THE UNIVARIATE TIME SERIES MODELLING OF EARNINGS: A REVIEW

bar p017

29-08-95 08:01:18

British Accounting Review (1995) 27, 187–210

THE UNIVARIATE TIME SERIES MODELLING OF EARNINGS: A REVIEW JOHN O’HANLON Lancaster University This paper reviews the literature on the time series modelling of earnings from its beginnings around 1970 through to the 1990s. The paper focuses first on the evidence reported in that part of the literature which analysed the time series properties of earnings without reference to an explicitly stated earnings-based valuation model and outlines the motivations underlying that literature. It then shows that recent theoretical developments in the construction of earnings-based valuation models incorporating the time series properties of earnings have provided an important new impetus to the time series modelling of earnings.  1995 Academic Press Limited

INTRODUCTION Unlike stock price changes which, in an efficient market, reflect immediately the randomly arising shocks to a company’s economic income series, accounting earnings tend only to recognize such shocks as they are gradually realized as series of cash (or near cash) flows. Time series dependence is thus likely to be a pervasive feature of accounting earnings series. Indeed, the time series processes generating companies’ accounting earnings series have long been an object of interest for accounting researchers. The area first became prominent in the accounting literature in the 1970s, following theoretical advances in time series modelling due to Box & Jenkins (1970), and continues to develop in the mid-1990s. Authors have cited a number of motivations for interest in the time series properties of accounting earnings. These have included: (i) the desire to understand the true earnings process in order to identify earnings smoothing practices; (ii) interest in the This paper was originally based on a presentation given at the Accounting Summer School on Market Based Accounting Research held at the University of Strathclyde in September 1989. The event was organized by Professor Peter Pope then of the University of Strathclyde and Professor Martin Walker of the University of Manchester and was sponsored by the Chartered Association of Certified Accountants and the Economic and Social Research Council. This paper has been substantially revised in the light of developments in the literature since 1989. The paper has benefited considerably from the comments of an anonymous referee. Correspondence should be addressed to: J. O’Hanlon, Department of Accounting and Finance, The Management School, Lancaster University, Lancaster LA1 4YX, UK. Received 16 November 1994; accepted 20 March 1995 0890–8389/95/030187+24 $12.00

 1995 Academic Press Limited

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usefulness of quarterly earnings in aiding the prediction of annual earnings; (iii) the desire to observe the impact of accounting policy changes on the earnings generating process; (iv) the need for measures of expected earnings for use in event studies; and (v) the role of earnings forecasts in equity valuation. A feature of the early literature on the time series properties of earnings was that, although equity valuation was often referred to as a motivation for interest in the topic, there was no supporting reference to explicitly stated and theoretically supported earnings-based equity valuation models employing the time series properties of earnings. In recent years, however, due to the growing acceptance on the part of capital market-based accounting researchers of the need for their research to be supported by theoretically rigorous accounting-based valuation models, such models have started to appear. Recent years have seen the development of residual income-based valuation models, inspired largely by the theoretical work of Ohlson (1991), which contain persistence terms based on the time series properties of residual income. In the empirical literature, the use of explicitly stated valuation models in support of research designs can be seen in the use of the persistence of earnings to explain earnings response coefficients. It can also be seen in the widespread adoption of a return/earnings regression methodology derived from the residual income-based framework popularized by Ohlson. The central role of persistence terms in the increasingly influential theoretical framework provides a powerful new impetus to interest in the time series properties of accounting earnings. This paper reviews the literature on the time series properties of earnings from the early 1970s through to the recent literature which incorporates the time series properties of earnings in explicitly stated valuation models. The second section provides a brief introductory summary of the properties of a number of simple time series processes. The third section reviews the evidence reported in that part of the literature on the time series properties of earnings which did not seek to justify itself by reference to an explicitly stated earnings-based valuation model. The fourth section reviews the various strands of the recent literature which have placed the analysis of the time series properties of earnings in a valuation context. The final section concludes the paper. THE PROPERTIES OF SOME SIMPLE TIME SERIES PROCESSES Autoregressive integrated moving average (ARIMA) modelling is a standard technique for modelling time series and has been extensively used as a basis for categorizing the time series properties of accounting earnings. This section provides an introductory illustration of the properties of some simple ARIMA processes. The general representation of an ARIMA ( p, d, q) generating process for a variable, Z, is as follows:

      c=p

] x (D Z

(DdZt−D¯ dZ)=

d

c

(t−c)

189

j=q

] he

−D¯ dZ)−

c=1

j (t−j)

+et

(1)

j=1

where xc is an autoregressive coefficient of order c, hj is a moving average coefficient of order j, et is a zero mean, independently and identically distributed disturbance term and p and q are orders of the autoregressive process and moving average process, respectively. DdZ denotes the dth difference of Z and D¯ dZ is the mean of the dth difference of Z, d being the number of times that Z needs to be differenced in order to make the series stationary. The special cases considered below are ARIMA (0, 0, 0), ARIMA (1, 0, 0), ARIMA (0, 0, 1), ARIMA (0, 1, 0), ARIMA (1, 1, 0) and ARIMA (0, 1, 1). ARIMA (0, 0, 0) (Zt−Z¯)=et

(1.i.a)

Zt=Z¯+et.

(1.i.b)

In this case, Z fluctuates randomly around its mean value of Z¯. Whatever value Zt takes, Et[Zt+k]=Z¯ for all k, where Et[.] denotes the expectation at time t. The autocorrelation of order k in the series Z (the correlation between Zt and Zt−k), denoted here by qt,t−k , is zero for all k. ARIMA (1, 0, 0) [AR(1)] (Zt−Z¯)=x(Zt−1−Z¯)+et

(1.ii.a)

Zt=l+xZt−1+et

(1.ii.b)

where l=Z¯(1−x). In this case, a deviation of Zt from Z¯ is expected to be followed by a series of subsequent deviations which decay at the rate of (1−x) per period. Et[Zt+k−Z¯]=xk[Zt−Z¯]. Here, qt,t−k is non-zero for k=1 and decays exponentially as k increases. ARIMA (0, 0, 1) [MA(1)] (Zt−Z¯)=−het−1+et

(1.iii.a)

Zt=Z¯−het−1+et.

