(X i X) 2. n 1 X X. s X. s 2 F (n 1),(m 1)

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3 X X X 10 n 5 X n X N(µ X, σx ) n s X = (X i X). n 1 (n 1)s X σ X n = (X i X) σ X χ n 1. t t χ t (X µ X )/ σ X n s X σx = X µ X σ X n σx s X = X µ X n s X t n 1. F F χ F F n (X i X) /(n 1) m (Y i Y ) /(m 1) = s X s F (n 1),(m 1) Y

4 ε = Y E[Y X 1,, X n ] = Y E[Y ]. ε = Y E[Y ] Y (; ω) = E[Y = ] + ε(; ω), or ε(; ω) = Y (; ω) E[Y = ] Y (; ω) Y = (ε(; )) Y i = β 0 + p β k X ik + ε i. Y t = β 0 + k=1 p β k X tk + ε t. k=1 p Y it = β 0 + β k X itk + ε it. k=1

5 n ˆβ 1 = (Y i Y )(X i X) n (X = ŝ XY i X) ŝ, ˆβ0 = Y ˆβ 1 X. X Y X Y = ˆβ 0 + ˆβ 1 X (X, Y ) y = ˆβ 0 + ˆβ 1 x ˆβ 0 ˆβ 1 Y Y Ŷ = ˆβ 0 + ˆβ 1 X = ˆβ 0 + ˆβ 1 X = Y ˆε = Y ( ˆβ 0 + ˆβ 1 X) = Y ( ˆβ 0 + ˆβ 1 X) = 0 Y n (Ŷi Y )ˆε i = 0 n ˆε ix i = 0 Z Z ( ˆσXk ˆσ Y ) Y i = β 0 +β 1 X i1 +β X i + +β p X ip +ε i ( ) ( ) ( ) ( ) ( ) ( ) Y i Y Xi1 X 1 ˆσX1 Xi X ˆσX Xip X p ˆσXp = β 1 + β + + β p + ˆε i. ˆσ Y ˆσ X1 ˆσ Y ˆσ X ˆσ Y ˆσ Xp ˆσ Y ˆσ Y 1 n n n n ESS = (Ŷi Y ), RSS = (Y i Ŷi) = ˆε i, T SS = (Y i Y ) = ESS + RSS.

6 R R = ESS T SS = 1 RSS T SS, R adj = 1 RSS n p 1 T SS n 1 n p. X i Y X i

7 E[ε X = x] = 0 x ε(x) = y (β 0 + β 1 x) (ε i k, ε i ) i=k+1 k 1 Y [ n n ] ˆβ 1 = c i (Y i Y ), ˆβ0 = Y c i (Y i Y ) X, X c i = i X n i = 1,, n (X i X) E[ ˆβ 1 ] = β 1 E[ ˆβ 0 ] = β 0 ( ˆβ 1 ) = σ ε n (X i X). X ε X N(0, σ ε) Y

8 ˆβ 1 = n (X n i X)(Y i Y ) n (X = (X n i X)(β 0 + β 1 X i + ε i Y ) i X) n (X = β i X) 1 + ε i(x i X) n (X i X) ) N (β 1, σ ˆβ1, σ ˆβ1 = σ ε n (X i X) ˆβ 0 = Y ˆβ 1 X = β 0 n ε i(x i X) ( ) n (X i X) X N β 0, σ ˆβ0. σ ˆβ0 = ( n X i )σ ε n n (X i X) t σε ˆσ ε = RSS n n p 1 = ˆε i n p 1. ˆσ ε ˆσ ˆβ1 = n, ˆσ ˆβ0 = (X i X) n X i n n (X i X) ˆσ ε. χ n p 1 k = 0, 1 ˆβ k β k ˆσ ˆβk t n p 1. Y Y X N(β 0 +β 1 X i1 +β X i +, σy ) ε X N(0, σε) H 0 : β 1 = β = = β p = 0, H 1 : H 0

9 F F = ESS p RSS n p 1 = R p (1 R ) (n p 1) F p,n p 1. p F F F = RSS r RSS ur q RSS ur n p 1 = ESS ur ESS r q RSS ur n p 1 F q,n p 1 RSS r RSS ur n p q F q = p F RSS r RSS ur X 0 Ŷ0 = ˆβ 0 + ˆβ 1 X 0 ) Ŷ 0 N (β 0 + β 1 X 0, σ Ŷ0 σ Ŷ 0 = σ ε [ 1 n + (X 0 X) ] n (X. i X) [ 1 ˆσ = ˆσ Ŷ 0 ε n + (X 0 X) ] n (X, i X) n p 1 Ŷ0 ± t α/,n p 1ˆσŶ0 X X

