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# 数据科学代写|金融统计代写Financial Statistics代考|MS&E349 Engle (2002) Dynamic Conditional Correlation Model

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## 数据科学代写|金融统计代写Financial Statistics代考|Engle (2002) Dynamic Conditional Correlation Model

Let $r_{t}=\left(r_{1 t}, \ldots, r_{n t}\right)$ represent a $(n \times 1)$ vector of financial assets returns at time $(\mathrm{t})$. Moreover, let $\varepsilon_{t}$ $=\left(\varepsilon_{1 t}, \ldots, \varepsilon_{n t}\right)$ be a $(n \times 1)$ vector of error terms obtained from an estimated system of mean equations for these return series.

Engle (2002) proposes the following decomposition for the conditional variance-covariance matrix of asset returns:
$$H_{t}=D_{t} R_{t} D_{t}$$
where $D_{t}$ is a $(n \times n)$ diagonal matrix of time-varying standard deviations from univariate Garch models, and $R_{t}$ is a $(n \times n)$ time-varying correlation matrix of asset returns $\left(\rho_{i j,} t\right)$.

The conditional variance-covariance matrix $\left(H_{t}\right)$ displayed in equation [1] is estimated in two steps. In the first step, univariate Garch $(1,1)$ models are applied to mean returns equations, thus obtaining conditional variance estimates for each financial asset $\left(\sigma^{2}\right.$ it; for $\left.i=1,2, \ldots ., n\right)$, namely:
$$\sigma_{i t}^{2}=\sigma^{2} U_{i t}\left(1-\lambda_{1 i}-\lambda_{2 i}\right)+\lambda_{1 i} \sigma_{i, t-1}^{2}+\lambda_{2 i} \varepsilon_{i, t-1}^{2}$$
where $\sigma^{2}$ uit is the unconditional variance of the ith asset return, $\lambda_{1 i}$ is the volatility persistence parameter, and $\lambda_{2 i}$ is the parameter capturing the influence of past errors on the conditional variance.
In the second step, the residuals vector obtained from the mean equations system $\left(\varepsilon_{t}\right)$ is divided by the corresponding estimated standard deviations, thus obtaining standardized residuals (i.e., $u_{i t}=$ $\varepsilon_{i t} / \sqrt{\sigma_{i, t}^{2}}$ for $\mathrm{i}=1,2, \ldots ., n$ ), which are subsequently used to estimate the parameters governing the time-varying correlation matrix.

More specifically, the dynamic conditional correlation matrix of asset returns may be expressed as:
$$Q_{t}=\left(1-\delta_{1}-\delta_{2}\right) \bar{Q}+\delta_{1} Q_{t-1}+\delta_{2}\left(u_{t-1} u_{t-1}^{\prime}\right)$$
where $\overline{\mathrm{Q}}=\mathrm{E}\left[u_{t} u_{t}^{\prime}\right]$ is the $(n \times n)$ unconditional covariance matrix of standardized residuals, and $\delta_{1}$ and $\delta_{2}$ are parameters (capturing, respectively, the persistence in correlation dynamics and the impact of past shocks on current conditional correlations). ${ }^{2}$

## 数据科学代写|金融统计代写Financial Statistics代考|Model Estimation and Dynamic Conditional Correlation Patterns

The econometric framework summarized by Equations (1)-(3) was applied to gold, oil, and exchange rate returns.

After a preliminary data inspection, and in line with many contributions relying on this approach (see e.g., Ding and Vo (2012) and Jain and Biswal (2016) with regards to “safe haven” assets), a VAR(1) specification was selected to model the mean returns equation system. Alternative filtering procedures (such as an $\mathrm{AR}(1)$ specification for return series) produced substantially identical results.

The VAR(1) specification was selected on the basis of the Akaike Information Criterion (AIC) and of Likelihood Ratio tests against higher-order VAR models. Diagnostic tests on residuals from the VAR(1) specification never rejected the null of absence of serial correlation, while rejecting the normality assumption. This rejection was consistent with the preliminary data analysis, where the Jarque and Bera (1980) statistics turned out to be strongly significant.

These departures from normality have relevant implications on the distributional assumptions underlying the Multivariate Garch DCC model. More specifically, instead of relying on the standard Gaussian assumption (as in Engle (2002) seminal model), it is advisable to assume a Multivariate $\mathrm{t}$-distribution in order to better capture the fat-tailed nature of asset returns. This is the approach taken in the present paper. ${ }^{3}$

With regards to parameters, conditional volatility coefficients were unrestricted, and assumed different for each asset (see Equation (2)). Conditional correlation coefficients were unrestricted as well, although a common correlation structure was imposed in model’s estimation (see Equation (3))).
The Maximum Likelihood estimator converged after 48 iterations and relied on 216 observations (20 observations were used to initialize the recursions).
Table 3 contains the results.

## 数据科学代写|金融统计代写FINANCIAL STATISTICS代 考|ENGLE (2002) DYNAMIC CONDITIONAL CORRELATION MODEL

$$H_{t}=D_{t} R_{t} D_{t}$$

$$\sigma_{i t}^{2}=\sigma^{2} U_{i t}\left(1-\lambda_{1 i}-\lambda_{2 i}\right)+\lambda_{1 i} \sigma_{i, t-1}^{2}+\lambda_{2 i} \varepsilon_{i, t-1}^{2}$$

Q_{t}=\left(1-\delta_{1}-\delta_{2}\right) \bar{Q}+\delta_{1} Q_{t-1}+\delta_{2}\left(u_{t-1} u_{t-1}^{\prime}\right)


capturing, respectively, thepersistenceincorrelationdynamicsandtheimpactofpastshocksoncurrentconditionalcorrelations ${ }^{2}$. $^{2}$.

## 数据科学代写|金融统计代写FINANCIAL STATISTICS代 考|MODEL ESTIMATION AND DYNAMIC CONDITIONAL CORRELATION PATTERNS

Equations 总结的计量经济学框妿1-3适用于昔金、石油和汇率回报。 程系统。萺代过滤程序suchasanSAR(1\$返回系列的规格) 产生了基本相同的结果。 VAR1规格是根据 Akaike 信息标准选择的$A I C$以及针对高阶 VAR 模型的似然比检验。VAR残差的诊断测试1规范从不拒绝不存在序列相关的零点，同时拒绝正态性假 设。这一拒绝与初步数据分析一致，其中 Jarque 和 Bera1980统计结果证明是非常重要的。 这些偏离正态性对多元 Garch DCC 模型的分布假设具有相关影响。更具体地说，而不是依赖于标准的高斯假设$a s i n E n g l e(2002\$ 开创性模型)，建议假设一个多变 量t-分配以更好地捕捉资产回报的肥尾性质。这是本文所采用的方法。

seeEquation(3) )。

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