(1.iii.b)

. ’

190

In this case, Et[Zt+1−Z¯] is a function of et. However, since Et[et+k]=0 for k>0, Et[Zt+k−Z¯]=0 for all k>1 regardless of the value of et. Here, qt,t−k is non-zero for k=1 and zero for k>1. ARIMA (0, 1, 0) [Random walk] (DZt−D¯ Z)=et

(1.iv.a)

Zt=D¯ Z+Zt−1+et.

(1.iv.b)

In this case, the series has a constant drift term of D¯ Z. et has the effect of shifting the expected level of Zt+k by et for all k. The effect of shocks does not die out as it does in the case of the three processes considered above in which d=0; qt,t−k does not decay rapidly as k increases. A random walk without drift is as follows: Zt=Zt−1+et. Note that as the x coefficient in an ARIMA (1, 0, 0) process (where d=0) approaches 1, the process becomes similar to a random walk (where d=1). A test of the null hypothesis that x=1, proposed by Dickey & Fuller (1979), is commonly used to categorize time series into those which are stationary in levels (d=0) and those which are stationary in first differences (d=1). ARIMA (1, 1, 0) with D¯ Z=0 DZt=xDZt−1+et

(1.v.a)

Zt=Zt−1+xDZt−1+et.

(1.v.b)

For x>0, the current first difference of Z is expected to be followed by a series of decaying differences of the same sign which will cause Z to asymptote to a new level. Here, there is a pattern of exponential decay in the autocorrelation function of the first difference of the series. ARIMA (0, 1, 1) with D¯ Z=0 DZt=−het−1+et

(1.vi.a)

Zt=Zt−1−het−1+et.

(1.vi.b)

For h<0 and et−1=0, a deviation of Zt from Zt−1 is expected to be followed by a single difference of the same sign which will take Z to a new level. Here, there is positive autocorrelation of order 1 in the first difference of the series, but the autocorrelation of higher orders is zero.

     

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The type of ARIMA generating process observed in the case of an accounting earnings variable such as accounting rate of return will depend largely upon the extent to which barriers to entry (or exit) permit the erosion of shifts in the earnings stream. ARIMA (0, 0, 0) and ARIMA (0, 0, 1) would be observed if such erosion were rapid; ARIMA (1, 0, 0) would be observed if such erosion were gradual. The processes for which d=1 would be observed if shifts were permanent. ARIMA modelling can also be applied to recognize both successive dependence, whereby Zt is related to Zt−1, and seasonal dependence, whereby Zt is related to Zt−S where S is the number of seasons in the cycle. Standard notation for such models is ARIMA (p, d, q)(P, D, Q)s, where P, D and Q are the orders of the autoregressive process, differencing and moving average process of the seasonally ordered items in the series.1 THE TIME SERIES PROPERTIES OF EARNINGS: EVIDENCE AND APPLICATIONS Work on the time series properties of accounting earnings started to appear about 1970. This section reviews the literature on the time series modelling of earnings that was not motivated by reference to an explicitly stated earnings-based valuation model. This literature is of interest both because of the research issues it sought to address and because of the evidence provided on the time series properties of earnings. The review deals first with studies of annual earnings and then with those of quarterly earnings. Annual Earnings Studies Beaver (1970) reported, inter alia, the results of an analysis of the autocorrelation properties of the annual accounting rate of return (earnings deflated by beginning of year book value) and the annual undeflated earnings series of 57 US companies over the period 1949 to 1968. He cited three motivations for interest in the time series properties of earnings. First, knowledge of the time series behaviour of earnings assists the study of earnings smoothing. Second, because forecastability of earnings is likely to be an artefact of undesirable distortions introduced by the accounting measurement process, forecastability of earnings might be a criterion by which to judge the quality of earnings. Third, knowledge of the time series properties of earnings might be useful in attempts to test the ability of quarterly earnings to forecast annual earnings. This issue of the forecasting ability of quarterly earnings was of interest at the time because of the SEC interim reporting requirement introduced in the mid-1960s and had been addressed empirically by Green & Segall (1967) and Brown & Niederhoffer (1968). Beaver found that the mean autocorrelation coefficient of order 1 for the level (first difference) of accounting rate of return was 0·48 (−0·06)

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and that the corresponding figures for the undeflated earnings data were higher at 0·68 (0·10). As might be expected, the likelihood of encountering non-stationarity was greater for the undeflated earnings series than for the deflated earnings series. Ball & Watts (1972) examined the time series properties of the annual accounting earnings of about 700 US companies for the period 1947 to 1966. The earnings variables considered were undeflated earnings, earnings per share (EPS) and earnings deflated by total assets. Ball & Watts motivated their interest in the time series properties of earnings by reference to income smoothing and the relevance of earnings changes to the probability of corporate failure. In the latter context, although no earnings-based corporate failure or valuation model was explicitly invoked, Ball & Watts stressed the contrast between the relatively low valuation impact of an earnings change where earnings are generated by a constant expectation finite variance process and the relatively high valuation impact where earnings are generated by a martingale process.2 As will be seen in a subsequent section of the paper, this key point underlies the recent use of ‘earnings persistence’ to explain the relationship between stock price changes and earnings innovations. Ball & Watts reported that undeflated earnings behaved as if generated by a submartingale or similar process and that EPS and earnings deflated by total assets behaved as though generated by martingale processes. The implication of their results is thus that the valuation impact of earnings innovations is likely to be relatively important. Dopuch & Watts (1972) suggested that comparing the time series properties of an earnings series before and after an accounting policy change would provide evidence as to whether the policy change had affected the behaviour of earnings. The authors identified3 the appropriate class of ARIMA process for both the unscaled earnings series and the accounting rate of return series for each of 11 companies for the period prior to a switch from straight line to accelerated depreciation. For each company/series, they then estimated the coefficients of the identified model for the period prior to the change, for the period after the change and for the combined period. The authors rejected the null hypothesis of no change in the earnings generating process in eight out of 11 companies in the case of unscaled earnings but in only one out of 11 companies in the case of accounting rate of return. The classes of ARIMA model identified by Dopuch & Watts, along with those identified by other studies of annual earnings series, are summarized in Table 1. The models are predominantly autoregressive models of order 1 or 2. In two of the 11 cases unscaled earnings required differencing; in no case did accounting rate of return require differencing. Watts & Leftwich (1977) carried out tests to establish whether ARIMA modelling techniques produced better earnings forecasts than the random walk model. In the course of this study, ARIMA models were identified and estimated for the unscaled annual earnings available for common stockholders series of 32 US companies for the period 1927 to 1964

