeearnings regression is unbiased when prices lead earnings by one period and an earnings-deflated-by-price specification for the price-earnings regression is employed. The explanatory power of the regression is R2 = b’var(X,/P,-,)/var(P,/P,-,) = var(x,/P,_1)/r2var(P,/P,-,).

(12)

The R2 thus depends on the relative variances of x, and Lx,+ 1. Variance of x, (Lx,+ 1) represents information about period t’s (t + l’s) earnings change that reaches the market in period t. The larger the variance of Lx, + 1, the greater the degree of market anticipation of t + 1 period’s earnings, and the lesser the surprise in period t’s earnings. Since var(P,/P,_ 1) is generated by the sum of the

S.P.

Kothari,

Price-earnings

regressions

183

information in (or variances of) x, and Lx, + 1, if the variances of x, and Lx, + 1 are assumed equal (see discussion below), then the R2 in (12) will be R2 = O.S[var{(x,

+ Lx,+~ )/Pt- l}/r2 var(P,/P,_

i)] = 0.5.

(13)

The price-earnings relation’s R2 is reduced to 50 percent from a deterministic relation because a contemporaneous price-earnings regression is estimated when the assumption that var(x,) = var(Lx,+,) means 50 percent of the variance in earnings changes is (assumed to be) anticipated one period ahead. In fact, the explanatory power in eq. (13) is given by var(x,)/[var(x,) + var(Lx,+ i)], which means the R2 will be lower than 50 percent if more of the variance in the earnings changes is anticipated one period ahead.’ While the assumptions that var(x,) = var(Lx,+ 1) and prices lead earnings by one period are made only to highlight the effect of prices leading earnings on the price-earnings association, both are conservative. Evidence in Kothari and Sloan (1992) and Collins et al. (1992) indicates returns anticipate more than 50 percent of the variation in annual earnings changes over three leading years. A 50 percent R2 in eq. (13) is still high compared to the typically observed lo- 15 percent R2 of a regression using annual price-earnings data. The difference can be attributed to the restrictive assumptions underlying the simple valuation model. First, as noted above, prices anticipate earnings changes more than one period ahead and they anticipate more than 50 percent of the variation in annual earnings changes. Second, the analysis ignores the effect of variation in the expected rates of returns and shocks to expected rates of returns, which affect stock returns [see, e.g., Fama and French (1988, 1989)], but are not reflected in contemporaneous annual earnings changes [Sloan (1993)]. Third, annual earnings changes exhibit a small degree of negative serial correlation [e.g., Ball and Watts (1972)], but a random-walk time-series property is assumed. Finally, the effect of accrual manipulation on accounting earnings changes is ignored. If these more realistic assumptions were modeled in the analysis, the R2 from a contemporaneous price-earnings regression is expected to be lower than 50 percent. Analysis in section 3.3, which assumes prices lead earnings by two periods, reveals that the explanatory power decreases well below 50 percent and the earnings response coefficient estimated from a contemporaneous regression is biased.

‘Since the anticipation period is of some fixed length, the contemporaneous measurement period’s length obviously will affect the degree of price-earnings association. The longer the contemporaneous measurement interval, the greater the association between stock prices and earnings. An intuitive discussion of the measurement interval’s effect on the degree of price-earnings association can be found in Easton, Harris, and Ohlson (1992) and Kothari and Sloan (1992). Fama (1990) offers a formal treatment of this issue in the context of the association between stock returns and aggregate output.

184

3.2.2.

S.P. Kotizari, Price-earnings regressions

Earnings-change-dejated-by-price

specijication

This subsection derives the earnings response coefficient estimate and R2 for the regression of contemporaneous return, P,/P, _ 1, on the earnings-changedeflated-by-price variable, AX, /P, _ 1. It then compares the results with those in the previous subsection, which are based on using X,/P,_, as the earnings variable. The earnings response coefficient estimate using the earnings-changedeflated-by-price specification is b = cov(dX,/P,_,, =

P,/P,_,)/var(dX,/P,_,)

covC(Lx,+ x,)/P,-

= cov(x,/P,-1,

1t

P,lP,- 1 llvarC(Lx,

P,/P,-,)l~var(Lx,IP,-l)

+

x,)/p,- 11

+ varWPt-l)),

(14)

where, because Lx, is known to the market during the period t - 1, I substitute cov( Lx,/P,_ 1, P,/Pl_ 1) = 0; and I make use of the assumption that Lx,/P,_ 1 and x,/P,_ 1 are mutually uncorrelated. If the variances of x, and Lx, are assumed equal, then (14) simplifies to the approximate equality

b = cov(x,IP,- I , P,lf’, - I )I2 * var(x,lP,- 1 h

(15)

and E(b) % 0.5 *(l/r). The equality is an approximation because var(Lx,/P,_,) = var(x,/P,_l), but the term in (14) is var(Lx,/P,_,). Since P,_l is affected due in part by the information in Lx,, var(Lx,/P,_l) is only approximately equal to var( Lx,/P, _ 2). The approximation is appropriate here because the difference, which can be shown to be equal to a product of two variances, is of second-order magnitude. Thus, the earnings response coefficient estimate in eq. (15) is approximately 50 percent biased if an earnings-changedeflated-by-price specification is used to estimate the price-earnings regression when prices lead earnings by one period. The R2 is given by R2 = b2 * var(dX,/P,_

l)/var(P,/P,_

= (0.25/r’) * 2var(x,/P,_ = 0.25,

1)

l)/var(P,/P,_

1) (16)

because var(x,/P,_ I)/r2var(P,/P,_ 1) = 0.5 from (12) and (13). Thus, when prices lead earnings, the R2 of the change specification is lower than that using the level specification.

