An Examination of Economic and Statistical Approaches that Address Sample Selection Bias, Inaccuracy, and Optimism in Analysts Earnings Forecasts

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1 An Examination of Economic and Statistical Approaches that Address Sample Selection Bias, Inaccuracy, and Optimism in Analysts Earnings Forecasts Mark Evans* (Indiana University) Kenneth Njoroge (University of Oregon) Kevin Ow Yong (Singapore Management University) This revision is preliminary and incomplete. Draft: February 2012 * Corresponding author. evansme@indiana.edu An earlier draft of this paper was entitled Bias and Accuracy in Long-Horizon Earnings Forecasts: Does a Cross-Sectional Model Improve Analysts Forecasts? and we received many helpful comments from the following sources: Christine Botosan, Asher Curtis, Linda Krull, Steve Matsunaga, Sarah McVay, Dale Morse, Stephen Penman, Marlene Plumlee, Katherine Schipper, Cathy Schrand, Teri Lombardi Yohn, and workshop participants at the University of Oregon, the University of Utah, the Midwest Accounting Research Conference, and the Northwest Accounting Research Conference. We also gratefully acknowledge financial support from Kelley School of Business at Indiana University, Lundquist College of Business at the University of Oregon and the School of Accountancy at Singapore Management University. 1

2 An Examination of Economic and Statistical Approaches that Address Sample Selection Bias, Inaccuracy, and Optimism in Analysts Earnings Forecasts ABSTRACT: This paper addresses concerns about forecasting accuracy and bias in analysts and extant model-based forecasting by proposing a cross-sectional earnings forecasting model that incorporates the mean reversion in earnings. We compare forecasts from our model to other models by evaluating out-of-sample accuracy, showing that forecasts from our model range from 7% more accurate at short horizons to 27% more accurate at long horizons. We compare forecasts from our model to consensus analysts forecasts by evaluating accuracy and bias, showing that forecasts from our model are more accurate than those of analysts from five quarters ahead to twenty quarters ahead. In addition, we find that consensus analysts forecasts are almost as optimistic as random walk forecasts are pessimistic; therefore, combining these forecasts yields relatively unbiased forecasts, compared to others. We also show that intrinsic values based on forecasts from our proposed model have comparable explanatory power for price when compared with intrinsic values based on analysts forecasts. Overall, our findings have important implications for market participants and researchers interested in forecasting earnings, especially over long horizons and for firms not covered by analysts. 2

3 1. Introduction This paper addresses concerns about bias and inaccuracy in analysts and model-based earnings forecasts. First, we propose a cross-sectional earnings forecasting model which incorporates the mean reversion of earnings and (1) compare its out-of-sample accuracy to that of analysts and extant models; and (2) examine whether intrinsic values based on forecasts from our model have comparable explanatory power for price, when compared with intrinsic values based on analysts forecasts. Our results show that forecasts from our model are more accurate and less biased than consensus analysts forecasts and extant models for five quarters to twenty quarters ahead; in addition, a valuation sample using forecasts from our model has approximately 50-60% greater coverage and comparable explanatory power for price than an IBES sample. These results suggest that our proposed model can be used to address the selection bias inherent in samples requiring analysts forecasts. Second, we develop a procedure that uses quarterly data to update the annual forecasts of statistical models, out-of-sample. This updating procedure is particularly helpful in comparing the accuracy of statistical forecasts derived using annual data to that of analysts forecasts, without discarding analysts forecasts that have a timing advantage. Third, we show that a simple yet effective procedure to undo the optimistic bias in consensus analysts forecasts is to combine them with random walk forecasts. Specifically, we document that analysts consensus forecasts are almost as optimistic as the random walk model s forecasts are pessimistic. Overall, our findings have important implications for market participants and researchers interested in forecasting earnings, especially over long horizons and for firms not covered by analysts. Forecasted earnings is important in accounting research because of its use as a key input in valuation and cost of capital models, and researchers frequently use analysts earnings 3

