Package 'mtarm'

Auxiliary function to specify the number of regimes and lag orders

Description

This auxiliary function defines the regime structure of a multivariate TAR model by specifying the number of regimes and the corresponding lag orders for the endogenous, exogenous, and threshold series in each regime.

Usage

ars(nregim = 1, p = 1, q = 0, d = 0)

Arguments

nregim	A positive integer indicating the total number of regimes.
p	A list of positive integers specifying the autoregressive order of the output series within each regime.
q	A list of non-negative integers specifying the maximum lag of the exogenous series within each regime.
d	A list of non-negative integers specifying the maximum lag of the threshold series within each regime.

Value

A list containing the number of regimes and the regime-specific lag-order specifications.

---

Coercion of mtar objects to mcmc objects

Description

This method converts an object of class mtar into a list of mcmc objects, each corresponding to a Markov chain produced during Bayesian estimation.

Usage

## S3 method for class 'mtar'
as.mcmc(x, ...)

Arguments

x	an object of class mtar obtained from a call to mtar().
...	additional arguments passed to specific coercion methods.

Value

A list of mcmc objects containing the posterior simulation draws generated by the mtar() routine.

See Also

as.mcmc

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2, ssvs=TRUE)
fit1.mcmc <- coda::as.mcmc(fit1)
summary(fit1.mcmc)
#plot(fit1.mcmc)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
fit2.mcmc <- coda::as.mcmc(fit2)
summary(fit2.mcmc)
#plot(fit2.mcmc)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
fit3.mcmc <- coda::as.mcmc(fit3)
summary(fit3.mcmc)
#plot(fit3.mcmc)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
fit4.mcmc <- coda::as.mcmc(fit4)
summary(fit4.mcmc)
#plot(fit4.mcmc)

---

coef method for objects of class mtar

Description

coef method for objects of class mtar

Usage

## S3 method for class 'mtar'
coef(object, ..., FUN = mean)

Arguments

object	an object of class mtar obtained from a call to the mtar() function.
...	additional arguments passed to FUN.
FUN	a function to be applied to the MCMC chains associated with each model parameter. By default, FUN is set to mean.

Value

A list containing the summary statistics obtained by applying FUN to the MCMC chains of each model parameter.

---

Deviance Information Criterion (DIC)

Description

Computes the Deviance Information Criterion (DIC), an adjusted within-sample measure of predictive accuracy, for models estimated using Bayesian methods.

Usage

DIC(...)

Arguments

...	one or more fitted model objects of the same class.

Value

A numeric matrix containing the DIC values corresponding to each fitted object supplied in ....

References

Spiegelhalter D.J., Best N.G., Carlin B.P. and Van Der Linde A. (2002) Bayesian Measures of Model Complexity and Fit. Journal of the Royal Statistical Society Series B (Statistical Methodology), 64(4), 583–639.

Spiegelhalter D.J., Best N.G., Carlin B.P. and Van der Linde A. (2014). The deviance information criterion: 12 years on. Journal of the Royal Statistical Society Series B (Statistical Methodology), 76(3), 485–493.

---

Deviance Information Criterion (DIC) for objects of class mtar

Description

This function computes the Deviance Information Criterion (DIC) for objects of class mtar.

Usage

## S3 method for class 'mtar'
DIC(...)

Arguments

...	one or several objects of the class mtar.

Value

A numeric matrix containing the DIC values corresponding to each mtar object in the input.

See Also

WAIC

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar_grid(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
                  subset={Date<="2015-12-07"}, dist=c("Gaussian","Student-t",
                  "Slash","Laplace"), nregim.min=2, nregim.max=3, p.min=2,
                  p.max=2, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
DIC(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar_grid(~ Bedon + LaPlata | Rainfall, data=riverflows,
                  row.names=Date, subset={Date<="2009-02-13"},dist="Laplace",
                  nregim.min=2, nregim.max=3, p.min=1, p.max=3,n.burnin=1000,
                  n.sim=2000, n.thin=2, plan_strategy="multisession")
DIC(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf,subset={Date<="1974-11-06"},row.names=Date,
                  dist=c("Slash","Student-t"), nregim.min=1, nregim.max=2,
                  p.min=15, p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,
                  n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
DIC(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar_grid(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
                  row.names=Date, dist=c("Laplace","Student-t","Slash"),
                  nregim.min=2, nregim.max=2, p.min=3, p.max=3, d.min=3,
                  d.max=3, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
DIC(fit4)

---

Effective sample size for mtar objects

Description

This function computes the effective sample size, adjusted for autocorrelation, of Markov chain Monte Carlo (MCMC) output obtained from the Bayesian estimation of multivariate TAR models. It serves as a wrapper around effectiveSize(), applying this function to the posterior chains returned by mtar().

Usage

effectiveSize_TAR(x)

Arguments

x	An object of class mtar produced by mtar().

Value

A list with the effective sample sizes for each parameter of the mtar model.

See Also

effectiveSize

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2)
effectiveSize_TAR(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
effectiveSize_TAR(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
effectiveSize_TAR(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
effectiveSize_TAR(fit4)

---

Geweke's convergence diagnostic for mtar objects

Description

This function computes Geweke's convergence diagnostic for Markov chain Monte Carlo (MCMC) output obtained from Bayesian estimation of multivariate TAR models. It is a wrapper around geweke.diag() that applies the diagnostic to the posterior chains returned by a call to mtar().

