Package 'BAYSTAR' reference manual

Title:	On Bayesian Analysis of Threshold Autoregressive Models
Description:	Fit two-regime threshold autoregressive (TAR) models by Markov chain Monte Carlo methods.
Authors:	Cathy W. S. Chen, Edward M.H. Lin, F.C. Liu, and Richard Gerlach
Maintainer:	Edward M.H. Lin <[email protected]>
License:	GPL (>= 2)
Version:	0.2-10
Built:	2025-02-16 05:16:11 UTC
Source:	https://github.com/cran/BAYSTAR

Threshold Autoregressive model: Bayesian approach

Description

Bayesian estimation and one-step-ahead forecasting for two-regime TAR model, as well as monitoring MCMC convergence. One may want to allow for higher-order AR models in the different regimes. Parsimonious subset AR could be assigned in each regime in the BAYSTAR function rather than a full AR model (i.e. the autoregressive orders could be not a continuous series).

Usage


BAYSTAR(x, lagp1, lagp2, Iteration, Burnin, constant, d0, 
        step.thv, thresVar, mu01, v01, mu02, v02, v0, lambda0, refresh,tplot)

BAYSTAR(x, lagp1, lagp2, Iteration, Burnin, constant, d0, 
        step.thv, thresVar, mu01, v01, mu02, v02, v0, lambda0, refresh,tplot)

Arguments

`x`	A vector of time series.
`lagp1`	A vector of non-zero autoregressive lags for the lower regime (regime one). For example, an AR model with p1=3 in lags 1,3, and 5 would be set as `lagp1<-c(1,3,5)`.
`lagp2`	A vector of non-zero autoregressive lags for the upper regime (regime two).
`Iteration`	The number of MCMC iterations.
`Burnin`	The number of burn-in iterations for the sampler.
`constant`	The intercepts include in the model for each regime, if `constant`=1. Otherwise, if `constant`=0. (Default: `constant`=1)
`d0`	The maximum delay lag considered. (Default: `d0` = 3)
`step.thv`	Step size of tuning parameter for the Metropolis-Hasting algorithm.
`thresVar`	A vector of time series for the threshold variable. (if missing, the series x is used.)
`mu01`	The prior mean of $\phi$ in regime one. This setting can be a scalar or a column vector with dimension equal to the number of $\phi$ . If this sets a scalar value, then the prior mean for all of $\phi$ are this value. (Default: a vector of zeros)
`v01`	The prior covariance matrix of $\phi$ in regime one. This setting can either be a scalar or a square matrix with dimensions equal to the number of $\phi$ . If this sets a scalar value, then prior covariance matrix of $\phi$ is that value times an identity matrix. (Default: a diagonal matrix are set to 0.1)
`mu02`	The prior mean of $\phi$ in regime two. This setting can be a scalar or a column vector with dimension equal to the number of $\phi$ . If this sets a scalar value, then the prior mean for all of $\phi$ are this value. (Default: a vector of zeros)
`v02`	The prior covariance matrix of $\phi$ in regime two. This setting can either be a scalar or a square matrix with dimensions equal to the number of $\phi$ . If this sets a scalar value, then prior covariance matrix of $\phi$ is that value times an identity matrix. (Default: a diagonal matrix are set to 0.1)
`v0`	`v0`/2 is the shape parameter for Inverse-Gamma prior of $\sigma^2$ . (Default: `v0` = 3)
`lambda0`	`lambda0`*`v0`/2 is the scale parameter for Inverse-Gamma prior of $\sigma^2$ . (Default: `lambda0` = the residual mean squared error of fitting an AR(p1) model to the data.)
`refresh`	Each `refresh` iteration for monitoring MCMC output. (Default: `refresh`=`Iteration`/2)
`tplot`	Trace plots and ACF plots for all parameter estimates. (Default: `tplot`=FALSE )

Details

Given the maximum AR orders p1 and p2, the two-regime SETAR(2:p1;p2) model is specified as:

$x_{t} = ( \phi _0^{(1)} + \phi _1^{(1)} x_{t - 1} + \ldots + \phi _{p1 }^{(1)} x_{t - p1 } + a_t^{(1)} ) I( z_{t-d} <= th) + ( \phi _0^{(2)} + \phi _1^{(2)} x_{t - 1} + \ldots + \phi _{p2 }^{(2)} x_{t - p2 } + a_t^{(2)} I( z_{t-d} > th)$

where $th$ is the threshold value for regime switching; $z_{t}$ is the threshold variable; $d$ is the delay lag of threshold variable; and the error term $a_t^{(j)}$ , $j, (j=1,2)$ , for each regime is assumed to be an i.i.d. Gaussian white noise process with mean zero and variance $sigma_j^2, j=1,2$ . I(A) is an indicator function. Event A will occur if I(A)=1 and otherwise if I(A)=0. One may want to allow parsimonious subset AR model in each regime rather than a full AR model.

