Overprecise forecasts

Abstract

We examine the properties of overprecise forecasts, i.e., forecasts with more digits after the decimal than the mode of forecasts issued by all analysts following a given firm in a given year. Contrary to conventional wisdom, we find that overprecise forecasts are less accurate than peer forecasts. The lower accuracy is related to inexperienced analysts, who tend to overweight their models and produce more specific, yet less accurate forecasts. Additional analyses indicate that analysts issue fewer overprecise forecasts as they gain experience and that experience mitigates the negative association between forecast overprecision and forecast accuracy. Forecast overprecision is also positively associated with forecast boldness and brokerage house prestige, two proxies for analyst overconfidence in their model outputs. We further document that the capital market partially sees through this inaccuracy, as stock prices react less to overprecise forecast revisions. By revealing a novel behavioral bias of sell-side analysts, our study challenges the view that form precision signals greater accuracy and informativeness.

“He who knows best knows how little he knows.” – Thomas Jefferson.

1 Introduction

It is widely contended that form precision is positively associated with accuracy and informativeness. Assuming costly information and rational agents, Grossman and Stiglitz (1980) model an information equilibrium wherein agents stop processing additional information when the expected benefit and the expected cost break even. An implication of this model is that prices, quotes, and forecasts with lower resolution (e.g., integers) are less informative. Prior research documents evidence consistent with this prediction for limit orders (Kuo et al. 2015), IPO issue prices (Bradley et al. 2004), and analyst forecasts (Herrmann and Thomas 2005). In this study, we incorporate individuals’ behavioral bias and address the following question: Are overly precise sell-side analysts’ forecasts more accurate and informative?

The above literature overlooks individuals’ overprecision, a behavioral bias that has long been documented in the psychology literature. Overprecision refers to the excessive faith that one knows the truth (Radzevick and Moore 2011; Moore et al. 2016). It has been observed across a wide spectrum of professions (e.g., clinical psychologists, physicians and nurses, engineers, lawyers, negotiators, and newsvendors).¹ Regarding the financial market, Odean (1998) illustrates that investors’ overprecision in stock valuation can lead to intensified differences of opinion and, therefore, to excessive trading volume. Daniel et al. (1998) propose that overprecision induces investors to overreact to their private signals and underreact to public signals. Relatedly, Adebambo and Yan (2018) show that investors who overestimate the precision of their information and underestimate risk exhibit stock overpricing. However, these studies indirectly infer individuals’ overprecision through observed outcomes such as trading volume and share prices.

In this study, we directly examine overprecision using the unique setting of sell-side analysts’ earnings forecasts. This setting is suitable for addressing overprecision for two reasons. First, analyst forecasts are made by individuals under uncertainty, a condition under which behavioral bias is most likely to manifest. Second, realized firm performance (reported earnings per share) can be observed ex post, thus facilitating an objective evaluation of forecast accuracy. Importantly, whether the behavioral bias of overprecision exists in the context of analyst forecasts remains an empirical question because the prior literature shows that forecasts of less form precision (e.g., rounded forecasts) exhibit lower accuracy (Herrmann and Thomas 2005). Given the above reasons, we focus on the fraction of analyst forecasts that are more specific than peer forecasts (i.e., forecasts with more digits after the decimal than the mode number of digits) to examine the implications of overprecision for analyst forecast properties. We refer to these forecasts as Specific forecasts.² We first document the existence of Specific forecasts, then explore their causes and assess their capital market consequences.

Using analyst forecasts from the Unadjusted Detail file in I/B/E/S over the period 1986–2015, we first study the distribution of the number of digits after the decimal in analyst forecasts. We find that during our sample period approximately 5% of analyst forecasts end with more digits than the mode number of digits of all analyst forecasts issued for the same firm-year (i.e., they are Specific forecasts). Furthermore, 35% of analysts in our sample exhibit such behavior at least once during their career, and 19% of firm-years include Specific forecasts, indicating the prevalence of such forecasts. Importantly, despite their form precision, Specific forecasts are less accurate than non-Specific forecasts (i.e., forecasts that contain the same or a smaller number of digits after the decimal than the mode). This significant, negative association with forecast accuracy is evident in the pooled sample and in 18 of the 30 years in our 1986–2015 sample period. Collectively, these results indicate that forecasts that are more precise in form are less accurate in realization than peer forecasts.

We rule out several alternative explanations for the negative association between overprecision and forecast accuracy. First, it is possible that the additional digits of Specific forecasts are the source of the inaccuracy. We address this concern by truncating/rounding the additional digits of Specific forecasts so that they have the same number of digits as the mode. We find that truncated/rounded Specific forecasts still have lower accuracy than non-Specific forecasts. Second, we examine whether analysts employ fixed templates or models which result in Specific forecasts. We control for brokerage house fixed effects and continue to find that Specific forecasts are less accurate. Third, management forecasts may affect the form and the value of analyst forecasts (Bamber et al. 2010). To address this concern, we exclude firm-years with management forecasts and show that our inference continues to hold. Lastly, Fang and Hope (2021) find that analyst teams issue forecasts that are more accurate. We exclude team forecasts and again establish the robustness of our finding.

Next, we explore reasons for analysts’ overprecision. We argue that analysts’ overprecision represents a Dunning-Kruger effect, i.e., less competent analysts are unaware of the boundary of their knowledge (Kruger and Dunning 1999). The literature proposes inexperience and lack of knowledge as two critical factors that induce overprecision (Kruger and Dunning 1999; Burson et al. 2006; Moore et al. 2016). Practical views echo this theory, suggesting that “new analysts” and “naïve analysts” are likely to be too certain about their forecasts or recommendations (Valentine 2010).

Empirically, we find that analysts become less likely to issue Specific forecasts as they gain experience. Whereas 6.81% of forecasts made by analysts with two years of experience are Specific forecasts, the percentage declines significantly with analyst experience, to a sample minimum of 1.24% for forecasts issued by analysts with 23 years of general experience.³ Further, using the last period’s forecast accuracy to proxy for an analyst’s competency, we find that less competent analysts are more likely to issue Specific forecasts. These results highlight the Dunning-Kruger effect in an analyst forecast setting.

Our findings are in line with practitioners’ suggestion not to “pretend to have a level of precision that doesn’t exist. … It (the precision) conveys the image you rely too much on your financial model output, without thinking through the big picture” (Valentine 2010). If Specific forecasts relate to analysts’ overconfidence in their forecasting models, these analysts are less likely to adjust their forecasts by referring to peers’ consensus. We therefore expect Specific forecasts to be bold rather than to herd towards the consensus. Consistent with this expectation, we find a positive and significant association between forecast overprecision and forecast boldness. Further, we show that Specific forecasts are more likely to be issued by analysts affiliated with more prestigious brokerages, plausibly another source of overconfidence in their financial model output (Clement and Tse 2005).

The analyses of the underlying reasons for overprecision yield additional implications for forecast accuracy. First, because overprecision represents inexperienced analysts’ behavioral bias, Specific forecasts issued by experienced analysts are more likely to result from additional information processing and less likely to be inaccurate. The empirical findings support this reasoning by showing that the negative association between overprecision and accuracy is weaker for experienced analysts. Second, the literature indicates that both forecast boldness and brokerage prestige positively impact forecast accuracy (Clement 1999; Clement and Tse 2005). We reason that boldness and brokerage prestige are less likely to improve forecast accuracy when they result in Specific forecasts. Empirically, we find that the positive association between forecast boldness or brokerage prestige and forecast accuracy is weaker for Specific forecasts.

Lastly, we examine the capital market implications of Specific forecasts. We find that investors adjust for the inherent inaccuracy of Specific forecasts, as evidenced by the weaker stock price reactions to these forecast revisions during the three-day event window centered on the forecast revision date. Such a discounting effect is not surprising, considering that forecast overprecision is an easily observable signal. However, investors’ adjustment proves incomplete, as stock prices during a delayed period (i.e., [2, 20]) continue to discount Specific forecasts. This result suggests the existence of market inefficiency and highlights the practical importance of viewing forecast overprecision as a public signal of forecast inaccuracy.

Our study makes several important contributions to the literature. First, we challenge the conventional wisdom that form precision implies greater accuracy and informativeness. Such a view has its theoretical foundation in Grossman and Stiglitz (1980), who derive an information equilibrium wherein more precise prices/quotes/forecasts result from additional information-processing, and is empirically supported by the evidence in Bradley et al. (2004), Herrmann and Thomas (2005), and Kuo et al. (2015). Unlike those studies, we show that the implicit assumption of agent rationality in trading off costs and benefits to gather and process information overlooks an important feature of individuals, i.e., their vulnerability to overprecision. We document the existence of overprecision in sell-side analysts’ forecasts and explore its causes and consequences. Together with the above studies, our study helps form a more complete picture of the link between form precision and real accuracy.