     

193

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(see Table 1). For each company the authors tested the null hypothesis that the identified and estimated processes were random walks. The null hypothesis was rejected at the 5% level in 17 of the 32 cases. Comparison of the forecasting ability of: (a) firm-specific identified ARIMA models; (b) the random walk model; and (c) the random walk with constant drift model, indicated that no approach dominated the others. The forecasting performance of the ARIMA models that were significantly different from a random walk was worse than that of the ARIMA models that were not significantly different from a random walk. This led the authors to suspect that earnings series could sometimes be concatenations of a number of series generated by different processes and to conclude that the random walk was a good description of the process generating unscaled annual earnings. Albrecht, Lookabill & McKeown (1977) identified and estimated ARIMA models for the unscaled annual earnings series and the accounting rate of return series of 49 US companies for the period 1947 to 1975 (see Table 1). For forecasting, a random walk model appeared to perform as well as an ARIMA model. The type of random walk model appeared to depend upon whether or not the earnings were deflated. As might be expected, a random walk with drift performed better on the unscaled earnings series, whereas a random walk without drift performed better on the accounting rate of return series. Lev (1983) explored the issue of whether the time series properties of earnings were related to firm-specific economic factors. For this purpose, he tested for the ability of ‘product type’, ‘barriers to entry’, ‘size’, ‘capital intensity’ and ‘inventory’ to explain the autocorrelation in: (i) the change in unscaled earnings; (ii) the change in accounting rate of return; and (iii) the change in sales. Justifying his interest in the time series properties of earnings, Lev mentioned their potential usefulness: (i) in detecting the effect of accounting policy change; (ii) in providing estimates of expected earnings for use in studies of the information content of accounting earnings; and (iii) in equity valuation. Data on 385 US companies for 15 to 27 years during the period 1947 to 1973 were used. As expected, it was found that producers of services and non-durable goods tended to have higher earnings autocorrelation than producers of durable goods and that companies in industries with high barriers to entry tended to have higher earnings autocorrelation than those in industries with low barriers to entry. The effects of size, capital intensity and inventory levels were generally unimportant. Callen, Cheung, Kwan & Yip (1993) re-addressed the analysis of the time series properties of annual earnings by using the maximum entropy method (MEM) to describe the autoregressive processes for the annual EPS series and the inflation-adjusted EPS series of 263 US companies for the period 1955 to 1984. The MEM technique suggested the predominance of low order (pΖ2) autoregressive processes in the levels of both series, as did the application of a standard ARIMA technique. The evidence suggested

     

195

that random walk forecasts beat those generated by both the MEM technique and the ARIMA technique for both EPS and inflation-adjusted EPS. There was also some evidence that the MEM technique was superior to the standard ARIMA technique in the case of the inflation-adjusted series. The summary of findings from the literature on the time series properties of annual earnings in Table 1 suggests that both earnings and accounting rate of return tend to be characterized by ARIMA processes in which pΖ2, dΖ1 and qΖ2, there being a predominance of AR effects in the case of those series that can be modelled in levels. In general, forecasts based on firm-specific ARIMA models do not out-perform those from a simple random walk model. In this context, however, it is important to mention the work of Dharan (1983). Using a model of the firm’s production, inventory and capital investment activities, this study showed that ARIMA processes with orders of p and q higher than those typically reported by empirical researchers might plausibly characterize the earnings generating processes of firms. Dharan suggested that the relatively small sample sizes available to empirical researchers might give rise to parsimonious but misspecified models. These might well give rise to misinterpretation of the results of tests requiring the identification of time series models of accounting earnings and to disagreement among researchers as to the true generating processes. Quarterly Earnings Studies Analysis of the time series properties of US quarterly earnings began to make an impact on the accounting literature in the late 1970s with the publication of three influential papers: Foster (1977), Griffin (1977) and Brown & Rozeff (1979). On the basis of empirical analysis, each paper proposed a uniform ARIMA (p, d, q)(P, D, Q)4 seasonal model for earnings. Although each paper proposed a different model, the models each contained both a successive quarter-to-quarter component and a seasonal component. The Foster (1977) paper motivated its interest in the time series properties of earnings by reference to earnings smoothing, valuation and, in the light of proposals that companies should publish seasonally adjusted quarterly accounting information, the need to document the properties of the unadjusted seasonal data. The study used the quarterly earnings of 69 US companies for the period 1946 to 1961 for model identification and estimation purposes and used the period 1962 to 1974 for testing the forecasts generated from the data of the earlier period. The quality of forecasts generated by six models was compared on each of two criteria: forecast accuracy and degree of association between forecast error and abnormal stock market return. The six models tested included five uniform models, one of which was ARIMA (1, 0, 0)(0, 1, 0)4 with drift,4 and a firm-specific ARIMA (p, d, q)(P, D, Q)4 model. The firm-specific models that Foster identified were predominantly either MA(1) with seasonal MA(1) or AR(1)

