Computing elasticities in a log-log model with stand-alone terms and interaction terms between categorical and log-transformed variables

[Gina Malek]

Dear Statlisters,
I am running a log-log model with the following specification: regress lnY lnX1 lnX2 lnT c. lnT # i.Urban c. lnT # i.Road
where,
Y= cost; T = traffic volume variable; Urban = binary variable for location (1= urban; 0 = rural); and Road = categorical variable for six roadway type categories (1= IN; 2= PA; 3= PAO; 4= MA; 5=MC; 6=LOC)

I would like to obtain the elasticity of the model above with respect to the T (traffic volume) variable. The elasticity should be based on urban/rural designations as well as the various categories of roadway type; that's why I have the interaction terms of those variables in the model. Assuming that the relevant coefficients estimated by the model are as follows, please consider my questions 1-6 below.

coefficient estimate of “lnT” variable = b₁ = 0.021123
coefficient estimate of the “c. lnT # i.Urban” variable) = b₂ = 0.003417
coefficient estimates of the “c. lnT # i.Road” variables:
category 2 (PA) = b₃ = -0.010111
category 3 (PAO) = b₄ = -0.010795
category 4 (MA) = b₅ = -0.019341
category 5 (MC) = b₆ = -0.020148
category 6 (LOC) = b₇ = -0.023651

Question 1:
Considering that in a log-log regression model, the coefficients are elasticities, can the elasticity with respect to the T (traffic volume) variable in my model for the case of “Urban = 1” and “Urban = 0” , respectively, be computed as:
b₁ + b₂ = 0.021123 + 0.003417 = 0.02454 (if Urban = 1)
b₁ + (0) = 0.021123 + 0 = 0.021123 (if Urban = 0”)

Question 2:
Can the elasticity with respect to the T (traffic volume) variable in the model for the case of roadway type category 2 (PA) be computed as:
b₁ + b₃ = 0.021123 + (-0.010111) = 0.011012

Question 3:
Can the elasticity with respect to the T (traffic volume) variable in the model for the case of roadway type category 1 (IN), which is the reference category for the “i.Road” factor variable, be computed as:
b₁ + 0 = 0.021123 + (0) = 0.021123

Question 4:
If the answers to Q2 and Q3 are “No”, then should the elasticity with respect to the T (traffic volume) variable be computed for each of the possible scenarios of the model separately? For example:
elasticity of [Urban = 1 & Road = 2 (PA)] = b₁ + b₂ + b₃ = 0.021123 + 0.003417 + (-0.010111) = 0.014429
elasticity of [Urban = 0 & Road = 2 (PA)] = b₁ + 0 + b₃ = 0.021123 + 0 + (-0.010111) = 0.011012
elasticity of [Urban = 1 & Road = 3 (PAO)] = b₁ + b₂ + b₄ = 0.021123 + 0.003417 + (-0.010795) = 0.013745
.
.
elasticity of [Urban = 0 & Road = 6 (LOC)] = b₁ + 0 + b₇ = 0.021123 + 0 + (-0.023651) = -0.002528

Question 5:
If the answer to Q4 is yes, how would we compute the elasticity with respect to the T (traffic volume) variable for the case of roadway type category 1 (IN), which is the reference category for the “i.Road” factor variable?

Question 6:
If the answer to Q4 is yes, how do we make sense of a negative elasticity (-0.002528) for the category 6 (LOC) of the i.Road factor variable in the model? (since this means that a 1% increase in the traffic volume on a category 6 roadway is associated with a 0.002528% decrease in the dependent variable Y, i.e., cost, which is counter-intuitive).

Any help is much appreciated!