The Validity of the Testing Procedure for Earnings Management

An unbiased test of earnings management requires that the measurement error in MEP, ( ), to be uncorrelated with the partitioning variable, PART (McNichols, 2000). Therefore, the omission of the relevant variable ( ) should bias the estimated coefficient on the variable PART ( ) if both variables were correlated. This would in turn lead to erroneous inferences about the existence of earnings management as the model will be misspecified.

Statistically, if the omitted variable (Xk) is correlated with the included variable (Xi), the

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coefficient (βi). Therefore, the expected value of ( i) would equal (βi) plus the bias (Gujarati,

2003). Symbolically,

E( i) = βi + (βk * bki) (5)

Clearly, the magnitude and sign of this bias equals multiplying (βk) by (bki)

Where,

βk : the slope coefficient in the regression of the dependent variable (Y) on the excluded

variable (Xk), and

bki : the slope coefficient in the regression of the excluded variable (Xk) on the included

variable (Xi).

In the earnings management context, Dechow et al. (1995, p.196) consider this statistical issue in their analysis as they identify two problems for statistical inference that arise from being ( ) a biased estimator of β when the correlation between PART and does exist. Recall that the relevant variable ( ) can represent the measurement error in MEP (i.e. the unmanaged earnings that are not extracted by any model) and/or omitted relevant variables influencing ME.

Problem 1: Incorrectly attributing earnings management to PART

This problem manifests itself in two ways that lead to committing type I error2. First, if earnings management that is hypothesised to be caused by PART does not take place (i.e. the true coefficient on PART is zero) and the measurement error in MEP is correlated with PART, then the estimated coefficient on PART will be biased away from zero. In other words, although earnings management does not take place, the non-extracted unmanaged

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earnings will be incorrectly considered as managed earnings caused by PART. Second, if earnings management does take place in response to other omitted relevant variables rather than PART and these omitted variables are correlated with PART, then the estimated coefficient on PART will be biased away from zero. That is, although earnings management are caused by other omitted variables, the model will correctly detect managed earnings but will incorrectly attribute them to PART.

Problem 2: Unintentionally extracting earnings management caused by PART

This problem arises when earnings management that is hypothesised to be caused by PART does take place (i.e. the true coefficient on PART is not zero) but the correlation between PART and ( ) is opposite in sign to the true coefficient on PART causing the estimated coefficient on PART to be biased toward zero. In other words, although earnings management does take place in response to PART, the model used to generate MEP will incorrectly consider some or all of the managed earnings as unmanaged earnings because of the negative correlation between MEP and its measurement error. This will increase the probability of committing type II error3.

So far, the above two problems have been found to arise from the bias in the estimated coefficient on PART caused by the omission of a correlated variable. Statistically, however, even if the included and excluded variables are uncorrelated (i.e. the estimated coefficient on PART, ( ), is unbiased), the estimated variance of the coefficient on the included variable, var( ), will remain a biased estimator of the true variance of the true coefficient ( ). To illustrate, recall equation No. (5), where in this case, the slope coefficient in the regression of the excluded variable (Xk) on the included variable (Xi), (bki), equals zero because there is no

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correlation between the omitted and included variables. Therefore, the estimated coefficient on PART, ( ), is unbiased. Yet because the slope coefficient in the regression of the dependent variable (Y) on the omitted relevant variable (Xk), (βk), does not equal zero, var( )

will remain a biased estimator of the true variance of the true coefficient ( ). That is, var( ) will always have a positive bias that overestimates the true variance of ( ) (Gujarati, 2006) 4. On this basis, Dechow et al. (1995, p.197) proceed with their analysis to cover a third statistical inference problem which is,

Problem 3: Low power test5

This problem is concerned with earnings management models’ ability to detect managed earnings when earnings management does take place. When the relevant variable ( ) is uncorrelated with the included variable PART, and is omitted from the estimated model, the variance of the coefficient on PART, var( ), will be overestimated. Consequently, the standard error of the estimated coefficient on PART, SE( ), will be inflated causing the confidence interval to be wider. As the confidence interval gets wider, the researcher may tend to accept the null hypothesis that the true value of the coefficient is zero more frequently than the true situation demands (i.e. increase the probability of committing type II error).

4

For statistical proof, consider Kmenta, J. (1986), Elements of Econometrics, 2nd Ed., Macmillan, New York.

5 _{Statistical power is the model's ability to reject a false null hypothesis (Gujarati, 2003). At length, since the}

probability of committing type II error (β) is about failing to detect earnings management when it genuinely exists, the statistical power is the opposite of not detecting earnings management (1- β).

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