# Why we use Ridge regression instead of Least squares in Multicollinearity?

Why do we use Ridge regression instead of Least squares in Multicollinearity?

Which one is correct:

a. lower bias and higher variance
b. lower bias with the same variance
c. higher bias with a lower variance
d. the same bias with lower variance

• Which is correct: I should eat an apple, or I should eat an orange? They are both correct, for different situations. Sometimes you don't mind a bit of bias, sometimes you do. – Jeremy Miles Jan 29 '17 at 19:36
• Is this an exercise/exam question? If so, see stats.stackexchange.com/tags/self-study/info – Juho Kokkala Jan 29 '17 at 19:42
• @JeremyMiles, in this situation the answers are not equally correct, because the situation is well defined: multicollinearity. – Richard Hardy Jan 29 '17 at 20:00
• Oh, thanks. You're right, I should have read the question more closely. – Jeremy Miles Jan 29 '17 at 20:06
• You do not need to put [solved] in the title. There is an accepted answer, it means that this is solved. – amoeba Jan 30 '17 at 11:25

Your OLS estimator is $$\hat{\beta}_{ols} = (X'X)^{-1}X'y,$$ while your ridge regression estimator is $$\hat{\beta}_{ridge} = (X'X + \lambda I)^{-1}X'y.$$ Take the expectation and variance of each one, and then compare your results.