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. ’

with seasonal MA(1), with either first differencing or seasonal differencing. According to the forecast accuracy criterion, the ARIMA (1, 0, 0)(0, 1, 0)4 model, with drift, clearly dominated; according to the market association criterion, a uniform ARIMA (0, 0, 0)(0, 1, 0)4 model with drift performed best with the ARIMA (1, 0, 0)(0, 1, 0)4 model with drift being the second best performer. Overall, the study suggested the appropriateness of a uniform model containing both a seasonal component and a successive quarter-toquarter component such as the ARIMA (1, 0, 0)(0, 1, 0)4 model with drift. Griffin (1977) motivated his interest in the time series properties of earnings by reference to valuation, smoothing, the need for expected earnings measures for use in event studies and the possibility of using earnings variability as a proxy for investment risk. The study used the unscaled quarterly earnings of 94 US companies quoted on the New York Stock Exchange for 1958 to 1971. The model identification, model estimation and diagnostic checking procedures employed by Griffin suggested the appropriateness of a uniform ARIMA (0, 1, 1)(0, 1, 1)4 representation. Brown & Rozeff (1979) used the quarterly EPS data of 23 US companies for the period 1951 to 1974. On the basis both of the identification/estimation of models and the analysis of forecast accuracy, they concluded that an ARIMA (1, 0, 0)(0, 1, 1)4 process was a good candidate for an assumed uniform generating process for quarterly EPS. Lorek (1979) evaluated the quality of a number of quarterly time series models of unscaled earnings by their ability to predict annual earnings. As well as testing the three premier uniform models due to Foster (1977), Griffin (1977) and Brown & Rozeff (1979), respectively, Lorek also tested a number of ‘simplistic’ models and the models identified from a firmspecific modelling approach. Generally, the Griffin and firm-specific models dominated. As for the classes of the firm-specific models, most earnings series required either successive or seasonal differencing and the most common models were MA(1) with seasonal MA(1) and MA(1) with seasonal AR(1). Bathke & Lorek (1984) compared various time series models, including the Foster, Griffin and Brown–Rozeff models, by reference to their forecast accuracy and to the market reaction to their forecast errors. Their results suggested the superiority of the Brown–Rozeff model. Bathke, Lorek & Willinger (1989) explored the issue of whether the time series properties of earnings varied according to the size of the firm. The models used for this study were again those due to Foster, Griffin and Brown–Rozeff. In general, it appeared that the Foster model fitted the data less well than the other two models. As regards the effect of firm size, whilst this was not a determinant of the class of earnings generating process, the AR(1) coefficient estimated in the Foster and Brown–Rozeff models was positively related to firm size. This suggested that large firms’ earnings were more persistent than those of small firms. Lorek & Bathke (1984) suggested that the belief in the suitability of uniform models of quarterly earnings involving seasonal differencing might

     

197

need to be tempered. The authors pointed out that some firms’ earnings series do not exhibit seasonal dependence and that superfluous seasonal differencing might cause the inclusion of unnecessary seasonal AR or MA terms which might introduce error into the forecasting process. A number of studies, including Brown & Rozeff (1978), Collins & Hopwood (1980), Imhoff & Pare´ (1982), Brown, Griffin, Hagerman & Zmijewski (1987) and O’Brien (1988), have compared the quality of forecasts based on time series models with that of analyst forecasts. The evidence generally suggests the superiority of analyst forecasts. The superiority of analyst forecasts is largely attributed to the fact that, unlike univariate ARIMA models, analysts employ a variety of inputs and can cope comfortably with structural change. Lee & Chen (1990), however, proposed a technique for dealing with structural change in ARIMA models of quarterly EPS and found that the application of their technique reduced the advantage of analyst forecasts over time series models to an insignificant level. The time series properties of the quarterly series of a number of earnings constructs besides historic cost earnings have also attracted attention. Motivated by the possibility that the reporting of income on a general purchasing power-adjusted (GPPA) basis might become a permanent requirement in the US and might therefore attract income smoothing activity, Hillison, Hopwood & Lorek (1983) identified the seasonal time series model that best characterized the historic cost (HC) and GPPA earnings of 24 US airline companies. The GPPA earnings series used in the analysis were constructed by the authors from HC earnings. The authors found that the time series properties of HC and GPPA earnings did not greatly differ and that, for both earnings constructs, uniform models of an ARIMA (1, 0, 0)(1, 0, 0)4 or ARIMA (1, 0, 0)(1, 1, 0)4 class could be proposed. Motivated by the theoretical valuation relevance of cash flows and by the lack of evidence on the time series properties of cash flow from operations and other funds flow items, Lorek, Schaeffer & Willinger (1993) examined the time series properties of cash flow from operations and working capital from operations for 109 US companies for the period 1976 to 1984. The results of the study are important for understanding the causes of time series dependence in accounting earnings. The authors reported that the working capital from operations series exhibited properties similar to those of quarterly earnings in that they tended to be generated by ARIMA processes with successive and seasonal dependence. However, the cash flow from operations series tended to exhibit seasonal dependence only. These results suggested the powerful influence of the accounting valuation process in inducing successive dependence in reported earnings. A further recent application of the time series modelling of quarterly earnings can be found in the study of the stock market’s response to earnings innovations. Bernard & Thomas (1990) observed that there is a positive but decaying dependence in successive quarterly seasonal earnings differences at lags 1, 2 and 3 and a substantial negative dependence at lag 4. In other

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words, year-on-year changes in quarterly earnings tend to be followed by similar but decaying year-on-year changes in the three subsequent quarters and by a substantial reversal four quarters ahead. The market, however, behaves as though it does not recognize this dependence and seems to be surprised by it. Bernard & Thomas remark that ‘if market prices fail to reflect fully the implications of information as freely available as earnings, how well do they reflect information that is not as well publicised?’ (p. 339). Other recent applications have included the use of forecast errors from time series models to observe inter-company information transfers around the time of earnings announcements (Freeman & Tse, 1992). To summarize the evidence on the time series properties of quarterly earnings, there is general agreement that uniform models containing both a seasonal component and a successive quarter-to-quarter component are required, although there is some danger that use of unnecessary seasonal components in the case of non-seasonal firms might introduce error into the modelling process. The results of Lorek et al. (1993) suggest that the seasonal dependence may largely be due to cash flow seasonality whereas the successive dependence may be mainly due to the accounting valuation process. Bernard & Thomas (1990) suggest there is substantial seasonal dependence in seasonally differenced quarterly earnings which the market fails to recognize in reacting to earnings innovations. THE USE OF THE TIME SERIES PROPERTIES OF EARNINGS WITHIN AN EXPLICITLY STATED VALUATION FRAMEWORK Some of the papers mentioned in the previous section referred to equity valuation as a context within which identification of the time series properties of earnings forecasts might potentially be of use. However, these papers did not explicitly describe an earnings-based valuation model within which the forecasts might be used. Recent years have seen a widespread realization, expressed among others by Bernard (1989), Lev (1989), Brennan (1991) and Penman (1991b), that much accounting research suffered from a lack of theoretical support for its research designs because of the lack of a model linking the outputs of accounting with the value of equity. Progress in this direction has brought a move towards the use of the time series properties of earnings within an explicitly stated valuation framework. There are a number of strands of the recent literature within which this move can be identified. First, the ‘earnings persistence’ literature, which is largely concerned with explaining stock price response to earnings innovations by reference to the permanence of the innovations, tends to employ a simple model in which it is effectively assumed that the value of equity capital is the discounted present value of accounting earnings. Second, Ohlson (1991) proposed a theoretically more rigorous earnings-based valuation model which developed the Peasnell (1982) result, that the value of equity capital

