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Implications of Data Restrictions on Performance Measurement and Tests of Rational Pricing

S.P. Kothari Sloan School of Management Massachusetts Institute of Technology 50 Memorial Drive, E52-325 Cambridge, MA 02142 617-253-0994, E-mail: [email protected]

Jowell S. Sabino Sloan School of Management Massachusetts Institute of Technology 50 Memorial Drive, E52-325 Cambridge, MA 02142 617-253-1959, E-mail: [email protected]

and Tzachi Zach William E. Simon Graduate School of Business Administration University of Rochester, Rochester, NY 14627 716-244-5863, E-mail: [email protected]

First draft: June 1998 Current draft: May 2000

We gratefully acknowledge comments by Ray Ball, Scott Keating, Adam Koch (discussant), Jon Lewellen, Jeff Pontiff, Terry Shevlin, and participants at the Financial Economics and Accounting Conference at the University of Texas at Austin, University of Washington, and the Australian Graduate School of Management. S.P. Kothari acknowledges financial support from the New Economy Value Research Lab at the MIT Sloan School of Management.

2 Abstract We argue that the previously documented association between ex ante information (e.g., earnings forecasts) and the subsequent, apparently predictable security price performance is potentially exaggerated. The exaggeration stems from non-random deletion of data, especially in highly right-skewed distributions of long-horizon security returns. Our simulations demonstrate that both forecast optimism and negative abnormal returns are induced when "extreme" observations of ex post long-horizon performance are truncated from samples of rationally priced, unbiased earnings forecasts. Our results suggest caution in interpreting the results of the accounting and finance research that examines the predictability of long-horizon performance based on ex ante information.

Implications of Data Restrictions on Performance Measurement and Tests of Rational Pricing 1. Introduction There is mounting evidence of long-horizon security price under-performance following corporate events like an initial public offering (IPO) and seasoned equity offerings (Loughran and Ritter, 1995, Ritter, 1991, and Spiess and Affleck-Graves, 1995). Systematic, predictable post-event under-performance violates semi-strong form of market efficiency. Several studies suggest caution in interpreting the evidence of long-horizon price performance for methodological reasons (see, for example, Barber and Lyon, 1997, Brav, 1999, Kothari and Warner, 1997, and Fama, 1998) and others question the economic significance of the evidence (e.g., Brav, Geczy, and Gompers, 1998, Brav and Gompers, 1997, and Fama, 1998). There is an emerging literature that seeks to strengthen the inference of systematic longhorizon return predictability. This literature documents a positive cross-sectional association between ex post long-horizon under-performance and management’s ex ante optimistic financial reporting or analysts’ optimistic forecasts at the time of events like an IPO. Examples of studies in the context of performance following initial or seasoned equity offerings include Teoh, Welch, and Wong (1998a and b), Rajan and Servaes (1997), Dechow, Hutton, and Sloan (1998), and Ali (1996). In the context of security-price performance following analysts’ forecasts examples of research are La Porta (1996), Dechow and Sloan (1997), and Frankel and Lee (1998).1 The underlying logic behind the inference of systematic mispricing due to the market’s naïve reliance on analysts’ optimistically biased forecasts is as follows. There is voluminous evidence that analysts’ forecasts are overly optimistic (see, for example, Brown, Foster, and Noreen, 1985, Brown, 1997, Lim, 1998, Abarbanell, 1991, and

1

Also see Lakonishok, Shliefer, and Vishny, 1994, and Sloan, 1996.

2 Stickel, 1990).2 The market naively utilizes these forecasts in setting security prices, particularly when analyst following in a firm is thin as in many IPOs and small market capitalization stocks. Thus, both prices and analysts’ forecasts are overly optimistic.

Future actual financial

performance reveals the optimism in the analysts’ forecasts, i.e., analysts’ forecast errors are negative. In response to this bad “news” in financial performance, prices decline. The postevent long-horizon price performance is negative.

Thus, both security price and financial

performances are predictably poor. In addition, there is a statistical association between ex ante proxies for the predictable biases in analysts' forecasts of financial variables (e.g., discretionary accruals) and ex post security price performance. This violates the semi-strong form of market efficiency.

The inference is that the cause of optimistic pricing and subsequent under-

performance is the market’s naïve reliance on analysts’ optimistic forecasts. Similar logic suggests management’s optimistic financial reporting through accruals

can also induce

systematic mispricing (e.g., Teoh et al., 1998a and b, and Sloan, 1996). Objective of the paper. In this paper, we argue that the strength of the previously documented association between ex ante information variables (e.g., analyst forecasts or management’s optimistic financial reporting) and the subsequent, apparently predictable security price performance is potentially exaggerated. The exaggeration stems from a combination of the statistical properties of security returns and financial performance measures, and certain research design features that are common and almost inevitable in previous studies. The statistical property we focus on is the right skewness of long-horizon stock-return and financial performance measures. The research design feature is inadvertent and/or deliberate truncation (i.e., deletion) of “extreme” observations from a sample used in a regression of longhorizon price performance on ex ante variables like analysts’ forecasts or financial variables that

2

New research indicates that in recent years there is evidence of either considerably muted analyst optimism (Brown, 1997) or a small degree of pessimism. The results also suggest analysts’ apparent optimism or pessimism is related to macro-economic conditions in the economy and that it is concentrated among economically less important small market-capitalization stocks (Lim, 1998). Overall, the evidence is consistent with analysts learning about their optimistic bias and/or the previous evidence of analyst optimism being a period-specific phenomenon.

3 proxy for over-optimism. Observations frequently are excluded simply because many firms do not survive the ex post long horizon or data are unavailable. Firms are delisted because of mergers, acquisitions, takeovers, bankruptcies, etc. For example, the sample of 1,649 IPOs in Teoh et al. (1998b) declines steadily to 739 (or less than 45% of the initial sample size) by year 6 because either firms do not survive or data are unavailable. Studies using analysts’ long-term forecasts from the Institutional Brokers Estimate System (IBES) lose 50% or more observations because of lack of availability of long-term growth forecasts and/or financial data on Compustat or Center for Research in Security Prices (CRSP) tapes. For example, Dechow and Sloan (1997), Frankel and Lee (1998), and Dechow, Hutton, and Sloan (1999) all suffer severe data attrition, although the researchers might not be in a position to avoid it. The lack of data availability is more common among smaller market capitalization stocks. Some of these stocks might be young, small, volatile firms, and others might be those who have experienced extreme poor performance. Deliberate deletion of observations arises because some observations are considered “outliers” in the sample. This active truncation of the data is more likely when analyzing IPO samples or samples of small firms. These firms’ ex post performance distribution is highly right skewed due to a few extreme performers. For example, among the recent IPO firms, America Online has earned over 11,000% return and Dell Computer Corporation has earned over 5,500% return in five years ending in October 1999. If included in a sample, these observations will likely be considered outliers based on non-arbitrary and well-known statistical techniques (e.g. Cook’s-D influence statistic). Therefore, truncation of such observations might be considered innocuous.

However, in setting prices at any point in time, an efficient market rationally

incorporates the possibility of firms earning extreme returns in future. If ex post extreme performers are truncated purely for statistical reasons, then observations containing information are non-randomly deleted from the sample. This kind of active truncation, coupled with the

4 passive truncation of non-surviving firms, can contribute to the apparent evidence of mispricing documented in the literature. The degree of systematic mispricing documented in past research is greater among the small market capitalization and relatively newly listed stocks. The evidence is often attributed to a lack of analyst and investor following, which creates a greater potential for the market being misled by analysts’ optimistic forecasting or management’s optimistic financial reporting. Our study suggests an alternative interpretation: the mispricing evidence might be an artifact of data problems and statistical techniques of data analysis.

There appears to be a need to

discriminate between the two competing explanations. In summary, if the sample firms’ survival and/or data deletion are not random with respect to the ex post performance variables, we show that statistical inferences are biased. We use simulation evidence to show that even a small degree of non-random data truncation (e.g., deletion of extreme ½% largest and smallest observations) can induce a statistically significant association between ex post performance and ex ante forecast variables of the magnitude reported in previous research to conclude market inefficiency. Summary of simulation results. The simulations examine the effect of data truncation (either due to survival, or to remove influential observations or outliers) in the presence of skewed distributions of economic variables for tests of rational pricing of securities. Specifically, we simulate rational analyst forecasts in the sense that they are unbiased estimates of future earnings. In the simulations, earnings realizations equal forecasted earnings plus a forecast error that is uncorrelated with the forecast. This means earnings realizations are more variable than the forecasts, which is a property of rational forecasts. Greater variability of earnings compared to their forecasts is also observed empirically. The pricing is also rational in the simulations. The price change from the time of forecast to the time of earnings realization is a noisy function of the forecast error. The forecast error and price change distributions are right

5 skewed with the parameters of the distributions being less extreme compared to those reported in the literature for IPOs and seasoned equity offerings.3 Consistent with rational analyst forecasts and rational pricing, when none of the observations is deleted, the simulated data do not exhibit systematic mispricing. In particular, ex post forecast errors are uncorrelated with the ex ante earnings forecasts. In addition, forecast errors are statistically indistinguishable from zero. Similarly, security-price performance or abnormal returns have a zero mean and they are uncorrelated with the ex ante analyst forecasts. We then examine the consequence of deleting up to 2% extreme earnings observations (i.e., largest and smallest 1% observations). The results are dramatically different from those obtained without data deletion.

