The Marginal Coefficient: A New Approach for Identifying Observation Level Sensitivities

Transcription

1 The Marginal Coefficient: A New Approach for Identifying Observation Level Sensitivities Presented by Dr Michael D Kimbrough Associate Professor University of Maryland #2017/18-10 The views and opinions expressed in this working paper are those of the author(s) and not necessarily those of the School of Accountancy, Singapore Management University.

2 The marginal coefficient: a new approach for identifying observation level sensitivities Ivy Ruyun Feng Oliver Kim Michael D. Kimbrough a University of Maryland Robert H. Smith School of Business September 2017 Abstract Many constructs of interest to accounting researchers relate to the sensitivity of one variable to another. Examples include the earnings response coefficient, earnings persistence, and earnings timeliness. These constructs are typically measured at the sample level using ordinary least squares (OLS). Because OLS quantifies sensitivity-based constructs at the sample level, it does not allow researchers to explore how variation in sensitivities within a sample affects economic outcomes of interest, which is a common research objective. Researchers have adopted a number of approaches to estimate observation level sensitivities. Each approach captures a portion of the total variation in sensitivities by capitalizing on one or more known sources of variation. We introduce an approach for estimating observation level sensitivities that reflects the variation from known sources captured by traditional approaches as well as the variation from unknown sources and randomness. This approach measures observation level sensitivities based on each observation s contribution to the sample coefficient. Using the earnings response coefficient context, we show that sensitivity estimates based on our approach subsume and capture substantially more of the within-sample variation in sensitivities than those from existing approaches, are associated in predicted ways with previously documented determinants, and explain economic outcomes. a Corresponding author : mkimbrough@rhsmith.umd.edu. We thank John Heater, Donald Lee, Alina Lerman, Frank Murphy, Jake Thomas, Jim Wahlen, and workshop participants at American University, University of Maryland, and Yale University for helpful comments and suggestions.

3 The marginal coefficient: a new approach for identifying observation level sensitivities Abstract Many constructs of interest to accounting researchers relate to the sensitivity of one variable to another. Examples include the earnings response coefficient, earnings persistence, and earnings timeliness. These constructs are typically measured at the sample level using ordinary least squares (OLS). Because OLS quantifies sensitivity-based constructs at the sample level, it does not allow researchers to explore how variation in sensitivities within a sample affects economic outcomes of interest, which is a common research objective. Researchers have adopted a number of approaches to estimate observation level sensitivities. Each approach captures a portion of the total variation in sensitivities by capitalizing on one or more known sources of variation. We introduce an approach for estimating observation level sensitivities that reflects the variation from known sources captured by traditional approaches as well as the variation from unknown sources and randomness. This approach measures observation level sensitivities based on each observation s contribution to the sample coefficient. Using the earnings response coefficient context, we show that sensitivity estimates based on our approach subsume and capture substantially more of the within-sample variation in sensitivities than those from existing approaches, are associated in predicted ways with previously documented determinants, and explain economic outcomes.

4 1. Introduction Many constructs commonly used in the accounting research, such as size or analyst following, can be directly measured at the observation level. Other constructs relate to the sensitivity of one variable to another and can only be measured at the sample level using ordinary least squares (OLS). Examples of sensitivity-based constructs include: (1) the earnings response coefficient, which captures the sensitivity of announcement returns to earnings surprises, (2) earnings persistence, which captures the sensitivity of future earnings to current earnings, and (3) earnings timeliness, which captures the sensitivity of earnings to returns. Because OLS quantifies sensitivity-based constructs at the sample level, it does not allow researchers to explore how variation in sensitivities within a sample affects economic outcomes of interest, which is a common research objective. Researchers have adopted a number of approaches to measure observation level sensitivities, including: (1) time-series estimation by firm, (2) cross-sectional estimation by time period, and (3) two-stage estimation. Each approach captures a portion of the total variation in sensitivities by capitalizing on one or more known sources of variation. In this paper, we develop a new approach for estimating observation level sensitivities that reflects the variation in sensitivities from known sources captured by traditional approaches as well as the variation from unknown sources and randomness. We compare our approach to existing approaches in terms of the ability to capture variation in sensitivities within a sample. We also document the ability of observation level sensitivities based on our approach to predict economic outcomes. As mentioned previously, there are several common approaches in the accounting literature to construct observation-specific sensitivities. Each approach exploits one or more known sources of variation in sensitivities. One approach is time series estimation for individual 1