S. P. Kothari. Price-earnings regressions

185

Comparison of the earnings response coefficient estimates and R*s from the earnings-deflated-by-price versus earnings-change-deflated-by-price specifications reveals the advantage of using X,/P,_, as the independent variable in a price-earnings regression, at least when prices are expected to lead earnings by one period. When the earnings-level-deflated-by-price variable is employed, deflation by P, _ 1 in effect eliminates the anticipated component of the earnings change, Lx,, leaving only the unanticipated component to explain the price change. As a result, the priceeearnings regression is well-specified. The earningschange-deflated-by-price variable, on the other hand, is not effective in eliminating the anticipated component of the earnings change. The independent variable, thus, contains a component that is irrelevant in explaining the price change variable. Presence of the irrelevant component in the earnings-changedeflated-by-price variable contributes to the errors-in-variables problem, biasing the estimated slope coefficient toward zero and adversely affecting the explanatory power. I next examine the implications of prices leading by more than one period for the price-earnings relation.

3.3. Price-earnings

regression

when prices lead earnings

by two periods

The motivation for examining a setting where prices lead earnings by up to two periods (years) is twofold: (i) previous evidence [e.g., Brown, Foster, and Noreen (1985) and Kothari and Sloan (1992)] suggests prices lead earnings by more than one period. and (ii) to examine whether the earnings response coefficient estimate is biased when the earnings-level-deflated-by-price variable is used. The market’s earnings expectation process is similar to (7), except a twoperiod lead is incorporated as

x, = x,-

1

+

x, + Lx, + L2x,,

(17)

where L*x, is the component of dx, that the market learns about in period t - 2 (i.e., a two-period lead). For analytical tractability and because it is sufficient to provide the intuition, (i) the market’s expectation is assumed to lead earnings by a maximum of two years and (ii) returns over the prior two years are assumed equally informative about the current year’s earnings change. The former assumption tends to understate the effect of prices leading earnings, whereas the latter tends to overstate the importance of prices leading earnings for the price-earnings relation. The market’s expectation of future earnings can be written as

E,(X,+i) = X, + Lx,+1 + L*x,+~

S.P. Kothari, Price-earnings regressions

186

and E,(X,+,) Under

= X, + Lx,+i

these assumptions

for

+ L*x,+~

k 2 2.

(18)

P, is

P, = 1(X, + Lx,+1 + z (X, + Lx,,1

+ L’x,+i

+

L*~,+I + L2x,+2)/rl- [L*x,+~/(~+ r)l

L2x,+I + L2x,+*)/r,

(19)

because the last term represents a discounted one-period cash flow from the earnings surprise L*x, +2 and thus makes only a small contribution to P,. The earnings response coefficient estimate from a contemporaneous priceearnings regression using the X,/P,_ 1 specification is + x, + Lx, + L2x,)IP,-1,

b = cov[(X,-,

var[(X,-,

P,/P,-1)

+ x, + Lx, + L2x,)/P,-1]

cov(x,Ip,-1,p,Ip,-1)

= var[(X,-,

+

Lx,+ L*x,)/P,-, + x,/P,-~]

=var[(X,_,

+

cov(x,IP,- 1, p,/pr-1) Lx,+ L2x,)/P,_,] + var[x,/P,-i]’

(20)

To simplify the expression for b, I focus on the var[(X,_ 1 + Lx,+ L2x,)/ P,_1]term in the denominator of eq. (20). Since (X,_ 1 + Lx,+ L*x,+ L'x,+ 1)/ z Y is a constant in eq. (IS), (X, _ 1 + Lx, + L2x,)/Pt _ 1 and L'x, + 1/P, _1 p,-11 are (almost) perfectly negatively correlated and, therefore, var[(X,_, + Lx, + L*x,)/P,_ 1] = var( L*x,+ 1/P,_ 1). Finally, since the variances of x,, Lx,, and _1)= var(x,/P, _ 1). Substituting this L*x,are assumed equal, var(L2x,+1 /P, result in eq. (20),

b = cov(x,IP, -

11

P,lP, - 1 l/2 * varb,lP, - 1 h

which means E(b) = 0.5 * (l/r). The earnings percent biased when prices lead earnings X,_1/P, _1 specification cf the price-earnings R* is even more dramatic as seen below:

R* = b’var[(X,_,

+ x, +

(21)

response coefficient estimate is 50 by up to two periods and the regression is used. The effect on

Lx,+ L2x,)/PI-l]/var(P,/P,-,)

= (0.25/r’) * 2 * var(x,/P,-

r)/var(P,/P,-

i).

(22)

S.P.

Kothari.

