Difference between a multiple regression and several linear regressions?

Question

Context: I'm an undergrad who took Intermediate Econometrics more than a year ago and I'm trying to brush up on some of the skills.

As I was reviewing multiple regression, I realized I didn't quite understand what makes the regressor coefficients different when doing a multiple regression vs doing a univariate regression. For instance, if I were to regress wages on years of education, why would the coefficient on years of education be different than when I regressed wages on years of education and test scores?

Answer 1

Beyond the assumption of multi-collinearity pointed out by @Mike J, the economist's goal is to infer that one variable (such as education) has a causal effect on another variable (such as wages). This is known as the notion of ceteris paribus, which means holding other relevant factors constant.

Multiple regression analysis is more amenable to ceteris paribus analysis

It allows us to explicitly control for many other factors that simultaneously affect the dependent variable. This explains why your coefficient on years of education in a simple regression is different from the one you get when you regress wages on years of education and test scores. In the multiple regression case you control for test scores, so the effect of education will be conditional on test scores. In other words, you are taking `test scores' out of the error term and put it explicitly into the equation.

Two important notes

1. The interpretation of the years of education estimate is now different. You are comparing the effect of an additional year of education for a given test score. For example, in the case of a simple regression, you compare Jane, who had 12 years of education and a score of 90, to John, who had 11 years of education and a score of 50. In the case of a multiple regression, you would compare a Jane and a John with the same test score but with a difference of one year of education.
2. Economists do not believe that multiple regression analysis buys you causality, but it is at least better than simple regression!

Answer 2

The data might be correlated which can lead to simple and multiple linear regression models giving different results.

Which goes against the assumption of no multi-collinearity: Multicollinearity occurs when independent variables in a regression model are correlated.

When modeling years of education to wages, the results may show significance but modeling both years of education and test scores could show years of education as not significant because years of education and test scores are correlated.

Try plotting years of education and test scores.