5 firms on a rolling basis, which exploits variation in sensitivities across firms (i.e. cross-sectional variation). Another approach is to estimate cross-sectional regressions each period for observations that fall in a meaningful cross-sectional unit, such as industry (Sloan 1996). One can then assign the same coefficient to all firms in the same cross-sectional unit each period. This approach exploits variation in sensitivities across time (i.e. time series variation). Finally, researchers have adopted a two-stage approach where the first stage is an estimation of a fully specified regression that incorporates interactions for known sources of variation in the coefficient of interest (e.g. firm characteristics), and the second stage is a linear projection of the first-stage coefficients on those sources. An example of this approach is the C-Score introduced by Khan and Watts (2009). This approach allows for both cross-sectional variation by incorporating known determinants and intertemporal variation through periodic estimation. In this paper, we propose an approach that combines and expands upon the merits of each of the approaches described above. Our approach relies on the basic intuition that the OLS coefficient represents an average association between the dependent variable and the independent variable. As with any sample average, some observations within the sample are above average (i.e. contribute positively to the sample average) and some are below average (i.e. contribute negatively to the sample average). Our approach measures the contribution of each observation to the overall average represented by the sample coefficient. We call the measure of each observation s contribution to the overall coefficient the marginal coefficient. We verify two mathematical properties of the marginal coefficient analytically and using simulated and archival data. First, we show that the weighted average of individual marginal coefficients is equal to the sample coefficient. Second, we show that classifying observations into those with marginal 2

6 coefficients that are greater than (less than) the sample coefficient correctly identifies those observations that contribute positively (negatively) to the overall coefficient. The marginal coefficient can be used to sort observations into groups based on their individual sensitivities. The marginal coefficient can be further transformed into a point estimate of observation level sensitivity by performing separate OLS regressions for groups based on the marginal coefficient and then assigning the resulting coefficient estimate for a given group to all observations in that group. Conceptually, the observation level sensitivities based on the marginal coefficient capture known sources of variation in sensitivities documented by prior studies, systematic sources of variation not yet documented by researchers, as well as randomness. By contrast, estimates based on the existing approaches described above capture only known sources of variation. Thus, we expect the sensitivity estimates based on the marginal coefficient to be more comprehensive than estimates based on existing approaches. That is, we expect sensitivity estimates based on the marginal coefficient to subsume and capture more of the variation in sensitivities within a sample than alternative sensitivity estimates. Moreover, we expect both the known and unknown sources of variation captured under the marginal coefficient to predict economic outcomes. We test these predictions using archival data on earnings response coefficients (ERC). We choose the ERC setting because the earnings-return relationship is a research area of longstanding interest. Previous literature has documented various cross-sectional and intertemporal sources of variation in ERC. The well-developed empirical literature allows us to assess whether our measure captures previously documented sources of variation in ERCs. We rank firms into deciles based on the marginal coefficient and then perform separate ERC regressions for each rank. We show that the decile-level ERC estimates increase monotonically. Moreover, the spread 3

7 in ERCs between the top and bottom deciles is consistently greater than the spread in ERCs based on sorting by sensitivities generated by the previously described alternative approaches. Thus, as expected, the marginal coefficient subsumes and captures more of the variation in ERCs than any of the existing approaches. We further find that the sensitivity estimates based on existing approaches collectively provide very little information over the marginal coefficient alone in capturing variation in observation level sensitivities. Thus, information in the sensitivity estimates based on our approach subsumes the information in the sensitivity estimates based on existing approaches. In addition, we find that the marginal coefficient-based earnings response coefficient (MERC) is associated in predicted ways with previously documented ERC determinants. This finding provides direct evidence that our measure captures known sources of variation. However, previously documented ERC determinants capture only a small portion of the overall variation in MERC. Our evidence that the marginal coefficient captures the most within-sample variation in ERC and that MERC captures but is not fully explained by known ERC determinants indicates that the marginal coefficient captures both known and unknown sources of variation in contrast to the existing approaches that only reflect known sources of variation. Finally, we examine the usefulness of our approach for estimating outcomes. First, we replicate Lennox and Park s (2006) finding of a positive relation between a firm s historical ERC (based on time-series estimation) and management s issuance of earnings forecasts. We document the same relationship using a firm s average quarterly MERC over the time-series estimation period. However, we provide the additional insight that more recent quarterly MERCs have a different impact on the likelihood of management forecast issuance than MERCs from more remote periods. This insight is not apparent when using the time-series estimate of ERC, 4

8 which constrains the ERC to be constant over the entire time series estimation period due to the intertemporal stationarity assumption. We also examine the relationship between ERCs and implied volatility. Prior studies have found that firms earning announcements convey new information that resolve investors uncertainty (Ball and Kothari 1991). We expect a more informative earnings number to reduce the uncertainty to a greater extent. We use MERC to measure the informativeness of the earnings number. We use declines in implied volatility around firms earnings announcements to assess the resolution of uncertainty (Hann, Kim and Zheng 2017; Patell and Wolfson 1979, 1981; Truong et al., 2012, Isakov and Perignon, 2001). Consistent with our expectation, we find that higher MERCs are associated with more pronounced decreases in implied volatility. In addition, MERC better explains subsequent changes in implied volatility than observation level ERC estimates based on alternative approaches. Finally, we find that both the portion of the MERC that can be explained by known determinants and the portion that is orthogonal to known determinants have predictive power for the change in implied volatility. The fact that the portion of MERC that is orthogonal to known components is also useful in predicting economic outcomes illustrates the importance of capturing unknown sources of variation when estimating observation level sensitivities. Thus, a key advantage of the marginal coefficient approach over existing approaches is that it captures such variation. In supplemental analysis, we compare the multicollinearity associated with two alternative approaches to documenting sensitivity determinants. Multicollinearity results in inflated standard errors of the regression coefficients, which increases the difficulty of rejecting a false null hypothesis of no effect for each explanatory variable. Based on variance inflation factors, we find that multicollinearity is significant in the traditional approach of incorporating 5

