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I. Introduction

According to modern finance textbooks, stock price fluctuations are explained by changes in the expected present value of future dividends. This subject received serious attention over two decades ago by the volatility tests of LeRoy and Porter (1981) and Shiller (1981). They found, based on a simple dividend discount model (DDM) with a constant discount rate, that stock market volatility was far greater than could be justified by subsequent changes in dividends. These studies were followed by papers by Flavin (1983), Kleidon (1986a, b), and Marsh and Merton (1986), which challenged the statistical validity of the volatility tests.

Still, a number of studies provide evidence that stock price fluctuations are too large to result solely from changes in the expected present discounted value of dividends. West (1988) devised a variance bounds test that is free from small‐sample bias and valid even when dividends are nonstationary. Campbell and Shiller (1987) derived testable implications of the present value model, taking into account the nonstationarity and cointegration of prices and dividends. Both West and Campbell‐Shiller found strong evidence against the simple present value relation. Although some argue that a time‐varying discount rate may help explain the failure of the simple dividend discount model, various tests found statistically significant excess volatility in stock prices.1

Given the failure of the DDM to explain stock price fluctuations, researchers pursued alternative models of stock valuation. Some researchers introduced behavioral finance models. For example, Cutler, Poterba, and Summers (1990) introduce some irrational traders such as feedback traders and the market's slow adjustment to fundamentals (see also Delong et al. 1990). Daniel, Hirshleifer, and Subrahmanyam (1998) developed a theory based on investor overconfidence and changes in confidence resulting from biased self‐attribution of investment outcomes.2

On the other hand, in accounting literature, an alternative valuation model, the discounted residual income model (RIM), has become popular primarily due to its formalization by Ohlson (1991, 1995) and Feltham and Ohlson (1995) (see also Lee 1999).3 The RIM assumes an accounting identity, the clean surplus relation (CSR), which posits that the change in book value of equity is equal to the difference between accounting earnings and dividends. The residual income (RI), or abnormal earnings, is defined as the difference between accounting earnings and the previous‐period book value multiplied by the cost of equity. The residual income model maintains that the current stock price equals the current book value of equity plus the present discounted value of expected future residual income. A very important implication of the RIM is that dividends, by way of the CSR, are defined broadly as the difference between earnings and the change in book value. As such, dividends in the RIM include not only conventional cash dividends but also other forms of cash payouts to shareholders (e.g., share repurchases, acquisitions). It is interesting to note that most previous studies of the DDM use narrow cash dividends.4

In this paper, as an alternative to the conventional DDM, we test the empirical validity of the accounting‐based RIM and compare the two models' performance in terms of their implications for volatility and their restrictions on the data. The volatility implication is tested by employing the West inequality (variance bounds) test. The models' restrictions on the data are tested by the Campbell‐Shiller VAR‐based cross‐equation restriction test.

The RIM, although derived from the DDM by replacing dividends with earnings and book value, seems particularly attractive and interesting to finance researchers for the following reasons. First, it is well recognized that regular cash dividends tend to be too smooth to be a sole fundamental variable for very volatile stock prices by, for example, various variance bounds tests.5 As such, the DDM seems not very promising from the econometric viewpoint. By replacing dividends with relatively more volatile earnings and book value, the RIM may be able to explain volatile stock price movements.

Second, many firms, especially high‐tech and high‐growth firms, do not pay regular cash dividends until much later in their life cycle. Fama and French (2001) document that the percentage of firms paying cash dividends among NYSE, AMEX, and NASDAQ nonfinancial, nonutility firms fell from 66.5 in 1978 to 20.8 in 1999.6 The conventional DDM that uses regular cash dividends may not be applied to these firms. The RIM, however, may still be implemented.

Third, the RIM incorporates a clean surplus accounting relation into the DDM, which enables us to use earnings and book value instead of dividends as fundamentals. By doing so, the RIM not only bridges the gap between equity value and accounting data but also allows us to employ a broad measure of dividends defined as the difference between earnings and the change in book values. As such, the RIM naturally reflects changes in share repurchases and other forms of cash payouts by way of the change in book value.7 Recently, share repurchase tends to be more widely used by managers as an alternative way of distributing cash to shareholders. However, the conventional DDM tends to use narrow cash dividends and thus ignores the potentially important role of share repurchase in the valuation model.

Fourth, some value‐relevant indicators, such as book‐to‐market‐value ratio (B/P) and earnings‐to‐price ratio (E/P), have been shown to have predictive ability for expected stock returns (see, e.g., Fama and French 1992, 1995; Campbell and Shiller 1988a; Lakonishok, Shleifer, and Vishny 1994; Lee 1996, 1998; Pontiff and Schall 1998; and Lamont 1998). The valuation models using these variables in previous studies are based mostly on regression equations, and thus they seem somewhat ad hoc. For example, Lakonishok et al. (1994) point out that, although the returns to the B/P strategy are impressive, B/P is not a “clean” variable uniquely associated with economically interpretable characteristics of the firms. However, the RIM incorporates these variables in a systematic manner in the form of a dynamic present value relation.