Second, our study contributes to the strand of research on overprecision. Moore et al. (2016) categorize overconfidence into three subcategories: (1) overestimation – thinking that you are better than you actually are; (2) overplacement – exaggerating the extent to which you are better than others; and (3) overprecision – having excessive faith that you know the truth. Of these three distinct forms of overconfidence, Moore et al. (2016) summarize that “overprecision in judgment is both the most durable and the least understood form of overconfidence.” Utilizing an analyst forecast setting, we document that inexperience and incompetence are two critical factors associated with overprecision. This evidence highlights a Dunning-Kruger effect, in which individuals who lack knowledge are also unaware of the boundaries of their knowledge. Our context of analyst forecasts overcomes two major challenges encountered by studies of individuals’ overprecision (Radzevick and Moore 2011): that statement accuracy is usually untestable, and that the degree of overprecision is difficult to measure.

Lastly, we add to the literature on analysts’ cognitive biases. Prior literature shows that analysts tend to overreact to positive information and underreact to negative information (Easterwood and Nutt 1999), that uncertainty amplifies analysts’ overreaction (De Bondt and Thaler 1990), and that analysts become overconfident due to past success (Hilary and Menzly 2006). Our study reveals a novel form of analysts’ cognitive bias – overprecision. We find that a proportion of analysts are overconfident in their model outputs and produce forecasts that are precise in form, yet low in accuracy.⁴

The remainder of our study is structured as follows. Section 2 develops testable hypotheses, Section 3 describes sample formation and variable construction, Section 4 discusses empirical findings, and Section 5 concludes the study.

2 Hypotheses development

The psychology literature defines overprecision as individuals’ excessive faith that they know the truth (see Moore et al. (2016) for a review). This behavioral bias has been documented in a variety of settings where individuals make decisions under uncertainty, such as clinical psychologists (Oskamp 1965), physicians and nurses (Baumann et al. 1991), engineers (Kidd 1970), lawyers (Wagenaar and Keren 1986), negotiators (Neale and Bazerman 1990), and newsvendors (Ren and Croson 2013; Li et al. 2017).

Extending the logic to sell-side analysts in the financial market, we posit that forecasting earnings per share (EPS) for listed firms is essentially a decision made by individuals under uncertainty. Prior literature shows that analysts exhibit cognitive biases when forecasting under uncertainty (De Bondt and Thaler 1990; Easterwood and Nutt 1999). Therefore, we argue that sell-side analysts could also be subject to the cognitive bias of overprecision. That is, when they issue forecasts with a level of precision that exceeds most peer forecasts, their forecasts are less likely to be accurate. In addition to the supportive academic evidence on overprecision, practitioners’ recommendations also confirm this notion. In his book Best Practices for Equity Research Analysts: Essentials for Buy-side and Sell-side Analysts, Valentine (2010) has the following suggestion for young analysts: “Don’t pretend to have a level of precision that doesn’t exist. … It (the precision) conveys the image you rely too much on your financial model output, without thinking through the big picture.”

Nonetheless, there is tension regarding whether the overprecision bias extends to sell-side analysts. The existing literature shows that forecasts of lower form precision (e.g., rounded forecasts) exhibit lower accuracy (Herrmann and Thomas 2005). Therefore, the implications of overprecision of analyst forecasts remain an empirical question. Building upon the above discussion, we state our first hypothesis as follows:

• Hypothesis 1: Forecasts ending with more digits after the decimal than the mode of forecasts for the same firm-year (i.e., Specific forecasts) are less accurate than peer forecasts (i.e., non-Specific forecasts).

Our second hypothesis concerns factors relating to sell-side analysts’ overprecision. The psychology literature proposes knowledge and experience as moderators that can deter overprecision (Kruger and Dunning 1999; Burson et al. 2006). These factors are linked to the influential theory of the Dunning-Kruger effect – that unskilled individuals are unaware of the boundaries of their knowledge. In their work, Kruger and Dunning (1999) state that unskilled individuals suffer from a dual burden because “[not] only do these people reach erroneous conclusions and make unfortunate choices, but their incompetence robs them of the metacognitive ability to realize it” (p. 1121).

The Dunning-Kruger effect is echoed by practitioners’ view that inexperience and incompetence can lead to analysts’ overprecision in forecasts or recommendations. Valentine (2010) states that “too often new analysts think they know the answer because they are overly confident in their models. Their mind can’t comprehend where they could be wrong” (p. 246). In the same book, an interviewed senior analyst expresses the following opinion: “I believe analysts who express a high degree of confidence in any recommendation or forecasts are usually naïve” (p. 49). Combining practitioners’ insights and academic evidence, we hypothesize the following:

• Hypothesis 2: Less experienced (competent) analysts are more likely to issue Specific forecasts than are more experienced (competent) analysts.

Our third hypothesis considers the implications of Specific forecasts for the capital market. We examine whether investors can rationally weight the information contained in overprecise forecasts in terms of its relation to forecast accuracy. Under rational expectations, investor responses to forecast revisions will be weaker when updated forecasts are Specific. This view is supported by Abarbanell et al. (1995), who illustrate that investor responses to forecasts increase in expected forecast accuracy. More specifically, investors can extract and utilize public information such as analyst-specific and forecast-specific characteristics that can affect forecast accuracy. Building upon this framework, investor responses to forecast revisions are likely to be a function of these characteristics. Gleason and Lee (2003), Park and Stice (2000), and Stickel (1992) report consistent empirical findings. Following this line of reasoning, we present the following hypothesis:

• Hypothesis 3: The stock market reacts less to revisions of Specific forecasts than to revisions of non-Specific forecasts.

We note that investors may not weight the information rationally in terms of its ability to predict future earnings. Existing studies also show that investors fail to fully understand the implications of accounting numbers when forecasting future earnings (Sloan 1996; Xie 2001). More pertinent to our context, Clement and Tse (2003) examine stock market reactions around forecast revisions and find that the implied weights on characteristics that are predictive of future earnings in the market reaction equation differ significantly from the weights on the same characteristics in the earnings forecast equation. Therefore, whether investors rationally incorporate public signals when responding to forecast revisions is open to empirical testing.

3 Sample, variables, and empirical specifications

3.1 Data and sample formation

Our sample comprises I/B/E/S forecasts of U.S.-listed firms’ annual earnings per share (EPS) from 1986 to 2015. Herrmann and Thomas (2005) employ a sample beginning in 1985. As we require information on lagged analyst/firm attributes in our empirical analyses, we exclude the initial year of data in 1985. Following Herrmann and Thomas (2005) and Dechow and You (2012), we employ the I/B/E/S Unadjusted Detail file to ensure that our analyses are not affected by the retroactive stock split rounding effect, also described in Baber and Kang (2002) and Payne and Thomas (2003).⁵ For an analyst i covering firm j, we retain analyst i’s last forecast of firm j’s annual earnings per share (EPS) for fiscal year t. Further, following Clement and Tse (2005), we retain only forecasts issued within the [−365, −30] window prior to the fiscal year-end. To obtain a meaningful comparison of analysts who provide forecasts for the same firm-year, we exclude firm-years followed by fewer than three analysts. We deflate forecast revisions and forecast errors by a firm’s security price two days prior to the forecast revision date using stock price data from the CRSP database. Following Clement and Tse (2005), we eliminate observations with price-deflated analyst forecast errors greater than 0.40 or lower than −0.40.⁶ We also eliminate observations without sufficient data to compute the variables used in our regression analyses. We winsorize all continuous variables at their 1st and 99th percentiles. For the transformed variables described below, we winsorize their corresponding raw values. These procedures yield a sample of 389,467 analyst-firm-year observations for 5760 (12,508) unique firms (analysts).

3.2 Variable construction and empirical specifications

Our empirical focus is on analyst forecasts that are more specific than forecasts issued by peer analysts. We identify these forecasts by creating an indicator, SpecificMode_ijt, that equals 1 if analyst i’s forecast for firm j in fiscal year t has more digits after the decimal than the mode number of digits of peer analysts’ forecasts for the same firm-year, and 0 otherwise. We term these forecasts Specific forecasts. The mode, by definition, refers to the value that appears most frequently in a set of data or observations. In our context, it equals the number of forecast digits that appears most frequently in analysts’ last forecasts for the annual EPS of a firm-year. The mode thus intuitively and methodologically allows us to capture the “normal” number of digits of analyst forecasts. For example, there are 20 analysts who followed Monsanto Company in fiscal year 2015. Appendix 1 shows the 20 analysts’ latest forecasts during [−365, −30] prior to the firm’s fiscal year end. Of these analysts, 17 forecast with two digits (i.e., the mode) and three forecast with three digits (i.e., Specific forecasts). The variable SpecificMode equals 1 for the three analyst forecasts with three digits, indicating their specificity compared with peer analysts, and 0 for the other forecasts. Monsanto Company eventually reported EPS ending with two digits.⁷ Appendix 1 discusses Specific forecasts in greater detail using two illustrative examples.

