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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Variance decomposition of forecast errors ： ウィキペディア英語版 Variance decomposition of forecast errors In econometrics and other applications of multivariate time series analysis, a variance decomposition or forecast error variance decomposition (FEVD) is used to aid in the interpretation of a vector autoregression (VAR) model once it has been fitted.〔Lütkepohl, H. (2007) ''New Introduction to Multiple Time Series Analysis'', Springer. p. 63.〕 The variance decomposition indicates the amount of information each variable contributes to the other variables in the autoregression. It determines how much of the forecast error variance of each of the variables can be explained by exogenous shocks to the other variables. == Calculating the forecast error variance == For the VAR (p) of form :$$ y_t=\nu +A_1y_+\dots+A_p y_+u_t . This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p)) :$$ Y_t=V+A Y_+U_t where ::$$ A=\begin A_1 & A_2 & \dots & A_ & A_p \\ \mathbf_k & 0 & \dots & 0 & 0 \\ 0 & \mathbf_k & & 0 & 0 \\ \vdots & & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & \mathbf_k & 0 \\ \end , $$ Y=\begin y_1 \\ \vdots \\ y_p \end , $V=\backslash begin$ \nu \\ 0 \\ \vdots \\ 0 \end and $$ U_t=\begin u_t \\ 0 \\ \vdots \\ 0 \end where $y\_t$, $\backslash nu$ and $u$ are $k$ dimensional column vectors, $A$ is $kp$ by $kp$ dimensional matrix and $Y$, $V$ and $U$ are $kp$ dimensional column vectors. The mean squared error of the h-step forecast of variable j is :$$ \mathbf()=\sum_^\sum_^(e_j'\Theta_ie_k)^2=\bigg(\sum_^\Theta_i\Theta_i'\bigg)_=\bigg(\sum_^\Phi_i\Sigma_u\Phi_i'\bigg)_, and where : *$e\_j$ is the jth column of $I\_K$ and the subscript $jj$ refers to that element of the matrix : *$\backslash Theta\_i=\backslash Phi\_i\; P\; ,$ where $P$ is a lower triangular matrix obtained by a Cholesky decomposition of $\backslash Sigma\_u$ such that $\backslash Sigma\_u\; =\; PP\text{'}$, where $\backslash Sigma\_u$ is the covariance matrix of the errors $u\_t$ : * $\backslash Phi\_i=J\; A^i\; J\text{'},$ where $$ J=\begin \mathbf_k &0 & \dots & 0\end , so that $J$ is a $k$ by $kp$ dimensional matrix. The amount of forecast error variance of variable $j$ accounted for by exogenous shocks to variable $k$ is given by $\backslash omega\_\; ,$ :$$ \omega_=\sum_^(e_j'\Theta_ie_k)^2/MSE() . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Variance decomposition of forecast errors」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.