Fifth, the RIM shifts the focus away from the distribution of wealth (dividends) to the creation of wealth (book value and abnormal earnings). Wealth creation depends solely on a firm's operation as opposed to the financing of those operations. As such, dividend policy irrelevance integrates naturally into the RIM. The key is that dividends are paid out of book value but not out of current earnings. That is, the residual income is invariant to changes in the dividend policy.

None of recent studies employs the RIM to formally test whether the model is consistent with dynamic stock price behavior and thus with stock market efficiency. Our goal is to directly test the dynamic implications of the accounting‐based RIM as an alternative to the DDM. In doing so, we employ the rational expectations econometrics methodology developed in finance and economics to handle expectations of future fundamentals. As such, we need not resort to the use of some proxy for ex ante earnings or of ex post data for forecasting, as in previous studies.8

To examine the implications of the DDM and the RIM, we focus on two aspects of the models. First, we examine volatility implications of the models by implementing the West (variance bounds) inequality test. The rejection of the RIM by the West test would imply that the model does not explain volatile stock price movements with earnings and book values. However, if the RIM is not rejected by the West test, we need to further evaluate other implications of the model, in part because the volatility implication is a necessary condition for the model to hold. Our view is that such dynamic models as the RIM and the DDM imply restrictions on the data. The hallmark of a dynamic rational expectations model is that the model's implications can be summarized by cross‐equation restrictions on a VAR system of the relevant variables. As such, we implement the Campbell‐Shiller VAR‐based cross‐equation restriction test taking into account possible cointegration among variables (see also Lee et al. 1999).

As noted already, previous studies of DDM tend to use narrow cash dividends, whereas the RIM anticipates the use of broad dividends by way of the CSR. To provide a comparison on an equal basis, we consider two versions of the DDM: one with narrow cash dividends (DDM1) and the other with broad dividends based on the CSR (DDM2). Then, the volatility test results of the DDM2 can be shown to be identical to those of the RIM (see Section III). As such, for volatility tests, we compare only DDM1 and RIM. However, for the cross‐equation restriction tests, we compare among DDM1, DDM2, and RIM.

Using the annual Dow Jones Industrial Average (DJIA) index data and the Standard and Poor's (S&P) index data, we find that the DDM1 is rejected but the RIM (or DDM2) is not rejected by the West volatility test. This implies that stock market price fluctuations may be too volatile to be explained by changes in the expected present discounted value of narrow cash dividends, but not either by broad dividends or changes in book value and the expected present discounted value of residual income. Regarding the cross‐equation restriction tests, both the DDM1 and the DDM2 are rejected, whereas the RIM is not rejected.

Using individual firm data from the DJIA components, we obtain mixed results. Again, the DDM1 is strongly rejected for all firms, whereas the RIM (or DDM2) is not rejected for the majority of firms by the West volatility tests. Regarding the cross‐equation restriction tests, the DDM2 is rejected for the majority of firms, whereas the DDM1 and the RIM are not rejected for the majority of firms. Still, the RIM appears to perform better than the DDM1. For several high‐tech (growth) firms that do not pay regular cash dividends, we again find mixed results. For most growth firms, the RIM is rejected by the West volatility tests, while the cross‐equation restrictions are not rejected. Using pooled S&P individual firm data, we obtain a similar result. We explore whether time‐varying discount rates help improve the performance of the RIM. For that purpose, we extend the RIM to a log‐linearized model that allows for time‐varying discount rates. We find that some firms perform better with a time‐varying discount rate, while others do not.

Our findings show that using broad dividends in the DDM (i.e., DDM2) helps improve the volatility implication of the model but at the expense of the cross‐equation restriction implications. In contrast, the RIM performs substantially better than the conventional DDM using narrow dividends (i.e., DDM1) in explaining volatile stock price movements. It also performs significantly better than the DDM using broad dividends (i.e., DDM2) in explaining cross‐equation restrictions implied from the model. The findings imply that book values and accounting earnings in the RIM contain more useful information than dividends alone whether they are defined narrowly or broadly. We believe that this result is related to the Miller‐Modigliani dividend irrelevance theorem in combination with the clean surplus relation.

The remainder of the paper is organized as follows. In the next section, we briefly review related literature. Section III introduces the RIM. Section IV discusses how we implement the West inequality test for the RIM. Section V presents how we implement the VAR‐based cross‐equation restriction test for the RIM. Section VI describes the data and sample used. Section VII reports the empirical results and compares the results of the RIM with those of the DDM, and Section VIII concludes the paper.

