# Bayesian Semiparametric Double Autoregressive Modeling

Authors: Min Wang; Ji-Chun Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper proposes a Bayesian semiparametric modeling approach for the return distribution in double autoregressive models. Monte Carlo investigation of finite sample properties and an empirical application are presented. The results indicate that the semiparametric model developed in this paper is valuable and competitive.


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## Body

## 1. Introduction

The original Bayesian theory is a parametric method. The parametric model has been long applied in classical statistical and Bayesian statistical inference studies, and its estimation is based on the unknown parameters of the overall distribution. In a review of Bayesian statistics, Lindley [1] identified nonparametric methods as a notable area in that it lacks Bayesian progress. Our purpose here is to present work with mixture priors that substantially contributes to both parametric and nonparametric density estimation. With the development of statistical theory, statisticians have found the need to reduce the overall distribution assumptions in many cases. We find that estimates of volatility using nonparametric models can differ dramatically from those using a normal return distribution if there is evidence of a heavy-tailed return distribution. When using a normal distribution or Weibull distribution in general, we often need to assume that the population distribution is unimodal (see [2]). The support set must be sufficiently large to capture the fat tail. We find that the Dirichlet process mixture (DPM) model is able to adapt to several frequently used distribution models and also accurately estimates the posterior distribution of the volatilities of the returns, without assuming any underlying distribution.Mixture models are noted for their flexibility and are widely used in the statistical literature. Nonparametric methods are usually combined Bayesian approaches with the development of more mature technologies in density estimation, regression, survival analysis, hierarchical models, and model validation [3]. The Dirichlet process (DP) is one type of nonparametric method often used in density estimation. Ferguson [4] took Dirichlet as a random probability measure and noted that if using the mixture distribution to approximate an unknown distribution, the posterior inference must be “controlled metaphysically” and the support set must be sufficiently large. Escobar and West [5] were the first to propose a modeling style involving the DP to introduce uncertainty in the number of components. The wide application of DPM priors is one of the major success stories of modern Bayesian statistics.A central property of economic time series that is common to many financial time series is that their volatility varies over time. Describing the volatility of an asset is a key issue in financial economics. The most popular class of models for time-varying volatility is represented by GARCH-type models [6]. GARCH models are commonly used to describe, estimate, and predict the dynamics of financial returns. Recent surveys of the existing GARCH model literature can be found in Davidson [7] and Rombouts et al. [8]. In contrast to Engle’s [9] ARCH model, a double autoregressive (DAR) model, which is a special case of the ARMA-ARCH models in Weiss [10] and an example of weak ARMA models in Francq and Zakoïan [11, 12], is also drawing attention from researchers. Under the assumption of the disturbance following a normal distribution, Ling [13] considered the structure and the maximum likelihood estimation. In addition, stochastic volatility (SV) models have enjoyed great popularity in analyzing financial data in the last couple of decades (see [14], etc.). Delatola and Griffin [15] presented a method for Bayesian nonparametric analysis of the return distribution in a stochastic volatility model, which allows noninformative prior distributions for the parameters and assumes that the mixing component variances are constant. In addition, they [16] extended the work and developed a Bayesian semiparametric model with a leverage effect allowing for stylized facts such as heavy tails of the distribution of returns and correlation between the returns and changes in volatility. Jensen [17] solved a semiparametric, Bayesian estimator of the long-memory stochastic volatility model using a Markov chain Monte Carlo (MCMC) algorithm. Jensen and Maheu [18] extended the existing fully parametric Bayesian literature on stochastic volatility to allow for more general return distributions.Jensen and Maheu [19, 20] proposed a Bayesian nonparametric modeling approach for the return distribution multivariate GARCH models (MGARCH). Virbickaite˙ [21] considered an asymmetric dynamic conditional correlation (ADCC) model for time-varying correlations with GJR-GARCH models for individual volatilities, denoted ADCC-GJR-GARCH, and provided a substantially more realistic evaluation of the comovements of the assets’ returns than standard symmetric MGARCH models. Ausín et al. [22] developed a Bayesian semiparametric approach based on modeling the innovations using the class of scale mixtures of Gaussian distributions with a DP prior on the mixing distribution. Jensen and Maheu [19, 20] extended the asymmetric, stochastic, and volatility models by modeling the return volatility distribution nonparametrically. The novelty of their paper was in modeling the distribution with an infinite mixture of normals, where the mixture of unknowns have a DPM prior. This motivated us to establish a DAR with a DPM model (DAR-DPM).The remainder of this paper is organized as follows. In Section2, we describe the DP mixed process and introduce the DAR model with a nonparametric DPM prior for the unknown distribution. An MCMC scheme is used to sample from the posteriors, and the parametric portions of the model are presented in Section 3. Section 4 provides some simulation studies regarding the performance of the proposed method. An empirical example is reported in Section 5, and Section 6 concludes the paper.


