Description Usage Arguments Details Value Control Arguments Author(s) References Examples
The function generates n
random numbers of a spatial GARCH process for given parameters and weighting schemes.
1 2 3 | sim.spGARCH(n = dim(W1)[1], rho, lambda, alpha, W1, W2,
b = 2, zeta = 0.5, theta = 0.5, type = "spGARCH",
control = list())
|
n |
number of observations. If |
rho |
spatial dependence parameter rho |
lambda |
spatial dependence parameter lambda |
alpha |
unconditional variance level alpha |
W1 |
|
W2 |
|
b |
parameter |
zeta |
parameter |
theta |
parameter |
type |
type of simulated spGARCH process (see details) |
control |
list of control arguments (see below) |
The function simulates n
observations Y = (Y_1, ..., Y_n)' of a spatial GARCH process, i.e.,
Y = diag(h)^(1/2) ε ,
where ε is a spatial White Noise process. The definition of h depends on the chosen type
. The following types are available.
type = "spGARCH"
- simulates ε from a truncated normal distribution on the interval [-a, a], such that h > 0 with
h = α + ρ W1 Y^(2) + λ W2 h and a = 1 / (ρ^2||W1^2||_1)^(1/4)
Note that the normal distribution is not trunctated (a = ∞), if W1 is a strictly triangular matrix, as it is ensured that h > 0. Generally, it is sufficient that if there exists a permutation such that W1 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 h = α + ρ W1 g(ε) + λ W2 log(h).
For the e-spGARCH process, the errors follow a standard normal distribution. The function g is given by
g(ε) = Θ ε + ζ (|ε| - E(|ε|)) .
type = "log-spGARCH"
- simulates a logarithmic spARCH process (log-spGARCH), i.e.,
ln h = α + ρ W1 g_b(ε) + λ W2 log(h).
For the log-spGARCH process, the errors follow a standard normal distribution. The function g is given by
g(ε) = (ln|h(ε_1)|^b, ..., ln|h(ε_n)|^b)' .
type = "complex-spGARCH"
- allows for complex solutions of h^(1/2) with
h = α + ρ W1 Y^(2) + λ W2 h .
The errors follow a standard normal distribution.
The functions returns a vector y.
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
, W is a triangular matrix and there are no checks to verify this assumption (default FALSE
)
Philipp Otto potto@europa-uni.de
Philipp Otto, Wolfgang Schmid (2019). Spatial GARCH Models - A Unified Approach. arXiv:1908.08320
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | 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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