II. A Brief Literature Review

Whether price movements in the stock market can be justified in terms of the simple efficient market model has long been debated, although there is no universally accepted definition of the term efficient market model. The conventional model has been the dividend discount model. Most financial economists believe that the DDM provides a good approximate description of stock price determination, at least, for the aggregate market. LeRoy and Porter (1981) and Shiller (1981) challenged this view by pointing out that aggregate stock prices appear to be too volatile to be measured by the fundamentals (i.e., dividends) in the DDM.

The variance bounds tests by Shiller (1981) and LeRoy and Porter (1981) have been debated and advanced by various critics. Kleidon (1986a, b) argues that variance bounds tests may give false signals because they are about cross‐sectional relations and not time‐series relations. Marsh and Merton (1986) object to the tests' assumption that dividends are stationary around a time trend and provide an example to show that both log dividends and log prices follow unit‐root processes. Flavin (1983) criticizes Shiller's econometric test for two reasons. First, both the variance of actual price, and the perfect‐foresight stock price, are estimated with a downward bias in small samples. Second, Shiller's procedure for calculating an observable version of also induces a bias toward rejection.

West (1988) developed a stock market volatility test that is not subject to these criticisms. His inequality test maintains that, if discount rates are constant, the variance of the innovation in the expected present discounted value of future dividends is larger when less information is used. West also finds that stock prices are too volatile to be the expected present discounted value of dividends, with a constant discount rate.

Campbell and Shiller (1987) notice that, when both dividends and prices are nonstationary, there exists a stationary linear combination of the two series under the DDM. Thus, the stock prices and dividends are cointegrated. They develop an approach in which the present value model implies testable restrictions on the coefficients of a bivariate vector autoregression of stock prices and dividends taking into account the cointegration.9 Using cointegration and the VAR framework, they also confirm Shiller's finding.

In accounting literature, an alternative valuation model, a present value of the residual income model, has become popularize. The idea of the RIM can be dated back to the work of Preinreich (1938), Edwards and Bell (1961), and Peasnell (1981, 1982) and its more recent popularity is primarily due to its formalization by Ohlson (1991, 1995) and Feltham and Ohlson (1995) (see also Lee 1999).

Lee et al. (1999) examine the relative performance of alternative empirical estimates of intrinsic value. In tracking ability and predictive power, they find that the intrinsic value‐to‐price (V/P) ratio, where the estimates of intrinsic value are based on the RIM with a time‐varying discount rate, outperforms the other market multiples, such as book to market (B/P), earnings to price (E/P), and dividend to price (D/P). Francis et al. (2000) compare the reliability of value estimates from the DDM, the RIM, and present value of the free cash flow model (FCFM). They find that the RIM value estimates are more accurate and explain more of the variation in stock prices than FCFM and DDM value estimates. They attribute the sources of the relative superiority of the RIM to the sufficiency of book value of equity as a measure of intrinsic value and perhaps the greater precision and predictability of residual income. In contrast, Penman and Sougiannis (1998) compare equity valuation in the RIM, DDM, and FCFM based on realized attributes. However, they find a similar result: The RIM performs better than the DDM and FCFM in terms of valuation errors. Dechow et al. (1999) implement the RIM by formalizing the information dynamics and find that the RIM provides minor improvements over the DDM.

Recent empirical studies that implement the RIM tend to examine the ability of the model to forecast cross‐sectional prices or expected returns (e.g., Abarbanell and Bernard 1995; Frankel and Lee 1998; Penman and Sougiannis 1998; and Dechow et al. 1999).10 In particular, a paper by Lee et al. (1999) is similar to this paper in that it examines the time‐series relation between value and price by employing the 30 stocks in the DJIA. However, the focus of the two papers is very different. Lee et al. (1999) focus on the valuation implications of various models by empirically evaluating several alternative measures for the intrinsic value of the 30 DJIA stocks. They find that, while traditional market multiples (e.g., B/P, E/P, and D/P ratios) have little predictive power, a V/P ratio, where V is an intrinsic value based on a residual income model, has statistically reliable predictive power for the DJIA index. They also find that time‐varying interest rates and analysts' forecasts are important to the success of V.11 Our paper focuses on testing overall performance of the RIM based on the model's implications on the volatility and dynamic relations between stock prices and fundamentals by cross‐equation restrictions emphasizing different measures of dividends.

