Working Paper Series Faculty of Finance. No. 11. Fixed-effects in Empirical Accounting Research

Transcription

1 Working Paper Series Faculty of Finance No. Fixed-effects in Empirical Accounting Research Eli Amir, Jose M. Carabias, Jonathan Jona, Gilad Livne

2 Fixed-effects in Empirical Accounting Research Eli Amir Tel Aviv Universy and Cy Universy of London Jose M. Carabias London School of Economics and Polical Science Jonathan Jona Universy of Melbourne Gilad Livne Universy of Exeter Business School 2 July 205 Comments welcome Abstract The fixed-effects specification is often used in panel datasets as a way of dealing wh correlated omted variables. A review of recent accounting publications reveals that while researchers are generally aware of the need to include fixed-effects in empirical models when using panel datasets (firm-time observations), many chose to replace firm fixed-effects wh other form of fixed-effects, mainly industry fixed-effects. We examine the consequences of using different specifications of fixed-effects and show analytically and using simulations that this can lead to biased estimates and wrong inferences. To illustrate the importance of properly including firm fixed-effects, we reexamine commonly used regression models in the accounting lerature. We show how inferences change when fixed-effects are properly included. We call for a more careful consideration wh regard to the use of fixed-effects specification. Address correspondence to Eli Amir at eliamir@post.tau.ac.il. We would like to thank Yakov Amihud, Eti Einhorn, Joanne Horton, Fani Kalogirou and seminar participants at the Universy of Exeter Business School and Tel Aviv Universy for helpful comments.

3 Fixed-effects in Empirical Accounting Research. Introduction Many empirical studies analyze panel datasets, where both cross-section and time-series observations are pooled together to obtain a larger and more powerful sample. The advantage of panel datasets is they allow the investigation of relations of interest both in the cross section and over time. However, examining panel data in accounting should carefully account for underlying statistical properties of the cross-section data, as well as of the time series. The challenge, however, is that such properties are not easily observed. Petersen (2009) argues that while researchers often use panel datasets, the coefficients standard errors are often wrongly estimated. Also, the methods used by researchers to correct for possible biases in standard errors vary widely, but are often wrong. To address this concern, he proposes clustering standard errors at the firm level, time dimension or both, when appropriate. However, he finds that clustering standard errors by both firm and time appears unnecessary in the finance applications he considers. Gow et al. (200) review and evaluate the procedures commonly used in estimating standard errors in accounting research and show that in the presence of both serial and cross-sectional dependence, existing methods often produce misspecified test statistics. Both Petersen (2009) and Gow et al. (200) provide important evidence on significant biases in standard errors and their implications for test statistics. However, both studies assume that the models estimated by researchers are well-specified and the coefficients themselves are not subject to the bias caused by correlated omted variables. Both studies focus on the problem of correlations among the error terms. We address a different problem than that addressed by Petersen (2009) and Gow et al. (200); namely the potential bias in the estimated coefficients

4 due to misspecified regression models. In particular, our study complements Petersen (2009) and Gow et al. (200) by investigating instead the effect of model misspecification (biased and inconsistent parameters) and s implications for statistical inference. Similar to Petersen (2009) and Gow et al. (200) we highlight the concern of incorrect statistical inferences. However, while Petersen (2009) and Gow et al. (200) consider bias in the denominator of the test statistic (standard errors), we consider the possible bias both in the numerator of the test statistic (the coefficient estimate) and s denominator (the standard error). To control for unobserved firm and time effects, researchers choose between two models. The first one, which has been widely used by accounting researchers, is the fixed-effects model. The maintained assumption under this model is that the unobserved firm and time effects are correlated wh the main explanatory variables. This is a reasonable assumption because financial information reflects firm characteristics, which are often time-invariant. An alternative model that is rarely used in accounting research, but is popular in other disciplines, when panel datasets are used, is the random-effects model. In this specification the unobserved firm and time effects are assumed uncorrelated wh the regression residuals. In datasets wh firms being the uns of analysis, the random-effect method estimates an unobserved effect that is drawn from an in i.i.d normal distribution, and is independent of the error term (Greene, 2003; Wooldridge, 2007). Consequently, the random-effects model consumes fewer degrees of freedom relative to the fixed-effects model. When the researcher believes that the model specification does not suffer from a correlated omted variable problem, then the random-effect model is the preferred specification, because will produce unbiased slope estimates and more efficient standard errors (Greene, 2003; Wooldridge, 2007). However, Mundlak (978) shows that the random-effects model is, in fact, a special case of the fixed-effects model. 2