Usage

geweke_diagTAR(x, frac1 = 0.1, frac2 = 0.5)

Arguments

x	An object of class mtar returned by the function mtar().
frac1	A numeric value in $(0,1)$ specifying the fraction of the initial part of each chain to be used in the diagnostic.
frac2	A numeric value in $(0,1)$ specifying the fraction of the final part of each chain to be used in the diagnostic.

Value

A list containing the Geweke z-scores for the parameters of the mtar model.

See Also

geweke.diag

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2)
geweke_diagTAR(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
geweke_diagTAR(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
geweke_diagTAR(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
geweke_diagTAR(fit4)

---

Geweke-Brooks plot for objects of class mtar

Description

This function is a wrapper around geweke.plot() that applies the Geweke-Brooks convergence diagnostic to the MCMC chains obtained from a fitted mtar model.

Usage

geweke_plotTAR(
  x,
  frac1 = 0.1,
  frac2 = 0.5,
  nbins = 20,
  pvalue = 0.05,
  auto.layout = TRUE,
  ask,
  ...
)

Arguments

x	An object of class mtar returned by a call to mtar().
frac1	fraction to use from beginning of chain
frac2	fraction to use from end of chain
nbins	Number of segments
pvalue	p-value used to plot confidence limits for the null hypothesis
auto.layout	If TRUE then, set up own layout for plots, otherwise use existing one
ask	If TRUE then prompt user before displaying each page of plots. Default is dev.interactive().
...	Additional graphical parameters passed to the plotting routines.

See Also

geweke.plot

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2)
geweke_plotTAR(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
geweke_plotTAR(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
geweke_plotTAR(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
geweke_plotTAR(fit4)

---

Highest Posterior Density intervals for objects of class mtar

Description

Highest Posterior Density intervals for objects of class mtar

Usage

## S3 method for class 'mtar'
HPDinterval(obj, prob = 0.95, ...)

Arguments

obj	an object of class mtar generated by a call to the function mtar().
prob	a numeric scalar in the interval $(0,1)$ giving the target probability content of the intervals. By default, prob is set to 0.95.
...	Optional additional arguments for methods. None are used at present.

See Also

HPDinterval

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2, ssvs=TRUE)
coda::HPDinterval(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
coda::HPDinterval(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
coda::HPDinterval(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
coda::HPDinterval(fit4)

---

Temperature, precipitation, and two river flows in Iceland

Description

This data set contains two daily river-flow time series, measured in cubic meters per second, for rivers in Iceland from January 1, 1972, to December 12, 1974. Additionally, daily measurements of precipitation (in millimeters) and temperature (in degrees Celsius) were recorded at the Hveravellir meteorological station. The precipitation values correspond to the accumulated precipitation up to 9:00 A.M. relative to the same time on the previous day.

Usage

data(iceland.rf)

Format

A data frame with 1,096 rows and 5 variables:

Vatnsdalsa

Numeric vector representing the daily flow of the Vatnsdalsá river.

Jokulsa

Numeric vector representing the daily flow of the Jökulsá Eystri river.

Precipitation

Numeric vector of daily precipitation amounts (millimeters).

Temperature

Numeric vector of daily temperature measurements (degrees Celsius).

Date

Vector indicating the calendar date of each observation.

References

Tong, Howell (1990) Non‑linear Time Series: A Dynamical System Approach. Oxford University Press. Oxford, UK.

Ruey S., Tsay (1998) Testing and Modeling Multivariate Threshold Models. Journal of the American Statistical Association, 93, 1188-1202.

Examples

data(iceland.rf)
dev.new()
plot(ts(as.matrix(iceland.rf[,-5])), main="Iceland")

---

Bayesian estimation of a multivariate Threshold Autoregressive (TAR) model.

Description

This function implements a Gibbs sampling algorithm to generate draws from the posterior distribution of the parameters of a multivariate Threshold Autoregressive (TAR) model, including special cases such as SETAR and VAR models. The approach accommodates a wide range of noise process distributions, including Gaussian, Student-$t$, Slash, symmetric hyperbolic, contaminated normal, Laplace, skew-normal, and skew-Student-$t$.

Usage

mtar(
  formula,
  data,
  subset,
  Intercept = TRUE,
  trend = c("none", "linear", "quadratic"),
  nseason = NULL,
  ars = ars(),
  row.names,
  dist = c("Gaussian", "Student-t", "Hyperbolic", "Laplace", "Slash",
    "Contaminated normal", "Skew-Student-t", "Skew-normal"),
  prior = list(),
  n.sim = 500,
  n.burnin = 100,
  n.thin = 1,
  ssvs = FALSE,
  setar = NULL,
  progress = TRUE,
  ...
)

