R: TARMA Modelling of Time Series

TARMA.fit2 {tseriesTARMA}

R Documentation

TARMA Modelling of Time Series

Description

Maximum Likelihood fit of a two-regime TARMA(p1,p2,q,q) model with common MA parameters, possible common AR parameters and possible covariates.

Usage

TARMA.fit2(
  x,
  ar.lags = NULL,
  tar1.lags = c(1),
  tar2.lags = c(1),
  ma.ord = 1,
  sma.ord = 0L,
  period = NA,
  estimate.thd = TRUE,
  threshold,
  d = 1,
  pa = 0.25,
  pb = 0.75,
  thd.var = NULL,
  include.int = TRUE,
  x.reg = NULL,
  optim.control = list(),
  ...
)

Arguments

`x`	A univariate time series.
`ar.lags`	Vector of common AR lags. Defaults to `NULL`. It can be a subset of lags.
`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)`.
`ma.ord`	Order of the MA part (also called `q` below).
`sma.ord`	Order of the seasonal MA part (also called `Q` below).
`period`	Period of the seasonal MA part (also called `s` below).
`estimate.thd`	Logical. If `TRUE` estimates the threshold over the threshold range specified by `pa` and `pb`.
`threshold`	Threshold parameter. Used only if `estimate.thd = FALSE`.
`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`
`thd.var`	Optional exogenous threshold variable. If `NULL` it is set to `lag(x,-d)`. If not `NULL` it has to be a `ts` object.
`include.int`	Logical. If `TRUE` includes the intercept terms in both regimes, a common intercept is included otherwise.
`x.reg`	Covariates to be included in the model. These are passed to `arima`. If they are not `ts` objects they must have the same length as `x`.
`optim.control`	List of control parameters for the optimization method.
`...`	Additional arguments passed to `arima`.

Details

Fits the following two-regime TARMA process with optional components: linear AR part, seasonal MA and covariates.
\[X_{t} = \phi_{0} + \sum_{h \in I} \phi_{h} X_{t-h} + \sum_{l=1}^Q \Theta_{l} \epsilon_{t-ls} + \sum_{j=1}^q \theta_{j} \epsilon_{t-j} + \sum_{k=1}^K \delta_{k} Z_{k} + \epsilon_{t} + \left\lbrace \begin{array}{ll} \phi_{1,0} + \sum_{i \in I_1} \phi_{1,i} X_{t-i} & \mathrm{if } X_{t-d} \leq \mathrm{thd} \\ &\\ \phi_{2,0} + \sum_{i \in I_2} \phi_{2,i} X_{t-i} & \mathrm{if } X_{t-d} > \mathrm{thd} \end{array} \right. \]

where \(\phi_h\) are the common AR parameters and \(h\) ranges in I = ar.lags. \(\theta_j\) are the common MA parameters and \(j = 1,\dots,q\) (q = ma.ord), \(\Theta_l\) are the common seasonal MA parameters and \(l = 1,\dots,Q\) (Q = sma.ord) \(\delta_k\) are the parameters for the covariates. Finally, \(\phi_{1,i}\) and \(\phi_{2,i}\) are the TAR parameters for the lower and upper regime, respectively and I1 = tar1.lags I2 = tar2.lags are the vector of TAR lags.

Value

A list of class TARMA with components:

fit - The output of the fit. It is a arima object.
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.
aic.v - Vector of values of the AIC over the threshold range.
thd.range - Vector of values of the threshold range.
d - Delay parameter.
thd - Estimated threshold.
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.
delta - Estimated parameters for the covariates x.reg.
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
arlag - Same as the input slot ar.lags
include.int - Same as the input slot include.int
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.
rss - Minimised residual sum of squares.
method - Estimation method.
call - The matched call.

Author(s)

Simone Giannerini, simone.giannerini@unibo.it

Greta Goracci, greta.goracci@unibz.it

References

Giannerini S, Goracci G (2021). “Estimating and Forecasting with TARMA models.” University of Bologna.
Chan K, Goracci G (2019). “On the Ergodicity of First-Order Threshold Autoregressive Moving-Average Processes.” J. Time Series Anal., 40(2), 256-264.

Examples

## a TARMA(1,1,1,1)
set.seed(127)
x    <- TARMA.sim(n=100, phi1=c(0.5,-0.5), phi2=c(0,0.8), theta1=0.5, theta2=0.5, d=1, thd=0.2)
fit1 <- TARMA.fit2(x, tar1.lags=1, tar2.lags=1, ma.ord=1, d=1)


## Showcase of the fit with covariates ---
## simulates from a TARMA(3,3,1,1) model with common MA parameter
## and common AR(1) and AR(2) parameters. Only the lag 3 parameter varies across regimes
set.seed(212)
n <- 300
x <- TARMA.sim(n=n, phi1=c(0.5,0.3,0.2,0.4), phi2=c(0.5,0.3,0.2,-0.2), theta1=0.4, theta2=0.4,
     d=1, thd=0.2, s1=1, s2=1)

## FIT 1: estimates lags 1,2,3 as threshold lags ---
fit1 <- TARMA.fit2(x, ma.ord=1, tar1.lags=c(1,2,3), tar2.lags=c(1,2,3), d=1)

## FIT 2: estimates lags 1 and 2 as fixed AR and lag 3 as the threshold lag
fit2 <- TARMA.fit2(x, ma.ord=1, tar1.lags=c(3),  tar2.lags=c(3), ar.lags=c(1,2), d=1)

## FIT 3: creates lag 1 and 2 and fits them as covariates ---
z1   <- lag(x,-1)
z2   <- lag(x,-2)
fit3 <- TARMA.fit2(x, ma.ord=1,  tar1.lags=c(3), tar2.lags=c(3), x.reg=ts.intersect(z1,z2), d=1)

## FIT 4: estimates lag 1 as a covariate, lag 2 as fixed AR and lag 3 as the threshold lag
fit4 <- TARMA.fit2(x, ma.ord = 1,  tar1.lags=c(3), tar2.lags=c(3), x.reg=z1, ar.lags=2, d=1)

[Package tseriesTARMA version 0.3-4 Index]