A new R package for TAR modeling

Transcription

1 XXVI Simposio Internacional de Estadística 2016 Sincelejo, Sucre, Colombia, 8 al 12 de Agosto de 2016 A new R package for TAR modeling Un nuevo paquete para el modelamiento TAR en R Hanwen Zhang 1,a, Fabio Humberto Nieto 1,b 1 Facultad de Estadística, División de Ciencias Económicas y Administrativas, Universidad Santo Tomás, Bogotá, Colombia 2 Departamento de Estadística, Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia Abstract In this document we present the advance of a new R package TAR for threshold autoregressive models. The package would consider the following distribution for the white noise process: Gaussian, Student s t, log-normal, and generalized error distribution. As for the scope of the package, this will include identifying TAR model and the model estimation using a Bayesian approach. Key words: TAR model, software, computer package. 1. Theoretical background A TAR (threshold autoregressive) model describes the relationship between two time series: the threshold series {Z t } and the output series {X t }. The specification of the model is given by: X t = a (j) 0 + k j i=1 a (j) i X t i + h (j) e t if r j 1 < Z t r j for some j = 1,, l. l is the number of regimes, r 1 < < r l 1 are the thresholds, k 1,, k l are the AR orders. a j i with j = 1,, l and i = 0,, k j are the AR coefficients. {e t } is the white noise process with variance equal to 1, and h (1),, h (l) are the variance weights. Nieto (2005) developed a Bayesian methodology for the identification and estimation of TAR models with Gaussian noise process, Zhang (2014) considered other distribuciones for the noise process. 2. Existing packages en R After look over the R s package website: packages_by_name.html, we found the following packages containing modeling of threshold models in time series. BAYSTAR package (Chen et al. 2013). Deals with SETAR (self-exciting threshold autoregressive model which is a special case of TAR model with Z t = X t d for some integer d 0)models with Gaussian white noise process with two regimes. The principal features of this package are: a Docente. hanwenzhang@usantotomas.edu.co b Profesor titular. fhnietos@unal.edu.co 1

2 2 Hanwen Zhang & Fabio Humberto Nieto Estimate the threshold value and the lag d using the Metropolis-Hastings algorithm. The user must provide the autoregressive orders. Provide the simulation of series from a SETAR model. TSA package. Contains R functions and datasets detailed in the book of Cryer & Chan (2008): Time Series Analysis with Applications in R. As well as the BAYSTAR package, this package only considers the case of two regimes. The principal features of this package are: The user must provide the value of d. Estimate the threshold value and using the minimum AIC criterion. Select the AR orders by minimizing AIC. Provide the simulation of series from a SETAR model. Provide likelihood test for threshold nonlinearity. Provide the prediction based on a fitted SETAR model. Revising these packages, it is clear that both packages consider the SETAR model with Gaussian noise process. There is no computational routine available for cases there the number of regimes is different from two. Also, there is no package for identification and estimation of all parameters of the TAR model. Also, distribution different from the Gaussian is not considered. 3. Package TAR The new package named TAR is planned to consider not only Gaussian distributions for the white noise process but also distributions as Student s t, log-normal and generalized error distribution. With this package, the user can identify the number of regimes, the thresholds and the AR orders, as well as the estimation of AR coefficients and the variance weights. We illustrate the use of the principal functions for Gaussian white noise process Function for the simulation of series from a TAR model simu.tar.norm This function simulates a serie from a TAR model with Gaussian distributed error given the parameters of the model from a given threshold process {Z t }. The arguments of the function are: Z: the threshold series, X the output series, l the number of regimes, r the thresholds, K the AR orders, theta the matrix of l rows containing the autoregressive orders and h the vector of l variance weights. For example, we can simulate a series from the following model { X t 1 0.3X t 2 + e t if Z t 0 X t = X t e t if Z t > 0 where {Z t } AR(1). The computation details are as follow: Z<-arima.sim(n=500,list(ar=c(0.5))) l <- 2 r <- 0 K <- c(2,1) theta <- matrix(c(1,-0.5,0.5,-0.7,-0.3,na),nrow=l) X <- simu.tar.norm(z, l, r, K, theta, H) ts.plot(x) The graphics of the simulated data are:

3 A new R package for TAR modeling 3 Z Time X Time 3.2. Function for the estimation of coefficients of a TAR model param.norm. This function estimate a TAR model using Gibbs Sampler given the structural parameters, i.e. the number of regimes, thresholds and autoregressive orders. The arguments of the function are: Z: the threshold series, X the output series, l the number of regimes, r the thresholds and K the AR orders. For example, we simulate a series with 2 regimes with threshold equal to 0, the AR orders are 2 and 1; nextly, we estimate the AR coefficients and the variance weghts. The computation details are as follow: Z<-arima.sim(n=500,list(ar=c(0.5))) l <- 2; r <- 0; K <- c(2,1) theta <- matrix(c(1,-0.5,0.5,-0.7,-0.3,na), nrow=l) X <- simu.tar.norm(z,l,r,k,theta,h) res <- param.norm(z,x,l,r,k) ============================================================ 100\% The autoregressive coefficients of the 1st regime are: Estimate Std.Err Limit inferior Limit superior lag lag lag The autoregressive coefficients of the 2nd regime are: Estimate Std.Err Limit inferior Limit superior lag

