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Evaluating the properties of analysts’ forecasts: A bootstrap approach
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The British Accounting Review 39 (2007) 3–13 www.elsevier.com/locate/bar
Evaluating the properties of analysts’ forecasts: A bootstrap approach Mark A. Clatworthya, David A. Peelb,, Peter F. Popec a
Accounting and Finance Section, Cardiff Business School, Cardiff University, Colum Drive, Cardiff, CF10 3EU, UK Department of Economics, Lancaster University Management School, Lancaster University, Lancaster, LA1 4YX, UK c Department of Accounting and Finance, Lancaster University Management School, Lancaster University, Lancaster, LA1 4YX, UK b
Received 8 April 2005; received in revised form 2 August 2006; accepted 16 August 2006
Abstract Previous research has reported that analysts’ forecasts of company profits are both optimistically biased and inefficient. However, many prior studies have applied ordinary least-squares regression to data where heteroskedasticity and nonnormality are common problems, potentially resulting in misleading inferences. Furthermore, most prior studies deflate earnings and forecasts in an attempt to correct for non-constant error variances, often changing the specification of the underlying regression equation. We describe and employ the wild bootstrap—a technique that is robust both to heteroskedasticity and non-normality—to assess the reliability of prior studies of analysts’ forecasts. Based on a large sample of 23,283 firm years covering the period 1981–2002, our main results confirm the findings of prior research. Our results also suggest that deflation may not be a successful method of correcting for heteroskedasticity, providing a strong rationale for using the wild bootstrap in future work in this, and other areas of accounting and finance research. r 2006 Elsevier Ltd. All rights reserved. Keywords: Analysts’ forecasts; Wild bootstrap; Deflation; Heteroskedasticity
1. Introduction The properties of analysts’ earnings forecasts have been studied extensively over more than 25 years (e.g., Brown et al., 1987; Brown, 1993; Capstaff et al., 2001; Das et al., 1998; Duru and Reeb, 2002; Kang et al., 1994; Kothari, 2001; O’Brien, 1988, 1990; O’Brien and Bhushan, 1990). Studies based on different data sets for various countries and time periods have reported that analysts’ forecasts are optimistically biased, i.e., on average forecast earnings are significantly higher than actual earnings realizations. Research also indicates that analysts’ forecasts do not incorporate information known at the forecast date efficiently. De Bondt and Thaler (1990) report that analysts’ forecast changes are greater than actual changes, suggesting that analysts systematically over-react. In contrast, Abarbanell and Bernard (1992) and Easterwood and Nutt (1999) find that analysts’ forecast errors (actual earnings minus forecast earnings) are positively related to the prior period Corresponding author.
E-mail address: [email protected] (D.A. Peel). 0890-8389/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.bar.2006.08.002
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M.A. Clatworthy et al. / The British Accounting Review 39 (2007) 3–13
earnings change, consistent with analyst under-reaction.1 Research into long run earnings forecasts provides stronger evidence of optimism and inefficiency: in a study of long run forecasts of growth in US companies’ earnings, Harris (1999) reports that forecast annual earnings growth exceeds actual growth by approximately 7% per annum. If analysts’ forecasts are biased and/or inefficient, this has important implications for their use as inputs to fundamental valuation models (e.g., Dechow et al., 1999; Francis et al., 2000; Choi et al., 2006) and other practical investment methodologies where forecasts (or forecast revisions) are an input to stock selection models. Similarly, forecast bias presents potential problems to capital markets researchers who rely on analysts’ forecasts as a proxy for the market’s expectations of earnings (e.g., Botosan, 1997). It is therefore essential that evidence on forecast bias and inefficiency is statistically robust. If forecasts are systematically biased and inefficient, it is then important to understand the reasons for, and consequences of that bias, and specifically to understand whether the bias is a rational consequence of analysts’ incentives (e.g., Lim, 2001) and/or the statistical properties of earnings (e.g., Helbok and Walker, 2004). Studies of analysts’ earnings forecasts typically examine forecasts pooled for different firms over several time periods using standard ordinary least squares (OLS) regression techniques (notable exceptions are Keane and Runkle, 1998; Capstaff et al., 2001; Basu and Markov, 2004). Basu and Markov (2004) suggest that OLS is inappropriate because analysts attempt to minimize mean absolute, rather than mean squared, forecast errors. Hence, if analysts weight large forecast errors proportionally the same as small errors, least absolute deviation (LAD) regression techniques are more appropriate.2 Like Basu and Markov (2004), we question the suitability of OLS for investigating the rationality of analysts’ forecasts. However, rather than motivate our research on the basis of the implied loss function faced by analysts, we show that the basic assumptions required for valid OLS estimation are highly likely to be violated in practise, potentially resulting in incorrect inferences being drawn. Specifically, we address the consequences of earnings forecast errors having non-constant variance in the cross-section and/or over time.3 A common practice in analyst forecast studies is to deflate earnings per share (EPS) and forecasts by a scalerelated variable, such as actual EPS or stock price. This procedure is designed to mitigate the problem of heteroskedasticity, where forecast errors are related to the level of actual EPS (Kothari, 2001). In this paper, we show that deflation can lead to invalid inferences concerning forecast bias. Inference problems are further exacerbated because the distribution of actual earnings and of forecast errors may be non-normal. Should this lead to non-normality of the residuals in estimating equations, incorrect inferences might be drawn concerning forecast properties.4 To mitigate these problems, we require a statistical estimation procedure that is robust to both heteroskedasticity and non-normality. We describe and illustrate such a technique—the wild bootstrap (Wu, 1986; Davidson and Flachaire, 2001). The wild bootstrap approach allows us to construct standard errors that can be employed for detecting bias and/or inefficiency in analysts’ forecasts when the assumptions required for OLS are not satisfied. This paper has two main objectives. The first is to describe the wild bootstrap approach and its benefits over estimation techniques commonly employed in accounting and finance research, where heteroskedasticity and non-normality are pervasive problems. The second is to employ the wild bootstrap to re-examine the findings of prior research into analyst forecast bias and inefficiency. We test for bias and inefficiency using familiar
1 Easterwood and Nutt (1999) attempt to reconcile these findings by arguing that analysts’ reaction to earnings is conditional on the nature of the information; they report that analysts are systematically optimistic because they under-react to bad news and over-react to good news. 2 Basu and Markov (2004) are not the first to argue in favour of a linear symmetric loss function (see Gu and Wu, 2003). However, the shape of analysts’ loss function remains an unresolved issue. For example, Lambert (2004, p. 220) points out that ‘‘there are no economic or psychology models that support why analysts (or investors) should have a linear loss function’’. 3 We note that if forecasts for different horizons are pooled, then this will also induce heteroskedasticity, with forecast error variances expected to be positively related to the forecast horizon. 4 Non-normality may be less problematic than heteroskedasticity in some studies of analysts’ forecasts because of the large sample sizes typically involved, especially since it is standard practise to remove the tails of the distribution, which are often responsible for non-normal forecast errors (Abarbanell and Lehavy, 2003); though we note that this procedure needs justification if not for reasons of data errors. We also note that many other areas of accounting and finance research often involve much smaller sample sizes; and here, non-normality may be a more important issue.
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model specifications (e.g., Abarbanell and Bernard, 1992; Capstaff et al., 1995, 2001; De Bondt and Thaler, 1990; Easterwood and Nutt, 1999) on a sample of over 22,000 analysts’ forecasts of annual EPS. The paper makes three important contributions. First, prior findings of bias and inefficiency are supported by our analysis. Second, we show that the practice of deflating actual and forecast EPS does not adequately address the problem of heteroskedasticity. Finally, our description and application of the wild bootstrap offers accounting researchers a useful and reliable alternative to existing estimation methods. One important advantage of the wild bootstrap is its ability to cope with heteroskedasticity without deflating actual and forecast EPS. Potentially problematic choices about which deflator to use (e.g., share price, absolute EPS or the standard deviation of EPS over a given period) are therefore avoided, as is the potential for the availability of data for deflation creating biased samples (cf. Christie, 1987; Durtschi and Easton, 2005). The remainder of the paper is structured as follows. In Section 2, we discuss statistical research design issues surrounding previous attempts to test for inefficiency and bias in analysts’ forecasts; we then describe the properties of the wild bootstrap. In Section 3, we describe the data used in the empirical analysis. Section 4 reports the results of our tests of bias and efficiency using test statistics based on the wild bootstrap. Our new empirical results confirm prior research that analysts’ forecasts of EPS exhibit bias and inefficiency. Section 5 presents our conclusions and the implications of our findings.
