Package 'tseriesTARMA'

Title: Analysis of Nonlinear Time Series Through Threshold Autoregressive Moving Average Models (TARMA) Models
Description: Routines for nonlinear time series analysis based on Threshold Autoregressive Moving Average (TARMA) models. It provides functions and methods for: TARMA model fitting and forecasting, including robust estimators, see Goracci et al. JBES (2025) <doi:10.1080/07350015.2024.2412011>; tests for threshold effects, see Giannerini et al. JoE (2024) <doi:10.1016/j.jeconom.2023.01.004>, Goracci et al. Statistica Sinica (2023) <doi:10.5705/ss.202021.0120>, Angelini et al. (2024) <doi:10.48550/arXiv.2308.00444>; unit-root tests based on TARMA models, see Chan et al. Statistica Sinica (2024) <doi:10.5705/ss.202022.0125>.
Authors: Simone Giannerini [aut, cre] , Greta Goracci [aut]
Maintainer: Simone Giannerini <[email protected]>
License: GPL (>= 2)
Version: 0.5-1
Built: 2024-10-09 09:26:43 UTC
Source: CRAN

Help Index

Andrews Tabulated Critical Values

Description

The data is taken from Table 1 of (Andrews 2003), which provides asymptotic critical values for sup Wald, LM, and LR tests for parameter instability. Critical values are given for degrees of freedom \(p=1,\dots,20\). They can be used as asymptotic critical values for the following tests:

Provided pb = 1- pa.

Usage

ACValues

Format

ACValues

A matrix with 13 rows and 62 columns. The first two columns contain the parameters, whereas the remaining 60 columns contain 3 critical values (at level 10%, 5%, and 1%) for each value of the parameter \(p\):

pi

\(\pi_0\) in the paper, corresponds to pa.

lambda

\(\lambda\) in the paper, not relevant here.

1-90

\(p=1\), critical level 10%.

1-95

\(p=1\), critical level 5%.

1-99

\(p=1\), critical level 1%.

...

20-90

\(p=20\), critical level 10%.

20-95

\(p=20\), critical level 5%.

20-99

\(p=20\), critical level 1%.

Note that \(p\) are the degrees of freedom and correspond to the total number of tested parameter in the above tests.

Source

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.

Plot from fitted/forecasted time series models.

Description

Plots a time series model fit, possibly together with its forecast and forecast bands.

Usage

## S3 method for class 'tsfit'
plot(
  x,
  fit,
  plot.fit = TRUE,
  fore = NULL,
  lcols = list(series = "lightblue", fit = 4, pred = "red4", band = "red"),
  ptype = list(series = 20, fit = 1, pred = 16, band = 20),
  ltype = list(series = 1, fit = 1, pred = 1, band = 1),
  lwdt = 2,
  ...
)

Arguments

x

A time series
fit

A time series model fitted upon x, e.g. an object obtained with TARMA.fit
plot.fit

Logical. If TRUE adds the fitted values from the model.
fore

Forecast derived from fit, e.g. an object obtained with predict.TARMA if not null it adds the prediction together with its confidence bands.
lcols

List of colours for the plots.
ptype

List of point types (pch) for the plots.
ltype

List of line types (lty) for the plots.
lwdt

A common line width for the plots.
...

Additional graphical parameters.

Value

No return value, called for side effects

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

See Also

TARMA.fit and TARMA.fit2 for TARMA modelling. predict.TARMA for prediction and forecasting.

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, threshold=0.2)
xp1  <- predict(fit1,x,n.ahead=5)

# plots both the fitted and the forecast
plot.tsfit(x,fit=fit1,fore=xp1);grid();

# plots only the forecast
plot.tsfit(x,fit=fit1,plot.fit=FALSE,fore=xp1);grid();

# plots only the fitted
plot.tsfit(x,fit=fit1);grid();

Forecast from fitted TARMA models.

Description

Forecasting with TARMA models

Usage

## S3 method for class 'TARMA'
predict(
  object,
  x,
  n.ahead = 0,
  n.sim = 1000,
  quant = c(0.05, 0.95),
  pred.matrix = FALSE,
  ...
)

Arguments

object

A TARMA fit upon x.
x

The fitted time series.
n.ahead

The number of steps ahead for which prediction is required.
n.sim

The number of Monte Carlo replications used to simulate the prediction density.
quant

Vector of quantiles (in the interval [0, 1]) to be computed upon the prediction density.
pred.matrix

Logical. if TRUE prints also the whole simulated prediction density for each prediction horizon from 1 to n.ahead.
...