9 additional interaction and main effect terms in a regression of abnormal returns on earnings surprises. However, multicollinearity is nonexistent in the alternative approach of regressing marginal coefficient-based sensitivity estimates against known and hypothesized determinants. This result indicates that, in addition to allowing researchers to examine the association between sensitivity-based constructs and economic outcomes, the marginal coefficient methodology facilitates the identification of previously undocumented determinants by mitigating multicollinearity issues. The approach we introduce meets researchers demand for a reliable way to measure observation-specific sensitivities in a variety of accounting contexts. By incorporating variation from all sources, our approach leads to more comprehensive estimates of observation level sensitivities than existing approaches. Thus, researchers can use marginal coefficient-based sensitivity estimates to better predict economic outcomes and to identify previously undocumented sensitivity determinants. Although we focus on the ERC in our empirical tests, the approach can be applied to any sensitivity-based construct. The rest of this paper proceeds as follows. Section 2 describes the demand for and existing approaches to estimating observation level sensitivities. Section 3 discusses and validates the new approach we propose for measuring observation level sensitivities and develops empirical predictions. Section 4 presents the results of testing the empirical predictions. Section 5 concludes. 6

10 2. Demand for and Existing Approaches to Estimating Observation Level Sensitivities 2.1. Demand for Measures of Observation Level Sensitivities Many important constructs in the accounting literature such as the earnings response coefficient, earnings persistence, and earnings timeliness refer to the sensitivity of one variable to another. Researchers estimate sensitivity constructs using ordinary least squares (OLS) regressions. Such regressions yield one sample coefficient that represents the average sensitivity. While some research questions may only require documenting average sensitivities, researchers are often interested in how variation in sensitivities within a sample affects some other outcomes of interest. For example, Lennox and Park (2006) ask whether variation in the earnings response coefficient affects the likelihood that a firm will issue a management forecast. Francis et al. (2004) ask whether variation in various sensitivity-based earnings attributes such as earnings timeliness and earnings persistence affect cost of capital. Donovan, Frankel and Martin (2015) explore how variation in conditional conservatism (incremental earnings timeliness for bad news) affects monitoring and bankruptcy outcomes. One sample coefficient from OLS estimation is not sufficient to address questions like these because researchers need to exploit variation in sensitivities within a sample. Researchers have adopted a number of approaches to estimate observation level sensitivities. These approaches include time series estimation, cross-sectional estimation and two-stage estimation. Each of these approaches exploits a different source of the total variation Description of Existing Approaches to Measuring Observation Level Sensitivities As illustrated in Figure 1, within-sample variation in sensitivities is comprised of crosssectional variation, intertemporal variation, and randomness. In time-series estimation, 7

11 researchers derive firm-period sensitivity estimates based on separate OLS regressions by firm for a fixed number of periods on a rolling basis. This approach exploits cross-sectional variation by allowing sensitivity estimates to vary across individual firms. In cross-sectional estimation, researchers derive observation level sensitivity estimates based on separate OLS regressions by cross-sectional unit (such as industry) each period. Because regressions are performed periodically, this approach primarily exploits time-series variation. Also, because regressions are performed separately by cross-sectional unit, this approach exploits variation across the cross-sectional unit. In the two-stage estimation, researchers derive observation level sensitivities by first estimating a fully specified regression each period that incorporates interactions for known sources of variation in the coefficient of interest (e.g. firm characteristics) and then applying these coefficients to the known sources of variation. As in the cross-sectional regression approach, this approach exploits variation across time because regressions are estimated periodically. This approach also exploits cross-sectional variation to a greater extent than the cross-sectional regression approach by incorporating all known sources of cross-sectional variation in the first stage model. Each of the above approaches has its own merits and captures a different portion of the total variation in sensitivities based on known sources. Researchers choose the best approach based on their assumptions about the relative importance of different sources of variation. While each approach is based on exploiting one or more known sources of variation, variation in sensitivities can also be due to unknown systematic sources as well as randomness. Variation from unknown systematic sources arises because researchers may not have documented all 8