4 forecasts rather than forecasts from statistical models. 1 Expanding analyst coverage over time and the imprecision of time-series models fit to individual firms makes analysts forecasts particularly appealing to researchers. However, as of 2010, IBES analysts issue one, two and three-years ahead forecasts for only 43%, 41% and 35% of Compustat firms (see Figure 1). 2 Therefore, because valuation and cost of capital models typically utilize long-horizon analysts forecasts as inputs, outputs from these models are subject to significant selection bias. Accordingly, recent research examines the out-of-sample performance of simple models in predicting earnings (see, e.g., Bradshaw, Drake, and Myers 2011; Hou, van Dijk, and Zhang 2012). In this paper, we show that applying more structure to the forecasting task by incorporating a standard economic argument the mean reversion of profits yields more accurate and less biased earnings expectations, when compared to both extant model-based forecasts and consensus analysts forecasts. 3 In addition to concerns about selection bias present in samples requiring analyst forecast data, a long string of prior research documents that incentive and behavioral factors contribute to bias in analyst forecasts (see Francis et al for a review). Despite these concerns, prior research suggests analysts should outperform statistical models in terms of forecast accuracy because of timing and information advantages. First, analyst coverage is nonrandom; that is, 1 As Kothari [2001, p. 142] explains, use of analysts forecasts violates the clean surplus assumption underlying the residual income model. However, use of analysts forecasts is guided by their usefulness in explaining and predicting empirical phenomena rather than whether they are consistent with the clean surplus assumption. 2 Other databases of analyst forecasts include: First Call (now incorporated into IBES), Zack s, and Value Line. These databases suffer from similar, and perhaps more severe, selection concerns. See Francis, Chen, Philbrick, and Willis [2004] for a more complete discussion of analyst forecast databases. 3 Economic theory posits that earnings should mean revert, and prior research provides empirical support for this notion, with regard to annual earnings (see, Brooks and Buckmaster [1976]; Freeman, Ohlson and Penman [1982]; Lev [1983]; Ramakrishnan and Thomas [1992]; Nissim and Pennman [2001]; Fairfield, Ramnath and Yohn [2009]). In particular, Fama and French [2000] argue that much of what is predictable about earnings is due to the mean reversion of profits, and they suggest that analysts forecasts should incorporate the mean reversion in profitability. In this study, we incorporate the mean reversion in earnings into our forecasting model and, by comparing the accuracy and bias of this model to analysts consensus forecasts, we examine the extent to which analysts anticipate mean reversion in profitability. 4

5 analysts choose both firm coverage and forecast horizon. Second, analysts have a timing advantage that allows updating of forecasts between earnings report dates and, thus, they can adjust forecasts in response to unexpected shocks (e.g., strikes, lawsuits, mergers, management turnover). In contrast, statistical models restrict coefficient estimates to be either constant over time, constant across firms, or both. Third, analysts have an information advantage in the sense that inputs to statistical models are likely a smaller subset of analysts information set. In particular, a thriving analyst profession provides evidence that analysts add value to statistical models, suggesting it would be surprising if such models outperform analysts. Despite these advantages, however, we find that out-of-sample forecasts from our proposed model are less biased and more accurate, on average, than analysts consensus forecasts for five quarters ahead through twenty quarters ahead. Moreover, our forecasts are more accurate than random walk forecasts (see, e.g., Bradshaw et al. 2011), autoregressive forecasts (i.e., AR1 forecasts) and forecasts from other extant cross-sectional models (see, e.g., Hou et al. 2012). Specifically, we show that accuracy improvements to extant models increase in forecast horizon, ranging from 7% improvement at the one-year horizon to 27% improvement at the five-year horizon. In addition, in valuation tests, we show that the ability to explain stock price by the intrinsic values derived using forecasts from our model is similar to that of intrinsic values derived using analysts consensus forecasts. This finding suggests that researchers can incorporate a significantly larger sample upwards of 50-60% larger by using our model-based forecasts for firms not covered by analysts. Overall, our results are statistically significant and economically meaningful, and suggest that the proposed model can address significant concerns about selection bias, optimistic bias and inaccuracy in analysts earnings forecasts. 5

6 Besides proposing a cross-sectional earnings forecasting model and comparing its out-ofsample accuracy to that of analysts and extant models, we also make a methodological contribution that addresses analysts timing advantage. Statistical models that use annual data as inputs yield a single forecast each fiscal year. Meanwhile, analysts use their timing advantage to update their initial forecasts on a monthly basis as new information becomes available. To offset analysts timing advantage, researchers often restrict analyst forecasts to only the ones issued immediately after the prior period s earnings are announced (e.g., analyst forecasts issued in March 2011, for a December 31 st year-end firm that announces its earnings on 28 th, February, 2011). While this approach eliminates the real timing advantage of analysts over statistical models, it also limits model applicability to specific months and restricts the generalizability of any accuracy gains from statistical models. Early research that fits time-series models separately to individual firms finds that forecast accuracy can be improved by expanding the conditioning information set to include quarterly earnings (Brown [1993]). We draw on this notion to develop a simple but effective procedure that uses interim quarterly announcements to make out-ofsample updates on the forecasts of statistical models that use annual data inputs. We use this updating procedure to offset some of the timing advantage in analysts forecasts when comparing statistical forecasts to all analysts forecasts. Finally, while prior research documents systemic optimism in analysts forecasts and examines its determinants, we are aware of only a few studies that address the extent to which this optimism can be undone, out of sample (e.g., Easton and Sommers 2007; Hughes, Liu, and Su 2008). In this paper, we document that analysts consensus forecasts are almost as optimistic as the random walk model s forecasts are pessimistic, which reveals a mirror image of analysts optimism with random walk s pessimism (see Figure 4). Accordingly, we show that a 6

