Interpretation of removed continuous variables in regression due to linear dependence

Question

I have created a standard OLS regression model to estimate the House Price and one group of variables describe the age group percentage of population in a particular neighborhood (ranging 0 to 100).

These variables are the percentage of the population in a particular neighborhood, belonging to an age group. For example Neighborhood Age 0-14 value of 23 would mean that there is 23% of people in a neighborhood, who are between 0-14 years old. The variables are presented below:

• Neighborhood Age 0-14 %
• Neighborhood Age 15-24 %
• Neighborhood Age 25-44 %
• Neighborhood Age 44-64 %
• Neighborhood Age >64 %

Now I know that since these are percentage values, I have to remove at least one of them due to perfect linear dependence, for example: Neighborhood Age 0-14 % = 1 - SUM(All of the other age %)

I have removed the Neighborhood Age >64 % variable and estimated the coefficients. The estimated coefficients for each variable are this (House price has been log-transformed so interpretation is ${\Delta}P\% = {\beta}_{i} * {\Delta}X_i\%$):

• Intercept: 11.1917
• Neighborhood Age 0-14 %: 0.0229
• Neighborhood Age 15-24 %: 0.0121
• Neighborhood Age 25-44 %: 0.0002
• Neighborhood Age 44-64 %: 0.008

As I removed one of the variables, how would I now interpret the Neighborhood Age >64 % effect on House Price? Note that these are continuous variables ranging 0-100.

Answer 1

The the house price of having lets say: Neighborhood Age 44-64% is 0.008 more than having Neighborhood Age >64%. Take note, that when adding dummy variables in a linear model:

$y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3D $ where $D$ is a dummy variable, can be re-written into :

$y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3 $ when D = 1

$y = \beta_0 + \beta_1X_1 + \beta_2X_2 $ when D = 0

So the difference in y in the model with and without the categorical predictor is simply just $\beta_3$ which is estimated.