TARMA.fit: TARMA Modelling of Time Series
In tseriesTARMA: Analysis of Nonlinear Time Series Through Threshold Autoregressive Moving Average Models (TARMA) Models

TARMA.fit

R Documentation

TARMA Modelling of Time Series

Description

\loadmathjax

Implements a Least Squares fit of full subset two-regime TARMA(p1,p2,q1,q2) model to a univariate time series

Usage

TARMA.fit(
  x,
  tar1.lags = c(1),
  tar2.lags = c(1),
  tma1.lags = c(1),
  tma2.lags = c(1),
  threshold = NULL,
  d = 1,
  pa = 0.25,
  pb = 0.75,
  method = c("L-BFGS-B", "solnp", "lbfgsb3c", "robust", "trimmed"),
  alpha = 0,
  qu = c(0.05, 0.95),
  innov = c("norm", "student"),
  optim.control = list(trace = 0),
  irls.control = list(maxiter = 100, tol = 1e-04),
  ...
)

Arguments

`x`	A univariate time series.
`tar1.lags`	Vector of AR lags for the lower regime. It can be a subset of `1 ... p1 = max(tar1.lags)`.
`tar2.lags`	Vector of AR lags for the upper regime. It can be a subset of `1 ... p2 = max(tar2.lags)`.
`tma1.lags`	Vector of MA lags for the lower regime. It can be a subset of `1 ... q1 = max(tma1.lags)`.
`tma2.lags`	Vector of MA lags for the upper regime. It can be a subset of `1 ... q2 = max(tma2.lags)`.
`threshold`	Threshold parameter. If `NULL` estimates the threshold over the threshold range specified by `pa` and `pb`.
`d`	Delay parameter. Defaults to `1`.
`pa`	Real number in `[0,1]`. Sets the lower limit for the threshold search to the `100*pa`-th sample percentile. The default is `0.25`
`pb`	Real number in `[0,1]`. Sets the upper limit for the threshold search to the `100*pb`-th sample percentile. The default is `0.75`
`method`	Optimization/fitting method, can be one of `"L-BFGS-B", "solnp", "lbfgsb3c", "robust", "trimmed"`.
`alpha`	Real positive number. Tuning parameter for robust estimation. Only used if `method` is `robust`.
`qu`	Quantiles for (initial) trimmed estimation. Tuning parameter for robust estimation. Only used if `method` is either `robust` or `trimmed`.
`innov`	Innovation density for robust estimation. can be one of `"norm", "student"`. Only used if `method` is `"robust"`.
`optim.control`	List of control parameters for the main optimization method.
`irls.control`	List of control parameters for the irls optimization method (see details).
`...`	Additional arguments.

Details

Implements the Least Squares fit of the following two-regime TARMA(p1,p2,q1,q2) process:
\mjdeqnX_t = \left\lbrace \beginarrayll \phi_1,0 + \sum_i \in I_1 \phi_1,i X_t-i + \sum_j \in M_1 \theta_1,j \varepsilon_t-j + \varepsilon_t & \mathrmif X_t-d \leq \mathrmthd \\\ &\\\ \phi_2,0 + \sum_i \in I_2 \phi_2,i X_t-i + \sum_j \in M_2 \theta_2,j \varepsilon_t-j + \varepsilon_t & \mathrmif X_t-d > \mathrmthd \endarray \right. X[t] = \phi[1,0] + \Sigma_i in I_1 \phi[1,i] X[t-i] + \Sigma_j in M_1 \theta[1,j] \epsilon[t-j] + \epsilon[t] – if X[t-d] <= thd \phi[2,0] + \Sigma_i in I_2 \phi[2,i] X[t-i] + \Sigma_j in M_2 \theta[2,j] \epsilon[t-j] + \epsilon[t] – if X[t-d] > thd where \mjeqn\phi_1,i\phi[1,i] and \mjeqn\phi_2,i\phi[2,i] are the TAR parameters for the lower and upper regime, respectively, and I1 = tar1.lags and I2 = tar2.lags are the corresponding vectors of TAR lags. \mjeqn\theta_1,j\theta[1,j] and \mjeqn\theta_2,j\theta[2,j] are the TMA parameters and \mjeqnj \in M_1, M_2j in M_1, M_2, where M1 = tma1.lags and M2 = tma2.lags, are the vectors of TMA lags.
The most demanding routines have been reimplemented in Fortran and dynamically loaded.

