Scaling-deflating

[Jo Smith]

Dear all,
My study examines effect of corporate governance on audit fees. Firms' size is a control variable because firms size (as measured by total assets) is main determinant of audit fees.
This suggests to scale (deflate) audit fees according to firm size in order to mitigate spurious correlations due to size and to reduce heteroscedasticity.
Could you please help me how can I do this.

Many thanks,

[Clyde Schechter]

You need to tell us more about your analytic approach. There are several ways of doing this, and different ones are suitable for different analyses.

[Jo Smith]

Thanks Clyde,
I am using panel data to test different corporate governance variables (board independece, number of board meetings, gender diversity in the board, audit committee independece..) on audit fees.
I found the coefficient of firm size is high (.35). This indicate that firm size has big effect on audit fees which in turn create the need for scaling audit fees on firm size.
Not sure if this helps.

[Clyde Schechter]

Well, I actually meant for you to tell us what kind of analysis you plan to do going forward.

But here are some general thoughts. If your outcome variable and the data lend themselves to analysis with a Poisson model (these models are not restricted to use with count data), then using assets as an -exposure()- variable would be one appropriate way of doing this.

If you are planning on doing linear regression then you could either change the outcome variable to audit fees/assets, or you could include assets as a covariate in the model. Those are not the same thing, and the latter is more flexible than the former in terms of allowing for a wider variety of relationships between assets and audit fees. For that matter, you might want to do some graphical exploration of the assets-audit fees relationship looking at a scatter plot, or perhaps a lowess plot. If you see substantial non-linearity then you might want to include some terms in your model that will express that non-linear relationship. (Think about quadratic or higher order terms, or fractional polynomials, or linear or cubic splines.)

[Jo Smith]

Thanks very much Clyde,
Actually I was thinking to divide audit fees on firm size, and your answer gave me more confidence to to so.
But I have 13 explanatory variables, and after the scaling process, two of them were had a significant relationship with audit fees are turned to be insignificant (I am originally employ GLS approach).
As a summary result should I consider those two variables as a significant variables to my study or should I ignore them because their previous significant relationship was due to firm size effect.

Appreciate your reply indeed.

[Clyde Schechter]

It sounds like any effect of those two variables was due to confounding with firm size. While you can say that they were statistically significantly associated with audit fees in an analysis that omitted assets, you would be obligated to also report that scaling for assets eliminated that association. Whether a reader would consider that interesting or not, I can't say--that's a finance/economics question.

[Jo Smith]

That's pretty clear, thanks very much Mr Clyde.

[Jo Smith]

Mr Clyde,

Given you helped me about this before, when I do scaling i.e fees/size , should I remove the size from the right hand side or it is acceptable to keep it?
Also, if I divide fees over square root of asset rather than fees/asset, is it acceptable?

Much appreciated

[Clyde Schechter]

Given you helped me about this before, when I do scaling i.e fees/size , should I remove the size from the right hand side or it is acceptable to keep it?

Yes, if you use fees/size as your outcome variable, you should definitely not have size among the predictors--there will be an automatic negative association between size and fees/size that has no connection to anything beyond the meaning of division, and it will just confuse your results, and if size is correlated with your other predictors probably introduce bias.

Also, if I divide fees over square root of asset rather than fees/asset, is it acceptable?

It is mathematically feasible, and you can certainly legally code it. The question is whether it is meaningiful and whether others would perceive it as such. If it would be credible among others in your discipline that accounting fees grow roughly in proportion to square root of assets, all else equal, then sure, go ahead. I don't have any expertise in finance or econometrics, so I can't comment on this issue myself. {On the other hand, if it's just a "trick" to squeeze a p < 0.05 out of your data that happened to work after you tried a bunch of other transformations, then, no, don't do it.}

[Jo Smith]

Thanks Professor Clyde, you are very kind. your comments directly to the point
For the rest of my work, the dependent variable will no longer be fees- (will become fees\asset)
It seems logic but just to make sure.