Our main analyses investigate the association between forecast specificity and forecast accuracy using the following model:

$$ {\displaystyle \begin{array}{c}{ForAccuracy}_{ijt}={\beta}_0+{\beta}_1{SpecificMode}_{ijt}+{\beta}_2{LagForAccuracy}_{ijt-1}+{\beta}_3{Bold}_{ijt}\\ {}+{\beta}_4{BrokerSize}_{it}+{\beta}_5{GenExp}_{it}+{\beta}_6{FirmExp}_{ijt}+{\beta}_7{Industries}_{it}\\ {}\begin{array}{c}+{\beta}_8{Companies}_{it}+{\beta}_9{ForFrequency}_{ijt}+{\beta}_{11}{DaysElapsed}_{ijt}\kern2.8em \\ {}+{\beta}_{11}{ForHorizon}_{ijt}+{\beta}_{10}{FyeDis}_{ijt}+{\varepsilon}_{ijt}\kern12em \end{array}\end{array}} $$

(1)

where the dependent variable ForAccuracy_ijt measures analyst i’s forecasting performance for firm j in year t relative to peer analysts’ forecasting performance for firm j in year t. Consistent with Clement and Tse (2005), we measure ForAccuracy using the following transformation:

$${ForAccuracy}_{ijt}=\frac{AFE\max_{jt}\;-\;{AFE}_{ijt}}{AFE\max_{jt}\;-\;AFE\min_{jt}}$$

(2)

In Eq. (2), AFE_ijt is the absolute forecast error of analyst i’s forecast for firm j in year t. We compute forecast error as firm j’s year t earnings minus analyst i’s forecast of firm j’s year t earnings. AFEmax_jt and AFEmin_jt are the maximum and minimum absolute forecast errors, respectively, of analysts that issue forecasts for firm j in year t. The transformed variable ForAccuracy increases in analyst forecast accuracy.

We include several covariates, identified in the existing literature, that could affect analyst forecast accuracy. These variables include last year’s forecast accuracy (LagForAccuracy), where ForAccuracy is defined as in Eq. (2); forecast boldness (Bold), defined as the distance between analyst i’s forecast for firm j in year t from the pre-revision (year-to-date) consensus forecast for firm j in year t (Clement and Tse 2005)⁸; brokerage firm size (BrokerSize), defined as an indicator that equals 1 if analyst i is employed by a brokerage firm in the top decile in terms of the number of analysts employed during year t, and 0 otherwise; general experience (GenExp), defined as the number of years since the analyst issued a forecast for any firm in the sample; firm-specific experience (FirmExp), defined as the number of years since the analyst began following the firm; number of industries followed (Industries), defined as the number of distinct two-digit SIC industries followed by the analyst during year t; number of companies followed (Companies), defined as the number of distinct companies followed by the analyst during year t; forecast frequency (ForFrequency), defined as the number of forecasts issued by the analyst for the firm during year t; number of days elapsed (DaysElapsed), defined as the number of days elapsed since the last forecast for the firm; forecast horizon (ForHorizon), defined as the number of days between the analyst’s forecast date and the firm’s fiscal-year-end date; and distance of the analyst forecast from the consensus forecast (FyeDis), defined as the absolute difference between the analyst’s forecast and the fiscal-year-end consensus forecast for the firm.

Consistent with Clement and Tse (2005), we transform these control variables (except for the indicator variable BrokerSize) as follows to facilitate comparison of regression model coefficients:

$${Char}_{ijt}=\frac{Char\_{Raw}_{ijt}- Char\_ Raw{\min}_{jt}}{Char\_ Raw{\max}_{jt}- Char\_ Raw{\min}_{jt}}$$

(3)

where Char_Raw_ijt denotes the raw value of one of the control variables and Char_Rawmax_jt and Char_Rawmin_jt are the original maximum and the original minimum of the variable for firm j in year t, respectively. A higher value of Char_ijt indicates that analyst i scores higher on that characteristic when issuing a forecast for firm j in year t. Therefore, the transformed variable preserves the relative distance of the raw variable. We present detailed variable definitions in Appendix 2.

3.3 Summary statistics and correlations of key variables

Table 1 Panel A reports descriptive statistics of the key variables (unscaled) employed in our analyses. SpecificMode averages 0.049, indicating that 4.9% of analyst forecasts end with more digits than the mode. Aggregated to the firm-year level, 19% of firm-years include Specific forecasts. Aggregated to the analyst level, 35% of analysts have provided Specific forecasts during their careers.⁹ These statistics indicate that overprecision of analyst forecasts is an economically important phenomenon.

Table 1 Summary statistics of empirical variables

An average analyst has roughly 11 years of general experience and five years of experience covering the firm of interest. Further, an analyst on average follows around 19 companies from four distinct industries. The descriptive statistics are comparable to those reported in the existing literature (e.g., Clement and Tse 2005; Dechow and You 2012). Because we scale and demean the variables before performing our regression analyses, we also show, in Panel B, the summary statistics of the transformed variables. The transformation procedures in Eq. (3) follow those of Clement and Tse (2005) and Herrmann and Thomas (2005). Panel C reports Pearson correlations among the variables used in our main regression analyses.

4 Empirical results

4.1 Specificity and accuracy

4.1.1 Baseline findings

In this section, we analyze whether Specific forecasts are more accurate than non-Specific forecasts by estimating the association between analyst forecast accuracy and SpecificMode, i.e., an indicator that equals 1 if a forecast ends with more digits after the decimal than the mode number of digits derived from all forecasts for the firm-year, and 0 otherwise. We estimate Eq. (1) by regressing forecast accuracy (ForAccuracy) on our variable of interest (SpecificMode) and a list of covariates known to affect analyst forecast accuracy.

Our panel stacks 30 years of data for the period 1986–2015. We report ordinary least squares (OLS) regression results in Table 2. To address potential correlations in error terms, we adjust standard errors of the coefficient estimates using two-way clustering by analyst and year (Petersen 2009; Luo and Nagarajan 2015).¹⁰ We find a negative and statistically significant coefficient on SpecificMode (−0.018, t = −4.86) in Column (1). This result indicates that, despite their form precision, Specific forecasts are less accurate than non-Specific forecasts.

Table 2 Forecast specificity and forecast accuracy

The coefficients on the control variables are largely in line with the results of prior literature (e.g., Clement 1999; Hong et al. 2000; Clement and Tse 2003, 2005). For example, consistent with Clement and Tse (2003, 2005), analysts who have better historical forecasting performance continue to have it in the current period, as reflected by the positive and significant coefficient on LagForAccuracy (0.056, t = 10.62). The significantly positive coefficient on Bold (0.076, t = 9.51) implies that bold forecasts are more accurate than herding forecasts. Analysts employed by larger brokerage houses (BrokerSize) provide more accurate forecasts (0.008, t = 3.68). An analyst’s general experience in the profession (GenExp) improves the forecast accuracy (0.010, t = 3.84). However, firm-specific experience (FirmExp) does not exhibit a significant relation with forecast accuracy (0.003, t = 1.28). Portfolio complexity adversely affects forecasting performance, as indicated by the negative and significant coefficient on the number of industries followed (Industries) (−0.014, t = −5.55). Frequent forecasters perform better, as reflected by the positive and significant coefficient on ForFrequency (0.017, t = 5.55). A longer period between the forecast revision date and the preceding forecast date for the same firm-year by any other analyst is associated with less accuracy, as revealed by the negative and significant coefficient on DaysElapsed (−0.022, t = −7.63). The negative coefficient on ForHorizon (−0.182, t = −15.58) suggests that forecasts more distant from the fiscal year-end have lower accuracy. Finally, the coefficient on FyeDis is negative and significant (−0.354, t = −36.57), suggesting that analysts who deviate more from the year-end consensus have less accurate forecasts.¹¹

Because the explanatory variables are transformed to range from 0 to 1, the regression coefficients allow us to assess the relative economic significance of these variables in explaining the variation in forecast accuracy (Clement and Tse 2005; Herrmann and Thomas 2005). The effect of overprecision on forecast accuracy is greater than that of analysts’ brokerage house size, general experience, and the number of industries followed, but less than that of forecast boldness.

Lastly, because loss firms are more difficult to value and exhibit larger forecast errors (Clement and Tse 2005), we perform corroborative analyses to examine whether our main findings are driven by forecasts for loss firms. We split our sample into subsamples of non-negative forecasts and negative forecasts. To avoid introducing a retroactive bias (i.e., analysts did not have information on realized EPS when making forecasts), we use forecasted losses instead of realized losses to identify observations for loss firms. Repeating our main analyses using the two subsamples, we have two primary findings. First, loss forecasts are fewer than non-loss forecasts, accounting for 8.23% and 91.77% of our total observations, respectively. Second, the negative association between SpecificMode and ForAccuracy is statistically significant in the subsample of non-negative forecasts (−0.018, t = −4.77 in Column (2)), but statistically insignificant in the subsample of negative forecasts (−0.015, t = −1.38 in Column (3)). Overall, these empirical findings support Hypothesis 1 – that seemingly precise Specific forecasts are less accurate – and suggest that this result is not attributable to forecasts for loss firms.¹²

4.1.2 Persistence in the association between specificity and accuracy

Our main results so far rely on the pooled sample covering 1986 to 2015. To examine whether these pooled sample results are driven by a few outlier years, we estimate Eq. (1) for each annual cross-section. Because each analyst-firm appears only once in each yearly subsample, such an analysis also mitigates potential concerns about correlated residual terms in the panel data.