Arguments

formula	A three-part expression of class Formula describing the TAR model to be fitted. The first part specifies the variables in the multivariate output series, the second part defines the threshold series, and the third part specifies the variables in the multivariate exogenous series.
data	A data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which mtar_grid() is called.
subset	An optional vector specifying a subset of observations to be used in the fitting process.
Intercept	An optional logical indicating whether an intercept should be included within each regime.
trend	An optional character string specifying the degree of deterministic time trend to be included in each regime. Available options are "linear", "quadratic", and "none". By default, trend is set to "none".
nseason	An optional integer, greater than or equal to 2, specifying the number of seasonal periods. When provided, nseason - 1 seasonal dummy variables are added to the regressors within each regime. By default, nseason is set to NULL, thereby indicating that the TAR model has no seasonal effects.
ars	A list defining the autoregressive structure of the model. It contains four components: the number of regimes (nregim), the autoregressive order within each regime (p), and the maximum lags for the exogenous (q) and threshold (d) series in each regime. The object can be validated using the helper function ars().
row.names	An optional variable in data labelling the time points corresponding to each row of the data set.
dist	A character string specifying the multivariate distributions used to model the noise process. Available options are "Gaussian", "Student-t", "Slash", "Hyperbolic", "Laplace", "Contaminated normal", "Skew-normal", and "Skew-Student-t". By default, dist is set to "Gaussian".
prior	An optional list specifying the hyperparameter values that define the prior distribution. This list can be validated using the priors() function. By default, prior is set to an empty list, thereby indicating that the hyperparameter values should be set so that a non-informative prior distribution is obtained.
n.sim	An optional positive integer specifying the number of simulation iterations after the burn-in period. By default, n.sim is set to 500.
n.burnin	An optional positive integer specifying the number of burn-in iterations. By default, n.burnin is set to 100.
n.thin	An optional positive integer specifying the thinning interval. By default, n.thin is set to 1.
ssvs	An optional logical indicating whether the Stochastic Search Variable Selection (SSVS) procedure should be applied to identify relevant lags of the output, exogenous, and threshold series. By default, ssvs is set to FALSE.
setar	An optional positive integer indicating the component of the output series used as the threshold variable. By default, setar is set to NULL, indicating that the fitted model is not a SETAR model.
progress	An optional logical indicating whether a progress bar should be displayed during execution. By default, progress is set to TRUE.
...	further arguments passed to or from other methods.

Value

an object of class mtar in which the main results of the model fitted to the data are stored, i.e., a list with components including

chains	list with several arrays, which store the values of each model parameter in each iteration of the simulation,
n.sim	number of iterations of the simulation after the burn-in period,
n.burnin	number of burn-in iterations in the simulation,
n.thin	thinning interval in the simulation,
ars	list composed of four objects, namely: nregim, p, q and d, each of which corresponds to a vector of non-negative integers with as many elements as there are regimes in the fitted TAR model,
dist	name of the multivariate distribution used to describe the behavior of the noise process,
threshold.series	vector with the values of the threshold series,
output.series	matrix with the values of the output series,
exogenous.series	matrix with the values of the exogenous series,
Intercept	If TRUE, then the model included an intercept term in each regime,
trend	the degree of the deterministic time trend, if any,
nseason	the number of seasonal periods, if any,
formula	the formula,
call	the original function call.

References

Nieto, F.H. (2005) Modeling Bivariate Threshold Autoregressive Processes in the Presence of Missing Data. Communications in Statistics - Theory and Methods, 34, 905-930.

Romero, L.V. and Calderon, S.A. (2021) Bayesian estimation of a multivariate TAR model when the noise process follows a Student-t distribution. Communications in Statistics - Theory and Methods, 50, 2508-2530.

Calderon, S.A. and Nieto, F.H. (2017) Bayesian analysis of multivariate threshold autoregressive models with missing data. Communications in Statistics - Theory and Methods, 46, 296-318.

Vanegas, L.H. and Calderón, S.A. and Rondón, L.M. (2025) Bayesian estimation of a multivariate tar model when the noise process distribution belongs to the class of gaussian variance mixtures. International Journal of Forecasting.

See Also

DIC, WAIC

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=1000, n.sim=2000,
             n.thin=2)
summary(fit1)
DIC(fit1)
WAIC(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
summary(fit2)
DIC(fit2)
WAIC(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
summary(fit3)
DIC(fit3)
WAIC(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
summary(fit4)
DIC(fit4)
WAIC(fit4)

---

Bayesian Estimation of Multivariate TAR Models

Description

This function is a wrapper that applies mtar() over a grid of model specifications defined by all combinations of the noise distribution (dist), the number of regimes (from nregim.min to nregim.max), the autoregressive order within each regime (from p.min to p.max), the maximum lag of the exogenous series within each regime (from q.min to q.max), and the maximum lag of the threshold series within each regime (from d.min to d.max). In all calls to mtar(), the same set of time points is used for model fitting. This is achieved by appropriately adjusting the subset argument of mtar() for each model specification, thereby ensuring comparability across models.

