February 4.docx

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Department
Statistical Sciences
Course
STA457H1
Professor
Zhou Zhou
Semester
Winter

Description
February 4, 2014 Chapter 2 – Stationary Time Series * Forecasting with optimality 1. The case of normal time series X 1 …, X ns a time series An easy-to-implement criterion for forecasting X n+1is to minimize the expected squared loss In other words, we will find a function f st E([X n+1– f(1 , …, n )] ) is as small as possible, where f is considered in some class of functions Theorem f should be E[X |X , …, X ] n+1 1 n ie. f = E[n+1X 1 …, X n a.s. Proof: Let g be any function of X 1 …, X n Let g* = E[X n+1,1…, X ]n E([X n+1 g(x 1 …, xn)] ) = E([(n+1– g*) + (g* – g)] ) = E[(n+1– g*) ] + 2E[(Xn+1– g*)(g* – g)] + E[(g* – g) ] # 2E[(X n+1– g*)(g* – g)] = 2E{E[(Xn+1– g*)(g* – g)|X1, …, Xn]} # (Law of iterated conditional expectation) Since (g* – g) is a function of X1, …, Xn # = 2E{(g* – g)2[X n+1– g*|X 2 …, X n} = 2E{(g* – g) 0} = 0 = E[(X n+1– g*) ] + E[(g* – g) ] => for all g, E[(n+1– g) ] ≥ E[(Xn+1– g*) ] Theorem X 1 μ 1 ∑ 11 ∑ 12 If (⃗ ) N ()( , ) X 2 ( μ 2 ∑ 21 ∑ 22) Def: Let X be a p-dimensional random vector T T T We say that X ~ N(µ, Σ) iff a X ~ N(a µ, a Σa) for any p-dimensional constant vector a (Σ is a pxp matrix, X and µ and a are vectors) Then the distribution -1 (X1|X2= a) is N(µ, Σ) where µ = µ 1 Σ Σ12 22 (a - 2 ) Σ = Σ 11Σ Σ12 22Σ 21 Corollary ⃗ µ ⃗ If X1, …, Xnis a normal time series st E[ X n ] = n and Cov[ X n ] = n X n γn Further assume that E[X ] n+1 n+1and Cov[X n+1 ] = X1 X = Here n … ()Xn Then the best forecast of X n+1is E[X n+1,nX ,n-1 X ] =1µ n+1+ γn TΣ n1[ X n – µn ] Proof: Plug in the above Theorem Observation: Optimal forecast depends critically on the covariance of your time series 2. What if the time series is not normal? One solution: Think of the best linear forecast In other words, try to find a function 0 + a1X1+ … + a Xnsn E([X n+1 (a0+ a 1 +1… + a X n] n** is as small as possible Find the solution! Let Y i X i µ i µ i E[X] i ** = E([Y + µ – a – a (Y – µ ) – … – a (Y – µ )] )2 n+1 n+1 o 1 1 2 n n n = E[(Y n+1– a0* – a1Y1– … – a Yn)n] where a 0 = a 0 a µ1+1… + a µ –nµn n+1 a 1 Let a= ⋮ (a n Y 1 Y n ⋮ () Yn a T Y 2 => ** = E[(Y n+1– a0* - n ) ] Open the squares 2 2 a T Y 2 a T Y a T Y ** = E[Yn+1] + (a0*) + E[( n ) ] – 2a0*E[Yn+1– 2E[Y n+1 n ] + 2a0* E[ n ] And to minimize **, a 0 should = 0 ⃗ ⃗ ** = E[Yn+1] + E[( a T Yn ) ] – 2 a TE[Y n+1 Y n ] γn ⃗X X Let = Cov[X ,n+1 n ] and ∑ n Cov[ n ] By the fact that E[Yi = 0 for all i, we have Y γ E[Y n+1 n ] = n and E[ Y n n T] = Σn ⃗ ⃗ ⃗ ⃗ ⃗ Observe that E[( a T Y n ) ] = E[ a T Y n ( a T Y n ) ] = E[ a T Y n Y n T a ] = a T Y Y T a a T a E[ n n ] = Σn ** = E[Yn+1] + a TΣ n a – 2 a T γn a a Since ** is a convex function of , differentiate ** wrt and set = 0 ∂**/∂ a = Σn a + Σn a – 2 γ n = 2Σ n a – 2 γn = 0 γ => a = Σ n1 n Finally 0 = 0 * =0a +1a1µ + … + n n – n+1 => a0+ a T µn – µn+1= 0
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