Measuring discretionary accruals: are ROA-matched models better than the original Jones-type models?

Abstract

Discretionary accruals estimated from Jones-type models are elevated or depressed for firms with extreme performance. Kothari et al. (J Acc Econ 39:163–197, 2005) propose performance matching to address the issue, that is, to difference discretionary accruals estimated from Jones-type models for treatment and control firms matched on current ROA. This study shows (1) performance matching will systematically cause discretionary accruals of either sign to be underestimated, and (2) the measurement error will be negatively correlated with the true discretionary accruals. As a result, using discretionary accruals estimated with performance matching to test whether certain events induce earnings management will increase the frequency of Type II errors, and using them as the dependent or an independent variable in regression analysis will bias the regression coefficient toward zero. The results of our empirical tests are consistent with these predictions.

Appendix: Estimated discretionary accruals as an independent variable

Let x be true discretionary accruals and y be the firm attribute that is explained by and correlated with x. That is,

$$ y = \alpha + \beta x + \varepsilon . $$

Not knowing x, the researcher uses discretionary accruals estimated from an original Jones-type model $\tilde{x}$ as a surrogate for x. The estimate contains error $\upsilon$, which is a zero-mean random variable uncorrelated with x if, as stated in the text, the sample firms’ distribution of (genuine) operating performance is unskewed. That is,

$$ \tilde{x} = x + \upsilon $$

And

$$ y = \alpha + \beta (\tilde{x} - \upsilon ) + \varepsilon $$

And

$$ y = \alpha + \beta \tilde{x} + (\varepsilon - \beta \upsilon ). $$

This shows that when one regresses y on $\tilde{x}$ to obtain an estimate of β, the error term in the regression equation $(\varepsilon - \beta \upsilon )$ contains some of the measurement error in $\tilde{x}$, thus recreating an endogeneity bias. The expected value of the OLS estimator in this case equals

$$ E(\hat{\beta }) = \frac{{\text{cov}} (\tilde{x},y)}{{\text{var} (\tilde{x})}} = \frac{{\text{cov} (x + \upsilon ,\alpha + \beta x + \varepsilon )}}{{\text{var} (x + \upsilon )}} = \frac{{\beta \sigma _{x}^{2} }}{{\sigma _{x}^{2} + \sigma _{\upsilon }^{2} }} $$

where $\sigma_{i}^{2}$ is the variance of any variable i.

Since 0 < $\sigma_{x}^{2}$/($\sigma_{x}^{2}$+$\sigma_{\eta }^{2}$) < 1, $E(\hat{\beta })$ is lower than the true coefficient β in absolute value regardless of the sign of β. In other words, the estimated coefficient is biased toward zero.

What would be the effect of performance matching on the estimated coefficient $\hat{\beta }$? As shown in the text, there are two measurement errors with performance matching, one (u) is negative and negatively correlated with true discretionary accruals (x), and the other (η) a zero-mean random variable. Thus we have

$$ \tilde{x} = x + u + \eta . $$

Let u = m + e, where m is the mean of u and e is a zero-mean variable negatively correlated with x. Then, the regression model $y = \alpha + \beta x + \varepsilon$ equals

$$ y = \alpha + \beta (\tilde{x} - u - \eta ) + \varepsilon $$

Or

$$ y = (\alpha - \beta m) + \beta \tilde{x} + (\varepsilon - \beta e - \beta \eta ). $$

Then the expected value of the OLS estimator of β is:

$$ E(\bar{\beta }) = \frac{{\text{cov} (\tilde{x},y)}}{{\text{var} (\tilde{x})}} = \frac{{\text{cov} (x + u + \eta ,\alpha + \beta x + \varepsilon )}}{{\text{var} (x + u + \eta )}} $$

Since u is the sum of m (a constant) and e, the expected value of $\bar{\beta }$ is

$$ E(\bar{\beta }) = \frac{{\beta \sigma_{x}^{2} + \beta \sigma_{xe} }}{{\sigma_{x}^{2} + \sigma_{e}^{2} + \sigma_{\eta }^{2} + 2\sigma_{xe} }} = \beta \frac{{\sigma_{x}^{2} + \sigma_{xe} }}{{\sigma_{x}^{2} + \sigma_{e}^{2} + \sigma_{\eta }^{2} + 2\sigma_{xe}^{{}} }} $$

(1)

where $\sigma_{ij}^{{}}$ is the covariance of two variables i and j.