     

199

can be stated as the sum of book value and the present value of expected future residual income, by superimposing upon it a particular set of assumptions about the process generating residual income. Following Ohlson (1991), Ramakrishnan & Thomas (1992) proposed a number of similar models which relied upon different assumptions about the time series properties of residual income. A third manifestation in the recent literature of the (at least implicit) use of assumptions about the time series properties of earnings is found in the growing use of a return/earnings regression specification, based on the Ohlson (1991) model, which uses both the level and the first difference of earnings as explanatory variables for stock returns. A fourth application in the accounting policy area is suggested by the analysis in Feltham & Ohlson (1993). Feltham & Ohlson showed that the difference between the market value of equity capital and its book value could be explained partly in terms of a mismatch between the time series properties of operating cash receipts and the depreciation rate applied to operating assets. This section of the paper considers these four strands of the literature in turn. The Earnings Persistence Literature A number of studies have taken the time series properties of earnings into a valuation setting by using them to explain cross-sectional variation in companies’ stock price response to earnings innovations. This literature tends to rely upon a rather simple valuation model which effectively expresses the value of equity as the discounted present value of current and expected future accounting earnings (or which expresses changes in the value of equity as the discounted present value of changes in current and future expected accounting earnings). However, it is an important step towards incorporating the time series properties of earnings into valuation models. Here, I will describe the basis for the general earnings persistence measure given in Collins & Kothari (1989) and then briefly summarize the findings of papers which have employed special cases of this measure. Assume the value of equity capital, denoted by P, and accounting earnings, denoted by x, are related by the following simple earnings capitalization model: s=x

Pt=

] E [x t

t+s

]R−s

(2)

s=0

where E is an expectations operator and R is one plus the cost of equity. We are interested in the effect of a current earnings surprise on equity value. From (2) this equals the discounted sum of revisions to current and future expected earnings induced by the earnings surprise. Assume that earnings are generated by the following ARIMA (p, 0, q) process:

. ’

200 c=p

j=q

] x (x

(xt−x¯)=

c

(t−c)

] he

−x¯)−

c=1

j (t−j)

+et

(3)

j=1

where x¯ denotes the mean of accounting earnings. Bearing in mind that et will have no effect on the mean of the series [note that, in (4), d denotes ‘change in’], x

dEt[xt+s]R−s dPt = det s=0 det

]

c=p

j=q

x

xc dEt[xt+s]R−s hj =1+ − c j R de R t c=1 s=0 j=1

] ]

x

dEt[xt+s]R det s=0

]

−s

]

A

c=p

xc

]R

1−

c=1

(4)

B

c

j=q

] dE [x t

t+s

s=0

]R−s=

A B hj

]R

1−

dPt=

j

j=1

j=q

x

hj

]R

=1−

j

j=1

c=p

xc 1− Rc c=1

det.

]

If, for an ARIMA (p, d, q) process, d=1 rather than 0, the entire effect needs to be multiplied by: 1+

R 1 = R−1 R−1

in order to reflect the fact that det is now impounded in Et[xt+s] for all s. Generalizing for any order of differencing, d, the expression needs to be multiplied by [R/(R−1)]d. Multiplying by [R/(R−1)]d and subtracting the current period effect of det gives the present value of the effect of det on expectations of future x when x is generated by an ARIMA (p, d, q) process. This is the earnings persistence measure given by Collins & Kothari (1989):

     

x

] dE [x t

s=1

t+s

AA

]R−s=

j=q

hj

]R

1−

j

j=1

BA

R−1 R

d

c=p

]

B

−1 det.

B

xc 1− Rc c=1

201

(5)

According to the assumptions used above, the earnings response coefficient (ERC), which is the price change induced by a unit earnings innovation, will be one plus the bracketed term on the right-hand side of (5). Special cases of (5) have been used by a number of studies to help explain cross-sectional differences in the ERC. Kormendi & Lipe (1987) applied the ARIMA (2, 1, 0) special case uniformly to the inflation-adjusted annual EPS of US firms and found the predicted positive relationship between the ERC and the earnings persistence term. Collins & Kothari (1989) and Ali & Zarowin (1992a) each applied the ARIMA (0, 1, 1) special case of (5) to the EPS of US firms and found the resulting measure of earnings persistence helped to explain the ERC. O’Hanlon, Poon & Yaansah (1992) used the parameters of firm-specific ARIMA ( p, d, q) models of the inflation-adjusted EPS of UK firms to compute persistence measures based on (5). It was reported that earnings innovations multiplied by one plus the bracketed term in (5) were more highly correlated with abnormal returns than were unadjusted earnings innovations. The Residual Income-based Valuation Framework Whilst it represents an important step forward, the earnings persistence literature referred to in the previous section is effectively based on a valuation model in which it is assumed that changes in earnings expectations can be capitalized as though they were changes in dividend expectations. However, this assumption is not justified in general. A major contribution to the development of more rigorous valuation models which employ information on the time series properties of earnings, and which are consistent with the fundamental dividend capitalization model, is found in Ohlson (1991) Ohlson started from the dividend capitalization model and noted, following Peasnell (1982), that this could be reformulated such that the value of equity was the sum of book value and the present value of expected future clean surplus residual income. He imposed upon this reformulation certain assumptions about the time series properties of clean surplus residual income. The result was a model in which the value of equity was stated in terms of a weighted average of book value and earnings where the weights included residual income persistence terms. This section of the paper briefly describes the basis for the weighted average model, highlighting the role of the time series properties of earnings.