[Jo Smith]

Further Clyde
The regression outcome is as following which makes concerns about the relationship between new outcome variable and other explanatory variables (very small coefficient):

xtreg scaled biratio rd bm bz gdratio aciratio acxratio acm acs insnonf insfinanc family gov foreignar fornonarab lev cox loss risk roa big4 naf industry, robust

Random-effects GLS regression Number of obs = 672
Group variable: code Number of groups = 114

R-sq: within = 0.0729 Obs per group: min = 4
between = 0.1351 avg = 5.9
overall = 0.1262 max = 6

Wald chi2(23) = 255.66
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

(Std. Err. adjusted for 114 clusters in code)

Robust
scaled Coef. Std. Err. z P>z [95% Conf. Interval]

biratio .0003578 .0002232 1.60 0.109 -.0000795 .0007952
rd -.0000597 .000057 -1.05 0.295 -.0001714 .0000521
bm -.0000303 .0000133 -2.28 0.023 -.0000564 -4.22e-06
bz - .0000709 .0000205 -3.46 0.001 -.0001111 -.0000307
gdratio -.0008492 .0005565 -1.53 0.127 -.0019399 .0002414
aciratio .0001853 .0001821 1.02 0.309 -.0001717 .0005423
acxratio -.0000858 .0002812 -0.31 0.760 -.0006369 .0004653
acm -.0000134 .000016 -0.84 0.402 -.0000447 .0000179
acs -.0000305 .0001616 -0.19 0.850 -.0003473 .0002862
insnonf -.0002511 .0003546 -0.71 0.479 -.000946 .0004439
insfinanc -.0001571 .0003157 -0.50 0.619 -.0007759 .0004617
family -.0002226 .0004267 -0.52 0.602 -.0010589 .0006137
gov -.0007915 .0006038 -1.31 0.190 -.001975 .000392
foreignar .0000241 6.88e-06 3.50 0.000 .0000106 .0000375
fornonarab -.0003264 .0002279 -1.43 0.152 -.000773 .0001203
lev -.0001945 .0001895 -1.03 0.305 -.0005659 .000177
cox -.0000254 . 0000233 -1.09 0.276 -.000071 .0000203
loss .0000585 .0000342 1.71 0.087 -8.46e-06 .0001254
risk -7.78e-07 9.40e-07 -0.83 0.408 -2.62e-06 1.06e-06
roa -1.23e-06 2.73e-07 -4.51 0.000 -1.77e-06 -6.97e-07
big4 .000045 .0000769 0.58 0.559 -.0001057 .0001957
naf .0000382 .000182 0.21 0.834 -.0003185 .0003949
industry .0000788 .0001272 0.62 0.536 -.0001706 .0003282
_cons .0013305 .0006638 2.00 0.045 .0000295 .0026315

sigma_u .00058318
sigma_e .00028852
rho .80336433 (fraction of variance due to u_i)


.



.

[Clyde Schechter]

I don't think this is anything to be concerned about. What does the output from -summ scaled- look like? If you defined scaled as the ratio of audit fees to assets, I would imagine that these ratios will be pretty small numbers. I mean, really, would you pay anything even as big as 50 cents per dollar of assets for audit fees? No, you'd probably not be willing to pay more than $1 in audit fees for every $10,000 in assets, or maybe even much less than that. I don't know what the other variables are and what they mean, but if the values of those variables are typically in the ballpark of 1 to 100, or are 0/1 indicator variables, then their coefficients will have to be vary small in order for the linear predictor to add up to an outcome variable that is probably something like 0.0001 or less on average. So these numbers look pretty sensible to me, absent more detail to persuade me otherwise.

If you are uncomfortable with coefficients this small, you can re-scale your outcome variable. Instead of making it fees:assets make it dollars of fees per $10,000 in assets or something like that.

[Jo Smith]

Thanks Clyde very much,
Should I use the new outcome variable (after scaling) as the main outcome variable in all regressions of my study Or use it only later in the sensitivity test to check effect of firms size on all other variables?
Thanks very much again

[Clyde Schechter]

Jo, I'd like to help, but I can't answer this one. It's not a Stata question, and it's not a statistics question. It's a question about what will be meaningful and useful to whoever is the target audience for your work. I don't know who those people are, and I don't know much about finance/econometrics. So I think you have to bounce this one off somebody who is familiar with those things, even if they know little to nothing about Stata or statistics.

[Jo Smith]

Thanks Clyde very much, I understand this.