12 sources of variation in sensitivities. Random variation, although not predictable ex-ante, can also be important since such variation may be an important driver of economic outcomes of interest. 1 In this paper, we seek to develop a single approach for estimating observation level sensitivities that captures all the known sources of variation captured by existing approaches as well as the variation in sensitivities due to unknown systematic sources and randomness. By combining variation from all these sources into a single measure, our approach offers researchers a comprehensive estimate of observation level sensitivities. Researchers can use this estimate to better how variation in sensitivities within a sample affects other outcomes of interest and to better identify previously undiscovered sources of variation. 3. A New Approach to Measuring Observation Level Sensitivities 3.1 A new way to look at the OLS coefficient estimator The OLS estimator of the slope of a classical linear regression model yy = bbbb + εε is: bb = (xx xx) 1 xx yy 2 OLS estimation uses the information from all observations to identify the line that minimizes the sum of the squared residuals. An equivalent but less common way to think about the OLS coefficient is as a composite of elementary estimators of the parameter. This concept was first discovered under the theorem of Jacobi (1841). Since the OLS coefficient captures the average slope between y and x for the sample population, the elementary building blocks of the sample-wide OLS coefficient are the 1 The random variation should be incorporated if one simply wants to know the effect of higher or lower sensitivity on another economic outcome of interest (i.e. one simply wants to use the variation as an explanatory variable in another context). 2 X is a N*2 matrix. X can be written as 1 xx 1 1 xx NN NN 2 9

13 pairwise slopes between two observations (Berman 1988; Sen 1968; Cressie and Keightley 1981; Herzberg 1984) 3. One can obtain the OLS coefficient by averaging and aggregating the information in all the pairwise slopes between any two observations. Specifically, as documented by Gelman and Park (2008), the slope coefficient for a simple linear model yy = bbbb + εε can also be expressed as bb = ( xx ii xx jj 2 ii jj bb iiii ) xx ii xx jj 2. 4 (1) ii jj Where bb iiii is estimated as the slope between observation i and observation j. bb iiii = (yy jj yy ii ) (xx jj xx ii ) The weight applied to the slope between two observations to arrive at the composite slope is proportional to the squared difference between the two observations along the x-axis. Jacobi (1841) proves that this weight guarantees that the resulting weighted average equals the OLS estimator and, consequently, minimizes the squared errors. Intuitively, giving more weight to the pairwise slopes of observations that are further away from each other makes sense because points that are further away from each other help to more precisely define the linear relationship. The pairwise slope (i.e. bb iiii ) varies across observations. If the association between x and y is uniform for the sample then the expected value of each pairwise slope is constant that is, each observation is expected to fall on the same line. Under this situation, the variation in bb iiii (i.e. deviations from a common line) represents pure randomness. However, the expected value of each pairwise slope is not expected to be constant when there is systematic variation. In this case, systematic variation as well as randomness will drive variation in pairwise slopes. 3 To the best of our knowledge: although this interesting insight has been documented for a long time, we are among the first to explore its empirical applications. 4 We provide the analytical proof in a supplemental appendix, which is available upon request. 10

14 3.2 Mathematical Derivation of the marginal coefficient Section 3.1 relates the insight from prior statistics literature that the pairwise slopes are the elementary building blocks in the estimation of the relationship between y and x. How one uses this insight depends on one s assumptions and objectives. If one assumes that the population is homogenous and that there is just one overall slope that describes the population then the focus is limited to deciding how to aggregate the individual pairwise slopes to arrive at the best estimate of the overall slope. Equation (1) shows how OLS estimation implicitly weights the pairwise slopes to arrive at the slope coefficient. 5 On the other hand, if one allows for the possibility that there is variation in the population then one need not be constrained to aggregating the pairwise slopes to one overall slope estimate. One can aggregate the pairwise slopes at a lower unit of analysis (e.g. at the observation level). If there is truly just one slope that describes the population then the expected sensitivity at the observation level is simply equal to the sample wide sensitivity. In this case, variation in sensitivities at the observation level will be purely random. However, prior empirical research that uses interaction terms or subsample analyses demonstrates the existence of systematic sources of variation in sensitivities. When there is such systematic variation then systematic variation as well as randomness will drive variation in sensitivities at the observation level. Since we are interested in exploring observation level sensitivity, we aggregate the pairwise slopes to the observation level to obtain the marginal coefficients of individual observations. Specifically, we use the same OLS weighting scheme as in equation (1) to construct the marginal coefficient of an observation as the weighted average of all pairwise slopes associated with that observation. Specifically, bb ii can be expressed as: 5 Alternatively, Theil-Sen estimator (Sen 1968; Theil 1950) uses the median pairwise slope as an estimate of the overall slope, which is another way to aggregate the information in the individual pairwise slopes. 11

15 bb ii = ( xx ii xx jj 2 jj bb iiii ) xx ii xx jj 2 jj (2) The marginal coefficient (bi) identifies each observation s contribution to the sample regression coefficient (b ). The overall sample coefficient can be expressed as the weighted average of all marginal coefficients where the weight is the square distance between observation i and all other observations along the x-axis divided by the total distance between any two observations within the sample, consistent with equation (1). An observation contributes positively (negatively) to the sample average if its marginal coefficient is greater than (less than) the sample coefficient. Stated alternatively, if bb ii > b then b > b ( ıı) and if bb ii < b then b < b ( ıı), where b ( ıı) refers to the OLS coefficient for the sample excluding observation i. 6 Thus, at a minimum, the marginal coefficient can be used to identify those observations with above average and below average sensitivities. How well the marginal coefficient allows observations to be grouped into finer partitions based on their sensitivities is ultimately an empirical question that we explore. 3.3 Verification of the Basic Mathematical Properties of the Marginal Coefficient The marginal coefficient derived in section 3.2 has two basic mathematical properties. First, the weighted average of all marginal coefficients in a sample equals the sample coefficient. Second, subsamples comprised of observations with marginal coefficients that are greater than (less than) the sample coefficient will have regression slopes that are greater than (less than) the 6 We include the proof in a supplemental appendix, which is available upon request. 12