7 simple and effective way to undo the optimism (bias) in consensus analysts forecasts is to combine them with random walk forecasts. Specifically, we show that forecasts from this combined model are more accurate and less biased than analysts forecasts and other modelbased forecasts. This result addresses suggestions in Bradshaw et al that researchers construct superior forecasts utilizing the relatively accurate long-horizon forecasts from a random walk model. Our findings suggest that a cross-sectional model that incorporates fundamental economic theory (mean reversion) yields out-of-sample earnings forecasts that are more accurate and less biased than consensus analysts earnings over forecast horizons of five to twenty quarters ahead. Also, valuation results suggest that both analysts forecasts and forecasts from our model are suitable proxies for earnings expectations. These results suggest that the costs of selection bias and over-optimism outweigh analysts information and timing advantages. In addition, our findings suggest that incorporating mean reversion yields more precise estimates of future earnings than a random walk forecast, an autoregressive forecast, and forecasts from the cross-sectional model proposed by Hou et al. [2012] for the following reasons. First, random walk and autoregressive forecasts are typically used in the interest of parsimony; accordingly, Bradshaw et al. [2011] motivate their model by showing that the simplest model can outperform analysts at long horizons. However, both random walk and the autoregressive models limit the conditioning information set to current-period earnings. In contrast, our model uses larger conditioning information set which is incremental to earnings. Second, Hou et al. s model largely follows Fama and French [2000], which essentially extends an AR(1) model by adding market value, dividends, and accruals as predictors. Also, Hou et al. do not scale regression variables since their stated focus is to predict dollar earnings; instead, they scale the predicted 7

8 dollar earnings by end-of-june market values, ex post. Our results suggest that this ex post scaling is costly with respect to both bias and accuracy. We make four contributions to extant literature. First, we develop an earnings forecasting model that is more accurate and less biased than consensus analysts forecasts, when the forecast horizon is five to twenty quarters ahead. We also show that this model outperforms random walk and autoregressive model benchmarks as well as an extant cross-sectional model (i.e., Hou et al.) at all forecast horizons up to twenty quarters ahead. Second, using valuation tests, we show that the intrinsic values derived using our model s forecasts are comparable to the intrinsic values derived using analysts forecasts, with respect to the ability to explain crosssectional variation in the stock price. Accordingly, we propose that researchers can use our model to make earnings forecasts for the huge sample of firms not covered by IBES. Third, we make a methodological contribution by developing a procedure which uses quarterly interim data to update forecasts from cross-sectional models that use annual data. This updating procedure alleviates the timing advantage of analysts and facilitates more relevant comparisons between statistical models and consensus analyst forecasts. Fourth, we show that averaging the consensus analysts forecast and random walk forecast effectively removes analyst bias, and creates the most accurate and unbiased forecast. In summary, our findings are relevant for researchers and market participants interested in forecasting earnings, especially over long horizons and for firms not covered by analysts. Our paper is organized as follows. In section 2, we discuss prior literature and competing forecasting models. In section 3, we discuss our sample and, in section 4, we present our main results. We conclude and discuss implications and caveats in section 5. 8

9 2. Mean-reversion model and competing forecasts In this section we describe and motivate our mean reversion model (section 2.1), as well as competing models in the extant literature (section 2.2): random walk model, autoregressive model, and a cross-sectional model proposed by Hou et al. [2012]. 2.1 Mean-reversion model Fama and French [2000] use a two-stage approach to estimate a partial adjustment model of profitability in the cross-section. The first stage estimates the expected level of profitability from the fitted value of a cross-sectional regression of return on assets (ROA) on the ratio of dividends to the book value of common equity, a dummy for dividend paying firms and the market value-to-total assets ratio. The second stage estimates the rate of mean reversion to the expected level of profitability from a regression of next year change in ROA on ROA, the fitted value of ROA (from the first-stage) and change in ROA. Under mean reversion in profitability, the coefficient on ROA should be positive while the coefficient on the fitted ROA should be negative, and the two should be of equal absolute magnitude if the model is properly specified. Meanwhile, the two coefficients provide an estimate of the average rate of mean reversion of profitability. We extend this approach by putting more structure to modeling the cross-sectional variation in earnings as Fama and French suggest. Specifically, we present stage 1 of our model below: Stage 1: Estimating Expected Profitability EPS t = α 0 + α 1 EPS t 1 + α 2 LOSS t 1 EPS t 1 + α 3 EPS t 1 + α 4 FIN_OBLIG t 1 + α 5 ΔEQUITY_DIST t 1 + α 6 ΔASSET_TO t 1 + α 7 SPLIT_DUM t 1 + α 8 DIV_DUM t 1 + α 9 SPEC_ITEMS t 1 + α 10 lnsize t 1 + ε t (1) 9