Price-earnings

regressions

187

Noting that var(x,/P,_ l)/r2var(P,/P,_ i) = 3 because xl, by assumption, represents only a third of the earnings information affecting current period’s price movement, the R2 = 0.25 * 4 z 16.7 percent. Thus, when prices lead earnings by up to two periods, the earnings response coefficient estimate is biased and the contemporaneous price-earnings regression R2 is only 16.7 percent. It can be shown that if AX,/P,_ 1 is used instead of X,/P,_,, the estimated earnings response coefficient is expected to be 0.33 * (l/r) and the explanatory power is expected to be about 11 percent. These results are similar to the earlier results in the context of prices leading earnings by one period in that the earningschange-deflated-by-price variable performs worse than the earnings-deflatedby-price variable. The result in this section, that the price-earnings regression’s R2 using the earnings-deflated-by-price variable is expected to be only 16.7 percent, is obviously driven by the assumption that as much as two thirds of the annual earnings change is anticipated and the anticipation occurs evenly over the two leading years. As noted earlier, to the extent the degree of earnings anticipation in prices is less than that assumed above, the price-earnings relation’s explanatory power will tend to be higher. However, with earnings anticipation more than two years in advance, variation in the expected rate of return through time, and shocks to expected rates of return, any deviation from the random-walk time-series property of annual earnings and accrual manipulation will collectively tend to lower the explanatory power. Overall, the observed explanatory power of the annual price-earnings regressions, which is about 10 to 15 percent, is likely to be due to a combination of prices leading earnings, mild deviations from a random walk in annual earnings, and accrual managment. 3.4. Priceeearnings

regression

using the market-unexpected

earnings

The motivation for examining the performance of an accurately measured market-unexpected-earnings variable in a price-earnings regression is that recent research questions whether the longstanding use of a proxy for unexpected earnings in the accounting literature has a logical/rigorous underpinning [see e.g., Ohlson (1991), Easton, Harris, and Ohlson (1992), and Ohlson and Shroff (1991)]. This research views earnings level as the ‘natural explanatory variable of returns’ [e.g., Easton, Harris, and Ohlson (1992)]. When prices lead earnings by one or two periods, if the current earnings’ unexpected component deflated by price is employed as the earnings variable, the expected value of the slope coefficient is

E(b) = cov(x,IP,-

1,

P,lp,-l)lvar(x,lP,-l)

= l/r,

(23)

which means there is no bias. It is easily shown that the explanatory power of the price-earnings regression using x,/P,_i as the earnings variable is a

188

S.P. Korhari, Price-earnings regressions

decreasing function of the extent to which prices anticipate future earnings changes. For example, if, as assumed in section 3.3, a third of the anticipation of an annual earnings change occurs over each of the previous two years, the R2 of the price-earnings regression using x,/P,_ 1is expected to be 33.3 percent. Thus, under the assumptions that annual earnings follow a random walk and prices anticipate earnings changes up to two periods ahead, the use of marketunexpected earnings, deflated by price, outperforms the earnings-deflated-byprice variable on both bias and explanatory power dimensions8 Recent evidence in Collins et al. (1992) is consistent with this result. They show that annual returns are better explained by more accurately capturing the current period’s earnings surprise (i.e., x,) and revision in expectations about the future periods’ earnings (i.e., Lx,+1,L*x, +*,and so on) than by (either only current or current and future periods’) earnings-level-deflated-by-price. Previous research also seeks to obtain a proxy for the market’s unexpected earnings. For example, Ball and Brown (1968, p. 161, emphasis added) state: ‘Specifically, we construct two alternative models of what the market expects income to be and then investigate the market’s reaction when its expectations prove false.’ However, as demonstrated above, since time-series-expectation-model-based proxies are noisy measures of the market’s expectations in the presence of prices leading earnings, the earnings-deflated-by-price variable can be preferable to a time-series-unexpected-earnings proxy. Evidence in Easton and Harris (1991) indicates that the annual-earnings-deflated-by-price variable outperforms the annual-earnings-change-deflated-by-price variable, which is a reasonable proxy for time-series unexpected earnings, in explaining annual returns. The analysis in this paper and the evidence in Easton and Harris, however, only suggest that earnings deflated by price is a less noisy proxy than some other proxies in the contexts examined here. It does not imply that seeking to use the marketunexpected earnings is less desirable or less fruitful [see evidence in Collins et al. (1992) which indicates that price-earnings association is substantially improved by controlling for the anticipated component of earnings]. It would be interesting to examine whether quarterly earnings level deflated by price outperforms analyst’s annual or quarterly earnings forecast error [Brown, Hagerman, Griffin, and Zmijewski (1987)] in explaining returns. The results of such an experiment will determine whether earnings level or analyst’s forecast error is a less noisy proxy for the market-unexpected earnings in long- and short-window research designs. If, as an extreme case, an analyst’s forecast released just prior to an earnings announcement precisely captures the market’s expectation, then the forecast error equals x,, and as seen from eq. (23) and earlier analysis, “By altering the assumptions, it is possible to construct examples in which the earningsdeflated-by-price variable would outperform the market-unexpected-earnings variable [see, e.g., Ohlson and Shroff (1991) for an example in the context of prices leading earnings by one period]. However, in my opinion, unless the new assumptions are plausible or empirically descriptive, such counter-examples generally are of limited economic significance.