They are now overwhelmingly consistent with systematic

mispricing. Since we simulate right-skewed distributions, data truncation imparts a negative bias to the analyst forecast errors, suggesting analyst over-optimism. More interestingly, there is a statistically reliable negative relation between ex post returns and ex ante analyst forecasts. The statistical association is documented using regressions of ex post returns on ex ante forecasts as well as Mishkin’s (1983) non-linear tests of rational pricing, which appear to have assumed the status of an industry standard to test market efficiency. These tests provide evidence consistent with systematic mispricing or return predictability arising from analyst optimism and market’s naïve reliance on analysts’ optimistic forecasts. In summary, the simulations demonstrate that even if analysts’ forecasts and pricing are rational, the deletion of a small number of extreme observations can create an appearance of highly significant systematic mispricing resulting from the market’s naïve reliance on analysts’ optimistic forecasts. Ironically, one motivation for truncating extreme data is to remove observations that represent possibly anomalous (but

3

The distributional properties of IPOs’ or seasoned equity offerings’ long-horizon stock returns reported in the literature likely suffer from a researcher-imposed data truncation and/or truncation due to survival- and dataavailability reasons and thus probably provide conservative estimates of the variance and skewness of the underlying distribution absent data truncation. Using these conservative parameter estimates, the simulations demonstrate the consequences of (further) data truncation.

6 unexplained) valuations. Our analysis, however, shows that truncation can in fact create the appearance of irrational pricing in the rest of the sample. Caveats.

While we demonstrate the potential for the degree of mispricing being

magnified as a result of the market’s naïve reliance on analysts’ optimistic forecasts and/or management’s optimistic financial reporting, we emphasize that our work is silent on longhorizon post-IPO or post-seasoned-equity offerings security price performance. In these contexts previous research is careful in avoiding survival bias and data truncation in estimating the security-price performance. Our work also does not directly touch upon examining biases in analysts’ forecasts. Our study has a limited objective of examining whether survival biases and data truncation can magnify the statistical association between ex ante forecast information and ex post security price performance, which violates semi-strong form of market efficiency. Section 2 presents a simple model that analytically demonstrates how correlation is induced between security price performance and rational forecasts as data are truncated at both ends of the distribution. The analytics assume a symmetric distribution, i.e., no skewness. We provide the intuition that the problems due to data truncation are aggravated with skewness. Section 3 describes the simulation procedure and presents the main results of the paper. We conclude in section 4. 2. Model We begin this section with an example that provides the intuition for how data truncation can induce spurious correlation between analyst forecasts and ex post performance.

We

formalize the intuition in an analytic model in sections 2.2 and 2.3. Section 2.2 presents variable definitions and other preliminaries, whereas section 2.3 derives the result of spurious association. 2.1 An example Assume an analyst makes either a low (= -$1) or a high (= +$1) earnings forecast, AF, with equal probability. Next, an earnings forecast error, FE, is realized independently and it is either low or high with equal probability. The forecast errors are therefore mean zero. Realized

7 earnings, X, which are the sum of forecasted earnings and the forecast error, will be either -$2, 0 or +$2. Since forecast errors are mean zero, expected realized earnings, E(X), for both low and high forecasts equal the forecasted level, low or high (see the table below).4 Variable Analyst’ forecast, AF

Low = -1

High = +1

Forecast error, FE

-1

+1

-1

+1

Realized earnings, X

-2

0

0

+2

E(Earnings), E(X)

-1

+1

In the stylized setting described above, analysts’ forecasts are rational. There is no correlation between the ex ante analyst forecast and the ex post forecast error. We next examine the consequence of data truncation. Assume a researcher deletes the two extreme earnings realizations, -$2 and +$2, from the analysis. This means 50% of the data are deleted in the stylized example here, which is an overly generous characterization of “extreme” observations. We address this issue below. Exclusion of the two extreme earnings observations induces a perfect negative correlation between the analyst forecast and the forecast error because only those realizations that mitigate each other (i.e., AF and FE) tend to be included in the sample. The negative correlation creates a spurious impression that analysts’ high forecasts are optimistic and low forecasts are pessimistic. The stylized example has several limitations. First, it exaggerates the extent of data truncation that is observed in practice. Second, it exaggerates the effect of data truncation by producing a perfect correlation between the forecasts and forecast errors, i.e., perfect

4

The stylized example never allows a zero forecast error. If forecast errors were to take on three values, -1, 0, and +1, each with 1/3 probability, the forecast error would still be mean zero and uncorrelated with the forecast, and thus rational. Realized earnings will be -$2, -$1, 0, +$1, or +$2, with extreme earnings, i.e., -$2 or +$2, observed with 1/3 probability instead of ½ probability in the example in the text. Truncation of the extreme earnings observations will still induce a (spurious) negative correlation between the analyst’s forecast and the forecast error, albeit the negative correlation is less than –100%.

8 predictability. Third, it employs a symmetric distribution of discrete values, whereas actual data are better described by skewed, continuous distributions. Finally, the example analyzes forecasts and forecast errors, but not post-forecast security price performance.

The security price

performance is expected to be less than perfectly correlated with the forecast error, which will weaken the correlation between ex ante forecasts and ex post price performance. Notwithstanding these limitations, the example helps convey the intuition that data truncation can produce spurious association. The analytical model below attempts to formalize the intuition and the simulation analysis seeks to demonstrate the result in a more realistic setting by relaxing all of the limitations discussed above. The degree of spurious correlation due to the truncation of a small fraction of the total sample is not large, but remains statistically highly significant, just as observed in the literature. 2.2 Model: The preliminaries Suppose AF is analysts’ consensus forecast of future earnings, X. The forecast is rational in that it is an unbiased estimate of future earnings, i.e., X = AF + FE

(1)

where FE is the forecast error, distributed normal with mean zero and variance σ2FE. For the moment, we assume that the forecast error, FE, is independent of AF.

Later on in the

simulations we allow for a mild form of dependence where the variance of the forecast errors is a function of the level of forecast AF. Earnings forecast AF is distributed normal with mean µAF and variance σ2AF. However, the succeeding analysis remains qualitatively unchanged using other continuous and symmetric distributions for AF like the uniform distribution with mean µAF. Pricing is also assumed to be rational. An earnings forecast determines a firm’s current price and realized future earnings determine its future price. Since information other than forecasted earnings and realized future earnings also influences prices, prices are expected to deviate from those based solely on forecasted and realized future earnings. The analytics below leave out this component in prices to keep the analysis as simple as possible, although the entire

9 analysis holds with the pricing function augmented to incorporate other information.

The

simulations in section 3 are based on the more realistic scenario. The current price is P0 = K + ERC*AF

(2)

where ERC is some earnings multiple that captures the implications of earnings for future cash flows and K captures the option value of limited liability, which is a consideration when we focus on the future price contingent on future earnings realization, X. The future price, modeled here simply as a function of X, is P = K + ERC*X.

(3)

In the pricing equations K is assumed “large enough” such that P is positive for all realizations of X. From eq. (2) and (3), the change in price is the abnormal price change, AR, as a result of earnings news, FE: AR = ∆P = ERC*FE.

(4)

We further simplify the model and the expression for AR by assuming that ERC = 1. 2.3 Consequences of data deletion Suppose the top and bottom α percent of realized earnings, X, are considered outliers. If earnings are distributed Φ(), which in our model is a normal distribution, truncation of the upper α percent observations corresponds to an upper bound XUB on the earnings realizations that remain in the distribution after truncation. In other words, Φ[(XUB - µAF)/σX] = 1 - α

(5)

where Φ[] is the standard normal cumulative distribution function and σX is the standard deviation of X. Also, the truncation of the lower α percent observations corresponds to a lower bound XLB on the earnings realizations that remain in the sample. Thus, Φ[(XLB - µAF)/σX] = α

(6)

and by symmetry, XUB - µAF = -(XLB - µAF). Truncation of the top and bottom α percent realizations of X reduces the sample to

10 Ω = {X: XLB ≤ AF + FE ≤ XUB}

(7)

Since X is assumed to be distributed normal, it has a symmetric distribution and truncation of extreme observations leaves the mean of the truncated distribution same as that before data truncation. In cases where X has a skewed distribution, the mean of the truncated sample will differ predictably from the original untruncated sample. The simulations employ both symmetric and skewed distributions. Here, we demonstrate the effect of truncation where X has a symmetric distribution. The simulations show that skewness exacerbates the consequences of data truncation. To test whether pricing is rational, researchers typically estimate the following crosssectional regression: ARi = β0 + β1 AFi + εi

(8)

where εi is regression error and i = 1, …, N is observation i in the untruncated sample of N observations. From (4), the residuals, εi, will be a function of the forecast error FEi. Thus, as long as E[FEi|AFi] = 0 for all i, the OLS estimator will be unbiased. Since we assume rational pricing and rational analyst forecasts, E[FEi|AFi] = 0 for the sample of N observations and thus the OLS estimator is unbiased. We now show that restricting the sample to Ω causes FEi and AFi to cease to be independent and thus the OLS estimator of β1 is biased. To demonstrate the bias in the estimated regression slope due to the induced dependence between the ex post forecast error and the ex ante forecast, we begin by standardizing the truncated sample bounds XLB and XUB:

and

Zi,LB = (XLB – AFi)/σFE

(9)

Zi,UB = (XUB – AFi)/σFE.