2. Statistical research design issues 2.1. Testing forecast bias and inefficiency Assume that yt f t;th þ vt ,
(1)
where yt is the actual realization of EPS at time t, f t;th is the forecast of yt using information available at time th, and vt is an error term. Empirical models estimated in the prior literature are typically based on one of the following regression equations: yt ¼ d0 þ d1 f t;th þ Zt ,
(2)
yt yt1 ¼ d00 þ d01 ðf t;th yt1 Þ þ t ,
(3)
yt f t;th ¼ d000 þ d001 ððyt1 yt2 Þ
4 1X ðy yti1 ÞÞ þ et , 3 i¼2 ti
(4)
where the forecast is measured by the average or median analysts’ forecast, d0 , d00 , d000 , d1 , d01 and d001 are regression coefficients, and Zt , t and et are regression error terms. Assuming that these error terms are normal, have a constant variance and are uncorrelated with right-hand side variables, regression coefficients can be interpreted as follows. In Eq. (2), forecasts are weakly inefficient if d1 is significantly different from unity. The intercept term (d0 ) in Eq. (2) typically captures any constant bias—forecasts are generally optimistic if d0 o0 and pessimistic if d0 40.5 In Eq. (3), a coefficient d00 that is different from zero generally indicates bias in forecasts of earnings changes (e.g., Capstaff et al., 2001), while an estimated coefficient of d01 that is less than unity is interpreted as evidence of analyst over-reaction (e.g., De Bondt and Thaler, 1990). In regression (4), forecasts are inefficient if d001 differs from zero. A significant negative coefficient is indicative of over-reaction, while a significant positive coefficient is consistent with under-reaction (Abarbanell and Bernard, 1992; Easterwood and Nutt, 1999). 5 We qualify this interpretation because, as pointed out by an anonymous reviewer, the interpretation of the intercept strictly depends on the value of d1 . It is possible for forecasts to be unbiased, even if d0 differs from zero; in particular, forecasts are unbiased if d0 ¼ ð1 d1 ÞEðyt Þ (see Holden and Peel, 1990).
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2.2. Deflating outcomes and forecasts Prior research involves the analysis of samples of data which are pooled outcomes and forecasts for different firms. Suppose expectations are rationally formed and analysts’ loss functions are quadratic. In these circumstances, the forecast error for each firm will be a random, unpredictable variable. However, unless the variance of the error in the time series process for earnings in each firm in the sample is identical, then the regression error will exhibit heteroskedasticity where, in principle, every forecast error is associated with a different error variance. It seems likely that the absolute magnitude of forecast errors is positively correlated across different firms with the level of earnings, so that the size of absolute forecast error is greater the higher EPS. As a consequence, it has become standard in empirical tests of the properties of analysts’ forecasts of EPS to deflate all variables by an instrument implicitly believed to be associated with the variance of the forecast error.6 Deflators used in prior research to mitigate potential heteroskedasticity include the absolute value of actual earnings (e.g., Capstaff et al., 1995, 2001), the standard deviation of prior earnings (De Bondt and Thaler, 1990) and share price (e.g., Duru and Reeb, 2002; Easterwood and Nutt, 1999; Keane and Runkle, 1998). However, prior research fails to acknowledge that this deflation procedure can have important effects on the properties of the regression and can affect the inferences from empirical tests. To illustrate this, assume that the forecast horizon is one period and that all relevant variables in a regression are scaled by a common deflator st1 in the information set at the forecast date th.7 It is straightforward to show that generally estimated coefficient values (denoted by * or ** in sequel) are influenced by the deflation process. For example, regression (2) can be rewritten as follows: yt st1
¼ d0 þ d1
f t;th þ t . st1
(5)
There is no guarantee that d0 ¼ d0 or d1 ¼ d1 . The actual coefficient estimates depend on the variance–covariance structure of the deflated variables and may bear no resemblance to the variance–covariance structure of the undeflated variables. For example, it is possible that d1 ¼ 0, i.e., Cov½yt ; f t;th ¼ 0, but for the deflator st1 to have such a high variance that it dominates the covariance term such that Cov½ðyt =st1 Þ; ðf t;th =st1 Þa0, so that the estimated value is on average non-zero. In summary, estimated coefficient values for Eq. (5) will not equal the ‘‘true’’ values obtained from (2) on average. Furthermore, the transformed model (5) is effectively changing the specification of the model. Eq. (5) can be rewritten as: yt ¼ d0 st1 þ d1 f t;th þ t .
(6)
Thus, the predicted level of earnings from (5) is a linear combination of the scaling variable and the level of EPS forecast by analysts. This is a different hypothesis to the one tested in Eq. (2), where the predicted level of EPS is equal to a constant,d0 , plus a multiple, d1 , of the level of EPS forecast by analysts. If the variable used to deflate yt and ft,th is taken at t– h (i.e., the period before the forecast is made), then under rational expectations, d0 is still expected to be zero. However, if the deflator contains information relevant to the prediction of earnings (for instance, contemporaneous share price) and such information is not yet contained in the forecast, then it is possible for the expected value of d0 to differ from zero. One potential resolution of the heteroskedasticity problem is to estimate a transformation of Eq. (2) using weighted least-squares regression. Thus, Eq. (2) can be rewritten as: f t;th yt 1 ¼ d þ d þ 0 1 t . st1 st1 st1 6
(7)
If deflation is to be consistent with weighted least squares regression, the standard deviation of the error term should be a linear function of the deflator. 7 Duru and Reeb (2002) deflate the ex-post forecast error by stock price. They attempt to explain this measure of the degree of ‘‘bias’’ by a variety of explanatory variables. Some of the most significant explanatory variables they report are deflated by the stock price. They state that their results are robust if they deflate by the mean of the consensus forecast. This robustness in fact suggests a statistical problem. Essentially the common deflator appears to be driving their results.
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Estimates of (7) will yield unbiased and efficient estimates of the standard errors of d^ 0 and d^ 1 if ðst1 Þ2 ¼ kðst1 Þ2 , where k, the error variance, is constant. However, in general, researchers have not reported formal tests to identify the nature of heteroskedasticity, nor have they reported whether the chosen deflator corrects for heteroskedasticity. This is despite reports in prior research that deflation can often exacerbate heteroskedasticity and coefficient bias (e.g., Barth and Kallapur, 1996). For instance, in valuation regressions, Lubberink and Pope (2005) find that deflation can lead to coefficients deviating from their predicted values by a factor of over 5 and conclude (2005, p. 12) that ‘‘deflating by a scale variable leads to seriously biased coefficient estimates.’’ Consequently, there is no way of knowing whether tests of equations such as (7) yield unbiased estimates of the true coefficients. In the event that the distribution of the error terms is non-normal, inference problems are further exacerbated. Similarly, although the well-known White (1980) method permits asymptotically correct inferences in the presence of heteroskedasticity of unknown form, it is known that in finite samples, t- and F-tests can be seriously biased.
2.3. The wild bootstrap approach Recent advances in computing offer an alternative approach to hypothesis testing when the error term in a regression is heteroskedastic and potentially non-normal.8 Under such conditions the wild bootstrap has been shown to be an appropriate method for determining appropriate critical values for t- and F-tests (Wu, 1986; Hardle and Mammen, 1993; Davidson and Flachaire, 2001). The intuition behind the approach is to identify the distributions of relevant test statistics when the null hypothesis holds and the distribution of regression residuals is based on the relevant sample distribution. We illustrate the bootstrap procedure with reference to Eq. (2), but in testing we apply it to the estimation of all equations of interest. The procedure is as follows: (i) Estimate (2) by OLS. We denote the residuals from this regression as Z^ . (ii) Create a new series of ‘‘pseudo’’ residuals based on Z^ : Z0i ¼ Z^ i ei , Z0i ¼ Z^ i oi , where ei and oi are drawn from one of the following two-point distributions: pffiffiffi 8 pffiffiffi 5þ1 51 > > p ffiffi ffi with probability ; > > < 2 2 5 pffiffiffi ei ¼ pffiffiffi > 5þ1 5þ1 > > p ffiffi ffi with probability 1 > : 2 2 5 or
( oi ¼
1 with probability 0:5; 1 with probability 0:5:
(8) (8a)
(9)
(10)
Both ei and oi are mutually independent drawings from a distribution independent of the original data and their distribution has the properties: Eei ¼ 0; Ee2i ¼ 1; Ee3i ¼ 1; Ee4i ¼ 2; Eoi ¼ 0; Eo2i ¼ 1; Eo3i ¼ 0; Eo4i ¼ 1: The rationale for using these specific distributions is that they preserve the heteroskedastic nature of the original regression residuals. In particular, since the errors are independent, any heteroskedasticity in the pseudo residuals, Z0i , matches that in the regression residuals, Z^ . This is because the variance of the pseudo residuals, Z0i , is a replica of the variance of the original regression residuals Z^ . Furthermore, the 8
Note that few prior studies in this area report tests for non-normality of residuals and employ standard errors modified for heteroskedasticity in their analysis.