10 where EPS is earning per share, LOSS is an indicator equal to 1 if EPS is negative and zero otherwise, ΔEPS is change in EPS, ΔFIN_OBLIG is change in net financial obligations, ΔEQUITY_DIST is equity distributions, ΔASSET_TO is change in asset turnover, (SPLIT_DUM) is a dummy for stock splits, DIV_DUM is a dummy for positive cash dividend payers, SPEC_ITEMS is special and extraordinary items, and lnsize is the natural log of size. Specific details on how we calculate each variable are presented in the Appendix. In this first stage, we model expected EPS as a function of levels and changes of past performance, as well as signals about future performance. First, we recognize the persistence of earnings (and differential persistence of losses) and include lagged EPS, LOSS*EPS, and ΔEPS as predictors of current performance. We expect a positive coefficient on EPS and a negative coefficient on LOSS*EPS and ΔEPS. Second, following Nissim and Penman [2001] and Dechow, Richardson, and Sloan [2008], we consider lagged equity distributions (ΔEQUITY_DIST) and lagged change in net financial obligations (ΔFIN_OBLIG) as predictors of future performance. The rationale is that distributions to equity and debt holders come from core earnings that management expects to persist to the future. We expect a positive coefficient for ΔEQUITY_DIST and a negative coefficient for ΔFIN_OBLIG. Third, we follow Nissim and Penman [2001] and include change in asset turnover, ΔASSET_TO, which is predicted to positively influence earnings. Fourth, we include indicator variables for lagged stock splits (SPLIT_DUM) and dividend payers (DIV_DUM), as credible management signals of future profitability, expecting positive coefficients for both (Fama and French 2000). Fifth, we include the lag of special items (SPEC_ITEMS), expecting a negative coefficient, consistent with managers using this classification to take a big bath in the current year, and realize more income in future years (Jones and Smith 2011). Finally, we also control for size using total 10

11 assets (lnsize), which is expected to be positively associated with future earnings (Fama and French 2006). We validate our Stage 1 regression s ability to explain cross-sectional variation in expected earnings using the Stage 2 partial adjustment model presented below: Stage 2: Partial Adjustment Model EPS t+1 EPS t = β 0 + β 1 E[EPS t ] + β 2 EPS t + β 3 (EPS t EPS t 1 ) + ε t+1 (2) E[EPS t ] is the fitted value from the Stage 1 regression, and it proxies for expected earnings. If the first stage is well specified in the sense that it reasonably captures the cross-sectional variation in expected EPS, mean reversion in EPS implies that β 1 = - β 2. Effectively, the stage 2 model provides the means to validate the extent to which our forecasting model captures crosssectional variation in expected EPS. Consistent with Fama and French [2000], we include (EPS t EPS t 1 ) as an explanatory variable to capture additional variation in earnings not captured by the partial adjustment term. While we are motivated by Fama and French [2000], our model differs from theirs in three ways. First, our model deliberately excludes market value of equity or stock returns as predictors for two reasons: (1) a primary objective of our study is to compare our model to analysts. To the extent that market values and stock returns are related to analyst forecasts, including market values in the model will bias the model toward outperforming analysts this is similar to using analyst forecasts as an input to our model; (2) an equally important objective is to compare the fundamental intrinsic valuation implied by our model s EPS forecast to that implied by analyst forecasts in explaining stock price variation in the cross-section. Using market values or returns in our model would complicate any inference we can draw from such tests. 11

12 Second, we add more structure to the model explaining future earnings by drawing on previous literature, such as Fama and French [2006], Nissim and Penman [2001], and Dechow, Richardson, and Sloan [2008]. In other words, we utilize various profitability drivers identified in previous literature to help explain expected profitability. Third, we scale earnings by weighted shares outstanding instead of assets. This yields earnings-per-share forecasts which we can use to compare to analysts and utilize in our valuation analysis, consistent with Bradshaw [2004] and Frankel and Lee [1998]. After validating our Stage 1 model, we use it to forecast earnings-per-share for h periods ahead using the following model, which re-states equation 1 in terms of multi-period forecasts: EPS t+h = α 0 + α 1 EPS t + α 2 LOSS t EPS t + α 3 EPS t + α 4 FIN_OBLIG t + α 5 ΔEQUITY_DIST t + α 6 ΔASSET_TO t + α 7 SPLIT_DUM t + α 8 DIV_DUM t + α 9 SPEC_ITEMS t + α 10 lnsize t + ε t+h 2.2 Competing Forecast Models (3) Random Walk Model If the earnings time series follows random walk, then current EPS is the best predictor of its future value. Specifically: EPS i,t+h = EPS i,t + ε t+h, where h {1,2,3,4,5}. The conditioning information set for the random walk model is limited to current EPS and the model implies that future change in EPS is unpredictable. Random walk forecasts are attractive since survivor bias is minimal and estimation is simple. One disadvantage of random walk is that it ignores growth, resulting in pessimistic forecasts for growth firms. To the extent that growth is difficult to predict, developing models that outperform random walk 12

13 forecasts out-of-sample is nontrivial, especially over long forecast horizons. Economics research frequently uses the random walk model as a benchmark in assessing the performance of competing forecast models. In addition, early accounting research that uses sophisticated timeseries models fit separately to individual firms is largely unable to reject the hypothesis that earnings follow random walk. Moreover, Bradshaw et al., [2011] show that random walk outperforms analysts over long forecast horizons and for small or high growth firms. For these reasons, we will use the random walk model as a benchmark in comparing forecasting models. Autoregressive model The autoregressive model (hereafter AR) estimates a slope coefficient that relates a variable s current value to its future values. Formally: EPS i,t+h = α 0 + α 1 EPS i,t + ε t+h (4) Like in the random walk, the conditioning information set for the AR model is limited to current EPS. Unlike random walk, however, the AR model implies that current EPS can predict future change in EPS. In particular, when the absolute value of the slope coefficient is between zero and one (i.e.,0 < α 1 < 1) then EPS is mean reverting. Effectively, EPS accommodates mean reversion in profitability, but assumes that current EPS subsumes all information that is useful in predicting future performance. In the pooled cross-section, the AR model is attractive for its minimal survivorship bias and simple estimation. To the extent that EPS subsumes other predictors, the AR model can be suitable. The AR model is also frequently used as a benchmark model in previous economics and accounting research and can accommodate additional lags of earnings. We find, however, that including additional lags of EPS slightly increases the model s 13