## 2. DAR Models with Dirichlet Process Mixture Prior

The usual structure of a DAR model [13] assumes that a return series, denoted by rt, can be written as follows:(1)rt=∑i=1pϕirt-i+ht1/2εt,ht=ω+∑i=1qβirt-i2,where ht are the conditional variances of rt given the past history, t∈-p,…,0,1,…, ω,βi>0; εt is a sequence of independent and identically distributed (IID) random variables with zero mean and unit variance; and rs is independent of εt:t≥1 for s≤0. Ling [13] gave the stationary ergodic conditions of the general model and obtained the necessary and sufficient conditions for the strict stationarity and weak stationarity of the DAR model. Let At be a random matrix of p×p:(2)ϕ1+β1ε1t⋯ϕp-1+βp-1εp-1,tϕp+βpεptIp-1Op-1×1where Ir is the unit matrix of r×r and Or×s is the zero matrix of r×s. To facilitate the presentation of results, we need to give the concept of the Lyapunov exponent here, which is defined as (3)r=inf⁡1nEln⁡A1⋯AnFrom εit~N(0,1), we know that there is Eln+⁡A1<∞, where ln+⁡x=max⁡x,0, subordinate to the Theorem of Subadditive Ergodicity [23], and (4)γ=limn→∞⁡1nA1⋯An.Ling [13] gave a theorem in which the necessary and sufficient condition for the existence of a strictly stationary solution of the DAR model is γ<0. When β=1, γ=Eln⁡ϕ1+β1εt, the condition is reduced to ϕ12+β1<1. From the study of Lu and Jiang [24], the sufficient condition for the weakest stationarity of the model is ∑i=1pϕi+Eξt∑i=1pβi<1.As noted by Ausín et al. [22], typical models for the innovation distribution include the Gaussian, Student, and Gaussian mixture models. The purpose of this paper is to construct a robust alternative to these distributional assumptions. Generally, the unit variance restriction of εt in (1) makes it difficult to undertake semiparametric Bayesian inference. To avoid this, we rewrite (1) as follows:(5)rt=∑i=1pϕirt-i+h~t1/2ξt,h~t=1+∑i=1qβ~irt-i2where h~t=ht/ω is a rescaled volatility and β~i=βi/ω, and ξt=ω1/2εt are sequences of IID random variables with mean 0 and variance ω.We relax all assumptions concerning the distribution ofξt in (5), allowing its distribution to be completely unknown and random as if the distribution were an unknown in addition to the parameters and latent volatilities of the DAR model. This strategy suggests using a Gaussian scale mixture model with a density function given by (6)fξξt∣G=∫ϕξt∣0,ΛtdGΛt,where ϕξt∣0,Λt denotes the density function of the Gaussian distribution with mean zero and variance Λt and G is a random probability measure of a DP; G0 is a measure function in probability space, and α is a strength parameter. The distribution of G is based on the unknown prior of the DP(α,G0). Employing a Lo [25] type DPM prior, assume that(7)ξt∣Λt~N0,Λt(8)Λt∣G~G(9)G∣α,G0~DPα,G0to model the unknown distribution, where G0~Inv-Gamma (v/2,c/2) and α~Gamma(a,b). In (7)-(9), ξt are normally distributed with a mean zero but a random variance Λt, distributed as G. The nonnegative value α is the strength parameter, and G0 is the base distribution. By the definition of DP(α,G0) in Sethuraman [26], G is almost necessarily equal to a discrete distribution G(dΛ)=∑j=1∞πjδσj(dΛ), where δσj(·) is a degenerative distribution on σj2 with the random variance extracted from the prior distribution G0 of DP. To ensure the conjugation, G0 is a gamma distribution with the parameters v/2 and c/2, i.e., the probability distribution of G0~Inv-Gamma (v/2,c/2). The probability of Λt being equal to a particular σj2 is πj, which satisfies(10)π1=π1′,πj=πj′∏l<j1-πl′under α>0, πj′~Beta(1,α).The DAR-DPM model above can also be introduced into the potential distribution of variablesS=(S1,S2,…,ST), while St=j, Λt equals a particular σj2. Under the prior of the DP, the distribution of St is St~∑j=1∞πjδj, where the probabilistic weight πj,j=1,2,…, defined by (10). Incorporating St into the definition of the DAR-DPM model, we propose rescaling the mode defined in (5), (7), (8), and (9) as follows: (11)rt=∑i=1pϕirt-i+ξt1+∑i=1qβirt-i2ξ∣St,Λt~N0,ΛStSt~∑j=1∞πjδjσSt2~G0.Finally, to complete the Bayesian inference of the model, we also need to deduce distributions for the parameters of the DAR-DPM-type models.


