Question about intepretation of constants in a Linear Regression

Question

Consider the number of passengers that, in 2009, used a given airport: $$D=[D_1, D_2, \ldots, D_m]^T$$ where $D_i$ represents the number of passengers that, in 2009, used the airport number $i$. I can access these values as part of my historical records.

Now, I want to see how these values depend on some other variables, which are:

1. Variable $x_1^{(i)}$ is the population of the city where airport $i$ is located.
2. Variable $x_2^{(i)}$ is the number of hotels within 100km from airport $i$.
3. Variable $x_3^{(i)}$ is the GDP (Gross Domestic Product) of the city where airport $i$ is located.

I have all those data, so they are given for each airport in 2009.

So I adjust a linear model, like this: $$\log{D_i}=\beta_0+\beta_1\log{x_1^{(i)}}+\beta_2\log{x_2^{(i)}}+\beta_3\log{x_3^{(i)}}$$ where, again, $i=1,2,\ldots,m$ (m is the total number of airports I have in my database). I successfully adjusted this model using MATLAB (but it can also be done by hand), and got the following values for our 4 parameters $\{\beta_0, \beta_1, \beta_2, \beta_3\}$:

• $\beta_0=-2$
• $\beta_1=0.15$
• $\beta_2=-0.1$
• $\beta_3=0.20$

Now I'm trying to understand the meaning of these values. More specifically, their sign. For example, $\beta_3=0.20>0$, so our linear model says that the greater the GDP in airport $i$ the more passengers its airport will experience in year 2009, which seems pretty logical to me.

I don't agree that $\beta_2<0$. Shouldn't air traffic increase in airport $i$ as the number of hotels increase?

And what I totally don't understand is the sign of $\beta_0$, which I can't seem to relate with the rest of the variables. Why is $\beta_0$ negative and what does this mean, qualitatively speaking?

Answer 1

Where $x_1 = x_2 = x_3 = 1$ than $\log D = \beta_0$ so $D=e^{\beta_0}$. I.e., you can view $\beta_0$ (intercept) as a baseline number of ($\log$) airports where all the other variables are $0$. So, in your case it leads to a less then one airport. Whether it is logical or not (as the negative sign of $\beta_2$) is a socioeconomic question rather than mathematical or statistical. From a technical point of view, biased estimators may occur in a case of model missspecification, particularity when you omit valuable explanatory variables. Therefore, if the signs of the estimators are wrong from a theoretical perspective, then you can try to collect more variables or/and estimate another model.

Answer 2

Too long for a comment.

I think that it could be good you provide the standard error associated to each parameter (and other statistical tests if you have them).

My main remark will be that what is measured is $D_i$ and not $\log(D_i)$. So, the first step you did provides "resonable" values for the $\beta$'s. What I should do from here is to start a nonlinear regression for the "true" model $${D_i}=\exp\left(\beta_0+\beta_1\log{x_1^{(i)}}+\beta_2\log{x_2^{(i)}}+\beta_3\log{x_3^{(i)}}\right)$$ This could lead to quite different results.

It could be interesting you try and report the results.