5 in the presence of correlated omted variables, which are time-invariant, the fixed-effects model is preferred. Our study deals only wh the empirical consequences of using the fixed-effects model. The maintained assumption under the fixed-effects model is that both dimensions (firm and time) of the panel are correlated wh the main regressors. Hence the research should include both time and firm controls. Nevertheless, in many accounting studies researchers replace firm fixed-effects wh industry fixed-effects. This replacement could lead to biased and inconsistent estimates, and hence incorrect inferences, when firm fixed-effects are correlated wh the main variables of interest. To assess the significance of this problem in accounting research, we reviewed articles published in six accounting journals over the period : Journal of Accounting and Economics, Journal of Accounting Research, The Accounting Review, Review of Accounting Studies, Contemporary Accounting Research, and European Accounting Review. Out of,842 articles,,52 (62.5%) can be classified as empirical studies (see Table, Panel A). While many of these empirical articles use more than one empirical methodology, 933 articles (50.6%) use some form of pooled regressions (see Table, Panel B). Many of these "pooled" studies use some form of fixed-effects. The most common fixed-effects specification used in these regressions are time and industry. Surprisingly, only 4 articles out of the 927 (2.2%) use firm fixed-effects, 75 of which in the recent three years. It seems that many researchers prefer using industry instead of firm fixed-effects. One plausible reason for researchers pervasive use of industry fixed-effects instead of firm fixed-effects is the belief that industry fixed-effects are sufficient and would not lead to incorrect inferences. That is, the researcher believes that whin-industry variation of the 3

6 variable of interest is negligible. Another reason may be that one or more of the explanatory variables are time-invariant which obviates the need to control for firm fixed-effects. A third possible explanation is that when empirical findings are not robust to the inclusion of firm fixedeffects the researcher may not know which set of results is more credible and opts to report results rather than no-results. The purpose of this study is to examine the implications of model misspecification due to omted unobserved fixed-effects in accounting settings. We start by analytically identifying the reasons why failing to control for firm fixed-effects could lead to wrong inferences. These include the correlation between the omted fixed-effects and the included explanatory variables, using incorrect degrees of freedom and biased estimate of the included coefficients standard error. Next, we use simulations of a simple model in which the main variable of interest is moderately correlated wh the unobserved firm fixed-effect. Even wh a relatively moderate correlation, we find that replacing firm fixed-effects wh industry fixed-effects, could lead to substantially wrong inferences. This is because the inclusion of industry fixed-effects does not fully remove cross-sectional correlations between the omted firm fixed-effects and the independent variables. These simulations also reveal that the distribution of the underlying coefficient of interest measured from OLS regressions wh industry and time fixed-effects do not overlap wh the distribution of this coefficient when measured from OLS regressions featuring both firm and time fixed-effects. This disconnect in the distributions indicates the severy of the inference problem that could result from failing to properly control for these fixed-effects. Specifically, omting fixed-effects results in a much higher rate of rejection of the null (a Type I error). However, the severy of the problem diminishes as the number of industries increases, or the number of firms per industry decreases. In general, however, 4

7 ignoring the whin-industry variation in the main explanatory variables results in biased estimated coefficients and wrong inferences. Furthermore, we show that the bias caused by omting firm fixed-effects decreases as the variance of the main regressor increases. The intuion is that firm fixed-effects become more crucial as the main regressor is more timeinvariant. We then examine the sensivy of commonly used capal markets-based accounting regressions to the omission of firm fixed-effects. First, we select Basu's (997) regression of the asymmetric response of earnings to good and bad news. We select this regression because the relation between earnings and stock returns may vary across firms and over time. Ball et al. (203) also argue that this is indeed the case because the relation between expected earnings and expected stock returns varies cross-sectionally. Another, not mutually exclusive, explanation is that the process of impounding economic news into earnings is firm-specific due to corporate governance mechanisms, internal controls, or relationships wh the audor, all of which are que stable over time. The crucial factor, however, is that the researcher eher cannot observe the underlying mechanisms, or cannot collect the full set of data, and hence needs to consider this in designing the research strategy. Wh similar motivation in mind, we also select the predictive regressions of accruals and cash flow components of earnings as explanatory variables for future earnings (Sloan, 996). In addion to the fixed-effects specifications, we also examine a number of alternative specifications, which have been used in the lerature. These include (i) first differencing of both the dependent and independent variables (full differencing), (ii) using first differences of only the dependent variable but not the independent variable, (iii) replacing the vectors of the dependent and independent variables by their means and estimating a single cross-sectional 5

8 regression, (iv) demeaning only the dependent variable, and (v) using the Fama and MacBeth (973) estimation approach, namely estimating the coefficient in question by averaging the coefficients and standard errors obtained from periodical cross-sectional regressions. Of these alternative specifications, only full differencing specification yields an overlapping distribution wh that of the correct specification (that includes both firm and time fixed-effects). However, this specification is somewhat less efficient, as the distribution of this alternative method is more dispersed than the distribution obtained from the correct regression specification. This highlights that incorrect inferences under the full differencing method are more likely than under the correct model. Our replication of the two studies, using firm fixed-effects, reveals that that the magnude and significance of the main coefficients of interest are que different from what is obtained under the original specifications. Although these replications still yield coefficients that are reliably different from zero, the differences in magnude suggest that the strength of the underlying economic phenomenon could be substantially weaker. For example, our fixed-effects estimate of the Basu s coefficient of timely loss recognion is 40-50% lower than the estimate obtained under the Basu (997) estimation procedure, depending on the estimation period. We also obtain qualatively similar results when we replicate Sloan s (996) study. Wh respect to Basu's (997) regression, Ball et al. (203) also acknowledge the need to control for fixed-effects. However, they use an alternative approach; they suggest that demeaning the dependent variable is equivalent to using firm fixed-effects. We show analytically that their procedure is not equivalent to using firm fixed-effects and that their approach is not free from bias. In fact, their approach tends to understate the magnude of the true coefficient. Furthermore, we show empirically that the magnude and the standard error of 6