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(iii) (iv) (v)
(vi)
distribution in Eq. (9) matches any skewness inherent in the original regression residuals (though it overstates the kurtosis); the distribution in Eq. (10), on the other hand, preserves the kurtosis in the distribution of the original residuals, though it does not preserve the skewness.9 Construct a series of simulated (or artificial) earnings outcomes, yst , imposing the null hypothesis d0 ¼ 0, d1 ¼ 1 in (2) and using the pseudo residual series created in step (ii), i.e., yst ¼ f t;th þ Z0i .10 Regress yst on f t;th and save the estimated intercept and slope coefficients. Repeat steps (ii)–(iv) a large number of times (e.g., 10,000) and create sets of residuals, simulated earnings realizations and regression intercepts and slope parameters from the regression of the artificial earnings realizations yst for a given bootstrap sample on the forecasts f t;th . The generated sequences of simulated data have a true intercept of zero and true slope of unity by construction. However, the estimated parameter values will, in general, differ from the true values. From the 10,000 iterations we have an empirical distribution for the estimates of d^ 0 ¼ 0; d^ 1 ¼ 1 together with their associated standard errors. This distribution is based solely on re-sampling the residuals of the original regression. It yields appropriate critical values of the pivotal test statistics such as the t- and Fstatistic consistent with not rejecting the null hypothesis at an appropriate level of significance (e.g., 5%). These critical values can then be employed to determine whether the estimates obtained from (2) employing real data are consistent with rejection of the null.
3. Data Our data set consists of 23,283 eight-month ahead median consensus forecasts of annual EPS for 3268 US companies taken from I/B/E/S.11 The sample covers the period 1981–2002. Actual EPS and share price data are also provided by I/B/E/S in order to ensure consistency with the forecast (Richardson et al., 2004).12 We require the consensus to comprise a minimum of four forecasts (Easterwood and Nutt, 1999) and also require each firm to have actual EPS data for at least six consecutive years in order to estimate Eq. (4). Our sample size and the time period covered by our data compare favourably with those in prior studies. For instance, Easterwood and Nutt’s (1999) analysis is based on 10,694 observations between 1982 and 1995, while Duru and Reeb’s (2002) sample comprises 3495 firm-year observations over the period 1995–1998. More recently, Helbok and Walker’s (2004) UK study consists of 4454 observations taken from the period 1990–1998. In line with prior research, we removed observations where earnings variables were greater than 100% of share price at the date of the forecast (for the scaled regressions) or greater than $10 (for the unscaled regressions). Even with simulation techniques, inclusion of high leverage observations may result in inefficient estimates (Davidson and Flachaire, 1996; Davison and Hinkley, 1997). Because it has been suggested that research in this area may be sensitive to the procedure used to delete outliers (Ahmed et al., 2000), we also 9 This preservation of any heteroskedasticity in regression errors is a central feature of the wild bootstrap; this contrasts with conventional bootstrapping with replacement, which may destroy the heteroskedastic nature of the residuals. Hence, the wild bootstrap is preferred to standard bootstrapping techniques on this basis (Goncalves and Kilian, 2004). We also note that, following the suggestion of an anonymous reviewer, we conducted further analysis using the standard bootstrap procedure built in to the STATA statistical software, and this confirmed our main conclusions. For bivariate regressions, this procedure involves drawing, with replacement, pairs of observations from the dataset, and re-running the regression (in our case) 10,000 times on the new dataset. However, Flachaire (2003) finds that the ‘pairs’ bootstrap is outperformed by the wild bootstrap; furthermore, Horowitz (2001) finds that inference based on the pairs bootstrap is not always accurate. 10 Note that for Eq. (4), the null hypothesis is that both d000 and d001 are zero. The artificial outcomes in this regression are therefore equal to Z0,i. 11 In line with prior research, we use earnings on a per share basis since it is EPS that analysts actually produce forecasts for. Furthermore, EPS is important to investors as part of the price/earnings ratio and as a variable in its own right: as noted by Durtschi and Easton (2005, p. 6) ‘‘particularly from the viewpoint of the individual investor, earnings per share is not a deflated variable—it is simply the income due to the owner of each share.’’ 12 We use the median consensus forecast for consistency with the majority of prior studies of analysts’ forecasts (e.g., Easterwood and Nutt, 1999; Lim, 2001; Doukas et al., 2002; Richardson et al., 2004); however, additional analysis employing the mean yielded virtually identical results to those reported. As Kothari (2001, p. 154) points out: ‘‘Even if the distribution of actual earnings might be skewed, the distribution of analysts forecasts for a given firm need not be skewed, so the use of the mean or median of analysts forecasts might not make much difference.’’
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Table 1 Summary statistics (n ¼ 23, 283) Variablea (label)
Meanb
Median
Std. dev.
Actual year t earnings (yt ) Eight-month ahead consensus forecast of year t earnings (f t;th ) Actual earnings change between time t – 1 and time t (yt yt1 ) Forecast change in time t – 1 earnings (f t;th yt1 ) Forecast error (yt f t;th ) P Unexpected prior period’s earnings change ððyt1 yt2 Þ 13 4i¼2 ðyti yti1 ÞÞ Share price at the time of the forecast (Pth )
1.2775 1.5184 0.0078 0.2487 0.2409 0.0284
1.1200 1.2700 0.0900 0.1600 0.0400 0.0133
21.4229
18.1300
a
Skewness
Kurtosis
1.3937 1.2469 0.9950 0.6528 0.8249 1.0726
0.36 1.44 –1.44 3.07 –3.87 –0.13
8.92 8.34 23.01 37.62 30.72 21.51
15.6341
2.41
18.32
All data are provided by I/B/E/S and cover the period 1981–2002. All figures are on a per share basis.
b
conducted our analysis with more (and less) stringent outlier deletion rules. Our results are not sensitive to these alternative procedures and we only report results for the full sample—the others are available on request. Summary statistics for the variables used in the regressions are presented in Table 1.13 Both the mean and median forecast are greater than the actual earnings realizations and, in line with prior research (e.g., Easterwood and Nutt, 1999), the mean forecast error is negative and greater (in absolute terms) than the median forecast error, consistent with the negative skewness coefficient of 3.87. Table 1 also shows that the mean and median forecasts of the change in earnings between time t1 and time t (0.25 and 0.16, respectively) are substantially higher than the actual change in earnings over this period (mean of 0.01 and median of 0.09). This finding is also consistent with prior research: for example, using UK data, Helbok and Walker (2004) find that the mean forecast change in earnings (1.93% of beginning of period share price) is higher than the mean actual change in earnings (0.58% of price). It is also noteworthy that a number of variables are non-normally distributed: there is evidence of both positive (e.g., the forecast change in EPS) and negative (e.g., forecast error) skewness and kurtosis, potentially resulting in OLS being an inefficient estimator.
4. Analysis Our main results are presented in Table 2. Columns two and three show the estimated parameters and tstatistics based on OLS and White’s (1980) standard errors in parentheses, while the critical values from the wild bootstrap are presented in square brackets.14 We note that the critical values obtained from the two wild bootstrap distributions differ only very slightly in our analysis.15 As column seven shows, the White F-test indicated significant heteroskedasticity in all specifications.16 This was also true of Eq. (iv) in which share price was used as a deflator (as employed in other studies to mitigate heteroskedasticity). Accordingly, we employed the wild bootstrap to obtain critical values of the ‘‘t’’ values corrected for heteroskedasticity employing the White (1980) method. As mentioned above, although the wellknown White method permits asymptotically correct inference in the presence of heteroskedasticity of unknown form, it is known that in finite samples, t- and F-tests can be seriously biased (see e.g., MacKinnon and White, 1985; Chesher and Jewitt, 1987). 13
As a consistency check, we compared our annual scaled mean forecast errors with those reported in Easterwood and Nutt (1999, p. 1781) for the common time period (1982–1995). Our results are virtually identical to theirs, with a correlation coefficient of 0.98 (p ¼ 0:000). 14 At the suggestion of an anonymous reviewer, we also employed least absolute deviation (LAD) regression to regressions (i)–(iii), in line with Basu and Markov (2004). With the exception of the intercept in regression (i), our conclusions remain unchanged. 15 The E-Views statistical software (version 5) was used to generate the main results. 16 We also conducted additional tests based on the Breusch-Pagan/Cook-Weisberg test for heteroskedasticity. In each case, the test results confirmed those of the White test.
