Management Earnings Forecasts and Subsequent Price Formation
SOMNATH DAS College of Business Administration University of Illinois-Chicago Chicago, IL 60607 Email:
[email protected] KYONGHEE KIM College of Business Administration University of Illinois-Chicago Chicago, IL 60607 Email:
[email protected] SUKESH PATRO College of Business Administration Kansas State University Manhattan, KS 66506 Email:
[email protected]
This version: October 2007
This paper has benefited from the comments and suggestions of Anand Desai, Harry Evans, Mei Feng, Donald Moser, Steven Orcutt, Ram Ramakrishnan, Lenny Soffer, Srini Sankaraguruswamy, Yoonseok Zang and seminar participants at Kansas State University, Singapore Management University, University of Illinois at Chicago and University of Pittsburgh. An earlier version of the paper was titled “Investor Conservatism and Long term Price Response to Management Earnings Forecasts”.
Management Earnings Forecasts and Subsequent Price Formation
ABSTRACT This paper examines price formation subsequent to management forecasts of quarterly earnings. Consistent with prior studies, we find that the short-term market reaction to bad news forecasts is much larger than that to good news forecasts. Examining returns in the post-guidance period, we find a significant upward drift for both good and bad news forecasts. The asymmetry in the initial market response and the subsequent upward drift in stock prices are consistent with a reversal of an initial overreaction to managers’ bad news forecasts and a continuation of an initial under-reaction to managers’ good news forecasts. This interpretation is supported by a negative (positive) relationship between the immediate market reaction and the post-guidance drift in the conservative (optimistic) group. The postguidance drift in turn is systematically associated with the market’s reaction to actual earnings announcements - it attenuates investors’ response to surprises in actual earnings. Examination of the robustness of the post-guidance drift shows that it persists after controlling for risk-based corrections, calendar time clustering of announcements and stock price drifts associated with prior or subsequent actual earnings announcements. Trading strategies exploiting the post-guidance drift suggest the existence of economically significant trading profits, net of trading costs. Evidence of a positive relationship between the magnitude of the drift and changes in trading volumes and costs suggests information asymmetry during the event period as a potential determinant of the drift. However, the magnitude of the drift and the profitability of the trading strategies remain a puzzle.
JEL Classification Code: G14, G30, M41 Keywords: Management forecasts, earnings guidance, stock price drift, overreaction, under-reaction.
Most prior studies of the price response to management forecasts examine returns in a short window, usually three days, surrounding the forecast announcement (Patell 1976, Penman 1980, Ajinkya and Gift 1984, Waymire 1984, Jennings 1987). In this paper we examine the price formation process subsequent to the issuance of a management forecast. We do so by examining the stock price response to management forecasts over the horizon of the forecast, i.e., from the time of management forecast announcement to the time of actual earnings announcement. Our investigation of the price formation over the forecast horizon is motivated by the extensive literature that documents drift in stock prices subsequent to releases of earnings-related information. The literature on post earnings announcement drift (PEAD) documents the tendency of stock prices to drift in the direction of the recent earnings surprise for several weeks following the announcement of earnings (e.g., Bernard and Thomas 1989, 1990, Livnat and Mendenhall 2006). Similarly, there is evidence of stock price drift associated with analyst revisions of earnings forecasts (e.g., Chan, Jegadeesh and Lakonishok 1996, Gleason and Lee 2003). These and other related studies suggest that such drifts are a consequence of a delayed response to the information revealed in the earnings-related event. Issuances of management forecasts of earnings,1 unlike actual earnings announcements, are choice decisions made by managers. Also, these forecasts are often aimed at guiding analysts’ forecasts to a beatable target (Cotter, Tuna, and Wysocki 2006). Given the discretionary nature of management forecasts and the horizon of the forecast, it is reasonable to expect that the immediate market reaction at the time of the forecast announcement is incomplete and that it may not be complete until the resolution of uncertainty at the time of the earnings announcement. Therefore, in this paper, we investigate the stock price consequences of management forecasts over this longer forecast horizon rather than in the short window surrounding the forecast announcement. Consistent with the above argument, evidence in McNichols (1989) suggests that the market takes a significantly long time to react fully to managers’ forecasts. While a detailed examination of the postforecast stock returns is not the primary focus in McNichols (1989), using managerial forecasts of annual 1
We use management forecasts and earnings guidance interchangeably in this paper.