Value

A list of output with containing the following components:

`mcmc`	All MCMC iterations.
`posterior`	The initial `Burnin` iterations are discarded as a burn-in sample, the final sample of (`Iteration-Burnin`) iterates is used for posterior inference.
`coef`	Summary Statistics of parameter estimation based on the final sample of (`Iteration-Burnin`) iterates.
`residual`	Residuals from the estimated model.
`lagd`	The mode of time delay lag of the threshold variable.
`DIC`	The deviance information criterion (DIC); a Bayesian method for model comparison (Spiegelhalter et al, 2002)

Author(s)

Cathy W. S. Chen, Edward M.H. Lin, F.C. Liu, and Richard Gerlach

Examples


set.seed(989981)
## Set the true values of all parameters
nob<- 200                ## No. of observations
lagd<- 1                  ## delay lag of threshold variable
r<- 0.4                   ## r is the threshold value
sig.1<- 0.8; sig.2<- 0.5  ## variances of error distributions for two regimes
p1<- 2; p2<- 1            ## No. of covariate in two regimes
ph.1<- c(0.1,-0.4,0.3)    ## mean coefficients for regime 1
ph.2<- c(0.2,0.6)     ## mean coefficients for regime 2
lagp1<-1:2
lagp2<-1:1

yt<- TAR.simu(nob,p1,p2,ph.1,ph.2,sig.1,sig.2,lagd,r,lagp1,lagp2)


## Total MCMC iterations and burn-in iterations
Iteration <- 500
Burnin    <- 200


## A RW (random walk) MH algorithm is used in simulating the threshold value
## Step size for the RW MH
step.thv<- 0.08

out <- BAYSTAR(yt,lagp1,lagp2,Iteration,Burnin,constant=1,step.thv=step.thv,tplot=TRUE)
set.seed(989981)
## Set the true values of all parameters
nob<- 200                ## No. of observations
lagd<- 1                  ## delay lag of threshold variable
r<- 0.4                   ## r is the threshold value
sig.1<- 0.8; sig.2<- 0.5  ## variances of error distributions for two regimes
p1<- 2; p2<- 1            ## No. of covariate in two regimes
ph.1<- c(0.1,-0.4,0.3)    ## mean coefficients for regime 1
ph.2<- c(0.2,0.6)     ## mean coefficients for regime 2
lagp1<-1:2
lagp2<-1:1

yt<- TAR.simu(nob,p1,p2,ph.1,ph.2,sig.1,sig.2,lagd,r,lagp1,lagp2)


## Total MCMC iterations and burn-in iterations
Iteration <- 500
Burnin    <- 200


## A RW (random walk) MH algorithm is used in simulating the threshold value
## Step size for the RW MH
step.thv<- 0.08

out <- BAYSTAR(yt,lagp1,lagp2,Iteration,Burnin,constant=1,step.thv=step.thv,tplot=TRUE)

Estimate AR coefficients

Description

We assume a normal prior for the AR coefficients and draw AR coefficients from a multivariate normal posterior distribution. Parsimonious subset AR could be assigned in each regime in the BAYSTAR function rather than a full AR model.

Usage

TAR.coeff(reg, ay, p1, p2, sig, lagd, 
          thres, mu0, v0, lagp1, lagp2, constant = 1, thresVar)
TAR.coeff(reg, ay, p1, p2, sig, lagd, 
          thres, mu0, v0, lagp1, lagp2, constant = 1, thresVar)

Arguments

A list containing:

`reg`	The regime is assigned. (equal to one or two)
`ay`	The real data set. (input)
`p1`	Number of AR coefficients in regime one.
`p2`	Number of AR coefficients in regime two.
`sig`	The error terms of TAR model.
`lagd`	The delay lag parameter.
`thres`	The threshold parameter.
`mu0`	Mean vector of conditional prior distribution in mean equation.
`v0`	Covariance matrix of conditional prior distribution in mean equation.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.
`thresVar`	Exogenous threshold variable. (if missing, the self series are used)

Author(s)

Cathy W.S. Chen, F.C. Liu

Identification of lag order of threshold variable

Description

The delay d has a discrete uniform prior over the integers: 1,2,..., d0, where d0 is a set maximum delay. We draw the delay lag of threshold variable from a multinomial distribution.