In evaluating the RIM's ability to forecast (or explain) cross‐sectional or time‐series price and expected returns, many of these studies implement the models by using some forecast of earnings. For example, Frankel and Lee (1998) and Lee et al. (1999) use I/B/E/S analysts' earnings forecasts, while Francis et al. (2000) use the market's expectations of the fundamental attributes from Value Line. Forecasts are usually made over finite horizons, 2–5 periods. Although using forecast data provides ex ante observations explicitly, this method has to specify the terminal value in order to use the forecast data, which appears sometimes ad hoc. On the other hand, Penman and Sougiannis (1998) implement the valuation models by using realized (ex post) attributes. They, however, have to deal with the problem that realized data contain unpredictable components of valuation errors. In this study, we test the RIM by employing the rational expectations econometrics methodologies developed in finance and economics to handle expectations of future fundamentals. As such, we need not resort to the use of either some proxy for ex ante earnings or ex post data for forecasting.

III. The Residual Income Model

In finance, the traditional stock valuation model has been the dividend discount model: stock price is determined by the present value of expected future dividends, where is the stock price at the beginning of time is the dividend paid during period is the constant discount factor, which is inverse of one plus the discount rate, and is the expected value operator conditional on the information set .

The RIM assumes an accounting identity, the clean surplus relation, to express equity value as a function of book value and residual income. The clean surplus accounting relation requires that all gains and losses affecting book value be included in earnings; that is, the change in book value is equal to earnings minus dividends: where is the book value of equity at time t, and is earnings for the period from t − 1 to t. An important implication of the CSR is that dividends reduce book value without affecting current earnings and dividends here may include cash dividends and other payouts (cash flows) to shareholders. We define residual income (or abnormal earnings) RI_t as earnings minus a charge for the use of capital as measured by beginning‐of‐period book value multiplied by the cost of capital: where r is the cost of equity (or capital). From (2) and (3), it follows that where . Solving forward, imposing the transversality condition that one obtains Substituting equation (5) into equation (1) yields which is the residual income model. It states that the current stock price equals the current book value of equity plus the present value of the expected future residual income. The model shows the relevance of residual income (or abnormal earnings) as a variable that influences a firm's value. Residual incomes bear on the difference between market and book values. That is, they bear on a firm's goodwill.12 It is noted that although the RIM does not include a measure of dividends explicitly in its relation, a broad measure of dividends still is captured by way of the CSR (i.e., . As such, it is more appropriate to use this broad measure of dividends for the DDM to provide a comparison on an equal basis.

IV. West Inequality Test

West (1988) derived an inequality test that is not only valid even if dividends are stationary but also free from small sample biases. As such, we adopt West's approach as a means of testing the excess volatility of stock prices implied by the RIM.

V. VAR‐Based Cross‐Equation Restriction Test

Nonrejection of the volatility test in Section IV is a necessary condition for the RIM to hold. We need to test other dynamic implications of the model. Campbell and Shiller (1987) provide cointegration and a VAR framework to test present value models. We now apply the Campbell and Shiller (1987) VAR‐based cross‐equation restriction test to the RIM. To apply the cointegration relation, define as the spread between (or linear combination of) : Then, we can rewrite equation (6) as

Consider estimating a bivariate VAR (BVAR) representation for : where the variables in the vector are demeaned, and the polynomials in the lag operator and are all of order k. This BVAR can be stacked into a first‐order VAR system as or where matrix A is sometimes called the companion matrix of the VAR. The first‐order VAR representation is useful because forecasts of future values of z_t are obtained as where includes current and past values of (i.e., and .

We define g1′ and g2′ as row vectors with 2k elements, all of which are zero except for the (k + 1)st element of g2′ and with the first element of g1′ being unity. Then, it follows that Thus, we can rewrite the restrictions in equation (23) by projecting the RIM onto the information set : Assuming a nonsingular variance‐covariance matrix of u_1t and u_2t, equation (27) can be written as or 13That is, the RIM in (23) (or [6]) is characterized by the cross‐equation restrictions on the VAR coefficients in (25). Thus, under the null hypothesis that the RIM holds, the restrictions in (28) (or [29]) should hold.

VI. Data and Sample Description

For empirical estimation and tests of the RIM and the DDM, we employ two types of aggregate index data and individual firm data. For the aggregate index data, we use both annual DJIA index data for the sample period of 1920–96 and annual S&P industrial index data for the sample period of 1946–96. The annual DJIA index data, including book values, are from the Value Line publication, A Long‐Term Perspective: Dow Jones Industrial Average, 1920–1996. For the annual S&P industrial index for the sample period of 1946–96, the index prices, book values, dividends, and earnings all come from the Standard and Poor's Statistical Service publication, Security Price Index Record. The price index is the December price. The index (narrow) dividend series are dividends per share, a 12‐month moving total adjusted to the index for the last quarter of the year. The index earnings series are earnings per share, adjusted to index, 4‐quarter total, fourth quarter. These index prices, dividends, and earnings are used by Pontiff and Schall (1998).14

For individual firm samples, we use quarterly data of individual firms in the Dow Jones Industrial Average for the sample period of 1990:1 to 1999:4. Price series are from the Center for Research in Securities Prices (CRSP) monthly tape, which is adjusted for all applicable splits and dividend distributions. Dividends, earnings and book values are taken from Compustat, which are also used by Lee et al. (1999). Dividends per share are defined as the cash dividends per share for which the ex‐dividend dates occur during the reporting quarter, adjusted for all stock splits and stock dividends that occurred during the period.15 The earnings per share are defined as basic earnings per share before extraordinary items and discontinued operations.16 The book values per share are defined as the common shareholders' equity divided by common shares outstanding at the end of the quarter. High‐tech (growth) firms are chosen from the most actively traded high‐tech stocks from the NASDAQ and the NYSE. All variables are deflated by the Bureau of Labor Statistics wholesale price index.