Value

A list of class TARMA with components:

fit - List with the following components
- coef - Vector of estimated parameters which can be extracted by the coef method.
- sigma2 - Estimated innovation variance.
- var.coef - The estimated variance matrix of the coefficients coef, which can be extracted by the vcov method
- residuals - Vector of residuals from the fit.
- nobs - Effective sample size used for fitting the model.
se - Standard errors for the parameters. Note that they are computed conditionally upon the threshold so that they are generally smaller than the true ones.
thd - Estimated threshold.
aic - Value of the AIC for the minimised least squares criterion over the threshold range.
bic - Value of the BIC for the minimised least squares criterion over the threshold range.
rss - Minimised value of the target function. Coincides with the residual sum of squares for ordinary least squares estimation.
rss.v - Vector of values of the rss over the threshold range.
thd.range - Vector of values of the threshold range.
d - Delay parameter.
phi1 - Estimated AR parameters for the lower regime.
phi2 - Estimated AR parameters for the upper regime.
theta1 - Estimated MA parameters for the lower regime.
theta2 - Estimated MA parameters for the upper regime.
tlag1 - TAR lags for the lower regime
tlag2 - TAR lags for the upper regime
mlag1 - TMA lags for the lower regime
mlag2 - TMA lags for the upper regime
method - Estimation method.
innov - Innovation density model.
alpha - Tuning parameter for robust estimation.
qu - Tuning parameter for robust estimation.
call - The matched call.
convergence - Convergence code from the optimization routine.
innovpar - Parameter vector for the innovation density. Defaults to NULL.

Fitting methods

method has the following options:

L-BFGS-B: Calls the corresponding method of optim. Linear ergodicity constraints are imposed.
solnp: Calls the function solnp. It is a nonlinear optimization using augmented Lagrange method with linear and nonlinear inequality bounds. This allows to impose all the ergodicity constraints so that in theory it always return an ergodic solution. In practice the solution should be checked since this is a local solver and there is no guarantee that the minimum has been reached.
lbfgsb3c: Calls the function lbfgsb3c in package lbfgsb3c. Improved version of the L-BFGS-B in optim.
robust: Robust M-estimator of Ferrari and La Vecchia \insertCiteFer12tseriesTARMA. Based on the L-BFGS-B in optim and an additional iterative re-weighted least squares step to estimate the robust weights. Uses the tuning parameters alpha and qu. Robust standard errors are derived from the sandwich estimator of the variance/covariance matrix of the estimates. The IRLS step can be controlled through the parameters maxiter (maximum number of iterations) and tol (target tolerance). These can be passed using irls.control.
trimmed: Experimental: Estimator based on trimming the sample using the tuning parameters qu (lower and upper quantile).

Where possible, the conditions for ergodicity and invertibility are imposed to the optimization routines but there is no guarantee that the solution will be ergodic and invertible so that it is advisable to check the fitted parameters.

Author(s)

Simone Giannerini, simone.giannerini@uniud.it

Greta Goracci, greta.goracci@unibz.it

References

\insertRef
Gia21tseriesTARMA
\insertRef
Cha19tseriesTARMA
\insertRef
Gor23btseriesTARMA
\insertRef
Fer12tseriesTARMA

Examples


## a TARMA(1,1,1,1) model
set.seed(13)
x    <- TARMA.sim(n=200, phi1=c(0.5,-0.5), phi2=c(0.0,0.5), theta1=-0.5, theta2=0.7, d=1, thd=0.2)
fit1 <- TARMA.fit(x,tar1.lags=1, tar2.lags=1, tma1.lags=1, tma2.lags=1, d=1)

## --------------------------------------------------------------------------
## In the following examples the threshold is fixed to speed up computations
## --------------------------------------------------------------------------

## --------------------------------------------------------------------------
## Least Squares fit
## --------------------------------------------------------------------------

set.seed(26)
n    <- 200
y    <- TARMA.sim(n=n, phi1=c(0.6,0.6), phi2=c(-1.0,0.4), theta1=-0.7, theta2=0.5, d=1, thd=0.2)

fit1 <- TARMA.fit(y,tar1.lags=1, tar2.lags=1, tma1.lags=1, tma2.lags=1, d=1, threshold=0.2)
fit1

## ---------------------------------------------------------------------------
## Contaminate the data with one additive outlier
## ---------------------------------------------------------------------------
x     <- y           # contaminated series
x[54] <- x[54] + 10

## ---------------------------------------------------------------------------
## Compare the non-robust LS fit with the robust fit
## ---------------------------------------------------------------------------

fitls  <- TARMA.fit(x,tar1.lags=1, tar2.lags=1, tma1.lags=1, tma2.lags=1, d=1, threshold=0.2)
fitrob <- TARMA.fit(x,tar1.lags=1, tar2.lags=1, tma1.lags=1, tma2.lags=1, d=1,
            method='robust',alpha=0.7,qu=c(0.1,0.95), threshold=0.2)

par.true <- c(0.6,0.6,-1,0.4,-0.7,0.5)
pnames   <- c("int.1", "ar1.1", "int.2", "ar2.1", "ma1.1", "ma2.1")
names(par.true) <- pnames

par.ls  <- round(fitls$fit$coef,2)  # Least Squares
par.rob <- round(fitrob$fit$coef,2) # robust

rbind(par.true,par.ls,par.rob)

tseriesTARMA documentation built on Oct. 8, 2024, 5:11 p.m.