. ’

202

The dividend capitalization model is as follows: x

Pt=

]R

−s

Et[Dt+s],

(6)

s=1

where Dt+s is the dividend expected to be paid at year end t+s. If earnings are stated in clean surplus form as xt=yt−yt−1+Dt

(7)

where yt is the book value of equity capital at year end t, and residual income for year t, xat, is defined as xat=xt−(R−1)yt−1 =yt+Dt−Ryt−1 ,

(8)

dividends can be written as follows: Dt=Ryt−1+xat−yt.

(9)

Substituting (9) into (6) gives Pt=Et[ytR+xat+1−yt+1]R−1+Et[yt+1R+xat+2−yt+2]R−2+. . . which collapses to x

] E [x

Pt=yt+

t

]R−s.

a t+s

(10)

s=1

Although this result has provided much of the impetus to the recent development of earnings-based valuation models, it is in fact an old result (see, for example, Edwards & Bell, 1961; Edey, 1962; Peasnell, 1982) which has been neglected. Its recent popularity is due in part to the imposition upon it by Ohlson of the following particular assumptions concerning the time series process generating residual income: xat+1=xxat+mt+e1,t+1

(11.i)

mt+1=cmt+e2,t+1

(11.ii)

where mt represents the effect of information, other than that contained in the current level of residual income, which influences the prediction of

     

203

expected future residual income. In (11.i) and (11.ii), 0
(11)

Assuming this generating process, the present value of future expected residual income per unit of current residual income is

AB AB AB 2

3

x x x + + +. . ., R R R

which collapses to x . (R−x) The value of equity can thus be expressed as Pt=yt+

A B

x xat R−x

which can be rearranged to give Pt=yt+[xt−(R−1)(yt−xt+Dt)]

A

B

A

BA

x R−x

(R−1)x Rxxt (R−1)xDt − + =yt 1− R−x R−x R−x

BC A B D

(R−1)x (R−1)x R + xt −Dt . =yt 1− R−x R−x R−1

(12)

Equation (12) expresses the value of equity as a weighted average of book value and an ex-div. earnings multiple, the weights being functions of the

. ’

204

autoregressive residual income persistence coefficient. This expression, with the addition of an ‘other information’ variable appeared in Ohlson (1991). It provides valuable insights into the role of the income statement and the balance sheet in equity valuation and suggests the role of the time series properties of earnings in determining their relative importance. If residual income is relatively transitory (x is low), the book value term becomes relatively important; if residual income is relatively permanent (x is high), the earnings term becomes relatively important. The particular residual income generating process underlying this model implies that residual income has an unconditional expected value of zero which implies that unrecorded goodwill is on average zero (i.e. accounting is unbiased). The assumption of unbiased accounting was relaxed by Feltham & Ohlson (1992) who allowed unrecorded goodwill to be positive on average (i.e. conservative accounting) as a result of the systematic under-valuation of operating assets. Following Ohlson (1991), Ramakrishnan & Thomas (1992) showed that the imposition of a variety of different sets of assumptions about the time series properties of residual income gave rise, after rearrangement, to a variety of different valuation models. They showed that the ARIMA (1, 0, 0) process with no constant, described in (11) and which provided the basis for the Ohlson weighted average model described above, also gave rise to an alternative weighted average model termed the ‘book value model’:

A

Pt+Dt=(yt−1+xt )

BA

BA B

R(1−x) R(1−x) R + 1− xt . R−x R−x R−1

(13)

The ARIMA (0, 1, 1) process with no constant: (xat−xat−1)=−het−1+et

(14)

gave rise to the following ‘market value model’:

A

Pt+Dt= xt

B

R (1−h)+h(Pt−1+xt). R−1

(15)

The ARIMA (1, 1, 0) process with no constant: (xat−xat−1)=x(xat−1−xat−2)+et

(16)

gave rise to the following ‘earnings model’:

A BA

Pt+Dt=xt

B

C

A B D

Rx Rx R R 1+ − xt+xt−1 −Dt−1 . R−1 R−x R−x R−1

(17)

     

205

Whilst any of the weighted average models described above may be open to criticism on the grounds of the restrictive assumptions made about the time series properties of residual income, they provide examples of how to deal with the permanence of earnings within a valuation framework which is consistent with the dividend capitalization model. They also demonstrate that, depending upon the assumptions made about the time series properties of earnings, different valuation structures result. The Use of the Residual Income-based Valuation Framework in Recent Empirical Literature Expression (12) can be expressed in returns form as follows, where r denotes return: Pt−Pt−1+Dt Pt−1

rt=

A

B

A B

A

B

(R−1)x Rx (R−1)x (yt−yt−1)+ (xt−xt−1)− (Dt−Dt−1)+Dt R−x R−x R−x

1−

=

Pt−1

A

B

A B

A

B

(R−1)x Rx (R−1)x (yt−yt−1+Dt)+ (xt−xt−1)+ Dt−1 R−x R−x R−x

1−

=

Pt−1

.

Substitution of xt=yt−yt−1+Dt gives an expression for returns containing the standardized level of clean surplus earnings, the standardized first difference of clean surplus earnings and a lagged dividend yield term:

A

B A B

(R−1)x 1− xt R−x

rt=

+

Pt−1

A

Rx (xt−xt−1) R−x

B

(R−1)x Dt−1 R−x

+

Pt−1

.

Pt−1

. ’

206

If the dividend yield term is dropped, this becomes an expression for return in terms of a weighted average of the price deflated level of earnings and the price deflated first difference of earnings:

A

B A B

(R−1)x 1− xt R−x

r t=

Rx (xt−xt−1) R−x

+

Pt−1

Pt−1

.