Usage

mtar_grid(
  formula,
  data,
  subset,
  Intercept = TRUE,
  trend = c("none", "linear", "quadratic"),
  nseason = NULL,
  nregim.min = 1,
  nregim.max = NULL,
  p.min = 1,
  p.max = NULL,
  q.min = 0,
  q.max = 0,
  d.min = 0,
  d.max = 0,
  row.names,
  dist = "Gaussian",
  prior = list(),
  n.sim = 500,
  n.burnin = 100,
  n.thin = 1,
  ssvs = FALSE,
  setar = NULL,
  plan_strategy = c("multisession", "sequential"),
  progress = TRUE
)

Arguments

formula	A three-part expression of class Formula describing the TAR model to be fitted. The first part specifies the variables in the multivariate output series, the second part defines the threshold series, and the third part specifies the variables in the multivariate exogenous series.
data	A data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which mtar_grid() is called.
subset	An optional vector specifying a subset of observations to be used in the fitting process.
Intercept	An optional logical indicating whether an intercept should be included within each regime.
trend	An optional character string specifying the degree of deterministic time trend to be included in each regime. Available options are "linear", "quadratic", and "none". By default, trend is set to "none".
nseason	An optional integer, greater than or equal to 2, specifying the number of seasonal periods. When provided, nseason - 1 seasonal dummy variables are added to the regressors within each regime. By default, nseason is set to NULL, thereby indicating that the TAR model has no seasonal effects.
nregim.min	An optional integer specifying the minimum number of regimes. By default, nregim.min is set to 1.
nregim.max	An integer specifying the maximum number of regimes.
p.min	An optional integer specifying the minimum autoregressive order within each regime. By default, p.min is set to 1.
p.max	An integer specifying the maximum autoregressive order within each regime.
q.min	An optional integer specifying the minimum value of the maximum lag of the exogenous series within each regime. By default, q.min is set to 0.
q.max	An optional integer specifying the maximum value of the maximum lag of the exogenous series within each regime. By default, q.max is set to 0.
d.min	An optional integer specifying the minimum value of the maximum lag of the threshold series within each regime. By default, d.min is set to 0.
d.max	An optional integer specifying the maximum value of the maximum lag of the threshold series within each regime. By default, d.max is set to 0.
row.names	An optional variable in data labelling the time points corresponding to each row of the data set.
dist	A character vector specifying the multivariate distributions used to model the noise process. Available options are "Gaussian", "Student-t", "Slash", "Hyperbolic", "Laplace", "Contaminated normal", "Skew-normal", and "Skew-Student-t". By default, dist is set to "Gaussian".
prior	An optional list specifying the hyperparameter values that define the prior distribution. This list can be validated using the priors() function. By default, prior is set to an empty list, thereby indicating that the hyperparameter values should be set so that a non-informative prior distribution is obtained.
n.sim	An optional positive integer specifying the number of simulation iterations after the burn-in period. By default, n.sim is set to 500.
n.burnin	An optional positive integer specifying the number of burn-in iterations. By default, n.burnin is set to 100.
n.thin	An optional positive integer specifying the thinning interval. By default, n.thin is set to 1.
ssvs	An optional logical indicating whether the Stochastic Search Variable Selection (SSVS) procedure should be applied to identify relevant lags of the output, exogenous, and threshold series. By default, ssvs is set to FALSE.
setar	An optional positive integer indicating the component of the output series used as the threshold variable. By default, setar is set to NULL, indicating that the fitted model is not a SETAR model.
plan_strategy	An optional character string specifying the execution strategy for parallel computation. Available options are "sequential" and "multisession". By default, plan_strategy is set to "sequential".
progress	An optional logical indicating whether a progress bar should be displayed during execution. By default, progress is set to TRUE.

Value

A list whose elements are objects of class mtar, each corresponding to a distinct model specification considered in the grid.

See Also

mtar

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar_grid(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
                  subset={Date<="2015-12-07"}, dist=c("Gaussian","Student-t",
                  "Slash","Laplace"), nregim.min=2, nregim.max=3, p.min=2,
                  p.max=2, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
summary(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar_grid(~ Bedon + LaPlata | Rainfall, data=riverflows,
                  row.names=Date, subset={Date<="2009-02-13"},dist="Laplace",
                  nregim.min=2, nregim.max=3, p.min=1, p.max=3,n.burnin=1000,
                  n.sim=2000, n.thin=2, plan_strategy="multisession")
fit2

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf,subset={Date<="1974-11-06"},row.names=Date,
                  dist=c("Slash","Student-t"), nregim.min=1, nregim.max=2,
                  p.min=15, p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,
                  n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
fit3

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar_grid(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
                  row.names=Date, dist=c("Laplace","Student-t","Slash"),
                  nregim.min=2, nregim.max=2, p.min=3, p.max=3, d.min=3,
                  d.max=3, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
fit4

---

Out-of-sample predictive accuracy measures

Description

Computes out-of-sample predictive accuracy measures for one or more fitted models of the same class, based on their predictive distributions.

Usage

out_of_sample(..., newdata, n.ahead)

Arguments

...	one or more fitted model objects of the same class.
newdata	a data frame containing the future values of the output series required to evaluate predictive performance.
n.ahead	a positive integer specifying the number of forecast steps ahead to use in the predictive performance evaluation.

---

Computing Out-of-Sample predictive accuracy measures

Description

Computes Out-of-Sample predictive accuracy measures for an object of class listmtar.

Usage

## S3 method for class 'listmtar'
out_of_sample(
  x,
  newdata,
  n.ahead = NULL,
  credible = 0.95,
  FUN = mean,
  rolling = NULL,
  ...
)

Arguments

x	An object of class listmtar returned by the routine mtar_grid().
newdata	A data.frame containing future values of the threshold series (if included in the fitted model), the exogenous series (if included in the fitted model), and the realized values of the output series.
n.ahead	A positive integer specifying the number of steps ahead to forecast.
credible	An optional numeric value in $(0,1)$ specifying the level of the required prediction intervals. By default, credible is set to 0.95.
FUN	An optional function used to summarize the n.ahead values computed for each predictive accuracy measure. By default, FUN is set to mean.
rolling	An optional positive integer specifying the rolling-window size used for forecasting. By default, rolling = NULL, indicating that rolling-window forecasting is not performed.
...	optional arguments to FUN.