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earnings from June 1979 through December 1983 she documents a significant upward and downward drift in stock prices, lasting over a hundred days after optimistic and pessimistic forecasts, respectively. Liu and Ziebart (1999), also using data from the 1980s, document a significant negative relation between the price response to bad-news forecasts of quarterly earnings and the price response to subsequent announcement of actual quarterly earnings, suggesting a correction to an initial overreaction to bad news forecasts.2 Overall, in contrast to the evidence in McNichols (1989), Liu and Ziebart (1999) conclude that they find no evidence of the “post-announcement-drift” phenomenon for management forecasts. Thus, the evidence to date on the nature and magnitude of the long-term response to management forecasts remains inconclusive. Concurrent with this paper, Ng, Tuna and Verdi (2007) examine the profitability of the twelvemonth abnormal return beginning with the month subsequent to annual and quarterly forecasts of earnings. Similar to PEAD, they find a positive price pattern subsequent to good news forecasts, but unlike PEAD they do not find any drift pattern for bad news forecasts. They then examine the effect of disclosure quality on these abnormal returns and find that superior forecast quality mitigates the observed patterns in abnormal returns. In contrast to Ng et al. (2007), our primary focus is on the price formation process subsequent to the announcement of a management forecast and how, if at all, the forecast helps unravel the uncertainty associated with actual earnings. Hence, we focus on examining daily returns in three periods – at the time the forecast is made, in the period subsequent to the forecast and leading up to the announcement of actual earnings, and at the time actual earnings are announced. First, we examine the immediate market response to forecasts. Prior literature has documented an asymmetry in the market reaction to good news and bad news forecasts. Second, we examine whether the asymmetric price response in the immediate short term can predict the price changes in the period subsequent to the forecast and preceding the announcement of actual earnings. Such an association would be expected if there is an immediate overreaction or under-reaction that is subsequently corrected as uncertainty about earnings information 2
However, they do not find such a reversal for the good news forecasts.
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resolves. Finally, to the extent the price changes subsequent to the forecast incorporate information about forthcoming actual earnings; the price reaction at the time of the actual earnings should be negatively correlated with the post-guidance price changes. By examining each of these stages we provide evidence on the process by which investors incorporate information contained in managers’ forecasts into prices. Given our focus on the price-formation process, we only use management forecasts of quarterly earnings because the literature on management forecasts suggests that interim or quarterly management forecasts are relatively more informative than annual forecasts (Pownall, Wasley, and Waymire, 1993, Anilowski, Feng, and Skinner 2007). Our use of quarterly forecasts is also motivated by their prevalence and the increasing propensity on the part of managers to use them as a tool for ‘walking down’ analysts to a beatable forecast (Cotter et al. 2006). As Anilowski et al. (2007) note, “….majority of earnings guidance is quarterly….” (p. 3). Our focus on daily returns is motivated by existing evidence on drift patterns in stock prices associated with earnings related information which suggests that such drift patterns typically are discernible over shorter intervals using daily returns (Bernard and Thomas 1989, 1990; Livnat and Mendenhall 2006). Furthermore, Liu and Ziebart (1999) using daily returns document a reversal in the drift patterns associated with management forecasts of quarterly earnings. Similarly, McNichols (1989), also using daily returns, documents stock price drifts associated with management forecasts of annual earnings. More fundamentally, our focus on daily returns helps to mitigate potential bad-model problems (Fama 1998) arising from the implicit assumption that expected returns remain stationary over longer horizons (Brown, Harlow and Tinic 1988, Lehmann 1990, Trueman, Wong and Zhang 2003) and hence is more likely to provide a stronger test of any associated inefficiencies. In addition to providing evidence on the price-formation process, our focus on a more detailed examination of the forecast horizon enables us to distinguish between PEAD and the post-guidance drift in stock prices. This is important given the inter-temporal mingling of management forecasts of quarterly earnings and announcements of actual quarterly earnings. Similarly, our approach also enables us to examine the effect that anticipation of earnings might have on the post-guidance drift. Thus, this paper 3
complements both the approach and the findings of Ng et al. (2007) by focusing upon daily returns over the forecast horizon. Consistent with prior literature (Jennings 1987, Rogers and Stocken 2005), we find that the abnormal return around the announcement of management forecasts is positively associated with the information in the forecasts, and that the immediate market reaction to bad news forecasts is much larger than that to good news forecasts. In addition, we provide new evidence that investors discount extremes of both conservative and optimistic3 disagreements as evidenced by a strong non-linear relation between the degree of disagreement and the cumulative abnormal return in the three days surrounding the forecast. Examining returns in the post-forecast period from two days after the announcement (day +2) until thirty days after (day +30), we document a significant upward drift in stock prices for both conservative and optimistic forecasts. In the optimistic group of forecasts the market-adjusted cumulative abnormal return (CAR) from day +2 to day +30 has a median (mean) value of 4.58% (5.45%). This is larger than the median (mean) CAR of 2.78% (3.90%) for the conservative group measured over the same holding period. The relatively smaller immediate reaction to the guidance in the optimistic group and the subsequent large upward drift suggests that the market initially under-reacts to managers’ good news forecasts about firms’ earnings. For the conservative group, the sharp downward price reaction at the time of guidance combined with the subsequent upward correction in the post-guidance period suggests that the market initially overreacts to managers’ negative news forecasts about firms’ earnings. The asymmetry in the long-term price reaction to managers’ forecasts in the two groups is in contrast to the symmetric drift patterns associated with PEAD (e.g., Bernard and Thomas 1989, 1990; Lamont and Frazzini, 2007). While the asymmetry we document is similar to the asymmetry documented in Ng et al.
