Categorical variable interpretation in "mixed" regression I have a linear regression with transformed variables:
log(y) = b0 + b1*log(X1) + b2*mid + b3*high

where "mid" and "high" are dummies from a 3-level categorical variable, "low"/"mid"/"high", with "low" left out.
Now since this a log-linear regression (at least in the X2 dummies), the interpretation of the impact of the 2 dummies on y should be the following:


*

*Suppose estimated coeff for b2 is -0.24.

*Then the impact of a "mid" case on y is to decrease it by 100*(exp(b2)-1), relative to a "low" case.

*Similarly for a "high" case - impact is to change in by 100*(exp(b3)-1), again relative to the "low" case.


Does this sound right?  Should I be accounting for the log(X1) impact in the impact of the dummies (as described above), especially in trying to demonstrate the extent of effect these dummies have on y (not log(y))?
 A: A) Your interpretations of the actual coefficients (how to convert them into a meaningful unit) is close, but not quite correct. Rather than include a long description here, you can read about interpreting coefficients in loglinear models here.
B) You do have to include X1 in your interpretation, assuming there is covariance between it and the predictor you are interpreting. The effect of any of your dummies is "controlling for" any effect of X1. Essentially, your effect of any dummy is any effect of your dummy variable on your DV that X1 cannot also explain.
An example is two variables that are perfectly co-linear. Let's say you want to predict height based on age of infants. If you didn't know statistics, you may include a variable for age expressed in days and a variable for age expressed in weeks. There is not difference here. They are just in different units. Age expressed in days cannot explain anything that age expressed in weeks can't also explain and vice versa. Therefore, the coefficient estimates would both be 0. That doesn't mean that age doesn't predict height. It just means your model doesn't make any sense.
Dummy variables also co-vary. However, you have already learned how that plays out. When you said that mid is the effect of being in the mid condition compared to the low condition, this is the correct interpretation. However, there is a longer way to think about it. Your mid variable is a series of 0's and 1's. This only contrasts two things: being in the mid group and being in any other group. However, you can control for being in the high group by including the high variable in your model. The comparison of being in the mid group against either of the other groups controlling for the effect of being in the high group versus any other group is correctly interpreted as the effect of being in the mid group against being in the low group. This may have added confusion, but the key is that you were interpreting dummies correctly, but you should know that whenever two IVs co-vary, it changes the interpretation of the individual coefficients slightly.