VII. Empirical Results

We test the implications of the RIM—volatility restriction and VAR‐based cross‐equation restriction—using two types of index data, DJIA index data and S&P index data, and individual firm data and compare the results with those of the DDM. For the DDM, we consider one with (narrow) cash dividends (i.e., DDM1) and the other with broad dividends (i.e., DDM2) by way of the CSR.

A. Empirical Results Using Aggregate Index Data

Table 1 reports the results of the West inequality tests for the DDM and the RIM using both annual DJIA data and annual S&P index data. Following West (1988), we report the test results of equation (21). The RIM (or, equivalently, the DDM2) is not rejected (i.e., estimate of equation [21] is negative) for any lags using the DJIA data and for the lag lengths of one using the S&P index data. However, the DDM1 is rejected for any lags regardless of whether we use the DJIA data or the S&P index data.17

Table 1 West Inequality Tests for Index Data

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Our finding implies that aggregate stock prices are not too volatile relative to fundamentals such as book value and residual income. The intuition is that the right‐hand side of the RIM provides more volatility than that of the DDM1, because earnings and book values are more volatile than cash dividends alone. When the lag length is selected by AIC or SIC as one, which seems reasonable, the RIM is not rejected (i.e., the estimate of equation [21] is negative) by the West inequality test whether we use the DJIA index data or the S&P index data. This provides strong support for stock prices not being too volatile to be the expected present discounted value of residual incomes even with a constant discount rate.

To help understand the volatility test results between the two measures of dividends, figure 1 illustrates historical behavior of stock market prices (P) and narrow (D1) and broad (D2) dividends using both annual DJIA data and annual S&P index data. In both cases, conventional narrow (cash) dividends (D1) appear too smooth relative to stock market prices, whereas broad dividends (D2) derived using the CSR appear sufficiently volatile relative to stock market prices.

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Fig. 1.— Price, narrow dividend (D1), and broad dividend (D2) series for real DJIA index and real S&P industrial index, respectively.

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A further investigation of the dynamic implications of the models requires cointegration and cross‐equation restriction tests based on a VAR framework (e.g., Campbell and Shiller 1987). As a preliminary step, table 2 reports the results of unit root tests for the variables used in the present value models employing both the augmented Dickey‐Fuller tests and the Phillips‐Perron tests. This is in part because the VAR framework anticipates that residual income, RI_t, is integrated of order one when the price adjusted for book value is nonstationary. We also examine whether the pair of series, adjusted price and residual income, is cointegrated with a vector (1, −1). Table 2 shows that index (narrow) dividend (D1), price (P), residual income (RI), and adjusted price series are all nonstationary for both the DJIA index data and the S&P index data. Broad dividend (D2) is nonstationary except for the DJIA index data by the Phillips−Perron tests. The spread for the RIM, S_t, appears to be stationary (i.e., the adjusted price and residual income are cointegrated) for the DJIA data but only marginally stationary for the S&P index data. Lamont (1998) finds similar evidence using both augmented Dickey−Fuller tests and Horvath and Watson (1995) tests.18 The spreads for the DDM1 (S01) and the DDM2 (S02), however, appear to be nonstationary for both the DJIA data and the S&P index data.

Table 2 Unit Root Tests

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In Table 3, we report the results for VAR‐based cross‐equation restriction tests for the DDM1, DDM2, and the RIM. The χ² test statistics (p‐values) for the cross−equation restrictions in equation (28) show that the RIM is not rejected for any lags at the 10% significance level regardless of whether we use the DJIA data or the S&P index data. However, the table shows that both the DDM1 and DDM2 are rejected by the cross‐equation restriction tests for all lags at the 10% significance level. The results are robust regardless of whether we use the DJIA index data or the S&P index data. Regarding the cross‐equation test results, figure 1 suggests that narrow cash dividends alone appear to be too smooth to reflect all the fundamental values of firms, whereas broad dividends appear to be too volatile by including some temporary cash payouts remotely related to the fundamental values of firms.