16 sample coefficient. We verify these two properties empirically using the following three samples 7 : (1) A simulated sample where each observation falls into one of two deterministic (i.e. no error term) linear data generating process with slopes of 0.1 or 0.3 and where the sample slope coefficient is 0.2. Figure 2a contains a plot of this sample. (2) A simulated sample generated from a completely random data generating process where the slope coefficient is zero. Figure 2b contains a plot of this sample. (3) An archival sample of actual returns and earnings data used to estimate earnings response coefficients. The sample contains firm-quarters from 2000 to Figure 2c contains a plot of this sample. Table 1 verifies that for all three samples, the weighted average of the marginal coefficient equals to the sample coefficient. For each of the three samples, we test the second property by first estimating the marginal coefficient and then grouping observations with above and below average coefficients into subsamples. We then estimate separate OLS regressions for each subsample. We plot the regression lines for the full sample and for each of the subsamples. Figures 2a 2c presents the plots for each of the three samples. Each plot illustrates that the marginal coefficient allows us to identify which observations make positive or negative contributions to the sample coefficient. 3.4 Transforming marginal coefficients into observation level sensitivity estimates The marginal coefficient allows observations to be grouped based on the contribution of each observation to the sample coefficient, but it does not yield a direct point estimate of each 7 We also verify these properties analytically. We include the proofs in a supplemental appendix, which is available upon request. 13

17 observation s sensitivity. A researcher who is interested in the ordinal relationship between observation level sensitivities and other outcomes can simply use the group rank (e.g. decile or quintile rank) based on the marginal coefficient. A researcher can further derive a point estimate of each observation s sensitivity by performing separate OLS regressions for each group rank and then assigning the resulting rank-level coefficient to all observations in that rank. 3.5 Conceptual Discussion and Empirical Predictions Our approach builds on the basic intuition that OLS coefficient that corresponds to sensitivity constructs represents a sample average. Like any sample average, each observation within a sample contributes to the overall average. Those observations that contribute positively (negatively) to the overall average are above (below) average. Our approach measures each observation s contribution to the sample average, which permits a classification of individual observations into those with above average or below average sensitivities. Our identification of each observation s contribution to the sample average captures all sources of variation in observation level sensitivities. 8 Thus, at a minimum, sensitivity estimates based on marginal coefficients should be associated with previously documented determinants. Thus, we test the following empirical prediction (EP). EP1: Observation level sensitivity estimates based on the marginal coefficient capture known sources of variation. 8 Our approach is analogous to a professor s ability to refer to an individual student s actual exam score to determine his relative exam performance without needing to know anything about the student s background that might have contributed to his performance. By contrast, existing approaches are analogous to a professor using predicted exam scores (based on known determinants) to determine his relative performance. Since predicted scores exclude unknown determinants as well as random variation, they provide a noisier basis for determining an individual student s relative performance. 14

18 As discussed previously, sensitivity estimates based on the marginal coefficient capture all potential sources of variation in sensitivities. By contrast, existing approaches capture only a portion of the total potential variation in sensitivities. Thus, sensitivity estimates based on the marginal coefficient are more comprehensive than those from existing approaches and should capture more of the variation in underlying sensitivities within a sample. In addition, sensitivity estimates based on the marginal coefficient should subsume any information captured by the sensitivity estimates based on existing approaches. We, therefore, test the following empirical prediction. EP2a: Observation level sensitivity estimates based on the marginal coefficient capture more of the within-sample variation in sensitivities than existing approaches. EP2b: Observation level sensitivity estimates based on existing approaches do not have incremental information over observation level sensitivity estimates based on the marginal coefficient about within-sample variation in sensitivities. Recall that the motivation for identifying observation level sensitivities is to predict economic outcomes. We expect sensitivities based on the marginal coefficient to be useful for this purpose because they capture all sources of variation in underlying sensitivities. Therefore, we test the following empirical prediction. EP3: Observation level sensitivity estimates based on the marginal coefficient explain economic outcomes. A key difference between our measure and existing approaches is that, in addition to known determinants, our approach incorporates unknown determinants and randomness, which existing approaches ignore. The incorporation of these additional sources of variation potentially contributes to the ability of sensitivity estimates based on the marginal coefficient to predict economic outcomes. For example, the portion of observation level sensitivities due to luck 15