Yearly regression results appear in Table 3. For brevity, we report only the coefficient on SpecificMode and its corresponding t-statistic, the adjusted R², and the number of observations in each annual regression. We find that the negative association between SpecificMode and ForAccuracy holds in 27 of the 30 years, the exceptions being 1988 (0.015, t = 1.52), 1992 (0.002, t = 0.19), and 1995 (0.004, t = 0.39). Further, the negative coefficients are statistically significant, at least at the 10% level, in 18 of the 27 years.

Table 3 Forecast specificity and forecast accuracy – evidence by year

Although we observe some variation in the magnitude of the coefficients over time, the yearly estimation results mitigate our concern that the pooled sample results may be driven by a small number of outlier years. We conclude that the observed negative relationship between forecast specificity and forecast accuracy persists during our sample period.¹³

4.1.3 Alternative measures of forecast specificity

Our primary measure of analyst forecast specificity, SpecificMode, compares an analyst’s forecast specificity with the specificity of forecasts issued by her peer analysts following the same firm in the same year. In this section, we perform sensitivity analyses employing three alternative measures of forecast specificity. SpecificCon_ijt is the number of overspecified digits compared to the mode number of digits; SpecificEps_ijt is an indicator that equals 1 if analyst i’s forecast for firm j in year t has more digits after the decimal than the actual earnings per share (EPS) of firm j in year t-1, and 0 otherwise; and SpecificMedian_ijt is an indicator that equals 1 if analyst i’s forecast for firm j in year t has more digits after the decimal than the median number of digits of analyst forecasts for the same firm-year, and 0 otherwise.

We re-estimate Eq. (1) after replacing SpecificMode with SpecificCon, SpecificEps, and SpecificMedian in succession. Table 4 reports the regression results. The coefficient on SpecificCon in Column (1) is significantly negative (−0.018, t = −6.00), confirming our main finding that more specific forecasts are less accurate. The results reported in Columns (2) and (3), employing SpecificEps and SpecificMedian, respectively, are qualitatively similar. We conclude that our main findings are robust to the measure of analyst forecast specificity used.

Table 4 Forecast specificity and forecast accuracy – using alternative constructs of forecast specificity

4.2 Why do analysts issue overprecise forecasts?

4.2.1 Inexperience and overprecision – The Dunning-Kruger effect

Our second hypothesis posits that overprecise forecasts reflect the Dunning-Kruger effect of sell-side analysts, i.e., incompetent analysts are unaware of the boundaries of their knowledge. Building on the academic evidence (e.g., Clement 1999; Bilinski et al. 2013; Bradshaw et al. 2019) and practitioners’ observations (Valentine 2010), we use analyst inexperience as a proxy for analyst incompetency. We conjecture that the behavioral bias of overprecision is greater for inexperienced analysts.

The graphical evidence in Fig. 1 confirms this line of reasoning. Specifically, we categorize sample observations by analysts’ general experience, i.e., the number of years since an analyst enters our database. For each subgroup of observations, we compute the group average of SpecificMode. In Fig. 1, the X-axis represents analysts’ general experience. It has a minimum of two years because our regression sample requires lagged values of forecast accuracy and therefore mechanically excludes analysts’ initial year forecasts. We aggregate all observations with general experience greater than or equal to 25 years in the subgroup “25″ because there are fewer observations that fall into these high-experience categories.¹⁴ The Y-axis denotes the mean SpecificMode, i.e., the proportion of forecasts that are Specific for each general experience subgroup. Figure 1 Panel A shows that analysts’ tendency to issue Specific forecasts declines with their general experience. For example, with corroborating statistics in Table 5, we find that 6.81% of the forecasts issued by analysts with two years of general experience are more specific than peer analysts’ forecasts. This ratio declines monotonically with analysts’ general experience, to a minimum of 1.24% for analysts with 23 years of experience.¹⁵ Because analysts who issue Specific forecasts are likely to differ from those who do not, we further retain only analysts who have issued at least one Specific forecast during our sample period and re-plot the pattern in Panel B. We observe the same relationship between analysts’ general experience and their tendency to issue Specific forecasts. The graphical evidence supports our Hypothesis 2 that inexperienced analysts produce disproportionally higher fractions of Specific forecasts.

Fig. 1

Forecast specificity by analysts’ general experience. A: Pooled sample for all analysts B: Analysts with at least one Specific forecast. This figure plots the average value of SpecificMode for each subgroup of observations based on analysts’ general experience. We categorize sample observations by analysts’ general experience. For each subgroup, we compute the group average of SpecificMode. SpecificMode is an indicator that equals 1 if an analyst’s EPS forecast for a firm-year has more digits than the most frequent number of digits (mode) of analysts’ forecasts for the same firm-year, and 0 otherwise. Analysts’ general experience (i.e., the raw value of GenExp) begins with 2 years because our empirical sample requires information of analysts’ lagged forecast accuracy, excluding analysts’ initial year forecasts. Further, we cluster observations corresponding to analysts with more than 25 years’ experience into the “25″ subgroup to the rightmost of the figures. Panel A presents the pattern for the pooled sample for all analysts. Panel B presents the pattern for analysts who have at least one observation with SpecificMode = 1 during the sample period. In both panels, the X-axis depicts analysts’ general experience, and the Y-axis presents the group average of SpecificMode

Table 5 Analyst general experience and forecast specificity

4.2.2 Specificity and analyst attributes: Multivariate evidence

To simultaneously consider determinants of forecast specificity, we examine the association between analysts’ attributes and analysts’ tendency to issue Specific forecasts using regression analysis. To do so, we construct an analyst-year-level dataset by aggregating each individual analyst’s forecasts within a year into a single observation and then estimate the following :

$$ {\displaystyle \begin{array}{c}{AvgSpecific}_{it}={\beta}_0+{\beta}_1{AvgBold}_{it}+{\beta}_2 Avg{\left| LagForError\right|}_{it}+{\beta}_3{AvgFirmExp}_{it}\\ {}+{\beta}_4{GenExp}_{it}+{\beta}_5{Companies}_{it}+{\beta}_6{Industries}_{it}+{\beta}_7{BrokerSize}_{it}\\ {}\begin{array}{c}\kern2.5em +{\beta}_8{AvgDaysElapsed}_{it}+{\beta}_9{AvgForHorizon}_{it}+{\beta}_{10}{AvgForFrequency}_{it}\\ {}+{\beta}_{11} Avg{FyeDis}_{it}+\textrm{Year}\ \textrm{Fixed}\ \textrm{Effects}+{\varepsilon}_{ijt}\kern9em \end{array}\end{array}} $$

(4)

where the dependent variable AvgSpecific_it is the percentage of analyst i’s forecasts that are Specific forecasts during year t, and the independent variables are yearly averages of (unscaled) forecast-level values of an analyst’s forecasts during the year. For example, AvgBold is the yearly average of Boldness for all the forecasts made during the fiscal year by a given analyst.

Table 6 reports the regression results. We have the following observations. First, the results confirm our earlier conjecture that inexperienced analysts are more likely to issue Specific forecasts, as evidenced by the significantly negative coefficient on general experience (GenExp) in Column (1) (−0.110, t = −5.13). A second proxy for experience, AvgFirmExp, exhibits an insignificant coefficient (0.003, t = 0.05).¹⁶ Given the high correlation between AvgFirmExp and GenExp (Corr. = 0.614, p < 0.01), we further estimate the regression after excluding either AvgFirmExp or GenExp from the control variables. We find that GenExp continues to have a significantly negative coefficient (−0.110, t = −4.88 in Column (2)) when AvgFirmExp is excluded. Further, without controlling for general experience, firm experience also reduces the frequency of Specific forecasts (−0.164, t = −2.49 on AvgFirmExp in Column (3)). Therefore, the negative and significant coefficient on AvgFirmExp in Column (3) likely manifests the effect of analysts’ general experience, a factor that is correlated with analysts’ firm experience.¹⁷ Comparing regression results across the three specifications further highlights the notion that an analyst’s general experience significantly reduces the analyst’s tendency to issue Specific forecasts.