Table 3 Cross‐Equation Restriction Tests for Index Data

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In sum, the results of both West inequality tests and VAR cross‐equation restriction tests provide strong evidence in favor of the RIM for the aggregate stock market represented by the DJIA index and the S&P index. Using broad dividends, the DDM2 may be able to explain volatile stock market prices but cannot explain the dynamic behavior of stock market prices. These findings imply that, first, stock price may be too volatile relative to the fundamentals such as cash dividend but not to the fundamentals such as earnings and book value; second, accounting data, such as earnings and book values, provide more useful information about stock price movements than dividends alone; and third, the RIM provides an explanation why the E/P ratio and B/P ratio may have predictive power for stock returns.

B. Empirical Results Using DJIA Individual Firm Data

Although various stock market index data have been extensively studied for the test of the present value relations, individual firm data have rarely been used, and thus relatively little is known about the empirical validity of the valuation models for individual stocks. This might be due to the failure of the DDM for aggregate data so that it was not necessary to look into individual firm data. However, we find that the RIM is not rejected by aggregate data. As such, we implement the West inequality test and the VAR‐based cross‐equation restriction test for individual stocks. Our sample firms are from the Dow Jones Industrial Average. Quarterly data from 1990:1 to 1999:4 are used. Since the DDM1 requires the presence of narrow (cash) dividends, Microsoft Corp., which has been a DJIA component since November 1, 1999, and does not pay cash dividends, is excluded from our sample. Therefore, our sample contains 29 firms.19

Table 4 presents the test results. For the RIM, the number of stocks not rejected by the West inequality test (i.e., the estimate of equation [21] is negative) for at least one lag is 19 out of 29 firms. For the DDM1, all 29 stocks considered are strongly rejected by the West inequality test (i.e., the estimate of equation [21] is positive for all firms) with any lags. This result is consistent with that of the aggregate index data and shows that individual firm cash dividends are indeed too smooth to explain volatile stock price movements based on the DDM, let alone firms that do not pay dividends. Instead, variations in book values and residual incomes can explain volatile stock prices for two‐thirds of the sample firms, which clearly indicates that the RIM performs better than the DDM1 in the context of the volatility of stock prices.

Table 4 Empirical Tests for the DJIA Individual Firms

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Regarding the cross‐equation restriction tests, the number of nonrejections of the RIM for at least one lag is 24 out of 29 firms. For the DDM2, the number of nonrejections with at least one lag is 15 out of 29 firms when we use a noncointegrated model but 9 out of 29 firms when we use a cointegrated model. For the DDM1, the number of nonrejections with at least one lag is 21 when we use a noncointegrated model but 20 when we use a cointegrated model.

This result is also consistent with that of the aggregate index data. The DDM2 may be able to explain volatile individual firm stock prices using a broad measure of dividends but cannot explain the dynamic behavior of individual firm stock. Between the RIM and the DDM1, the former still performs better than the latter for the cross‐equation restriction tests. Overall, these findings imply that accounting data, such as earnings and book values, provide more useful information about both the volatility and dynamics of individual firm stock price movements than dividends alone whether they are narrowly or broadly defined.

Given that the RIMs for five firms are rejected by the cross‐equation restriction test, we look further into the data of these firms. Among the five firms, we find that only one firm's, Philip Morris's, price and residual income are cointegrated. That is, prices and residual incomes for the other four firms are not moving together over time, which contradicts the spirit of the cointegration implied by the model. Although four of these firms are not rejected by the West inequality tests, we find that quarterly earnings of these firms are very irregular and noisy, containing both substantial seasonal patterns and big write‐offs with negative earnings. We try to remedy the irregular earnings by smoothing them somewhat. For example, we use previous earnings to replace negative earnings. Then, we use 1‐year moving‐average earnings to replace irregular and seasonal quarterly raw earnings. After smoothing earnings series, we implement the tests for the five firms rejected by the cross‐equation test for all lags. Table 5 reports the test results with the new adjusted data. Each of them, although improved somewhat, is still rejected, which shows the limitations of the RIM. However, given that the RIM is not rejected by aggregate data such as the DJIA index data and the S&P index data or by the majority of individual firms in the DJIA, the RIM seems to provide a reasonable description of the data.

Table 5 Cross‐Equation Tests Using Modified Data

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D. Empirical Results Using High‐Tech and High‐Growth Firm Data

The DDM1 cannot be applied to some high‐tech (growth) firms when they do not pay cash dividends, in particular in the early stage of their life cycle (see Fama and French 2001). This, however, does not preclude the RIM (or DDM2) from being applied, because the model does not directly require regular cash dividends. We conduct the West inequality test and the VAR test of the RIM and DDM for the most actively traded high‐tech (growth) firms in the NASDAQ and the NYSE. The sample includes America Online Inc. (AOL), Cisco Systems Inc. (CSCO), Dell Computer Corporation (DELL), Intel Inc. (INTC), Motorola Inc. (MOT), Microsoft Corporation (MSFT), Oracle Corporation (ORCL), and WorldCom Inc. (WCOM).

