Class Notes (839,195)
STA457H1 (9)
Zhou Zhou (7)
Lecture

# 2.5 Forecasting.docx

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Department
Statistical Sciences
Course Code
STA457H1
Professor
Zhou Zhou

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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 =] [U −a W W ]=E [U W ]−a E [W W ]   aT(W−E W [ ] =E W W a =E W W [́ ́ 8Tá   E [(P(U ∣W )) =]ov(U
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