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar_grid(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
                  subset={Date<="2015-12-07"}, dist=c("Gaussian","Student-t",
                  "Slash","Laplace"), nregim.min=2, nregim.max=3, p.min=2,
                  p.max=2, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
oos1 <- out_of_sample(fit1, newdata=subset(returns, Date>"2015-12-07"),
                      n.ahead=75, FUN=median)
oos1

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar_grid(~ Bedon + LaPlata | Rainfall, data=riverflows,
                  row.names=Date, subset={Date<="2009-02-13"},dist="Laplace",
                  nregim.min=2, nregim.max=3, p.min=1, p.max=3,n.burnin=1000,
                  n.sim=2000, n.thin=2, plan_strategy="multisession")
oos2 <- out_of_sample(fit2, newdata=subset(riverflows, Date>"2009-02-13"),
                      n.ahead=60, FUN=median)
oos2

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf,subset={Date<="1974-11-06"},row.names=Date,
                  dist=c("Slash","Student-t"), nregim.min=1, nregim.max=2,
                  p.min=15, p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,
                  n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
oos3 <- out_of_sample(fit3, newdata=subset(iceland.rf, Date>"1974-11-06"),
                      n.ahead=55, FUN=median)
oos3

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar_grid(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
                  row.names=Date, dist=c("Laplace","Student-t","Slash"),
                  nregim.min=2, nregim.max=2, p.min=3, p.max=3, d.min=3,
                  d.max=3, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
oos4 <- out_of_sample(fit4, newdata=subset(US.returns, Date>"2025-11-28"),
                      n.ahead=100, FUN=median)
oos4

---

Computing Out-of-Sample predictive accuracy measures

Description

Computes Out-of-Sample predictive accuracy measures for two or more objects of class mtar.

Usage

## S3 method for class 'mtar'
out_of_sample(
  ...,
  newdata,
  n.ahead = NULL,
  credible = 0.95,
  FUN = mean,
  rolling = NULL
)

Arguments

...	one or more objects of class mtar.
newdata	A data.frame containing future values of the threshold series (if included in the fitted model), the exogenous series (if included in the fitted model), and the realized values of the output series.
n.ahead	A positive integer specifying the number of steps ahead to forecast.
credible	An optional numeric value in $(0,1)$ specifying the level of the required prediction intervals. By default, credible is set to 0.95.
FUN	An optional function used to summarize the n.ahead values computed for each predictive accuracy measure. By default, FUN is set to mean.
rolling	An optional positive integer specifying the rolling-window size used for forecasting. By default, rolling = NULL, indicating that rolling-window forecasting is not performed.

---

Forecasting for multivariate TAR models

Description

Computes forecasts from a fitted multivariate Threshold Autoregressive (TAR) model.

Usage

## S3 method for class 'mtar'
predict(
  object,
  ...,
  newdata,
  n.ahead = NULL,
  row.names,
  credible = 0.95,
  out.of.sample = FALSE,
  rolling = NULL
)

Arguments

object	An object of class mtar obtained from a call to mtar().
...	Additional arguments that may affect the prediction method.
newdata	An optional data.frame containing future values of the threshold series (if included in the fitted model), the exogenous series (if included in the fitted model), and, when out.of.sample = TRUE, the realized values of the output series.
n.ahead	A positive integer specifying the number of steps ahead to forecast.
row.names	An optional variable in newdata specifying labels for the time
credible	An optional numeric value in $(0,1)$ specifying the level of the required credible intervals. By default, credible is set to 0.95.
out.of.sample	An optional logical indicator. If TRUE then the log-score, Energy-Score (ES), Absolute Error (AE), Absolute Percentage Error (APE), Squared Error (SE), are computed as measures of predictive accuracy. In this case, newdata must include the observed values of the output series.
rolling	An optional positive integer specifying the rolling-window size used for forecasting with fixed parameters. By default, rolling = NULL, indicating that rolling-window forecasting is not performed.

References

Nieto, F.H. (2005) Modeling Bivariate Threshold Autoregressive Processes in the Presence of Missing Data. Communications in Statistics - Theory and Methods, 34, 905-930.

Romero, L.V. and Calderon, S.A. (2021) Bayesian estimation of a multivariate TAR model when the noise process follows a Student-t distribution. Communications in Statistics - Theory and Methods, 50, 2508-2530.

Calderon, S.A. and Nieto, F.H. (2017) Bayesian analysis of multivariate threshold autoregressive models with missing data. Communications in Statistics - Theory and Methods, 46, 296-318.

Karlsson, S. (2013) Chapter 15-Forecasting with Bayesian Vector Autoregression. In Elliott, G. and Timmermann, A. Handbook of Economic Forecasting, Volume 2, 791–89, Elsevier.

Vanegas, L.H. and Calderón, S.A. and Rondón, L.M. (2025) Bayesian estimation of a multivariate tar model when the noise process distribution belongs to the class of gaussian variance mixtures. International Journal of Forecasting.

