State-Space Modeling

The VARMAX Procedure

State-Space Modeling

Another representation of the VARMAX(p,q,s) model is in the form of a state-variable or a state-space model, which consists of a state equation

$z_{t}=F{z}_{t-1} +K{x}_{t} +G{\epsilon}_{t}$

and an observation equation

y_t = H z_t

where

$z_{t}=[\matrix{ y_{t} \cr \vdots \cr y_{t-p+1} \cr x_{t} \cr \vdots \cr x_{t... ...\cr \vdots \cr 0_{rx k} \cr I_{kx k} \cr 0_{kx k} \cr \vdots \cr 0_{kx k} } ]$

$F=[\matrix{ \Phi_{1} & ... & \Phi_{p-1}& \Phi_{p} & \Theta^*_{1} & ... &... ...dots \cr 0 & ... & 0 & 0 & 0 & ... & 0 & 0 & 0 & ... & I_k & 0 } ]$

and

H = [I_k, 0_k×k, ... , 0_k×k, 0_k×r, ... , 0_k×r, 0_k×k, ... , 0_k×k].

On the other hand, it is assumed that x_t follows a VARMA(p,q) model

$x_t=\sum_{i=1}^p A_i{x}_{t-i} + a_t - \sum_{i=1}^qC_i{a}_{t-i}$

or A(B)x_t = C(B)a_t, where A(B) = I_r - A₁B - ... - A_pB^p and C(B) = I_r - C₁B - ... - C_qB^q are matrix polynomials in B, and the A_i and C_i are r×r matrices. Without loss of generality, the AR and MA orders can be taken to be the same as the VARMAX(p,q,s) model, and a_t and ${\epsilon}_t$ are independent white noise processes.

Under suitable (such as stationarity) conditions, x_t is represented by an infinite order moving-average process

$x_t=A(B)^{-1}C(B)a_t=\Psi^x(B) a_t=\sum_{j=0}^{\infty}\Psi^x_{j}a_{t-j}$

where $\Psi^x(B)=A(B)^{-1}C(B)=\sum_{j=0}^{\infty} \Psi^x_j B^j$ .

The optimal (Minimum Mean Squared Error, MMSE) i-step-ahead forecast of x_t+i is

$x_{t+i| t} &=& \sum_{j=i}^{\infty}\Psi^x_{j}a_{t+i-j} \x_{t+i| t+1} &=& x_{t+i| t} + \Psi^x_{i-1}a_{t+1}.$

For i>q,

$x_{t+i| t}=\sum_{j=1}^p A_j x_{t+i-j| t}.$

The VARMAX(p,q,s) model has an absolutely convergent representation as

$y_t &=& \Phi(B)^{-1}\Theta^*(B)x_t + \Phi(B)^{-1}\Theta(B){\epsilon}_{t} \ &... ...+ \Phi(B)^{-1}\Theta(B){\epsilon}_{t} \ &=& V(B) a_t + \Psi(B) {\epsilon}_{t}$

$y_t=\sum_{j=0}^{\infty}V_{j} a_{t-j} + \sum_{j=0}^{\infty}\Psi_j {\epsilon}_{t-j}$

where $\Psi(B)=\Phi(B)^{-1}\Theta(B)=\sum_{j=0}^{\infty}\Psi_j B^j$ , $\Psi^{*}(B)=\Phi(B)^{-1}\Theta^*(B)$ , and $V(B)=\Psi^{*}(B)\Psi^x(B)=\sum_{j=0}^{\infty} V_j B^j$ .

The optimal (MMSE) i-step-ahead forecast of y_t+i is

$y_{t+i| t} &=& \sum_{j=i}^{\infty}V_{j}a_{t+i-j} + \sum_{j=i}^{\infty}\Psi_{j}{... ...j}\y_{t+i| t+1} &=& y_{t+i| t} + V_{i-1} a_{t+1} + \Psi_{i-1} {\epsilon}_{t+1}$

for i = 1, ... ,v with v = max(p,q+1). For i>q,

$y_{t+i| t} &=& \sum_{j=1}^p \Phi_j y_{t+i-j| t} + \sum_{j=0}^s \Theta_j^* x_{t... ..._j y_{t+i-j| t} + \sum_{j=1}^u ( \Theta_0^* A_j + \Theta_j^* ) x_{t+i-j| t} \$

where u = max(p,s).

Define $\Pi_j=\Theta_0^* A_j + \Theta_j^*$ .For i=v>q with v = max(p,q+1), you obtain

$y_{t+v| t} &=& \sum_{j=1}^p \Phi_j y_{t+v-j| t} + \sum_{j=1}^u \Pi_j{x}_{t+v-j... ...j=1}^p \Phi_j y_{t+v-j| t} + \sum_{j=1}^r \Pi_j{x}_{t+v-j| t} {\rm for} u\gt v$

From the preceding relations, a state equation is

z_t+1 = Fz_t + Kx_t^* + Ge_t+1

and an observation equation is

y_t = Hz_t

where

$z_{t}=[\matrix{ y_{t} \cr y_{t+1| t} \cr {\vdots} \cr y_{t+v-1| t} \cr x_... ...dots} \cr x_{t-1} }], e_{t+1}=[\matrix{ a_{t+1} \cr {\epsilon}_{t+1} }]$

$F=[\matrix{ 0 & I_k & 0 & { ... } & 0 & 0 & 0 & 0 & { ... } & 0 \cr 0 & 0 ... ...\cr 0 & 0 & 0 & { ... } & 0 & A_v & A_{v-1} & A_{v-2} & { ... } & A_1 \cr } ]$

$K=[\matrix{ 0 & 0 & { ... } & 0 \cr 0 & 0 & { ... } & 0 \cr {\vdots} & {\vd... ...si^x_{1} & 0_{rx k} \cr {\vdots} & {\vdots} \cr \Psi^x_{v-1} & 0_{rx k} \cr }]$

and

H = [I_k, 0_k×k, ... , 0_k×k, 0_k×r, ... , 0_k×r].

Note that the matrix K and the input vector x_t^* are defined only when u>v.

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