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Given prior evidence that in the short run investors react differently to bad news and good news forecasts (e.g. Jennings 1987, McNichols 1989), we condition our analysis on the nature and magnitude of the disagreement between managers’ forecasts and analysts’ estimates. Forecasts are classified as optimistic when management forecasts are higher than outstanding analysts’ forecasts and conservative when they are lower than outstanding analysts’ forecasts. Management forecasts that are close to analyst consensus estimates are classified as neutral and serve as a control group.
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(2007), there is an important difference – they find no drift (neither upward nor downward) for the bad news group, while we document an upward drift following the immediate reaction. To examine the robustness of this post-guidance drift we investigate competing factors that could potentially explain away the observed drift. First, based on Brown and Warner (1985) and Fama (1998) we examine the effect of calendar time clustering of management forecasts. Second, we account for changes in risk premiums around the announcement of the forecast. Third, we test whether the observed drift is merely reflecting the PEAD from the prior quarter earnings announcement or, alternatively, the anticipation of the current quarter’s actual earnings announcement. Our results show that the postguidance drift in stock price persists even after controlling for these factors. Examination of the cross-section of post-guidance drift shows that the upward drift is negatively (positively) associated with the market’s initial reaction to conservative (optimistic) forecasts. This suggests that stock prices continue to adjust to information in managers forecasts in the post-guidance period. Also, this result is consistent with a correction of an initial overreaction to bad news and an initial under-reaction to good news in managers’ forecasts. To examine the final stage in the price-formation process we relate the post-guidance drift to the market response to the surprise in earnings when they are announced, a time when uncertainty about both the information content and credibility of the management forecast is resolved. We find that the magnitude of price drift in the post-guidance period is negatively associated with the market reaction at the time of actual earnings announcement for both the conservative and optimistic groups of forecasts. This result is consistent with the explanation that the greater the earnings information impounded into prices in the post-guidance period the smaller the market reaction when actual earnings are announced. Our examination of the price formation process over the horizon of management forecasts suggests that the process by which earnings information is impounded into stock prices differs depending on the type of earnings information. Specifically, investors initially under-react to good news forecasts, while they overreact to bad news forecasts. However, in both cases, investors correct their initial
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mispricing over time in the period subsequent to the management forecasts and eventually at the time of the earnings announcement, when all uncertainty is resolved. We also examine the profitability of a trading strategy that exploits these predictable price patterns. Using Fama-French/ Carhart four-factor regressions, we find that excess returns for portfolios of conservative (optimistic) forecasts yield annualized alphas of 10.25% (9.20%), net of estimated bid-ask spreads, for holding periods of 30 trading days. This result supports the economic significance of the phenomenon in that the post-guidance drift in stock prices during our sample period had the potential to yield profitable trading strategies. While the economically significant portfolio profitability for the optimistic group is consistent with Ng et al. (2007), our result for the conservative group is inconsistent with their result of insignificant portfolio profits from bad news forecasts (analogous to our conservative group). To examine the source of this difference we replicate their estimation of alpha using a twelve month horizon (Ng et al, 2007, Table 5 Panel B). Our finding of a positive but insignificant annualized alpha of 2.32% for the conservative group and a positive and significant annualized alpha of 10.32% for the optimistic group is qualitatively similar to their results. The similarity in behavior over the twelvemonth horizon between the two samples suggests that the horizon over which the post-guidance drift is measured can significantly alter the patterns in the post-guidance drift. Having confirmed the robustness and economic significance of the price drift in the postguidance period, we explore potential explanations for the drift patterns. Under-reaction may arise due to investor conservatism (e.g., Daniel, Hirshleifer and Subrahmanyam, 1998 and Barberis, Shleifer and Vishny, 1998) and behavioral biases, such as the attribution or representativeness biases, could explain overreaction (Daniel, Hirshleifer and Subrahmanyam, 1998). Alternatively, these patterns may exist as a result of limits to arbitrage (Shleifer and Vishny 1997) or structural uncertainty in information (Brav and Heaton 2002).4 Examining changes in the trading activity and trading costs surrounding the forecast date,
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Structural uncertainty argues that investors may rationally put less weight when they perceive that there is greater uncertainty associated with the new information. Conservatism, on the other hand, implies that investors are not fully rational in the sense that they may not fully update their beliefs although they may act in accordance with Bayesian belief revision.
