Earnings co-movements and earnings manipulation

Abstract

This study develops a theory that predicts the lower the degree to which firms’ earnings are correlated with the industry the greater the probability a firm will issue a biased signal of firm performance. The theory provides for causal predictions in our empirical tests in which we examine the probability a firm will be subject to an Accounting and Auditing Enforcement Release (AAER). The empirical findings provide support for the theory, even after controlling for various predictive variables from the literature, indicating the degree of earnings co-movements with the industry is in fact a causal factor in managers decisions to bias earnings reports. We further illustrate that low co-movement firms are less conservative than high co-movement firms, which provides an application of our theory to a broader setting. Overall, we provide both a theory and an empirical validation of the theory helping to discipline the thinking about earnings management and allowing for causal relations to be uncovered.

Appendix 1: Proof of Proposition 1

Proof

Note that

$$\begin{array}{@{}rcl@{}} \alpha_{r} &=&\frac{{\sigma_{u}^{2}}}{\left[ {\sigma_{u}^{2}}+\sigma_{\varepsilon }^{2}+{\lambda_{x}^{2}}{\sigma_{x}^{2}}\right] }=\frac{\sigma^{2}-\beta^{2}{\sigma_{E}^{2}}}{\left[ \sigma^{2}-\beta^{2}\sigma_{E}^{2}+\sigma_{\varepsilon }^{2}+{\lambda_{x}^{2}}{\sigma_{x}^{2}}\right] } <1,\\ \alpha_{E} &=&\frac{\left[ \sigma_{\varepsilon }^{2}+\lambda_{x}^{2}{\sigma_{x}^{2}}\right] }{\left[ {\sigma_{u}^{2}}+\sigma_{\varepsilon }^{2}+{\lambda_{x}^{2}}{\sigma_{x}^{2}}\right] },\text{ and } \\ \alpha_{0} &=&-\lambda_{x}\mu_{x}\alpha_{r}. \end{array} $$

(7)

With $\lambda _{x}=\frac {\alpha _{r}}{c},$

$$\begin{array}{@{}rcl@{}} \alpha_{r} &=&\frac{\left[ \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}\right] }{ \left[ \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}+\sigma_{\varepsilon }^{2}+{\lambda_{x}^{2}}{\sigma_{x}^{2}}\right] } \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} &=&\frac{\sigma^{2}-\beta^{2}{\sigma_{E}^{2}}}{\left[ \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}+\sigma_{\varepsilon }^{2}+\left( \frac{\alpha_{r}}{c} \right)^{2}{\sigma_{x}^{2}}\right] } \\ &\Leftrightarrow & \\ &&\frac{{\sigma_{x}^{2}}}{c^{2}}{\alpha_{r}^{3}}+\left( \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}+\sigma_{\varepsilon }^{2}\right) \alpha_{r}-\left( \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}\right) =0, \end{array} $$

(9)

The optimal reporting bias is given by Eq. 5:

$$b(x,s)=\frac{\alpha_{r}}{c}x. $$

Our interestis to sign $\frac {db(x,s)}{d\beta }:$

$$\frac{db(x,s)}{d\beta }=\frac{x}{c}\frac{d\alpha_{r}}{d\beta }. $$

Differentiating Eq. 9with respect to β,

$$\left[ \frac{3{\sigma_{x}^{2}}}{c^{2}}{\alpha_{r}^{2}}+\left[ \sigma^{2}-\beta^{2}{\sigma_{E}^{2}}+\sigma_{\varepsilon }^{2}\right] \right] \frac{d\alpha_{r}}{d\beta }+2\beta {\sigma_{E}^{2}}(1-\alpha_{r})=0. $$

Notingthat $\sigma ^{2}-\beta ^{2}{\sigma _{E}^{2}}={\sigma _{u}^{2}}>0$(where ${\sigma _{u}^{2}}$,the firm-specific risk component is always positive by assumption—see footnote 3), the coefficient of d α _r /d β is positive.With 0 < α _r < 1,the term$2\beta \sigma ^{2}\left (1-\alpha _{r}\right )$is positive if β > 0, and it mustbe that $\frac { d\alpha _{r}}{d\beta }<0$, whichyields $\frac {db(x,s)}{d\beta }<0$. The term$2\beta \sigma ^{2}\left (1-\alpha _{r}\right )$is negative if β < 0, in which case it mustbe that $\frac {d\alpha _{r}}{d\beta }>0$, which yields$\frac {db(x,s)}{d\beta }>0$. Combining these twoeffects, it can be seen that $\frac {d\alpha _{r}}{d\left \vert \beta \right \vert }<0$,and therefore $\frac {db(x,s)}{d\left \vert \beta \right \vert }<0$,i.e., the reporting bias is decreasing in the magnitude of the earnings co-movement. It can also be seenthat $\frac {d\alpha _{r}}{d{\sigma _{u}^{2}}} >0$and $\frac {db(x,s)}{d{\sigma _{u}^{2}}}>0$. □