x 1 = x i1 x i2 y = x 1 β x K β K + ε, x i =

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3 x k T x k k = 1,, K T K X X x 1 = 1 β 1 y T y 1 y T ε T T 1 x i1 x i2 y = x 1 β x K β K + ε, x i = y T 1 = X T K β K 1 + ε T 1. x it T 1 y x 1 x K y = Xβ + ε X T K K E[ε i x j1, x j2,, x jk ] = 0 i ε i E[ε X] = 0. ε i σ 2 ε j E[εε X] = σ 2 I. (x j1, x j2,, x jk ) X ε X N(0, σ 2 I). β 1 β 2 β K ( ) 2 T T K RSS = ˆε ˆε = ˆε 2 t = y t x it β i t=1 t=1 i=1

4 ˆβ 1 ˆβ = ˆβ 2 = (X X) 1 X y. ˆβ k σ 2 s 2 = RSS T T K = t=1 ˆε2 t T K K K K T K ( ˆβ) = s 2 (X X) 1. ˆβ = (X X) 1 X y = β + (X X) 1 X ε s 2 T t=1 = ˆε2 t T K ( ˆβ) = s 2 (X X) 1. t ˆβ i SE( ˆβ t(t K) i ) SE( ˆβ i ) = ( ˆβ) ii F F β F F RRSS URSS URSS T K m F (m, T K) URSS RRSS m T K

5 RRSS URSS t F t 2 (T K) F (1, T K). (X X) x k x j r kj = s kj s k s j. t 1 V IF k = 1 Rk 2 R 2 k x k

6 t R 2 y

7 E[ε 2 i X] T T 1 T 2 s 2 1 = ˆε 1 ˆε 1/(T 1 K), s 2 2 = ˆε 2 ˆε 2/(T 2 K) GQ GQ = s2 1 s 2 2 s 2 1 > s 2 2 F (T 1 K, T 2 K) GQ y t = β 1 + β 2 x 2t + β 3 x 3t + ε t. (ε t ) = σ 2 ˆε t ˆε 2 t = α 1 + α 2 x 2t + α 3 x 3t + α 4 x 2 2t + α 5 x 2 3t + α 6 x 2t x 3t + ν t. (ε t ) = E[ε 2 t ] E[ε t ] = 0 ˆε 2 t F ˆε 2 t F R 2 R 2 R 2 R 2 T T R 2 χ 2 (m) m F

8 α 2 = α 3 = α 4 = α 5 = α 6 = 0 χ 2 x k σε 2 t = σεx 2 α kt y t = β 1 + β 2 x 2t + + β K x Kt + ε t ˆε 2 t ˆε 2 t = γ + α x kt + ν t α t t = ˆαˆσ ˆα α σ 2 σ 2 z t (ε t ) = σ 2 z 2 t. z t y t 1 x 2t x 3t = β 1 + β 2 + β 3 + ν t z t z t z t z t ν t = εt z t

9 se( ˆβ T 1 ) HC = t=1 (x t x) 2ˆε 2 t ( T ) 2. t=1 (x t x) 2 se( ˆβ T k ) HC = t=1 ˆω2 tk ˆε2 t ) 2 ( T t=1 ˆω2 tk ˆω tk 2 x k y t = β 1 +β 2 x 2t + +β K x Kt +ε t ˆε 2 t K 1 T (K 1) (ˆω tpk 2 ) se( ˆβ T k ) HC = t=1 ˆω2 tk ˆε2 t ) 2. ( T t=1 ˆω2 tk ˆε t ˆε t 1 ˆε t (ˆε t 1, ˆε t ) (ˆε t 1, ˆε t ) ˆε t

10 µ r ± 1.96σ r µ r σ r µ r = 2T 1T 2 2T 1 T 2 (2T 1 T 2 T 1 T 2 ) + 1, σ r = T 1 + T 2 (T 1 + T 2 ) 2 (T 1 + T 2 1) r T 1 T 2 T t ε t = ρε t 1 + ν t ν t N(0, σ 2 nu) H 0 : ρ = 0, H 1 : ρ 0. T t=2 DW = (ˆε t ˆε t 1 ) 2 T t=2 ˆε2 t 2(1 ˆρ) ˆρ t 1 t d U d L AR(1)

11 r ε t = ρ 1 ε t 1 + ρ 2 ε t ρ r ε t r + ν t, ν t N(0, σ 2 ν). H 0 : ρ 1 = ρ 2 = = ρ r = 0, H 1 : ρ 1 0 ρ 2 0 ρ r 0. ˆε t R 2 ˆε t = γ 1 + K γ i x it + i=2 r ρ j ˆε t j + ν t, ν t N(0, σν). 2 T j=1 (T r)r 2 χ 2 r. (T r) R 2 T r r (T r) r r R 2 K y t = β 1 + β i x it + ε t, ε t = ρε t 1 + ν t. i=2 1 < ρ < 1 ν t E[ν t ε t 1 ] = 0 (ν t ε t 1 ) = σ 2 ν (ν t, ν s ) = 0 t s ε t = ν t + ρν t 1 + ρ 2 ν t 2 + ρ 3 ν t 3 +.

