sim.spGARCH: Simulation of spatial ARCH models

Description

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

Usage

sim.spGARCH(n = dim(W1)[1], rho, lambda, alpha, W1, W2,
            b = 2, zeta = 0.5, theta = 0.5, type = "spGARCH",
            control = list())

Value

The functions returns a vector $\boldsymbol{y}$.

Arguments

n: number of observations. If length(n) $> 1$, the length is taken to be the number required. Default dim(W1)[1]
rho: spatial dependence parameter rho
lambda: spatial dependence parameter lambda
alpha: unconditional variance level alpha
W1: n times n spatial weight matrix (ARCH component, parameter rho)
W2: n times n spatial weight matrix (GARCH component, parameter lambda)
b: parameter b for logarithmic spatial GARCH (only needed if type = "log-spGARCH"). Default 2.
zeta: parameter zeta for exponential spatial GARCH (only needed if type = "e-spGARCH"). Default 0.5.
theta: parameter theta for exponential spatial GARCH (only needed if type = "e-spGARCH"). Default 0.5.
type: type of simulated spGARCH process (see details)
control: list of control arguments (see below)

Author

Philipp Otto philipp.otto@glasgow.ac.uk

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)

Details

The function simulates n observations $Y = (Y_1, ..., Y_n)'$ of a spatial GARCH 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 = "spGARCH" - simulates $\boldsymbol{\varepsilon}$ from a truncated normal distribution on the interval $[-a, a]$, such that $\boldsymbol{h} > 0$ with $$ \boldsymbol{h} = \alpha + \rho \mathbf{W}_1 \boldsymbol{Y}^{(2)} + \lambda \mathbf{W}_2 \boldsymbol{h} \; \mbox{and} \; a = 1 / ||\rho^2\mathbf{W}_1^2||_1^{1/4}. $$ Note that the normal distribution is not trunctated ($a = \infty$), if $\mathbf{W}_1$ 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 $\mathbf{W}_1$ is strictly triangular. In this case, the process is called oriented spGARCH process.
type = "e-spGARCH" - simulates an exponential spARCH process (e-spGARCH), i.e., $$ \ln\boldsymbol{h} = \alpha + \rho \mathbf{W}_1 g(\boldsymbol{\varepsilon}) + \lambda \mathbf{W}_2 log(\boldsymbol{h}) \, . $$ For the e-spGARCH process, the errors follow a standard normal distribution. The function $g$ is given by $$ g(\boldsymbol{\varepsilon}) = \Theta {\varepsilon} + \zeta (|\varepsilon| - E(|\varepsilon|)) \, . $$
type = "log-spGARCH" - simulates a logarithmic spARCH process (log-spGARCH), i.e., $$ \ln\boldsymbol{h} = \alpha + \rho \mathbf{W}_1 g(\boldsymbol{\varepsilon}) + \lambda \mathbf{W}_2 log(\boldsymbol{h}) \, . $$ For the log-spGARCH process, the errors follow a standard normal distribution. The function $g$ is given by $$ g(\boldsymbol{\varepsilon}) = (\ln|\varepsilon(\boldsymbol{s}_1)|^{b}, \ldots, \ln|\varepsilon(\boldsymbol{s}_n)|^{b})' \, . $$
type = "complex-spGARCH" - allows for complex solutions of $\boldsymbol{h}^{1/2}$ with $$ \boldsymbol{h} = \alpha + \rho \mathbf{W}_1 \boldsymbol{Y}^{(2)} + \lambda \mathbf{W}_2 \boldsymbol{h} \, . $$ The errors follow a standard normal distribution.

References

Philipp Otto, Wolfgang Schmid (2019). Spatial GARCH Models - A Unified Approach. arXiv: tools:::Rd_expr_doi("10.48550/arXiv.1908.08320")

Examples

Run this code

require("spdep")

# 1st example (spatial GARCH)
##############

# parameters

rho    <- 0.5
lambda <- 0.3
alpha  <- 1
d      <- 5

nblist <- cell2nb(d, d, type = "rook") # lattice process with Rook's contiguity matrix
W_1    <- nb2mat(nblist)
W_2    <- W_1

# simulation

Y      <- sim.spGARCH(rho = rho, lambda = lambda, alpha = alpha,
                      W1 = W_1, W2 = W_2, type = "spGARCH")

# visualization

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

# 2nd example (exponential spatial GARCH)
##############

# parameters

rho    <- 0.5
lambda <- 0.3
alpha  <- 1
zeta   <- 0.5
theta  <- 0.5
d      <- 5

nblist <- cell2nb(d, d, type = "rook") # lattice process with Rook's contiguity matrix
W_1    <- nb2mat(nblist)
W_2    <- W_1

# simulation

Y      <- sim.spGARCH(rho = rho, lambda = lambda, alpha = alpha,
                      W1 = W_1, W2 = W_2, zeta = zeta, theta = 0.5, type = "e-spGARCH")

# visualization

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

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