9 the slope coefficient on negative stock returns in the Basu (997) model are smaller using the Ball et al. (203) approach than when firm and time fixed-effects are included. Nevertheless, the Basu (997) results hold, albe wh lower magnude and significance, which is broadly consistent wh Ball et al. (203). 2 It is important to note that including firm fixed-effects does not solve all correlated omted variable problems. In particular, does not solve the problem of omting a correlated variable that varies across time. However, in such cases including firm and time fixed-effects would not exacerbate the underlying problem. We therefore recommend the use of firm and time fixed-effects because there is no harm to doing so. Including firm and time fixed-effects when this is not necessary would nevertheless yield correct inferences, but excluding firm or time fixed-effects would lead to incorrect inferences when they are correlated wh the explanatory variables. If including firm fixed-effects is not feasible, then the second-best approach is first differencing of both the dependent variable and independent variables. While first differencing yields unbiased coefficients, is less efficient owing to loss of data (e.g., the first year in the panel is lost). Eher way, replacing firm wh industry fixed-effects is likely to yield biased coefficients. Section 2 summarizes the main insights from our analytical framework. Section 3 presents results of simulations aimed at quantifying the potential bias caused by correlated omted variables in panel datasets. Section 4 uses the empirical models employed by Basu (997) and Sloan (996) to demonstrate the effect of correlated omted variables on estimated coefficients. Section 5 provides concluding remarks. 2 Patatoukas and Thomas (205) argue that the conservatism coefficient in Ball et al. (203) is still upward biased. Our study focuses on the role of fixed effects in regression specifications and does not address the issues raised by Patatoukas and Thomas (205). 7

10 2. Analytical Derivations An appendix to this study provides detailed calculations, which are the basis for the following text. Here we only present the essential elements. 3 Let D = [D F, D T ] be a matrix of indicator (dummy) variables for firm fixed-effects (D F ) and time fixed-effects (D T ), and the unobserved fixed-effects be denoted by [, ] where the subscripts F and T denote firm and time fixed-effects, respectively. Then for the panel data the model becomes F T y X D () The variance of ε is denoted 2. Estimating this model will yield y XbDa e (2) where b and a are the coefficient estimates of β and α, respectively and e is the estimated regression residual. The vector of estimated slope coefficients b can be expressed (using partioned matrix conventions) as: b[ XM X] [ XM y] [ XX ] [ Xy ] (3) D D * * * * where ; X* MDX ; y* MDy; * MD. (4) M D I D[ DD] D M D can be thought of as a particular process of demeaning the independent and dependent variables. It is straightforward to show that b is unbiased (that is, Eb [ X] ). Furthermore, in the case where the fixed effects are uncorrelated wh X, employing (3) would generate an unbiased estimate of β, which will be the same as the estimate one would obtain from regressing y only on X. Crucially, the demeaning process captured in the matrix M is dependent on the researcher s choice of which fixed-effects to include. Specifically, assume that the researcher 3 The full analytical appendix is available from the authors upon request. 8

11 uses instead industry fixed-effects and time fixed-effects. Let D* [ D, D ] and * [, ] where D H stands for the matrix of industry dummies and industry fixed-effects. Then for the panel data the estimated model is H T H is the matrix of unobserved yx D * * (5) Importantly, the disturbance term μ involves firm fixed-effects that have not been removed by the inclusion of industry fixed-effects. Now, the estimated coefficient b, from this model can be expressed similarly to (3) H T b [ XM X] [ XM y] [ X X ] [ X y ] D* D* ** ** ** ** [ X X ] X M [ X X ] X ** ** ** D* ** ** ** ** (6) where M I D[ DD ] D; D* * * * * X** MD * X ; y** MD * y; M (7) ** D * Because the disturbance term still includes firm fixed-effects (since using industry fixedeffects has not fully controlled for cross-sectional variations), follows that E[ b X] [ X X ] X E[ X] ** ** ** ** [ X X ] X [ D ] ** ** ** F* F* (8) It follows from (8) that b is biased. The magnude of the bias is related to the covariance element, X **[ DF* F* ], and the scaling factor, [ X ** X** ], which can be thought of as a measure of the variabily in the undemeaned regressor X (see Figure 2). Hence, including industry fixedeffects instead of firm fixed-effects will affect inferences. Furthermore, t-statistics are also a function of the estimated coefficients standard error. Note also that and 2 Var b X X X (9) * * 2 Var b X X X (0) ** ** 9

12 2 Since is unknown to the researcher, has to be estimated from the data. Let T denote the number of years, F the number of firms and H the number of industries. The unbiased 2 estimator of is s 2, whereby: 2 e y XbDa y XbDa () 2 ( )( ) s FT ( F ) ( T ) K FT ( F ) ( T ) K Hence, the condional variance of the k-th coefficient b k is based on the diagonal element kk in the matrix X * X * as follows: 2 2 sˆ b Est. Var b k k X s X * X * kk. (2) Wh industry and time fixed-effects, the equivalent expressions are e ( y Xb Da )( y Xb Da ) Fx T ( H ) (T) K Fx T ( H ) (T) K (3) 2 2 * * * * s and sˆ Est. Var[ b X] s X X (4) 2 2 b k ** ** kk Observation 3 in the Appendix states that a t-test based on the (misspecified) regression coefficients b, t b k b k, and a t-test based on the correct regression coefficients b, 2 sb k t bk b k s 2 bk, are identical if and only if t b k b b 2 k[ X ** X** ] k k, k s e /( Fx T ( H ) ( T ) K ) 2 2 b k b 2 k[ X * X* ] k, k bk t e /( Fx T ( F ) ( T ) K ) s 2 2 bk bk (5) 0