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Table 2 Properties of analysts’ forecasts (n ¼ 23, 283) Equationa (estimator)
d0 (t-value)
d1 (t-value)
R2
Fb
J–Bc
Hetd
(i) yt ¼ d0 þ d1 f t;th þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical valuee Wild bootstrap (II) 0.05 critical value
0.0980 (11.62) (9.27) [2.14, 2.01] [2.11, 2.11]
0.9059 (21.94) (12.60) [2.03, 1.82] [1.96, 1.96]
0.66
1025
767,583
267
(ii) yt yt1 ¼ d0 þ d1 ðf t;th yt1 Þ þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
0.2069 (35.98) (31.56) [1.52, 1.52] [1.52, 1.52]
0.8631 (16.62) (6.68) [2.05, 1.88] [1.96, 1.96]
0.32
771,213
308
P (iii) yt f t;th ¼ d0 þ d1 ððyt1 yt2 Þ 13 4i¼2 ðyti yti1 ÞÞ þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
0.2388
0.0742
0.01
799,205
320
(44.37) (44.53) [1.91, 1.96] [1.95, 1.99]
(14.79) (6.64) [2.05, 1.93] [1.98, 1.99]
0.0382
0.6523
2,188,987
245
(17.49) (8.78) [2.16, 1.75] [2.00, 1.90]
(43.81) (12.17) [1.84, 2.07] [1.93, 1.95]
(iv)
ðyt yt1 Þ Pth
¼ Pd0 þ d1 th
ðf t;th yt1 Þ Pth
þ mt
OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
[6.4] [6.6] 1009
[11.9] [10.2] 1006
[10.1] [10.2] 0.22
158
[9.4] [26.4]
a Variable definitions are provided in Table 1; OLS and White’s corrected t-statistics are in parentheses, while critical values for t- and Fstatistics are in square brackets. b All equations except (iii) test the null that d0 ¼ 0; d1 ¼ 1; equation (iii) tests the null that d0 ¼ 0; d1 ¼ 0. c Jarques–Bera test for normality of residuals. d White’s F-test for heteroskedasticity. pffiffiffi pffiffiffi ð 5 1Þ ð 5 þ 1Þ e ; distribution; wild bootstrap II uses the (1, –1) distribution. Wild bootstrap I uses the 2 2
The Jarque–Bera test results in column six indicate severe departures from normality in all regression residuals. In fact, although the test statistics obtained from the wild bootstrap methods differ somewhat from those obtained from the White (1980) method, it is clear from the results reported in Table 2 that they are sufficiently close in this sample not to change any inferences based on the standard White adjusted ‘‘t’’ values. Even though the degree of non-normality is great, the critical values are close to the asymptotic values. Indeed, in some cases (for example, in the case of the intercept term in regression (ii)), the simulation results suggest that the reported rejections of the null hypothesis are marginally stronger than implied by the usual tests. Naturally this was not known ex-ante. The evidence in Table 2 is consistent with analyst optimism and inefficiency. For example, the intercept terms in regressions (i) and (ii) are both significantly negative, suggesting that analysts’ forecasts are optimistic (consistent with Capstaff et al., 1995, 2001). Moreover, regression (iii) demonstrates that information contained in the prior year’s unexpected earnings change is reflected in the time t forecast error, indicative of analyst under-reaction. In this regard, our results offer confirmation of previous results reported in the literature (e.g., De Bondt and Thaler, 1990; Abarbanell and Bernard, 1992; Easterwood and Nutt, 1999), though obtained employing a very different and more reliable methodology. The results from regression (iv) show that even when the equation is correctly specified (by deflating the constant term d0 ), deflating by share price does not appear to ‘‘solve’’ the heteroskedasticity problem. The White test statistic is only marginally lower than the corresponding value for the undeflated regression (ii). Furthermore, the t-statistics obtained using White’s correction in regression (iv) are substantially lower than the OLS t-statistics. Also noteworthy from Table 2 are the differences between the estimated coefficients in regressions (ii) and (iv). The undeflated equation (ii) has a slope parameter much closer to the expected value
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of unity (0.86 compared to 0.65 in Eq. (iv)), although the intercept is indicative of a greater degree of optimistic bias. A possible limitation of our analysis is cross-section correlation in forecast errors due to macro-economic factors that affect all firms similarly. Keane and Runkle (1998) control for this potential problem by employing a generalized method of moments (GMM) estimator. However, they assume homoskedasticity and rely on deflation by price—a procedure that our results suggest is unreliable. In order to test for the effects of cross-section correlation, we estimated our main regressions using the delete-group jackknife method proposed by Shao and Rao (1993) and by using Rogers (1993) standard errors clustered by year.17 We also conducted Shao Rao jackknife tests and estimated Rogers (1993) standard errors clustered by company to control for the possibility of errors being correlated within firms over time. In each case, our results were supported. 5. Conclusions Forecast errors can exhibit non-normality and heteroskedasticity under the null of rationality. In such circumstances, inferences based on pivotal test statistics using the conventional critical values are suspect, i.e., they tend to deviate from the true size of the test. The wild bootstrap is an appropriate and particularly useful method for testing the properties of forecasts in these circumstances. Our main results confirm the findings of previous research: across various model specifications, we find evidence consistent with analysts’ forecasts of corporate profits being biased and inefficient. However, this may not be the case in studies employing different data sets and/or different model specifications. Since there are persuasive arguments for moving beyond potentially unreliable estimation methods, we suggest that the wild bootstrap represents a dependable alternative. Our results also suggest that the common method of deflation is not successful at reducing heteroskedasticity. This is in line with Barth and Kallapur (1996) and implies that even after deflation, future research could usefully report results of tests for heteroskedasticity and, where appropriate, employ techniques (such as the wild bootstrap) that are robust to heteroskedasticity. The approach adopted in this paper permits the estimation of the ‘‘true’’ undeflated regression, thereby improving the comparability of research on analysts’ forecasts (e.g., see Kothari, 2001) and eliminating the problems associated with selecting an appropriate deflator (e.g., Christie, 1987). For example, using the wild bootstrap on the undeflated regression means that it is unnecessary to remove data due to the ‘small deflator’ problem (e.g., see Gu and Wu, 2003). Sample selection bias induced by data providers including data for commonly used deflators in a non-random fashion can also be avoided by using the wild bootstrap (cf. Durtschi and Easton, 2005). Many areas of accounting and finance often rely on smaller sample sizes where non-normality is more problematic and where commonly used corrections for heteroskedasticity such as White’s (1980) standard errors are less effective. For example, samples of long run earnings forecasts are typically much smaller than those in research into short term (i.e., annual and quarterly) forecasts (e.g., Harris, 1999). In such circumstances, the wild bootstrap may yield different, and more dependable, results than conventional regression methods. Acknowledgements The authors gratefully acknowledge the financial support of INQUIRE (UK) and I/B/E/S for providing analyst forecast data. The authors are grateful to Ivan Paya for writing the programme code and to two anonymous reviewers for helpful comments. References Abarbanell, J.S., Bernard, V.L., 1992. Tests of analysts’ overreaction/underreaction to earnings information as an explanation for anomalous stock price behaviour. Journal of Finance 47 (3), 1181–1207. 17
See Petersen (2005) and Vuolteenaho (2002) for further discussion of these methods.