Table 6 Analyst attributes and forecast specificity – analyst-level analyses

Second, we observe additional evidence that is consistent with the Dunning-Kruger effect. We find that analysts who issue Specific forecasts have poorer historical performance (28.391, t = 3.53 on Avg|LagForError|) and lower forecasting frequency (−0.140, t = −1.68 on AvgForFrequency). We argue that competent analysts provide better forecasts historically. The finding that an analyst’s historical forecasting performance impacts the analyst’s tendency to issue Specific forecasts further highlights the importance of controlling for prior forecast accuracy when examining the association between overprecision and current forecast accuracy in Table 2. Regarding forecasting frequency, the literature shows that analysts who forecast more frequently work harder and are better able to incorporate the latest information in their forecasts (Jacob et al. 1999; Clement and Tse 2005; Loh and Stulz 2018).

Third, we obtain suggestive evidence that analysts who issue Specific forecasts may have over-relied on their model output (Valentine 2010). Although such an activity is unobservable, we argue that it can be captured to some degree by the following two proxies: (1) forecast boldness, because analysts who over-rely on their model output are likely to issue bolder forecasts and not to herd; and (2) brokerage house prestige, because analysts employed by large brokerage houses are likely to be overconfident about their forecasting models. We find that the coefficient on analyst boldness is positive and significant (4.638, t = 4.64 on AvgBold in Column (1)). Further, analysts employed by prestigious brokerage houses are more likely to issue Specific forecasts (1.918, t = 3.52 on BrokerSize). Although only suggestive, these results are consistent with the notion that analysts who issue Specific forecasts overly rely on their forecasting models.¹⁸

Lastly, we examine whether incorporating brokerage house fixed effects affects the association between analyst attributes and analysts’ tendency to issue Specific forecasts. Such a test will alleviate the concern that Specific forecasts may result from the forecast model setup because analysts in the same brokerage house are likely to employ similar models. We augment the original regression model by adding brokerage house fixed effects. The results in Panel B show negative and significant coefficients on GenExp (−0.057, t = −4.11 in Column (1); −0.055, t = −3.95 in Column (2)) and positive and significant coefficients on Avg|LagForError| and AvgBold across all specifications.¹⁹ Therefore, controlling for brokerage fixed effects, analyst attributes continue to explain analysts’ tendencies to issue Specific forecasts. Overall, the empirical results support Hypothesis 2.²⁰

4.3 Specificity and accuracy – Implications of inexperience and behavioral bias

In the preceding subsection, we documented that, consistent with the Dunning-Kruger effect, inexperience is a critical determinant of analysts’ overprecision. Further, the significant effects of forecast boldness and brokerage house prestige confirm that analysts may over-rely on their model output. Importantly, these causes for analyst forecast specificity also have implications for triangulating the relationships between forecast specificity, analyst attributes, and forecast accuracy. In this section, we design and provide related tests to further address this notion.

Our analyses here comprise two levels. First, we are interested in whether experience can mitigate the negative association between forecast specificity and forecast accuracy. If, as we have argued, inexperienced analysts exhibit behavioral bias by overly relying on their models, we expect the negative association to be weaker for experienced analysts. We find empirical results that are consistent with this expectation. In Table 7, we estimate the association between forecast specificity and forecast accuracy after allowing the relationship to differ across analysts with varying levels of experience. We do so by including the interaction between SpecificMode and GenExp. In Column (1), the coefficient on SpecificMode*GenExp is positive and significant (0.013, t = 1.87). This result suggests a smaller reduction in forecast accuracy when analysts with more general experience issue Specific forecasts.

Table 7 Forecast specificity and forecast accuracy – moderating effects of analyst attributes

Second, we leverage the two factors that capture analysts’ tendency to overweight their model outputs, i.e., analyst boldness (Bold) and brokerage house size (BrokerSize). The existing literature shows that boldness and brokerage house prestige positively impact forecast accuracy (Hong et al. 2000; Clement and Tse 2005).²¹ Drawing upon our earlier findings that Specific forecasts are more likely to be bold forecasts and issued by analysts employed by prestigious brokerage houses, we conjecture that the positive impact of boldness and brokerage house prestige on forecast accuracy may be weaker when forecasts are Specific. Such a line of reasoning is similar to that in Yin and Zhang (2014), who show differential associations between forecast boldness and forecast accuracy due to analysts’ tournament incentives, and to that in Fang and Yasuda (2009), who find that the association between bank prestige and affiliated analysts’ forecast accuracy depends on analysts’ individual reputations.

Columns (2) and (3) in Table 7 report positive and significant coefficients on Bold and BrokerSize, confirming the notions in the existing literature for non-Specific forecasts. More importantly, Column (2) reports a negative and significant coefficient on SpecificMode*Bold (−0.029, t = −2.45), suggesting that boldness is less likely to improve forecast accuracy for Specific forecasts. Further, Column (3) reports a negative and weakly significant coefficient on SpecificMode*BrokerSize (−0.009, t = −1.65), suggesting that brokerage house prestige also is less likely to improve forecast accuracy for Specific forecasts.

4.4 Capital market implications of more specific forecasts

Having documented the existence of overprecision and suggested its causes, we next consider the implications of Specific forecasts for the capital market. We explore whether investors can rationally weight the information contained in form specificity in terms of its relation to forecast accuracy, i.e., Hypothesis 3. Rational expectations predict that investors’ responses to forecast revisions will be weaker when updated forecasts are Specific (Stickel 1992; Park and Stice 2000; Gleason and Lee 2003).

4.4.1 Analyses of price reactions during the short-term event window

To analyze investors’ reactions to Specific forecasts, we begin with the following baseline model of the relation between stock returns and forecast revisions:

$${CAR}_{ijt}={\alpha}_0+{\alpha}_1{REV}_{ijt}+{\varepsilon}_{ijt}$$

(5)

where CAR_ijt is the cumulative abnormal return during the three-day window centered on the forecast revision date (i.e., days −1 to +1), and REV_ijt is analyst i’s forecast revision for firm j in year t, computed as analyst i’s revised forecast for firm j in year t less analyst i’s prior forecast for firm j in year t, divided by firm i’s stock price two days prior to the forecast revision.²²

We then examine whether investors incorporate information in analyst-specific and forecast-specific characteristics that are related to analyst forecast accuracy in stock prices. If investors consider forecast specificity (SpecificMode) and the other explanatory variables of forecast accuracy in Eq. (1), they will incorporate them in their responses to forecast revisions, and the coefficients (i.e., α₀ and α₁) in Eq. (5) will be a function of these variables, as shown in Eqs. (6a) and (6b) below:

$$ {\displaystyle \begin{array}{c}{\alpha}_0={\beta}_0+{\beta}_1{SpecificMode}_{ij t}+{\beta}_2{LagForAccuracy}_{ij,t-1}+{\beta}_3{Bold}_{ij t}\\ {}+{\beta}_4{BrokerSize}_{it}+{\beta}_5{GenExp}_{it}\kern13.5em \\ {}\begin{array}{c}+{\beta}_6{FirmExp}_{ij t}+{\beta}_7{Industries}_{it}\kern12.75em \\ {}+{\beta}_8{Companies}_{it}+{\beta}_9{ForFrequency}_{ij t}\kern10.75em \\ {}\begin{array}{c}+{\beta}_{10}{DaysElapsed}_{ij t}+{\beta}_{11}{ForHorizon}_{ij t}\kern9.5em \\ {}+{\beta}_{12}{FyeDis}_{ij t}\kern20em \end{array}\end{array}\end{array}} $$

(6a)

$${\alpha}_1={\gamma}_0+{\gamma}_1{SpecificMode}_{ij t}+{\gamma}_2{LagForAccuracy}_{ij,t-1}+{\gamma}_3{Bold}_{ij t}+{\gamma}_4{BrokerSize}_{it}+{\gamma}_5{GenExp}_{it}+{\gamma}_6{FirmExp}_{ij t}+{\gamma}_7{Industries}_{it}+{\gamma}_8{Companies}_{it}+{\gamma}_9{ForFrequency}_{ij t}+{\gamma}_{10}{DaysElapsed}_{ij t}+{\gamma}_{11}{ForHorizon}_{ij t}+{\gamma}_{12}{FyeDis}_{ij t}$$

(6b)

Substituting Eqs. (6a) and (6b) into Eq. (5), we obtain the following regression model:

$$ {\displaystyle \begin{array}{c}{CAR}_{ij t}={\beta}_0+{REV}_{ij t}\left\{{\gamma}_0+{\gamma}_1{SpecificMode}_{ij t}+{\gamma}_2{LagForAccuracy}_{ij,t-1}+{\gamma}_3{Bold}_{ij t}\right.\\ {}+{\gamma}_4{BrokerSize}_{it}+{\gamma}_5{GenExp}_{it}+{\gamma}_6{FirmExp}_{ij t}+{\gamma}_7{Industries}_{it}\kern0.75em \\ {}\begin{array}{c}+{\gamma}_8{Companies}_{it}+{\gamma}_9{ForFrequency}_{ij t}+{\gamma}_{10}{DaysElapsed}_{ij t}\kern3em \\ {}\left.+{\gamma}_{11}{ForHorizon}_{ij t}+{\gamma}_{12}{FyeDis}_{ij t}\right\}+{\beta}_1{SpecificMode}_{ij t}\kern4.75em \\ {}\begin{array}{c}\kern2em +{\beta}_2{LagForAccuracy}_{ij,t-1}+{\beta}_3{Bold}_{ij t}+{\beta}_4{BrokerSize}_{it}+{\beta}_5{GenExp}_{it}\\ {}\kern3em +{\beta}_6{FirmExp}_{ij t}+{\beta}_7{Industries}_{it}+{\beta}_8{Companies}_{it}+{\beta}_9{ForFrequency}_{ij t}\\ {}+{\beta}_{10}{DaysElapsed}_{ij t}+{\beta}_{11}{ForHorizon}_{ij t}+{\beta}_{12}{FyeDis}_{ij t}+{\varepsilon}_{ij t}\kern2em \end{array}\end{array}\end{array}} $$