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we find that larger increases in information asymmetry5 during the event period are associated with larger magnitudes of post-guidance drift. Similarly, greater increases in trading volume, quoted spread, and effective spreads are associated with greater magnitudes of post-guidance drift. Broadly, this suggests that the documented drift patterns are at least partially consistent with Brav and Heaton’s (2002) structural uncertainty hypothesis.6 The rest of the paper is organized as follows. Section I describe the sample selection criteria and research design, and provide descriptive statistics. Section II presents evidence on the price formation process in the post-guidance period. In Section III we carry out additional tests that examine the economic significance of the post-guidance drift and explore potential causes for its existence. Section IV concludes the paper. I. Sample Selection, Research Design, and Descriptive Statistics A. Data and Sample Selection We begin with an initial sample of management forecasts of quarterly earnings from the First Call database for the period 1995-2004. We limit the sample to management forecasts of earnings per share and to those forecasts where a point estimate is provided or a point estimate can be calculated as the midpoint of provided lower and upper bounds. This yields an initial sample of 34,252 management forecasts of quarterly earnings. We eliminate management forecasts issued prior to the end of the previous fiscal quarter since the subsequent announcement of actual earnings may contaminate the returns associated with the forecast. From this sample we pick the last guidance issued within each quarter. We impose this requirement because multiple guidance issuances for the same quarter could have overlapping periods thereby reducing the independence of stock return observations. We pick the last rather than the first (or an intermediate) guidance because managers cannot “undo” the effects of this guidance. For this sample of
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As measured by the information asymmetry component of the spread using the Lin, Sanger and Booth (1995) model. 6 It is beyond the scope of this paper to distinguish among the possible alternative explanations for the post-guidance drift.
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forecasts we obtain corresponding analyst consensus estimates from the I/B/E/S database. Specifically, we obtain the most recent analyst consensus estimate prior to guidance for each of the management forecasts. We also require that an analyst consensus estimate be available after the guidance is issued. This allows us to examine the market reaction to the announcement of actual earnings in the traditional framework, i.e., using analyst forecast errors.7 These restrictions reduce the sample to 13,414 quarterly management forecasts. Consistent with prior research, we next eliminate observations for which the forecast is issued within two days of the announcement of the actual earnings figures for any fiscal period. Application of this restriction results in a further significant reduction from 13,414 to 8,386 observations. This is the baseline sample used in the analyses reported in the paper.8 B. Research Design In the absence of prior beliefs on the duration over which the price response to the forecast may persist, we conservatively conjecture that the stock price effects of managers’ forecasts could potentially last as long as uncertainty about the guidance provided by managers remains unresolved, i.e., till actual earnings are announced. Therefore, we examine the price formation process in three windows – around the announcement of the manager’s forecast, in the period subsequent to the forecast and around the announcement of actual earnings. Figure 1 provides a time line of the sequence of events that we examine. For simplicity we assume that all days are trading days. Managers forecast earnings (CIG for company issued guidance) on September 1st in the context of an outstanding analyst consensus estimate (Pre_AF for prior analyst forecast). We measure the degree of disagreement between managers and analysts as the difference between CIG and Pre_AF. We classify the sample into three groups based on this degree of disagreement. If CIG exceeds the existing consensus forecast (Pre_AF) by more than $0.01, the guidance is classified as optimistic (relative to analysts). If CIG is smaller than Pre_AF by more than $0.01, the guidance is 7
This requirement also ensures that the guidance date and the actual earnings announcement date are separated by at least a few days, thereby reducing the effect that anticipation of announcement of actual earnings may have on the post-guidance drift. 8 We note, however, that the qualitative nature of the results does not change when all the analyses are replicated using the less restricted sample of 13,414 observations. These results are available from the authors on request.