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Abarbanell, J., Lehavy, R., 2003. Biased forecasts or biased earnings? The role of reported earnings in explaining apparent bias and over/ underreaction in analysts’ earnings forecasts. Journal of Accounting and Economics 36, 105–146. Ahmed, A.S., Lobo G.J., Zhang, X., 2000. Do analysts under-react to bad news and over-react to good news? Working Paper, Syracuse University and University of Chicago. Barth, M.E., Kallapur, S., 1996. The effects of cross-sectional scale differences on regression results in empirical accounting research. Contemporary Accounting Research 13 (2), 527–567. Basu, S., Markov, S., 2004. Loss function assumptions in rational expectations tests on financial analysts’ earnings forecasts. Journal of Accounting and Economics 38, 171–203. Botosan, C.A., 1997. Disclosure level and the cost of capital. The Accounting Review 72 (3), 323–349. Brown, L.D., 1993. Earnings forecasting research: its implications for capital market research. International Journal of Forecasting 9, 295–320. Brown, L.D., Hagerman, R., Griffin, P., Zmijewski, M., 1987. Security analyst superiority relative to univariate time-series models in forecasting quarterly earnings. Journal of Accounting and Economics 9, 61–87. Capstaff, J., Paudyal, K., Rees, W., 1995. The accuracy and rationality of earnings forecasts by UK analysts. Journal of Business Finance and Accounting 22 (1), 69–87. Capstaff, J., Paudyal, K., Rees, W., 2001. A comparative analysis of earnings forecasts in Europe. Journal of Business Finance and Accounting 28, 531–562. Chesher, A., Jewitt, I., 1987. The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica 55, 1217–1222. Christie, A.A., 1987. On cross-sectional analysis in accounting research. Journal of Accounting and Economics 9, 231–258. Choi, Y-S., O’Hanlon, J., Pope, P.F., 2006. Conservative accounting and linear information models. Contemporary Accounting Research 23 (1), 73–101. Das, S., Levine, C., Sivaramakrishnan, K., 1998. Earnings predictability and bias in analysts’ forecasts. The Accounting Review 73, 277–294. Davidson, R., Flachaire, E., 1996. Monte Carlo evidence of the behaviour of the wild bootstrap in the presence of leverage points. Working Paper, Department of Economics, Queen’s University, Kingston, Ontario. Davidson, R., Flachaire, E., 2001. The wild bootstrap, tamed at last. Working Paper, Department of Economics, Queen’s University, Kingston, Ontario. Davison, A.C., Hinkley, D.V., 1997. Bootstrap Methods and Their Applications. Cambridge University Press, Cambridge. De Bondt, W.F.M., Thaler, R.H., 1990. Do security analysts overreact? American Economic Review 80 (2), 52–57. Dechow, P., Hutton, A., Sloan, R., 1999. An empirical assessment of the residual income valuation model. Journal of Accounting and Economics 26, 1–34. Doukas, J.A., Kim, C.F., Pantzalis, C., 2002. A test of the errors-in-expectations explanation of the value/glamour stock returns performance: evidence from analysts’ forecasts. Journal of Finance 57 (5), 2143–2165. Durtschi, C., Easton, P., 2005. Earnings management? The shapes of the frequency distributions of earnings metrics are not evidence ipso facto. Journal of Accounting Research 43 (4), 557–592. Duru, A., Reeb, D.M., 2002. International diversification and analysts’ forecast accuracy and bias. The Accounting Review 77, 415–433. Easterwood, J.C., Nutt, S.R., 1999. Inefficiency in analysts’ earnings forecasts: systematic misreaction or systematic optimism? Journal of Finance 54 (5), 1777–1797. Flachaire, E., 2003. Bootstrapping heteroskedastic regression models: wild bootstrap vs. pairs bootstrap. Working Paper, Universite´ Paris 1 Panthe´on-Sorbonne. Francis, J., Olsson, P., Oswald, D., 2000. Comparing the accuracy and explainability of dividend, free cash flow and abnormal earnings value estimates. 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The British Accounting Review 39 (2007) 3–13 www.elsevier.com/locate/bar
Evaluating the properties of analysts’ forecasts: A bootstrap approach Mark A. Clatworthya, David A. Peelb,, Peter F. Popec a
Accounting and Finance Section, Cardiff Business School, Cardiff University, Colum Drive, Cardiff, CF10 3EU, UK Department of Economics, Lancaster University Management School, Lancaster University, Lancaster, LA1 4YX, UK c Department of Accounting and Finance, Lancaster University Management School, Lancaster University, Lancaster, LA1 4YX, UK b
Received 8 April 2005; received in revised form 2 August 2006; accepted 16 August 2006
Abstract Previous research has reported that analysts’ forecasts of company profits are both optimistically biased and inefficient. However, many prior studies have applied ordinary least-squares regression to data where heteroskedasticity and nonnormality are common problems, potentially resulting in misleading inferences. Furthermore, most prior studies deflate earnings and forecasts in an attempt to correct for non-constant error variances, often changing the specification of the underlying regression equation. We describe and employ the wild bootstrap—a technique that is robust both to heteroskedasticity and non-normality—to assess the reliability of prior studies of analysts’ forecasts. Based on a large sample of 23,283 firm years covering the period 1981–2002, our main results confirm the findings of prior research. Our results also suggest that deflation may not be a successful method of correcting for heteroskedasticity, providing a strong rationale for using the wild bootstrap in future work in this, and other areas of accounting and finance research. r 2006 Elsevier Ltd. All rights reserved. Keywords: Analysts’ forecasts; Wild bootstrap; Deflation; Heteroskedasticity
1. Introduction The properties of analysts’ earnings forecasts have been studied extensively over more than 25 years (e.g., Brown et al., 1987; Brown, 1993; Capstaff et al., 2001; Das et al., 1998; Duru and Reeb, 2002; Kang et al., 1994; Kothari, 2001; O’Brien, 1988, 1990; O’Brien and Bhushan, 1990). Studies based on different data sets for various countries and time periods have reported that analysts’ forecasts are optimistically biased, i.e., on average forecast earnings are significantly higher than actual earnings realizations. Research also indicates that analysts’ forecasts do not incorporate information known at the forecast date efficiently. De Bondt and Thaler (1990) report that analysts’ forecast changes are greater than actual changes, suggesting that analysts systematically over-react. In contrast, Abarbanell and Bernard (1992) and Easterwood and Nutt (1999) find that analysts’ forecast errors (actual earnings minus forecast earnings) are positively related to the prior period Corresponding author.
E-mail address: [email protected] (D.A. Peel). 0890-8389/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.bar.2006.08.002
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earnings change, consistent with analyst under-reaction.1 Research into long run earnings forecasts provides stronger evidence of optimism and inefficiency: in a study of long run forecasts of growth in US companies’ earnings, Harris (1999) reports that forecast annual earnings growth exceeds actual growth by approximately 7% per annum. If analysts’ forecasts are biased and/or inefficient, this has important implications for their use as inputs to fundamental valuation models (e.g., Dechow et al., 1999; Francis et al., 2000; Choi et al., 2006) and other practical investment methodologies where forecasts (or forecast revisions) are an input to stock selection models. Similarly, forecast bias presents potential problems to capital markets researchers who rely on analysts’ forecasts as a proxy for the market’s expectations of earnings (e.g., Botosan, 1997). It is therefore essential that evidence on forecast bias and inefficiency is statistically robust. If forecasts are systematically biased and inefficient, it is then important to understand the reasons for, and consequences of that bias, and specifically to understand whether the bias is a rational consequence of analysts’ incentives (e.g., Lim, 2001) and/or the statistical properties of earnings (e.g., Helbok and Walker, 2004). Studies of analysts’ earnings forecasts typically examine forecasts pooled for different firms over several time periods using standard ordinary least squares (OLS) regression techniques (notable exceptions are Keane and Runkle, 1998; Capstaff et al., 2001; Basu and Markov, 2004). Basu and Markov (2004) suggest that OLS is inappropriate because analysts attempt to minimize mean absolute, rather than mean squared, forecast errors. Hence, if analysts weight large forecast errors proportionally the same as small errors, least absolute deviation (LAD) regression techniques are more appropriate.2 Like Basu and Markov (2004), we question the suitability of OLS for investigating the rationality of analysts’ forecasts. However, rather than motivate our research on the basis of the implied loss function faced by analysts, we show that the basic assumptions required for valid OLS estimation are highly likely to be violated in practise, potentially resulting in incorrect inferences being drawn. Specifically, we address the consequences of earnings forecast errors having non-constant variance in the cross-section and/or over time.3 A common practice in analyst forecast studies is to deflate earnings per share (EPS) and forecasts by a scalerelated variable, such as actual EPS or stock price. This procedure is designed to mitigate the problem of heteroskedasticity, where forecast errors are related to the level of actual EPS (Kothari, 2001). In this paper, we show that deflation can lead to invalid inferences concerning forecast bias. Inference problems are further exacerbated because the distribution of actual earnings and of forecast errors may be non-normal. Should this lead to non-normality of the residuals in estimating equations, incorrect inferences might be drawn concerning forecast properties.4 To mitigate these problems, we require a statistical estimation procedure that is robust to both heteroskedasticity and non-normality. We describe and illustrate such a technique—the wild bootstrap (Wu, 1986; Davidson and Flachaire, 2001). The wild bootstrap approach allows us to construct standard errors that can be employed for detecting bias and/or inefficiency in analysts’ forecasts when the assumptions required for OLS are not satisfied. This paper has two main objectives. The first is to describe the wild bootstrap approach and its benefits over estimation techniques commonly employed in accounting and finance research, where heteroskedasticity and non-normality are pervasive problems. The second is to employ the wild bootstrap to re-examine the findings of prior research into analyst forecast bias and inefficiency. We test for bias and inefficiency using familiar
1 Easterwood and Nutt (1999) attempt to reconcile these findings by arguing that analysts’ reaction to earnings is conditional on the nature of the information; they report that analysts are systematically optimistic because they under-react to bad news and over-react to good news. 2 Basu and Markov (2004) are not the first to argue in favour of a linear symmetric loss function (see Gu and Wu, 2003). However, the shape of analysts’ loss function remains an unresolved issue. For example, Lambert (2004, p. 220) points out that ‘‘there are no economic or psychology models that support why analysts (or investors) should have a linear loss function’’. 3 We note that if forecasts for different horizons are pooled, then this will also induce heteroskedasticity, with forecast error variances expected to be positively related to the forecast horizon. 4 Non-normality may be less problematic than heteroskedasticity in some studies of analysts’ forecasts because of the large sample sizes typically involved, especially since it is standard practise to remove the tails of the distribution, which are often responsible for non-normal forecast errors (Abarbanell and Lehavy, 2003); though we note that this procedure needs justification if not for reasons of data errors. We also note that many other areas of accounting and finance research often involve much smaller sample sizes; and here, non-normality may be a more important issue.
