PPMiss (version 0.1.2)

d.fit: Long memory parameter estimation

Description

Let \(\theta_h\) be the copula parameter associated to \((X_t,X_{t+h})\) and \(\hat\theta_h\) be an estimate of \(\theta_h\) based on pseudo observations. The long memory parameter \(d\) is estimated by

$$\hat d:=\mathrm{argmin}_{|d|<0.5}\bigg\{\sum_{h=s}^m \bigg|{\hat K_1}(\hat\theta_h-a)-\frac{\Gamma(1-d)}{\Gamma(d)}h^{2d-1}\bigg|^r\bigg\}, \quad r > 0$$.

Usage

d.fit(xt, copula = gauss, dCdtheta = dCtheta_gauss, theta.lower = -1,
  theta.upper = 1, optim.method = "Brent", method = "mpl", s = 1,
  m = 24, theta.start = 0.1, empirical = TRUE, r = 2, a = 0,
  d.interval = c(-0.5, 0.5))

Value

\(\hat d\), the estimated value of \(d\).

Arguments

xt

a vector with the observed time series. Missing observations are allowed.

copula

an object of class ‘copula’. Readily available options are frank, amh, fgm and gauss. Other copulas can be used but the user must provide the corresponding dCdtheta. Default is gauss.

dCdtheta

a two parameter function that returns the limit of the copula derivative, with respect to \(\theta\), as \(\theta\) goes to \(a\), where \(a\) is such that \(C_a(u,v)=uv\). Readily available options are dCtheta_frank, dCtheta_amh, dCtheta_fgm and dCtheta_gauss. Default is dCtheta_gauss.

theta.lower

the lower bound for \(\theta\). Default is -1.

theta.upper

the upper bound for \(\theta\). Default is 1.

optim.method

a character string specifying the optimization method. For all available options see optim. Default is ‘Brent’. See fitCopula for more details.

method

a character string specifying the copula parameter estimator used. This can be one of: ‘mpl’, ‘itau’, ‘irho’, ‘itau.mpl’ or ‘ml’. See fitCopula for details. Default is ‘mpl’.

s

integer. The smallest lag \(h\) considered in the estimation. Default is 1.

m

integer. The largest lag \(h\) considered in the estimation. Default is 24.

theta.start

starting value for the parameter optimization via optim.

empirical

logical. If TRUE, the sample estimators for the density and quantile functions are considered. Otherwhise, the gaussian density and quantile functions are used. Default is TRUE

r

the exponent used in the Minkowski distance used to calculate \(\hat d\). Default is 2, the Euclidean distance.

a

the value of \(\theta\) such that \(\lim_{\theta \to a} C_\theta(u,v)=uv\), is the product (or independence) copula. Default is 0, which is the common value for the available copulas, namely, frank, amh, fgm and gauss.

d.interval

a vector of size 2 giving the lower and upper bound for the long memory parameter \(d\). Default is c(-0.5,0.5).

Examples

# \donttest{
#-------------------------
# ARFIMA(0,d,0) process
#-------------------------
trunc <- 50000
n = 1000
cks <- arfima.coefs(d = 0.25, trunc = trunc)
eps <- rnorm(trunc+n)
x <- sapply(1:n, function(t) sum(cks*rev(eps[t:(t+trunc)])))

#----------------------
# Original time series
#-----------------------
# For Frank copula, -Inf < theta < Inf. However, "Brent" requires
# finite lower and upper bounds so we use c(-100, 100) here
d_frank <- d.fit(xt = x, copula = frank, dCdtheta = dCtheta_frank,
                 theta.lower = -100, theta.upper = 100)
d_amh <- d.fit(xt = x, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm <- d.fit(xt = x, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss <- d.fit(xt = x, copula = gauss, dCdtheta = dCtheta_gauss,
                 theta.lower = -1, theta.upper = 1)

c(FRANK = d_frank, AMH = d_amh, FGM = d_fgm, GAUSS = d_gauss)

#----------------------------
# Adding some missing values
#----------------------------
index <- sample(1:n, size = round(0.2*n))
xt <- x
xt[index] <- NA

d_frank_m <- d.fit(xt = xt, copula = frank,
                   dCdtheta = dCtheta_frank,
                   theta.lower = -100, theta.upper = 100)
d_amh_m <- d.fit(xt = xt, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm_m <- d.fit(xt = xt, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss_m <- d.fit(xt = xt, copula = gauss,
                   dCdtheta = dCtheta_gauss,
                   theta.lower = -1, theta.upper = 1)

data.frame(
  series = c("Complete", "Missing"),
  FRANK = c(d_frank, d_frank_m),
  AMH = c(d_amh, d_amh_m),
  FGM = c(d_fgm, d_fgm_m),
  GAUSS = c(d_gauss, d_gauss_m))

#-------------------------
# ARFIMA(1,d,1) process
#-------------------------
# For a faster algorithm to generate ARFIMA processes,
# see the package "arfima"
trunc <- 50000
cks <- arfima.coefs(d = 0.35, trunc = trunc, ar = -0.2, ma = 0.4)
n = 1000
eps <- rnorm(trunc+n)
x <- sapply(1:n, function(t) sum(cks*rev(eps[t:(t+trunc)])))

#----------------------
# Original time series
#-----------------------
# For Frank copula, -Inf < theta < Inf. However, "Brent" requires
# finite lower and upper bounds so we use c(-100, 100) here
d_frank <- d.fit(xt = x, copula = frank, dCdtheta = dCtheta_frank,
                 theta.lower = -100, theta.upper = 100)
d_amh <- d.fit(xt = x, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm <- d.fit(xt = x, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss <- d.fit(xt = x, copula = gauss, dCdtheta = dCtheta_gauss,
                 theta.lower = -1, theta.upper = 1)

c(FRANK = d_frank, AMH = d_amh, FGM = d_fgm, GAUSS = d_gauss)

#----------------------------
# Adding some missing values
#----------------------------
n = 1000
index <- sample(1:n, size = round(0.2*n))
xt <- x
xt[index] <- NA

d_frank_m <- d.fit(xt = xt, copula = frank,
                   dCdtheta = dCtheta_frank,
                   theta.lower = -100, theta.upper = 100)
d_amh_m <- d.fit(xt = xt, copula = amh, dCdtheta = dCtheta_amh,
                 theta.lower = -1, theta.upper = 1)
d_fgm_m <- d.fit(xt = xt, copula = fgm, dCdtheta = dCtheta_fgm,
                 theta.lower = -1, theta.upper = 1)
d_gauss_m <- d.fit(xt = xt, copula = gauss,
                   dCdtheta = dCtheta_gauss,
                   theta.lower = -1, theta.upper = 1)

data.frame(
  series = c("Complete", "Missing"),
  FRANK = c(d_frank, d_frank_m),
  AMH = c(d_amh, d_amh_m),
  FGM = c(d_fgm, d_fgm_m),
  GAUSS = c(d_gauss, d_gauss_m))
# }