Usage

TAR.lagd(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, 
         thres, lagp1, lagp2, constant = 1, d0, thresVar)
TAR.lagd(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, 
         thres, lagp1, lagp2, constant = 1, d0, thresVar)

Arguments

A list containing:

`ay`	The real data set. (input)
`p1`	Number of AR coefficients in regime one.
`p2`	Number of AR coefficients in regime two.
`ph.1`	The vector of AR parameters in regime one.
`ph.2`	The vector of AR parameters in regime two.
`sig.1`	The error terms of AR model in the regime one.
`sig.2`	The error terms of AR model in the regime two.
`thres`	The threshold parameter.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.
`d0`	The maximum delay lag could be selected.
`thresVar`	Exogenous threshold variable. (if missing, the series x is used)

Author(s)

Cathy W.S. Chen, Edward Lin

Log-likelihood function

Description

To calculate the model log-likelihood function.

Usage

TAR.lik(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, 
        lagd, thres, lagp1, lagp2, constant = 1, thresVar)
TAR.lik(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, 
        lagd, thres, lagp1, lagp2, constant = 1, thresVar)

Arguments

A list containing:

`ay`	The real data set. (input)
`p1`	Number of AR coefficients in regime one.
`p2`	Number of AR coefficients in regime two.
`ph.1`	The vector of AR parameters in regime one.
`ph.2`	The vector of AR parameters in regime two.
`sig.1`	The error terms of AR model in the regime one.
`sig.2`	The error terms of AR model in the regime two.
`lagd`	The delay lag parameter.
`thres`	The threshold parameter.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.
`thresVar`	Exogenous threshold variable. (if missing, the series x is used)

Author(s)

Cathy W.S. Chen, F.C. Liu

To draw the variance of error distribution.

Description

We employ a conjugate prior, Inverse-Gamma distribution, for sigma squared in regime j, j=1,2. To draw the variance of error distribution from an Inverse-Gamma posterior distribution.

Usage

TAR.sigma(reg, ay, thres, lagd, p1, p2, ph, v, 
          lambda, lagp1, lagp2, constant = 1, thresVar)
TAR.sigma(reg, ay, thres, lagd, p1, p2, ph, v, 
          lambda, lagp1, lagp2, constant = 1, thresVar)

Arguments

A list containing:

`reg`	The regime is assigned. (equal to one or two)
`thres`	The threshold parameter.
`lagd`	The delay lag parameter.
`p1`	Number of AR coefficient in regime one.
`p2`	Number of AR coefficient in regime two.
`ph`	The vector of AR parameters in regime `reg`.
`ay`	The real data set. (input)
`v`, `lambda`	The hyper-parameter of Inverse Gamma distribution for priors of variance. (i.e. IG(v/2,lambda/2))
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.
`thresVar`	Exogenous threshold variable. (if missing, the series x is used)

Author(s)

Cathy W.S. Chen, Edward Lin

Simulated data from TAR model

Description

To generate the simulated data from TAR(2;p1,p2) model.

Usage

TAR.simu(nob, p1, p2, ph.1, ph.2, sig.1, sig.2, lagd, thres, lagp1, lagp2)
TAR.simu(nob, p1, p2, ph.1, ph.2, sig.1, sig.2, lagd, thres, lagp1, lagp2)

Arguments

`nob`	Number of observations that we want to simulate.
`p1`	Number of AR coefficient in regime one.
`p2`	Number of AR coefficient in regime two.
`ph.1`	The vector of AR parameters in regime one.
`ph.2`	The vector of AR parameters in regime two.
`sig.1`	The error terms in regime one.
`sig.2`	The error terms in regime two.
`lagd`	The delay lag parameter.
`thres`	The threshold parameter.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)

Author(s)

Cathy W.S. Chen, Edward Lin

Examples

## Set the true values of all parameters
nob<- 2000                ## No. of observations
lagd<- 1                  ## delay lag of threshold variable
r<- 0.4                   ## r is the threshold value
sig.1<- 0.8; sig.2<- 0.5  ## variances of error distributions for two regimes
p1<- 2; p2<- 2            ## No. of covariate in two regimes
ph.1<- c(0.1,-0.4,0.3)    ## mean coefficients for regime 1
ph.2<- c(0.2,0.3,0.3)     ## mean coefficients for regime 2
lagp1<-1:2
lagp2<-1:2