Table 6 summarizes the test results. It is not surprising that the results are somewhat weaker than individual stocks in the DJIA. Among the eight firms we consider, only WorldCom Inc. is not rejected by the West inequality tests, showing that these growth firms' stock prices are too volatile relative to their broad dividends. Regarding the cross‐equation restriction tests, there is no significant difference between the RIM and the DDM2. However, noncointegrated models tend to perform better than cointegrated models. For the former model, six firms are not rejected for both the DDM2 and the RIM, whereas for the latter model, four firms are not rejected for both the DDM2 and the RIM.

Table 6 Empirical Tests for High‐Tech and High‐Growth Firms

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As pointed out by Penman and Sougiannis (1998), the poor performance of the RIM may be partially due to low E/P and B/P ratios. Comparing the sample of high‐tech (growth) firms with that of the S&P firms, the E/P and B/P ratios in the former group of firms are about 60% and 23% of the ratios of the latter, respectively. This suggests that we need to further explore the role of E/P and B/P ratios.20 In addition, we find that the E/P and B/P ratios of the high‐tech (growth) firms tend to decline substantially over the sample period.21 This suggests potential changes in the risk of these firms, which prompts us to formally examine the role of time‐varying discount rates.

E. Empirical Results Using S&P Individual Firm Data

Given the mixed performance of the RIM for individual firms data, in particular for high‐tech and high‐growth firms, and the Penman and Sougiannis's finding that the RIM performs poorly for low E/P and low B/P firms in the context of forecasting, we look further into the role of E/P and B/P ratios by using more extensive individual firm data and partitioning the firms into groups based on E/P and B/P ratios (see Penman and Sougiannis 1988). For this, we use annual data of individual firms in the S&P 500 index for the sample period of 1981 to 2000.22

Table 7 presents the test results using pooled S&P individual firm data.23 We use weighted least squares (WLS) to allow for the heteroscedasticity across firms, which is similar to the methodology used by Vuolteenahoo (2000). In the West inequality test, the DDM1 is rejected by all groups while the RIM is not rejected for most groups. Regarding the cross‐equation restriction tests, the DDM1 and DDM2 are strongly rejected while the RIM is rejected for some groups. Overall, the S&P individual firm result is consistent with that of aggregate index data: Accounting data, such as earnings and book values, seem to provide more useful information about both the volatility and dynamics of individual firm stock price movements than dividends alone.

Table 7 Empirical Tests for the S&P Individual Firms

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Regarding the role of B/P and E/P ratios, for the volatility (West) test, the RIM performs better for the high B/P and E/P group than the low B/P and E/P group. This finding is consistent with Penman and Sougiannis (1998), who find that the RIM tends to perform poorly for low B/P and E/P firms in forecasting equity valuation. When book value deviates from the market price substantially, the RIM has a greater chance to fail. From equation (6) of the RIM, low B/P and E/P ratios imply that we have a relatively high left‐hand‐side value (i.e., price) and a relatively low right‐hand‐side value (i.e., book value and residual income), which helps explain the failure of the RIM, in particular in the West test.

For cross‐equation restriction tests, we obtain somewhat different results. For E/P grouping, the RIM tends to perform better for a high E/P group than a low E/P group, but the difference is not significant. For B/P grouping, we observe that the RIM model performs better for a low B/P group than a high B/P group. This indicates that high B/P or high E/P firms are not necessarily better in explaining cross‐equation restrictions. This may be because the cross‐equation restriction is a complicated nonlinear functional relation between market value and earnings (or book value), and even a lower book value can explain the relation better. If, in particular, conservatism in accounting is quite consistent, the RIM model could explain the cross‐equation restriction better for low B/P firms than high B/P firms. Still the most important and robust finding is that the RIM performs substantially better than the DDM1 and the DDM2 regardless of B/P and E/P ratios, as we can see by the values of chi‐square statistics.

F. A Log‐Linear Rim with Time‐Varying Discount Rates

As mentioned already, one possible explanation for the failure of the RIM for some high‐tech (growth) firms may be due to the assumption of a time‐constant discount rate. To address this issue in a more formal manner, we employ a version of the log‐linearized RIM allowing for time‐varying discount rates recently developed by Vuolteenaho (2000). He derives a book‐to‐market‐ratio‐based version of the approximate present value model using the clean surplus relation.24 This may turn out to be an important extension of the model, because unlike in the case of the DDM, the time‐varying discount rate in the RIM may affect not only the discount factor for the expected future residual income (see equation [6]) but also the level of residual income itself (see equation [3]).

