Wald Decomposition Theorem

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    Wald Decomposition Theorem
    Any zero mean covariance stationary processxt can be represented in the form of
    xt =
    1X
    j =0
    dj t j + j ; where d0 = 1 ;
    1X
    j =0
    d
    2
    j < 1
    The termt is white noise and represents the prediction error defined to bet = xt P[xt j xt 1;].
    The value oft is uncorrelated witht j for anyj, thought can be predicted perfectly byxt 1;xt 2;.
    Proof
    Lett = xt P[xt j xt 1;], whereP[xt j xt 1;] is the prediction ofxt based on a linear function
    ofxt 1;xt 2;. In order thatP[xt j xt 1;] is a linear projection,t and xt 1;xt 2;must not be
    correlated with one another so that we have,
    E(txt j ) = 0 for j 1 (1)
    And since t s = xt s P[xt s j xt s 1;], then we get
    E(tt s) = E
    t(xt s P[xt s j xt s 1;])
    = E(txt s) E(t P[xt s j xt s 1;])
    = 0 from Eq.(1)
    (2)
    Hence we have proved thatftg is serially uncorrelated process.
    Now consider the projection ofxt against t;t 1;;t m for sufficiently largem. Letting ˆxt
    (m)
    denote the linear projection ofxt againstt;;t m, the typical projection ofxt is given by,
    ˆxt
    (m) =
    mX
    j =0
    dj t j
    Applying Hamilton[1994](Chap4, Eq.(4.1.13)) to this problem and noticing thatt are serially uncorre-
    lated, each coefficientsdj is given by,
    dj =
    E(xtt j )
    E(2
    t)
    (3)
    Now since t = xt P[xt j xt 1;]; E(txt j ) = 0(j 1), we have
    E(
    2
    t) = E
    t(xt P[xt j xt 1;])
    = E(txt) E(t P[xt j xt 1;])
    = E(txt) from Eq.(1)
    ) d0 =
    E(xtt)
    E(2
    t)
    =
    E(
    2
    t)
    E(2
    t)
    = 1
    (4)
    1
    And letting E(
    2
    t) =
    2, the variance of the prediction error can be calculated as,
    E
    xt
    mX
    j =0
    dj t j
    2
    = E

    資料の原本内容( この資料を購入すると、テキストデータがみえます。 )

    Wald Decomposition Theorem
    Any zero mean covariance stationary processxt can be represented in the form of
    xt =
    1X
    j =0
    dj t j + j ; where d0 = 1 ;
    1X
    j =0
    d
    2
    j < 1
    The termt is white noise and represents the prediction error defined to bet = xt P[xt j xt 1;].
    The value oft is uncorrelated witht j for anyj, thought can be predicted perfectly byxt 1;xt 2;.
    Proof
    Lett = xt P[xt j xt 1;], whereP[xt j xt 1;] is the prediction ofxt based on a linear function
    ofxt 1;xt 2;. In order thatP[xt j xt 1;] i..

    コメント9件

    manaryu 購入
     
    2006/12/21 22:46 (10年前)

    yoshinori 購入
    good
    2006/12/22 0:06 (10年前)

    dragstar 購入
    参考になりました
    2006/12/22 7:33 (10年前)

    syu_30 購入
    good
    2006/12/25 0:40 (10年前)

    kanotch 購入
    参考になりました!
    2006/12/29 11:02 (10年前)

    shu600507 購入
    good!
    2007/01/02 12:35 (9年11ヶ月前)

    cticdire 購入
    b
    2007/01/08 22:04 (9年11ヶ月前)

    leehikaru 購入
    good!
    2007/01/17 8:59 (9年11ヶ月前)

    kentaro1214 購入
    参考になりました。
    2007/02/03 18:32 (9年10ヶ月前)

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