## 3. Bayesian Inference for DAR-DPM-Type Models

This section describes how to perform Bayesian inference for DAR-DPM-type models. Given an observed time seriesr=r1,r2,…,rT and the priors defined in the previous section, we consider the joint posterior distribution of the model parameters. Unfortunately, the joint posterior distribution does not have a closed analytical expression. To this end, the MCMC algorithms of the DPM model in Escobar and West [5], which produce samples from a Markov chain whose stationary distribution is the joint posterior distribution of the model parameters, are employed.LetΘ=(ϕ,β,Λt,α), where ϕ=(ϕ1,ϕ2,…,ϕp) and β=(β1,β2,…,βq) are the full model parameters and Θ∖θ represent a set of parameters that do not contain the parameter θ. We block the unknown parameters such that the conditional posterior distribution of the controllable block is either known or given by a manageable form. (12)πϕ,β,Λt,α∣r∝fr∣ϕ,β,Λt,απϕ∣β,απβ∣απΛt∣απα.Assuming that other parameters and latent variables have been obtained, we can construct a Markov chain on each piece of the posterior distributions for iterative sampling.


### 3.1. Sample fromΛt∣r,ϕ,β,α

{ Λ t } ∣ r , ϕ , β , α can be sampled using the two-step method based on the Polya method [27]. Set ξ=ξ1,ξ2,…,ξT′, where ξt=rt-∑i=1pϕirt-i/1+∑i=1qβirt-i2. The posterior distribution of Λt can be written as(13)Λt∣Λt′,t′≠t,ξt,α~αα+T+1gξtGdΛ∣ξt+1α+T-1∑t′≠tfNξt∣Λt′δΛt′dΛtwhere g(ξt)≡∫fN(ξt∣0,Λ)G0(Λ)dΛ. Under the conditional probability rule, (14)GdΛ∣ξt∝fNξt∣0,ΛG0dΛwhose density function is (15)gΛ∣ξt∝Λ-1/2exp⁡-ξt22Λ·Λ-1+v/2exp⁡-c2Λ=Λ-1+v+1/2exp⁡-ξt2+c2Λ.Assuming known ϕ,β,α, the posterior distribution is (16)πσj,S∣ξ,α≡πσj,S∣r,ϕ,β,α,where σj,j=1,2,…,k, k≤T, are the distinct Λt,t=1,2,…,T.The two steps are sampled as follows.Step 1.
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Discardingσ in the extraction of Step 1, new σj,j=1,2,…,k, are extracted using the sample S and k iteration, and the posterior distribution is as follows: (18)πσj∣ξ,S,k∝∏t:St=jfNξt∣0,σjG0dσj∝σj-1+v/2exp⁡-c2σj·∏t:St=jσj-1/2exp⁡-ξt22σj=σj-∑t:St=j1+v/2+1exp⁡-∑t:St=jξt2+c2σj~IG∑t:St=j1+v2,∑t:St=jξt2+c2.