10 ˆσ ε ˆσ ˆβk = n (X i X) (1 Rk ) R k X k ˆσ ε R k Ŷ i Ŷi 3 4 Ŷi X Y i = β 0 + β 1 X i1 + + ε i Y i = β 0 + β 1 X i1 + + αŷ i + γŷ i 3 + δŷ i 4 + ε i Y i α γ δ F

11 A B RSS r n p 1 A RSS ur,a n A p 1 B RSS ur,b n B p 1 F F = RSS r (RSS ur,a +RSS ur,b ) p+1 RSS ur,a +RSS ur,b. n p F F F F F = RSS r RSS ur q RSS ur n p 1 = ESS ur ESS r q RSS ur n p 1 F q,n p 1 RSS r RSS ur n p q D

12 t σ ˆβk = ˆσ ε (Xik X k ) (1 R k ), ˆσ ε Rk X k X Rk t t k = ˆβ k ˆσ ˆβk t

13 X k X j r kj = s kj s k s j. t 1 V IF k = 1 Rk R k X k t t

14 V ar(ε i ) = σε = i i = 1,,, N i V ar(ε i ) = σiε i = 1,,, N t Y i = β 0 + β 1 X i + ε i σ ε V ar( ˆβ 1 ) = T SS X T SS X = (X i X) β 1 V ar( ˆβ (Xi X) σiε 1 ) = T SSX σiε t t = β β. t σiε X X ε i = α 0 + α 1 X i1 + + α p X ip + u i ε i ˆε i ε i ˆε i = α 0 + α 1 X i1 + + α p X ip + u i ˆε i = δ 0 + δ 1 Ŷ i Ŷi = ˆβ 0 + ˆβ 1 X i1 + + ˆβ p X ip Y i = β 0 + β 1 X i1 + + β p X ip + ε i Y (Ŷi) ˆε i = δ 0 + δ 1 Ŷ i R ˆε F F = R ˆε 1 (1 R ˆε ) n χ = nr ˆε ˆε i = α 0 + α 1 X i1 + + α p X ip + α p+1 X i1 + + α p X ip + α p+1 (X i1 X i ) + + u i,

15 ˆε i ε i ˆε i = δ 0 + δ 1 Ŷ i + δ Ŷi Ŷi Ŷi = ˆβ 0 + ˆβ 1 X i1 + + ˆβ p X ip Y i = β 0 + β 1 X i1 + + β p X ip + ε i Y (Ŷi) ˆε i = δ 0 + δ 1 Ŷ i + δ Ŷi (R ˆε ) F F = R ˆε (1 R ˆε ) n 3 χ = nr ˆε RSS F RSS A RSS B F F = RSS A n p 1 RSS B n p 1 F (X k ) σiε = σ εxik α Y i = β 0 + β 1 X i1 + + β p X ip + ε i ˆε i ˆε i = γ + α X ik + u i α t t = ˆαˆσ ˆα α V ar(ε i ) = σ εh( i ) (Y i ˆβ 0 ˆβ 1 X i1 ˆβ p X ip ). (Y i ˆβ 0 ˆβ 1 X i1 ˆβ ) p X ip. h( i ) h( i ) V ar(ε i ) = σε (α 0 + α 1 X i1 + + α p X ip ) Y i = β 0 + β 1 X i1 + + β p X ip + ε i ˆε i ˆε i ˆε i = γ + δ 1X i1 + + δ p X ip + v i ˆε i = γ + ϕ 1Ŷi + ϕ Ŷi + u i ĝ i = ˆγ + ˆϕ 1 Ŷ i + ˆϕ Ŷi (ĝ i ) ĥi Y i = β 0 + β 1 X i1 + + β p X ip + ε i ĥi.

16 V ar( ˆβ 1 ) = σε c i V ar( ˆβ i ) = c i σ iε (ˆε i ) σ iε se( ˆβ i ) HC = (Xi X) ˆε i ( (Xi X) ). se( ˆβ k ) HC = ˆω ik ˆε i ( ˆω ik ) ˆω ik X j Y i = β 0 + β 1 X i1 + + β p X ip + ε i ˆε i p p (ˆω ik ) se( ˆβ ˆω k ) HC = ik ˆε i ( ˆω. ik ) Cov(ε t, ε s ) = 0 Corr(ε t, ε s ) = 0 t s t Y t = β 0 + p β i X ti + ε t