S.P. Kothari,

Price-earnings

regressions

189

analyst’s forecast error will ‘outperform’ the earnings-level variable. Yet, regardless of which is empirically the superior variable, in principle, ‘true’ unexpected earnings will be preferred. A second reason for focusing on the market-unexpected earnings is that accounting research is often interested in assessing the usefulness of accounting information to investors, as judged by the stock price reaction to earnings announcements [e.g., Beaver (1968), Ball and Brown (1968), and research thereafter]. If usefulness is being assessed by some portfolio-stock-price-performance metric [e.g., abnormal performance index (API) or portfolio cumulative abnormal return], then a researcher’s ability to assess accounting information’s usefulness critically hinges on her/his ability to correctly classify firms into various portfolios on the basis of unexpected earnings as of the beginning of the return measurement period [e.g., Pate11 (1979)]. Which proxy (e.g., time series or analyst’s forecast or earnings level) performs the best in this context, however, is an empirical issue. 3.5. Summary This section demonstrates that, in the presence of prices leading earnings, the estimated earnings response coefficient is biased and the price-earnings regression’s R2 is likely to be relatively low. The analysis also shows that the bias in the earnings response coefficient estimate is greater and the R2 is lower when the explanatory variable is earnings-change-deflated-by-price, AX,/P,_ l , than earnings-level-deflated-by-price, X,/P,_ 1. Finally, the longstanding use of the market-unexpected earnings in price-earnings regressions is logical. Empirically, however, an accurate proxy for the market’s expectation may be difficult to obtain and earnings deflated by price may therefore be the least noisy (and hence the best) proxy for unexpected earnings. 4. Earnings versus price as deflator in price-earnings

regressions

Previous research has focused on the choice between X,-i and P,_l as deflator in the price-earnings regression. Christie (1987) concludes that P,-l is the appropriate deflator in a regression of returns on deflated earnings change. The rationale is that the dependent variable, return, has P,_ 1 as the deflator and the use of price as the deflator avoids a potential correlated-omitted-variables problem in a cross-sectional analysis. Landsman and Magliolo (1988) conclude that the choice of price as a deflator is not unambiguous, particularly when the earnings variable in a price-earnings regression contains ‘measurement error’.’ ‘Since it is not central source or nature of the sources are: the use of a and earnings time-series and section 6 considers

to their analysis, Landsman and Magliolo (1988, p. 598) do not explain the errors-in-variables problem in using X,_ 1 as the deflator. Two potential time-series earnings expectation proxy, instead of the market’s expectation, process that differs from a random walk. This section focuses on the former the latter.

190

S.P. Kothari,

Price-earnings

regressions

Neither study examines the role of price and earnings as deflators in the presence of prices leading earnings. This section examines the effect of alternative deflators first assuming prices do not lead earnings, and then allowing stock prices to lead earnings. The analysis shows that the earnings response coefficient estimate is less biased and the R2 is higher when P,_ 1 is the deflator than when X,_ 1 is the deflator. These conclusions also hold when the market-unexpected earnings, x,, deflated by X,_ 1, is the explanatory variable in a price-earnings regression. The superior performance of price (P,_ 1) as the deflator is attributable to the fact that price impounds the market’s expectations better than earnings (X,_ 1) when prices lead earnings.

4.1. Earnings

as the dejator

The price-earnings

p,/pr-

1

=

assuming prices do not lead earnings

using X,_ 1 as the deflator

regression

2 +

px,/x,-

1

+

is

E,.

This specification is obviously problematic when X,_ I is negative, which is one reason against X,_ 1 as the deflator. Attempts to empirically deal with negative X,_ 1 are discussed in section 5. Using the valuation model (l), the predicted value of p in model (24) is one. The estimated coefficient,

b = coW,/X, - 1, p,lp, - 1 )lvar(X,lX, - 1 ),

(25)

also has an expected value of one, i.e., E(b) = /I = 1. Thus, when earnings follow a random walk and prices do not lead earnings, X,_ 1 as a deflator yields an unbiased earnings response coefficient estimate. In addition, since the difference between AX,/X,_ 1 and X,/X,_ 1 is a constant equal to one, which does not affect the regression slope coefficient, use of X,/X,_ 1 is equivalent to using AX,/X,_ 1. Comparison of the result using earnings as the deflator with that in section 2.1 based on price as the deflator suggests that the two differ only by a factor of l/r, a constant. The explanatory power of the model in (25) is one because the price-earnings relation is deterministic, as in section 2.1. I next examine the implications of prices leading earnings by up to one period.

4.2. Earnings

as the dejator

assuming prices lead earnings

The assumptions with respect to prices leading earnings given in section 3.1. Under these assumptions, the estimated

by one period

by one period are earnings response

191

S.P. Korhari, Price-earnings regressions

coefficient

of the price-earnings

regression

b = cov(X,/X, =

in (24) will be

)lvar(X,lX, - 1 )

covC(X,-1 + Lx, + x,)lx,-l,p,Ip,-l1 /varCW-1 + Lx, + x,)/X,-11

=

COV(X~/X,-~,

P,/p,-l)l{var(Lx,lX,-l)

+ varklXt-l)).

