CHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE 7.3 Structural change Examples of events that led to structural change Oil shock of 1973 Emergency of Deng XiaoPing ata to Y2, X2: data for period 2 We want to test whether coefficient vector undergoes changes from period 1 to period 2 1. Unrestricted model that allows structural change X1 0B 0X2|A2 E2 b=(X'X)Xy XIXi 0 0X2X2(X2 (X1X1) 0 91 0 (X2X2) Thus -x=(0)( X161 X.bo el unrestricted residual ee=betel 2. Restricted model XI + El 2 e"=y-Xb: restricted residual The null of no structural change can be written as RB=rCHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE 5 7.3 Structural change Examples of events that led to structural change Oil shock of 1973 Emergency of Deng XiaoPing Y1, X1 : data for period 1 Y2, X2 : data for period 2 We want to test whether coefficient vector undergoes changes from period 1 to period 2. 1. Unrestricted model that allows structural change y1 y2  = X1 0 0 X2  β1 β2  + ε1 ε2  b = (X ′X) −1 X ′ y =  X′ 1X1 0 0 X′ 2X2   X′ 1 y1 X′ 2 y2  =  (X′ 1X1) −1 0 0 (X′ 2X2) −1   X′ 1 y1 X′ 2 y2  =  b1 b2  Thus e = y − Xb =  y1 y2  −  X1b1 X2b2  =  e1 e2  : unrestricted residual and e ′ e = e ′ 1 e1 + e ′ 2 e2. 2. Restricted model y1 y2  = X1 X2  β + ε1 ε2  or y = X ∗β + ε b ∗ = (X ∗′X ∗ ) −1 X ∗′y, e ∗ = y − X ∗ b ∗ : restricted residual The null of no structural change can be written as H0 : Rβ = r