13 Otherwise, the sign and significance of the t-test of b would be different than that of b. Note that wh non-zero correlation between the firm fixed-effects and the independent variables the two expressions for the t-statistics differ along four dimensions. The first is the difference point estimates b and b. The second difference relates to the specific kk-th element in the square bracket in the numerator. The third is the two sums of the regression squared residuals. The fourth and last is the difference in the degrees of freedom. Wh respect to the latter, note that as H approaches F the difference in the degrees of freedom becomes smaller in magnude. In the extreme, when H = F the two t-statistics will be identical, as firm and industry fixed-effects coincide. The Ball et al. (203) specification Ball et al. (203) argue that the Basu (997) model suffers from a correlated omted variable problem. They suggest the problem could be solved by using a fixed-effects specification. However, due to computation infeasibily, they suggest an alternative approach to the standard fixed-effects specification in which the dependent variable, earnings, is adjusted for average earnings (where averages are taken at a firm level over time). Demeaning only the dependent variable is not identical to including firm and time fixed-effects. 4 It is important to note that Ball et al. (203) employ several measures of returns. The important specification for us appears in Table 5, and for this table they use size and book-to-market adjusted returns. Importantly these returns are not zero mean. This would lead to biased coefficients. 4 Ball et al. (203) compute unexpected returns by subtracting from firm-specific returns the market return (or the return on the corresponding size and book-to-market portfolios). If the average unexpected return is zero, the full fixed-effects model and Ball et al. (203) approach yields unbiased estimators in the Basu (997) framework.

14 To see this, let y i denote the firm level average of y (that is, T yi y / T ). If only the dependent variable is demeaned, the model can be wrten as (assuming includes time fixedeffects): y y y X u (6) i * t u C y where f i i C f=i denotes the time-invariant firm fixed-effects that are not explicly modelled and hence are absorbed in the disturbance term. Using matrix algebra, we get 5 y D X u; * T M I D[ DD] D y M y M ( X u) X u * (7) The estimated vector of coefficients is therefore b [ X X ] Xy Taking expectations, and bearing in mind that the correct model is given by equation () above and that EC [ X] D : F F F Eb X E XX Xy X [ ] [[ ] ] [ XX] XE[ M( y yi ) X] [ XX] X( X MD T MDF F X MD T MDF F) [ XX] XM AX [ I [ XX] XM AX] (8) where M A is the firm-level average-creating matrix. That is, demeaning the dependent variable but whout demeaning the right-hand side variables leads to a biased coefficient. Only in the case where X + is zero-mean, we obtain unbiased coefficients since M A X + = 0. Importantly, the bias in (8) is unrelated to the correlation between the firm fixed effects and the other u C y. 5 Wh matrix algebra F 2

15 explanatory variables. Equation (8) suggests that the Ball et al s (203) coefficients are the true coefficients scaled down by the expression [ I [ XX ] XM X ]. A The Ball et al. s (203) estimates are therefore biased and they would differ from a full fixed-effect model. To stress, this is because including time and firm fixed-effects works to demean the panel variables in a different way. Under this specification all variables (dependent and independent) are transformed as follows: v v v v v * i t (9) where * v is the demeaned variable for firm i and year t, v is the original observation, v i is the average across all annual observations for firm i, v t is the average of the underlying variable v in year t across all firms, and v is the grand average of v. For the dependent variable this is different from the Ball et al. s (203) transformation v * v vi. This also affects inferences, because the standard error of the t-test is derived from the residuals sum of squares, which also differs between the two approaches. Addionally, a research design using the Ball et al. s (203) approach is likely to incorrectly calculate the degrees of freedom. The typical statistical software used in regression analysis will identify the transformed variable v v v as a * i single variable and hence instead of using FT T K degrees of freedom. 6 would use FT F T K degrees of freedom The Fama-MacBeth (973) approach According to Fama-MacBeth (973) regression coefficients are not calculated from a panel, but rather from periodical cross-sectional regressions. Specifically, the overall coefficient 6 Ball et al. (203) also estimate a model in which the dependent variable (earnings) is adjusted by subtracting the lagged dependent variable (but not the independent variable). It can be shown analytically that this approach also leads to biased estimates. For brevy we do not include the proof here, but simulate this specification later. 3