(18)

This result has had a major impact on research designs employed in market-based accounting research. It provides a justification for regressing stock returns on the level and first difference of earnings which has been adopted in tests of association between accounting earnings and stock prices (see, for example, Easton & Harris, 1991; Penman, 1991a; Ali & Zarowin, 1992b; Strong & Walker, 1993). The growing use of empirical research designs based on an earnings-based valuation model which can be clearly traced back to the dividend capitalization model is a welcome reaction to the criticism that return/earnings studies have lacked theoretical support for their research designs. Accounting Policy and the Time Series Properties of an Earnings Construct A recent analysis by Feltham & Ohlson (1993) suggests that a potential use of time series modelling in accounting lies in the setting of accounting policy. The analysis developed the residual income-based valuation framework described above into a valuation model expressing unrecorded goodwill as being made up partly by future positive net present value opportunities not yet captured by the accounting model and partly by a mismatch between the depreciation policy and the time series properties of operating cash flows. The cash flow generating process and the cash flow-based valuation model underlying the Feltham & Ohlson (1993) analysis is: crt+1=ccrt+jcit+e1,t+1 cit+1=xcit+e2,t+1 crt+1−cit+1=ct+1 x

Pt=

]R

−s

Et[ct+s]

s=1

10

     

207

where cr (ci)=operating cash receipts (cash investment) and the e terms are zero mean random disturbance terms. Net financial assets are assumed to be zero. After much rearrangement involving manipulation of the clean surplus earnings relationship, the following expression results:

A B

Pt=oat+

c oxat R−c

A B

+

CA B DA B

R j R (c−d)oat−1+ −1 cit R−c R−c R−x

(19)

where oat=operating assets at time t, oxat=residual operating income for period t and d is one minus the reducing balance depreciation rate applied to operating assets. The fourth term, being the effect of omitted positive net present value opportunities, disappears if j =1 R−c which holds if the net present value of new investments is zero. The third term reflects the relationship between one minus the reducing balance depreciation rate (d) and the term which captures the persistence in operating cash receipts (c). Note that this term disappears if c=d. In other words, if the rate of decay in the operating cash flows earned from an investment is matched by the reducing balance depreciation rate applied to operating assets, the contribution to unrecorded goodwill of ‘over-depreciation’ is zero. This analysis suggests directions for the use of time series modelling in the determination of accounting policy. CONCLUSION Analysis of the time series properties of earnings has a long history and there is a wealth of evidence on these properties. A number of applications of knowledge of the time series properties of earnings have been demonstrated or suggested. Although much of the early literature cited ‘valuation’ as a motivation for interest in the time series properties of earnings, this was less convincing than it might have been because of the lack of an explicitly stated, theoretically supported earnings-based valuation model employing the time series properties of earnings. In recent years, however, such models have appeared and their influence can be seen in a number of strands of the literature. The theoretical literature has seen the ‘weighted average’ model of Ohlson (1991), which suggests the joint role of earnings,

208

. ’

book value and residual income persistence in equity valuation, and the Feltham & Ohlson (1993) analysis, which suggests the potential for the use of time series concepts in the setting of accounting policy. The empirical literature has seen the use of ‘earnings persistence’ to explain earnings response coefficients and the adoption of a return/earnings regression methodology which is a development of the Ohlson ‘weighted average’ model of the value of equity capital. The influence of such valuation models provides an important new impetus to interest in the time series properties of earnings. N 1. Note that the notation, D, is used later to denote dividend. 2. The terms ‘martingale’ and ‘submartingale’ refer to processes which are similar to a random walk without drift and with drift, respectively. The martingale and submartingale differ from the random walk in that they do not require the disturbance term to be independently and identically distributed. 3. In time series modelling, the term ‘identification’ refers to the task of identifying the order of p, d and q which best characterizes the generating process of the series. The ‘estimation’ task involves the estimation of the coefficients of the class of model identified. 4. Using a standard algebraic representation, the ARIMA (1, 0, 0)(0, 1, 0)4 process for quarterly earnings (denoted by Q) is (Qt−Qt−4)=l+x(Qt−1−Qt−5)+et. 5. Note, however, that the intuition underlying the mt term in (11.i) and (11.ii) is reflected in studies such as Kothari & Sloan (1992) in which leading period returns are included in return/earnings regressions.

R Albrecht, W., Lookabill, L. & McKeown, J. (1977). ‘The time series properties of annual earnings’, Journal of Accounting Research, pp. 226–244. Ali, A. & Zarowin, P. (1992a). ‘Permanent versus transitory components of annual earnings and estimation error in earnings response coefficients’, Journal of Accounting and Economics, pp. 249–264. Ali, A. & Zarowin, P. (1992b). ‘The role of earnings levels in annual earnings-return studies’, Journal of Accounting Research, pp. 286–296. Ball, R. & Watts, R. (1972). ‘Some time series properties of accounting income’, Journal of Finance, pp. 663–682. Bathke, A. & Lorek, K. (1984). ‘The relationship between time series models and the security market’s expectation of quarterly earnings’, The Accounting Review, pp. 163–176. Bathke, A., Lorek, K. & Willinger, G. (1989). ‘Firm size and the predictive ability of quarterly earnings data’, The Accounting Review, pp. 49–68. Beaver, W. (1970). ‘The time series behaviour of earnings’, Empirical research in accounting: Selected studies. Journal of Accounting Research (Suppl.), Vol. 8, pp. 62–99. Bernard, V. (1989). ‘Capital markets research in accounting during the 1980s: A critical review’. In T. Frecka (ed.), The State of Accounting Research as We Enter the 1990’s, pp. 72–100, University of Illinois.