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model specifications (e.g., Abarbanell and Bernard, 1992; Capstaff et al., 1995, 2001; De Bondt and Thaler, 1990; Easterwood and Nutt, 1999) on a sample of over 22,000 analysts’ forecasts of annual EPS. The paper makes three important contributions. First, prior findings of bias and inefficiency are supported by our analysis. Second, we show that the practice of deflating actual and forecast EPS does not adequately address the problem of heteroskedasticity. Finally, our description and application of the wild bootstrap offers accounting researchers a useful and reliable alternative to existing estimation methods. One important advantage of the wild bootstrap is its ability to cope with heteroskedasticity without deflating actual and forecast EPS. Potentially problematic choices about which deflator to use (e.g., share price, absolute EPS or the standard deviation of EPS over a given period) are therefore avoided, as is the potential for the availability of data for deflation creating biased samples (cf. Christie, 1987; Durtschi and Easton, 2005). The remainder of the paper is structured as follows. In Section 2, we discuss statistical research design issues surrounding previous attempts to test for inefficiency and bias in analysts’ forecasts; we then describe the properties of the wild bootstrap. In Section 3, we describe the data used in the empirical analysis. Section 4 reports the results of our tests of bias and efficiency using test statistics based on the wild bootstrap. Our new empirical results confirm prior research that analysts’ forecasts of EPS exhibit bias and inefficiency. Section 5 presents our conclusions and the implications of our findings.
2. Statistical research design issues 2.1. Testing forecast bias and inefficiency Assume that yt f t;th þ vt ,
(1)
where yt is the actual realization of EPS at time t, f t;th is the forecast of yt using information available at time th, and vt is an error term. Empirical models estimated in the prior literature are typically based on one of the following regression equations: yt ¼ d0 þ d1 f t;th þ Zt ,
(2)
yt yt1 ¼ d00 þ d01 ðf t;th yt1 Þ þ t ,
(3)
yt f t;th ¼ d000 þ d001 ððyt1 yt2 Þ
4 1X ðy yti1 ÞÞ þ et , 3 i¼2 ti
(4)
where the forecast is measured by the average or median analysts’ forecast, d0 , d00 , d000 , d1 , d01 and d001 are regression coefficients, and Zt , t and et are regression error terms. Assuming that these error terms are normal, have a constant variance and are uncorrelated with right-hand side variables, regression coefficients can be interpreted as follows. In Eq. (2), forecasts are weakly inefficient if d1 is significantly different from unity. The intercept term (d0 ) in Eq. (2) typically captures any constant bias—forecasts are generally optimistic if d0 o0 and pessimistic if d0 40.5 In Eq. (3), a coefficient d00 that is different from zero generally indicates bias in forecasts of earnings changes (e.g., Capstaff et al., 2001), while an estimated coefficient of d01 that is less than unity is interpreted as evidence of analyst over-reaction (e.g., De Bondt and Thaler, 1990). In regression (4), forecasts are inefficient if d001 differs from zero. A significant negative coefficient is indicative of over-reaction, while a significant positive coefficient is consistent with under-reaction (Abarbanell and Bernard, 1992; Easterwood and Nutt, 1999). 5 We qualify this interpretation because, as pointed out by an anonymous reviewer, the interpretation of the intercept strictly depends on the value of d1 . It is possible for forecasts to be unbiased, even if d0 differs from zero; in particular, forecasts are unbiased if d0 ¼ ð1 d1 ÞEðyt Þ (see Holden and Peel, 1990).
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2.2. Deflating outcomes and forecasts Prior research involves the analysis of samples of data which are pooled outcomes and forecasts for different firms. Suppose expectations are rationally formed and analysts’ loss functions are quadratic. In these circumstances, the forecast error for each firm will be a random, unpredictable variable. However, unless the variance of the error in the time series process for earnings in each firm in the sample is identical, then the regression error will exhibit heteroskedasticity where, in principle, every forecast error is associated with a different error variance. It seems likely that the absolute magnitude of forecast errors is positively correlated across different firms with the level of earnings, so that the size of absolute forecast error is greater the higher EPS. As a consequence, it has become standard in empirical tests of the properties of analysts’ forecasts of EPS to deflate all variables by an instrument implicitly believed to be associated with the variance of the forecast error.6 Deflators used in prior research to mitigate potential heteroskedasticity include the absolute value of actual earnings (e.g., Capstaff et al., 1995, 2001), the standard deviation of prior earnings (De Bondt and Thaler, 1990) and share price (e.g., Duru and Reeb, 2002; Easterwood and Nutt, 1999; Keane and Runkle, 1998). However, prior research fails to acknowledge that this deflation procedure can have important effects on the properties of the regression and can affect the inferences from empirical tests. To illustrate this, assume that the forecast horizon is one period and that all relevant variables in a regression are scaled by a common deflator st1 in the information set at the forecast date th.7 It is straightforward to show that generally estimated coefficient values (denoted by * or ** in sequel) are influenced by the deflation process. For example, regression (2) can be rewritten as follows: yt st1
¼ d0 þ d1
f t;th þ t . st1
(5)
There is no guarantee that d0 ¼ d0 or d1 ¼ d1 . The actual coefficient estimates depend on the variance–covariance structure of the deflated variables and may bear no resemblance to the variance–covariance structure of the undeflated variables. For example, it is possible that d1 ¼ 0, i.e., Cov½yt ; f t;th ¼ 0, but for the deflator st1 to have such a high variance that it dominates the covariance term such that Cov½ðyt =st1 Þ; ðf t;th =st1 Þa0, so that the estimated value is on average non-zero. In summary, estimated coefficient values for Eq. (5) will not equal the ‘‘true’’ values obtained from (2) on average. Furthermore, the transformed model (5) is effectively changing the specification of the model. Eq. (5) can be rewritten as: yt ¼ d0 st1 þ d1 f t;th þ t .
(6)
Thus, the predicted level of earnings from (5) is a linear combination of the scaling variable and the level of EPS forecast by analysts. This is a different hypothesis to the one tested in Eq. (2), where the predicted level of EPS is equal to a constant,d0 , plus a multiple, d1 , of the level of EPS forecast by analysts. If the variable used to deflate yt and ft,th is taken at t– h (i.e., the period before the forecast is made), then under rational expectations, d0 is still expected to be zero. However, if the deflator contains information relevant to the prediction of earnings (for instance, contemporaneous share price) and such information is not yet contained in the forecast, then it is possible for the expected value of d0 to differ from zero. One potential resolution of the heteroskedasticity problem is to estimate a transformation of Eq. (2) using weighted least-squares regression. Thus, Eq. (2) can be rewritten as: f t;th yt 1 ¼ d þ d þ 0 1 t . st1 st1 st1 6
(7)
If deflation is to be consistent with weighted least squares regression, the standard deviation of the error term should be a linear function of the deflator. 7 Duru and Reeb (2002) deflate the ex-post forecast error by stock price. They attempt to explain this measure of the degree of ‘‘bias’’ by a variety of explanatory variables. Some of the most significant explanatory variables they report are deflated by the stock price. They state that their results are robust if they deflate by the mean of the consensus forecast. This robustness in fact suggests a statistical problem. Essentially the common deflator appears to be driving their results.
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Estimates of (7) will yield unbiased and efficient estimates of the standard errors of d^ 0 and d^ 1 if ðst1 Þ2 ¼ kðst1 Þ2 , where k, the error variance, is constant. However, in general, researchers have not reported formal tests to identify the nature of heteroskedasticity, nor have they reported whether the chosen deflator corrects for heteroskedasticity. This is despite reports in prior research that deflation can often exacerbate heteroskedasticity and coefficient bias (e.g., Barth and Kallapur, 1996). For instance, in valuation regressions, Lubberink and Pope (2005) find that deflation can lead to coefficients deviating from their predicted values by a factor of over 5 and conclude (2005, p. 12) that ‘‘deflating by a scale variable leads to seriously biased coefficient estimates.’’ Consequently, there is no way of knowing whether tests of equations such as (7) yield unbiased estimates of the true coefficients. In the event that the distribution of the error terms is non-normal, inference problems are further exacerbated. Similarly, although the well-known White (1980) method permits asymptotically correct inferences in the presence of heteroskedasticity of unknown form, it is known that in finite samples, t- and F-tests can be seriously biased.