16 is the average coefficient over the T annual regressions and the standard error is derived from the distribution of the individual (periodical) coefficients. Because periodical regressions are not tooled to accommodate fixed firm or annual effects, this method is still prone to the same problem. Each annual underlying model can be wrten as: yt Xt t (20) where X is the matrix of annual explanatory variables. However, the individual disturbance term incorporates the fixed-effects implying Cf i Ct. The estimated coefficient for a single year t therefore can be expressed as in a simple OLS setup: b [ XX ] [ Xy ] [ XX ] X (2) t t t t t t t t t Averaging over T annual regressions yields the Fama-MacBeth coefficient: (22) T T [ ] FM T t T t t t t t b X X X The assumption of correlated fixed-effects implies that EC [ C X] 0. Hence we obtain that in expectation f t t T [ ] [ ] [ FM t t t t t ] T t Eb X XX EX X. (23) In other words, the Fama-MacBeth (973) procedure is prone to the omted fixed-effects problem under the assumption that the true model is as expressed in equation (). 3. Simulations To illustrate the potential bias in estimated slope coefficients that is caused by omting fixed-effects, we simulate a panel dataset according to the following specification: y a a a x e i t I 4

17 X Xi t and, e ~ N0, t 0 Xa i t Xa i i Xa i I 0 Xa 0 0 i t Xi, at, ai, ai ~ NΜ, Σ, Μ, Σ 0 Xa 0 0 i i 0 Xa 0 0 i I. The variable, is the dependent variable,,, are firm, time and industry fixed-effects, respectively, and is the independent variable. The values of the true parameters are,, {0.5,0.25,0, 0.25, 0.50} and. The null hypothesis is X i t X i i X i I. Since the bias in depends on the correlation between the omted fixed-effects and the regressor X, we impose five different levels of correlations between the fixed-effects and X: 0.50, 0.25, 0, and We expect a posive, zero and negative bias for the posive, zero and negative correlations, respectively, when the model oms the fixed-effects. 7 Using the above specification, we simulate a panel of 8,000 observations made of 0 periods, 20 industries and 40 firms per industry. We repeat this process 8,000 times, applying the following eight specifications: () Firm and time fixed-effects (FE) This model includes both firm and time fixed-effects. We expect this specification to yield an unbiased slope estimate (b = ). We label this model as FE. (2) Industry and time fixed-effects (IE) Here, we include time and industry fixed-effects, by replacing wh. This specification ignores whin-industry variations at the firm level and hence we expect the slope estimate to be biased. Notice that this specification approaches the full fixed-effects model as the number of firms per industry decreases. At 7 In a single variable setting, is easy to sign the bias; In a multivariate setting, however, the sign of the bias depends on the correlation matrix of the regressors (see the Appendix). 5

18 the extreme case where there is one firm per industry, this specification is identical to the full fixed-effects model. To show this, we conduct sensivy analysis where we sequentially increase the number of industries (and reduce the number of firms per industry) keeping the number of observations constant (see Table 3). We label this model as IE. (3) Misspecified model (MS) Here we om all fixed-effects; hence, we expect the estimated slope coefficient (β) to be biased to a greater degree than the previous model. However, given the small number of time periods (0) relative to the number of firms (800), the bias from omting time effects is expected to be small. This suation is similar to that in many studies that use archival data, as the number of firms is much larger than the number of periods. We label this model MS. (4) First differences model for both the dependent and independent variables (FD) In this model, we use first differences instead of current values (that is, current value minus the lagged one). This leads to an unbiased estimated slope coefficient, and one could om the fixed-effects from the model. However, the differencing process involves loss of information, as the first period in the panel is lost. We label this method FD. (5) Using first differences for the dependent variable only (LY) Here, the researcher firstdifferences the dependent variable but not the independent variables. We expect this specification to yield biased results. However, in this case the bias in induced not only by the covariance between the independent variable and the fixed-effects, but also by the exclusion of the variable βx - from the model. Hence, the bias in this case is also a function of the true β; when β is posive the bias is negative and when β is negative the bias is posive. We label this model LY (Lag Y). 6

19 (6) Using the time-series means of the independent and the dependent variables (MYX) Here we convert the panel dataset into a single cross-sectional regression by using the means of the independent and dependent variables as the main variables in the regression (see for instance, Aghion et al., 200). Using time-level means implies that the error term still includes firm fixed effects and hence the coefficient estimates are biased. (7) Demeaning the dependent variable Y (MY) - Similar to the LY case and motivated by Ball et al. (203), this specification only adjusts the Ys by subtracting the firm level averages. However, this specification is expected to yield biased estimates, as argued above. The bias is induced not by the covariance between the independent variable and the fixedeffects, but by the failure to demean the dependent variable at the firm level. Equation (8) suggests that coefficient estimate under this specification, is a scaled-down estimate of the true parameter. We therefore expect to be smaller than, regardless of the sign of the correlation between the fixed-effects and independent variable. We label this model as MY. (8) Fama-MacBeth (FM) We also estimate the model (whout fixed-effects) using the Fama-MacBeth (973) procedure; that is, estimating 0 periodical regressions and reporting the average slope coefficient. Equation (23) suggests the FM specification would yield biased estimate, under the true model that includes firm and time effects. We label this model as FM. For each of the eight specifications, we obtain 8,000 slope estimates. We also vary the magnude of the correlation between the fixed-effects and the regressor X. Table 2 reports the means of the estimated slope coefficients, standard errors, t-statistics of the distance from the true coefficient (β = ), and R 2 s for five levels of correlations: 0.5, 0.25, 0, -0.25, and We 7