Additional arguments.

Details

If n.ahead = 0 it gives the fitted values from the model. If the fit is from TARMA.fit2 and includes covariates, these are ignored.

Value

A list with components pred.matrix, pred, and pred.interval. The latter two are ts objects that contain the prediction and the quantiles of the prediction density, respectively. If pred.matrix = TRUE then the prediction density from which the quantiles are computed is also returned.

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

References

  • Giannerini S, Goracci G (2021). “Estimating and Forecasting with TARMA models.” University of Bologna.

See Also

TARMA.fit and TARMA.fit2 for TARMA modelling. plot.tsfit for plotting TARMA fits and forecasts.

Examples

## a TARMA(1,1,1,1) model
set.seed(13)
x1   <- 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(x1, method='L-BFGS-B',tar1.lags = 1, tar2.lags = 1, tma1.lags = 1, 
        tma2.lags = 1, d=1, threshold=0.2)
xp1  <- predict(fit1,x1,n.ahead=2)
xp1

Methods for TARMA fits

Description

Methods for TARMA fits

Usage

## S3 method for class 'TARMA'
print(x, digits = max(3L, getOption("digits") - 3L), se = TRUE, ...)

## S3 method for class 'TARMA'
coef(object, ...)

## S3 method for class 'TARMA'
vcov(object, ...)

## S3 method for class 'TARMA'
residuals(object, ...)

Arguments

x

A TARMA fit.
digits

Number of decimal digits for the output.
se

Logical. if TRUE (the default) prints also the standard errors.
...

Further parameters.
object

A TARMA fit.

Value

No return value, called for side effects

See Also

TARMA.fit and TARMA.fit2 for TARMA modelling, plot.tsfit for plotting TARMA fits and forecasts.

Methods for TARMA tests

Description

Methods for TARMA tests

Usage

## S3 method for class 'TARMAtest'
print(x, ...)

Arguments

x

A TARMAtest object.
...

Further parameters passed to print.htest.

Details

Print method for TARMAtest objects. Prints the results using the method for class htest and adds critical values extracted from ACValues for the test for threshold nonlinearity and supLMQur for the unit root test against the TARMA alternative. Note that the bootstrap version of the tests also print the bootstrap p-value.

Value

No return value, called for side effects

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

See Also

print.htest

Tabulated Critical Values for the Unit Root IMA vs TARMA test

Description

The data provides asymptotic null critical values for the unit root supLM test described in (Chan et al. 2024). They are used with the following tests:

Provided pb = 1- pa.

Usage

supLMQur

Format

supLMQur

A 4-dimensional array that contains 4 critical values (at level 10%, 5%, 1%, 0.1%) for each combination of

pa

Lower bound for the threshold range. From 0.01 to 0.4

th

MA(1) parameter.

n

Sample size.

Source

Chan K, Giannerini S, Goracci G, Tong H (2024). “Testing for threshold regulation in presence of measurement error.” Statistica Sinica, 34(3), 1413-1434. doi:10.5705/ss.202022.0125, https://doi.org/10.5705/ss.202022.0125.

AR versus TARMA supLM robust test for nonlinearity

Description

Heteroskedasticity robust supremum Lagrange Multiplier test for a AR specification versus a TAR specification. Includes the classic (non robust) AR versus TAR test.

Usage

TAR.test(x, pa = 0.25, pb = 0.75, ar.ord, d = 1)

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.
d

Delay parameter. Defaults to 1.

Details

Implements a heteroskedasticity robust asymptotic supremum Lagrange Multiplier test to test an AR specification versus a TAR specification. 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) or (Andrews 2003). Both the non-robust supLM and the robust supLMh statistics are returned.

Value

An object of class TARMAtest with components:

statistic

A named vector with the values of the classic supLM and robust supLMh statistics.

parameter

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

test.v

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

thd.range

Range of values of the threshold.

fit

The null model: AR fit over x.

sigma2

Estimated innovation variance from the AR 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, [email protected]

Greta Goracci, [email protected]

References

  • Goracci G, Giannerini S, Chan K, Tong H (2023). “Testing for threshold effects in the TARMA framework.” Statistica Sinica, 33(3), 1879-1901. doi: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. TARMA.test for the (robust) ARMA vs TARMA asymptotic version of the test, which includes also the AR vs TAR test, with different defaults. TARMAGARCH.test for the robust version of the ARMA vs TARMA test that assumes GARCH innovations. TARMA.sim to simulate from a TARMA process.