2.3. The wild bootstrap approach Recent advances in computing offer an alternative approach to hypothesis testing when the error term in a regression is heteroskedastic and potentially non-normal.8 Under such conditions the wild bootstrap has been shown to be an appropriate method for determining appropriate critical values for t- and F-tests (Wu, 1986; Hardle and Mammen, 1993; Davidson and Flachaire, 2001). The intuition behind the approach is to identify the distributions of relevant test statistics when the null hypothesis holds and the distribution of regression residuals is based on the relevant sample distribution. We illustrate the bootstrap procedure with reference to Eq. (2), but in testing we apply it to the estimation of all equations of interest. The procedure is as follows: (i) Estimate (2) by OLS. We denote the residuals from this regression as Z^ . (ii) Create a new series of ‘‘pseudo’’ residuals based on Z^ : Z0i ¼ Z^ i ei , Z0i ¼ Z^ i oi , where ei and oi are drawn from one of the following two-point distributions: pffiffiffi 8 pffiffiffi 5þ1 51 > > p ffiffi ffi with probability ; > > < 2 2 5 pffiffiffi ei ¼ pffiffiffi > 5þ1 5þ1 > > p ffiffi ffi with probability 1 > : 2 2 5 or
( oi ¼
1 with probability 0:5; 1 with probability 0:5:
(8) (8a)
(9)
(10)
Both ei and oi are mutually independent drawings from a distribution independent of the original data and their distribution has the properties: Eei ¼ 0; Ee2i ¼ 1; Ee3i ¼ 1; Ee4i ¼ 2; Eoi ¼ 0; Eo2i ¼ 1; Eo3i ¼ 0; Eo4i ¼ 1: The rationale for using these specific distributions is that they preserve the heteroskedastic nature of the original regression residuals. In particular, since the errors are independent, any heteroskedasticity in the pseudo residuals, Z0i , matches that in the regression residuals, Z^ . This is because the variance of the pseudo residuals, Z0i , is a replica of the variance of the original regression residuals Z^ . Furthermore, the 8
Note that few prior studies in this area report tests for non-normality of residuals and employ standard errors modified for heteroskedasticity in their analysis.
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(iii) (iv) (v)
(vi)
distribution in Eq. (9) matches any skewness inherent in the original regression residuals (though it overstates the kurtosis); the distribution in Eq. (10), on the other hand, preserves the kurtosis in the distribution of the original residuals, though it does not preserve the skewness.9 Construct a series of simulated (or artificial) earnings outcomes, yst , imposing the null hypothesis d0 ¼ 0, d1 ¼ 1 in (2) and using the pseudo residual series created in step (ii), i.e., yst ¼ f t;th þ Z0i .10 Regress yst on f t;th and save the estimated intercept and slope coefficients. Repeat steps (ii)–(iv) a large number of times (e.g., 10,000) and create sets of residuals, simulated earnings realizations and regression intercepts and slope parameters from the regression of the artificial earnings realizations yst for a given bootstrap sample on the forecasts f t;th . The generated sequences of simulated data have a true intercept of zero and true slope of unity by construction. However, the estimated parameter values will, in general, differ from the true values. From the 10,000 iterations we have an empirical distribution for the estimates of d^ 0 ¼ 0; d^ 1 ¼ 1 together with their associated standard errors. This distribution is based solely on re-sampling the residuals of the original regression. It yields appropriate critical values of the pivotal test statistics such as the t- and Fstatistic consistent with not rejecting the null hypothesis at an appropriate level of significance (e.g., 5%). These critical values can then be employed to determine whether the estimates obtained from (2) employing real data are consistent with rejection of the null.
3. Data Our data set consists of 23,283 eight-month ahead median consensus forecasts of annual EPS for 3268 US companies taken from I/B/E/S.11 The sample covers the period 1981–2002. Actual EPS and share price data are also provided by I/B/E/S in order to ensure consistency with the forecast (Richardson et al., 2004).12 We require the consensus to comprise a minimum of four forecasts (Easterwood and Nutt, 1999) and also require each firm to have actual EPS data for at least six consecutive years in order to estimate Eq. (4). Our sample size and the time period covered by our data compare favourably with those in prior studies. For instance, Easterwood and Nutt’s (1999) analysis is based on 10,694 observations between 1982 and 1995, while Duru and Reeb’s (2002) sample comprises 3495 firm-year observations over the period 1995–1998. More recently, Helbok and Walker’s (2004) UK study consists of 4454 observations taken from the period 1990–1998. In line with prior research, we removed observations where earnings variables were greater than 100% of share price at the date of the forecast (for the scaled regressions) or greater than $10 (for the unscaled regressions). Even with simulation techniques, inclusion of high leverage observations may result in inefficient estimates (Davidson and Flachaire, 1996; Davison and Hinkley, 1997). Because it has been suggested that research in this area may be sensitive to the procedure used to delete outliers (Ahmed et al., 2000), we also 9 This preservation of any heteroskedasticity in regression errors is a central feature of the wild bootstrap; this contrasts with conventional bootstrapping with replacement, which may destroy the heteroskedastic nature of the residuals. Hence, the wild bootstrap is preferred to standard bootstrapping techniques on this basis (Goncalves and Kilian, 2004). We also note that, following the suggestion of an anonymous reviewer, we conducted further analysis using the standard bootstrap procedure built in to the STATA statistical software, and this confirmed our main conclusions. For bivariate regressions, this procedure involves drawing, with replacement, pairs of observations from the dataset, and re-running the regression (in our case) 10,000 times on the new dataset. However, Flachaire (2003) finds that the ‘pairs’ bootstrap is outperformed by the wild bootstrap; furthermore, Horowitz (2001) finds that inference based on the pairs bootstrap is not always accurate. 10 Note that for Eq. (4), the null hypothesis is that both d000 and d001 are zero. The artificial outcomes in this regression are therefore equal to Z0,i. 11 In line with prior research, we use earnings on a per share basis since it is EPS that analysts actually produce forecasts for. Furthermore, EPS is important to investors as part of the price/earnings ratio and as a variable in its own right: as noted by Durtschi and Easton (2005, p. 6) ‘‘particularly from the viewpoint of the individual investor, earnings per share is not a deflated variable—it is simply the income due to the owner of each share.’’ 12 We use the median consensus forecast for consistency with the majority of prior studies of analysts’ forecasts (e.g., Easterwood and Nutt, 1999; Lim, 2001; Doukas et al., 2002; Richardson et al., 2004); however, additional analysis employing the mean yielded virtually identical results to those reported. As Kothari (2001, p. 154) points out: ‘‘Even if the distribution of actual earnings might be skewed, the distribution of analysts forecasts for a given firm need not be skewed, so the use of the mean or median of analysts forecasts might not make much difference.’’
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Table 1 Summary statistics (n ¼ 23, 283) Variablea (label)
Meanb
Median
Std. dev.
Actual year t earnings (yt ) Eight-month ahead consensus forecast of year t earnings (f t;th ) Actual earnings change between time t – 1 and time t (yt yt1 ) Forecast change in time t – 1 earnings (f t;th yt1 ) Forecast error (yt f t;th ) P Unexpected prior period’s earnings change ððyt1 yt2 Þ 13 4i¼2 ðyti yti1 ÞÞ Share price at the time of the forecast (Pth )
1.2775 1.5184 0.0078 0.2487 0.2409 0.0284
1.1200 1.2700 0.0900 0.1600 0.0400 0.0133
21.4229
18.1300
a
Skewness
Kurtosis
1.3937 1.2469 0.9950 0.6528 0.8249 1.0726
0.36 1.44 –1.44 3.07 –3.87 –0.13
8.92 8.34 23.01 37.62 30.72 21.51
15.6341
2.41
18.32
All data are provided by I/B/E/S and cover the period 1981–2002. All figures are on a per share basis.
b
conducted our analysis with more (and less) stringent outlier deletion rules. Our results are not sensitive to these alternative procedures and we only report results for the full sample—the others are available on request. Summary statistics for the variables used in the regressions are presented in Table 1.13 Both the mean and median forecast are greater than the actual earnings realizations and, in line with prior research (e.g., Easterwood and Nutt, 1999), the mean forecast error is negative and greater (in absolute terms) than the median forecast error, consistent with the negative skewness coefficient of 3.87. Table 1 also shows that the mean and median forecasts of the change in earnings between time t1 and time t (0.25 and 0.16, respectively) are substantially higher than the actual change in earnings over this period (mean of 0.01 and median of 0.09). This finding is also consistent with prior research: for example, using UK data, Helbok and Walker (2004) find that the mean forecast change in earnings (1.93% of beginning of period share price) is higher than the mean actual change in earnings (0.58% of price). It is also noteworthy that a number of variables are non-normally distributed: there is evidence of both positive (e.g., the forecast change in EPS) and negative (e.g., forecast error) skewness and kurtosis, potentially resulting in OLS being an inefficient estimator.