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
             subset={Date<="2015-12-07"}, dist="Student-t",
             ars=ars(nregim=3,p=c(1,1,2)), n.burnin=100, n.sim=200,
             n.thin=2)
p1 <- predict(fit1, newdata=subset(returns,Date>"2015-12-07"), n.ahead=75,
              credible=0.8, out.of.sample=TRUE)
with(p1,summary)
with(p1,cbind(LS,ES,APE,CR))
plot(p1,last=100)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar(~ Bedon + LaPlata | Rainfall, data=riverflows, row.names=Date,
             subset={Date<="2009-02-13"}, dist="Laplace",
             ars=ars(nregim=3,p=5), n.burnin=1000, n.sim=2000, n.thin=2)
p2 <- predict(fit2, newdata=subset(riverflows,Date>"2009-02-13"), n.ahead=60,
              credible=0.8, out.of.sample=TRUE)
with(p2,summary)
with(p2,cbind(LS,ES,APE,CR))
plot(p2,last=100)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
             data=iceland.rf, subset={Date<="1974-11-06"}, row.names=Date,
             ars=ars(nregim=2,p=15,q=4,d=2), n.burnin=1000, n.sim=2000,
             n.thin=2, dist="Slash")
p3 <- predict(fit3, newdata=subset(iceland.rf,Date>"1974-11-06"), n.ahead=55,
              credible=0.8, out.of.sample=TRUE)
with(p3,summary)
with(p3,cbind(LS,ES,APE,CR))
plot(p3,last=100)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
             row.names=Date, ars=ars(nregim=2,p=3,d=3), n.burnin=1000,
             n.sim=2000, n.thin=2, dist="Student-t")
p4 <- predict(fit4, newdata=subset(US.returns,Date>"2025-11-28"),n.ahead=100,
              credible=0.8, out.of.sample=TRUE)
with(p4,summary)
with(p4,cbind(LS,ES,APE,CR))
plot(p4,last=100)

---

Auxiliary function for setting hyperparameter values

Description

This function constructs and validates the list of hyperparameter values used to define the prior distributions of the model parameters. Hyperparameters not explicitly provided by the user are assigned their default values, which define non-informative prior distributions.

Usage

priors(prior, regim, k, dist, setar, ssvs)

Arguments

prior	A list specifying user-defined values for the hyperparameters. Any hyperparameters not included in this list are set to their default values.
regim	A positive integer indicating the number of regimes in the model.
k	A positive integer indicating the dimension of the multivariate output series.
dist	A character string specifying the distribution chosen to model the noise process.
setar	A positive integer indicating the component of the output series that acts as the threshold variable in a SETAR specification. If NULL is specified then the model is not a SETAR.
ssvs	A logical indicating whether the Stochastic Search Variable Selection (SSVS) procedure should be applied.

Value

A list containing the hyperparameter values defining the prior distributions of all model parameters.

---

Returns of the closing prices of three financial indexes

Description

This dataset contains daily returns computed from the closing prices of the COLCAP, BOVESPA, and S&P 500 stock market indexes over the period from February 10, 2010, to March 31, 2016, comprising 1,505 observations. The COLCAP index reflects the price dynamics of the 20 most liquid stocks traded on the Colombian stock market. The BOVESPA index represents the Brazilian stock market, the largest and most important exchange in Latin America and among the largest worldwide. The Standard & Poor's 500 (S&P 500) index tracks the performance of 500 large-cap companies listed in the United States.

Usage

data(returns)

Format

A data frame with 1,505 rows and 4 variables:

Date

A vector indicating the date of each observation.

COLCAP

A numeric vector containing the returns of the COLCAP index.

SP500

A numeric vector containing the returns of the S&P 500 index.

BOVESPA

A numeric vector containing the returns of the BOVESPA index.

References

Romero, L.V. and Calderon, S.A. (2021) Bayesian estimation of a multivariate TAR model when the noise process follows a Student-t distribution. Communications in Statistics - Theory and Methods, 50, 2508-2530.

Examples

data(returns)
dev.new()
plot(ts(as.matrix(returns[,-1])), main="Returns")

---

Rainfall and two river flows in Colombia

Description

This dataset contains daily observations of rainfall (in millimeters) and river flows (in $m^3$/s) for two rivers in southern Colombia. Rainfall was recorded at a meteorological station located at an altitude of 2400 meters above sea level. River flows were measured at two hydrological stations: El Trébol station, which monitors the Bedón River at an altitude of 1720 meters, and Villalosada station, which monitors the La Plata River at an altitude of 1300 meters. The stations are located near the equator, a geographic feature that helps reduce seasonal distortions and facilitates the analysis of the dynamic relationship between rainfall and river flows. The sample period spans from January 1, 2006, to April 14, 2009.

Usage

data(riverflows)

Format

A data frame with 1200 rows and 4 variables:

Date

A vector indicating the date of each observation.

Bedon

A numeric vector giving the daily flow of the Bedón River.

LaPlata

A numeric vector giving the daily flow of the La Plata River.

Rainfall

A numeric vector indicating daily rainfall amounts.

References

Calderon, S.A. and Nieto, F.H. (2017) Bayesian analysis of multivariate threshold autoregressive models with missing data. Communications in Statistics - Theory and Methods, 46, 296-318.