For the time‐varying discount rates, we use commercial paper rates from Datastream. Since the log‐linear model is also in a present‐value form, both the West inequality test and the VAR‐based cross‐equation restriction test can be implemented in a manner similar to those in Sections IV and V. The test results are summarized in table 8. With a time‐varying discount rate, Motorola performs significantly better, some firms (e.g., America Online, Dell, and Intel) perform marginally better, and WorldCom performs marginally worse in terms of the West volatility test. Regarding the VAR‐based cross‐equation restriction test, the time‐varying discount rate model performs significantly better for Microsoft (with a cointegrated model) and Dell (with a noncointegrated model) but performs significantly worse for Cisco (with a cointegrated model) and marginally worse for Microsoft (with a noncointegrated model). In sum, the results are mixed. With a time‐varying discount rate, we find some firms show an improved performance, while others do not (see also West 1988; Campbell and Shiller 1988b; Cochrane 1992; and Lee 1998).

Table 8 Log‐Linear Models for High‐Tech and High‐Growth Firms

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VIII. Concluding Remarks

The failure of the conventional dividend discount model to explain volatile stock prices may be related to issues such as the appropriate measure of dividends and discount factors or investors' rationality. In this paper, we introduce the accounting‐based residual income model as an alternative and examine the model's valuation implications for aggregate markets and individual firms. For this purpose, we examined the models' volatility implication by extending and applying the West inequality (or variance bounds) tests and the models' dynamic relations between stock prices and fundamentals by the VAR‐based cross‐equation restriction tests.

The RIM provides a different measure of dividends from the conventional regular cash dividends, and we log‐linearize the model to take into account time‐varying discount factors. The model is attractive in many ways relative to the DDM. It shifts the value analysis away from dividends to book value plus the present value of expected abnormal earnings by using the clean surplus relation. As a result, the model is still applicable even for firms that do not pay regular cash dividends, which increasingly seems to be the case (e.g., Fama and French 2001). And the model is in line with the importance of book values and accounting earnings found independently of the model in finance literature.

The RIM is found to be not rejected by either test, using either the S&P index data or the DJIA index data. The model is not rejected by a majority of individual firms of the Dow Jones Industrial Average by either test. The model, in particular its volatility implication, tends to perform better for high B/P and E/P ratio firms. When we extend the RIM by log‐linearizing it, allowing for time‐varying discount rates, we find that some firms perform better while others do not. Regarding the DDM, we have considered two versions. The DDM with narrow cash dividends (DDM1) is strongly rejected by the West volatility tests regardless of whether we use aggregate market index data or individual firm data, confirming previous test results. The DDM with broad dividends (DDM2) by way of the clean surplus relation is not rejected by the West volatility tests but is strongly rejected by the cross‐equation tests regardless of whether we use aggregate market index data or individual firm data. That is, using broad dividends, the DDM2 may be able to explain volatile stock prices but cannot explain the dynamic behavior of stock prices.

The RIM may be an algebraic transformation of the DDM, and the two models may merely reflect alternative pro forma accounting systems (e.g., Lee 1999). However, given the superior performance of the RIM to the DDM2, we believe the difference is related to the idea of the Miller‐Modigliani dividend irrelevance theorem in combination with the clean surplus relation. These findings imply that accounting data, such as earnings and book values, provide more useful information about stock price movements than dividends alone and provide an explanation as to why the E/P and B/P ratios may have predictive power for stock returns. Given the better performance of the RIM relative to the conventional DDM, it is not surprising to see some recent studies, mostly in the accounting literature, employ the RIM in forecasting.

The failure of the RIM for a small number of DJIA firms, some high‐tech (growth) firms, and the low B/P and E/P group of S&P firms may result from poor specification of the model, if we take the position that stock prices always reflect the fundamental (or intrinsic) value of the company. However, the failure could be related to the irrationality of investors. That is, it may be due to mispricing (i.e., over‐ or underpricing in the market). The latter possibility seems to have received more attention recently (for recent evidence, see, e.g., Dechow et al. 2001; Lamont and Thaler 2000; and Schill and Zhou 2000).25 This possibility seems more plausible in view of recent corrections and collapses of some high‐tech firms, which deserves more examination. By March 2001, shares across the tech sector were mired in a very real bear market, down an average of 60% from the previous March's frothy highs.26 The mispricing possibility makes the recent behavioral finance approach more attractive (see the Introduction).

Our analyses and results confirm that, while the accounting data are incomplete indicators of value, the weighted average of capitalized earnings and book value still provides the core of the valuation function.27 Our contention is simple and clear: earnings and book value act as complementary value indicators, and the accounting‐based RIM works better than the conventional DDM. Therefore, it deserves more attention by finance researchers. It should be discussed in finance textbooks along with the conventional DDM. The RIM has become even more relevant than the DDM in recent years given the trend that cash‐dividend‐paying firms are declining across all types of firms.28