LS.whittle: Whittle estimator to Locally Stationary Time Series

Description

This function computes Whittle estimator to LS-ARMA and LS-ARFIMA models.

Usage

LS.whittle(
  series,
  start,
  order = c(p = 0, q = 0),
  ar.order = NULL,
  ma.order = NULL,
  sd.order = NULL,
  d.order = NULL,
  include.d = FALSE,
  N = NULL,
  S = NULL,
  include.taper = TRUE,
  control = list(),
  lower = -Inf,
  upper = Inf,
  m = NULL,
  n.ahead = 0
)

Arguments

series

(type: numeric) univariate time series.

start

(type: numeric) numeric vector, initial values for parameters to run the model.

order

(type: numeric) vector corresponding to ARMA model entered.

ar.order

(type: numeric) AR polimonial order.

ma.order

(type: numeric) MA polimonial order.

sd.order

(type: numeric) polinomial order noise scale factor.

d.order

(type: numeric) d polinomial order, where d is the ARFIMA parameter.

include.d

(type: numeric) logical argument for ARFIMA models. If include.d=FALSE then the model is an ARMA process.

(type: numeric) value corresponding to the length of the window to compute periodogram. If N=NULL then the function will use $N = \textrm{trunc}(n^{0.8})$, see Dahlhaus (1998) where $n$ is the length of the y vector.

(type: numeric) value corresponding to the lag with which will go taking the blocks or windows.

include.taper

(type: logical) logical argument that by default is TRUE. See periodogram.

control

(type: list) A list of control parameters. More details in nlminb .

lower

(type: numeric) lower bound, replicated to be as long as start. If unspecified, all parameters are assumed to be lower unconstrained.

upper

(type: numeric) upper bound, replicated to be as long as start. If unspecified, all parameters are assumed to be upper unconstrained.

(type: numeric) truncation order of the MA infinity process, by default $m = 0.25n^{0.8}$. Parameter used in LSTS_kalman.

n.ahead

(type: numeric) The number of steps ahead for which prediction is required. By default is zero.

Value

A list with the following components:

coef

The best set of parameters found.

var.coef

covariance matrix approximated for maximum likelihood estimator $\hat{\theta}$ of $\theta:=(\theta_1,\ldots,\theta_k)^{\prime}$. This matrix is approximated by $H^{-1}/n$, where $H$ is the Hessian matrix $[\partial^2 \ell(\theta)/\partial\theta_i \partial\theta_j]_{i,j=1}^{k}$.

loglik

log-likelihood of coef, calculated with LS.whittle.

aic

Akaike'S `An Information Criterion', for one fitted model LS-ARMA or LS-ARFIMA. The formula is $-2L + 2k/n$, where L represents the log-likelihood, k represents the number of parameters in the fitted model and n is equal to the length of the series.

series

original time serie.

residuals

standard residuals.

fitted.values

model fitted values.

pred

predictions of the model.

the estimated standard errors.

model

A list representing the fitted model.

Details

This function estimates the parameters in models: LS-ARMA $$\Phi(t/T, \, B)\, Y_{t, T} = \Theta(t/T,\, B)\,\sigma(t/T)\, \varepsilon_t$$ and LS-ARFIMA $$\Phi(t/T, \, B)\, Y_{t, T} = \Theta(t/T,\, B)\, (1-B)^{-d(t/T)}\, \sigma(t/T)\, \varepsilon_t,$$ with infinite moving average expansion $$Y_{t, T} = \sigma(t/T)\, \sum_{j=0}^{\infty} \psi(t/T)\,\varepsilon_t,$$ for $t = 1,\ldots, T$, where for $u = t/T \in [0,1]$, $\Phi(u,B)=1+\phi_1(u)B +\cdots+\phi_p(u)B^p$ is an autoregressive polynomial, $\Theta(u, B) = 1 + \theta_1(u)B + \cdots + \theta_q(u)B^q$ is a moving average polynomial, $d(u)$ is a long-memory parameter, $\sigma(u)$ is a noise scale factor and $\{\varepsilon_t \}$ is a Gaussian white noise sequence with zero mean and unit variance. This class of models extends the well-known ARMA and ARFIMA process, which is obtained when the components $\Phi(u, B)$, $\Theta(u, B)$, $d(u)$ and $\sigma(u)$ do not depend on $u$. The evolution of these models can be specified in terms of a general class of functions. For example, let $\{g_j(u)\}$, $j = 1, 2, \ldots$, be a basis for a space of smoothly varying functions and let $d_{\theta}(u)$ be the time-varying long-memory parameter in model LS-ARFIMA. Then we could write $d_{\theta}(u)$ in terms of the basis $\{g_j(u) = u^j\}$ as follows $d_{\theta}(u) = \sum_{j=0}^{k} \alpha_j\,g_j(u)$ for unknown values of $k$ and $\theta = (\alpha_0,\,\alpha_1,\,\ldots, \,\alpha_k)^{\prime}$. In this situation, estimating $\theta$ involves determining $k$ and estimating the coefficients $\alpha_0,\,\alpha_1,\,\ldots, \,\alpha_k$. LS.whittle optimizes LS.whittle.loglik as objective function using nlminb function, for both LS-ARMA (include.d=FALSE) and LS-ARFIMA (include.d=TRUE) models. Also computes Kalman filter with LS.kalman and this values are given in var.coef in the output.

Examples

Run this code

# NOT RUN {
# Analysis by blocks of phi and sigma parameters
N <- 200
S <- 100
M <- trunc((length(malleco) - N) / S + 1)
table <- c()
for (j in 1:M) {
  x <- malleco[(1 + S * (j - 1)):(N + S * (j - 1))]
  table <- rbind(table, nlminb(
    start = c(0.65, 0.15), N = N,
    objective = LS.whittle.loglik,
    series = x, order = c(p = 1, q = 0)
  )$par)
}
u <- (N / 2 + S * (1:M - 1)) / length(malleco)
table <- as.data.frame(cbind(u, table))
colnames(table) <- c("u", "phi", "sigma")
# Start parameters
phi <- smooth.spline(table$phi, spar = 1, tol = 0.01)$y
fit.1 <- nls(phi ~ a0 + a1 * u, start = list(a0 = 0.65, a1 = 0.00))
sigma <- smooth.spline(table$sigma, spar = 1)$y
fit.2 <- nls(sigma ~ b0 + b1 * u, start = list(b0 = 0.65, b1 = 0.00))
fit_whittle <- LS.whittle(
  series = malleco, start = c(coef(fit.1), coef(fit.2)), order = c(p = 1, q = 0),
  ar.order = 1, sd.order = 1, N = 180, n.ahead = 10
)
# }