Eq. (26) is simplified by substituting for P, and P,_ 1 using eq. (9) and noting Lx,/X, _ 1 and x,/X, _ I are uncorrelated:

b = cov{x,/X,-l,x,/(X,-l

+ Lx,)}l{WLx,lX,-l)

(26) that

+ var(x,lX,-,)}. (27)

The bias in b can be approximated

by setting

cov{xt/Xt-~,xt/(Xt-~ + Lx,)}lvar(x,lX,-,), =

cov(x,/X,- 1, x,/X,- l)lvarb,lX,- I)

=p=1,

(28)

and therefore E(b) z p*

1 1 + var(Lx,/X,-,)/var(x,/X,-,)’

can at best reflect the time-series earnings expectation, which is inferior to the market’s earnings expectation when prices lead earnings. The R2 for the regression is R2 = b2var(X,/X,_,)/var(P,/P,-l) z 0.25*var[(Lx,

+ x,)/X,-,]/var(P,/P,-,)

= 0.25.

(30)

S.P.

192

Kothuri.

Price-eumings

regressions

The price-earnings regression R2 too is lower when X,_ 1, instead of P,_ 1, is the deflator. Recall from the analysis in section 3.2.1 that R* using X,/P,_, as the earnings variable is 50 percent. 4.3. Use of the market-unexpected

earnings

If the market-unexpected earnings, x,, deflated by X,_ 1, are employed as the earnings variable, the earnings response coefficient estimate is given by b = cov(x,/X,_

1, P,/P,-

Eq. (31) is simplified by substituting x, and Lx, are uncorrelated: b = cov{~,/X,~~,x,/(X,~~

,)/var(x,/X,-

i).

(31)

for P, and P,_ 1 using eq. (9) and noting

that

+ Lx,)}/var(x,/X,-i).

(32)

The earnings response coefficient estimate in eq. (32) is biased because x,/(X,_ 1 + Lx,) appears in the covariance term, instead of x,/X,_, . The source of the bias is that the dependent and the independent variables are deflated by variables that reflect different information sets. Since prices (are assumed to) anticipate only one-period-ahead earnings changes, the bias is likely to be relatively small compared to that using X,/X,_, [see eq. (29)]. The bias will, however, increase as prices anticipate more than one-period-ahead earnings changes. Note also that, if P,_ 1 is the deflator, the earnings response coefficient will be unbiased, as seen in section 3.4. 5. Empirical evidence This section provides evidence on the earnings response coefficient estimates and explanatory power of the three alternative price-earnings regression specifications analyzed above. Firm-level time-series analysis is performed using annual earnings and return data. The results suggest that the R’s are higher and earnings response coefficient estimates are less biased when earnings level is used and price is the deflator than when earnings change is used and earnings is the deflator. 5.1.

Data,

sample

selection,

and descriptive

statistics

I use data from the Compustat Annual Industrial and the Annual Research tapes and the Center for Research in Security Prices (CRSP) monthly tape. Annual earnings and return data extending from 1950 through 1988 are used. All firms that have at least five consecutive annual earnings and return observations are included so that time-series regressions can be estimated for each

S.P. Kothari,

Price-earnings Table

Descriptive Variableb

Mean

statistics

for returns

Std. error

Panel A: All observations PtiP,- 1 X,/P,- 1 AXJP,- I AX,,‘X, 1

1.1656 0.0700 0.0148 80.1760

0.0023 0.0014 0.0016 7.8330

regressions

1 and earnings

AX,IP,+ AX,/X,-

I I

0.084 0.010 0.147

0.0005 0.0005 0.0063

variables.”

Med.

Min.

Max.

0.04 - 14.13 - 11.96 - 652101

13.17 12.05 24.40 83 599

included, N = 50,322 1.097 0.085 0.008 0.102

Panel B: Extreme one percent large and small obseraations XtiP,-,

193

0.085 0.008 0.102

excluded, N = 49,326 - 0.819 - 0.600 - 9.143

0.450 0.727 12.795

“Sample selection: Any firm that has earnings data on the Compustat annual industrial or the Compustat annual research tape and return data on the Center for Research in Security Prices monthly tape for the five consecutive years 1951 to 1988 is included. A total of 2,720 firms are included in the sample. bP,/P,_ I is one plus the buy-and-hold return, inclusive of dividends, over fiscal year t. X,/P,_ 1 is primary annual earnings per share, excluding extraordinary items and discontinued operations for year r, divided by the price at the beginning of year t. AX,/P,_ , is the change in annual earnings divided by the beginning of the year price. AX,/X,_ , is the annual earnings change divided by the previous year’s earnings; if X,_ I is negative, its absolute value is used.

firm. Annual earnings excluding the extraordinary items and earnings from discontinued operations are used. The earnings variable in the price-earnings regressions is either earnings, X,, deflated by price at the beginning of the year (Pt_ 1), or earnings change, AX,, deflated by P, - 1, or AX,, deflated by X,_ 1. If similar the deflator, X, _ 1, is negative, its absolute value is used. Qualitatively results are obtained by deleting the negative X,-i observations from the analysis. Annual buy-and-hold returns, inclusive of dividends, over the firm’s fiscal year are used. Since a time-series analysis is performed, no restriction is placed on the firms’ fiscal year ends, provided it does not change over time for a given firm. Annual price relatives (Pt/P,_ , ) are annual buy-and-hold returns plus one. I obtain earnings and return data for 50,322 firm-years by applying the above criteria. Table 1 reports descriptive statistics for P,/P,_l and the three earnings variables. The sample mean P,/P,_ 1 is 1.1656, which means the average annual return for the sample firms is 16.56 percent. This is comparable to the average annual return of 16.43 percent on the CRSP equal-weighted portfolio from 1951 to 1988. Descriptive statistics for the three earnings variables are provided before and after excluding extreme one percent large and small values for the respective variables. The effect of the extreme values is most apparent for the ,4X,/X,-, variable: the mean percentage earnings change is 8017.6 percent compared to the median of 10.2 percent. The descriptive statistics for the sample

