3-oxoacyl-(acyl-carrier-protein) reductase: Difference between revisions
en>ZéroBot m r2.7.1) (Robot: Adding zh:3-氧酰基-(酰基载体蛋白)还原酶 |
en>Christian75 →References: Clean up using AWB |
||
Line 1: | Line 1: | ||
{{Howto|date=August 2013}} | |||
This page shows the details for different matrix notations of a [[vector autoregression]] process with ''k'' variables. | |||
==Var(''p'')== | |||
{{Main|Vector autoregression}} | |||
:<math>y_t =c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t, \, </math> | |||
Where each <math>y_{i}</math> is a ''k'' × 1 vector and each <math> A_i </math> is a ''k'' × ''k'' matrix. | |||
==Large matrix notation== | |||
:<math>\begin{bmatrix}y_{1,t} \\ y_{2,t}\\ \vdots \\ y_{k,t}\end{bmatrix}=\begin{bmatrix}c_{1} \\ c_{2}\\ \vdots \\ c_{k}\end{bmatrix}+ | |||
\begin{bmatrix} | |||
a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1\\ | |||
a_{2,1}^1&a_{2,2}^1 & \cdots & a_{2,k}^1\\ | |||
\vdots& \vdots& \ddots& \vdots\\ | |||
a_{k,1}^1&a_{k,2}^1 & \cdots & a_{k,k}^1 | |||
\end{bmatrix} | |||
\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\\ \vdots \\ y_{k,t-1}\end{bmatrix} | |||
+ \cdots + | |||
\begin{bmatrix} | |||
a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ | |||
a_{2,1}^p&a_{2,2}^p & \cdots & a_{2,k}^p\\ | |||
\vdots& \vdots& \ddots& \vdots\\ | |||
a_{k,1}^p&a_{k,2}^p & \cdots & a_{k,k}^p | |||
\end{bmatrix} | |||
\begin{bmatrix}y_{1,t-p} \\ y_{2,t-p}\\ \vdots \\ y_{k,t-p}\end{bmatrix} | |||
+ \begin{bmatrix}e_{1,t} \\ e_{2,t}\\ \vdots \\ e_{k,t}\end{bmatrix}</math> | |||
==Equation by equation notation== | |||
Rewriting the ''y'' variables one to one gives: | |||
<math>y_{1,t} = c_{1} + a_{1,1}^1y_{1,t-1} + a_{1,2}^1y_{2,t-1} +\cdots + a_{1,k}^1y_{k,t-1}+\cdots+a_{1,1}^py_{1,t-p}+a_{1,2}^py_{2,t-p}+ \cdots +a_{1,k}^py_{k,t-p} + e_{1,t}\,</math> | |||
<math>y_{2,t} = c_{2} + a_{2,1}^1y_{1,t-1} + a_{2,2}^1y_{2,t-1} +\cdots + a_{2,k}^1y_{k,t-1}+\cdots+a_{2,1}^py_{1,t-p}+a_{2,2}^py_{2,t-p}+ \cdots +a_{2,k}^py_{k,t-p} + e_{2,t}\,</math> | |||
<math>\qquad\vdots</math> | |||
<math>y_{k,t} = c_{k} + a_{k,1}^1y_{1,t-1} + a_{k,2}^1y_{2,t-1} +\cdots + a_{k,k}^1y_{k,t-1}+\cdots+a_{k,1}^py_{1,t-p}+a_{k,2}^py_{2,t-p}+ \cdots +a_{k,k}^py_{k,t-p} + e_{k,t}\,</math> | |||
==Concise matrix notation== | |||
One can rewrite a VAR(''p'') with ''k'' variables in a general way which includes ''T+1'' observations <math>y_0</math> through <math>y_T</math> | |||
:<math> Y=BZ +U \, </math> | |||
Where: | |||
:<math> Y= | |||
\begin{bmatrix}y_{p} & y_{p+1} & \cdots & y_{T}\end{bmatrix} = | |||
\begin{bmatrix}y_{1,p} & y_{1,p+1} & \cdots & y_{1,T} \\ y_{2,p} &y_{2,p+1} & \cdots & y_{2,T}\\ | |||
\vdots& \vdots &\vdots &\vdots \\ y_{k,p} &y_{k,p+1} & \cdots & y_{k,T}\end{bmatrix} </math> | |||
:<math> B= | |||
\begin{bmatrix} c & A_{1} & A_{2} & \cdots & A_{p} \end{bmatrix} = | |||
\begin{bmatrix} | |||