4. Analysis Our main results are presented in Table 2. Columns two and three show the estimated parameters and tstatistics based on OLS and White’s (1980) standard errors in parentheses, while the critical values from the wild bootstrap are presented in square brackets.14 We note that the critical values obtained from the two wild bootstrap distributions differ only very slightly in our analysis.15 As column seven shows, the White F-test indicated significant heteroskedasticity in all specifications.16 This was also true of Eq. (iv) in which share price was used as a deflator (as employed in other studies to mitigate heteroskedasticity). Accordingly, we employed the wild bootstrap to obtain critical values of the ‘‘t’’ values corrected for heteroskedasticity employing the White (1980) method. As mentioned above, although the wellknown White method permits asymptotically correct inference in the presence of heteroskedasticity of unknown form, it is known that in finite samples, t- and F-tests can be seriously biased (see e.g., MacKinnon and White, 1985; Chesher and Jewitt, 1987). 13
As a consistency check, we compared our annual scaled mean forecast errors with those reported in Easterwood and Nutt (1999, p. 1781) for the common time period (1982–1995). Our results are virtually identical to theirs, with a correlation coefficient of 0.98 (p ¼ 0:000). 14 At the suggestion of an anonymous reviewer, we also employed least absolute deviation (LAD) regression to regressions (i)–(iii), in line with Basu and Markov (2004). With the exception of the intercept in regression (i), our conclusions remain unchanged. 15 The E-Views statistical software (version 5) was used to generate the main results. 16 We also conducted additional tests based on the Breusch-Pagan/Cook-Weisberg test for heteroskedasticity. In each case, the test results confirmed those of the White test.
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Table 2 Properties of analysts’ forecasts (n ¼ 23, 283) Equationa (estimator)
d0 (t-value)
d1 (t-value)
R2
Fb
J–Bc
Hetd
(i) yt ¼ d0 þ d1 f t;th þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical valuee Wild bootstrap (II) 0.05 critical value
0.0980 (11.62) (9.27) [2.14, 2.01] [2.11, 2.11]
0.9059 (21.94) (12.60) [2.03, 1.82] [1.96, 1.96]
0.66
1025
767,583
267
(ii) yt yt1 ¼ d0 þ d1 ðf t;th yt1 Þ þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
0.2069 (35.98) (31.56) [1.52, 1.52] [1.52, 1.52]
0.8631 (16.62) (6.68) [2.05, 1.88] [1.96, 1.96]
0.32
771,213
308
P (iii) yt f t;th ¼ d0 þ d1 ððyt1 yt2 Þ 13 4i¼2 ðyti yti1 ÞÞ þ mt OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
0.2388
0.0742
0.01
799,205
320
(44.37) (44.53) [1.91, 1.96] [1.95, 1.99]
(14.79) (6.64) [2.05, 1.93] [1.98, 1.99]
0.0382
0.6523
2,188,987
245
(17.49) (8.78) [2.16, 1.75] [2.00, 1.90]
(43.81) (12.17) [1.84, 2.07] [1.93, 1.95]
(iv)
ðyt yt1 Þ Pth
¼ Pd0 þ d1 th
ðf t;th yt1 Þ Pth
þ mt
OLS White’s corrected Wild bootstrap (I) 0.05 critical value Wild bootstrap (II) 0.05 critical value
[6.4] [6.6] 1009
[11.9] [10.2] 1006
[10.1] [10.2] 0.22
158
[9.4] [26.4]
a Variable definitions are provided in Table 1; OLS and White’s corrected t-statistics are in parentheses, while critical values for t- and Fstatistics are in square brackets. b All equations except (iii) test the null that d0 ¼ 0; d1 ¼ 1; equation (iii) tests the null that d0 ¼ 0; d1 ¼ 0. c Jarques–Bera test for normality of residuals. d White’s F-test for heteroskedasticity. pffiffiffi pffiffiffi ð 5 1Þ ð 5 þ 1Þ e ; distribution; wild bootstrap II uses the (1, –1) distribution. Wild bootstrap I uses the 2 2
The Jarque–Bera test results in column six indicate severe departures from normality in all regression residuals. In fact, although the test statistics obtained from the wild bootstrap methods differ somewhat from those obtained from the White (1980) method, it is clear from the results reported in Table 2 that they are sufficiently close in this sample not to change any inferences based on the standard White adjusted ‘‘t’’ values. Even though the degree of non-normality is great, the critical values are close to the asymptotic values. Indeed, in some cases (for example, in the case of the intercept term in regression (ii)), the simulation results suggest that the reported rejections of the null hypothesis are marginally stronger than implied by the usual tests. Naturally this was not known ex-ante. The evidence in Table 2 is consistent with analyst optimism and inefficiency. For example, the intercept terms in regressions (i) and (ii) are both significantly negative, suggesting that analysts’ forecasts are optimistic (consistent with Capstaff et al., 1995, 2001). Moreover, regression (iii) demonstrates that information contained in the prior year’s unexpected earnings change is reflected in the time t forecast error, indicative of analyst under-reaction. In this regard, our results offer confirmation of previous results reported in the literature (e.g., De Bondt and Thaler, 1990; Abarbanell and Bernard, 1992; Easterwood and Nutt, 1999), though obtained employing a very different and more reliable methodology. The results from regression (iv) show that even when the equation is correctly specified (by deflating the constant term d0 ), deflating by share price does not appear to ‘‘solve’’ the heteroskedasticity problem. The White test statistic is only marginally lower than the corresponding value for the undeflated regression (ii). Furthermore, the t-statistics obtained using White’s correction in regression (iv) are substantially lower than the OLS t-statistics. Also noteworthy from Table 2 are the differences between the estimated coefficients in regressions (ii) and (iv). The undeflated equation (ii) has a slope parameter much closer to the expected value
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of unity (0.86 compared to 0.65 in Eq. (iv)), although the intercept is indicative of a greater degree of optimistic bias. A possible limitation of our analysis is cross-section correlation in forecast errors due to macro-economic factors that affect all firms similarly. Keane and Runkle (1998) control for this potential problem by employing a generalized method of moments (GMM) estimator. However, they assume homoskedasticity and rely on deflation by price—a procedure that our results suggest is unreliable. In order to test for the effects of cross-section correlation, we estimated our main regressions using the delete-group jackknife method proposed by Shao and Rao (1993) and by using Rogers (1993) standard errors clustered by year.17 We also conducted Shao Rao jackknife tests and estimated Rogers (1993) standard errors clustered by company to control for the possibility of errors being correlated within firms over time. In each case, our results were supported. 5. Conclusions Forecast errors can exhibit non-normality and heteroskedasticity under the null of rationality. In such circumstances, inferences based on pivotal test statistics using the conventional critical values are suspect, i.e., they tend to deviate from the true size of the test. The wild bootstrap is an appropriate and particularly useful method for testing the properties of forecasts in these circumstances. Our main results confirm the findings of previous research: across various model specifications, we find evidence consistent with analysts’ forecasts of corporate profits being biased and inefficient. However, this may not be the case in studies employing different data sets and/or different model specifications. Since there are persuasive arguments for moving beyond potentially unreliable estimation methods, we suggest that the wild bootstrap represents a dependable alternative. Our results also suggest that the common method of deflation is not successful at reducing heteroskedasticity. This is in line with Barth and Kallapur (1996) and implies that even after deflation, future research could usefully report results of tests for heteroskedasticity and, where appropriate, employ techniques (such as the wild bootstrap) that are robust to heteroskedasticity. The approach adopted in this paper permits the estimation of the ‘‘true’’ undeflated regression, thereby improving the comparability of research on analysts’ forecasts (e.g., see Kothari, 2001) and eliminating the problems associated with selecting an appropriate deflator (e.g., Christie, 1987). For example, using the wild bootstrap on the undeflated regression means that it is unnecessary to remove data due to the ‘small deflator’ problem (e.g., see Gu and Wu, 2003). Sample selection bias induced by data providers including data for commonly used deflators in a non-random fashion can also be avoided by using the wild bootstrap (cf. Durtschi and Easton, 2005). Many areas of accounting and finance often rely on smaller sample sizes where non-normality is more problematic and where commonly used corrections for heteroskedasticity such as White’s (1980) standard errors are less effective. For example, samples of long run earnings forecasts are typically much smaller than those in research into short term (i.e., annual and quarterly) forecasts (e.g., Harris, 1999). In such circumstances, the wild bootstrap may yield different, and more dependable, results than conventional regression methods. Acknowledgements The authors gratefully acknowledge the financial support of INQUIRE (UK) and I/B/E/S for providing analyst forecast data. The authors are grateful to Ivan Paya for writing the programme code and to two anonymous reviewers for helpful comments. References Abarbanell, J.S., Bernard, V.L., 1992. Tests of analysts’ overreaction/underreaction to earnings information as an explanation for anomalous stock price behaviour. Journal of Finance 47 (3), 1181–1207. 17
See Petersen (2005) and Vuolteenaho (2002) for further discussion of these methods.
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