S.P. Kothari,

194

regressions Table

Selected fractiles from

R’s earnings response three specifications the earnings

the distribution adjusted price-arnings regressions

Regression

Expl. variables

1 2 3

X,/P,- 1 AX,/P,_, AX,IX,- L

Mean”

Min.

Ql

estimates

Med.

43

Max.

12.0 5.5 2.5

26.4 21.5 15.7

99.0 99.9 96.8

Panel A: Adjusted R’s in percent 15.0 11.0 7.9

- 83.1 - 96.5 - 83.6

- 0.6 - 3.5 - 4.2

Panel B: Earninys response coeficienr estimares I 2 3

x,/p,AX,if’-

I

1 AX,,‘X,m I

2.61 3.31 0.26

0.070 0.117 0.010

- 21.81 - 28.69 - 4.63

0.97 0.59 0.43

2.00 1.82 0.13

3.40 4.25 0.32

60.62 92.50 8.08

“Sample selection: Any firm that has earnings data on the Compustat annual industrial or the Compustat annual research tape and return data on the Center for Research in Security Prices monthly tape for the five consecutive years 1951 to 1988 is included. Firm-specific time-series price-earnings regressions are estimated by regressing P,.‘P,_ I ( = one plus the buy-and-hold return, inclusive of dividends. over fiscal year t) on one of the following three earnings variables: X,/P,_, ( = primary annual earnings per share, excluding extraordinary items and discontinued operations for year t, divided by the price at the beginning of year r), AX,/P,_ , ( = change in annual earnings divided by the beginning of the year price), and AX,/‘X,_ , ( = the annual earnings change divided by the previous year’s earnings). If X,_ 1 is negative, its absolute value is used. Extreme one percent small and large values of the above three variables are excluded when estimating regressions. bMean, median, and fractiles of the distribution of adjusted R’s and earnings response coefficient estimates are based on 2,720 firm-specific regressions.

after excluding extreme values reveal that 8.4 percent and the average percentage percent. The standard errors of the means considerably on excluding extreme values

the mean earnings yield (X,/P,_ I) is earnings change (dX,/X,_,) is 14.7 for all three earnings variables decline from the respective samples.

5.2. Evidence Table 2 reports the results of price-earnings regressions employing the three alternative earnings variables. To avoid the excessive influence of extreme observations on the averages of the estimated parameters and since the medians are virtually unaffected by the decision to exclude or include extreme observations, results are reported for the samples that exclude one percent extreme large- and small-valued observations for the respective earnings variables. Sample mean, median, and selected fractiles of the distributions of adjusted R’s and earnings response coefficients from time-series regressions for 2,720 sample firms are reported in panels A and B. The average adjusted R2 of the regressions using the X,/P,_ 1 variable is 15.0 percent and the median is 12.0 percent. The corresponding values for the AX,/P, _ 1 variable are 11 .O percent and 5.5 percent

S. P. Korhari. Price-earnings regressions

195

and those for the dX,/X,_, variable are 7.9 percent and 2.5 percent. The explanatory power using X,/P,_ 1 exceeds that using AX,/P,_ 1 in 62 percent of the firms and that using AX,/X,_, in 66 percent of the firms. The R* using AX,/P,_, is greater than that using AX,/X,_, in 59 percent of the firms. Assuming independence, the null hypothesis that these percentages equal to 50 percent can be rejected. The Z-statistic to test whether the fraction of the times R* using X,/P,_ 1 exceeds that using AX,/P,_ 1 (AX,/X,_ 1) is 12.04 (16.49). The Z-statistic to test whether the fraction of the times R* using AX,/P,_ 1 exceeds that using Ax,/X,_ 1 is 9.51. To the extent observations are not independent, the Z-statistics are overstated. Overall, the evidence is consistent with the prediction that in the presence of prices leading earnings, the explanatory power of the regression using X,/P,_ 1 as the explanatory variable will be the higher than that using AX,/P,_ 1 or AX,/X,_ 1. The mean and median earnings response coefficient estimates using X,/P,_ 1 and AX,/P, _ 1 variables are similar. The sample mean coefficient estimate using X,/P,_, is 2.61 compared to 3.31 using AX,/P,_, and the corresponding medians are 2.00 and 1.82. If one focuses on the medians, because the means are influenced by extreme values, there is weak evidence that X,/P,- 1 yields less biased earnings response coefficient estimates. The median pairwise difference, however, is only 0.01, which is not reliably positive. The mean and median earnings response coefficient estimates are not directly comparable across the two deflators because the predicted value using the X,/P,_ 1 and AX,/P,_ 1 variables is (1 + l/r) whereas it is (1 + Y) using AX,/P,- 1 [see, e.g., Watts and Zimmerman (1986, p. 31) and Colloins and Kothari (1989, table l)]. The reason the earnings response coefficient’s predicted value using either X,/P, _ 1 or AX,/P, _ 1 is (1 + l/r), compared to l/r as noted in the previous sections, is that returns inclusive of dividends are used. Similarly, when returns inclusive of dividends are used, the coefficient on AX,/X, _ 1 is (1 + r), and not one. Since the magnitudes of the predicted coefficients on the earnings variable deflated by price versus earnings are not identical, I compare the medians of the estimated coefficients’ percentage deviations from their respective predicted values. The median earnings response coefficient estimate using AX,/P,_ 1 is relatively less biased than that using AX,/X,_,. Using X,/P,- 1, the median earnings response coefficient estimate of 2.00 is 70.7 percent smaller than the predicted value of 6.83 ( = 1 + l/0.1715, where 0.1715 is the average annual return on the CRSP equal-weighted portfolio from 1926 to 1988 as a proxy for r). The bias is greater using AX,/X,_ 1: the median coefficient estimate of 0.13 is 88.6 percent smaller than its predicted value. Comparison using average values of the earnings response coefficient estimates reported in table 2 yields qualitively similar results. However, average of the estimated earnings response coefficients for X,/P,_, is an upward-biased measure of the earnings response coefficient of the average security because