17 ε t = ρε t 1 + u t, 1 < ρ < 1 u t E[u t ε t 1 ] = 0 V ar(u t ε t 1 ) = σ u Cov(u t, u s ) = 0 t s ε t = u t + ρu t 1 + ρ u t + ρ 3 u t 3 +. E[ε t ] = 0, V ar(ε t ) = σ u + ρ σ u + ρ 4 σ u + = σ u 1 ρ. ρ < 1 ρ = 0 σε = σ u 1 ρ ρ µ r ± 1.96σ r µ r σ r µ r = T 1T T 1 T (T 1 T T 1 T ) + 1, σ r = T 1 + T (T 1 + T ) (T 1 + T 1) r T 1 T T AR(1) AR(1) Y t = β 0 + p β i X ti + ε t, ε t = ρε t 1 + u t. T t= d = (ˆε t ˆε t 1 ) T t=1 ˆε t T t= = ˆε t T t=1 ˆε t + T t= ˆε t 1 T t=1 ˆε t T t= ˆε tˆε t 1 T t=1 ˆε t ˆρˆσ u 1 ˆρ ˆσ u 1 ˆρ (1 ˆρ). T d (1 ˆρ) d d d

18 d L d u (p + 1) H 0 ρ > 0 0 < d < d L d L d d U H 0 d U < d < 4 d L 4 d U d 4 d L H 0 ρ < 0 4 d L < d < 4. d AR(1) d AR(q) AR(q) Y t = β 0 + p β i X ti + ε t, ε t = q ρ j ε 1 j + u t 1 q < T q = 1 j=1 Y t = β 0 + p β ix ti + ε t ˆε t ˆε t = α 0 + p α ix ti + q Rˆε j=1 ρ j ˆε t j + u t F ˆρ 1 ˆρ ˆρ q χ = (n q)rˆε q F H 0 : ρ 1 = ρ = = ρ q = 0 AR(1) ε t = ρε t 1 + u t Y t = β 0 + p β ix ti + ε t ˆε t ρ T t= ˆρ = ˆε t ˆε t 1 T t=1 ˆε t ˆρ = 1 d ˆε t = ρˆε t 1 +u t ˆρ ρ ρ ρ ˆρ ˆρ ρ Y t ρy t 1 = β 0 (1 ρ) + p β i (X ti X (t 1)i ) + u t

19 Yt = Y t ρy t 1 ε t = u t Y t = β 0 + p βi Xti + ε t. Y 1 = ( 1 ρ )Y 1, X 1 = ( 1 ρ )X 1, ε t = ( 1 ρ )ε 1. AR(1) AR(q) Y t = β 0 + p β ix ti + ε t ˆε t X t1 = α 0 + p i= α ix ti + r t ˆr t ˆα t = ˆr tˆε t ˆv = T t=1 ˆα t + [ ] ( g T h=1 1 h g+1 t=h+1 ˆα t ˆα t h ) g X 1 se( ˆβ 1 ) HAC = ( se( ˆβ 1 ) ˆσ ε ) ˆv. X X p Y i = β 0 +β 1 X i +ε i Y Ŷ = ˆβ 0 + ˆβ 1 X E[Y X] ˆP i 0.5 Y = 1 ˆP i < 0.5 Y = 0

20 ˆP i Y Y = 1 ˆP i < Y Y = 0 ˆP i 0.5 Y = 1 ˆP i < 0.5 Y = 0 Y = 0 Y = 1 ˆP i Y Y = 1 ˆP i < Y Y = 0 Y = 0 Y = 1 t F V ar(ε i ) = (β 0 + β 1 X i )(1 β 0 β 1 X i ). E[Y X i ] = P (Y = 1 X i ) = F (β 0 + β 1 X i ), F (0, 1) F F (x) = 1 x / π dξ F (x) = e ξ ex 1+e x β β 0,β 1 ( Y 1 Y n ) = ˆβ 0 ˆL ˆβ 1 [ F ( ˆβ 0 + ˆβ 1 X i ) Yi F ( ˆβ 0 + ˆβ 1 X i ) F ( ˆβ 0 + ˆβ 1 X i ) β 0,β 1 n F (β 0 + β 1 X i ) Yi [1 F (β 0 + β 1 X i )] 1 Yi. ˆβ ˆL = [ ] n Yi F ( ˆβ 0 + ˆβ 1 X i ) (1 Y i)f ( ˆβ 0 + ˆβ 1 X i ) 1 F ( ˆβ 0 + ˆβ = 0 1 X i ) = ] n X i = 0 (1 Y i)f ( ˆβ 0 + ˆβ 1 X i ) 1 F ( ˆβ 0 + ˆβ 1 X i ) R = 1 ˆL ur ˆL 0 L ur L 0 Y