19 may be just as important as the portion due to systematic factors. Therefore, we test the following empirical predictions. EP4a: Observation level sensitivity estimates based on the marginal coefficient better explain economic outcomes than observation level sensitivity estimates based on existing approaches. EP4b: The portion of observation level sensitivity estimates based on the marginal coefficient that is attributable to known sources of variation explains economic outcomes. EP4c: The portion of observation level sensitivity estimates based on the marginal coefficient that is orthogonal to known sources of variation explains economic outcomes. 4. Tests of Empirical Predictions 4.1. The Earnings Response Coefficient Context Although the approach we propose can be applied to any sensitivity-based construct, we choose to test our empirical predictions in the ERC context because the earnings-return relation is of long-standing academic interest. Considerable research has documented various crosssectional and intertemporal sources of variation in the ERC (Beaver, Lambert and Morse 1980; Collins and Kothari 1989; Kormendi and Lipe 1987; Easton and Zmijewski 1989). The welldeveloped empirical literature allows us to benchmark our findings with existing evidence Sample Selection We extract quarterly data from Compustat for firm-quarters from 2000 to We obtain quarterly earnings forecasts from the I/B/E/S historical database. We measure the earnings surprise as the difference between the actual earnings per share before extraordinary items and the most recent consensus analyst forecast, deflated by quarter-end stock price 9. We measure the abnormal return as the three-day market adjusted return around firms earnings announcements. 9 The earnings surprise constructed using the seasoned random walk model yields similar empirical results. 16

20 We delete firm quarters with missing data for abnormal return or earnings surprise, and firm quarters with negative total asset. We further delete firm quarters with price per share less than $1. These filters result in a sample of 194,736 firm-quarters from 2000 to Table 2 provides descriptive statistics. Panel A presents the descriptive statistics of variables used in testing EP1 and EP2. The mean and median earnings surprise and marketadjusted returns for sample firms are around 0, indicating that market s forecast of earnings is relatively accurate. Most firms in our sample are profit firms with small earnings surprises. Around 25% of the observations have consecutive earnings increase. Panel B presents the descriptive statistics of variables used in testing EP3 and EP4. For the implied volatility test, firms implied volatilities based on options with 30-day duration decrease on average by after earnings announcements, consistent with the arrival of earnings information reducing market uncertainties. For the management forecast test, about 36.6% of firms issue management earnings forecast during the current quarter. We note that firms historical ERC based on time series estimation is much larger and more volatile than the MERC. The extreme estimate is a result of the limited numbers of observations used in each time series regression. Table 3 presents correlations. Panel A presents the correlation of variables used in testing EP1 and EP2. The market adjusted return is significantly positively associated with the earnings surprise. Panel B presents the correlation of variables used in testing EP3. Analyst following is higher and earnings volatility is lower in quarters when managers issue earnings forecasts. The issuance of management forecasts and the average historical ERC are also significantly positively correlated. By contrast, the correlation between the recent MERC and the issuance of earnings forecast is negative. 17

21 4.3. Test of EP1 For our test of EP1, we estimate the following model, which regresses the rank and continuous ERC estimates based on the marginal coefficient (hereafter referred to as MERCs) against previously documented determinants. MMMMMMMM tt = bb 0 + bb 1 LLLLLLLL tt + bb 2 SSSSSSSS tt bb 3 LLLLLLLLLL SSSSSSSSSSSSSSSS tt + bb 4 CCCCCCCCCCCCCCCCCCCCCC IIIIIIIIIIIIIIII tt + bb 5 VVVVVV tt + bb 6 BBBBBBBB tt + bb 7 GGGGGGGGGGGGGGGGGGGG tt + bb 8 BBBBBB tt + bb 9 PPPPPPPPPP EEEEEEEEEEEEEEEE RRRRRRRRRRRR tt + bb 10 LLLLLL tt + bb 11 SSSSSSSSSSSSSS IIIIIIIIII tt + bb 12 RRIISSSS FFFFFFFF RRRRRRRR tt + ee (1) See Appendix A for variable definitions and Appendix B for a summary of the previously documented ERC determinants. Table 4 reports the results of the determinant test. Column (1) uses the continuous ERC estimate as the independent variable. Column (2) uses the decile rank estimate as the independent variable. Panel A presents the analysis under pooled regression, and Panel B under Fama-Macbeth regression. We find that most signs are consistent with previous findings, except for the risk free rate and risk beta 10, which are not significant. The findings provide direct evidence that our measure captures variation in ERC from known sources documented by the previous literature. We note that the adjusted R 2 from the determinant test is relatively low (15.0% and 0.8% for the continuous and ranked MERCs, respectively), which indicates that a substantial portion of the within-sample variation in ERCs is not explained by known determinants. This finding indicates that the MERC also captures sources of variation not previously documented in the 10 The beta is transformed to 1/β, and the risk-free rate to 1/(1+r) 18