196

S.P. Kothari, Price-earnings

regressions

(l/N)~( l/ri) > l/r, where N is the number of securities, ri is the expected return on security i, and r is the average expected return for the securities. Therefore, a comparison based on the average values of earnings response coefficient estimates is biased in favor of the X,/P,_ 1 variable. 6. Implications

of negative serial correlation in annual earnings changes

This section examines whether the results in the earlier sections are sensitive to the presence of negative serial correlation in annual earnings changes. Since the primary objective of the analysis in the previous sections was to demonstrate the implications of prices leading earnings, it ignored the negative serial dependence in annual earnings changes [e.g., Ball and Watts (1972), Brooks and Buckmaster (1976), and Watts and Leftwich (1977)] to keep the analysis simple. This section shows that the tenor of the results from the previous sections is unaffected by allowing annual earnings changes to exhibit first-order negative serial correlation. The analysis below compares price and earnings as deflators as well as the level and change specifications of the price-earnings regressions. For analytical simplicity, the analysis assumes zero valuation effect of the negative-serialcorrelation-inducing component of earnings and contemporaneous returns. While the negative-serial-correlation-inducing component could reflect opportunistic accrual manipulation, accruals, in general, are not viewed as contributing to an errors-in-variables problem. The maintained view is that accounting accruals transform cash flows into earnings as a more meaningful measure of firm/managerial performance, and therefore it is more highly associated with contemporaneous stock returns than cash flows [see, e.g., Dechow (1992)]. The assumption that the negative-serial-correlation-inducing component of earnings is valuation irrelevant, however, is not essential. Without making such an assumption, Ali and Zarowin (1992) show that coefficient estimate on earnings change deflated by price is biased. Suppose accounting earnings, X,, have an accrual component v,, that is (1992)]. Earnings without negatively serially correlated [e.g., Dechow q,, X, - q,, follows a random walk and is denoted by x. Formally,

x, = r, + qf=

y-1

+ a, + qr,

(33)

where a, is a serially uncorrelated disturbance term. For this earnings process, the earnings response coefficient estimate from a levels regression, using price as the deflator, is

b = covC(y,-1 + a,)lP,-,,P,IP,-,I

/Cvar{(K-l + a,)lPt-l) + varhlP,-1)1,

(34)

S.P.

Kothari.

Price-earnings

regressions

197

where the presence of var(a,/P,_l) in the denominator of (34) suggests b is biased. Similarly, the earnings response coefficient estimate from a change regression is

b = COVC~,I~,-~,~,I~,-~~/~~~C~,I~,-~

z co~C~,I~,-~,~,I~,-~I/Cvaro -

2*covwp,-l>

v,-l/P,-,)I,

+ 4rlP,-ll

+ 2*varhlP,-1) (35)

where the equality is an approximation because var(qt/P,_ i) z var(v,_ l/P,_ 1). Comparison of eq. (34) with (35) reveals that the earnings response coefficient estimate from the change regression will be smaller than that from the levels regression because twice the var(q,/Pr_i) appears in the denominator [Landsman and Magliolo (1988) make a similar point]. Additionally, negative serial correlation in v], [i.e., cov(q,, q,_ i) < 0] exacerbates the bias, as seen from the covariance term in the denominator of eq. (35). The expression for the earnings response coefficient estimate from a change regression using earnings as the deflator is similar to eq. (35), except that the deflator is X,-i. This estimate will be more biased than that from the levels regression using price as the deflator, once again, because of the negative covariance between qr and ?ll_ 1. In summary, negative dependence in annual earnings biases the estimated slope coefficient of a price-earnings regression and the degree of bias is more pronounced for the change than the levels specification, and for earnings than price as the deflator. For the sample analyzed earlier, the average and median first-order serial correlation in annual earnings changes is - 0.11, which is small as expected.” As a simple test of the predictions, I partition the sample into quartiles ranked on their serial correlations. As noted earlier, earnings response coefficient estimates of the firms whose earnings are more negatively serially correlated are expected to be relatively more biased. For the firms within each quartile, I estimate price-earnings regressions in the level and change specifications, and using price or earnings as the deflator. The results are summarized below. First, the pattern of results within each of the four quartile subsamples is similar to that reported in table 2. That is, within each quartile, the average and median adjusted R2 of the price-earnings regressions are lower, and the earnings

“‘I focus only on the first-order serial correlation in earnings changes. Lipe and Kormendi (1991) and Kendall and Zarowin (1990) examine the implications of higher-order time-series properties of earnings changes for earnings response coefficient estimates.

