- Research
- Open Access
Quasi-maximum exponential likelihood estimator and portmanteau test of double \(\operatorname{AR}(p)\) model based on \(\operatorname{Laplace}(a,b)\)
- Haiyan Xuan^{1, 2}Email author,
- Lixin Song^{1}Email author,
- Un Cig Ji^{3},
- Yan Sun^{4} and
- Tianjiao Dai^{2}
https://doi.org/10.1186/s13660-018-1769-9
© The Author(s) 2018
- Received: 10 October 2017
- Accepted: 4 June 2018
- Published: 11 September 2018
Abstract
The paper studies the estimation and the portmanteau test for double \(\operatorname{AR}(p)\) model with \(\operatorname{Laplace}(a,b)\) distribution. The double \(\operatorname{AR}(p)\) model is investigated to propose firstly the quasi-maximum exponential likelihood estimator, design a portmanteau test of double \(\operatorname{AR}(p)\) on the basis of autocorrelation function, and then establish some asymptotic results. Finally, an empirical study shows that the estimation and the portmanteau test obtained in this paper are very feasible and more effective.
Keywords
- Double \(\operatorname{AR}(p)\) model
- Quasi-maximum exponential likelihood estimator
- Portmanteau test
- Autocorrelations
MSC
- 62M10
- 91G70
1 Introduction
In 1982, Engle proposed the ARCH model, and used it to analyze the volatility clustering of the inflation index. Weiss [1] considered a model called double \(\operatorname{AR}(p)\) as an extension of the ARCH model. After parameter estimation and test he showed that there are various available test means to be used to test this model. The double \(\operatorname{AR}(p)\) model is an \(\operatorname{AR}(p)\) model with conditional heteroscedasticity. Then Francq and Zakoian gave an example of weak ARMA model in 1998 and 2000, respctively. Ling [2] applied quasi-maximum likelihood estimation (QMLE) to give a parameter estimation, then used two ways to test the stationary of double \(\operatorname{AR}(1)\) model under weak conditions, finally presented an empirical study. Chan and Peng [3] presented a locally weighted least absolute deviation estimation for the double \(\operatorname{AR}(1)\) model and established its asymptotic theory. Wang et al. [4] studied the heteroscedastic mixture double AR model to simulate the nonlinear time series, and proposed some stability conditions of the model. Ling and Li [5] performed the diagnostic tests for non-stationary double \(\operatorname{AR}(1)\) model. Zhu and Ling [6] proposed the quasi-maximum exponential likelihood estimator for the double \(\operatorname{AR}(p)\) model, and made a comparison with the weighted least squares under the finite sample condition.
The research of a diagnostic test, often accompanied by the development of a model, plays an important role in the research of the model. McLeod and Li [7] presented a diagnostic test using squared-residual autocorrelation function. Dufour and Roy [8] gave a nonparametric portmanteau test. Monti [9] proposed a portmanteau test based on residual partial autocorrelation function. Wong and Li [10] made the portmanteau test for multivariate conditional heteroscedasticity model. Francq et al. [11] proposed a diagnosis test method for weak ARMA model. Kwan et al. [12] studied the portmanteau test under the condition of finite sample. Francq [13] aiming at autoregressive models with uncorrelated but non-independent errors made the multivariate portmanteau test. And then Mainassara [14] also made a multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms. Kwan et al. [15] defined two portmanteau tests based on residual autocorrelation function and square residual autocorrelation function, respectively. Fisher and Gallagher [16] proposed a new weighted portmanteau statistic for the goodness of fit for time series. Zhu and Ling [17] presented a Ljung–Box portmanteau test based on symbolic function in order to test the properties of ARMA model with fat-tailed noise. Zhu [18] used the random weighting method to make a bootstrap portmanteau test on the basis of residual autocorrelation function and residual partial autocorrelation function of weak ARMA model. Recently, Xuan [19] made a portmanteau test aiming at ARFIMA–GARCH model. Stefanos [20] studied time-varying parameter regression models with stochastic volatility and made a semiparametric Bayesian inference.
The structure of this paper is as follows. The second part focuses on the parameter estimation method and the portmanteau test statistic of double \(\operatorname{AR}(p)\) model derived from this method. The quasi-maximum exponential likelihood estimator and portmanteau test statistic based on residual autocorrelation function will be given in this section. In the third part, there is an empirical study of CSI 800 which applies the portmanteau test to check the double \(\operatorname{AR}(p)\) model. Conclusions are given in the final section.