(10)

The data truncation rule to obtain Ω is equivalent to Ω = {FEi: Zi,LB ≤ FEi/σFE ≤ Zi,UB}.

(11)

11 We next use the following property for the mean of a truncated Normal distribution (see Greene, 1997, pp. 951-952): for x ~ N[µ, σ2] E[x | truncation] = µ + σλ(α),

(12)

where α = (a − µ)/σ, λ(α) = φ(α)/[1 − Φ(α)] if truncation is x > a, and

λ(α) = −φ(α)/[ Φ(α)] if truncation is x < a.

Therefore, in the context of the truncated sample Ω, E[FEi|Ω(AF)] = σFE[(φ(Zi,LB) − φ(Zi,UB))/(Φ(Zi,UB) − Φ(Zi,LB))].

(13)

In eq. (13), note that σFE and Φ(Zi,UB) − Φ(Zi,LB) are always positive. This fact is helpful in signing the correlation between analysts’ forecasts and subsequent forecast errors.

The

correlation’s sign is determined by the covariance between analysts’ forecasts and subsequent forecast errors. However, note that

and

E[FEi|Ω(AF)] ≤ 0 and φ(Zi,LB) ≤ φ(Zi,UB) if and only if AFi ≥ µAF

(14)

E[FEi|Ω(AF)] ≥ 0 and φ(Zi,LB) ≥ φ(Zi,UB) if and only if AFi ≤ µAF.

(15)

In other words, FEi is negatively correlated with AFi in the truncated sample Ω. Thus, truncation based on realized earnings X = AF + FE induces a negative correlation between analysts’ forecasts and the subsequent forecast errors because only those realizations that mitigate each other (i.e., AF and FE) tend to be included in the sample. This means that the firms more likely to be included in the sample are: (1) firms that were predicted to exhibit high growth, but failed, and (2) firms with low growth forecasts that succeeded spectacularly. Conversely, high growth forecast firms that do extremely well or low forecasted growth firms that do poorly are more likely to be truncated out of the sample.

The result is apparent predictability of future

performance on the basis of cross-sectional variation in forecasted growth. The intuitive example and the model both demonstrate spurious negative association between ex ante forecasts and realized earnings as a result of data truncation assuming a

12 symmetric distribution of earnings.

The intuition, however, is applicable also for skewed

distributions. The risky, high growth firms that exhibit tremendous success and thus fall in the right-tail region of a right-skewed distribution of future performance, are likely to be deleted. This imparts a negative (spurious) correlation between ex ante forecasts and future performance. In addition, unlike the symmetric distribution scenario, truncation of extreme observations from a right-skewed distribution will downward bias the mean of the future performance variable. This is seen from the simulation results in the next section. 3. Results Summary.

This section presents results of simulation analysis that examines the

consequences of data truncation. We describe the simulation procedure in section 3.1. We state our assumptions about the distributions and the parameter values of the distributions used to generate simulated data. The distributions’ parameter values are chosen to be comparable to the summary statistics reported in previous literature for IPOs’ long-horizon stock returns and analysts’ long-term forecasts of earnings growth. Section 3.2 presents descriptive statistics for the samples of simulated data without and with truncation up to 2% of the sample size. The descriptive statistics demonstrate that the simulations are quite successful in matching the distributional characteristics of actual IPOs’ long-horizon returns and analysts’ forecasts of long-term and short-term earnings growth. Descriptive statistics for the untruncated samples are consistent with rational forecasting and rational pricing. In contrast, but not surprisingly, truncated samples exhibit biased analysts’ forecasts and optimistic pricing. The main purpose of the descriptive statistics is to provide the reader with an appreciation for the magnitude of bias due to data truncation in the context of realistic skewed distributions of IPO stocks or other samples. Section 3.3 presents the results of regressing ex post price performance on ex ante forecasts of long-term earnings growth using untruncated and truncated samples. Consistent with rational pricing and a lack of predictability, regressions using untruncated samples do not

13 exhibit a statistically significant coefficient on forecasted earnings growth. However, even a mild degree of truncation of ½% of the sample observations produces statistically significant negative association between forecasted growth and ex post price performance, which suggests irrational pricing and market inefficiency. The regressions explore the results’ sensitivity to the degree of truncation, skewness, and sample size. Section 3.4 presents results of the Mishkin (1983) test of rational pricing. Recently the Mishkin test has gained popularity in both accounting and finance literatures examining market efficiency (see Dechow and Sloan, 1997, Sloan, 1996, Collins and Hribar, 1999, and Dechow, Hutton, and Sloan, 1999, Thomas, 1999). Section 3.4 briefly describes the basic idea behind the Mishkin test. We then show that the Mishkin test is well specified when we use untruncated samples. Data truncation dramatically increases the Mishkin test’s rejection rate. It rises to almost 100% with 1-2% extreme observations truncated. The results suggest that the Mishkin test is powerful. However, the results also suggest that an inference of market inefficiency based on a Mishkin test is likely to be fragile because the rejection could be due to non-random data truncation. 3.1 Simulation procedure We report each simulation result using 100 independent samples of 6,000 observations each. The choice of 6,000 observations in each sample is not unreasonable. Some studies using long-term analysts’ forecasts have sample sizes much larger than 6,000 (e.g., Dechow and Sloan, 1997, and Frankel and Lee, 1998). Others like Teoh et al. (1998a, b) have 1,000 to 2,000 observations.

We therefore perform sensitivity analysis using fewer and more than 6,000

observations in each sample. The distributions of the simulated samples of five-year earnings growth, forecast errors, and abnormal returns are either right-skewed or symmetric. We start by generating forecasts of annualized long-term earnings growth rates from a uniform distribution with a support of [5%, 45%] so that the average growth rate is 25% per year. Average forecasted annual earnings growth of 25% is in line with analysts’ average annual long-term growth

14 forecast for IPOs. Rajan and Servaes (1997, table IV) report that the average of the consensus analysts’ long-term growth forecasts listed on IBES within six months of an IPO for a sample of over 500 IPOs is 23.7% per year. We calculate the five-year forecasted growth rates, FGR, by compounding the annualized growth rates. Next, we simulate right-skewed (symmetric) actual five-year growth rates by adding log-normally (normally) distributed zero-mean noise to the forecasted growth rates. Thus, by construction, forecasted growth rates are unbiased estimates of actual growth rates. Forecast errors are the difference between the actual and forecasted growth rates, which should have a zero mean and a skewed (symmetric) distribution. Finally, we calculate abnormal returns by adding normally distributed mean-zero noise to the forecast errors. The annualized long-term growth rates for the simulated firms vary considerably from 5 to 45% per annum. It is reasonable to expect that there is greater uncertainty associated with the actual outcomes of the high- compared to the low-growth firms. We simulate this phenomenon by setting the standard deviation of the zero-mean noise term that is added to the forecasted earnings growth to be proportional to the forecasted growth. Specifically, assume the standard deviation of the noise term, i.e., forecast error, for a firm with the sample mean forecasted annualized growth rate of 25% is 100%.

To make the standard deviation proportional to

forecasted growth rate, we linearly increase the standard deviation of the noise from 80% for the 5% annualized forecasted growth rate firm to 120% for the 45% annualized forecasted growth rate firm. While a positive dependence between the forecasted earnings growth and the standard deviation of the earnings forecast errors seems economically realistic, we also report results using data simulated without such dependence. 3.2 Descriptive statistics In tables 1-4 we report descriptive statistics for simulated samples from skewed and symmetric distributions to demonstrate the consequences of data truncation on the basis of extreme values of a number of variables. Each table provides descriptive statistics for 100

15 untruncated samples of 6,000 observations each, and for 100 progressively truncated samples that contain a subset of the 6,000-observation samples. For each untruncated and truncated sample, we calculate summary statistics like the mean and standard deviation and report the average of each summary statistic in the table. Panel A of table 1 reports summary statistics for abnormal returns for samples truncated on the basis of realized five-year earnings growth. The first row in each panel is for untruncated samples. Consistent with rational pricing, average abnormal return is only –0.16% with an average t-statistic of –0.07. Average standard deviation for the 100 samples is 235% and the distribution is significantly right skewed as seen from the average skewness of 2.72. The average of the maximum five-year abnormal returns is 2,984%. Buy-and-hold abnormal returns of this magnitude are rare, but certainly not unheard of. From July 1994, America Online, Inc.’s five-year return is 12,300% and Dell Computer Company’s five-year return exceeds 9,000%. The average of the minimum abnormal returns is –557%.