(7)

We report the estimation results in Table 8. In the first column, where we employ CAR[−1, 1] (i.e., cumulative abnormal stock return during the three-day event window) as the dependent variable, the coefficient on REV*SpecificMode is negative and significant (−0.409, t = −6.16), suggesting that investors, on average, discount forecasts that are more specific than peer analyst forecasts. This result is consistent with the view of investors’ rational expectations that is modeled in Abarbanell et al. (1995) and empirically supported in studies such as Gleason and Lee (2003), Park and Stice (2000), and Stickel (1992).²³

Table 8 Market reactions to Specific forecasts

4.4.2 Analyses of price reactions during the subsequent period

It is also possible that investor reactions to analyst forecast revisions are incomplete during the three-day event window. To test this conjecture, we re-estimate Eq. (6) using CAR[2, 20], the cumulative abnormal return during the period from days 2 to 20 after the revision date (i.e., day 0), as the dependent variable. Extending the window to the 20th trading day after the forecast revision date also sees the aggregated window, including the short-term and the delayed periods, approximating one calendar month (Jegadeesh and Kim 2010). We again find a negative and significant coefficient on REV*SpecificMode (−0.105, t = −2.19), indicating that investors do not exhibit full rationality, as they only partially incorporate the lower accuracy of Specific forecasts during the three-day event window and continue to discount these forecasts in a delayed period.²⁴

Although not of primary interest, the results for the control variables also suggest partial rationality of investors (Clement and Tse 2003). In Table 8 Column (1), stock price responses to forecast revisions are stronger when the analyst (1) has better historical forecasting performance, (2) has greater boldness, (3) has a higher forecasting frequency, and (4) is employed by a larger brokerage house. Our earlier analyses show that these factors are positively related to forecast accuracy. However, the implied weights on some other factors, such as analysts’ general experience and the number of industries followed, are inconsistent with the associations between these factors and forecast accuracy.

What might investors be learning during the delayed period? We conjecture that subsequent non-Specific forecasts issued by peer analysts, which are of different form but more accurate, may lead investors to further adjust share prices. We use the following procedures to test this conjecture. For each Specific revision, we retrieve peer analysts’ non-Specific forecasts of annual EPS issued during the [2, 20] window. Of the 18,930 Specific revisions, 9735 can be matched with at least one non-Specific forecast subsequently issued by peer analysts. Untabulated t-test results show that the subsequent non-Specific peer forecasts are 6.1% more accurate than the treated Specific forecast.

Next, we split the Specific forecast revisions into two groups: (1) Specific revisions where peer analysts’ subsequent non-Specific forecasts (for multiple forecasts, we use the mean) are more accurate, and (2) Specific revisions where peer analysts’ subsequent non-Specific forecasts are not more accurate. We exclude 9195 Specific revisions without available information on subsequent peer analysts’ forecasts.²⁵ We create two indicators, SpecificMode_Corrected and SpecificMode_NonCorrected, that equal 1 for the first group and the second group, respectively, and 0 otherwise.

We then repeat our market reaction test for the [2, 20] window after replacing REV×SpecificMode in Table 8 with REV×SpecificMode_Corrected and REV×SpecificMode_nonCorrected. Column (3) shows that the coefficient on REV×SpecificMode_Corrected is negative and significant (−0.256, t = −2.09). By contrast, the coefficient on REV×SpecificMode_nonCorrected is smaller in magnitude and statistically insignificant. These results support our conjecture that investors learn from peer analysts’ forecasts issued during the delayed period, which leads to further price adjustment.²⁶

Overall, the empirical findings support Hypothesis 3 that investors on average react less to Specific forecasts. Further, stock price responses to Specific forecast revisions during the delayed period are consistent with the view that investors – and therefore security prices – exhibit incomplete adjustments when responding to public information such as accounting accruals (Sloan 1996; Xie 2001) and forecast revisions (Gleason and Lee 2003).

4.5 Additional analyses and discussions

4.5.1 Rounding and overprecision – A reconciliation

Herrmann and Thomas (2005) show that analyst forecasts of earnings per share occur in nickel intervals approximately 55% of the time; i.e., these are rounded forecasts. By comparison, actual earnings occur in nickel intervals only 22.75% of the time. They also document a negative association between forecast rounding and forecast accuracy and attribute the negative association to analysts’ lack of information and resources to ascertain a more precise forecast number.

We note that our focus on forecast specificity differs from the rounding phenomenon analyzed in Herrmann and Thomas (2005) in two important ways. First, rounding refers to individuals’ tendency to place a zero or a five in the penny location. Forecast specificity, however, can appear in both rounded and non-rounded forecasts. Regarding the former category, a forecast of 2.35 is more specific than one of 2.3.²⁷ As for the latter category, a forecast of 2.345 is more specific than one of 2.34. Second, if specificity is simply the complement of rounding, the existing literature would predict a positive association between forecast specificity and forecast accuracy. The reason is that rounded forecasts are shown to be less accurate than non-rounded forecasts (Herrmann and Thomas 2005).

Building on this discussion, we perform empirical analyses to revisit the rounding effect documented in prior research and to establish the robustness of our finding. We begin by replicating the main finding in Herrmann and Thomas (2005), i.e., that rounded forecasts are less accurate. We employ the same sample period used in Herrmann and Thomas (2005): 1986–2001.²⁸ We construct an indicator variable, Round, that equals 1 for forecasts with zero or five in the penny location, and 0 otherwise.

The results in Table 9 Panel A Column (1) confirm the finding in Herrmann and Thomas (2005) that rounded forecasts are less accurate. The coefficient on Round is negative and significant (−0.006, t = −2.28). Performing the same analysis in the 2002–2015 period, we find an insignificant association between Round and ForAccuracy (−0.001, t = −0.38 in Column (2)). The changing association between forecast rounding and forecast accuracy is of interest in its own right. One possible reason is that, as argued in Herrmann and Thomas (2005) and Dechow and You (2012), rounding is a public signal that can be observed by investors at a low cost. To the extent that investors learn the inherent inaccuracy of rounded forecasts and discount these forecasts, analysts will gradually have less incentives to perform rounding.

Table 9 Forecast specificity and forecast rounding

In untabulated analyses, we find that the fraction of rounded forecasts in the 1986–2001 period is 54.3% but drops to approximately half that (27.8%) in the 2002–2015 period, with the difference being statistically significant at the 1% level. Aggregating the two subperiods, Column (3) shows that the rounding effect is also statistically insignificant for the combined period. This evidence relates to the following question that Dechow and You (2012, p.1,963) posed when suggesting avenues for future research: “As analysts’ incentives change, do we observe changes in rounding behavior?”

More importantly, Columns (4)–(6) in Table 9 Panel A establish the robustness of our results. We find that, after controlling for Round, the negative association between SpecificMode and ForAccuracy continues to hold in the 1986–2001 subsample (−0.017, t = −3.35), the 2002–2015 subsample (−0.027, t = −7.03), and the full sample period (−0.018, t = −4.83).²⁹

A related observation, that the negative association between overprecision and forecast accuracy appears more pronounced in the second half of our sample period (in both Table 9 Panel A and our earlier Table 3), warrants further discussion because it raises the logical question of whether the overprecision bias becomes stronger over time. We draw on two sets of evidence to address this question. First, the declining trend of rounding, discussed above, suggests a higher probability that some Specific forecasts may result from their issuing analysts’ non-rounding decisions. If these analysts’ peers make their rounding choices to signal higher uncertainty or a lack of effort (Herrmann and Thomas 2005; Dechow and You 2012), then Specific forecasts will have higher accuracy. The results reported in Table 9 confirm this effect, which has been proposed in the earlier literature. This effect, however, runs counter to the overprecision effect and would weaken the estimated association between SpecificMode and ForAccuracy.

