# Why can we vary parameters in linear regression?

I want to understand what the formal justification is for relating the response variable to new values of explanatory variables in linear regression. I see linear regression models defined as $$Y=X\beta +\epsilon$$, where $$Y$$ is a random vector, $$X$$ is a given matrix of constants, $$\beta$$ is a vector of unobservable fixed parameters, and $$\epsilon$$ is a random vector of errors with mean 0.

What in this formulation implies that we can generalize the relationship between $$Y$$ and the given rows of $$X$$ to "new rows" or observations? I basically understand intuitively that we're approximating the relationship between $$Y$$ and a some "quantity" $$x$$ with a linear function of $$x$$, but where is the idea that we can vary $$x$$ built into the concept of linear regression?

• The title appears misleading. You are asking about varying the regressors not the parameters. Commented Nov 26, 2016 at 12:55

The $$X$$'s that you refer to as constants are values of the explanatory variables. Linear regression assumes that the outcome variable $$Y$$ is a linear (in the betas) function of the $$X$$'s. But the result is subject to a small amount of random noise which is represented by the epsilons in the model.