12 E[ε t ] = 0, (ε t ) = σ 2 ν + ρ 2 σ 2 ν + = σ2 ν 1 ρ 2. ρ < 1 ρ = 0 σε 2 = σ2 ν 1 ρ 2 CO y t = β 1 + K β i x it + ε t i=2 ε t = ρε t 1 + ν t. ˆε t = ρˆε t 1 + ν t. ˆρ yt = y t ˆρy t 1 β1 = (1 ˆρ)β 1 x 2t = (x 2t ˆρx 2(t 1) K yt = β1 + β i x it + ν t y t = β 1 + i=2 K β i x it + ν t i=2 ˆρ AR(1) AR(q) P W

13 y t = β 1 + K i=2 β ix it + ε t ˆε t x 2t = α 1 + K i=3 α ix it + r t ˆr t ˆα t = ˆr tˆε t 4(T /100) 2/9 ˆv = T t=1 ˆα2 t + 2 [ ] ( g T h=1 1 h g+1 t=h+1 ˆα t ˆα t h ) g x 2 se( ˆβ 2 ) HAC = ( se( ˆβ ) 2 2 ) ˆv. ˆσ ε x 3 x K y

14 y t W = T [ b (b 2 3) 2 ] 24 T b 1 b 2 b 1 = E[ε3 ] (σ 2 ) 3/2, b 2 = E[ε4 ] (σ 2 ) 2. W χ 2 (2) b 1 b 2 ˆε

15 ŷ 2 t ŷ 3 t ŷ 4 t y y t = β 1 + β 2 x 1t + + β K x Kt + ε t y t = α 1 + α 2 ŷ 2 t + + α p ŷ p t + K β i x it + ν t. y t F T R 2 χ 2 (p 1) R 2 A B RSS r T 2K A RSS ur,a T A K B RSS ur,b T B K F F = i=1 RSS r (RSS ur,a +RSS ur,b ) K RSS ur,a +RSS ur,b. T 2K F F F (K, T 2K) F F

16 y y RSS RSS RSS 1 T 1 RSS RSS 1 RSS T 1 K T 2 T 2 F (T 2, T 1 K) F F F F F ±2 ±2

17 y = Xβ + ε E[ε X] = 0 E[εε X] = σ 2 Ω = Σ, Ω n T ˆβ = (X X) 1 X y = β + (X X) 1 X ε F t ˆβ X [ ˆβ X] = E[( ˆβ β)( ˆβ β) X] = E[(X X) 1 X εε X(X X) 1 X] = (X X) 1 X (σ 2 Ω)X(X X) 1 ( ) 1 ( ) ( ) 1 = σ n n X X n X ΩX n X X ( ) 1 ( ) ( ) = n X X n Φ n X X. ( ) Φ = σ2 n X 1 ΩX = n X (y Xβ) E X [[b X]] ˆβ ε ε ˆβ X N(β, σ 2 (X X) 1 (X ΩX)(X X) 1 ) [ ˆβ X] ˆβ (X X/n) 1 (σ 2 /n)(x ΩX/n) Q = p (X X/n) p (X ΩX/n) ˆβ β p ˆβ = β.

18 X Ω ˆβ Ω [ ˆβ] = σ2 n Q 1 p ( 1 n X ΩX ) Q 1. Ω Ω Ω Ω Ω ˆβ σ 2 Ω ˆβ V OLS = 1 n ( ) 1 ( ) ( ) n X X n X [σ 2 Ω]X n X X. σ 2 Ω = E[εε X] (Ω) = n σ 2 Ω = σ 2 I Σ = (σ ij ) i,j = σ 2 Ω = σ 2 (ω ij ) i,j K(K + 1)/2 Q = 1 n X ΣX = 1 n σ ij x i x n j. x i i X Q i = 1,, K i,j=1 x i1 x i2 X = [x 1,, x K ], x i = x i = [x 1i, x 2i,, x Ki ] x j = [x 1j, x 2j,, x Kj ], 1 i, j n. x in x 1 x 2 X =, X = [ x 1, x 2,..., x n ], X ΣX = x n n σ ij x i x j i,j=1 ˆβ β ˆε i ε i X ˆε Q

19 S 0 = 1 n ˆε 2 i n x i x i i=1 p S 0 = p Q. [ ˆβ] = 1 ( ) ( ) n ( ) 1 1 n n X X ˆε 2 i n x i x i n X X = n (X X) 1 S 0 (X X) 1. i=1 ˆβ Q = 1 n σ ij x i x j n ˆQ = 1 n i,j=1 n i,j=1 ˆε iˆε j x i x j ˆQ 1/n n 2 ˆQ X ˆQ ˆQ = S n L n l=1 t=l+1 w lˆε tˆε t l ( x t x t l + x t l x t), w l = 1 l L + 1. L L T 1/4

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