14 in-sample R 2, but does not significantly improve its out-of-sample accuracy. For this reason, we exclude additional lags of EPS from the model. Hou, Dijk and Zhang [2012] model Hou et al., [2012] point out that use of analysts forecasts in estimating implied cost of capital is problematic for two reasons. First, despite their wide application in research, analysts forecasts are characterized by well-documented optimism bias. Second, limited analyst coverage gives rise to survivor bias for example, coverage in the 1970s is scarce and forecasts beyond the second year is unavailable for many of the covered firms. To address these concerns, Hou et al. [2012] draw on Fama and French [2000] and propose a pooled cross-sectional model to forecast the earnings of individual firms. They find that, on average, relative to analysts forecasts, the forecasts from their cross-sectional model have: (1) lower bias; (2) lower accuracy; and (3) higher earnings response coefficients. The model proposed by Hou et al., [2012] is: E i,t+h = α 0 + α 1 E i,t + α 2 NegE i,t + α 3 AC i,t + α 4 V i,t + α 5 A i,t + α 6 D i,t +α 7 DD i,t + ε t+h, (5) where E i,t+h is unscaled dollar earnings and the forecast horizon is h {1,2,3,4,5}. Meanwhile, NegE i,t is a dummy for negative earnings, AC i,t is operating accruals, V i,t is the market value of the firm, A i,t is total assets, D i,t is the dividend payment and DD i,t is a dummy for dividend payers. The model is estimated using a 10-year rolling window to predict, out-of-sample, dollar earnings according to the authors, analysts EPS forecasts are comparable with dollar earnings, rather than with profitability (i.e., return on assets or return on equity). The authors then scale the model s dollar earnings forecasts by each firm s end-of-june market equity and divide analysts EPS forecasts by end-of-june stock price and compare the two. 14

15 The conditioning information set for Hou et al. [2012] model includes non earnings information dividends, total assets and market value. In addition the model allows the persistence of negative earnings to be different from that of positive earnings and allows persistence of accruals to be different from that of cash flows. To the extent that these variables have incremental predictive ability over earnings then this model should, in expectation, outperform both random walk and the AR model. The model s in-sample R 2 is unsurprisingly high, because the variables are unscaled. However, whether this high R 2 combined with scaling predicted dollar earnings with the end-of-june market value translates to superior out-of-sample performance is an empirical question. In particular, since it is not the focus of their study, Hou et al., [2012] do not compare their model s performance to either the random walk or the AR model, out-of-sample. 3. Sample In Table 1, Panel A, we report the number and percentage of Compustat firms covered by IBES analysts. We provide this descriptive analysis because we are motivated by the optimism and sample selection concerns inherent in samples of analysts forecasts. We show that, for oneyear (two-year) [three-year] forecasts, IBES coverage increased from 25.58% (24.07%) [0.00%] of Compustat in 1980 to 42.51% (41.09%) [26.50%] in Due to these coverage concerns especially for long-horizon forecasts previous accounting research typically enlarges the sample by extracting long-run earnings forecasts from analysts two-year-ahead earnings forecasts and IBES analysts forecasts of earnings long-term growth rates (e.g., Frankel and Lee 1998; Bradshaw et al. 2011; Hou et al. 2012). However, analysts forecasts of long-term earnings growth rates (LTG) are problematic because their forecast horizons, and the corresponding growth forecast errors, are difficult to specify. In addition, it is problematic to interpret LTG 15

16 when the base fiscal years are loss years. This problem forces researchers to drop loss years from the sample, leading to additional selection bias. Finally, Harris (1999) and Chan et al. (2003) show that analyst forecasts of long-term growth lack predictive ability and are overly optimistic. Accordingly, studies exclusively using analysts forecasts face a severe selection concern, especially for forecasts with greater than two year horizons, which reveals the importance for researchers to develop forecasts of earnings using a greater proportion of the Compustat population. Table 1, Panel B, presents descriptive statistics for our Stage 1 mean-reversion estimation model sample. The estimation sample includes data from with required Compustat variables and excludes utilities, banks, and public service firms. 4. Results 4.1 In-Sample Estimation of Mean-Reversion Model In-sample estimation results are presented in Table 2. To alleviate the effect of outliers, regression variables beyond the 1 st and 99 th percentiles are treated as missing and estimation in both the first and second stage uses least absolute deviations (LAD) as opposed to OLS. In Table 2, Panel A, the first column shows results for the stage 1 regression, which estimates expected EPS. The model has significant explanatory power (R-squared = 37%) and each of the coefficients are significant in the expected direction. The results for the Stage 2 model are presented in the second column. The coefficient estimates on E[EPS t ] and EPS t are and , respectively, each significant at the 1% level and the null hypothesis that these coefficients have different absolute values is rejected at the 1% level. In addition, the coefficient on the lagged change in EPS is positive and significant at the 1% level. These results have the following implications. First, this model suggests that, on average, EPS reverts to an economywide mean at a rate of about 56% per year that is, on average, it takes less than 2 years for EPS 16