20 also present the distribution of the estimated slope coefficients in Figure, using three different correlations: Figures a, b, and c present the distribution for a correlation of 0.5, 0, and -0.5, respectively. By construction, the full fixed-effects model (FE) yields an unbiased estimate (b = ) and high R 2 s for all levels of correlation. Also, the distribution of the bs is the tightest among all alternative distributions, as can be seen from the figures. The second model (industry and time effects, denoted IE) yields a posive bias (b =.25) when the correlation between the fixedeffects and the regressor is 0.5; zero bias when the correlation is zero and negative bias when the correlation is -0.5 (b = 0.75). We would incorrectly reject the null hypothesis that the slope is equal to in all cases, except for the case of zero correlation between the fixed-effects and the regressor. Also, the regression R 2 s are lower relatively to the FE model. When both firm and time effects are omted (MS), the pattern of bias is similar to that observed for model IE. That is, wh 20 industries and 800 different firms, controlling for industry fixed-effects performs equally poorly as the fully misspecified model. Moreover, from Figures a-c we note that the distributions of the slope coefficient under both the MS and IE are completely disjoint from the distribution of b under the FE specification. This suggests that is very unlikely that a slope estimate from these two specifications would fall whin a conventional confidence interval obtained under the full fixed-effect model. Using first differences for both the dependent and independent variables (FD) yields an unbiased slope estimate (b =.00) for all five correlations, but the estimate is less efficient as reflected by the larger standard errors, lower t-statistic, and lower adjusted R 2 and the larger tails seen in Figures a-c. Using a first difference only for the dependent variable (LY) yields a biased and less efficient estimate (b = 0.50; Adjusted R 2 = 0.09), regardless of the correlation 8

21 between the fixed-effects and regressor X. This is because the model is not sensive to the correlations between the fixed-effects and the independent variable. In model MYX, we use means of X and Y and estimate a cross-sectional regression. This model yields a large posive (negative) bias when the correlation between the fixed effects and the regressor is posive (negative). However, in the case of demeaning the dependent variable only (MY), the bias is negative regardless of the correlation between the fixed effects and the regressor. The reason the two models LY and MY lie to the left of the FE distribution is consistent wh the theoretical prediction derived in the previous section, which states that wh one explanatory variable the estimated coefficient will be smaller in magnude. Since β is posive here, they yield values smaller than. 8 Using the Fama-MacBeth (973) specification (FM) yields qualatively similar results as the misspecified model, wh even larger tails of the distribution. This is seen in Figures a-c where the FM parameter distribution obscures the parameter distribution of the MS model. Finally, when the correlation between the fixed-effects and the independent variable is zero, omting the fixed-effects is not expected to cause any bias. Indeed, the results show that the slope coefficients in models IE, MS, and FM are unbiased. 9 8 We also find from addional simulations (not tabulated) that when β = -, the LY and MY distributions are whin negative value range and lie to the right of the FE and FD distributions. This, again, is consistent wh lower magnude relative to the true value. 9 Note that the distributions of the slope coefficient depicted in the three charts of Figure do not correspond to the average standard errors reported in Table 2. For example, the distribution of β under the FM specification features larger tails than that of the FE model, although the standard error for the FM model (0.008) is smaller than that of the FE model (0.02). To demonstrate this issue, assume that we run 5 simulations of 8,000 observations each and obtain the following coefficient estimates for the FM model: -2, -, 0,, and 2. Also, suppose the output for the OLS regression is such that each coefficient is estimated wh standard error of. In contrast, for the FE model assume we obtain coefficients of -, -0.5, 0, 0.5 and, but each coefficient is estimated wh a standard error of 2. Then, if we were to chart these outcomes, the distribution of the FM (FE) will be wider (narrower), but the average standard error tabulated would be smaller (larger) for the FM (FE) specification. 9

22 The conclusion from this analysis is that omting firm fixed-effects will result in biased slope coefficients unless the fixed-effects are uncorrelated wh the independent regressor. Using firm fixed-effects is a safe approach in that will generate unbiased coefficients even when the data generating process does not contain unobserved correlated fixed-effects. An alternative approach would be to conduct the Hausman (978) test procedure to identify whether fixedeffects should be employed. However, since the Hausman test practically runs a fixed-effect model against a model wh no fixed-effects, we see no clear advantage over routinely including firm fixed-effects. 0 (Table 2 and Figure about here) Since industry fixed-effects is often used instead of firm fixed-effects, we examine the effect of firm distribution across industries on the results by changing the number of firms per industry while keeping the total number of observations constant at 8,000. We consider the following cases: (i) 0 industries wh 80 companies in each industry; (ii) 20 industries wh 40 companies in each industry (the baseline used above); (iii) 40 industries wh 20 companies in each industry; (iv) 60 industries wh 5 companies in each industry; and (v) 400 industries wh 2 companies in each industry. We expect the bias to decline as the number of industries increases. In the extreme case of one firm per industry, there will not be any bias, as this case coincides wh the full fixed-effects specification. Table 3 contains the results of this analysis. As the number of industries increases and the number of companies per industry decreases, the bias declines. However, the decline in the bias is rather small. For example, the bias is 24% when we use 40 industries and 20 companies per industry; declines to 7% when using 400 industries and two companies per industry. Hence, 0 Another advantage for using fixed effects specifications is that typical software output can report the fixed effects coefficients, if the researcher is interested in exploring or reporting these coefficients. 20