Examples

data(riverflows)
dev.new()
plot(ts(as.matrix(riverflows[,-1])), main="Rainfall and river flows")

---

Simulation of multivariate time series from a TAR model

Description

This function simulates multivariate time series generated by a user-specified Threshold Autoregressive (TAR) model.

Usage

simtar(
  n,
  k = 2,
  ars = ars(),
  Intercept = TRUE,
  trend = c("none", "linear", "quadratic"),
  nseason = NULL,
  parms,
  delay = 0,
  thresholds = NULL,
  t.series = NULL,
  ex.series = NULL,
  dist = c("Gaussian", "Student-t", "Hyperbolic", "Laplace", "Slash",
    "Contaminated normal", "Skew-Student-t", "Skew-normal"),
  skewness = NULL,
  extra = NULL,
  setar = NULL,
  Verbose = TRUE
)

Arguments

n	A positive integer specifying the length of the simulated output series.
k	A positive integer specifying the dimension of the multivariate output series.
ars	A list defining the TAR model structure, composed of four elements: the number of regimes (nregim), the autoregressive order (p), and the maximum lags of the exogenous (q) and threshold (d) series within each regime. This object can be validated using the ars() function.
Intercept	An optional logical indicating whether an intercept term is included in each regime.
trend	An optional character string specifying the degree of deterministic time trend to be included in each regime. Available options are a linear trend ("linear"), a quadratic trend ("quadratic"), or no trend ("none"). By default, trend is set to "none".
nseason	An optional integer, greater than or equal to 2, specifying the number of seasonal periods. When provided, nseason - 1 seasonal dummy variables are added to the regressors within each regime. By default, nseason is set to NULL, thereby indicating that the TAR model has no seasonal effects.
parms	A list with one sublist per regime. Each sublist contains two matrices: the first matrix corresponds to the location parameters, and the second matrix corresponds to the scale parameters of the model.
delay	An optional non-negative integer specifying the delay parameter of the threshold series. By default, delay is set to 0.
thresholds	A numeric vector of length nregim-1 containing the threshold values, sorted in ascending order.
t.series	A matrix containing the values of the threshold series.
ex.series	A matrix containing the values of the multivariate exogenous series.
dist	An optional character string specifying the multivariate distribution chosen to model the noise process. Supported options include Gaussian ("Gaussian"), Student-$t$ ("Student-t"), Slash ("Slash"), Symmetric Hyperbolic ("Hyperbolic"), Laplace ("Laplace"), and Contaminated Normal ("Contaminated normal"). By default, dist is set to "Gaussian".
skewness	An optional numeric vector specifying the skewness parameters of the noise distribution, when applicable.
extra	An optional value specifying the extra parameter of the noise distribution, when required.
setar	An optional positive integer indicating which component of the output series is used as the threshold variable. By default, setar is set to NULL, indicating that the model is not of SETAR type.
Verbose	An optional logical indicating whether a description of the simulated TAR model should be printed. By default, Verbose is set to TRUE.

Value

A data.frame containing the simulated multivariate output series, and, if specified, the threshold series and multivariate exogenous series.

References

Vanegas, L.H. and Calderón, S.A. and Rondón, L.M. (2025) Bayesian estimation of a multivariate tar model when the noise process distribution belongs to the class of gaussian variance mixtures. International Journal of Forecasting.

Examples

###### Simulation of a trivariate TAR model with two regimes
n <- 2000
k <- 3
myars <- ars(nregim=2,p=c(1,2))
Z <- as.matrix(arima.sim(n=n+max(myars$p),list(ar=c(0.5))))
probs <- sort((0.6 + runif(myars$nregim-1)*0.8)*c(1:(myars$nregim-1))/myars$nregim)
dist <- "Student-t"; extra <- 6
parms <- list()
for(j in 1:myars$nregim){
    np <- 1 + myars$p[j]*k
    parms[[j]] <- list()
    parms[[j]]$location <- c(ifelse(runif(np*k)<=0.5,1,-1)*rbeta(np*k,shape1=4,shape2=16))
    parms[[j]]$location <- matrix(parms[[j]]$location,np,k)
    parms[[j]]$scale <- rgamma(k,shape=1,scale=1)*diag(k)
}
thresholds <- quantile(Z,probs=probs)
out1 <- simtar(n=n, k=k, ars=myars, parms=parms, thresholds=thresholds,
               t.series=Z, dist=dist, extra=extra)
str(out1)

fit1 <- mtar(~ Y1 + Y2 + Y3 | Z, data=out1, ars=myars, dist=dist,
             n.burn=2000, n.sim=3000, n.thin=2)
summary(fit1)

###### Simulation of a trivariate VAR model
n <- 2000
k <- 3
myars <- ars(nregim=1,p=2)
dist <- "Slash"; extra <- 2
parms <- list()
for(j in 1:myars$nregim){
    np <- 1 + myars$p[j]*k
    parms[[j]] <- list()
    parms[[j]]$location <- c(ifelse(runif(np*k)<=0.5,1,-1)*rbeta(np*k,shape1=4,shape2=16))
    parms[[j]]$location <- matrix(parms[[j]]$location,np,k)
    parms[[j]]$scale <- rgamma(k,shape=1,scale=1)*diag(k)
}
out2 <- simtar(n=n, k=k, ars=myars, parms=parms, dist=dist, extra=extra)
str(out2)