17 to mean revert. Fama and French [2000] report that profitability reverts to an economy-wide mean at a rate of roughly 38%. These rates are substantially different and could be due to different stage 1 models, different deflators, or both. Second, the significant coefficient for the lagged change in EPS in the stage 2 model suggests that the partial adjustment term (deviation from expected earnings) does not fully capture all variation in earnings. Third, the equality in absolute values for β 1 and β 2 suggest that our model for expected EPS is well specified and measured with relatively little error. This result provides empirical support for the validity of our model for expected earnings. In Panel B of Table 2, we present results of the in-sample estimation of expected EPS over longer horizons; specifically, for two, three, four, and five years ahead. All of the coefficients remain significant in the expected direction, while the R-squared decreases monotonically from 37% (one year ahead) to 15% (five years ahead). These results provide support for the stability of our model and its success in explained future earnings, up to five years ahead. 4.2 Out-of-Sample Comparison of Mean-Reversion Model vs. Other Models Table 3 reports results comparing the out-of-sample accuracy of the mean reversion model that we propose with the other models described in section 2 random walk, AR(1), and Hou et al. s model. Forecasts from our model are based on expanding windows using data beginning in 1964, with out-of-sample forecasts beginning in In order to make comparisons, in Panel A, we first compute the relative mean squared forecast error (RMSFE) as follows. Let eps i,t+h be firm i s actual GAAP earnings per share, h-years-ahead. Further, let k eps i,t+h t be model k s out-of-sample forecast of eps i,t+h, made at time t. On a firm-specific 4 Rolling 10-year windows yield similar results. 17

18 basis, we calculate the mean squared forecast error by model k as: MSFE i k = mean(( eps i,t+h k i,t+h t eps ) 2 ), where at least three out-of-sample forecasts are required to calculate the firmspecific mean. To compare the accuracy of model 1 to that of model 2, out-of-sample, we calculate their firm-specific relative mean squared error as follows: RMSFE 1,2 i = MSFE 1 i 2 MSFE i In this table we divide, on a firm-specific basis, the MSFE of the model in the top row by the MSFE of the model in the first column and report the cross-sectional median of this ratio. If the reported median RMSFE is smaller (bigger) than one, then the model in the row is more (less) accurate than model in the first column if it is equal to one, the two models are equally accurate. Results in Panel A indicate that the mean reversion model is more accurate than each of the three other models over every forecast horizon. In addition, in most cases, the mean reversion model yields greater improvement the longer the forecast horizon. When compared to random walk, the mean reversion model is 11.7% more accurate ( ) for one year ahead forecasts, and 20.5% ( ) more accurate for five year ahead forecasts. When compared to AR(1), the mean reversion model is 7.4% more accurate for one year ahead and 10% more accurate for five years ahead. When compared to Hou et al. model, the mean reversion model is 10.9% more accurate for one year ahead and 27.9% more accurate for five years ahead. The results are depicted graphically in Figure 2. It is apparent from the graph that the mean-reversion model yields the greatest forecast accuracy improvement, when compared to the random walk model and the Hou et al. model, and this improvement is especially greater over longer forecast horizons. While the AR(1) model performs quite well, our model is more accurate over every forecast horizon. 18

19 Results in Panel B show results from tests concerning accuracy using mean absolute forecast error (MAFE). Let eps i,t+h be firm i s actual GAAP earnings per share, h-years-ahead. k Further, let eps i,t+h t be model k s out-of-sample forecast of eps i,t+h, made at time t. We calculate the absolute forecast error by model k as: AFE i k = (eps i,t+h eps k i,t+h t ) eps i,t+h. Results for AFE are similar to those in Panel A. The mean reversion model is more accurate than each of the other three models over every forecast horizon, and the accuracy improvement increases monotonically in forecast horizon. Results are significant at the 1% level using the Wilcoxon signed-rank test and t-test for difference in means. Results are depicted graphically in Figure 3. Results are difference in means is similar to results for mean squared error Hou et al. performs the worst, followed by random walk, and AR(1). However, median results suggest that the random walk model is more accurate than the AR(1) model, while the Hou et al. model still performs the worst. 4.3 Out-of-Sample Comparison of Mean-Reversion Model vs. Analysts Our next analysis involves comparing the bias and accuracy from the mean reversion model to the bias and accuracy from analysts. 5 In addition, we examine the random walk model in this section to represent a naïve baseline. In order to set the mean reversion and random walk models on the same footing (in terms of timing) with analysts, we begin by implementing the following quarterly updating procedure, using the quarterly EPS seasonal difference to update annual forecasts. Specifically, let EPS t+h q=1,t+1 be the out-of-sample forecast for fiscal year t + h, made in the first quarter of fiscal year t +1, after a firm reports the previous fiscal year EPS. Recall that the newly reported EPS is used in a forecasting model, such that: EPS t+h q=1,t+1 = 5 Our main results utilize GAAP EPS as the benchmark. In robustness tests (section 4.5), we use IBES EPS as the benchmark, which yields qualitatively similar results. 19