### 3.2. Sampleϕ and β

The sampling ofπϕ∣r,β,Λt and πβ∣r,ϕ,Λt is based on the Metropolis-Hastings (MH) method [28]. The use of R language for this method is mentioned in a previous article [29], which proposed that the distribution is the conditional distribution of the parameters.Letp(ϕi),i=1,2,…,p, denote the prior distribution of ϕi assuming known S=S1,S2,…,ST′ and σ=(σ1,σ2,…,σk), ξt/σSt~N(0,1). In addition, write ht≔1+∑j=1qβjrt-j2. Then, ξt=(rt-∑i=1pϕirt-i)/ht; therefore, the posterior distribution of ϕi is(19)πϕi∣r,S,Θ∖ϕi∝∏t=1T12πσSthtexp⁡-rt-∑j=1pϕirt-j22σSthtpϕi∝exp⁡-ϕi-μϕi22σϕi2·pϕiwhere σϕi2=∑i=1T(rt-i2/σStht)-1, μϕi=∑t=1T(rt-i(rt-∑j≠iϕjrt-i2)/σStht)σϕi2. Then, our proposed distribution for ϕi is N(μϕi,σϕi2),i=1,2,…,p. By MH sampling, the accepted probability of ϕi′, α(ϕi′,ϕi)=min⁡gϕi′fNϕi∣μϕi,σϕi2/gϕifNϕi′∣μϕi,σϕi2,1, where g(ϕi)∝πϕi∣r,S,Θ∖ϕi. If the suggested value ϕi′ has been refused, then the prior ϕi remains as the current value drawn from πϕi∣r,S,Θ∖ϕi.Sampling fromπ(β∣r,ϕ,{Λt}), set p(βi) as the prior of βi,i=1,2,…,q, ϵt≔rt-∑j=1pϕirt-j=ξtht1/2, and Ft≔σ(rs,s≤t). Then, under Ft-1, (20)ut≔ϵt2-σStht=ϵt2σStht-1σStht=χ21-1σStht.The conditional mean of ut is Eut∣Ft-1=0, and the conditional variance is varut∣Ft-1=2σStht2. Nakatsuma [30] suggests replacing this ut with N(0,2σSt2ht2). Then, the posterior distribution of βi is (21)πβi∣r,S,Θ∖βi∝∏t=1T12π·2σSt2ht2exp⁡-ϵt2-1+∑j=1qβjrt-j2σSt24σSt2ht2·pβi∝exp-βi-μβi22σβi2pβigh1:Twhere σβi2=∑t=1Trt-i4/2ht2-1, μβi=σβi2∑t=1T(rt-i2(ϵt2/σSt-1-∑j≠iβjrt-j2)/2ht2). The suggested distribution of βi is N(μβi,σβi2),i=1,…,q. Extracting βi, the idea of rejecting or accepting is similar to ϕi.