21 Y i = β 0 + β 1 X i + ε i, ε N(0, σ ε), Y i = { Yi b Y i < b Y i b. n { ( ) [ ( )]} β0 + β 1 X i b 1 L = F + F Yi β 0 β 1 X i σ ε σ ε F F { Yi Yi < b Y i = β 0 + β 1 X i + ε i, ε N(0, σ ε), Y i = σ ε Y i b. L = n (πσ ε) 1 σ ε n (Y i β 0 β 1 X i ) n ( ) b β0 β 1 X i F F F σ ε

22 Y i = β 0 + β 1 X i + ε i, ε N(0, σ ε) S i = γ 0 + γ 1 W i1 + γ W i + + u i, S i = { 1 Yi 0 Yi, u N(0, 1), Corr(ε, u) = ρ. { [ ] n ((γ 0 + γ 1 W i1 + γ W i + ) + (Yi β 0 β 1 X i )ρ)/σ ε L = F 1 ρ 1 ( ) Y i β 0 β 1 X i ( πσ ε ) + F ( γ 0 γ 1 W i1 γ W i )} σ ε F S i = γ 0 + γ 1 W i1 + γ W i + + u ˆλ i = F (ˆγ 0 + ˆγ 1 W i1 + ˆγ W i + ) F (ˆγ 0 + ˆγ 1 W i1 + ˆγ W i + ) F Y i = β 0 + β 1 X i + β ˆλi + ε i F Y t = α + δ 0 X t + δ 1 X t 1 + δ X t + + δ r X t r + ε t.

23 Y t = α + δx t + γy t 1 + ε t d d Y t = α 0 +α 1 t+ε t Y t = α 0 +α 1 t+ε t Y t = ˆα 0 + ˆα 1 t + ˆε ty X tk = ˆα 0k + ˆα 1k t + ˆε txk k k ˆε ty = β 0 + β 1ˆε txk + u t Y t = α 0 + α 1 S 1 + α S + + ε t S Y t = β 0 + p β i X ti + q λ i S i + ε t. j=1

24 Y t = ˆα 0 = ˆα 0 + q j=1 ˆα js j + ˆε ty X tk = ˆα 0k + q j=1 ˆα jks j + ˆε txk k k ˆε ty = β 0 + β 1ˆε txk + u t Y i = β 0 + β 1 X i1 + β X i + + δ 1 R i1 + δ R i + + ε i. δ ˆδ Y i = β 0 + β 1 X i1 + β X i + + δ 0 R i + δ 1 (X 1 R) i + δ (X R) i + + ε i. F δ(δ 0, δ 1, δ, ) Y it = β 0 + β 1 X it + β w it + ε it X ω Y

25 ω it = ω i t Y it = β 0 + β 1 X it + β w it + ε it Y it 1 = δ 0 + β 1 X it + β w it + ε it Y i = Y it Y it 1 = (β 0 δ 0 ) + β 1 (X it X it 1 ) + β (ω it ω it 1 ) + (ε it ε it 1 ) = α 0 + β 1 X i + ε i. Y it = n α i0 A i + p β k X it,k + ε it A = 1 i 0 Y it = β 0 + β 1 X it + β ω it + ε it, ω it = ω i Y it = β 0 + β 1 X it + v it v i t = ω it + ε it (ω i ) ε it ω i v it = ω i + ε it (ω i ) k=1 H = ( ˆβ 1(F E) ˆβ 1(RE) ) σ ˆβ1(F E) σ ˆβ1(RE) χ 1

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29 n ˆβ 1 = (Y i Y )(X i X) n (X, ˆβ0 = Y ˆβ i X) 1 X. n ˆσ ε = ˆε i n p 1. n ˆσ ε ˆσ ˆβ1 = n, ˆσ ˆβ0 = X i (X i X) n n (X i X) ˆσ ε. ESS = n (Ŷi Y ), RSS = n (Y i Ŷi) = R t n ˆε i, T SS = R = ESS T SS = 1 RSS T SS. t = ˆβ k ˆσ ˆβk. ˆβ k ± t α/ ˆσ ˆβk n (Y i Y ) = ESS + RSS.

30 Y i = β 0 + β 1 X i + β X i + ε i. Y i = β 0 + β 1 X i + β X i + β 3 X 3 i + ε i. Y i = β 0 + β 1 1 X i + ε i. Y i = β 0 + β 1 X i + ε i. Y i = β 0 + β 1 X i + ε i. Y i = β 0 + β 1 X i + ε i. t