The average of the minimum

abnormal returns is –557%. Abnormal returns can be less than –100% because the normal fiveyear buy-and-hold return that is subtracted from the raw return can be greater than 100% and frequently so for high-growth firms. For example, an annual normal rate of return of 25% corresponds to over 200% buy-and-hold five-year normal rate of return. Table 1 The distributional characteristics of the simulated samples reported in table 1 are less extreme compared to those reported in the literature for IPOs. The standard deviation of fiveyear abnormal returns for a sample of 1,521 IPOs analyzed in Brav (1998b, table 8) is 245% and the skewness is 11.9. Brav (1998b) concludes that the reported standard deviation understates the true standard deviation because long-horizon returns are cross-sectionally dependent. The simulated samples’ return distributions in table 1 are not as right skewed or as dispersed as the post-IPO return distributions. We show that the greater the variance and skewness of simulated

16 samples’ return distributions, the more sensitive the tests of rational pricing are to data truncation. The effect of truncation is seen from the 2nd through 5th rows. Because the distribution is right skewed, mean abnormal return of the truncated samples quickly turns negative and significant. Truncation of a mere 1% extreme observations (i.e., 0.5% smallest and 0.5% largest values) leaves the average abnormal return to be highly significant -7.18%, with an average tstatistic of –2.70.

The standard deviation also declines dramatically from 234% for the

untruncated samples to 206% for the 1% truncation samples. Panel B shows that truncation has a negligible effect on the distributional properties of forecasted 5-year earnings growth. The average 5-year compounded growth is about 231%. The average annual earnings growth (not reported in the table) is about 25%. Like abnormal return, the average earnings forecast error in panel C declines rapidly with data truncation. The actual five-year earnings growth in panel D also declines from almost 231% for the untruncated sample to 219.5% for the sample with 2% of the observations truncated. The average maximum fiveyear earnings growth is 3,436% or an annualized growth rate of 105%. While the maximum growth rate strikes as extremely high, such growth rates are indeed observed regularly. For example, the May 31, 1999 issue of Business Week reports 100 “Hot Growth Companies,” with 23 of those reporting more than 100% annual earnings growth over three years. Most of the companies’ May 1999 market capitalization is more than $500 millions and the four smallest of the 23 companies had market capitalization between $50 and $100 millions. Overall, descriptive statistics in table 1 demonstrate that the untruncated samples are consistent with rational forecasts and rational pricing. For the untruncated samples, average abnormal return is zero and the actual earnings growth equals forecasted growth.

The

descriptive statistics also show that even a small degree of data truncation has statistically and economically significant effects on average abnormal return and forecast error in the presence of skewed distributions.

17 Table 2 reports descriptive statistics similar to those in table 1 except that data truncation in table 2 is on the basis of earnings forecast errors. The effect of truncating extreme earnings forecast errors on the distributional properties is similar to that we observed when realized earnings growth observations were truncated. Table 2 We report descriptive statistics for data truncation by forecasted five-year earnings growth in table 3. Unlike tables 1 and 2, truncation by forecasted earnings growth does not affect the mean abnormal return, forecast error, or actual earnings growth. Since forecasted earnings growth is an ex ante, symmetrically distributed information variable, truncation on that does not affect the average values of the ex post realizations. This is consistent with rationality and the absence of predictability using ex ante information. In tables 1 and 2, data truncation was on the basis of ex post values of the variables, which imparts a bias in the presence of skewness. Table 3 Table 4 provides descriptive statistics for untruncated and truncated samples of symmetric distributions. Not surprisingly, even though truncation is by realized earnings growth and regardless of whether the variance of the distributions is increasing in forecasted growth or not, mean abnormal returns is zero and mean earnings forecast errors is zero. Table 4 Summary. Descriptive statistics in tables 1-4 show the following. (1) The distributional properties of the simulated data are less extreme compared to those of real-life IPO return data. (2) The distributional properties of untruncated samples are consistent with unbiased earnings forecasts and rational pricing. (3) Data truncation in the presence of skewed distributions of long-horizon returns and earnings results in biased performance that is apparently consistent with systematic mispricing. We now turn to formal tests of the consequences of data truncation on tests of rational pricing.

18 3.3 Linear regression tests of rational pricing To demonstrate the effect of data truncation on tests of rational pricing, we estimate the following regression model: ARit+1,t+5 = β0 + β1 FGRit + εit+1,t+5

(16)

where ARit+1,t+5 is five-year abnormal return for years t+1 to t+5, FGRit is five-year forecasted earnings growth for years t+1 to t+5 with the forecast made at the end of year t, β0 and β1 are regression coefficients, and εit+1,t+5 is the regression error term. Regression model (16) or some variation of the model is commonly employed in the literature to test market efficiency (see, for example, Frankel and Lee, 1998, Dechow, Hutton, and Sloan, 1999, and Teoh et al., 1998b). A negative coefficient on FGR in eq. (16) suggests the market naively prices optimistic long-term forecasts and that subsequent returns are more negative for high, more optimistic forecasts. This violates market efficiency. Table 5 reports results of estimating regression model (16).

The regressions are

estimated using 100 independent samples of simulated data whose properties are as reported in table 1.

Specifically, simulated five-year abnormal returns are right-skewed with a cross-

sectional standard deviation of 235%. Simulated data are generated under the assumption of rational pricing and rational forecasts. We report average values of the estimated regression parameters, t-statistics, and adjusted r-squares from the 100 regressions. The first row in each panel of table 5 reports results using untruncated samples, whereas the remaining ten rows in each panel report results for progressively truncated samples. Results in the first row of table 5, panel A confirm that simulated data exhibit rational pricing. The average coefficient on FGR is zero and the average t-statistic is only –0.16. The model’s average explanatory power is a negligible 0.01%. Thus, ex ante forecasted earnings growth does not have predictive power with respect to future abnormal returns. This scenario changes quickly with data truncation. For example, when 1% of the extreme observations are truncated (i.e., 0.5% largest and 0.5% smallest observations are deleted), both intercept and slope

19 coefficients are reliably different from zero, with average t-statistics of 2.23 and –4.37, and the average explanatory power is 0.33%.5 Forecasted earnings growth reliably negatively predicts future return performance, which is consistent with market inefficiency. The predictive ability improves with increased data truncation and the average explanatory power rises to 0.91% when 2% of the data are truncated. The estimated average slope coefficient is –0.13 with an average tstatistic of –7.31. Table 5 Table 5, panel B reports results using samples truncated on the basis of earnings forecast errors. Since earnings forecast errors are simulated to be uncorrelated with forecasted earnings growth, truncation by earnings forecast errors has a weaker effect on the relation between forecasted growth and future returns than when truncation is on the basis of actual earnings growth. When 2% of the sample observations are deleted, there is a statistically significant negative relation between forecasted earnings growth and future returns, with an average explanatory power of 0.2%. While the results in table 5 clearly indicate statistical significance, how do we assess the results’ economic significance?

We offer two related pieces of evidence that suggest the

spurious association between forecasted growth and future performance, if taken at face value, would be economically highly significant. First, following Fama and MacBeth (1974) and Bernard and Thomas (1990), we estimate a regression of future performance on scaled decile ranks assigned to the sample observations on the basis of forecasted earnings growth. The regression model is similar to model (16) except that scaled decile ranks of FGR are used. Decile ranks of FGR are reduced by one and then scaled by 10 so that they attain values that range from 0 to 0.9. The appeal of scaled decile ranks is that the slope coefficient in model (16) is an estimate of the abnormal return to the highest-minus-lowest ranked decile portfolios, even 5

Note that there is no cross-sectional dependence in the data, so that it cannot be a source of bias in the t-statistics. In actual long-horizon data, there is generally positive cross-sectional dependence, which tends to bias results in favor of rejecting the null hypothesis (see Brav, 1998a).

20 though the regression itself is estimated using firm-level data. Another desirable property of ranks that is frequently discussed in the context of regressions in the capital markets literature is that the “use of ranks avoids assigning a large weight to the small number of outlying observations” (Dechow and Sloan, 1997, p. 23). Table 6 reports results of estimating model (16) using scaled decile ranks. Panel A contains results using samples truncated on the basis of actual earnings growth whereas in panel B truncation is on the basis of actual forecast errors. Panel A shows that a zero-investment portfolio that is long in the highest decile of forecasted growth firms and short in the lowest decile of forecasted growth firms progressively loses more as truncation is increased from 0 to 2% of the sample size. This is consistent with high earnings growth forecast firms being predictably overpriced at the beginning of the five-year performance measurement period. The last row of panel A shows that if extreme 2% actual earnings growth observations are deleted, then the zero-investment portfolio’s five-year abnormal return is about –60%. The t-statistic on the slope coefficient is significant at a p-value of 0.05 for regressions using 0.4% or more data truncated. Results in panel B exhibit less extreme abnormal return on the zero-investment portfolio. The performance nonetheless remains economically as well as statistically significant. Table 6 We next present the second piece of evidence that suggests the abnormal performance induced by data truncation is economically significant. We form quintile portfolios on the basis of ranking stocks on forecasted earnings growth. Table 7 reports average abnormal returns on the quintile portfolios with no data truncation and one and two percent data truncation. Because we simulate data consistent with rational pricing, in the absence of data truncation all the quintile portfolios earn abnormal returns that are statistically and economically indistinguishable from zero. The highest growth forecast quintile earns a five-year abnormal return of 0.01%, and the lowest quintile portfolio’s abnormal return is a mere –0.04%. Truncation of extreme earnings growth observations rapidly creates the appearance of systematic mispricing. With 2% of the