### 3.3. Sampleα

Drawing from the conditional posterior distribution ofα can be performed, as explained in Escobar and West [5], by first sampling an auxiliary variable ζ from a beta distribution, ζ~Beta(α+1,T), and then sampling α from a gamma mixture, πζG(a+S∗,b-log⁡ζ)+(1-πζ)G(a+S∗-1,b-log⁡ζ), where S∗=max⁡Stt=1T and πζ/(1-πζ)=(a+S∗-1)/(T(b-log⁡ζ)).


### 3.4. MCMC Algorithm

This part briefly discusses the steps of the MCMC used to fit the DAR-DPM model. Given the priors, we define the following MCMC algorithm:Initialize ϕ , α , β , k , S , a , b , ΛSample ϕ ∣ α , β , k , S , a , b , Λ , rSample β ∣ ϕ , α , k , S , a , b , Λ , rSample α ∣ ϕ , β , k , S , a , b , Λ , rSample k , S ∣ ϕ , α , β , a , b , Λ , rSample a , b , Λ ∣ ϕ , α , β , k , S , r


## 3.1. Sample fromΛt∣r,ϕ,β,α

{ Λ t } ∣ r , ϕ , β , α can be sampled using the two-step method based on the Polya method [27]. Set ξ=ξ1,ξ2,…,ξT′, where ξt=rt-∑i=1pϕirt-i/1+∑i=1qβirt-i2. The posterior distribution of Λt can be written as(13)Λt∣Λt′,t′≠t,ξt,α~αα+T+1gξtGdΛ∣ξt+1α+T-1∑t′≠tfNξt∣Λt′δΛt′dΛtwhere g(ξt)≡∫fN(ξt∣0,Λ)G0(Λ)dΛ. Under the conditional probability rule, (14)GdΛ∣ξt∝fNξt∣0,ΛG0dΛwhose density function is (15)gΛ∣ξt∝Λ-1/2exp⁡-ξt22Λ·Λ-1+v/2exp⁡-c2Λ=Λ-1+v+1/2exp⁡-ξt2+c2Λ.Assuming known ϕ,β,α, the posterior distribution is (16)πσj,S∣ξ,α≡πσj,S∣r,ϕ,β,α,where σj,j=1,2,…,k, k≤T, are the distinct Λt,t=1,2,…,T.The two steps are sampled as follows.Step 1.
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tk(t) is the distinct number of Λt′,t′≠t presenting different σj. If St is drawn to be 0, k increases by 1, St is set for the new value k, and the new σk is extracted from (15); otherwise, St is the random sample value j.Step 2.
Discardingσ in the extraction of Step 1, new σj,j=1,2,…,k, are extracted using the sample S and k iteration, and the posterior distribution is as follows: (18)πσj∣ξ,S,k∝∏t:St=jfNξt∣0,σjG0dσj∝σj-1+v/2exp⁡-c2σj·∏t:St=jσj-1/2exp⁡-ξt22σj=σj-∑t:St=j1+v/2+1exp⁡-∑t:St=jξt2+c2σj~IG∑t:St=j1+v2,∑t:St=jξt2+c2.