2 Quasi-maximum exponential likelihood estimator and portmanteau test
In this section, the double \(\operatorname{AR}(p)\) model with \(\operatorname{Laplace}(a,b)\) distribution will be investigated to propose the quasi-maximum exponential likelihood estimator, establish some asymptotic results, and design the portmanteau test based on autocorrelation function.
2.1 Quasi-maximum exponential likelihood estimator based on \(\operatorname{Laplace}(a,b)\)
Let \(\theta = (r',\delta ')'\) be the unknown parameters of the model, and the true value is \(\theta_{0} = (r'_{0},\delta '_{0})\), \(r = (\phi _{1},\phi_{2}, \ldots,\phi_{p})'\), \(\delta = (\omega,\alpha_{1},\alpha _{2}, \ldots,\alpha_{p})'\). Define the parameter space \(\Theta = \Theta_{r} \times \Theta_{\delta } \), and \(\Theta_{r} \in \mathbb{R} ^{p}\), \(\Theta_{\delta } \in \mathbb{R}_{0}^{p + 1}\), \(\mathbb{R} = ( - \infty, + \infty)\), \(\mathbb{R}_{0} = [0,\infty)\).
Let X obey \(\operatorname{Laplace}(a,b)\) distribution, where a is a positional parameter and b is a scale parameter. After the transformation \(Y = \frac{X - a}{b}\) is drawn, Y follows the \(\operatorname{Laplace}(0,1)\) distribution. Therefore, we only discuss the situation of \(\operatorname{Laplace}(0,1)\) distribution as follows. In order to carry out the calculation successfully, we also need the following three assumptions.
Assumption 1
Θ is a compact set, \(\theta_{0}\) is the inner point of Θ. \(\underline{\omega} \le \omega \le \bar{ \omega } \) and \(\underline{\alpha }_{i} \le \alpha_{i} \le \bar{ \alpha }_{i}\) (\(i = 1, \ldots,p\)), \(\underline{\omega }\), ω̄, \(\underline{ \alpha }_{i}\), \(\bar{\alpha }_{i}\) (\(i = 1, \ldots,p\)) are positive constants.
Assumption 2
For \(l > 0\) and \(E \vert y_{t} \vert ^{l} < \infty,\{ y_{t}:t = 1 - p, \ldots,0,1,2, \ldots \}\) is a strictly stationary and ergodic sequence.
Assumption 3
In the situation of \(E\eta_{t}^{2} < \infty \), the median of \(\eta_{t}\) is zero, and it has a bounded continuous density function \(f(x)\) in \(\mathbb{R}\) which satisfies the range of density function \((0, + \infty )\).
Based on the conditions of the three assumptions, in what follows we will derive asymptotic distribution of the estimators.
Then \(\hat{\theta }_{n}\) is called the quasi-maximum exponential likelihood estimator of \(\theta_{0}\). It follows from Assumptions 1–3 that \(\hat{\theta }_{n}\) is obtained immediately, and the asymptotic properties of the estimators are derived as follows.
Theorem 1
It is easy to obtain the proof of Theorem 1 by using compact set theory, Markov theorem, and ergodicity theorem.
2.2 Portmanteau test based on autocorrelation function
Theorem 2
Proof
The decision rule is to reject \(H_{0}\) if \(\bar{Q}_{m} > \chi_{\alpha }^{2}\), where \(\chi_{\alpha }^{2}\) denotes the \(100 ( 1 - \alpha ) \) percentile of a chi-squared distribution with m degrees of freedom.
According to Theorem 2, we can directly get the exact asymptotic distribution of the portmanteau statistics.
Theorem 3
It follows from Theorem 3 that \(\bar{Q}_{m}\) is always a portmanteau test statistic of residual autocorrelation function under the condition of quasi-maximum exponential likelihood estimator. The conclusion is derived that the double \(\operatorname{AR}(p)\) model with the quasi-maximum exponential likelihood estimator can be used to test the diagnostic results of the portmanteau test statistic.
3 An empirical study
To study the law of financial market development, researchers generally select some indices to investigate the features of comprehensive economics and reflect the overall rather than one-sided trend of economic development in order to ensure the conclusions proposed appropriately for most phenomena. The CSI 300 index is one of indexes with these characteristics which can reflect the situation of Chinese stock market.