21 observations truncated, the highest forecasted earnings growth quintile has an economically and statistically significant abnormal return of –46.2% (t-statistic = -59.2) and the lowest forecasted earnings growth quintile’s abnormal return is a modest 4.3% (t-statistic = 14.4). Table 7 In summary, evidence in tables 6 and 7 suggests that, when working with long-horizon financial data that is right skewed, even a small amount of data truncation can generate economically large magnitudes of apparent abnormal returns consistent with market inefficiency. The magnitudes of estimated β1 coefficients (i.e., returns to zero-investment portfolios) and the average explanatory power of the regression model (16) reported in table 6 and the portfolio abnormal returns in table 7 to are comparable to those reported in the literature documenting evidence consistent with market inefficiency. For example, Dechow and Sloan (1997, table 5) regress five-year ahead returns on scaled decile ranks of forecasted earnings per share growth. Using a sample of 23,203 observations from 1981 to 1992, they report a five-year return to the zero-investment portfolio (i.e., the equivalent of the estimated slope coefficient in model 18), 63% and an adjusted R2 of 1.57%. Frankel and Lee (1998, table 8) report results of regressing three-year returns on scaled decile ranks of a composite measure of market mispricing, using information available ex ante, that is labeled Perr. The Perr measure uses analysts’ long-term earnings forecast, an optimism measure, and other financial information. The return on a zeroinvestment Perr portfolio over three years is –27.7% and the adjusted R2 of the regression model is 1.03%. As a final example, consider the regression In Teoh, Welch, and Wong (1998b) of threeyear buy-and-hold post-IPO returns for a sample of 1,649 IPOs from 1980 to 1982 on 13 independent variables and 12 year dummies. The independent variable of interest in testing the predictability of post-IPO returns is discretionary current accruals. Teoh et al. (1998, p. 1955) report “a significantly negative coefficient of –0.227 (p-value 0.03)” on discretionary current accruals and conclude “firms with high earnings management proxy to boost earnings in the year

22 of the IPO subsequently show greater underperformance.” The regression model has a 6.37 percent explanatory power, but, as noted above, the model includes 22 other variables including three-year market return. 3.4 Mishkin (1983) tests of rational pricing Description of the test. Mishkin (1983) is a test of rationality of market expectations. It tests whether the market acts as if it optimally uses all available information in forming the expectations.

Mishkin’s approach yields directly testable implications of rationally formed

expectations, making it quite popular with researchers testing market inefficiency. To illustrate, suppose we have a linear pricing equation where departures from earnings expectations determine abnormal stock returns: ARt = β0 + β1 (Earnt – E[Earnt| θt-1]) + νt

(17)

where ARt is abnormal return in period t, Earnt is realized earnings in period t, E[Earnt| θt-1] are expected earnings conditional on the information set at time t-1, θt-1, β0 and β1 are regression parameters, and νt is the regression error. Market rationality of earnings expectations requires abnormal returns to be independent of the information set θt-1. The Mishkin test examines whether a particular expression of the expectation E[Earnt|θt-1] is rational in this “informationefficiency” sense. In our context, we apply the Mishkin text to examine whether the market rationally uses information in growth forecasts in setting current prices so that future performance is related to unexpected earnings growth, but not forecasted growth. Specifically, suppose FEt+5 = α0 + α1 FGRt + εt+5

(18)

If analysts’ expectations of growth, FGR, are rational, then forecast errors, FE, would be uncorrelated with FGR. In addition, we can test if the market’s pricing is rational, by estimating the following system of equations: FEt+5 = α0 + α1 FGRt + εt+5

(19)

23 ARt+5 = β0 + β1(FEt+5 - α∗0 − α∗1 FGRt) + ν∗t+5

(20)

Rational expectations require that α0 = α∗0 and α1 = α∗1, which is a non-linear constraint in the system of above two equations. The Mishkin test estimates the above system with and without the constraint of rationality. Whether the constraint is binding or not is ascertained on the basis of the log-likelihood ratio of the constrained and unconstrained sum of squared errors, which has a chi-square distribution. In the case of linear pricing models with normally distributed earnings variables, the expectation models are also linear. Our model and subsequent simulations allow us to examine the effect of skewness on the specification of the Mishkin test when we impose the null of rationality with no data truncation. We believe this is uncharted territory. When we apply the Mishkin test on truncated samples, the null hypothesis is false, and thus the application of the Mishkin test on truncated samples enables us to comment on the power of the Mishkin test. Results. Table 8 reports results of the Mishkin test on 100 independent, untruncated and truncated samples of 6,000 observations. The untruncated sample is simulated assuming rational forecasts and rational pricing, and the distribution is right skewed. Truncation is on the basis of extreme values of realized earnings growth. In the last two columns of table 8 we report the percentage of the times out of 100 the Mishkin test rejects the null hypothesis of rational pricing at the 0.01 and 0.05 level of significance for each level of data truncation. Columns 2 and 3 report the average test statistic on the slope coefficient and the average explanatory power of the predictability regression model (16). The Mishkin test rejection frequencies for untruncated samples are 4 and 11% at the 0.01 and 0.05 levels of significance. At these significance levels, the expected rejection rates are 1 and 5%. Assuming independent samples and a Binomial distribution of the rejection frequencies under the null hypothesis, the actual rejection rates significantly exceed the expected rejection rates at the 0.05 level of significance. However, the rejection frequency is not too large in absolute terms and thus the test is not too badly misspecified.

The modest degree of

24 misspecification appears to be a consequence of skewed distributions. Unreported simulations using symmetric distributions confirm our conjecture. Table 8 The striking result in table 8 is the consequence of truncation on the Mishkin test’s rejection frequency. The rejection rate rises quickly with truncation and it reaches almost 100% when 1% of the extreme observations (i.e., ½% extreme negative and ½% extreme positive observations) are deleted. The Mishkin test thus appears to be extremely powerful in detecting even small (apparent or real) departures from rationality. The question, however, is whether data truncation has in part contributed to the departures from rationality in an empirical experiment. 3.5 Sensitivity tests We examine the sensitivity of the Mishkin test to two parameters of interest: sample size and the variance of abnormal returns. Increasing the sample size in statistical hypothesis testing yields more powerful tests. In our model, a larger sample size decreases the estimation variance of the regression parameters, α0, α1, β0 and β1, making the Mishkin test more powerful in detecting mispricing. However, simulations in the preceding section suggest that increased power will come at a price: any spurious correlation induced by extreme data truncation will also be detected with above normal frequency. In addition to sensitivity to sample size, we examine how the variance of abnormal returns affects the specification and power of the Mishkin test. Although we believe that our parameter value for the variance of abnormal returns is on the conservative side in the context of IPOs, the Mishkin test is used in a wide variety of test scenarios. We expect that a larger variance of abnormal returns will lead to relatively more extreme observations being truncated from the sample, thus aggravating the bias induced by truncation. We examine the effects of sample size by replicating the simulation procedure using sample sizes from 2,000 through 10,000 (in increments of 2,000). To benchmark the results with those of the previous section, we hold the truncation at 1% of the extreme earnings growth

25 observations in the sample and the variance of abnormal returns is maintained at 235%. Table 9, panel A reports the results. As expected, increases in sample size make the Mishkin test more powerful. Compared to 90% rejection rate at the 0.01 level for samples of 6,000 observations, the rejection rate is 57% for samples of 2,000 observations and 100% for samples of 10,000 observations. To examine the effect of sample size on the regression test of rationality, we report the average t-statistic on the estimated slope coefficient and the average explanatory power of the regression of abnormal returns on earnings forecasts. The average t-statistic is –2.60 for a sample size of 2,000 and decreases to -5.70 for a sample size of 10,000. The adjusted R2 of the regressions slightly decreases as sample size increases, from 0.40% for a sample size of 2,000 to 0.33% for a sample size of 10,000. Overall, the results indicate significant rejection of the null hypothesis of rationality for all sample sizes examined. Table 9 Panel B of table 9 reports the effect of variance on the rejection rates of the Mishkin test. Intuitively, increasing the variance of abnormal returns will make the truncated observations on the right side of the distribution more extreme, increasing the bias induced by truncation. Again to benchmark the results, we use samples of 6,000 observations and the truncation rate is maintained at 1% of the extreme earnings growth observations in the sample. We increase the variance of abnormal returns from 80% to 500%. Consistent with the intuition, the results suggest that increasing the variance of realized earnings growth increases the rejection rate of the Mishkin test. Probabilistic truncation of extreme observations. So far we examine the effect of truncating each of the α% extreme observations. Perhaps a more realistic setting is one in which extreme observations are more likely to be deleted, but not certain to be deleted. To simulate this case, we construct truncated samples in which each of the α% extreme observations has a 50% probability of being truncated out of the sample. The tenor of unreported results from these simulations is similar to those reported earlier. Once again, the conclusion is that tests of

26 rationality are extremely sensitive to non-random data deletion and truncation of extreme observations. Summary. The results in this section show that, with only a modest amount of data truncation, the regression test and the Mishkin test both would indicate a significant rejection of the null hypothesis of rationality for a wide range of sample sizes and variance of realized future earnings growth. The results are not sensitive to the choice of parameter values. 4. Conclusions and Implications for Future Research We argue that the previously documented association between ex ante information variables (e.g., analyst forecasts or management’s optimistic financial reporting) and the subsequent, apparently predictable security price performance is potentially exaggerated. The exaggeration stems from a combination of (i) the right-skewness of long-horizon security returns and financial performance measures, and (ii) data truncation of extreme observations. These are common and almost inevitable research design features of previous studies and in many cases beyond the control of a researcher. In particular, we show that if the sample firms’ survival and/or data deletion are not random with respect to the ex post performance variables, statistical inferences are biased. We demonstrate this in our paper using a simple model where the market rationally prices unbiased earnings forecasts. The results suggest a more careful economic analysis in linking ex ante information to ex post performance, rather than using the statistical properties of the data to guide sample selection and data truncation. Simulations demonstrate that both forecast optimism and negative abnormal returns are induced when “extreme” observations of ex post long-horizon performance are truncated from the sample. As a consequence of data truncation, a statistically and economically significant association between ex ante earnings growth forecast and ex post abnormal performance is induced. We also show that the Mishkin test, which has become popular in the market efficiency literature, will almost certainly reject the null hypothesis of rational pricing in the presence of just one or two percent of extreme observations being truncated. The Mishkin test is thus well

27 specified and extremely powerful, provided that the sample does not suffer from non-random survival biases and/or data truncation. Overall, our results suggest caution should be exercised in interpreting the results of the accounting and finance research that examines the predictability of long-horizon performance based on ex-ante information.