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response coefficient estimates relatively more biased, when a change specification is used and when the deflator is X,_ 1. Second, the pattern of results across the four quartiles reveals that, for each specification, the earnings response coefficient estimates and adjusted R’s of the price-earnings regressions are lower for the firms with a more negative serial correlation in annual earnings changes. Since the sample firms across the four quartiles, by construction, differ with respect to their earnings persistence, any variation in the estimated earnings response coefficient magnitudes across the four quartiles in itself cannot be unambiguously interpreted as supportive of the analysis in this section. Kormendi and Lipe (1987), Collins and Kothari (1989), and easton and Zmijewski (1989) show that the earnings response coefficient increases in earnings persistence and decreases in the systematic risk of equity. Average and median relative risks of the firms in the four quartiles, however, exhibit limited variation. The average betas for the four quartiles range from 0.97 to 1.05 and the medians fall between 0.87 and 0.96. Thus, differences in relative risks are unlikely to have materially contributed to the variation in the earnings response coefficient estimates across the four quartiles. Overall, the within- and across-quartile results together lend support to the analysis in this section. 7. Summary and conclusions The paper evaluates alternative price-earnings specifications in the presence of prices leading earnings. The criteria for evaluation are the degree of bias in the earnings response coefficient estimates and the explanatory power of a model. The specifications considered include price-earnings relation in levels and changes, and the earnings variable deflated by price or earnings. The paper demonstrates that, if one assumes prices do not lead earnings, the alternative specifications rank identically on the above criteria. However, once a more plausible assumption that prices lead earnings is invoked, the levels specification is predicted to yield a less biased earnings response coefficient estimate and higher R2 than the change specification. The intuition for the result is that when the earnings-deflated-by-price variable is used, in e&et, the anticipated component of earnings, in part at least, is eliminated from the explanatory variable. As a result, a less biased slope coefficient and higher explanatory power are expected. Similar predictions based on a similar reasoning are obtained for price compared to earnings as the deflator. The paper also focuses on the recent debate over the use of an unexpected earnings proxy in price-earnings regressions. I argue that unexpected earnings is the logical variable to use in price-earnings regressions. The analysis indicates that an accurate proxy for the market-unexpected earnings outperforms both earnings-leveland change-deflated-by-price variables. Empirically, however, an accurate proxy for the market’s expectation may be difficult to obtain and other

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proxies, including earnings deflated by price, may be the ‘best’ available variable for use in a price-earnings regression. This appears to be the case when the performances of earnings level and earnings change are compared [see Easton and Harris (1991)]. However, it is an unresolved issue in other contexts. Empirical results in the paper are consistent with the econometric analysis of the priceeearnings regressions. A sample of approximately 50,000 firm-year observations is analyzed. The average and median explanatory power of firmspecific time-series regressions is higher when returns are regressed on the earnings-deflated-by-price variable (i.e., levels specification) compared to the earnings-change-deflated-by-price variable (i.e., change specification), and the earnings response coefficient estimates are relatively less biased in the former than the latter regressions. The results also indicate that price outperforms earnings as a deflator. These conclusions are unaffected by the observed small degree of negative serial correlation in annual earnings changes. Easton and Harris (199 1) report qualitatively similar results using cross-sectional regressions. The analysis in this paper suggests that the observed low contemporaneous correlation between returns and earnings changes cannot necessarily be interpreted as a ‘weakness’ of the historical-cost accounting measurement process. Objectivity, conservatism, and other conventions that underlie the historicalcost accounting measurement process limit its ability to reflect shareholders’ expectations of future net cash flows in the earnings number for a period. In an efficient market, stock prices, on the other hand, provide an unbiased assessment of the present value of the expected net cash flows. As a result, the information sets impounded in earnings and returns over a period are not identical and the typically estimated contemporaneous price-earnings correlation will be low. The presence of a post-earnings-announcement drift in apparent abnormal returns raises the possibility that earnings, in part, lead prices, which would be an additional reason for the low contemporaneous association between returns and earnings changes. The analysis in this paper assumes markets are efficient, thus ignoring the earnings-lead-prices scenario. Whether low contemporaneous price-earnings association is evidence of a weakness in the accounting measurement process depends on the objective of earnings measurement. If the information sets impounded in earnings and returns over a period should be identical, then the earnings measurement process will have to be modified to fully reflect the market’s unbiased expectations of future cash flows. However, since this will probably entail relying on the managers’ estimates of future cash flows, objectivity and verifiability of accounting numbers will be compromised, and perhaps accounting earnings will be less correlated with returns [Dechow (1992)]. While this paper does not resolve these issues, the objective here has been to highlight the importance of prices leading earnings on the degree of price-earnings association.

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