Examples

set.seed(123)
## a TAR(1,1) ---------------
x1   <- 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)
TAR.test(x1, ar.ord=1, d=1)

## a AR(1)    ----------------
x2   <- arima.sim(n=100, model=list(order=c(1,0,0), ar=0.5))
TAR.test(x2, ar.ord=1, d=1)

AR versus TAR bootstrap supLM test for nonlinearity

Description

Implements various bootstrap supremum Lagrange Multiplier tests for a AR specification versus a TAR specification.

Usage

TAR.test.B(
  x,
  B = 1000,
  pa = 0.25,
  pb = 0.75,
  ar.ord,
  d = 1,
  btype = c("iid", "wb.h", "wb.r", "wb.n"),
  ...
)

Arguments

x

A univariate time series.
B

Integer. Number of bootstrap resamples. Defaults to 1000.
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.
d

Delay parameter. Defaults to 1.
btype

Bootstrap type, can be one of 'iid','wb.h','wb.r','wb.n', see Details.
...

Additional arguments to be passed to arima.

Details

Implements the bootstrap version of TAR.test the supremum Lagrange Multiplier test to test an AR specification versus a TARMA specification. The option btype specifies the type of bootstrap as follows:

iid

Residual iid bootstrap. See (Giannerini et al. 2022), (Giannerini et al. 2024).

wb.h

Stochastic permutation of (Hansen 1996).

wb.r

Residual wild bootstrap with Rademacher auxiliary distribution. See (Giannerini et al. 2022), (Giannerini et al. 2024).

wb.n

Residual wild bootstrap with Normal auxiliary distribution. See (Giannerini et al. 2022), (Giannerini et al. 2024).

Value

A list of class htest 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

The null model: AR fit over x.

sigma2

Estimated innovation variance from the AR 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 bootstrap p-value of the test.

method

A character string indicating the type of test performed.

Tb

The bootstrap null distribution.

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

References

  • Giannerini S, Goracci G, Rahbek A (2022). “The validity of bootstrap testing in the threshold framework.” doi:10.48550/ARXIV.2201.00028, https://arxiv.org/abs/2201.00028.

  • Giannerini S, Goracci G, Rahbek A (2024). “The validity of bootstrap testing in the threshold framework.” Journal of Econometrics, 239(1), 105379. ISSN 0304-4076, doi:10.1016/j.jeconom.2023.01.004, Climate Econometrics.

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

  • Giannerini S, Goracci G (2021). “Estimating and Forecasting with TARMA models.” University of Bologna.

  • Hansen BE (1996). “Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis.” Econometrica, 64(2), 413–430. ISSN 00129682, 14680262, https://doi.org/10.2307/2171789.

See Also

TAR.test for the heteroskedastic robust asymptotic 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 TAR(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, theta2=0, d=1, thd=0.2)
TAR.test.B(x1, ar.ord=1, d=1)
TAR.test.B(x1, ar.ord=1, d=1, btype='wb.r')
TAR.test.B(x1, ar.ord=1, d=1, btype='wb.h')

## a AR(1)
x2 <- arima.sim(n=100, model=list(order = c(1,0,0),ar=0.5))
TAR.test.B(x2, ar.ord=1, d=1)
TAR.test.B(x2, ar.ord=1, d=1, btype='wb.r')
TAR.test.B(x2, ar.ord=1, d=1, btype='wb.h')

TARMA Modelling of Time Series

Description

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:
\[X_{t} = \left\lbrace \begin{array}{ll} \phi_{1,0} + \sum_{i \in I_1} \phi_{1,i} X_{t-i} + \sum_{j \in M_1} \theta_{1,j} \varepsilon_{t-j} + \varepsilon_{t} & \mathrm{if } X_{t-d} \leq \mathrm{thd} \\ &\\ \phi_{2,0} + \sum_{i \in I_2} \phi_{2,i} X_{t-i} + \sum_{j \in M_2} \theta_{2,j} \varepsilon_{t-j} + \varepsilon_{t} & \mathrm{if } X_{t-d} > \mathrm{thd} \end{array} \right. \] where \(\phi_{1,i}\) and \(\phi_{2,i}\) 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. \(\theta_{1,j}\) and \(\theta_{2,j}\) are the TMA parameters and \(j \in M_1, M_2\), 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 (Ferrari and La-Vecchia 2011). 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, [email protected]

Greta Goracci, [email protected]

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.

