Measure based on absolute residuals of an OLS regression with/without intercept

[Harald Leber]

A typical measure for firms’ use of earnings management in the finance literature is based on the absolute value of the residual (|e|) of a OLS regression estimated separately for firms in each industry/year-group.

However, literature uses two versions of this model:

without intercept: Accruals(t)/TotalAssets(t-1) = b1 * 1/TotalAssets(t-1) + b2 * ChangeInSales/TotalAssets(t-1)+ b3 * FixedAssets/TotalAssets(t-1) + e

or with intercept (b0): Accruals(t)/TotalAssets(t-1) = b0 + b1 * 1/TotalAssets(t-1) + b2 * ChangeInSales/TotalAssets(t-1)+ b3 * FixedAssets/TotalAssets(t-1) + e

My question(s) - how does the exclusion/inclusion of the intercept affect the measure |e|? And more specifically, how does the inclusion/exclusion affect a comparison of the measure |e| across industry/year-groups (e.g. comparison of the mean of |e| between Industry1 in 1997 and 1998)?

[Carlo Lazzaro]

Harald:
for Stata users. a mathodologically mandatory stop-by concerning the use (or the suppression) of constant term in an OLS dealing with financial data is: Baum CF. An Introduction to Modern Econometrics Using Stata. College Station, TX: Stata Press, 2006: pages 81-82.
With a handful of exceptions, Kit Baum recommends to retain the constant in OLS, as Stata will take itself care of any perfect linear combination of the regressors,

...and indicate a coefficient value of (dropped).

[Harald Leber]

Thanks Carlo for the fast answer - I read the pages you suggested, and I agree that econometrically it is most likely better to include the constant in this situation. However, many studies in this area run the model without an intercept as in the first model (although some studies state that the first term 1/TotalAssets(t-1) is a scaled constant?!). The problem is that including/excluding the constant to derive this measure (|e|) makes a huge difference. If, for example, I run the first model for all firms within each industry/year grouping (100 industries, 2 years) to derive firm-specifc residuals and I then test |e| in year 1 vs. year 2, the difference is signficant (high t) while with |e| derived from model 2 it is not...

[Carlo Lazzaro]

Harald:
does the -abs(res) calculation give you back normal distributions (which are one of the prerequisites of classic ttest)? If not, might this explain the difference in model 1 and 2?
Does this scenario change if you perform a bootstrap ttest?

[Harald Leber]

I reran the analysis and in the first case (without constant) the difference is not significant, while in the second case it is... i bootstrapped the t-test (not sure if i did that right) - result seems to be the same..

ABSRES DERIVED FROM A MODEL WITHOUT CONSTANT

. ttest ABSRES, by(POST)

Two-sample t test with equal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 870 .0765144 .0025404 .0749296 .0715285 .0815004
1 | 857 .079812 .0027721 .0811509 .0743712 .0852529
---------+--------------------------------------------------------------------
combined | 1727 .0781508 .0018787 .0780736 .074466 .0818356
---------+--------------------------------------------------------------------
diff | -.0032976 .0037578 -.0106678 .0040727
------------------------------------------------------------------------------
diff = mean(0) - mean(1) t = -0.8775
Ho: diff = 0 degrees of freedom = 1725

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Pr(T < t) = 0.1902 Pr(|T| > |t|) = 0.3803 Pr(T > t) = 0.8098

. bootstrap t=r(t), rep(1000): ttest ABSRES, by(POST)

Bootstrap results Number of obs = 1727
Replications = 1000

command: ttest ABSRES, by(POST)
t: r(t)

------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
t | -.8775392 1.021092 -0.86 0.390 -2.878842 1.123764
------------------------------------------------------------------------------

ABSRES DERIVED FROM A MODEL WITH CONSTANT

. ttest ABSRES, by(POST)

Two-sample t test with equal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 870 .0723909 .002351 .0693449 .0677765 .0770052
1 | 857 .0796018 .0027221 .0796871 .0742591 .0849445
---------+--------------------------------------------------------------------
combined | 1727 .0759692 .001798 .0747217 .0724426 .0794958
---------+--------------------------------------------------------------------
diff | -.0072109 .003593 -.0142581 -.0001637
------------------------------------------------------------------------------
diff = mean(0) - mean(1) t = -2.0069
Ho: diff = 0 degrees of freedom = 1725

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Pr(T < t) = 0.0225 Pr(|T| > |t|) = 0.0449 Pr(T > t) = 0.9775

. bootstrap t=r(t), rep(1000): ttest ABSRES, by(POST)

Bootstrap results Number of obs = 1727
Replications = 1000

command: ttest ABSRES, by(POST)
t: r(t)

------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
t | -2.006909 .9871237 -2.03 0.042 -3.941635 -.0721817
------------------------------------------------------------------------------