23 replacing firm fixed-effects wh industry fixed-effects and increasing the number of industries will not eliminate the bias in the coefficients, although using a finer industry classification might reduce the bias. For example, using the Fama-French 48-industry classification in estimating panel datasets is expected to yield less biased results than using the 2-industry classification. (Table 3 about here) The bias caused by omting firm and time fixed-effects depends also on the time-variance of the main regressor X. As the time-variance of the regressor X increases, the bias caused by omting the firm fixed-effects is expected to decline. To see this, we let the variance of X (i.e., of the parameter - t ) to decrease from 2.0 to 0.25 in intervals of As Figure 2 shows, the bias in the slope coefficient increases as the variance of X decreases. In other words, omting firm fixed-effects results in a larger bias as the main regressors become more time-invariant. In contrast, when the regressor X varies over time, omting firm fixed-effects is likely to result in ltle bias if at all. (Figure 2 about here) Overall we draw several conclusions from the simulation analysis: (i) Omting firm fixed-effects may generate biased estimates and overstated t-statistics, hence, wrong inferences. Replacing firm wh industry fixed-effects is not a valid approach as does not eliminate the coefficient bias, if the purpose is to control for unobserved correlated omted variables that are time-invariant. While increasing the number of industries is likely to reduce the bias, this approach is unlikely to eliminate the bias. (ii) Using means of the dependent and independent variables, or the approach taken by Ball et al. (203) are not equivalent to using firm fixed-effects. These methods yield biased estimates. Same holds for lagging just the dependent variable. 2

24 (iii) Using first differences (for both the dependent variable and independent variables) is a valid, but less efficient, estimation strategy. (iv) The coefficient distributions of several specifications may be so disjoint from the coefficient distribution of a fixed-effect model that respective confidence intervals may be entirely non-overlapping. That is, the chance of correct inference under the wrong specification may be que slim. 4. Implications for Empirical Accounting Research We now examine the effects of using different model specifications on the results of commonly used regression models in accounting research. We chose two regressions that have gained wide recognion: Basu's (997) model of asymmetric timeliness of earnings and Sloan s (996) differential persistence of accruals and cash flow components of earnings. 4. The Asymmetric Timeliness of Earnings Basu (997) The Basu (997) model highlights the differential reaction of earnings to good and bad news, where stock returns serve as a proxy for news. The regression is: X / P 0 D( R 0) 0R D( R 0) R where denotes firm i's annual stock returns for the 2 months starting nine months prior to fiscal year-end until three months after the fiscal year-end, a period that roughly corresponds to the period between earnings announcements. denotes firm i's earnings per share for year t, The problem of excluding fixed-effects applies to any dynamic panel data models where the dependent variable is a function of lagged values of the independent variables. Suppose that the data generating process is y ai at y e. Then follows that the dependent variable y ai at y2 e. If the researcher estimates instead the regression y y u where u e ai a, follows that t Eu [ y y ] 0and therefore the estimate b will be biased. 22

25 denotes firm i s share price at the beginning of fiscal year t, and D(R < 0) is an indicator variable obtaining the value "" if stock returns are negative, and "0" otherwise. Like Basu (997), we use all firm-year observations from 963 to 990 for which stock returns are available on the CRSP monthly files, and the necessary accounting data available on Compustat. Similarly, we deflate earnings by the beginning-of-year share price and eliminate observations falling in the top or bottom 0.5% of opening price-deflated earnings in each calendar year to reduce the effects of outliers on the results. Like Ball et al. (203) we consider two addional definions of stock returns: market adjusted returns, and size and book-to-market adjusted returns. The size and book-to-market adjusted returns are computed by forming 5x5 portfolios based on annual sorts on market capalization and on the book-to-market ratio (at the end of year t-). We then calculate monthly value-weighted mean returns for each size and book-to-market portfolio and subtract the portfolio returns from the same size and book-to-market quintiles raw returns. Market adjusted returns are raw returns minus the value-weighted market returns. To save space, we report results only for raw returns; results for market-adjusted returns and size and book to market-adjusted returns are similar. We collect data for all US firms that trade on the NYSE, AMEX and NASDAQ. Our sample contains 4,75 firm-year observations for the period , and 42,546 firm-year observations for the period For comparison, Basu (997) reports results for a sample of 43,32 firm-year observations over the same period. Panel A of Table 4 reports summary statistics for the regression variables for the period These statistics are consistent wh those reported in Patatoukas and Thomas (20) and Ball et al. (203). For instance, X / P is left-skewed and R is right-skewed. The median value of X / P in our study is 23