  • Goracci G, Ferrari D, Giannerini S, Ravazzolo F (2023). “Robust estimation for Threshold Autoregressive Moving-Average models.” Free University of Bolzano, University of Bologna. doi:10.48550/ARXIV.2211.08205.

  • Ferrari D, La-Vecchia D (2011). “On robust estimation via pseudo-additive information.” Biometrika, 99(1), 238-244. ISSN 0006-3444, doi:10.1093/biomet/asr061.

See Also

TARMA.fit2 for Maximum Likelihood estimation of TARMA models with common MA part. print.TARMA for print methods for TARMA fits. predict.TARMA for prediction and forecasting. plot.tsfit for plotting TARMA fits and forecasts.

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)

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(),
  ...
)

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.

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, [email protected]

Greta Goracci, [email protected]

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.

See Also

TARMA.fit for Least Square estimation of full subset TARMA models. print.TARMA for print methods for TARMA fits. predict.TARMA for prediction and forecasting.

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)

Simulation of a two-regime TARMA(p1,p2,q1,q2) process

Description

Simulates from the following two-regime TARMA(p1,p2,q1,q2) process:

\[X_{t} = \left\lbrace \begin{array}{ll} \phi_{1,0} + \sum_{i=1}^{p_1} \phi_{1,i} X_{t-i} + \sum_{j=1}^{q_1} \theta_{1,j} \varepsilon_{t-j} + \varepsilon_{t}, & \quad\mathrm{if}\quad X_{t-d} \leq \mathrm{thd} \\ &\\ \phi_{2,0} + \sum_{i=1}^{p_2} \phi_{2,i} X_{t-i} + \sum_{j=1}^{q_2} \theta_{2,j} \varepsilon_{t-j} + \varepsilon_{t}, & \quad\mathrm{if}\quad X_{t-d} > \mathrm{thd} \end{array} \right. \]

Usage

TARMA.sim(
  n,
  phi1,
  phi2,
  theta1,
  theta2,
  d = 1,
  thd = 0,
  s1 = 1,
  s2 = 1,
  rand.gen = rnorm,
  innov = rand.gen(n, ...),
  n.start = 500,
  xstart,
  start.innov = rand.gen(n.start, ...),
  ...
)

Arguments

n

Length of the series.
phi1

Vector of p1+1 Autoregressive parameters of the lower regime. The first element is the intercept.
phi2

Vector of p2+1 Autoregressive parameters of the upper regime. The first element is the intercept.
theta1

Vector of q1 Moving Average parameters of the lower regime.
theta2

Vector of q2 Moving Average parameters of the upper regime.
d

Delay parameter. Defaults to 1.
thd

Threshold parameter. Defaults to 0.
s1

Innovation variance for the lower regime. Defaults to 1.
s2

Innovation variance for the upper regime. Defaults to 1.
rand.gen

Optional: a function to generate the innovations. Defaults to rnorm.
innov

Optional: a time series of innovations. If not provided, rand.gen is used.
n.start

Length of the burn-in period. Defaults to 500.
xstart

Initial condition as a named list:
$ar: AR part length k = max(p1,p2,d), X[k], X[k-1], ... ,X[1];
$ma: MA part length q = ma.ord, e[q], ... , e[1].
start.innov

Optional: a time series of innovations for the burn-in period.
...

Additional arguments for rand.gen.

Details

Note that the parameters are not checked for ergodicity.

Value

A time series object of class ts generated from the above model.

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

References

  • Giannerini S, Goracci G (2021). “Estimating and Forecasting with TARMA models.” University of Bologna.

