|
|
| Line 1: |
Line 1: |
| 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.<ref>Lütkepohl, H. (2007) ''New Introduction to Multiple Time Series Analysis'', Springer. p. 63.</ref> 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.
| | Oscar is how he's known as and he [http://Www.Onhealth.com/genital_herpes_in_women/article.htm completely loves] this name. North Dakota is her beginning location but she will have to transfer one day or an additional. The favorite pastime for my children [http://reinodoprazer.com.br/index.php?do=/blog/326/tips-about-how-to-overcome-candidiasis-easily/ std testing at home] std home test and me is to play baseball and I'm attempting to make it at home std testing [http://muzikoman.com/profile-682/info/ std home test] a occupation. Managing people has been his working home std test day occupation for a whilst.<br><br>Feel free to visit my website - [http://faculty.jonahmancini.com/groups/solid-advice-for-dealing-with-a-candida-albicans/members/ jonahmancini.com] |
| | |
| == Calculating the forecast error variance ==
| |
| For the VAR (p) of form
| |
| | |
| :<math>
| |
| y_t=\nu +A_1y_{t-1}+\dots+A_p y_{t-p}+u_t
| |
| </math> .
| |
| | |
| This can be changed to a VAR(1) structure by writing it in companion form (see [[general matrix notation of a VAR(p)]])
| |
| :<math> | |
| Y_t=\mathbf{\nu} +A Y_{t-1}+U_t
| |
| </math> where
| |
| | |
| ::<math>
| |
| A=\begin{bmatrix}
| |
| A_1 & A_2 & \dots & A_{p-1} & A_p \\
| |
| \mathbf{I}_k & 0 & \dots & 0 & 0 \\
| |
| 0 & \mathbf{I}_k & & 0 & 0 \\
| |
| \vdots & & \ddots & \vdots & \vdots \\
| |
| 0 & 0 & \dots & \mathbf{I}_k & 0 \\
| |
| \end{bmatrix}
| |
| </math> , <math>
| |
| Y=\begin{bmatrix}
| |
| y_1 \\ \vdots \\ y_p \end{bmatrix}
| |
| </math>, <math>V=\begin{bmatrix}
| |
| \nu \\ 0 \\ \vdots \\ 0 \end{bmatrix}
| |
| </math> and <math>
| |
| U_t=\begin{bmatrix}
| |
| u_t \\ 0 \\ \vdots \\ 0 \end{bmatrix}
| |
| </math>
| |
| | |
| where <math>y_t</math>, <math>\nu</math> and <math>u</math> are <math>k</math> dimensional column vectors, <math>A</math> is <math>kp</math> by <math>kp</math> dimensional matrix and <math>Y</math>, <math>V</math> and <math>U</math> are <math>kp</math> dimensional column vectors.
| |
| | |
| The mean squared error of the h-step forecast of variable j is
| |
| :<math>
| |
| \mathbf{MSE}[y_{j,t}(h)]=\sum_{i=0}^{h-1}\sum_{k=1}^{K}(e_j'\Theta_ie_k)^2=\bigg(\sum_{i=0}^{h-1}\Theta_i\Theta_i'\bigg)_{jj}=\bigg(\sum_{i=0}^{h-1}\Phi_i\Sigma_u\Phi_i'\bigg)_{jj},
| |
| </math>
| |
| and where | |
| :*<math> e_j </math> is the j<sup>th</sup> column of <math> I_K </math> and the subscript <math>jj</math> refers to that element of the matrix
| |
| | |
| :*<math> \Theta_i=\Phi_i P ,</math> where <math>P</math> is a lower triangular matrix obtained by a [[Cholesky decomposition]] of <math> \Sigma_u </math> such that <math> \Sigma_u = PP'</math>, where <math> \Sigma_u </math> is the covariance matrix of the errors <math>u_t</math> | |
| | |
| :* <math>\Phi_i=J A^i J',</math> where <math> | |
| J=\begin{bmatrix}
| |
| \mathbf{I}_k &0 & \dots & 0\end{bmatrix} ,
| |
| </math> so that <math>J</math> is a <math>k</math> by <math>kp</math> dimensional matrix.
| |
| | |
| The amount of forecast error variance of variable <math>j</math> accounted for by exogenous shocks to variable <math>k</math> is given by <math>\omega_{jk,h} ,</math>
| |
| | |
| :<math>
| |
| \omega_{jk,h}=\sum_{i=0}^{h-1}(e_j'\Theta_ie_k)^2/MSE[y_{j,t}(h)] .
| |
| </math>
| |
| | |
| {{Refimprove|date=March 2011}}
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
| | |
| [[Category:Econometrics]]
| |
| [[Category:Multivariate time series analysis]]
| |
Oscar is how he's known as and he completely loves this name. North Dakota is her beginning location but she will have to transfer one day or an additional. The favorite pastime for my children std testing at home std home test and me is to play baseball and I'm attempting to make it at home std testing std home test a occupation. Managing people has been his working home std test day occupation for a whilst.
Feel free to visit my website - jonahmancini.com