yt<- TAR.simu(nob,p1,p2,ph.1,ph.2,sig.1,sig.2,lagd,r,lagp1,lagp2)
## Set the true values of all parameters
nob<- 2000                ## No. of observations
lagd<- 1                  ## delay lag of threshold variable
r<- 0.4                   ## r is the threshold value
sig.1<- 0.8; sig.2<- 0.5  ## variances of error distributions for two regimes
p1<- 2; p2<- 2            ## No. of covariate in two regimes
ph.1<- c(0.1,-0.4,0.3)    ## mean coefficients for regime 1
ph.2<- c(0.2,0.3,0.3)     ## mean coefficients for regime 2
lagp1<-1:2
lagp2<-1:2

yt<- TAR.simu(nob,p1,p2,ph.1,ph.2,sig.1,sig.2,lagd,r,lagp1,lagp2)

Calculate summary statistics for all parameters

Description

A summary of the MCMC output can be obtained via the function TAR.summary. TAR.summary returns the posterior mean, median, standard deviation and the lower and upper bound of the 95% Bayes posterior interval for all parameters, all botained from the sampling period only, after burn-in.

Usage

TAR.summary(x, lagp1, lagp2, constant = 1)
TAR.summary(x, lagp1, lagp2, constant = 1)

Arguments

A list containing:

`x`	A matrix of the MCMC output of estimater parameters.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.

Author(s)

Cathy W.S. Chen, F.C. Liu

To draw a threshold value.

Description

The prior for the threshold parameter $thres$ , follows a uniform prior on a range (l,u), where l and u can be set as relevant percentiles of the observed threshold variable. This prior could be considered to correspond to an empirical Bayes approach, rather than a fully Bayesian one. The posterior distribution of $thres$ is not of a standard distributional form, thus requiring us to use the Metropolis-Hastings (MH) method to achieve the desired sample for $thres$ .

Usage

TAR.thres(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, lagd, thres, 
          step.r = 0.02, bound, lagp1, lagp2, constant = 1, thresVar)
TAR.thres(ay, p1, p2, ph.1, ph.2, sig.1, sig.2, lagd, thres, 
          step.r = 0.02, bound, lagp1, lagp2, constant = 1, thresVar)

Arguments

A list containing:

`ay`	The real data set. (input)
`p1`	Number of AR coefficients in regime one.
`p2`	Number of AR coefficients in regime two.
`ph.1`	The vector of AR parameters in regime one.
`ph.2`	The vector of AR parameters in regime two.
`sig.1`	The error terms of AR model in the regime one.
`sig.2`	The error terms of AR model in the regime two.
`lagd`	The delay lag parameter.
`thres`	The threshold parameter.
`step.r`	Step size of threshold variable for the MH algorithm are controlled the proposal variance.
`bound`	The bound of threshold parameter.
`lagp1`	The vector of non-zero autoregressive lags for the lower regime. (regime one); e.g. An AR model with p1=3, it could be non-zero lags 1,3, and 5 would set lagp1<-c(1,3,5).
`lagp2`	The vector of non-zero autoregressive lags for the upper regime. (regime two)
`constant`	Use the CONSTANT option to fit a model with/without a constant term (1/0). By default CONSTANT=1.
`thresVar`	Exogenous threshold variable. (if missing, the series x is used)

Author(s)

Cathy W.S. Chen, F.C. Liu

U.S. monthly civilian unemployment rate

Description

U.S. monthly civilian unemployment rate, seasonally adjusted and measured in percentage, from January 1948 to March 2004. The data set is available in Tsay (2005).

Usage

data(unemployrate)data(unemployrate)

Format

unemployrate is a data frame with 675 observations

Source

The data are obtained from the Bureau of Labor statistics, Department of Labor.

References

Tsay, R.S. (2005) Analysis of Financial Time Series. Second ed. Wiley, Hoboken.

Examples

data(unemployrate)
plot.ts(unemployrate)
data(unemployrate)
plot.ts(unemployrate)

Package 'BAYSTAR'

Help Index

Threshold Autoregressive model: Bayesian approach

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Estimate AR coefficients

Description

Usage

Arguments

Author(s)

Identification of lag order of threshold variable

Description

Usage

Arguments

Author(s)

Log-likelihood function

Description

Usage

Arguments

Author(s)

To draw the variance of error distribution.

Description

Usage

Arguments

Author(s)

Simulated data from TAR model

Description

Usage

Arguments

Author(s)

Examples

Calculate summary statistics for all parameters

Description

Usage

Arguments

Author(s)

To draw a threshold value.

Description

Usage

Arguments

Author(s)

U.S. monthly civilian unemployment rate

Description

Usage

Format

Source

References

Examples