CHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE But we should not use =B1+BXt+61D1t+62D2+63D3+64D+t since we run into a multicollinearity problem Example 3(Threshold effects and categorical variables) income=B,+B2age+SBB+SMM+OpP+E n{ 1, Bachelor's degree only 0. otherwise M=,Master's and Bachelor's degrees 0. otherwise 1, Ph. D, Master's and Bachelor's degrees 0. otherwise High school E Incomelage, HS=B,+B2age Bachelor 's E[Incomelage, B=B1+B2age +8B Master's EIncomelage, M=B1+B2age +8B+SM E [Incomelage, P=B,+B2age+8B+5M+8p SB: marginal effect of Bachelor's degree SM: marginal effect of Master's degree Sp: marginal effect of Ph. D's degree Example 4(Spine regression Suppose that E[ncomelagel= a0+Bo if t*>age>t E[ncomelagel=a+B if age >t2(t1<t2)CHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE 2 But we should NOT use yt = β1 + β ′Xt + δ1D1t + δ2D2t + δ3D3t + δ4D4t + εt since D1t + D2t + D3t + D4t = 1, we run into a multicollinearity problem. Example 3 (Threshold effects and categorical variables) income = β1 + β2age + δBB + δMM + δP P + ε B
= 1, Bachelor’s degree only = 0, otherwise M
= 1, Master’s and Bachelor’s degrees = 0, otherwise P
= 1, Ph.D, Master’s and Bachelor’s degrees = 0, otherwise High school E [Income|age, HS] = β1 + β2age Bachelor’s E [Income|age, B] = β1 + β2age + δB Master’s E [Income|age, M] = β1 + β2age + δB + δM Ph.D E [Income|age, P] = β1 + β2age + δB + δM + δP δB : marginal effect of Bachelor’s degree δM : marginal effect of Master’s degree δP : marginal effect of Ph.D’s degree Example 4 (Spine regression) Suppose that E [Income|age] = α 0 + β 0 if t ∗ 2 > age ≥ t ∗ 1 E [Income|age] = α 1 + β 1 if age ≥ t ∗ 2 (t1 < t∗ 2 )