We now turn to multiple regression and, in particular, the three-variable regression model.

The population regression function is:

Y_i = β₀ + β₁X_1i + β₂X_2i + i

Consider what happens if X2i = 1 + 2X1_i.

Yi = β₀ + β₁X_1i + β₂X_2i + i
Yi = β₀ + β₁X_1i + β₂(1 + 2X_1i) + i
Yi = β₀ + β₁X_1i + β₂ + β₂2X_1i + i
Yi = (β₀ + β₂) + (β₁ + 2β₂)X_1i + i
Yi = β₀0 + β₀₁X_1i + i

You can see that we're only getting two parameters (β₀ and β₀¹) back even though the original model speciβcation had three (β₀, β1, and β2).

There are multiple values of β₀, β₁, and β₂ which will solve the two equations:

β0¹ = β₁ + 2β₂

β0⁰ = β₀ + β₂

The bottom line is that we cannot estimate the separate in uence of X₁ and X₂ on Y.

What we have been discussing so far is really perfect multicollinearity.

In the case where we have two independent variables, this occurs when the correlation, r, between two variables is 1.

In the case where we have more than two independent variables, this occurs when your independent variables, or combinations of your independent variables, are not linearly independent, i.e., your matrix of independent variables is not of full rank.