# Threshold Autoregressive model: Bayesian approach 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). ```r 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 character(0) is the threshold value for regime switching; character(0) is the threshold variable; character(0) is the delay lag of threshold variable; and the error term character(0), character(0), for each regime is assumed to be an i.i.d. Gaussian white noise process with mean zero and variance character(0). 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. ## Returns 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 ```r 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) ```