20 α EPS t + β X t, where α is the EPS coefficient estimate, β is a vector representing all other model coefficient estimates and X t represents all other regressors. 6 Let YTDEPS q=k,t+1 be the year-todate EPS in quarter k of year t+1. We use the following algorithm to update the out-of-sample EPS forecast for fiscal year t + h, made in quarter q of year t + 1: EPS t+h q=1,t+1 if q = 1 EPS EPS t+h q=1,t+1 + α YTDEPS q=1,t+1 YTDEPS q=1,t if q = 2 t+h q=1,t+1 = EPS t+h q=1,t+1 + α YTDEPS q=2,t+1 YTDEPS q=2,t if q = 3 EPS t+h q=1,t+1 + α YTDEPS q=3,t+1 YTDEPS q=3,t if q = 4 We now illustrate how this updating works using the random walk model, in which α = 1 and β = 0. Consider the earnings announcements of a December year-end firm as illustrated below. HISTORICAL EPS FIGURES FOR CATERPILLAR INC. Q1 Q2 Q3 Q4 Year 2010 (FY = cal. year) (FY = cal. year) * Seasonal change (A) Update factor (B) Update value (A x B) * 2009 EPS, announced on 1/27/10 The table above presents actual quarterly EPS amounts for all four quarters of 2009 and the first three quarters of Seasonal changes are indicated by row A and, because this example utilizes random walk forecasts, the update factor in row B is The seasonal change is 6 For example, in random walk, α = 1 and β = 0; in an AR1 model α is the AR1 slope coefficient estimate and β = 0; in the proposed model, α is the EPS coefficient estimate and β is the vector of coefficient estimates for the other RHS variables. 20

21 multiplied by the update factor to produce an update value, which is utilized to update the forecasts, as in the figure below. QUARTERLY FORECASTS OF ANNUAL EPS FOR CATERPILLAR INC Q earnings announcements 4/26/10 7/22/10 10/21/10 Forecast Forecast Date /28/10 Initial forecast /27/10 First update /23/10 Second update /22/10 Third update This tables presents the updating procedure. The initial forecast (without updating) can be made after 2009 earnings are announced. This is the initial forecast of $1.44 (2009 EPS). The first update incorporates the update value from the first quarter of 2010 ($ $0.56 = $2.00). The second update incorporates the update value from the second quarter of 2010 ($ $0.51 = $2.51), while the third update incorporates the update value from the third quarter of 2010 ($ $0.60 = $3.11). Forecasts using the random walk model and the mean-reversion model are updated in a similar fashion. Table 4, Panel A, reports results comparing bias among the random walk model, analysts, and our mean-reversion model. In all analyses, we use consensus analysts forecasts rather than individual analyst forecasts because cost of capital and valuation studies commonly use consensus forecasts in their tests. Tables 4, Panel B, reports results comparing accuracy among the models. Bias is calculated as the median percentage difference between actual EPS and forecast EPS. Panel A shows that random walk forecasts are consistently pessimistic, while consensus analysts forecasts consistently optimistic. In addition, forecasts from our model are less biased than either analysts or random walk, beginning pessimistic for long horizon forecasts 21

22 before becoming optimistic seven quarters ahead. Figure 4 presents results graphically along with the bias a forecast which simply averages forecasts from random walk and analysts. These forecasts are the least biased among those considered. Panel B shows that forecasts from our model are more accurate than analysts for every forecast horizon except one, when both forecasts are equally accurate. In addition, the accuracy improvement increases in forecast horizon. In addition, the combination forecast averaging random walk and analysts is more accurate than forecasts from our model at every horizon, and the accuracy improvements are the greatest the shorter the forecast horizon. Finally, we show that analysts forecasts are more accurate than the combination forecasts at very short horizons (1 4 quarters ahead), but less accurate at longer horizons. 4.4 Relevance of Forecast-Based Intrinsic Values (Model vs. Analysts) In this section, we discuss tests comparing different forecasts intrinsic values to market price. Evidence that forecasts from our model (when compared to analysts) yield similar explanatory power for market price suggests that such forecasts can be used in samples of firms not covered by analysts. Following Bradshaw [2004] and others, we calculate intrinsic value as follows: Value t = BVPS t + E t[ri t+τ ] (1 + r) τ 3 τ=1 + E t[ri t+3 ] [r + (1 + r) 3 ] where RI = E[EPS] minus (BVPS * r), where r is cost of capital calculated on an industry basis (Fama and French 1997); BVPS is book value of common equity per share; and E[EPS] is either forecasted earnings-per-share by analysts, or forecasted earnings-per-share by our proposed model. Table 5 presents results from OLS regressions of market price on intrinsic value. Panel A for varying samples. The first two columns present results for two IBES-based samples. The 22

