CHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE 2. Test of structural change with unequal variances Suppose that Eli N iid (0, 02) and E2i N iid (0, 02)(02+02).In additions, assume (X1, E1) and(X2, E2) are independent. Wald test for the null hypothesis B1=B2 is defined by a)(xx)-2+(XX2)](-A2) asn→∞,W→x2(K2). The formula follows because under the null E B. Am()=()+m() The second equality uses the independence assumption 3. Insufficient observation Suppose that n2 K. This implies that e2 cannot be calculated. In this case, use e1e1)/n2 F=飞e1/(mn1-K) We are using e =el Fisher(1970, Econometrica)show thatCHAPTER 7 FUNCTIONAL FORM AND STRUCTURAL CHANGE 7 2. Test of structural change with unequal variances Suppose that ε1i ∼ iid (0, σ2 1 ) and ε2i ∼ iid (0, σ2 2 ) (σ 2 1 = σ 2 2 ). In additions, assume (X1, ε1) and (X2, ε2) are independent. Wald test for the null hypothesis H0 : β1 = β2 is defined by W = βˆ 1 − βˆ 2 ′  s 2 1 (X ′ 1X1) −1 + s 2 2 (X ′ 2X2) −1 −1 βˆ 1 − βˆ 2 as n → ∞, W d→ χ 2 (K2). The formula follows because under the null E βˆ 1 − βˆ 2 = 0 Asy.V ar βˆ 1 − βˆ 2 = σ 2 1plim  X′ 1X1 n1 −1 + σ 2 2plim  X′ 2X2 n2 −1 . The second equality uses the independence assumption. 3. Insufficient observation Suppose that n2 < K. This implies that e2 cannot be calculated. In this case, use F = (e ∗′e ∗ − e ′ 1 e1) /n2 e ′ 1e1/ (n1 − K) . (We are using e = e1) Fisher (1970, Econometrica) show that F ∼ F (n2, n1 − K)