Second, we perform additional analyses that are less affected by the rounding trend and the ensuing sample composition issue. We repeat the subperiod regressions after excluding all rounded forecasts. Columns (1) and (2) in Table 9 Panel B report negative and significant coefficients on SpecificMode that are of similar magnitude (−0.024, t = −3.97 for 1986–2001 and − 0.027, t = −6.38 for 2002–2015). Alternatively, we employ the full sample but replace SpecificMode with SpecificEps, which is less affected by peer analysts’ rounding decisions. Columns (3) and (4) in Panel B show that the coefficients on SpecificEps are also of similar magnitude (−0.009, t = −2.40 for 1986–2001; −0.008, t = −2.72 for 2002–2015). Collectively, these results do not provide consistent evidence of stronger overprecision biases over time.

4.5.2 Level of EPS, overprecision, and forecast accuracy

Another issue related to our main results is that the link between overprecision and forecast accuracy may be driven by low EPS (in absolute terms) firms. As the EPS magnitude increases, the economic importance of overprecision declines. Furthermore, Roger et al. (2018) build on the neuropsychology research and show that individual analysts process small and large numbers differently. This issue is relevant, as it speaks to the generalizability of our findings.

To address this concern, we first examine the distribution of the numbers of digits of analyst forecasts for firms with varying levels of absolute EPS. We categorize sample observations (i.e., analyst forecasts) into the following groups based on the absolute value of EPS (in $) of each firm-year: (1) 0 < =|EPS| < $0.1, (2) $0.1 < =|EPS| < $1, (3) $1 < =|EPS| < $5, (4) $5 < =|EPS| < $10, (5) $10 < =|EPS| < $100 and (6) |EPS| > =$100.

Panel A of Table 10 tabulates the frequency and the percentage of analyst forecasts ending with specific numbers of digits (i.e., from one digit to four digits) for each of the above subsamples. We find that the patterns of forecast digits vary with EPS magnitude. As |EPS| increases, the percentage of forecasts ending with one digit after the decimal increases, almost doubling from the [0, 0.1) subsample to the [10, 100) subsample (the > = 100 subsample includes only 17 observations and its statistics should be interpreted with caution); however, the percentage of forecasts ending with two digits significantly declines from 81.74% for the [0, 0.1) subsample to 67.36% for the [10, 100) subsample. Forecasts with zero digits after the decimal also increase with |EPS|, except for the [0, 0.1) subsample, which has a relatively higher percentage of 4.83%.³⁰ These patterns are generally consistent with additional digits being of greater importance for small |EPS| firms.

Table 10 |EPS| level, forecast digits, and overprecision

Given the above forecast distributions, it is important to investigate whether small |EPS| firms, whose analysts may be more likely to misunderstand the importance of the penny part, drive the observed negative association between SpecificMode and ForAccuracy. We conduct subsample regressions to address this question. For each of the subsamples categorized based on |EPS|, we perform the multivariate regression relating SpecificMode to ForAccuracy in Eq. (1).³¹

Panel B of Table 10 shows that our findings are not driven by the small |EPS| firms. Within the five subsamples, the four subsamples with the largest |EPS| exhibit negative and significant coefficients on SpecificMode in Columns (2)–(5). The coefficient is positive and statistically insignificant for the smallest |EPS| subsample in Column (1). The results show that it is precisely for these small |EPS| firms that analysts pay the closest attention to the additional digits. Consequently, Specific forecasts for such firms do not exhibit lower accuracy. Collectively, the results of the analyses in this section alleviate the concern that our main findings may be driven by small |EPS| firms.

4.5.3 Are additional digits the source of inaccuracy?

A potential concern with our interpretation of the negative association between forecast specificity and forecast accuracy is that the additional digits of Specific forecasts could be the source of inaccuracy. In the extreme case, if the majority of forecasts are precisely equal to the reported EPS number, a Specific forecast will be mechanically inaccurate. As an illustration, if a firm’s reported EPS is $1.23 and the majority of analysts forecast with two digits and precisely at $1.23, the Specific analyst who forecasts at $1.235 has to be less accurate.

We address this potential concern by forcing all Specific forecasts to have the same number of digits as the mode forecast using two alternative approaches. First, we truncate the additional digits in Specific forecasts so that these forecasts have the same number of digits as the mode forecast. We then repeat our main regression analyses on this subsample. The results in Table 11 Panel A show that (truncated) Specific forecasts are less accurate than non-Specific forecasts. The coefficient on SpecificMode (−0.021, t = −4.71) is negative and significant in Column (1) for the full sample. This effect manifests for non-negative forecasts (Column (2)) but not for negative forecasts (Column (3)), consistent with our baseline findings.

Table 11 Additional analyses

Second, we round all Specific forecasts to the nearest digit as the mode forecast. For example, if the mode forecast has two digits, we round a Specific forecast in the form of X.123 (X.127) to X.12 (X.13), again removing additional digits for Specific forecasts. The results in Columns (4)–(6) of Table 11 Panel A confirm that (rounded) Specific forecasts are less accurate than non-Specific forecasts. The coefficient on SpecificMode (−0.008, t = −2.35) is negative and significant in Column (4) for the full sample. This effect again holds for non-negative forecasts (Column (5)) but not for negative forecasts (Column (6)), consistent with our earlier evidence. Collectively, we conclude that our findings that Specific forecasts are less accurate are not attributable to the additional digits.³²

4.5.4 Specific forecasts with too many digits

Table 11 Panel B presents the matrix of distributions for the number of forecast digits and the mode number of digits for each firm-year. The majority of Specific forecasts have no more than two digits ((235 + 12,082)/18,930 = 65.07%). The remaining 34.93% of Specific forecasts have three or four digits after the decimal point. The latter group requires additional attention because analysts who are conforming to corporate reporting forms may round such forecasts to the second digit after the decimal point in their analyst reports. Our reading of several randomly selected analyst reports corresponding to observations of Specific forecasts confirms that Specific forecasts with no more than two digits appear with the same value in analyst reports, while those with more than two digits are rounded to the second digit.

Notably, investors subscribing to I/B/E/S (which indeed is widely subscribed to by practitioners such as money managers) can observe and react to the I/B/E/S forecasts (Ertimur et al. 2011).³³ Nonetheless, the observation of analysts rounding forecasts with more than two digits in their analyst reports raises the concern that some investors may only have access to, or rely on, the rounded values of Specific forecasts with more than two digits in the reports.³⁴ To further alleviate this concern, we perform additional analyses by transforming the analyst forecast data through rounding all forecasts with more than two digits to their second digit (accounting for 2.97% of our total sample forecasts). With the transformed data, we repeat our main analyses related to the forecast accuracy of, and investors’ reactions to, Specific forecasts. Panels C and D of Table 11 report empirical results that are both economically and statistically similar to those presented earlier in Tables 2 and 8, respectively.

4.5.5 Overprecision and forecast accuracy – Controlling for brokerage house fixed effects

Earlier in Section 4.2.2, we noted that Specific forecasts may result from brokerage house effects. We alleviate this concern by showing that, after controlling for brokerage house fixed effects, analyst attributes (experience, boldness, and forecasting ability) continue to explain the analysts’ tendency to issue Specific forecasts (as in Table 6). Here, we further corroborate this result by examining whether Specific forecasts are less accurate after controlling for brokerage house fixed effects.

Empirically, we augment Eq. (1) by adding brokerage house fixed effects. Econometrically, such a test retains the within-brokerage variations in forecast accuracy and overprecision, but controls for potential forecasting model heterogeneity across brokerage houses. In Table 11 Panel E, we find a significantly negative coefficient on SpecificMode (−0.015, t = −3.97), an estimate that is economically and statistically similar with the estimate in our main specification.

4.5.6 The potential role of management forecasts in shaping analyst forecasts

It is possible that analysts, when issuing forecasts, are affected by both the form and the value of management forecasts, which raises a correlated omitted variable concern (Bamber et al. 2010). To address this concern, we collect data on management forecasts from the I/B/E/S Guidance database and perform additional analyses. To begin with, our primary sample of 389,467 firm-analyst-year observations corresponds to 40,523 unique firm-years. Matching these firm-years to the management forecast data, we find that 121,493 (31.19%) firm-analyst-year observations have matched management forecasts (point or range) for annual EPS.

We exclude these 121,493 observations from this analysis to ensure that the form of analyst forecasts is not influenced by management forecasts. We re-estimate our main regression (Eq. 1) and report the results in Panel F Column (1) of Table 11. The coefficient on SpecificMode continues to be negative and significant (−0.018, t = −4.02).³⁵

Next, we focus on the 121,493 observations of firm-years with matched management forecasts. We divide these observations into two groups: (1) 76,452 analyst forecasts with specificity (number of digits) equal to that of the management forecasts,³⁶ and (2) 45,041 analyst forecasts with specificity different from that of the management forecasts. The former (latter) subgroup comprises analyst forecasts that are more (less) likely to be influenced by the form of management forecasts. Re-estimating our main regression, we find negative and significant coefficients on SpecificMode for both subsamples (−0.024, t = −2.43 in Column (2); −0.023, t = −3.02 in Column (3)). Collectively, the empirical evidence suggests that the association between overprecision and forecast accuracy is not explained by management forecasts.