Examples

## a TARMA(1,1,1,1) model
set.seed(123)
x <- 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)

## a TARMA(1,2,1,1) model
x <- TARMA.sim(n=100,phi1=c(0.5,-0.5,0),phi2=c(0,0.5,0.3),theta1=-0.5,theta2=0.5,d=1,thd=0.2)

ARMA versus TARMA (and AR versus TAR) supLM tests for nonlinearity

Description

Heteroskedasticity robust supremum Lagrange Multiplier tests 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, either a ts or zoo object.
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. Both the non-robust supLM and the robust supLMh statistics are returned. 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). These are automatically computed and printed by print.TARMAtest. 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 and its robust version supLMh.

parameter

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

test.v

Vector of values of the two LM statistics 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, [email protected]

Greta Goracci, [email protected]

References

  • Goracci G, Giannerini S, Chan K, Tong H (2023). “Testing for threshold effects in the TARMA framework.” Statistica Sinica, 33(3), 1879-1901. doi: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 for the AR vs TAR asymptotic version of the test with different defaults. TAR.test.B for the bootstrap version of the AR vs TAR test. TARMAGARCH.test for the robust version of the test that assumes 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)

ARMA GARCH versus TARMA GARCH supLM test for nonlinearity

Description

Implements a supremum Lagrange Multiplier test for a ARMA-GARCH specification versus a TARMA-GARCH specification. Both the AR and MA parameters are tested. Also includes the ARCH case.

Usage

TARMAGARCH.test(
  x,
  pa = 0.25,
  pb = 0.75,
  ar.ord = 1,
  ma.ord = 1,
  arch.ord = 1,
  garch.ord = 1,
  d = 1,
  thd.range,
  ...
)

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. It must be a positive integer
ma.ord

Order of the MA part. It must be a positive integer
arch.ord

Order of the ARCH part. It must be a positive integer
garch.ord

Order of the GARCH part. It must be a non negative integer.
d

Delay parameter. Defaults to 1.
thd.range

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

Additional arguments to be passed to ugarchfit.

Details

Implements an asymptotic supremum Lagrange Multiplier test to test an ARMA-GARCH specification versus a TARMA-GARCH specification. In other words, the test is robust with respect to heteroskedasticity. Both the AR parameters 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 reported in the output and taken from (Andrews 2003). It includes the ARCH case if garch.ord=0. The null ARMA-GARCH model is estimated in one step with the function ugarchfit from the package rugarch. The estimated AR and MA polynomials are checked for stationarity and invertibility.

Value

A list of class htest 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

The null model: ARMA-GARCH fit using rugarch.

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, [email protected]

Greta Goracci, [email protected]

References

  • Angelini F, Castellani M, Giannerini S, Goracci G (2023). “Testing for Threshold Effects in Presence of Heteroskedasticity and Measurement Error with an application to Italian Strikes.” University of Bologna and Free University of Bolzano. 2308.00444, https://arxiv.org/abs/2308.00444.

  • Goracci G, Giannerini S, Chan K, Tong H (2021). “Testing for threshold effects in the TARMA framework.” University of Bologna, Free University of Bolzano, University of Iowa, London School of Economics. https://arxiv.org/abs/2103.13977.

  • 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

TARMA.test and TAR.test.B for the asymptotic and bootstrap test without the GARCH component. TARMA.sim to simulate from a TARMA process. TARMA.fit and TARMA.fit2 for TARMA modelling.

Examples

## Function to simulate from a ARMA-GARCH process

arma11.garch11 <- function(n, ph, th, a, b, a0=1, rand.gen = rnorm, innov = rand.gen(n, ...),
n.start = 500, start.innov = rand.gen(n.start, ...),...){

  #  Simulates a ARMA(1,1)-GARCH(1,1) process
  #  with parameters ph, th, a, b, a0.
  #         x[t] <- ph*x[t-1] + th*eps[t-1] + eps[t]
  #       eps[t] <- e[t]*sqrt(v[t])
  #         v[t] <- a0 + a*eps[t-1]^2 + b*v[t-1];
  # ph  : AR
  # th  : MA
  # a   : ARCH
  # b   : GARCH

  # checks
  if(abs(a+b)>=1)   stop("model is not stationary")
  if(b/(1-a)>=1) stop("model has infinite fourth moments")

  if (!missing(start.innov) && length(start.innov) < n.start)
    stop(gettextf("'start.innov' is too short: need %d points", n.start), domain = NA)
  e <- c(start.innov[1L:n.start], innov[1L:n])
  ntot <- length(e)
  x <- v <- eps <- double(ntot)
  v[1]   <- a0/(1.0-a-b);
  eps[1] <- e[1]*sqrt(v[1])
  x[1]   <- eps[1];
  for(i in 2:ntot){
    v[i]   <- a0 + a*eps[i-1]^2 + b*v[i-1];
    eps[i] <- e[i]*sqrt(v[i])
    x[i]   <- ph*x[i-1] + th*eps[i-1] + eps[i]
  }
  if (n.start > 0)  x <- x[-(1L:n.start)]
  return(ts(x));
}

