# Multiple Regression with Predictors that Restrict other Predictors

I'm not even sure if the title of my question makes sense at first sight, so let me try to explain it. I'd like to fit a parametric multiple regression model to data. But depending on the value chosen for a given predictor (it actually should only be a discrete integer, but maybe I can also consider it as continuous), the possible values for another predictor are restricted to a subset of values.

Would this make sense in a regression exercise? Would it incur multicollinearity? Any suggestions of how to go about this problem?

Thanks!

In general, deterministically related regressors are not prohibited in a regression model.
In introductory econometrics textbooks one may encounter a model of the form

$$y=\beta_0+\beta_1 x+\beta_2 x^2+\varepsilon$$

This is one special case of what you are concerned about, if I understand correctly.

A restriction in the regression model is that the regressors cannot be linearly dependent. (For example, you cannot have $x_1=\alpha_0+\alpha_1 x_2$.) If they are, the marginal effects of the linearly dependent regressors cannot be disentangled; the $\beta$ coefficient vector taken from the regression $y=X \beta + \varepsilon$ is not unique ($X$ is the design matrix where each column is a different regressor). Other than that, you may have deterministically related regressors.

However, you have to be careful interpreting the estimated coefficients when the regressors are deterministically related. If you have both $x$ and $x^2$ as regressors, you cannot say that "keeping all other regressors constant, a unit change in $x$ brings a $\beta_1$ change in $y$" -- because when $x$ changes, $x^2$ also changes.

• Thanks, Richard. In my case, I could think of a conditional statement between predictors. For example, if $x_1$ takes on a particular value, say $a$, then another predictor $x_2$ should only take on a subset of values. I'm afraid that if I just consider these two predictors independently in the regression model, I won't capture that relationship. So I'm wondering if that can be done, and how. Apr 27, 2015 at 11:40