23 first (second) column shows that analyst-based intrinsic values have explanatory power of 50.8% (48.6%) for samples comprised of firms with available data in IBES, ignoring model-based requirements (with available data in IBES, but without necessary data to compute model-based forecasts). Coefficients on the analyst-based intrinsic value are less than one (0.773 and 0.756, p-values <.001), consistent with analyst optimism. The next two columns present results without regard for analyst coverage. The third (fourth) column shows that model-based intrinsic values have explanatory power of 48.9% (46.3%) for samples comprised of firms with available data to computed model-based forecasts, ignoring IBES requirements (with available data to compute model-based forecasts, but not IBES-based forecasts). Coefficients on model-based intrinsic values are greater than one (1.116 and 1.265, p-values <.001), consistent with forecast pessimism in our proposed model, consistent with Figure 4. These results suggest that using a sample of firms covered by analysts yields only a two percentage point improvement in explanatory power over forecasts from our model. Accordingly, our results suggest that excluding close to half of firm-year observations in favor of a small incremental explanatory power improvement is potentially unwarrantable. Panel B reports results from a constant sample at the intersection of data available in IBES and data available to compute model-based forecasts. The first and second columns report that analyst-based intrinsic values have 48.0% explanatory power, while model-based intrinsic values have 43.1% explanatory power. This result suggests that consensus analysts forecasts are suitable proxies for earnings expectations when utilizing samples requiring analyst coverage. Analyst coefficients (model-based coefficients) are less than one (greater than one) consistent with results in Panel A. The third and fourth columns include both intrinsic values in the same regression, and show that model-based intrinsic values have incremental information content 23

24 beyond that contained in analysts forecasts. In particular, coefficients for both values are positive and significant, and the R-squared increases from 48.0% (analysts alone) to 52.0% (combined model). In addition, the coefficient for analyst-based intrinsic value (0.505) is not significantly different from the coefficient on model-based intrinsic value (0.577); that is, information in model-based forecasts is not subsumed by information in analysts forecasts. 4.5 IBES Actual EPS vs. Compustat Actual EPS One potential concern with our accuracy and bias results regarding consensus analysts forecasts is that analysts do not forecast earnings per share, but a performance measure excluding certain revenue and expense items that they determine to be non-recurring. We address this concern with results reported in Table 6. Panel A shows that, when using IBES actual EPS, analysts and random walk display a similar mirror image of pessimism and optimism. Panel B shows that our proposed model is more accurate than consensus analysts forecasts seven quarters ahead to twenty quarters ahead. This evidence suggest that our results concerning inaccuracy and bias in consensus analysts forecasts is not driven by the measure used for actual EPS. 5. Conclusion We propose a cross-sectional earnings forecasting model which incorporates meanreversion, presenting evidence consistent with its usefulness in explaining market price, and its relative accuracy compared to other extant models and analysts. Specifically, we show that our model is significantly more accurate than a random walk model, an AR(1) model, and an existing cross-sectional model proposed by Hou et al. [2012]. In addition, improvements are increasing in forecasts horizon, ranging from 7% improvement for one year ahead to 27% improvement for five years ahead. We also show that forecasts from our proposed model are 24

25 more accurate than consensus analysts forecasts for five quarters ahead through twenty quarters ahead. In addition, we show that random walk forecasts are roughly as pessimistic as consensus analysts forecasts are optimistic, and averaging forecasts from these models yields the most unbiased prediction. Finally, we show that intrinsic values computed based on forecasts from our model have similar explanatory power for market price as intrinsic values computed based on consensus analysts forecasts. In addition, when using a sample of firms not covered by analysts, our model has significant explanatory power for price, which is comparable to that of analysts for an IBES sample. These valuation results suggest that using a sample of firms with analyst coverage unnecessarily restricts the sample by close to half and is potentially unjustifiable, given comparable explanatory power for price when using forecasts from our proposed model. We contribute to the literature in the following ways. First, we show that building an earnings forecasting model which incorporates mean reversion is more accurate and less biased than consensus analysts forecasts and extant model-based forecasts, and this improvement increases in forecast horizon. Second, we show that using forecasts from this model results in similar explanatory power for market prices as consensus analysts forecasts. Accordingly, we propose that researchers can use our model to make earnings forecasts for a sample of firms not covered by IBES. Third, we make a methodological contribution by implementing a quarterly updating procedure which adjusts forecasts from our cross-sectional model to provide more relevant comparisons with consensus analysts forecasts. Fourth, we show that averaging the consensus analysts forecast and random walk forecast effectively removes analyst bias, and creates the most accurate and unbiased forecast. In summary, our findings are relevant for 25

26 researchers and market participants interested in forecasting earnings, especially over long horizons and for firms not covered by analysts. In the interest of parsimony, our model is relatively simple and assumes that earnings revert to an economy-wide mean. Recent research suggests a role for industry information in predicting sales and earnings growth (Fairfield, Ramnath, and Yohn 2009). In addition, Fama and French [2000] show that profitability reverts to the mean at different speeds depending on the distance from actual to expected earnings. We do not address these refinements to our model but acknowledge these are fruitful avenues for future research. In addition, while our results suggest that using forecasts from our proposed cross-sectional forecasting model can significantly increase sample size without sacrificing relevance, analysts forecasts also contain significant information content as evidenced by the explanatory power of analyst-based intrinsic values for price. 26

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