28 References Abarbanell, J. S., 1991, Do analysts’ earnings forecasts incorporate information in prior stock price changes? Journal of Accounting & Economics 14, 147-165. Ali, A., 1996, Bias in analysts’ earnings forecasts as an explanation for the long-run underperformance of stocks following equity offerings, working paper, University of Arizona. Barber, B. and J. Lyon, 1997, Detecting long-run abnormal stock returns: The empirical power and specification of test statistics, Journal of Financial Economics 43, 341-372. Bernard, V., 1984, The use of market data and accounting data in hedging against consumer price inflation, Journal of Accounting Research 22, 445-466. Bernard, V. and J. Thomas, 1990, Evidence that stock prices do not fully reflect the implications of current earnings for future earnings, Journal of Accounting & Economics 13, 305-340. Brav, A., 1998a, Inference in long-horizon event studies: A re-evaluation of the evidence, working paper, Duke University, Fuqua School of Business. Brav A., 1998b, Inferences in long-horizon event studies: A Bayesian approach with application to initial public offerings, working paper, Duke University, Fuqua School of Business. Brav A., C. Geczy, and P. Gompers, 1998, The equity in equity issues, working paper, Duke University, Fuqua School of Business. Brav, A. and P. Gompers, 1997, Myth or reality? The long-run underperformance of initial public offerings: Evidence from venture and nonventure capital-backed companies, Journal of Finance 52, 1791-1821. Brown, L., 1997, Analyst forecasting errors: Additional evidence, Financial Analysts Journal 53, 81-88. Brown, P., G. Foster, and E. Noreen, 1985, Security analyst multi-year earnings forecasts and the capital markets, Studies in Accounting Research 21, American Accounting Association, Sarasota, Florida. Collins, D. and P. Hribar, 1999, Earnings-based and accrual-based market anomalies: one effect or two? Working paper, University of Iowa. Dechow, P. and R. Sloan, 1997, Returns to contrarian investment strategies: tests of naïve expectation hypotheses, Journal of Financial Economics 43, 3-27. Dechow, P., A. Hutton, and R. Sloan, 1999, The relation between analysts’ forecasts of longterm earnings growth and stock price performance following equity offerings, working paper, University of Michigan Business School.

29

Fama, E., 1998, Market efficiency, long-term returns, and behavioral finance, Journal of Financial Economics 49, 283-306. Fama, E. and J. MacBeth, 1973, Risk, return, and equilibrium: Empirical tests, Journal of Political Economy 38, 607-636. Frankel, R. and C. Lee, 1998, Accounting valuation, market expectation, and cross-sectional stock returns, Journal of Accounting & Economics 25, 283-319. Greene, W., 1997, Econometric Analysis, Prentice Hall, Upper Saddle River, New Jersey 07458. Kothari, S.P. and J. Warner, 1997, Measuring long-horizon security price performance, Journal of Financial Economics 43, 301-339. Lakonishok, J., A. Shleifer, and R. Vishny, 1994, Contrarian investment, extrapolation, and risk, Journal of Finance 45, 455-477. La Porta, R., 1996, Expectations and the cross-section of stock returns, Journal of Finance 51, 1715-1742. Lim, T., 1998, Are analysts’ earnings forecasts optimistically biased? Working paper, Amos Tuck School, Dartmouth University. Loughran, T. and J. Ritter, 1995, The new issues puzzle, Journal of Finance 50, 23-51. Mishkin, F., 1983, A rational expectations approach to macroeconomics: Testing policy effectiveness and efficient markets models, University of Chicago Press, Chicago, IL, for the National Bureau of Economic Research. Rajan, R. and H. Servaes, 1997, Analyst following of initial public offerings, Journal of Finance 52, 507-529. Ritter, J., 1991, The long-run performance of initial public offerings, Journal of Finance 42, 365394. Sloan, R., 1996, Do stock prices fully reflect information in accruals and cash flows about future earnings? The Accounting Review 71, 289-315. Spiess, Katherine and J. Affleck-Graves, 1995, The long-run performance following seasoned equity issues, Journal of Financial Economics 38, 243-267. Stickel, S., 1990, Predicting individual analyst earnings forecast, Journal of Accounting Research 28, 409-417.

30 Teoh, S., I. Welch, and T. Wong, 1998a, Earnings management and the long-run underperformance of seasoned equity offerings, Journal of Financial Economics 50, 63100. Teoh, S., I. Welch, and T. Wong, 1998b, Earnings management and the long-run underperformance of initial public offerings, Journal of Finance 53, 1935-1974.

Thomas, W., 1999, A test of the market’s (mis)pricing of domestic and foreign earnings, working paper, University of Utah.

31 Table 1 Descriptive statistics: Data truncation by extreme values of realized earnings growth Descriptive statistics for untruncated and truncated samples of abnormal returns, five-year forecasted and realized earnings growth, and forecast errors. Simulations use skewed distributions with variance of realized growth increasing in forecasted growth and the standard deviation of five-year abnormal returns is equal to 235%. Descriptive statistics are simple averages of the summary statistics for 100 independent samples of 6,000 observations each with zero data truncation and progressive subsamples with up to 2% of the extreme realized fiveyear earnings growth observations truncated.

Truncation

Mean

T-statistic

Min

Max

0.0%

-0.16%

-0.07

-557%

2984%

0.5%

-4.42%

-1.61

214.6%

1.65

5.56

-557%

1450%

1.0%

-7.18%

-2.70

206.0%

1.38

3.90

-557%

1215%

1.5%

-9.46%

-3.65

200.0%

1.21

3.08

-557%

1109%

2.0%

-11.45%

-4.51

195.1%

1.09

2.56

-557%

1031%

0.0%

Panel B: Forecasted 5-year earnings growth 231.4% 121.80 147.2% 0.43 -1.01

28%

541%

0.5%

231.4%

121.72

146.8%

0.43

-1.00

28%

541%

1.0%

231.2%

121.61

146.5%

0.43

-1.00

28%

541%

1.5%

231.1%

121.49

146.3%

0.43

-0.99

28%

541%

2.0%

231.0%

121.35

146.0%

0.44

-0.99

28%

541%

0.0%

Std. Dvn. Skewness Kurtosis Panel A: Abnormal return 234.1% 2.72 18.66

Panel C: Forecast error = (Actual – Forecasted) 5-year earnings growth -0.28% -0.12 228.7% 2.91 20.44 -511%

2986%

0.5%

-4.53%

-1.70

208.6%

1.79

6.20

-511%

1431%

1.0%

-7.29%

-2.83

199.8%

1.51

4.39

-511%

1201%

1.5%

-9.57%

-3.82

193.6%

1.33

3.50

-511%

1085%

2.0%

-11.56%

-4.72

188.6%

1.20

2.92

-511%

1009%

0.0%

231.1%

Panel D: Actual five-year earnings growth 65.87 271.9% 2.80 15.18

-77%

3436%

0.5%

226.7%

69.79

251.1%

2.00

5.48

-61%

1786%

1.0%

223.9%

71.61

241.1%

1.80

4.04

-55%

1504%

1.5%

221.5%

72.97

233.5%

1.68

3.27

-51%

1358%

2.0%

219.5%

74.11

227.1%

1.58

2.75

-48%

1252%

32 Table 2 Descriptive statistics: Data truncation by extreme values of earnings forecast errors Descriptive statistics for untruncated and truncated samples of abnormal returns, five-year forecasted and realized earnings growth, and forecast errors. Simulations use skewed distributions with variance of realized growth increasing in forecasted growth and the standard deviation of five-year abnormal returns is equal to 235%. Descriptive statistics are simple averages of the summary statistics for 100 independent samples of 6,000 observations each with zero data truncation and progressive subsamples with up to 2% of the extreme five-year earnings forecast error observations truncated.