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classified as conservative (relative to analysts). All other cases are classified as neutral. In regression analyses we scale this difference by the stock price at the end of the previous fiscal quarter (henceforth DIFF). Because the $0.01 cut-off is somewhat arbitrary we replicate our tests using two other classification rules. First, we classify all issuances with CIG (Pre_AF) greater than Pre_AF (CIG) as optimistic (conservative) thus avoiding the arbitrary choice of $0.01. Second, we classify all issuances with CIG (Pre_AF) greater than Pre_AF (CIG) by at least 10% as optimistic (conservative). In both cases there is no significant change in the results. One advantage of our classification schemes is that they do not depend on the contemporary or prior distribution of DIFF. The prior distribution of DIFF is noisy due to sample heterogeneity from year to year. The contemporary distribution of DIFF is inappropriate because it induces a look-ahead bias (Foster, Olsen, and Shevlin, 1984) We begin our analysis by first examining the immediate market response to the issuance of guidance (CIGAR for CIG Abnormal Return). We do this in a three-day window around September 1st. We then examine the magnitude and duration of the post-guidance movement in stock prices using two approaches. First, similar to McNichols (1989) we use secular windows of varying length beginning on day +2 (September 3rd in our example), where day 0 is the day of the announcement. We specifically exclude day +1 because a large fraction of management forecasts are made after the main markets are closed.9 The windows we use extend from day +5 to day +30. Second, similar to Soffer, Thiagarajan, and Walther (2000) we calculate abnormal returns over the entire post-guidance period. This period lasts from day +2 till 5 trading days prior to actual earnings are announced, i.e., October 20th. We calculate DRIFT as the average daily abnormal return during the post-guidance period. We stop at 5 trading days prior to October 20th (therefore, the last day of returns used to calculate DRIFT is October 15th) because returns after this stopping point may be significantly affected by anticipation of actual earnings as well as potential leakage of earnings numbers. By analyzing returns from the announcement of the forecast to the announcement of actual earnings we aim to examine the full market reaction to managers’ forecasts. Finally, we examine the immediate market response to the announcement of actual earnings (ACTAR for 9
This causes the impact of the forecast to be absorbed in the stock’s CRSP return on the next trading day i.e. day+1.
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Actual (Earnings) Abnormal Return). Our approach is similar to the literature that examines market reactions to earnings announcements with the added feature that we examine ACTAR in the context of the post-guidance drift. FIGURE 1 ABOUT HERE C. Descriptive statistics Table I, Panel A shows the year-wise frequency of conservative, optimistic and neutral forecasts. The fraction of neutral forecasts increases from 10% in 1995 to around 20% in the early 2000’s. This is consistent with the increased accuracy of analyst forecasts and also the increasing efforts of managers to meet or beat analyst forecasts. The increase in the fraction of neutral estimates seems to come from a decrease in the fraction of conservative forecasts from over 55% in the late 1990’s to around 45% towards the end of the sample period. The fraction of optimistic forecasts remains fairly stable at around 30% till 2001 and thereafter increases to around 35%. Table I, Panel B reports descriptive statistics of the firms that constitute the conservative, neutral and optimistic groups. On average, there are around 19 days (Horizon) between the date of the last consensus analyst forecast prior to management forecasts (CIG) and the date of CIG announcement. This period is short enough to suggest that managers are not voluntarily disclosing earnings information solely to correct stale information in outdated analyst forecasts. Rather, it is more likely that there is some disagreement between managers and analysts on the earnings outlook. The mean (median) book