22 literature. In section 4.6, we test whether the portion of MERC that is orthogonal to known determinants is important for predicting economic outcomes Test of EP2 We test EP2 by first deriving firm-quarter ERC measures based on the marginal coefficient approach as well as each of the existing approaches: (1) rolling time series, (2) crosssectional, and (3) two-stage estimation (linear projection). For the rolling time series approach, we generate a time-series sensitivity estimation for each firm-quarter based on the preceding 16 quarters starting from year t-5 and ending in year t-1 (requiring a minimum of 5 observations). The coefficient from the firm-level regression serves as the firm-quarter estimate. For cross-sectional estimation, we estimate the following equation quarterly for each twodigit SIC code. rrrrrr tt = bb 0 + bb 1tt EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt + ee (2) The coefficient b1 from estimating equation (2) is the same for all firms for a given industry quarter. 11 For the two-stage linear projection approach, we estimate the following equation each quarter. rrrrrr tt = bb 0 + bb 1tt EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt + bb iiii EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt xx iiii + xx iiii + ee (3) We then apply the coefficient estimates on the interaction terms to the firm-quarter realizations of xi to obtain our firm-quarter estimate under this approach. 11 Barth et al. (2016) use an extended cross-sectional approach. That is, instead of just using one cross-sectional unit (industry membership), they use two (industry membership and membership in portfolios based on residuals from industry level regressions). This expanded approach will still not be comprehensive if there are more than just these two determinants. In untabulated analysis, we find that observation level ERC estimates based on the Barth et al. (2016) methodology measure does not significantly outperform the basic cross-sectional approach based on industry membership alone. 19

23 EEEEEE iiii = bb 1tt + bb iiii xx iiii + ee xx iiii is the vector of determinants of ERC used in equation (1). For each approach, we rank observations into deciles each quarter based on the firmquarter ERC estimate. We then estimate separate regressions for each decile-quarter 12. Each quarter, we examine the relation between the decile rank and the decile-level ERC. If a measure captures variation in underlying sensitivities, then the decile-level ERC should increase monotonically across the decile ranks. A measure captures more of the underlying variation in ERC if the spread between the ERC for the top decile and the bottom decile is larger. We summarize the average for the quarterly decile-level ERCs as well as the average of the quarterly spreads between the ERCs for the top and bottom deciles. We also compare the amount of within-sample variation in ERCs captured by each approach by estimating the following regression each quarter for each approach. rrrrrr tt = bb 0 + bb 1 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt + bb 2 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEee tt rrrrrrrr tt + bb 3 rrrrrrrr tt + ee (4) rankt is the quarterly decile rank for the indicated observation level sensitivity. The bb 2 coefficient as well as R 2 from estimating equation (4) are alternative measures of the amount of variation in ERC captured by each approach. We examine whether regressions based on MERC have the biggest bb 2 and R 2 to provide further evidence on whether the marginal coefficient captures more variation in ERCs than any of the existing approaches. Table 5 reports the results of the comparative analyses. Panel A presents the average of 60 quarterly ERC estimates for each decile. The estimated ERCs increase monotonically across deciles under all approaches. However, the spread in the ERC based on sorting by the marginal coefficient is nearly twice the spread in ERCs based on sorting by estimated observation level ERCs based on the alternative approaches. 12 We also form quintile quarter estimates. The results are similar. 20

24 Panel B of Table 5 reports the average of the quarterly regression analyses of estimating equation (4). Both the average R 2 from the regressions and the coefficient on the interaction term between earnings surprise and rank are the greatest by a substantial margin under marginal coefficient approach, which is consistent with this approach yielding estimates that capture more of the variation in ERCs than any of the existing approaches. In fact, the R 2 from the marginal coefficient approach dominates the R 2 from the three other approaches in all 60 quarters. We also note from the results presented in Panel B of Table 5 that the average R 2 from estimating equation (4) using ranks based on the marginal coefficient only is 39.5% whereas the average R 2 from estimating an expanded version of equation (4) that includes ranks based on all four sensitivity estimates is 41.5%. Thus, the sensitivity estimates based on the existing approaches collectively provide very little information over the marginal coefficient alone in capturing variation in observation level sensitivities, providing preliminary support for EP2b. To formally test whether observation level sensitivity estimates based on existing approaches have incremental information over observation level sensitivity estimates based on the marginal coefficient, we estimate the following expanded version of equation (4). rrrrrr tt = bb 0 + bb 1 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt + bb 2 EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt rrrrrrrr tt + bb 3 rrrrrrrr tt + bb 2aa EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE tt rrrrrrrr_aaaaaa tt + bb 3aa rrrrrrrr_aaaaaa tt + ee (4a) rank_altt is the quarterly decile rank for the indicated observation level sensitivity under the indicated alternative approach. We use the marginal coefficient-based sensitivity estimates as the benchmark against which to assess the incremental information in sensitivity estimates under each of the alternative existing approaches. The incremental R 2 from estimating equation (4a) over estimating equation (4) measures the amount of incremental information contained in the alternative marginal sensitivity estimate. Panel C of Table 5 presents the results of this analysis. 21

