Simulate ARFIMA Process

Description

Simulates a series under the given ARFIMA model by applying an MA filter to a series of innovations.

Usage

arfima.sim(
  n,
  d = 0,
  ar = numeric(),
  ma = numeric(),
  mu = 0,
  sig2 = 1,
  stat.int = FALSE,
  n.burn,
  innov,
  exact.innov = TRUE
)

Arguments

Details

The model is defined by values for the AR and MA parameters (\phi and \theta, respectively), along with the fractional differencing parameter d. When d\geq 0.5, then the integer part is taken as m=\lfloor d+0.5\rfloor, and the remainder (between -0.5 and 0.5) stored as d. For m=0, the model is:

\left(1 - \sum_{i=1}^p \phi_i B^i\right)\left(1 - B\right)^d (y_t - \mu)=\left(1 + \sum_{i=1}^q \theta_i B^i\right) \epsilon_t

where B is the backshift operator (B y_t = y_{t-1}) and \epsilon_t is the innovation series. When m > 0, the model is defined by:

y_t = (1 - B)^{-m}x_t

\left(1 - \sum_{i=1}^p \phi_i B^i\right)(1 - B)^d (x_t - \mu)=\left(1 + \sum_{i=1}^q \theta_i B^i\right) \epsilon_t

When stat.int = FALSE, the differencing filter applied to the innovations is not split into parts, and the series model follows the first equation regardless of the value of d. This means that \mu is added to the series after filtering and before any truncation. When stat.int = TRUE, x_t - \mu is generated from filtered residuals, \mu is added, and the result is cumulatively summed m times. For non-zero mean and m>0, this will yield a polynomial trend in the resulting data.

Note that the burn-in length may affect the distribution of the sample mean, variance, and autocovariance. Consider this when generating ensembles of simulated data

Value

A numeric vector of length n.

Examples

## Generate ARFIMA(1,d,0) series with Gaussian innovations
x <- arfima.sim(1000, d=0.6, ar=c(-0.4)) 

## Generate ARFIMA(1,d,0) series with uniform innovations.
innov.series <- runif(1000, -1, 1)
x <- arfima.sim(1000, d=0.6, ar=c(-0.4), innov=innov.series, exact.innov=TRUE)

Minimum Distance Estimation of ARFIMA Model

Description

Fits an ARFIMA(p,d,q) model to a time series using a minimum distance estimator. For details see Mayoral (2007).

Usage

mde.arfima(
  y,
  p = 1,
  q = 0,
  d.range = c(0, 1),
  start,
  lag.max = floor(sqrt(length(y))),
  incl.mean = TRUE,
  verbose = FALSE,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  control = list()
)

Arguments

Value

A list containing:

Note

This method makes no assumptions on the distribution of the innovation series, and the innovation variance does not factor into the covariance matrix of parameter estimates. For this reason, it is not included in the results, but can be estimated from the residuals—see Mayoral (2007).

References

Mayoral, L. (2007). Minimum distance estimation of stationary and non-stationary ARFIMA processes. The Econometrics Journal, 10, 124-148. doi: 10.1111/j.1368-423X.2007.00202.x

See Also

mle.arfima for psuedo-maximum likelihood estimation.

Examples

set.seed(1)
x <- arfima.sim(1000, d=0.6, ar=c(-0.4))
fit <- mde.arfima(x, p=1, incl.mean=FALSE, verbose=TRUE) 


## Fit Summary
## --------------------
## Ljung-Box p-val:  0.276 
##           d     ar.1
## est 0.55031 -0.39261
## err 0.03145  0.03673
##
## Correlation Matrix for ARFIMA Parameters
##           d   ar.1
## d    1.0000 0.6108
## ar.1 0.6108 1.0000

Pseudo-Maximum Likelihood Estimation of ARFIMA Model

Description

Fits an ARFIMA(p,d,q) model to a time series using a pseudo-maximum likelihood estimator. For details see Beran (1995).

Usage

mle.arfima(
  y,
  p = 1,
  q = 0,
  d.range = c(0, 1),
  start,
  incl.mean = TRUE,
  verbose = FALSE,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  control = list()
)

Arguments

Value

A list containing:

References

Beran, J. (1995). Maximum Likelihood Estimation of the Differencing Parameter for Short and Long Memory Autoregressive Integrated Moving Average Models. Journal of the Royal Statistical Society. Series B (Methodological), 57, No. 4, 659-672. doi: 10.1111/j.2517-6161.1995.tb02054.x

See Also

mde.arfima for minimum distance estimation.

Examples

set.seed(1)
x <- arfima.sim(1000, d=0.6, ar=c(-0.4))
fit <- mle.arfima(x, p=1, incl.mean=FALSE, verbose=TRUE) 

## Fit Summary
## --------------------
## Ljung-Box p-val:  0.263 
##        sig2       d     ar.1
## est 1.09351 0.53933 -0.37582
## err 0.05343 0.04442  0.05538
##
## Correlation Matrix for ARFIMA Parameters
##         sig2       d    ar.1
## sig2  1.0000 -0.3349  0.4027
## d    -0.3349  1.0000 -0.8318
## ar.1  0.4027 -0.8318  1.0000