value of assets is $4,869 million ($594 million). Firms in the optimistic and neutral groups are similar in size (mean size of $6,022 and $6,028 million respectively) and are much larger than firm in the conservative group (mean size of $3,828 million). Supporting managers’ optimistic outlook on earnings, firms in the optimistic group perform better and have bigger increases in performance (as measured by return on book equity) than those in the conservative group. TABLE I ABOUT HERE Table I, Panel C reports descriptive statistics of managers’ dollar disagreement with analysts (DIFF_RAW) for each of the conservative, neutral and optimistic groups. The mean of (median) 10
DIFF_RAW is -$0.113 (-$0.070) for the conservative group. The mean (median) for the optimistic group of $0.191 ($0.110) suggests that these forecasters have larger dollar disagreements than the conservative forecasters. After deflating DIFF_RAW by the price at the end of the previous fiscal quarter we see that the deflated difference (DIFF) is more comparable in magnitude across the optimistic (mean =0.008, median = 0.005) and conservative (mean =-0.011, median = -0.005) groups. II. Post-Guidance Price Formation A. Immediate Market Reaction to Conservative versus Optimistic Forecasts We begin our examination of the price formation process with an analysis of the short-term market reaction to managers’ earnings guidance, CIGAR, measured as the cumulative abnormal return over the three-day window surrounding the issuance of guidance (using the CRSP value-weighted index as the benchmark). Table II, Panel A shows that CIGAR is statistically different from zero for both the conservative group and the optimistic group (p 0.01 where Pre_AF is the last consensus analyst forecast prior to CIG issuance
Total 276
8,386
Panel B – Descriptive statistics Horizon is the number of days between the last consensus analyst forecast prior to CIG issuance (Pre_AF) and the date of CIG issuance. Assets ($ mil) is book value of total assets. MTB Ratio is the ratio of market value of assets to book value of assets. Pre_Std is standard deviation of analysts’ forecasts corresponding to Pre_AF. Pre_Num is the number of analyst estimates corresponding to Pre_AF. ROEq-1 is the prior quarter return on equity measured as income before extraordinary items divided by book value of equity. ∆ROEq-1 indicates changes in ROE and is measured as [ROE for quarter q-1 of year t – ROE for quarter q-1 in year t-1]. Full sample
Conservative
Neutral
Optimistic
Horizon Assets MTB Ratio Pre_Std Pre_Num
Mean 19.245 4869 1.369 0.020 7.346
Median 20.000 594 0.961 0.010 6.000
Mean 18.881 3828 1.325 0.024 6.575
Median 20.000 464 0.931 0.010 5.000
Mean 18.873 6028 1.429 0.013 8.735
Median 20.000 710 1.023 0.010 7.000
Mean 20.011 6022 1.412 0.018 7.956
Median 21.000 843 0.984 0.010 6.000
ROE q-1
0.004
0.028
-0.002
0.025
0.001
0.026
0.016
0.034
∆ROE q-1
0.015
0.000
0.015
-0.001
0.019
0.000
0.011
0.001
N
8386
4415
32
1263
2708
Table I – (contd.) Panel C – Managers’ Guidance on Earnings per Share CIG is managements’ forecast of quarterly earnings per share. DIFF_RAW is the difference between CIG and the last consensus analyst forecasts prior to CIG issuance. DIFF is DIFF_RAW deflated by price at the end of the previous fiscal quarter. Conservative CIG
Neutral CIG
Optimistic CIG
DIFF_RAW Mean Median Standard deviation
-0.113 -0.070 0.160
-0.001 0.000 0.005
0.191 0.110 0.261
DIFF Mean Median Standard deviation
-0.011 -0.005 0.019
0.000 0.000 0.000
0.008 0.005 0.010
N
4,415
1,263
2708
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Table II Panel A – Univariate Tests of the Immediate Market Reaction to Management Guidance CIGAR is the cumulative abnormal returns for the 3 day window around issuance of management earnings forecasts: -1, 0 and +1 where 0 is CIG issuance day. The CRSP value weighted index serves as the benchmark return. P-values for tests of the median CIGAR are reported using the signed rank test.
Conservative CIG
Neutral CIG
Optimistic CIG
Mean Median Standard deviation
-0.115 -0.083 0.154
-0.007 -0.002 0.112
0.014 0.015 0.121
N
4,415
1,263
2708
Mean test: (H0: CIGAR = 0)
t = -49.66 (p < 0.00)
t = -2.27 (p < 0.02)
t = 6.18 (p < 0.00)
Median test (H0: CIGAR = 0)
p < 0.00
p=0.45
p
