2.5 Forecasting (update).docx

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Department
Statistical Sciences
Course
STA457H1
Professor
Zhou Zhou
Semester
Winter

Description
2.5 Forecasting W n Framework: Let  W= ⋮  and  U  be random variables (W 1 a +a W +a W +…+a W We want to find the best linear combination 0 1 1 2 2 n n  such that the mean squared  2 error (MSE) of  E [( a +( 0 +1 W 1…+2 W 2 n n))]  is minimized Denote the best linear combination by (U W) , where P stands for “projection” P(U ∣W) Properties of  : 1. P U W =E U +a[ ] ́T(W−E W [ ] )  where  Cov(W)a ́=Cov(U ,W )́ −1 If  Cov(W)́  is non­singular, then = [ov(W) Co](U ,W) ́ Proof: HW 2. E [(P U ∣Ẃ )W ]0  and  E [−P(U ∣W)=0] ́ T ́ ́ Proof: Note  P U W =E U +a ] ́ (W−E W [ ] )   ( ∣ ́ ) [ ] ́T ́ [ ] ¿ ́ ¿ Hence  U−P U W =U−E U −a (W−E W =U −W   ( ∣ ́ ) ́ ( ¿ ́T ́ ¿)́ ¿ ́ ́T ́ ¿ ́ E [(U−P U W )W ]E [U −a W W ]=E U W −a E [W W ]   a TW−E W [ ] =E W W a =E W W [́ ́ 8T]a   E [−P(U ∣W )W)=C]v(U ,W)−Cov(W)a ́ ́=0 Hence   by property 1 E U−P U W( ∣́ )]=E U −E a W =E U −a E W =0 ́T [ ]¿ On the other hand,  E [(P(U ∣W ))=]ar [U −a ́ Cov (U ,W ) 3. T −1 ¿Var U ]− [ov(U ,W) Co] [ ) Cov(U],W) ́ ́ ¿ T ́ ¿ Proof: According to the proof of 2, −P U W =U −a W ́ ́ 2 ¿ T́ ¿2 ¿2 T ¿ ́¿ T ́¿2 Therefore,  E [(P(U ∣W ))=] [(U −á W )]=E U −2a ́ E [U W ]+E [́ W )] ¿2 ¿́ ¿ ́ Note:  E U =Var U [ ]  and  E [U W =Cov(U ,W )  and  T ¿2 T T T T ¿ ¿T T ¿ ¿T T E [́ Ẃ )]=E á Ẃ(́ Ẃ ) =E [́ W W a ́]=á E [W W ]́=á Cov (W)a ́   ́ ́ By 1,  Cov(W)a ́=Cov(U ,W ) MSE=Var U −2a Cov U ,W +a Cov U ,W =Var U −a Cov U ,W ) ́ T ( ́ )   P ( 1+α V 2β W =∣ P) 1 (U Ẃ +α 2 (V Ẃ +β α 1α 2 β 4.  where   are constants Proof: HW n n 5. P (∑ αiW +i W∣= ) ∑ α i +i i=1 i=1 Proof: HW P(U ∣W)=E [U ] Cov(U ,W)=0 6.  if  Proof: Use property 1 Properties of  PnX n+h : Here  PnX n+h  denotes the best linear forecast of h  based on  X 1 X 2…, X n n a1 P n n+hμ+ ∑ a i n+1−iμ ) an= ⋮ Γnán=γ nh) 1. i=1  where  (a  satisfies  n Xn Here  Γ nCov X ( n =́ n ⋮  and  γnh = [γ h) … γ n+h−1 )] ()X 1 Proof: Use previous property 1 2 T 2. (MSE of forecast)  E [ n+hP Xn n+h)]γ(0)−a ́nγn(h) γ(0)=Var(X ) Reminder:  i Proof: Use previous property 3 3. E [ n+h−P n (n+h)0 Proof: Use previous property 2 4. E [ n+hP Xn n+h)X j0, j=1,2,…,n Proof: Use previous property 2 Example: Forecasting AR(1) Xt=aX t−1+Z t 2 Where  ZtWN 0,σ )  and  ∣a∣<1 PnX n+h=? h h−1 Note:  X n+ha X +n n+h+aZ n+h−1+…+a Zn+1   h h h h h−i h h−i h h−i h PnX n+h=P n[X + n ∑ a Z n+]=P n[X +Pn] n[∑ a Z n+](5)a X +n6) ∑ a P n n+iX n i=1 i=1
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