Truncation

Mean

T-statistic

Std. Dvn. Skewness Kurtosis Panel A: Abnormal return 234.1% 2.72 18.66

Min

Max

0.0%

-0.16%

-0.07

-557%

2984%

0.5%

-3.59%

-1.32

213.3%

1.68

5.56

-522%

1388%

1.0%

-5.62%

-2.14

203.6%

1.43

3.91

-501%

1136%

1.5%

-7.22%

-2.84

196.4%

1.28

3.07

-486%

1006%

2.0%

-8.60%

-3.47

190.5%

1.17

2.51

-478%

916%

28%

541%

Panel B: Forecasted 5-year earnings growth 121.80 147.2% 0.43 -1.01

0.0%

231.4%

0.5%

230.2%

121.42

146.5%

0.44

-1.00

28%

541%

1.0%

229.1%

121.01

145.9%

0.44

-0.98

28%

541%

1.5%

228.1%

120.57

145.4%

0.45

-0.97

28%

541%

2.0%

227.1%

120.13

145.0%

0.46

-0.96

28%

541%

Panel C: Forecast error = (Actual – Forecasted) 5-year earnings growth 0.0% -0.28% -0.12 228.7% 2.91 20.44 -511%

2986%

0.5%

-3.71%

-1.40

207.2%

1.83

6.22

-433%

1362%

1.0%

-5.74%

-2.26

197.3%

1.57

4.42

-407%

1090%

1.5%

-7.35%

-2.99

189.9%

1.42

3.50

-389%

947%

2.0%

-8.71%

-3.65

183.8%

1.31

2.89

-375%

854%

-77%

3436%

0.0%

231.1%

Panel D: Actual five-year earnings growth 65.87 271.9% 2.80 15.18

0.5%

226.5%

69.60

251.5%

2.01

5.53

-77%

1839%

1.0%

223.4%

71.19

241.9%

1.80

4.10

-77%

1575%

1.5%

220.7%

72.28

234.8%

1.68

3.36

-77%

1442%

2.0%

218.4%

73.13

229.0%

1.59

2.86

-77%

1350%

33 Table 3 Descriptive statistics: Data truncation by extreme values of forecasted five-year earnings growth Descriptive statistics for untruncated and truncated samples of abnormal returns, five-year forecasted and realized earnings growth, and forecast errors. Simulations use skewed distributions with variance of realized growth increasing in forecasted growth and the standard deviation of five-year abnormal returns is equal to 235%. Descriptive statistics are simple averages of the summary statistics for 100 independent samples of 6,000 observations each with zero data truncation and progressive subsamples with up to 2% of the extreme forecasted five-year earnings growth observations truncated.

Truncation

Mean

T-statistic

0.0%

-0.2%

Std. Dvn. Skewness Kurtosis Panel A: Abnormal return -0.07 234.1% 2.72 18.66

Min

Max

-557%

2984%

0.5%

-0.1%

-0.06

233.8%

2.72

18.63

-557%

2979%

1.0%

-0.1%

-0.06

233.4%

2.71

18.60

-555%

2971%

1.5%

-0.2%

-0.09

232.8%

2.69

18.41

-554%

2947%

2.0%

-0.2%

-0.09

232.4%

2.69

18.37

-553%

2929%

0.0%

Panel B: Forecasted 5-year earnings growth 231.4% 121.80 147.2% 0.43 -1.01

28%

541%

0.5%

231.1%

122.02

146.4%

0.43

-1.01

28%

539%

1.0%

230.9%

122.24

145.6%

0.43

-1.01

29%

537%

1.5%

230.6%

122.46

144.8%

0.42

-1.01

29%

534%

2.0%

230.3%

122.69

144.0%

0.42

-1.02

30%

532%

Panel C: Forecast error = (Actual – Forecasted) 5-year earnings growth 0.0% -0.3% -0.12 228.7% 2.91 20.44 -511%

2986%

0.5%

-0.2%

-0.10

228.3%

2.91

20.41

-507%

2980%

1.0%

-0.3%

-0.11

227.9%

2.91

20.39

-505%

2973%

1.5%

-0.3%

-0.14

227.2%

2.89

20.19

-503%

2949%

2.0%

-0.3%

-0.14

226.9%

2.89

20.15

-502%

2931%

Panel D: Actual five-year earnings growth 65.87 271.9% 2.80 15.18

-77%

3436%

0.0%

231.1%

0.5%

230.9%

65.81

271.3%

2.80

15.19

-77%

3428%

1.0%

230.6%

65.76

270.4%

2.79

15.21

-77%

3420%

1.5%

230.3%

65.75

269.4%

2.78

15.08

-77%

3391%

2.0%

230.0%

65.68

268.7%

2.78

15.08

-77%

3369%

34 Table 4 Descriptive statistics: Symmetric distributions, data truncation by extreme values of actual earnings growth Descriptive statistics for untruncated and truncated samples of abnormal returns and five-year realized earnings growth: Simulations use symmetric distributions with the variance of realized growth either constant (panel A) or increasing in forecasted growth (panel B) and the standard deviation of five-year abnormal returns is equal to 235%. Descriptive statistics are simple averages of the summary statistics for 100 independent samples of 6,000 observations each with zero data truncation and progressive subsamples with up to 2% of the extreme actual fiveyear earnings growth observations truncated.

Truncation

Mean

T-statistic Std. Dvn. Skewness Kurtosis Min Panel A: Variance constant in forecasted growth Abnormal return 0.10 240.4% 0.00 0.01 -914%

Max

0.0%

0.3%

0.5%

0.4%

0.14

236.2%

0.01

-0.12

-828%

835%

1.0%

0.5%

0.17

233.4%

0.01

-0.16

-811%

817%

1.5%

0.6%

0.21

230.9%

0.02

-0.19

-790%

797%

2.0%

0.7%

0.24

228.7%

0.02

-0.21

-777%

780%

28%

541%

896%

0.0%

231.5%

Forecasted 5-year earnings growth 121.74 147.3% 0.43 -1.01

0.5%

231.3%

121.59

147.0%

0.43

-1.00

28%

541%

1.0%

231.1%

121.41

146.7%

0.43

-1.00

28%

541%

1.5%

230.9%

121.21

146.5%

0.43

-1.00

28%

541%

2.0%

230.8%

121.01

146.2%

0.43

-0.99

28%

541%

Panel B: Variance increasing in forecasted growth Abnormal return 0.01 240.5% 0.00 0.28 -979%

0.0%

0.0%

971%

0.5%

-0.2%

-0.05

235.6%

-0.01

0.06

-870%

825%

1.0%

-0.3%

-0.10

232.3%

-0.02

0.00

-848%

790%

1.5%

-0.4%

-0.14

229.4%

-0.03

-0.05

-831%

771%

2.0%

-0.5%

-0.18

226.9%

-0.03

-0.08

-814%

758%

28%

541%

0.0%

231.5%

Forecasted 5-year earnings growth 121.86 147.2% 0.43 -1.01

0.5%

231.0%

121.52

146.9%

0.43

-1.00

28%

541%

1.0%

230.7%

121.21

146.7%

0.44

-1.00

28%

541%

1.5%

230.3%

120.90

146.4%

0.44

-0.99

28%

541%

2.0%

230.0%

120.59

146.2%

0.44

-0.99

28%

541%

35 Table 5 Regression results: Data truncation by extreme values of realized earnings growth and earnings forecast errors Average values of the estimated coefficients, t-statistics, and adjusted r-squares from the following regression model: ARit+1,t+5 = β0 + β1 FGRit + εit+1,t+5 where ARit+1,t+5 is five-year abnormal return for years t+1 to t+5, FGRit is five-year forecasted earnings growth for years t+1 to t+5 with the forecast made at the end of year t, β0 and β1 are regression coefficients, and εit+1,t+5 is the regression error term. The regressions are estimated using untruncated and truncated samples of abnormal returns and five-year forecasted earnings growth. Samples are truncated either by deleting observations with extreme values of actual earnings growth (panel A) or earnings forecast errors (panel B). Data for the regressions are simulated using parameters for right-skewed distributions with variance of realized earnings growth increasing in forecasted growth and the standard deviation of five-year abnormal returns equal to 235% (see table 1). Regression results are based on 100 independent samples of 6,000 observations each.

Truncation

Average β 0

Average tstatistic

Average β 1

Average tstatistic

Average Adj. R2 in %

Panel A: Truncation by actual earnings growth 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0%

0.00 0.04 0.06 0.08 0.10 0.11 0.13 0.14 0.15 0.17 0.18

0.08 0.72 1.16 1.54 1.90 2.23 2.55 2.85 3.16 3.45 3.74

0.00 -0.03 -0.04 -0.06 -0.07 -0.08 -0.09 -0.10 -0.11 -0.12 -0.13

-0.16 -1.38 -2.25 -3.01 -3.71 -4.37 -5.00 -5.60 -6.19 -6.76 -7.31

0.01 0.04 0.09 0.16 0.24 0.33 0.43 0.54 0.66 0.78 0.91

Panel B: Truncation by earnings forecast errors 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0%

0.00 0.02 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05 0.05

0.00 0.36 0.53 0.64 0.74 0.81 0.87 0.93 0.96 0.97 0.99

0.00 -0.02 -0.03 -0.03 -0.04 -0.04 -0.05 -0.05 -0.05 -0.06 -0.06

-0.01 -0.85 -1.33 -1.69 -2.03 -2.31 -2.57 -2.82 -3.02 -3.20 -3.38

0.01 0.02 0.04 0.05 0.08 0.10 0.12 0.14 0.16 0.18 0.20

36 Table 6 Regression results using scaled decile ranks of forecasted earnings growth: Data truncation by extreme values of realized earnings growth and earnings forecast errors Average values of the estimated coefficients, t-statistics, and adjusted r-squares from the following regression model: ARit+1,t+5 = β0 + β1 RFGRit + εit+1,t+5 where ARit+1,t+5 is five-year abnormal return for years t+1 to t+5, RFGRit is the scaled decile rank of five-year forecasted earnings growth for years t+1 to t+5 with the forecast made at the end of year t, β0 and β1 are regression coefficients, and εit+1,t+5 is the regression error term. All sample observations are ranked according to forecasted five-year earnings growth and assigned decile ranks. Scaled decile ranks are decile ranks minus one divided by 10. The regressions are estimated using untruncated and truncated samples of abnormal returns and five-year forecasted earnings growth. Samples are truncated either by deleting observations with extreme values of actual earnings growth (panel A) or earnings forecast errors (panel B). Data for the regressions are simulated using parameters for right-skewed distributions with variance of realized earnings growth increasing in forecasted growth and the standard deviation of five-year abnormal returns equal to 235%. Regression results are based on 100 independent samples of 6,000 observations each.

