sim.spARCH {spGARCH}

Simulation of spatial ARCH models

Description

The function generates n random numbers of a spatial ARCH process for given parameters and weighting schemes.

Usage

sim.spARCH(n = dim(W)[1], rho, alpha, W, b = 2, type = "spARCH", control = list())

Arguments

n

number of observations. If length(n) > 1, the length is taken to be the number required. Default dim(W)[1]

rho

spatial dependence parameter rho

alpha

unconditional variance level alpha

W

n times n spatial weight matrix

b

parameter b for logarithmic spatial ARCH (only needed if type = "log-spARCH"). Default 2.

type

type of simulated spARCH process (see details)

control

list of control arguments (see below)

Details

The function simulates n observations Y = (Y_1, ..., Y_n)' of a spatial ARCH process, i.e.,

\boldsymbol{Y} = diag(\boldsymbol{h})^{1/2} \boldsymbol{\varepsilon} \, ,

where \boldsymbol{\varepsilon} is a spatial White Noise process. The definition of \boldsymbol{h} depends on the chosen type. The following types are available.

• type = "spARCH" - simulates \boldsymbol{\varepsilon} from a truncated normal distribution on the interval [-a, a], such that \boldsymbol{h} > 0 with

  \boldsymbol{h} = \alpha + \rho \mathbf{W} \boldsymbol{Y}^{(2)} \; \mbox{and} \; a = 1 / ||\rho^2\mathbf{W}^2||_1^{1/4}.

  Note that the normal distribution is not trunctated (a = \infty), if \mathbf{W} is a strictly triangular matrix, as it is ensured that \boldsymbol{h} > \boldsymbol{0}. Generally, it is sufficient that if there exists a permutation such that \mathbf{W} is strictly triangular. In this case, the process is called oriented spARCH process.

• type = "log-spARCH" - simulates a logarithmic spARCH process (log-spARCH), i.e.,

  \ln\boldsymbol{h} = \alpha + \rho \mathbf{W} g(\boldsymbol{\varepsilon}) \, .

  For the log-spARCH process, the errors follow a standard normal distribution. The function g_b is given by

  g_b(\boldsymbol{\varepsilon}) = (\ln|\varepsilon(\boldsymbol{s}_1)|^{b}, \ldots, \ln|\varepsilon(\boldsymbol{s}_n)|^{b})' \, .

• type = "complex-spARCH" - allows for complex solutions of \boldsymbol{h}^{1/2} with

  \boldsymbol{h} = \alpha + \rho \mathbf{W} \boldsymbol{Y}^{(2)} \, .

  The errors follow a standard normal distribution.

Value

The functions returns a vector \boldsymbol{y}.

Control Arguments

• seed - positive integer to initialize the random number generator (RNG), default value is a random integer in [1, 10^6]

• silent - if FALSE, current random seed is reported

• triangular - if TRUE, \mathbf{W} is a triangular matrix and there are no checks to verify this assumption (default FALSE)

Author(s)

Philipp Otto [e-mail]

References

Philipp Otto, Wolfgang Schmid, Robert Garthoff (2018). Generalised Spatial and Spatiotemporal Autoregressive Conditional Heteroscedasticity. Spatial Statistics 26, pp. 125-145. arXiv:1609.00711

Examples

require("spdep")

# 1st example
##############

# parameters

rho <- 0.5
alpha <- 1
d <- 2

nblist <- cell2nb(d, d, type = "queen")
W <- nb2mat(nblist)

# simulation

Y <- sim.spARCH(rho = rho, alpha = alpha, W = W, type = "log-spARCH")

# visualization

image(1:d, 1:d, array(Y, dim = c(d,d)), xlab = expression(s[1]), ylab = expression(s[2]))

# 2nd example
##############

# two spatial weighting matrices W_1 and W_2
# h = alpha + rho_1 W_1 Y^2 + rho_2 W_2 Y^2

W_1 <- W
nblist <- cell2nb(d, d, type = "rook")
W_2 <- nb2mat(nblist)

rho_1 <- 0.3
rho_2 <- 0.7

W <- rho_1 * W_1 + rho_2 * W_2
rho <- 1

Y <- sim.spARCH(n = d^2, rho = rho, alpha = alpha, W = W, type = "log-spARCH")
image(1:d, 1:d, array(Y, dim = c(d,d)), xlab = expression(s[1]), ylab = expression(s[2]))