4 4 Hanwen Zhang & Fabio Humberto Nieto lag The variance weights are: Estimate Std.Err Limit inferior Limit superior Regime Regime Function for the identification of AR orders of a TAR model ARorder.norm. This function identifies the autoregressive orders for a TAR model with Gaussian noise process given the number of regimes and thresholds. The arguments of the function are: Z: the threshold series, X the output series, l the number of regimes and r the thresholds. For example, we simulate a series with 2 regimes with threshold equal to 0, the AR orders are 2 and 1; nextly, we estimate the AR orders. The computation details are as follow: Z<-arima.sim(n=300,list(ar=c(0.5))) l <- 2; r <- 0; K <- c(2,1) theta <- matrix(c(1,-0.5,0.5,-0.7,-0.3,na), nrow=l) X <- simu.tar.norm(z,l,r,k,theta,h) res <- ARorder.norm(Z,X,l,r) ================================================================ 100\% The identified AR order in the 1st regime is: 2 The identified AR order in the 2nd regime is: 1 A posteriori probability of the autoregressive orders are: Regime e Regime e Function for the identification of number of regimes and thresholds of a TAR model reg.thr.norm. This function identify the number of regimes and the corresponding thresholds for a TAR model with Gaussian noise process. The arguments of the function are: Z: the threshold series and X the output series. Here we present 2 examples: 1. We simulate a series with 2 regimes with threshold equal to 0, the AR orders are 2 and 1; nextly, we estimate the number of regimes and the thresholds. The computation details are as follow: Z<-arima.sim(n=300,list(ar=c(0.5))) l <- 2 r <- 0 K <- c(2,1) theta <- matrix(c(1,-0.5,0.5,-0.7,-0.3,na), nrow=l) X <- simu.tar.norm(z,l,r,k,theta,h) res <- Reg.Thr.norm(Z,X)

5 A new R package for TAR modeling 5 [1] Creating all possible thresholds for 2 regimes... ================================================== 100\% [1] Creating all possible thresholds for 3 regimes... ================================================== 100\% [1] Running Gibbs Sampler... ================================================== 100\% The identified regime number is: 2. A posteriori probability of the number of regimes are: 2 regimes 3 regimes Posterior probability e-130 The estimated threshold(s) is(are): The 95\% credible interval of the 1st threshold is: ( , ) 2. We simulate a series with 3 regimes with thresholds equal to -0.6 and 0.6, the AR orders are 1, 2 and 1; nextly, we estimate the number of regimes and the thresholds. The computation details are as follow: Z<-arima.sim(n=300, list(ar=c(0.5))) l <- 3; r <- c(-0.6, 0.6); K <- c(1, 2, 1) theta <- matrix(c(1,0.5,-0.5,-0.5,0.2,-0.7,na, 0.5,NA), nrow=l) H <- c(1, 1.5, 2) X <- simu.tar.norm(z, l, r, K, theta, H) res <- reg.thr.norm(z, X) [1] Creating all possible thresholds for 2 regimes... ============================================== 100% [1] Creating all possible thresholds for 3 regimes... ============================================== 100% [1] Running Gibbs Sampler... ============================================== 100% The identified regime number is: 3. A posteriori probability of the number of regimes are: 2 regimes 3 regimes Posterior probability e The estimated threshold(s) is(are): The 95% credible interval of the 1st threshold is: ( , ). The 95% credible interval of the 2nd threshold is: ( , )

6 6 Hanwen Zhang & Fabio Humberto Nieto 4. Conclusion In this document, we present some basic function of a new R package for TAR modeling with Gaussian white noise process. The package also includes other distributions for the noise process. References Chen, C. W. S., Lin, E. M., Liu, F. & Gerlach, R. (2013), BAYSTAR: On Bayesian analysis of Threshold autoregressive model (BAYSTAR). R package version * Cryer, J. & Chan, K.-S. (2008), Time Series Analysis with Applications in R (second edition), Springer. Nieto, F. H. (2005), Modeling bivariate threshold autoregressive processes in the presence of missing data, Communications in Statistics. Theory and Methods. 34, Zhang, H. (2014), TAR modeling with missing data when the white noise process is not Gaussian, Tesis de doctorado, Universidad Nacional de Colombia, BogotÃą, Colombia.