## 3.2. Sampleϕ and β

The sampling ofπϕ∣r,β,Λt and πβ∣r,ϕ,Λt is based on the Metropolis-Hastings (MH) method [28]. The use of R language for this method is mentioned in a previous article [29], which proposed that the distribution is the conditional distribution of the parameters.Letp(ϕi),i=1,2,…,p, denote the prior distribution of ϕi assuming known S=S1,S2,…,ST′ and σ=(σ1,σ2,…,σk), ξt/σSt~N(0,1). In addition, write ht≔1+∑j=1qβjrt-j2. Then, ξt=(rt-∑i=1pϕirt-i)/ht; therefore, the posterior distribution of ϕi is(19)πϕi∣r,S,Θ∖ϕi∝∏t=1T12πσSthtexp⁡-rt-∑j=1pϕirt-j22σSthtpϕi∝exp⁡-ϕi-μϕi22σϕi2·pϕiwhere σϕi2=∑i=1T(rt-i2/σStht)-1, μϕi=∑t=1T(rt-i(rt-∑j≠iϕjrt-i2)/σStht)σϕi2. Then, our proposed distribution for ϕi is N(μϕi,σϕi2),i=1,2,…,p. By MH sampling, the accepted probability of ϕi′, α(ϕi′,ϕi)=min⁡gϕi′fNϕi∣μϕi,σϕi2/gϕifNϕi′∣μϕi,σϕi2,1, where g(ϕi)∝πϕi∣r,S,Θ∖ϕi. If the suggested value ϕi′ has been refused, then the prior ϕi remains as the current value drawn from πϕi∣r,S,Θ∖ϕi.Sampling fromπ(β∣r,ϕ,{Λt}), set p(βi) as the prior of βi,i=1,2,…,q, ϵt≔rt-∑j=1pϕirt-j=ξtht1/2, and Ft≔σ(rs,s≤t). Then, under Ft-1, (20)ut≔ϵt2-σStht=ϵt2σStht-1σStht=χ21-1σStht.The conditional mean of ut is Eut∣Ft-1=0, and the conditional variance is varut∣Ft-1=2σStht2. Nakatsuma [30] suggests replacing this ut with N(0,2σSt2ht2). Then, the posterior distribution of βi is (21)πβi∣r,S,Θ∖βi∝∏t=1T12π·2σSt2ht2exp⁡-ϵt2-1+∑j=1qβjrt-j2σSt24σSt2ht2·pβi∝exp-βi-μβi22σβi2pβigh1:Twhere σβi2=∑t=1Trt-i4/2ht2-1, μβi=σβi2∑t=1T(rt-i2(ϵt2/σSt-1-∑j≠iβjrt-j2)/2ht2). The suggested distribution of βi is N(μβi,σβi2),i=1,…,q. Extracting βi, the idea of rejecting or accepting is similar to ϕi.


## 3.3. Sampleα

Drawing from the conditional posterior distribution ofα can be performed, as explained in Escobar and West [5], by first sampling an auxiliary variable ζ from a beta distribution, ζ~Beta(α+1,T), and then sampling α from a gamma mixture, πζG(a+S∗,b-log⁡ζ)+(1-πζ)G(a+S∗-1,b-log⁡ζ), where S∗=max⁡Stt=1T and πζ/(1-πζ)=(a+S∗-1)/(T(b-log⁡ζ)).


## 3.4. MCMC Algorithm

This part briefly discusses the steps of the MCMC used to fit the DAR-DPM model. Given the priors, we define the following MCMC algorithm:Initialize ϕ , α , β , k , S , a , b , ΛSample ϕ ∣ α , β , k , S , a , b , Λ , rSample β ∣ ϕ , α , k , S , a , b , Λ , rSample α ∣ ϕ , β , k , S , a , b , Λ , rSample k , S ∣ ϕ , α , β , a , b , Λ , rSample a , b , Λ ∣ ϕ , α , β , k , S , r


