SimModel1 {SMFilter}

Simulate from the type one state-space Model on Stiefel manifold.

Description

This function simulates from the type one model on Stiefel manifold. See Details part below.

Usage

SimModel1(iT, mX = NULL, mZ = NULL, mY = NULL, alpha_0, beta,
  mB = NULL, Omega = NULL, vD, burnin = 100)

Arguments

iT

the sample size.

mX

the matrix containing X_t with dimension T \times q_1.

mZ

the matrix containing Z_t with dimension T \times q_2.

mY

initial values of the dependent variable for ik-1 up to 0. If mY = NULL, then no lagged dependent variables in regressors.

alpha_0

the initial alpha, p \times r.

beta

the \beta matrix, iqx+ip*ik, y_1,t-1,y_1,t-2,...,y_2,t-1,y_2,t-2,...

mB

the coefficient matrix \boldsymbol{B} before mZ with dimension p \times q_2.

Omega

covariance matrix of the errors.

vD

vector of the diagonals of D.

burnin

burn-in sample size (matrix Langevin).

Details

The type one model on Stiefel manifold takes the form:

\boldsymbol{y}_t \quad = \quad \boldsymbol{\alpha}_t \boldsymbol{\beta} ' \boldsymbol{x}_t + \boldsymbol{B} \boldsymbol{z}_t + \boldsymbol{\varepsilon}_t

\boldsymbol{\alpha}_{t+1} | \boldsymbol{\alpha}_{t} \quad \sim \quad ML (p, r, \boldsymbol{\alpha}_{t} \boldsymbol{D})

where \boldsymbol{y}_t is a p-vector of the dependent variable, \boldsymbol{x}_t and \boldsymbol{z}_t are explanatory variables wit dimension q_1 and q_2, \boldsymbol{x}_t and \boldsymbol{z}_t have no overlap, matrix \boldsymbol{B} is the coefficients for \boldsymbol{z}_t, \boldsymbol{\varepsilon}_t is the error vector.

The matrices \boldsymbol{\alpha}_t and \boldsymbol{\beta} have dimensions p \times r and q_1 \times r, respectively. Note that r is strictly smaller than both p and q_1. \boldsymbol{\alpha}_t and \boldsymbol{\beta} are both non-singular matrices. \boldsymbol{\alpha}_t is time-varying while \boldsymbol{\beta} is time-invariant.

Furthermore, \boldsymbol{\alpha}_t fulfills the condition \boldsymbol{\alpha}_t' \boldsymbol{\alpha}_t = \boldsymbol{I}_r, and therefor it evolves on the Stiefel manifold.

ML (p, r, \boldsymbol{\alpha}_{t} \boldsymbol{D}) denotes the Matrix Langevin distribution or matrix von Mises-Fisher distribution on the Stiefel manifold. Its density function takes the form

f(\boldsymbol{\alpha_{t+1}} ) = \frac{ \mathrm{etr} \left\{ \boldsymbol{D} \boldsymbol{\alpha}_{t}' \boldsymbol{\alpha_{t+1}} \right\} }{ _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) }

where \mathrm{etr} denotes \mathrm{exp}(\mathrm{tr}()), and _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) is the (0,1)-type hypergeometric function for matrix.

Note that the function does not add intercept automatically.

Value

A list containing the sampled data and the dynamics of alpha.

The object is a list containing the following components:

dData

a data.frame of the sampled data

aAlpha

an array of the \boldsymbol{\alpha}_{t} with the dimension T \times p \times r

Author(s)

Yukai Yang, [e-mail]

Examples

iT = 50 # sample size
ip = 2 # dimension of the dependent variable
ir = 1 # rank number
iqx=2 # number of variables in X
iqz=2 # number of variables in Z
ik = 1 # lag length

if(iqx==0) mX=NULL else mX = matrix(rnorm(iT*iqx),iT, iqx)
if(iqz==0) mZ=NULL else mZ = matrix(rnorm(iT*iqz),iT, iqz)
if(ik==0) mY=NULL else mY = matrix(0, ik, ip)

alpha_0 = matrix(c(runif_sm(num=1,ip=ip,ir=ir)), ip, ir)
beta = matrix(c(runif_sm(num=1,ip=ip*ik+iqx,ir=ir)), ip*ik+iqx, ir)
if(ip*ik+iqz==0) mB=NULL else mB = matrix(c(runif_sm(num=1,ip=(ip*ik+iqz)*ip,ir=1)), ip, ip*ik+iqz)
vD = 50

ret = SimModel1(iT=iT, mX=mX, mZ=mZ, mY=mY, alpha_0=alpha_0, beta=beta, mB=mB, vD=vD)