## **************************************************************************
## Comparison between the robust and the non-robust test in presence of GARCH errors
## Simulates from the ARMA(1,1)-GARCH(1,1)

set.seed(12)
x1 <- arma11.garch11(n=100, ph=0.9, th=0.5, a=0.85, b=0.1, a0=1,n.start=500)
TARMAGARCH.test(x1, ar.ord=1, ma.ord=1, arch.ord=1, garch.ord=1, d=1)
TARMA.test(x1, ar.ord=1, ma.ord=1, d=1, ma.fixed=FALSE)

## a TARMA(1,1,1,1) where the threshold effect is on the AR parameters
set.seed(123)
x2  <- 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)
TARMAGARCH.test(x2, ar.ord=1, ma.ord=1, d=1)

Unit root supLM test for an integrated MA versus a stationary TARMA process

Description

Implements a supremum Lagrange Multiplier unit root test for the null hypiythesis of a integrated MA process versus a stationary TARMA process.

Usage

TARMAur.test(x, pa = 0.25, pb = 0.75, thd.range, method = "ML", ...)

Arguments

x

A univariate vector or 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
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 integrate MA(1) specification versus a stationary TARMA(1,1) specification. This is an asymptotic test and the value of the test statistic has to be compared with the critical values tabulated in (Chan et al. 2024) and available in supLMQur. The relevant critical values are automatically shown upon printing the test, see print.TARMAtest.

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: IMA(1) fit over x.

sigma2

Estimated innovation variance from the IMA fit.

data.name

A character string giving the name of the data.

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.

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

References

See Also

TARMAur.test.B for the bootstrap version of the test. print.TARMAtest for the print method.

Examples

## a TARMA(1,1,1,1) 
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)
TARMAur.test(x1)


## a IMA(1,1)
x2   <- arima.sim(n=100, model=list(order = c(0,1,1),ma=0.6))
TARMAur.test(x2)

Unit root supLM test for an integrated MA versus a stationary TARMA process

Description

Implements a supremum Lagrange Multiplier unit root test for the null hypothesis of a integrated MA process versus a stationary TARMA process.

Usage

TARMAur.test.B(
  x,
  B = 1000,
  pa = 0.25,
  pb = 0.75,
  thd.range,
  method = "ML",
  btype = c("wb.r", "wb.n", "iid"),
  ...
)

Arguments

x

A univariate vector or time series.
B

Integer. Number of bootstrap resamples. Defaults to 1000.
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.range

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

Fitting method to be passed to arima.
btype

Bootstrap type, can be one of 'iid','wb.r','wb.n', see Details.
...

Additional arguments to be passed to arima.

Details

Implements the bootstrap version of TARMAur.test the supremum Lagrange Multiplier test to test an integrate MA(1) specification versus a stationary TARMA(1,1) specification. The option btype specifies the type of bootstrap as follows:

wb.r

Residual wild bootstrap with Rademacher auxiliary distribution. See (Giannerini et al. 2022).

wb.n

Residual wild bootstrap with Normal auxiliary distribution. See (Giannerini et al. 2022).

iid

Residual iid bootstrap. See (Goracci et al. 2021).

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: IMA(1) fit over x.

sigma2

Estimated innovation variance from the IMA fit.

data.name

A character string giving the name of the data.

p.value

The bootstrap p-value of the test.

method

A character string indicating the type of test performed.

d

The delay parameter.

pa

Lower threshold quantile.

Tb

The bootstrap null distribution.

Author(s)

Simone Giannerini, [email protected]

Greta Goracci, [email protected]

References

See Also

TARMAur.test for the asymptotic version of the test. print.TARMAtest for the print method.

Examples

## a TARMA(1,1,1,1) 
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)
TARMAur.test.B(x1, B=100) # B=100 for speedup


## a IMA(1,1)
x2   <- arima.sim(n=100, model=list(order = c(0,1,1),ma=0.6))
TARMAur.test.B(x2, B=100) # B=100 for speedup