## 4. Simulation Study

In this section, the proposed methodology is illustrated using an artificial time series of sizeT=1000. The simulated data are given by yt=0.2yt-1+ξt1+0.3yt-12, where we set ϕ=0.2 and β=0.3. The following three scenarios are considered: (1) ξt~DP(α,G0); (2) ξt~N(0,1); and (3) ξt~0.9N(0,0.6)+0.1N(0,4.6) (Ausín, 2014). The MCMC algorithm is run for a burn-in of 30000 iterations and subsequently a further 30000 iterations, storing one in 1500 iterations, thus obtaining a total of 1500 realizations from the posterior distribution.Sensitivity Analysis, Comparison of Different Prior Effects.The set of hyperparameters has been set as a=2,b=4,v=10,c=30. Note that we set the prior α~Gamma(2,4) initially because there is a fair degree of support for values near α=1 (see [5]), which leads to a rather noninformative prior.Under the seven different priors in Table1, we change only the hyperparameters a,b (Priors 2, 3, and 4), simply change the hyperparameters v,c (Prior 5 and 6), or change them all (Prior 7). Table 1 shows that all the estimated values of ϕ and β are closer to the true values of the parameters, with a smaller standard deviation. Therefore, we believe that the parameter estimation of DAR-DPM is not very sensitive to the setting of the hyperparameters. In the following analysis, a significant loss of parameter estimation accuracy is not expected when using the hyperparameters of prior 7.Table 1
Sensitivity analysis of the choice of priors.
Prior 1 Prior 2 Prior 3 Prior 4 Prior 5 Prior 6 Prior 7 a= 2, b= 4 a= 6, b= 6 a= 0.1, b= 20 a= 2, b= 4 a= 2, b= 4 a= 2, b= 4 a= 1, b= 5 v= 10, c= 30 v= 10, c= 30 v= 10, c= 30 v= 3, c= 0.003 v= 10, c= 0.01 v= 10, c= 1 v= 3, c= 6 ϕ 0.2008 0.2008 0.2033 0.1999 0.2042 0.1949 0.1919 0.0017 0.0025 0.0015 0.0022 0.0018 0.0023 0.0018 β 0.3164 0.3151 0.3092 0.3089 0.3319 0.3407 0.3146 0.0019 0.0037 0.0026 0.0019 0.0036 0.0036 0.0034 α 0.9625 0.9780 0.9289 0.9187 0.8949 0.9096 0.9492 0.0025 0.0011 0.0048 0.0033 0.0062 0.0045 0.0025 k 5.8512 5.9677 5.8917 3.5473 3.2172 3.7628 5.5979 1.3858 1.4954 2.1101 0.6853 0.4840 0.4602 1.3408 Note. This table presents the posterior mean and standard deviations of the model parameters for prior 1 setsa=2, b=4, v=10, c=30, and k=6. The remaining prior uses the same choices but changes the values of the indicated hyperparameters in the first rows.Normal and Finite Normal State Compared with Infinite Normal State. Table 2 shows that when estimating ϕ, the standard normal and finite mixture models behave well. However, both models are unable to estimate β. Clearly, the DPM method exhibits its comparative advantage in all cases and can accurately estimate the two parameters ϕ and β simultaneously, regardless of the data being from the standard Gaussian, finite mixture, or DPM mixed models.Table 2
Full sample estimates.
Model Parameter True values DPM Standard Normal Finite mixture Post.Mean Post.Mean Post.Mean Post.std.dev Post.std.dev Post.std.dev DPM ϕ 0.2 0.1909 0.2046 0.1973 0.0364 0.0441 0.0492 β 0.3 0.3095 0.3557 0.3209 0.0367 0.3765 0.0621 α 1 0.9413 0.0472 k  5.4095 1.1744 Standard Normal ϕ 0.2 0.1930 0.1957 0.2048 0.0367 0.0357 0.0344 β 0.3 0.2951 0.2989 0.3071 0.0532 0.0386 0.0704 α  0.9124 0.0491 k  1.5485 0.1827 Finite mixture ϕ 0.2 0.2088 0.1885 0.1956 0.0377 0.0694 0.0647 β 0.3 0.2957 0.6394 0.3283 0.0679 0.3066 0.1954 α  0.9219 0.0370 k  2.4693 0.0774 Note. This table displays the posterior means and standard deviations of the model parameters obtained from the DPM model, Gaussian distribution and mixture of two zero-mean Gaussians.We have drawn postdensity plots for each parameter based on simulation data for clarity; see Figure1.Figure 1
The postdensity plots of each parameter.