Truncation

Average β 0

Average tstatistic

Average β 1

Average tstatistic

Average Adj. R2 in %

Panel A: Truncation by actual earnings growth 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0%

-0.00 0.03 0.05 0.06 0.08 0.09 0.10 0.12 0.13 0.14 0.15

-0.03 0.48 0.86 1.20 1.52 1.81 2.10 2.38 2.65 2.91 3.16

0.00 -0.11 -0.18 -0.25 -0.31 -0.36 -0.41 -0.46 -0.50 -0.55 -0.59

-0.02 -1.09 -1.88 -2.59 -3.26 -3.87 -4.56 -5.03 -5.57 -6.10 -6.61

0.01 0.03 0.07 0.12 0.19 0.26 0.34 0.43 0.53 0.64 0.75

Panel B: Truncation by earnings forecast errors 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0%

-0.00 0.01 0.02 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04

-0.03 0.25 0.39 0.51 0.58 0.66 0.71 0.74 0.78 0.82 0.83

0.00 -0.07 -0.11 -0.15 -0.17 -0.20 -0.22 -0.24 -0.25 -0.27 -0.28

-0.02 -0.73 -1.19 -1.56 -1.86 -2.15 -2.40 -2.62 -2.84 -3.05 -3.23

0.01 0.02 0.03 0.05 0.07 0.09 0.11 0.12 0.14 0.17 0.19

37 Table 7 Five-year average abnormal returns on quintile portfolios formed on the basis of ranking stocks on forecasted earnings growth: Data truncation by extreme values of realized earnings growth Quintile portfolios are formed by ranking all observations according to forecasted five-year earnings growth. Abnormal returns are measured over five years following the earnings growth forecast. Samples are truncated by deleting observations with extreme values of actual earnings growth. Data are simulated using parameters for rightskewed distributions with variance of realized earnings growth increasing in forecasted growth and the standard deviation of five-year abnormal returns equal to 235%. Abnormal returns and t-statistics for the quintile portfolios reported below are simple averages of the abnormal returns and t-statistics for the 100 independent samples of 6,000 observations each. T-statistic for each sample of 6,000 observations is calculated as the ratio of the sample mean average abnormal return to the cross-sectional standard deviation divided by (N – 1)1/2, where N = the sample size = 6,000 if no data are truncated.

Five-year average abnormal returns on quintile portfolios formed on the basis of forecasted earnings growth (t-statistic) Truncation Lowest II III IV Highest 0%

1%

2%

-0.04%

-0.51%

-0.45%

0.21%

0.01%

(-0.14)

(-1.25)

(-0.80)

(0.27)

(0.01)

2.27%

-0.05%

-1.61%

-7.43%

-29.08%

(7.67)

(-0.13)

(-2.92)

(-10.51)

(-34.04)

4.28%

0.56%

-2.77%

-13.10%

-46.20%

(14.44)

(1.39)

(-5.13)

(-19.45)

(-59.24)

38 Table 8 Mishkin test results: Data truncation by extreme values of realized earnings growth Mishkin test is performed by estimating the following system of two cross-sectional equations: FEt+5 = α0 + α1 FGRt + εt+5 ARt+5 = β0 + β1(FEt+5 - α∗0 − α∗1 FGRt) + ν∗t+5 where FEt+5 is the earnings forecast error over a five-year horizon, ARit+1,t+5 is five-year abnormal return for years t+1 to t+5, FGRit is five-year forecasted earnings growth for years t+1 to t+5 with the forecast made at the end of year t, α0, α1, β0 and β1 are regression coefficients, and εt+5 and νt+5 are regression errors. The Mishkin test imposes the constraints that α0 = -α∗0 and α1 = -α∗1. The Mishkin test statistic of rationality is the log-likelihood ratio 2N*Ln(SSRc/SSRu) ~ χ2(q), where N = the number of observations, q = 1, the number of constraints, and SSR is the sum of squared residuals from the constrained or unconstrained regression system. We report the number of times out of 100 estimations the Mishkin test indicates rejection of rationality at 0.01 and 0.05 level of significance. The analysis is performed using untruncated samples and samples truncated by deleting up to 2% of the observations with extreme values of actual earnings growth. Using these samples, we also estimate the following regression model: ARit+1,t+5 = β0 + β1 FGRit + εit+1,t+5 where β0 and β1 are regression coefficients, and εit+1,t+5 is the regression error term. We report average values of tstatistics of the estimated slope coefficient, and adjusted r-squares. Data for the tests are simulated using parameters for right-skewed distributions with variance of realized earnings growth increasing in forecasted growth and the standard deviation of five-year abnormal returns equal to 235% (see table 1). Results are based on 100 independent samples of 6,000 observations each.

Data truncation in % of observations

Average tstatistic for β1

Adj-R2 in %

Mishkin test rejection frequency at 0.01 level, in %

Mishkin test rejection frequency at 0.05 level, in %

Truncation by actual earnings growth 0.00

-0.16

0.01

4

11

0.20

-1.38

0.04

20

37

0.40

-2.25

0.09

43

60

0.60

-3.01

0.16

65

79

0.80

-3.71

0.24

80

91

1.00

-4.37

0.33

90

99

1.20

-5.00

0.43

99

100

1.40

-5.60

0.54

100

100

1.60

-6.19

0.66

100

100

1.80

-6.76

0.78

100

100

2.00

-7.31

0.91

100

100

39 Table 9 Regression and Mishkin tests’ sensitivity to sample size and variability of future performance: Data truncation by extreme values of realized earnings growth Mishkin test is performed by estimating the following system of two cross-sectional equations: FEt+5 = α0 + α1 FGRt + εt+5 ARt+5 = β0 + β1(FEt+5 - α∗0 − α∗1 FGRt) + ν∗t+5 where FEt+5 is the earnings forecast error over a five-year horizon, ARit+1,t+5 is five-year abnormal return for years t+1 to t+5, FGRit is five-year forecasted earnings growth for years t+1 to t+5 with the forecast made at the end of year t, α0, α1, β0 and β1 are regression coefficients, and εt+5 and νt+5 are regression errors. The Mishkin test imposes the constraints that α0 = -α∗0 and α1 = -α∗1. The Mishkin test statistic of rationality is the log-likelihood ratio 2N*Ln(SSRc/SSRu) ~ χ2(q), where N = the number of observations, q = 1, the number of constraints, and SSR is the sum of squared residuals from the constrained or unconstrained regression system. We report the number of times out of 100 estimations the Mishkin test indicates rejection of rationality at 0.01 and 0.05 level of significance. The analysis is performed using untruncated samples and samples truncated by deleting 1% of the observations with extreme values of actual earnings growth. Using these samples, we also estimate the following regression model: ARit+1,t+5 = β0 + β1 FGRit + εit+1,t+5 where β0 and β1 are regression coefficients, and εit+1,t+5 is the regression error term. We report average values of tstatistics of the estimated slope coefficient, and adjusted r-squares. Data for the tests are simulated using parameters for right-skewed distributions with variance of realized earnings growth increasing in forecasted growth. In panel A, the standard deviation of five-year abnormal returns is set equal to 235% (see table 1) and the sample size varies from 2,000 to 10,000. In panel B, the sample size is 6,000 observations and the standard deviation of five-year abnormal returns is from 80% to 500%. All the results are based on 100 independent samples each.

Sample size or abnormal return variability

Average tstatistic for β1

Adj-R2 in %

Mishkin test rejection frequency at 0.01 level, in %

Mishkin test rejection frequency at 0.05 level, in %

Panel A: Sensitivity to sample size One percent extreme observations truncated and abnormal return variability = 235% 2000

-2.60

0.40

57

69

4000

-3.60

0.34

80

90

6000

-4.37

0.33

90

99

8000

-5.09

0.33

96

99

10000

-5.65

0.33

100

100

Panel B: Sensitivity to future abnormal return variability One percent extreme earnings observations truncated and sample size = 6000 ≅ 80%

-2.68

0.12

61

77

≅ 150%

-3.70

0.24

83

93

≅ 235%

-4.37

0.33

90

99

≅ 350%

-5.00

0.43

98

99

≅ 500%

-5.81

0.57

100

100

40