# How to interpret and determine the reference levels of independent variables in a regression problem?

I encountered several regression problems in the field of engineering that are formulated similar to the following form (I will ommit the error terms):

$$y=a_0+a_1(x_1-c_{ref})+a_2\ln(x_2/d_{ref})\,,$$

where $y$ is the dependent variable; $x_1$ and $x_2$ are independent variables; $a_0$, $a_1$ and $a_2$ are unknown coefficients need to be estimated. What I have difficulties in the kind of regression problems is the interpretation of the two reference parameters $c_{ref}$ and $d_{ref}\,$. In papers, these two reference levels are usually given (i.e., they are preset values and not estimated together with $a_0$, $a_1$ and $a_2$) without any explanations on why they are considered and how their values are chosen...

In my view, all these preset reference parameters can be merged with the constant term $a_0$ into a new constant, say $\tilde{a}_0$, i.e.,

$$\tilde{a}_0=a_0-a_1c_{ref}-a_2\ln(d_{ref})$$

and the original regression problem becomes to

$$y=\tilde{a}_0+a_1x_1+a_2\ln(x_2)\,.$$

Therefore, I do not understand why these reference parameters are involved in the very first place, except that they represent some important information. For this reason, I wonder if anyone dealt with this kind regression problem and can share some ideas on

1. Why are these reference parameters needed? What roles do these parameters play in a regression problem? How can I interpret them in terms of the corresponding independent variables?

2. How can I determine their values in advance? Is there any formal way to decide their values?

• I have an idea on Cref. If I am regressing data on the years 1971, 1972, 1973, etc. but what I really want is years since 1970 I would use (year - 1970) as my regression variable, so that the regression is instead on 1, 2, 3, etc. This is common in some types of regression. – James Phillips Aug 18 '18 at 18:24
• Dref might be a scaling factor for unit conversion, for example converting data in units of meters to data in in units of kilometers. – James Phillips Aug 18 '18 at 18:31

Same for the second term. Taking the ln implies there is a belief that term is of the form $e^{x_2/d_{ref}}$. Maybe this is the current in a wire attached to a large battery with an inductor in it. The $X_2$ would be time, and $d_{ref}$ would be a known quantity for different-sized inductors.