TARMA.fit2: 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,
  threshold = NULL,
  d = 1,
  pa = 0.25,
  pb = 0.75,
  thd.var = NULL,
  include.int = TRUE,
  x.reg = NULL,
  optim.control = list(),
  ...
)

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.

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).
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
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.

Author

Simone Giannerini, simone.giannerini@uniud.it

Greta Goracci, greta.goracci@unibz.it

Details

Fits the following two-regime TARMA process with optional components: linear AR part, seasonal MA and covariates.
X_t = _0 + _h I _h X_t-h + _l=1^Q _l _t-ls + _j=1^q _j _t-j + _k=1^K _k Z_k + _t + arrayll _1,0 + _i I_1 _1,i X_t-i & if X_t-d thd \\\ &\\\ _2,0 + _i I_2 _2,i X_t-i & if X_t-d > thd array . X[t] = [0] + _h in I [h] X[t-h] + _j = 1,..,q [j] [t-j] + _j = 1,..,Q [j] [t-js] + _k = 1,..,K [k] Z[k] + [t] + + [1,0] + _i in I_1 [1,i] X[t-i] -- if X[t-d] <= thd + [2,0] + _i in I_2 [2,i] X[t-i] -- if X[t-d] > thd

where _h[h] are the common AR parameters and h ranges in I = ar.lags. _j[j] are the common MA parameters and j = 1,...,qj = 1,...,q (q = ma.ord), _l[l] are the common seasonal MA parameters and l = 1,...,Ql = 1,...,Q (Q = sma.ord) _k[k] are the parameters for the covariates. Finally, _1,i[1,i] and _2,i[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.

References

Gia21tseriesTARMA
Cha19tseriesTARMA

Examples

Run this code

## 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)

# \donttest{
## 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)

# }

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