This article selects recent closing price data of CSI 300 index (399300) from December 1, 2016 to March 16, 2018, 315 sample observations in total. We used statistical software MATLAB to conduct research and analysis. The data can be downloaded from Netease Finance.
Main digital feature of CSI 300 index
Mean | Std. | Skewness | Kurtosis | Minimum | Maximum | JB statistic |
---|---|---|---|---|---|---|
0.0004 | 0.0077 | −1.1129 | 7.5885 | −0.0437 | 0.0214 | 340.2778 |
Stationary test table
Test method | 1% critical value | 5% critical value | 10% critical value | t-statistic | Probability |
---|---|---|---|---|---|
ADF test | −3.451078 | −2.870561 | −2.571647 | −16.06732 | 0 |
PP test | −3.451078 | −2.870561 | −2.571647 | −16.07588 | 0 |
The autocorrelation function value and the partial autocorrelation function value of the sequence
Lag | ACF | PACF | Q-Statistic | P |
---|---|---|---|---|
1 | −0.463 | −0.463 | 67.786 | 0.0000 |
2 | −0.068 | −0.359 | 69.232 | 0.0000 |
3 | 0.172 | −0.054 | 78.664 | 0.0000 |
4 | −0.18 | −0.175 | 88.993 | 0.0000 |
5 | −0.036 | −0.253 | 89.407 | 0.0000 |
6 | 0.11 | −0.158 | 93.264 | 0.0000 |
7 | −0.087 | −0.183 | 95.725 | 0.0000 |
8 | 0.039 | −0.146 | 96.224 | 0.0000 |
9 | 0.032 | −0.138 | 96.555 | 0.0000 |
10 | 0.028 | −0.024 | 96.803 | 0.0000 |
11 | 0.036 | 0.092 | 97.237 | 0.0000 |
12 | −0.163 | −0.126 | 105.99 | 0.0000 |
13 | 0.104 | −0.057 | 109.51 | 0.0000 |
14 | −0.004 | −0.014 | 109.52 | 0.0000 |
In order to verify the usefulness of the double \(\operatorname{AR}(p)\) model under autocorrelation function, it is necessary to estimate the parameters for the given model under the actual data. The effect of model fitting has a direct impact on the accuracy of data prediction. Thus, model parameter estimation must be carried out firstly.
Regarding the approaches of parameter estimation, scholars put forward many methods of estimation for model parameters, such as moment estimation, least squares estimation, maximum likelihood estimation, and so on. In application, however, the quasi-maximum exponential likelihood estimator is used widely because of its excellent properties. Thus, we use quasi-maximum exponential likelihood estimator to make parameter estimation in this part.
On the basis of quasi-maximum exponential likelihood estimation, this paper uses the nonlinear multivariate function to determine the initial value. Finally, we can obtain model parameter estimation: \(\alpha_{0} = 0.5093\), \(\beta_{1} = 0.2501\). For \(\beta_{1}\) reflects the correlation between the observed data. This data indicates that the CSI 300 index has a weak sequence correlation in this period. The results show that the volatility of the CSI 300 index has a short duration.
Simulation results of portmanteau test statistic
Statistic | Lag n = 7 | Lag n = 14 |
---|---|---|
\(\bar{Q}_{m}\) | 5.376 | 22.896 |
From Table 4, it is obvious that no matter the lag is 7 or 14, the portmanteau test statistic \(\bar{Q}_{m}\) is always less than the \(\chi^{2}\) statistic with different degrees of freedom in the same lag. Therefore, we can judge that the \(\operatorname{AR}(1)\) model is tested by portmanteau test based on the quasi-maximum exponential likelihood, and thus, the model fitting is reasonable.
4 Conclusions
- (i)
The CSI 300 index return sequence has weak correlation in the selected time period, with a short duration and no long memory.
- (ii)
On the basis of quasi-maximum exponential likelihood estimation method, the double \(\operatorname{AR}(p)\) model is fitted. Then a diagnostic test for this model is conducted by using portmanteau test statistic based on residual partial autocorrelation function. It is concluded that the double \(\operatorname{AR}(p)\) model is reasonable in practical application.
Declarations
Funding
This work is supported by the National Natural Science Foundation of China (No. 11371077).
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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