$E(\bar{\beta })$ can have a different sign from β in theory since $\sigma_{xe}$ is negative. But that is probably unlikely. If $E(\bar{\beta })$ has the same sign as β, it will be lower in absolute value than $E(\tilde{\beta })$, the expected value of the estimated coefficient with no performance matching:

$$ \left| {\beta \frac{{\sigma_{x}^{2} + \sigma_{xe} }}{{\sigma_{x}^{2} + \sigma_{e}^{2} + \sigma_{\eta }^{2} + 2\sigma_{xe} }}} \right| < \left| {\beta \frac{{\sigma_{x}^{2} }}{{\sigma_{x}^{2} + \sigma_{\upsilon }^{2} }}} \right|. $$

The inequality in absolute value results from two facts. First, $\sigma_{x}^{2} + \sigma_{xe}$ < $\sigma_{x}^{2}$ since $\sigma_{xe}$ is negative. Second, empirical results in this paper show that performance (ROA) matching increases the variance of estimated discretionary accruals, that is, $\sigma_{x}^{2} + \sigma_{e}^{2} + \sigma_{\eta }^{2} + 2\sigma_{xe}^{{}}$ > $\sigma_{x}^{2} + \sigma_{\upsilon }^{2} .$

We assume that discretionary accruals are the only independent variable in the regression equation. The results will hold, however, in cases where there are multiple independent variables unless estimated discretionary accruals are highly correlated with the other independent variables.

1.1 Estimated discretionary accruals as the dependent variable

If discretionary accruals estimated from ROA-matched models appear on the lefthand side of the regression equation, coefficients on all the righthand side variables are likely to be affected. We use matrix notation to show the result. Let n be the sample size and the relation of discretionary accruals x with k explanatory variables be specified by the following equation

$$ X = Z\beta + E $$

where X is a n × 1 vector of observations on x (discretionary accruals), Z a n × k matrix of observations on the k independent variables, β a k × 1 vector of the k coefficients, and E a vector of zero-mean random errors.

Not knowing X, the researcher uses discretionary accruals estimated with performance matching ($\tilde{X}$) instead. Discretionary accruals estimated with performance matching contains measurement error negatively correlated with the true discretionary accruals. Thus we have $\tilde{X} = X + U + {\rm H}$, where U is a n × 1 vector of measurement error and H is a n × 1 vector of random error. Then, the OLS estimator of the coefficient vector in $X = Z\beta + E$ is:

$$ \hat{\beta } = (Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} \tilde{X} = (Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} (X + U + H). $$

Since measurement error in discretionary accruals estimated with performance matching is negatively correlated with true discretionary accruals, assume $U = \tau X + G$ (for simplicity and without loss of generality), where τ is a negative scalar and G is a n × 1 vector of random errors. Then, we have

$$ \tilde{X} = X + U + H = (1 + \tau )X + G + H = (1 + \tau )X + L $$

where L = G + H.

Then

$$ \hat{\beta } = (Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} \tilde{X} = (Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} [(1 + \tau )X + L] = (1 + \tau )(Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} X + (Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} L. $$

Since the expected value of $(Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} X$ is β and that of $(Z^{{\prime }} Z)^{ - 1} Z^{{\prime }} L$ is zero, we have

$$ E(\hat{\beta }) = (1 + \tau )\beta . $$

Since τ is a negative, the effect of performance matching is to bias coefficients on all the independent variables toward zero, except for those coefficients that equal zero.

It is straightforward to show that, if one uses discretionary accruals with no performance matching as the dependent variable in a regression, the expected values of estimated coefficients on all the independent variables equal the true coefficients if the sample firms’ distribution of (genuine) operating performance is unskewed. That is, the vector $\tilde{\beta }$ is an unbiased estimator of β.