fit2 <- mtar(~ Y1 + Y2 + Y3, data=out2, ars=myars, dist=dist,
             n.burn=2000, n.sim=3000, n.thin=2)
summary(fit2)

n <- 5000
k <- 3
myars <- ars(nregim=2,p=c(1,2))
dist <- "Laplace"
parms <- list()
for(j in 1:myars$nregim){
    np <- 1 + myars$p[j]*k
    parms[[j]] <- list()
    parms[[j]]$location <- c(ifelse(runif(np*k)<=0.5,1,-1)*rbeta(np*k,shape1=4,shape2=16))
    parms[[j]]$location <- matrix(parms[[j]]$location,np,k)
    parms[[j]]$scale <- rgamma(k,shape=1,scale=1)*diag(k)
}
out3 <- simtar(n=n, k=k, ars=myars, parms=parms, delay=2,
               thresholds=-1, dist=dist, setar=2)
str(out3)

fit3 <- mtar(~ Y1 + Y2 + Y3, data=out3, ars=myars, dist=dist,
             n.burn=2000, n.sim=3000, n.thin=2, setar=2)
summary(fit3)

---

U.S. Stock Returns

Description

The dataset comprises observations of both continuously compounded and simple returns derived from the S&P 500 index, along with the difference of the Chicago Board Options Exchange Market Volatility Index (VIX). The sample period spans from January 5, 2005, to April 24, 2026.

Usage

data(US.returns)

Format

A data frame with 5420 rows and 6 variables:

Date

A vector indicating the date of each observation.

SP500

A numeric vector giving the S&P500 index.

VIX

A numeric vector giving the Chicago Board Options Exchange Market Volatility Index (VIX).

CCR

A numeric vector giving the continuously compounded returns.

SR

A numeric vector giving the simple returns.

dVIX

A numeric vector giving the difference $VIX_{t-1}-VIX_{t-2}$.

References

Massacci, D. (2014) A two-regime threshold model with conditional skewed student-t distributions for stock returns. Economic Modelling, 43, 9-20.

Examples

data(US.returns)
dev.new()
plot(ts(as.matrix(US.returns[,-1])))

---

vcov method for objects of class mtar

Description

Computes estimates of the variance–covariance matrices for the scale parameters of a fitted multivariate TAR model.

Usage

## S3 method for class 'mtar'
vcov(object, ..., FUN = mean)

Arguments

object	an object of class mtar, typically obtained from a call to mtar().
...	additional arguments passed to FUN.
FUN	a function to be applied to the MCMC chains of the scale parameters in order to obtain point estimates. By default, FUN is set to mean().

Value

A list containing the variance–covariance estimates obtained by applying FUN to the MCMC chains associated with the scale parameters.

---

Watanabe-Akaike or Widely Available Information Criterion (WAIC)

Description

Computes Watanabe-Akaike or Widely Available Information Criterion (WAIC), an adjusted within-sample measure of predictive accuracy, for models estimated using Bayesian methods.

Usage

WAIC(...)

Arguments

...	one or more fitted model objects of the same class.

Value

A numeric matrix containing the WAIC values corresponding to each fitted object supplied in ....

References

Watanabe S. (2010). Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. The Journal of Machine Learning Research, 11, 3571–3594.

---

Watanabe-Akaike or Widely Available Information Criterion (WAIC) for objects of class mtar

Description

This function computes the Watanabe-Akaike or Widely Available Information Criterion (WAIC), for objects of class mtar.

Usage

## S3 method for class 'mtar'
WAIC(...)

Arguments

...	one or several objects of the class mtar.

Value

A numeric matrix containing the WAIC values corresponding to each mtar object in the input.

See Also

DIC

Examples

###### Example 1: Returns of the closing prices of three financial indexes
data(returns)
fit1 <- mtar_grid(~ COLCAP + BOVESPA | SP500, data=returns, row.names=Date,
                  subset={Date<="2015-12-07"}, dist=c("Gaussian","Student-t",
                  "Slash","Laplace"), nregim.min=2, nregim.max=3, p.min=2,
                  p.max=2, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
WAIC(fit1)

###### Example 2: Rainfall and two river flows in Colombia
data(riverflows)
fit2 <- mtar_grid(~ Bedon + LaPlata | Rainfall, data=riverflows,
                  row.names=Date, subset={Date<="2009-02-13"},dist="Laplace",
                  nregim.min=2, nregim.max=3, p.min=1, p.max=3,n.burnin=1000,
                  n.sim=2000, n.thin=2, plan_strategy="multisession")
WAIC(fit2)

###### Example 3: Temperature, precipitation, and two river flows in Iceland
data(iceland.rf)
fit3 <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf,subset={Date<="1974-11-06"},row.names=Date,
                  dist=c("Slash","Student-t"), nregim.min=1, nregim.max=2,
                  p.min=15, p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,
                  n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
WAIC(fit3)

###### Example 4: U.S. stock returns
data(US.returns)
fit4 <- mtar_grid(~ CCR | dVIX, data=US.returns, subset={Date<="2025-11-28"},
                  row.names=Date, dist=c("Laplace","Student-t","Slash"),
                  nregim.min=2, nregim.max=2, p.min=3, p.max=3, d.min=3,
                  d.max=3, n.burnin=1000, n.sim=2000, n.thin=2,
                  plan_strategy="multisession")
WAIC(fit4)

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