TARMA.test {tseriesTARMA}

ARMA versus TARMA supLM test for nonlinearity

Description

Implements a supremum Lagrange Multiplier test for a ARMA specification versus a TARMA specification. Includes the AR versus TAR test.

Usage

TARMA.test(
  x,
  pa = 0.25,
  pb = 0.75,
  ar.ord,
  ma.ord,
  ma.fixed = TRUE,
  d,
  thd.range,
  method = "CSS-ML",
  ...
)

Arguments

x

A univariate time series.

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

ar.ord

Order of the AR part.

ma.ord

Order of the MA part.

ma.fixed

Logical. Only applies to testing ARMA vs TARMA. If TRUE computes the test where only the AR parameters are tested, see (Goracci et al. 2021) for details.

d

Delay parameter. Defaults to 1.

thd.range

Vector of optional user defined threshold range. If missing then pa and pb are used.

method

Fitting method to be passed to arima.

...

Additional arguments to be passed to arima.

Details

Implements an asymptotic supremum Lagrange Multiplier test to test an ARMA specification versus a TARMA specification. If ma.fixed=TRUE (the default), the AR parameters are tested whereas the MA parameters are fixed. If ma.fixed=FALSE both the AR and the MA parameters are tested. This is an asymptotic test and the value of the test statistic has to be compared with the critical values tabulated in (Goracci et al. 2021) and (Andrews 2003). If ma.ord=0 then the AR versus TAR test is used. Note that when method='CSS', this is equivalent to TAR.test, which uses least squares.

Value

An object of class TARMAtest with components:

statistic

The value of the supLM statistic.

parameter

A named vector: threshold is the value that maximises the Lagrange Multiplier values.

test.v

Vector of values of the LM statistic for each threshold given in thd.range.

thd.range

Range of values of the threshold.

fit.ARMA

The null model: ARMA fit over x.

sigma2

Estimated innovation variance from the ARMA fit.

data.name

A character string giving the name of the data.

prop

Proportion of values of the series that fall in the lower regime.

p.value

The p-value of the test. It is NULL for the asymptotic test.

method

A character string indicating the type of test performed.

d

The delay parameter.

pa

Lower threshold quantile.

dfree

Effective degrees of freedom. It is the number of tested parameters.

Author(s)

Simone Giannerini, [e-mail]

Greta Goracci, [e-mail]

References

• Goracci G, Giannerini S, Chan K, Tong H (2023). “Testing for threshold effects in the TARMA framework.” Statistica Sinica, 33(3), 1879-1901. https://doi.org/10.5705/ss.202021.0120.

• Andrews DWK (2003). “Tests for Parameter Instability and Structural Change with Unknown Change Point: A Corrigendum.” Econometrica, 71(1), 395-397. doi:10.1111/1468-0262.00405.

See Also

TAR.test.B for the bootstrap version of the test. TARMAGARCH.test for the robust version of the test with respect to GARCH innovations. TARMA.sim to simulate from a TARMA process.

Examples

## a TARMA(1,1,1,1) where the threshold effect is on the AR parameters
set.seed(123)
x1    <- TARMA.sim(n=100, phi1=c(0.5,-0.5), phi2=c(0.0,0.8), theta1=0.5, theta2=0.5, d=1, thd=0.2)
TARMA.test(x1, ar.ord=1, ma.ord=1, d=1)
TARMA.test(x1, ar.ord=1, ma.ord=1, d=1, ma.fixed=FALSE) # full TARMA test

## a TARMA(1,1,1,1) where the threshold effect is on the MA parameters
set.seed(212)
x2    <- TARMA.sim(n=100, phi1=c(0.5,0.2), phi2=c(0.5,0.2), theta1=0.6, theta2=-0.6, d=1, thd=0.2)
TARMA.test(x2, ar.ord=1, ma.ord=1, d=1)
TARMA.test(x2, ar.ord=1, ma.ord=1, d=1, ma.fixed=FALSE) # full TARMA test

## a ARMA(1,1)
x3   <- arima.sim(n=100, model=list(order = c(1,0,1),ar=0.5, ma=0.5))
TARMA.test(x3, ar.ord=1, ma.ord=1, d=1)

## a TAR(1,1)
x4   <- TARMA.sim(n=100, phi1=c(0.5,-0.5), phi2=c(0.0,0.8), theta1=0, theta2=0, d=1, thd=0.2)
TARMA.test(x4, ar.ord=1, ma.ord=0, d=1)

## a AR(1)
x5   <- arima.sim(n=100, model=list(order = c(1,0,0),ar=0.5))
TARMA.test(x5, ar.ord=1, ma.ord=0, d=1)