4.5.7 Market sentiment and behavioral biases of analysts

Does the overprecision bias become more or less pronounced in periods of “bad times”? We address the potential interaction between market sentiment and analysts’ behavioral biases. Two countervailing forces exist. First, Loh and Stulz (2018) find that analysts’ research is more valuable during bad times. Second, using a different context (investors), Zhang (2006) contends that behavioral biases manifest more during periods of high uncertainty.

Empirically, we follow Loh and Stulz (2018) and identify the following periods as bad times: (1) the crisis periods of September–November 1987 (1987 market crash crisis), August–December 1998 (LTCM crisis), and July 2007–March 2009 (credit crisis); (2) the NBER recessions in July 1990–March 1991, March–November 2001, and December 2007–July 2009; and (3) the periods of high policy uncertainty, as indicated by the Baker et al. (2016) index (i.e., when the index is in the top tercile during our sample period). We note that the three categories above have overlapping periods of bad times. We then form a “BadTime = 1” subsample, which includes observations that fall in any of the above periods. The remaining observations form the “BadTime = 0” subsample.

We estimate our regression model separately for the two subsamples. The results in Panel G indicate that the negative association between overprecision and forecast accuracy holds for both subsamples (−0.016, t = −4.00 for BadTime = 1 in Column (1); −0.022, t = −3.25 for BadTime = 0 in Column (2)). Furthermore, we estimate a single regression that adds the BadTime indicator and the interaction variable SpecificMode×BadTime to our main regression model. We find an insignificant coefficient on the interaction term in Column (3). In brief, our finding that overprecise forecasts are less accurate holds in periods of varying market sentiments.

4.5.8 Excluding team forecasts

Using a hand-collected sample of analyst forecast reports over the period 2013–2016, Fang and Hope (2021) find that many forecasts are issued by analyst teams and that team forecasts are more accurate than individual forecasts. In our context, one might argue that team forecasts are less applicable to our investigation of individuals’ overprecision bias. We re-estimate our main regression in Table 2 after excluding observations for analysts who cover more than 30 (or alternatively 50) firms in a given fiscal year (Kaustia and Rantala 2015). In Table 11 Panel H, we continue to find significantly negative coefficients on SpecificMode.

4.5.9 Forecasting task complexity

Because behavioral biases may be more pronounced for firms that are harder to value, in this section we explore whether firms with greater business complexity exhibit more Specific forecasts. Before we proceed, we note that such a test may have limited power because our primary construct of overprecision (SpecificMode) relies upon intra-firm comparison in analysts’ forecasts (i.e., benchmarked against peer analysts following the same firm). Nonetheless, we attempt to shed some light on this relevant issue.

Empirically, we construct a firm-year-level variable, MoreSpecific%, defined as the ratio of the number of Specific forecasts to the total number of forecasts issued by analysts following the firm during the year. For independent variables, we follow Dechow and You (2012) and use the two proxies of firm complexity, namely firm size (Size) and the number of business segments (LnNumsegment). In addition, we control for three proxies for firm-level uncertainty: no management forecasts (MissGuidance), stock return volatility (Volatility), and institutional ownership (ISHolding). Detailed variable definitions appear in Appendix 2. We include industry and year fixed effects and cluster standard errors by firm and year.

Table 11 Panel I reports the regression results. Across all specifications, we find positive and significant coefficients on Size and LnNumsegment. These results are consistent with the notion that firms of greater size or with more segments have greater business complexity (Dechow and You 2012), which induces more analysts to exhibit the overprecision bias. There is evidence that high volatility firms also exhibit more Specific forecasts. However, the coefficients on MissGuidance and ISHoldings are insignificant. Lastly, we control for a firm’s analyst coverage (LnNumAnalyst), suggested by Fang and Hope (2021) as another candidate for uncertainty (or “task complexity”). Column (6) shows that firms with greater analyst coverage exhibit a lower percentage of Specific forecasts.

5 Conclusion

In this study, we identify a novel behavioral bias of sell-side analysts – overprecision. Overprecision has been identified as the most durable and least understood form of overconfidence (Moore et al. 2016). Using sell-side analysts’ forecasts to examine the properties of overprecise forecasts, we document that forecasts that have more digits than the mode are less accurate, despite their form precision. We find that these forecasts exhibit a Dunning-Kruger effect, i.e., that incompetent (proxied by inexperience) analysts over-rely on their financial model output when producing overprecise forecasts. Lastly, we show that the stock market appears to recognize that these overprecise forecasts are less informative.

An emerging literature incorporates psychology research into financial decision-making by investors and intermediaries (Hilary and Menzly 2006; Hribar and McInnis 2012; Hirshleifer et al. 2018). Similarly, we explore individuals’ overprecision in the context of sell-side analyst forecasts. By doing so, we challenge the conventional wisdom in the finance and accounting literatures that numbers that appear less precise imply higher uncertainty and less information-processing and, therefore, lower accuracy (e.g., Bradley et al. 2004; Herrmann and Thomas 2005; Kuo et al. 2015). The results of our study add to those findings and deepen our understanding of the link between the form and the substance of earnings forecasts. By examining the relationship between forecast specificity and forecast accuracy, we are able to provide direct evidence on the existence, causes, and consequences of overprecision in the financial market. This evidence validates the implicit assumption in existing studies that agents can be subject to overprecision (Daniel et al. 1998; Odean 1998; Barber and Odean 2000; Adebambo and Yan 2018).

We conclude our study with a discussion of potential future research. An individual’s overprecision bias, like other behavioral biases, emerges when judgments are made under uncertainty (Tversky and Kahneman 1974). The capital market is close to a natural laboratory for such a scenario. Aside from our setting of analyst forecasts, there are other relevant contexts, such as the issuance of management forecasts, investors determining bidding prices for equity offerings, and contractual parties setting targets/benchmarks/covenants that include the necessary ingredients to explore the overprecision bias (uncertainty, judgment manifested in numerical output, and sufficient peer observations as benchmarks). We believe that future research exploiting these contexts would bring valuable evidence to advance our understanding of individuals’ overprecision bias in the capital market.

Appendices

Appendix 1 Illustrations of Specific forecasts

In this appendix, we illustrate our definition of SpecificMode using examples of original analyst forecast data retrieved from I/B/E/S. We discuss two companies – Monsanto Company in Illustration 1 and Wynn Resorts in Illustration 2. Both examples explain our definition of SpecificMode. The main distinction between the two examples is that analyst forecasts for Wynn Resorts also include forecasts that end with less digits than the mode number of digits after the decimal point. Both examples are included in our empirical sample.

1.1 Illustration 1 Monsanto Company

Monsanto Company (ticker: MNO) is a company in the industry of agricultural chemicals. The company was founded in 2000 and is based in Saint Louis, Missouri. It provides agricultural products such as corn, soybean, and vegetable seeds.

Table 12 shows original analyst forecasts for the annual earnings per share (EPS) of Monsanto Company during the fiscal year ending on August 31, 2015. The table presents information on the company name, the fiscal year end date, the forecast date, the analyst ID in I/B/E/S, the original forecast value (Forecast), the realized EPS value (Actual EPS), and the value of the indicator for Specific forecasts (SpecificMode). For each analyst following the company, we retain the analyst’s latest forecast within the [−365, −30] window before the fiscal year end date. We sort these observations by analyst forecast value.

Out of the 20 forecasts (provided by the 20 analysts), 17 end with two digits after the decimal point and three end with three digits after the decimal point (issued by analysts 110,142, 155,176, and 5469). The mode number of digits of analyst forecasts for Monsanto’s 2015 EPS therefore equals two. The three forecasts ending with three digits after the decimal point are identified as Specific forecasts and have SpecificMode = 1.

Table 12 Analyst forecasts for Monsanto Company

1.2 Illustration 2 Wynn Resorts

Wynn Resorts Ltd. (ticker: WYNN) is a company in the industry of Resorts & Casinos. The company was founded in 2002 and is based in Las Vegas, Nevada. It owns and operates high end hotels and casino resorts.

Table 13 shows original analyst forecasts for the annual earnings per share (EPS) of Wynn Resorts during the fiscal year ending on December 31, 2015. Information presented in the table is similar to that in Table 12 for the Monsanto Company. We again sort observations by analyst forecast value.

Out of the 20 forecasts, 14 end with two digits after the decimal point. The mode number of digits after the decimal point therefore equals two. There are three forecasts ending with three digits after the decimal point, issued by analysts 118,417, 86,522, and 132,128. These three forecasts are identified as Specific forecasts and have SpecificMode = 1. The remaining three forecasts have less digits after the decimal point, compared with the mode (analysts 81,050, 109,229, and 124,805). In our main analyses, these three rounded forecasts are pooled with the 14 forecasts ending with two digits after the decimal point, and benchmarked against our Specific forecasts.

Table 13 Analyst forecasts for Wynn Resorts

Appendix 2

Table 14 Variable definitions