26 0.063 whereas Patatoukas and Thomas (20) report median value of and Ball et al. (203) report median value of The median value of annual stock returns R in our study is 0.094, whereas Patatoukas and Thomas (20) report and Ball et al. (203) report a value of (Table 4 about here) Table 5 presents the sensivy of the asymmetric timeliness of earnings regression to the inclusion of fixed-effects. Looking at the sample, our results show that when pooled OLS is used, the coefficient is posive (0.98) and significant at the 0.0 level (t-statistic of 24.2). Basu (997) reports a somewhat larger coefficient of (t-statistic of 27.4). The common interpretation of this result is that contemporaneous earnings reflect negative news in a timelier manner than posive news (accounting earnings are condionally conservative). When firm and time fixed-effects are included in the model the slope coefficient is significantly lower than that reported by Basu (997). Specifically, the coefficient is 0.09 (t- statistic = 0.88) for the period and 0.45 (t-statistic = 28.45) for the period. When industry effects are included in the model instead of firm effects, is 0.63 (t- statistic = 2.72) for the period and (t-statistic = 44.90) for the period, very similar to those reported by Basu (997). Furthermore, using the Fama-MacBeth (973) methodology yields equal to 0.2 (t-statistic = 9.26) for the period and 0.26 (t-statistic = 4.5) for the period, again very close to the results reported by Basu (997). In sum, adding firm fixed-effects reduces the coefficient on conservatism ( ) substantially, while using industry fixed-effects or the Fama-MacBeth (973) methodology 24

27 yields a much higher conservatism coefficient. These results are in line wh our theoretical predictions and highlight the fact that when the data generating process contains firm fixedeffects, the inclusion of industry fixed-effects does not help in dealing wh unobserved firm heterogeney. Similarly, as predicted, the Fama-MacBeth (973) approach is also subject to unobserved heterogeney biases. Aghion et al. (200) use a different approach to estimating a panel dataset. Instead of adding firm and time fixed-effect, they convert the panel dataset into a cross-sectional regression whereby instead of the vectors of dependent and independent variables, they use time-level means (denoted here MYX). Using this method yields conservatism coefficients ( ) equal to (t-statistic = 9.03) for the period and (t-statistic = 0.43) for the period. Ball et al. (203) acknowledge that in order to deal wh this problem, the researcher could use a fixed-effects specification. However, in their empirical approach and due to computational constraints, they use a different approach: demeaning the dependent variable (MY). To assess the effect of their approach, the last specification in Table 5 employs the Ball et al. (203) specification (denoted, MY). As can be seen, is (t-statistic = 6.65) for the period and (t-statistic = 8.96) for the period. These values are similar to the estimates reported in Table 5 (row 5) in Ball et al. (203). We therefore conclude that under the Ball et al. (203) specification is significantly lower than the coefficient obtained from the full fixed-effects specification. This result suggests that the MY specification provides a lower bound for the condional conservatism coefficient. This is likely to be the case as the mean value of the dependent variable is posive, as suggested by our simulations. 25

28 Overall, our results suggest that there is substantial unobserved heterogeney at the firm level that seems to be an important determinant for explaining price-deflated earnings. These findings are consistent wh the Ball et al. (203) finding that the Basu (997) is affected by correlated omted variables due to the expected components of earnings being correlated wh the expected components of stock returns. However, we argue that the empirical specification of Ball et al. (203) is not an appropriate substute for firm and time fixed-effects (FE). This notwhstanding, from a qualative standpoint, our results confirm the presence of condional conservatism in earnings. (Table 5 about here) 4.2 The differential persistence of accruals and cash flow components of earnings Sloan (996) explores the association between future income and previous year s accruals and cash flows. He finds that the persistence of cash flows exceeds that of accruals, which is consistent wh the reversal property of accruals. Sloan (996) estimates the following regression, allowing the persistence coefficient on the accruals and cash flow components of earnings to be different. The model is: OI / TA ACC / TA CF / TA e 0 0 where OI / TA is operating income divided by average total assets, ACC / TA denotes operating accruals divided by average total assets, and CF TA denotes operating cash flows divided by average total assets. / The accrual component of operating income is measured as ACC = ( CA - Cash ) - ( CL - STD - TP ) Dep, where CA is the change in current assets; Cash is the change in cash and cash equivalents; CL is the change in current liabilies; STD is the 26

29 change in debt included in current liabilies; TP is the change in income taxes payable; and Dep is the depreciation and amortization expense. The cash flow component of earnings (CF ) is measured as the difference between operating income and the accrual component of earnings. We collect a sample, which includes all firm-year observations wh the necessary accounting and stock return data available on Compustat and CRSP monthly file between 962 and 99. This is the sample analyzed by Sloan (996). We also collect data for an extended sample for We only sample US firms that trade on NYSE, AMEX or NASDAQ. As before, we eliminate the top and bottom 0.5% of observations. Table 4, Panel B, contains descriptive statistics for the main variables for the extended period Median operating income over average total assets is 0.3. This figure is made of a median accrual component of 0.0 and a median cash flow component of 0.2. Table 6 presents results of estimating the differential persistence regressions for the full period and for the sub-period For the sub-period, using a pooled regression whout fixed-effects, the coefficient on the accrual component ( ) is 0.62 (t- 0 statistic = 4.29), lower, as expected, than the coefficient on the cash flow component ( ), which is (t-statistic = ). These coefficients are somewhat lower than those reported by Sloan (996) in Table 3 ( = and 0 = 0.855). However, similar to Sloan (996), our results also show that the accrual component of earnings is less persistent than the cash flow component (0.62 vs. 0.73) and that the difference between the two coefficients is significant at the 0.0 level. Adding firm and year fixed-effects reduces the persistence coefficients que significantly. The coefficient on the accrual component ( ) is (t-statistic = 28.33) and the coefficient 0 on the cash flow component ( ) is (t-statistic = 97.53) for the period. The 27