     

209

Bernard, V. & Thomas, J. (1990). ‘Evidence that stock prices do not fully reflect the implications of current earnings for future earnings’, Journal of Accounting and Economics, pp. 305–340. Box, G. & Jenkins, G. (1970). Time Series Analysis: Forecasting and Control, Holden-Day. Brennan, M. (1991). ‘A perspective on accounting and stock prices’, The Accounting Review, pp. 67–79. Brown, L. & Rozeff, M. (1978). ‘The superiority of analyst forecasts as measures of expectations’, Journal of Finance, pp. 1–16. Brown, L. & Rozeff, M. (1979). ‘Univariate time-series models of quarterly accounting earnings per share: A proposed model’, Journal of Accounting Research, pp. 179–189. Brown, P. & Niederhoffer, V. (1968). ‘The predictive content of quarterly earnings’, Journal of Business, pp. 488–497. Brown, L., Griffin, P. Hagerman, R. & Zmijewski, M. (1987). ‘Security analyst superiority relative to univariate time-series models in forecasting quarterly earnings’, Journal of Accounting and Economics, pp. 61–87. Callen, J., Cheung, C., Kwan, C. & Yip, R. (1993). ‘An empirical investigation of the random character of annual earnings’, Journal of Accounting, Auditing and Finance, pp. 151–162. Collins, W. & Hopwood, W. (1980). ‘A multivariate analysis of annual earnings forecasts generated from quarterly forecasts of financial analysts and univariate time series models’, Journal of Accounting Research, pp. 390–406. Collins, D.W. & Kothari, S.P. (1989). ‘An analysis of intertemporal and cross-sectional determinants of earnings response coefficients’, Journal of Accounting and Economics, pp. 143–181. Dharan, B. (1983). ‘Identification and estimation issues for a causal earnings model’, Journal of Accounting Research, pp. 18–41. Dickey, D.A. & Fuller, W.A. (1979). ‘Distribution of the estimators for autoregressive time series with a unit root’, Journal of the American Statistical Association, June, pp. 427–431. Dopuch, N. & Watts, R. (1972). ‘Using time-series models to assess the significance of accounting changes’, Journal of Accounting Research, pp. 180–194. Easton, P. & Harris, T. (1991). ‘Earnings as an explanatory variable for returns’, Journal of Accounting Research, pp. 19–36. Edey, H. (1961). ‘Business valuation, goodwill and the super-profit method’. In W. Baxter & S. Davidson (eds), Studies in Accounting Theory, Sweet and Maxwell. Edwards, E. & Bell, P. (1961). The Theory and Measurement of Business Income, University of California Press. Feltham, G. & Ohlson, J. (1992). ‘Valuation and clean surplus accounting for operating and financial activities’, Working paper, University of British Columbia and Columbia University. Feltham, G. & Ohlson, J. (1993). ‘Uncertainty resolution and the theory of depreciation measurement’, Working paper, University of British Columbia and Columbia University. Foster, G. (1977). ‘Quarterly accounting data: Time series properties and predictive ability results’, The Accounting Review, pp. 1–21. Freeman, R. & Tse, S. (1992). ‘An earnings prediction approach to examining intercompany information transfers’, Journal of Accounting and Economics, pp. 509–523. Green, D. & Segall, J. (1967). ‘The predictive power of first quarter earnings reports’, Journal of Business, pp. 44–55. Griffin, P. (1977). ‘The time series behaviour of quarterly earnings: Preliminary evidence’, Journal of Accounting Research, pp. 71–83. Hillison, W., Hopwood, W. & Lorek, K. (1983). ‘Quarterly GPPA earnings data: Time series properties and predictive ability results in the airlines industry’, Journal of Forecasting, pp. 363–375. Imhoff, E. & Pare´, P. (1982). ‘Analysis and comparison of earnings forecast agents’, Journal of Accounting Research, pp. 429–439.

210

. ’

Kormendi, R. & Lipe, R. (1987). ‘Earnings innovations, earnings persistence and stock returns’, Journal of Business, pp. 323–345. Kothari, S.P. & Sloan, R. (1992). ‘Information in prices about future earnings’, Journal of Accounting and Economics, pp. 143–171. Lee, C. & Chen, C. (1990). ‘Structural changes and the forecasting of quarterly accounting earnings in the utility industry’, Journal of Accounting and Economics, pp. 93–122. Lev, B. (1983). ‘Some economic determinants of time series properties of earnings’, Journal of Accounting and Economics, pp. 31–48. Lev, B. (1989). ‘On the usefulness of earnings and earnings research: Lessons and directions from two decades of empirical research’, Journal of Accounting Research (Suppl.), pp. 153–201. Lorek, K. (1979). ‘Predicting annual net earnings with quarterly earnings time series models’, Journal of Accounting Research, pp. 190–204. Lorek, K. & Bathke, A. (1984). ‘A time series analysis of nonseasonal quarterly earnings data’, Journal of Accounting Research, pp. 369–379. Lorek, K., Schaefer, T. & Willinger, G. (1993). ‘Time series properties and predictive ability of funds flow variables’, The Accounting Review, pp. 151–163. O’Brien, P. (1988). ‘Analysts’ forecasts as earnings expectations’, Journal of Accounting and Economics, pp. 53–83. O’Hanlon, J., Poon, S.H. & Yaansah, R.A. (1992). ‘Market recognition of differences in earnings persistence: U.K. evidence’, Journal of Business Finance and Accounting, pp. 625–639. Ohlson, J. (1991). ‘Earnings, book value and dividends in security valuation’, Working paper, University of British Columbia and Columbia University. Peasnell, K. (1982). ‘Some formal connections between economic values and yields and accounting numbers’, Journal of Business Finance and Accounting, pp. 361–381. Penman, S. (1991a). ‘An evaluation of accounting rate of return’, Journal of Accounting, Auditing and Finance, pp. 233–255. Penman, S. (1991b). ‘Return to fundamentals’, Working paper, University of Berkeley. Ramakrishnan, R. & Thomas, J. (1992). ‘What matters from the past: Market value, book value or earnings? Earnings valuation and sufficient statistics for prior information’, Journal of Accounting, Auditing and Finance, pp. 423–464. Strong, N. & Walker, M. (1993). ‘The explanatory power of earnings for stock returns’, The Accounting Review, pp. 385–399. Watts, R. & Leftwich, R. (1977). ‘The time series of annual accounting earnings’, Journal of Accounting Research, pp. 253–271.