## 5. Empirical Application

This section applies the above method to actual financial data to fit the model. The utilized data are a series of US weekly observations of 3-month Treasury bill data: 1989/03-2019/03 (1566 observations, Data Source: Federal Reserve Economic Data). To make the data smooth, we perform some processing of the original data. Letxt denote the 90-day T-bill weekly returns, and let yt denote the rate of return of the order difference, i.e., yt=xt-xt-1; see Figure 2.Figure 2
Plots of the observations (a) and the order difference of returns (b) of the weekly observations of American 3-month treasury bills. As shown in (b), the transformed data are smooth.
(a) (b)To evaluate the performance of the different models, first, we give some descriptive statistics of the premodeling data and their autocorrelation function and partial autocorrelation function; see Table3 and Figure 3.Table 3
Descriptive statistics.
Min. 1st Qu. Median Mean 3rd Qu. Max. Variance Sd y -0.590000 -0.03000 0.000000 0.004051 0.020000 0.980000 0.007110 0.084323 Note. This table shows the summary of the order difference.Figure 3
(a) is the ACF of y; the first-order lag autocorrelation coefficient is very significant. The PACF diagram (b) shows that we cannot build a simple autoregressive model.We now illustrate the four models using the above data.First, assume thatξt~DP(α,G); we may obtain the following fitted DAR(1,1)-DPM model: yt=0.2507yt-1+ξt1+0.7043yt-12, where the standard deviation of each coefficient is 0.0361 and 0.2460 and AIC=-2.1361. Then, assume that ξt~N(0,σ2); the fitted model is yt=0.1776yt-1+ξt1+0.2162yt-12, the standard deviations of the coefficients are 0.2162 and 0.1210, and AIC=-1.7912. Again, suppose that ξt~δN(0,σ12)+(1-δ)N(0,σ22) to arrive at yt=0.2517yt-1+ξt1+0.6161yt-12, where the standard deviations of the coefficients are 0.0385 and 0.2505 and AIC=-2.0394.The predictive distribution of the series showcases the exibility of the DPM. We computed and plotted the predictive densities for the above models; see Figure4.Figure 4
The predictive densities for both the parametric and nonparametric approaches.According to the three models obtained above, we can see that the parameter values with the DPM model have a smaller variance and minimum value ofAIC. In the DPM model, we calculate that α is 0.8628 and that the mean of k is 2.7956. Obviously, under these figures, the disturbance tends to form several groups; therefore, using the DPM model to match these figures is satisfactory. To obtain the above results, we perform 20000 iterations; see Figure 5.Figure 5
MCMC results for DAR-DPM: (a) the estimation results ofϕ and (b) the estimation of β; (c) and (d) are their respective density maps. It can be observed that β is left-biased, so we take the median as the estimated value because the median is less affected by skewness.
(a) (b) (c) (d)Finally, we also compare our model with that of Jensen and Maheu [19, 20] using the same datasets. The fitted GARCH(1,1)-DPM model is (22)yt=ht1/2ξtht=0.2613+0.0619yt-12+0.0462ht-1where ξt~DP(α,G). The standard deviations of the coefficients are 0.6421, 0.1288, and 0.1073, and AIC=-2.0979. This model was also applied in Ausín, Galeano, & Ghosh [22].Based on this example, the results demonstrate that the DAR-DPM model developed in this paper is valuable and competitive. The support for u is always confined to values below 0.7, as is evident in the posterior density of u plotted in Figure6 for the equity data, where u=α/(α+1), BF(u=u0)=π(u=u0∣r,DPM)/π(u=u0)[19, 20]. This indicates that the equity data strongly support a nonparametric specification with only a few.Figure 6
Posterior density ofu≡α/(1+α).


## 6. Conclusion

In this article, a semiparametric Bayesian approach has been developed. The innovation distribution has been modeled using a scale mixture of a Gaussian model with a DP prior for the mixed distribution. An MCMC algorithm based on a combination of retrospective and slice sampling has been constructed to obtain samples from posterior distributions of the model parameters. The results that we achieved in each of our experiments in both a simulation study and a real data application are quite encouraging.



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