c_{1} & a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1 &\cdots & a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ | |||
c_{2} & a_{2,1}^1&a_{2,2}^1 & \cdots & a_{2,k}^1 &\cdots & a_{2,1}^p&a_{2,2}^p & \cdots & a_{2,k}^p \\ | |||
\vdots & \vdots& \vdots& \ddots& \vdots & \cdots & \vdots& \vdots& \ddots& \vdots\\ | |||
c_{k} & a_{k,1}^1&a_{k,2}^1 & \cdots & a_{k,k}^1 &\cdots & a_{k,1}^p&a_{k,2}^p & \cdots & a_{k,k}^p | |||
\end{bmatrix} | |||
</math> | |||
:<math> | |||
Z= | |||
\begin{bmatrix} | |||
1 & 1 & \cdots & 1 \\ | |||
y_{p-1} & y_{p} & \cdots & y_{T-1}\\ | |||
y_{p-2} & y_{p-1} & \cdots & y_{T-2}\\ | |||
\vdots & \vdots & \ddots & \vdots\\ | |||
y_{0} & y_{1} & \cdots & y_{T-p} | |||
\end{bmatrix} = | |||
\begin{bmatrix} | |||
1 & 1 & \cdots & 1 \\ | |||
y_{1,p-1} & y_{1,p} & \cdots & y_{1,T-1} \\ | |||
y_{2,p-1} & y_{2,p} & \cdots & y_{2,T-1} \\ | |||
\vdots & \vdots & \ddots & \vdots\\ | |||
y_{k,p-1} & y_{k,p} & \cdots & y_{k,T-1} \\ | |||
y_{1,p-2} & y_{1,p-1} & \cdots & y_{1,T-2} \\ | |||
y_{2,p-2} & y_{2,p-1} & \cdots & y_{2,T-2} \\ | |||
\vdots & \vdots & \ddots & \vdots\\ | |||
y_{k,p-2} & y_{k,p-1} & \cdots & y_{k,T-2} \\ | |||
\vdots & \vdots & \ddots & \vdots\\ | |||
y_{1,0} & y_{1,1} & \cdots & y_{1,T-p} \\ | |||
y_{2,0} & y_{2,1} & \cdots & y_{2,T-p} \\ | |||
\vdots & \vdots & \ddots & \vdots\\ | |||
y_{k,0} & y_{k,1} & \cdots & y_{k,T-p} | |||
\end{bmatrix} | |||
</math> | |||
and | |||
:<math>U= | |||
\begin{bmatrix} | |||
e_{p} & e_{p+1} & \cdots & e_{T} | |||
\end{bmatrix}= | |||
\begin{bmatrix} | |||
e_{1,p} & e_{1,p+1} & \cdots & e_{1,T} \\ | |||
e_{2,p} & e_{2,p+1} & \cdots & e_{2,T} \\ | |||
\vdots & \vdots & \ddots & \vdots \\ | |||
e_{k,p} & e_{k,p+1} & \cdots & e_{k,T} | |||
\end{bmatrix}. | |||
</math> | |||
One can then solve for the coefficient matrix ''B'' (e.g. using an [[ordinary least squares]] estimation of <math> Y \approx BZ</math>) | |||
==References== | |||
{{Reflist}} | |||
*{{Cite book |first=Helmut |last=Lütkepohl |title=New Introduction to Multiple Time Series Analysis |publisher=Springer |location=Berlin |year=2005 |isbn=3540401725 }} | |||
[[Category:Econometrics]] | |||
[[Category:Multivariate time series analysis]] |
Latest revision as of 22:44, 24 August 2013
This page shows the details for different matrix notations of a vector autoregression process with k variables.
Var(p)
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Where each is a k × 1 vector and each is a k × k matrix.
Large matrix notation
Equation by equation notation
Rewriting the y variables one to one gives:
Concise matrix notation
One can rewrite a VAR(p) with k variables in a general way which includes T+1 observations through
Where:
and
One can then solve for the coefficient matrix B (e.g. using an ordinary least squares estimation of )
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534