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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.

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 [

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 [

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 [

Jensen and Maheu [

The remainder of this paper is organized as follows. In Section

The usual structure of a DAR model [

As noted by Ausín et al. [

We relax all assumptions concerning the distribution of

The DAR-DPM model above can also be introduced into the potential distribution of variables

This section describes how to perform Bayesian inference for DAR-DPM-type models. Given an observed time series

Let

The two steps are sampled as follows.

Discarding

The sampling of

Let

Sampling from

Drawing from the conditional posterior distribution of

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:

In this section, the proposed methodology is illustrated using an artificial time series of size

Under the seven different priors in Table

Sensitivity analysis of the choice of priors.

Prior 1 | Prior 2 | Prior 3 | Prior 4 | Prior 5 | Prior 6 | Prior 7 | |
---|---|---|---|---|---|---|---|

| | | | | | | |

| | | | | | | |

| 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 | |

| 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 | |

| 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 | |

| 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 sets

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 | | 0.2 | 0.1909 | 0.2046 | 0.1973 |

0.0364 | 0.0441 | 0.0492 | |||

| 0.3 | 0.3095 | 0.3557 | 0.3209 | |

0.0367 | 0.3765 | 0.0621 | |||

| 1 | 0.9413 | |||

0.0472 | |||||

| 5.4095 | ||||

1.1744 | |||||

Standard Normal | | 0.2 | 0.1930 | 0.1957 | 0.2048 |

0.0367 | 0.0357 | 0.0344 | |||

| 0.3 | 0.2951 | 0.2989 | 0.3071 | |

0.0532 | 0.0386 | 0.0704 | |||

| 0.9124 | ||||

0.0491 | |||||

| 1.5485 | ||||

0.1827 | |||||

Finite mixture | | 0.2 | 0.2088 | 0.1885 | 0.1956 |

0.0377 | 0.0694 | 0.0647 | |||

| 0.3 | 0.2957 | 0.6394 | 0.3283 | |

0.0679 | 0.3066 | 0.1954 | |||

| 0.9219 | ||||

0.0370 | |||||

| 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 Figure

The postdensity plots of each parameter.

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. Let

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.

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 Table

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.

(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 that

The predictive distribution of the series showcases the exibility of the DPM. We computed and plotted the predictive densities for the above models; see Figure

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 of

MCMC results for DAR-DPM: (a) the estimation results of

Finally, we also compare our model with that of Jensen and Maheu [

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 Figure

Posterior density of

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.

The text data used to support the findings of this study are included within the supplementary information file. Data Sources: Federal Reserve Economic Data can be obtained by visiting the following websites:

The authors declare that they have no conflicts of interest.

This research was partially supported by NSFC (11071202&11861025) & Science and Technology Foundation of Guizhou Province (LKS