In a regression is the Y-intercept a measure of unaccounted biases?

Question

I have been working on a set of data which contains information on the width, age, weight of statues and relate them to the price (I am not actually working on that, but I cannot disclose the topic of my work).

I came up with the following regression:

$Price = -9 -width + 4 log10(age) + 8 height$

the minimum-maximum interval for each parameter are:

width = [0,1[
age = [200,4000]
height = [0.65,0.89]

My price therefore varies between about 4 and 12.

As the value of the constant is -9, which is of the same order of magnitude as my price range, I am wondering if this regression could be criticized for being to generic, with a lot of the price variation unaccounted for and "hidden" inside the constant.

Am I trying to over interpret my data here? Could my dataset be missing a crucial variable, for example the weight?

Answer 1

The simple answer is no. Take a look at the following example. It shows a simple linear regression results between two variables $X$ and $Y$. The only difference between the black points and regression line, and the red ones, is that in the second case I changed $Y$ to $Y + 5$. In the "black scenario, the parameters are $\beta_0 = 0.1256, \beta_1 = 4.9122$, while in the "red" scenario $\beta_0 = 5.1256, \beta_1 = 4.9122$. Answer yourself: did the amount of unaccounted biases changed by five between the two scenarios? Is the "black" model worse because it has lower intercept?

The answer to both questions is "no". The only thing that has changed between the two models is that the mean of $Y$ has changed by five, what has led to intercept changing by the same value. That is exactly the role of intercept: to "shift" the regression line upwards, or downwards, on the $y$-axis, so to correct for the mean of the dependent variable. In fact, in most cases we do not expect the intercept to be anything close to zero, because this would mean regression line going through the origin what in many cases leads to inferior models as compared to the models that include the intercept (i.e. it is non-zero).

I guess that what you are looking for is rather the $R^2$ statistic, but beware that it can be misleading as describes in Is $R^2$ useful or dangerous?

Answer 2

We rarely report or interpret the intercept in a linear regression model. In your case it is an extrapolation of the data. The intercept would be interpreted as a expected price for a product with width 0, an age of 1, and a height of 0. That is nonsense. A value of -9 is an artifact of the projection.

If you want a more interpretable intercept, center the covariates. Then the intercept is the average price for all covariates taking their average values.

Answer 3

Who is the audience that you expect will criticize the intercept? If it is a "business audience", they are much more likely to be concerned about model accuracy, which you haven't mentioned here. The value of the parameter coefficients - showing the relative impact of each variable on price - is where you should focus your interpretation/discussion.