25 The incremental adjusted R 2 for all alternative measures are all close to 0, which further supports the superior informativeness of the marginal coefficient. A noteworthy empirical finding from Panel A of Table 5 is that the group-level ERC for firms in the lowest two MERC deciles are negative, which is at odds with the general expectation of positive ERCs. To provide insight into this finding, we present evidence on the percentage of observations in each MERC decile where the abnormal return and the earnings surprise differ in sign in Panel C of Table 5. This percentage is 91% in the lowest MERC decile and decreases monotonically as the MERC deciles increase. The fact that nearly all of the observations in the lowest MERC decile have abnormal returns and earnings surprises that differ in sign is consistent with what a negative ERC implies. Although a negative ERC is unexpected, it is important to note that theory calls for a positive ERC on average, which does not necessarily mean a uniformly positive ERC. Thus, it is theoretically possible for negative ERCs to exist. Assuming no measurement error, a negative ERC indicates that the capital market believes the actual news conveyed to be opposite of what is reported, possibly because the market distrusts the earnings number 13. It is also possible that a negative ERC is due to measurement error as a result of improperly specified expectations or to other data errors. In these cases, it is important to note that our approach does not introduce distortions in the data but simply reflects them. Thus, as with any statistical approach, the data needs to be clean before being subjected to our procedure. 4.5 Test of EP3 - Historical ERCs and Management Forecast Issuance Lennox and Park (2006) show that historical ERCs based on time-series estimation are positively associated with a firm s decision to issue a forecast. They interpret this finding as 13 We test EP3 and EP4 by excluding firms with negative ERCs. The empirical results are similar. 22

26 indicating that a manager is more likely to issue an earnings forecast if investors perceive that the firm s earnings are more informative (i.e. if the firm s historical ERC is higher). Therefore, our first test of EP3 examines the ability of MERCs to predict the management forecast decision. Specifically, we estimate the following logit regression from Lennox and Park (2006). FFFFFFFFFFAAAAAA tt = bb 0 + bb 1 EEEEEE tt 1 + bb 2 XX tt + ee (5) ERCt-1 is the firm-quarter ERC estimate for quarter t-1 and Xt is the vector control of variables used by Lennox and Park (2006). See Appendix A for the list of control variables and variable definitions. Following Lennox and Park (2006), we estimate equation (5) using firm-quarter ERC estimates based on time-series regressions for each firm using observations from the 16 quarters ending in quarter t-1. We then estimate equation (5) using the average of the firm-quarter ERC estimates based on the marginal coefficient approach (MERC) for the same 16 quarters. In addition, we provide insights into the effects of constraining the ERC to be constant over the estimation period under the time-series approach used by Lennox and Park (2006). Because ERC estimates under the marginal coefficient approach do not have this constraint, we examine whether the relation between historical ERCs and the likelihood of forecast issuance differs for more recent vs older ERCs. Specifically, we estimate the following version of equation (5) that uses MERC from different periods within the rolling window as explanatory variables. FFFFFFFFFFFFFFFF tt = bb 0 + bb 1 MMMMMMMM_RRRRRRRRRRRR tt 1 + bb 2 XX tt + ee FFFFFFFFFFFFFFFF tt = bb 0 + bb 1 MMMMMMMM_RRRRRRRRRRRR tt 1 + bb 2 XX tt + ee (5a) (5b) MERC_RECENTt-1 is the marginal coefficient-based ERC estimate for quarter t-1 and MERC_REMOTEt-1 is the average marginal coefficient-based ERC estimate from quarter t-16 to 23

27 quarter t-2. We expect b1 to be different from b1 if the relationship between the dependent and independent variables varies across the rolling time-series estimation period. The fact that the marginal coefficient approach does not constrain the ERC to be constant over the estimation period also allows us to examine how periodic changes in the observation level ERC affects the likelihood of management forecast issuance. Therefore, we estimate the following regression. FFFFFFFFFFFFFFFF tt = bb 0 + bb 1 ΔΔΔΔΔΔΔΔΔΔ tt 1 + bb 2 XX tt + ee (5c) ΔMERCt-1 is the marginal coefficient-based ERC estimate for quarter t-1 minus the marginal coefficient-based ERC estimate for quarter t-2. Panel A of Table 6 reports the results of estimating equation (5). Consistent with the results from Lennox and Park (2006), the average relationship between a firm s historical ERC and the likelihood of management earnings forecast issuance is positive for the time-series ERC and for the ranked and continuous estimates of MERC. Panels B and C of Table 6 presents the results of estimating equations (5a) through (5c) using ranked and continuous versions of MERC, respectively. Consistent with the results from Lennox and Park (2006), the coefficient on MERC_REMOTEt-1 is significantly positive. However, the coefficient on both versions of MERC_RECENTt-1 is significantly negative, which indicates that more recent ERCs are associated with a lower probability of management forecast issuance. In addition, the coefficient on ΔMERCt-1 is significantly negative, which indicates that increases in the ERC are associated with a lower probability of management forecast issuance. The latter two insights highlights the improvements of our approach over the time-series approach by relaxing the intertemporal stationarity constraint. 24