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# 统计代写|时间序列分析代写Time-Series Analysis代考|The two-step estimation method

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## 统计代写|时间序列分析代写Time-Series Analysis代考|The two-step estimation method

When the GO-GARCH model was first proposed by van der Weide (2002), he used the singular value decomposition of the linkage matrix as a parameterization, that is, $\boldsymbol{\Omega}=\mathbf{U} \mathbf{\Lambda}^{1 / 2} \mathbf{V}^{\prime}$, where $\mathbf{U}$ is the orthogonal matrix containing the orthogonal eigenvectors of $\Omega^{\prime}, \boldsymbol{\Lambda}=\operatorname{diag}\left(\lambda_1, \ldots, \lambda_m\right)$ contains the corresponding eigenvalues $\left(\lambda_i>0\right.$, for all $\left.i\right)$ and $\mathbf{V}$ is the orthogonal matrix of eigenvectors of $\boldsymbol{\Omega}^{\prime} \boldsymbol{\Omega}$. He proposed a two-step estimation method. In the first step, $\mathbf{U}$ and $\boldsymbol{\Lambda}$ are consistently estimated through PCA of the unconditional sample covariance:
$$\hat{\mathbf{\Sigma}}=T^{-1 / 2} \Sigma_{t=1}^T \varepsilon_t \varepsilon_t^{\prime}=\hat{\mathbf{U}} \hat{\mathbf{\Lambda}} \hat{\mathbf{U}}^{\prime}$$
where $\hat{\mathbf{U}}$ contains the orthogonal eigenvectors of $\hat{\mathbf{\Sigma}}, \hat{\boldsymbol{\Lambda}}=\left(\hat{\lambda}_1, \ldots, \hat{\lambda}_m\right)$ contains the corresponding eigenvalues, and $T$ is the length of the series. In the second step, $\mathbf{V}$ and the univariate GARCH parameters are estimated by maximizing the following log likelihood:
\begin{aligned} L(\boldsymbol{\theta}) & =-\frac{1}{2} \sum_{t=1}^T\left{m \log (2 \pi)+\log \left|\hat{\boldsymbol{\Sigma}}t\right|+\boldsymbol{\varepsilon}_t^{\prime} \hat{\boldsymbol{\Sigma}}_t^{-1} \boldsymbol{\varepsilon}_t\right} \ & =-\frac{1}{2} \sum{t=1}^T\left{m \log (2 \pi)+\log \left|\hat{\boldsymbol{\Omega}} \boldsymbol{\Gamma}{\mathbf{t}} \hat{\mathbf{\Omega}}^{\prime}\right|+\boldsymbol{\varepsilon}_t^{\prime}\left(\hat{\boldsymbol{\Omega}} \boldsymbol{\Gamma}{\mathbf{t}} \hat{\boldsymbol{\Omega}}^{\prime}\right)^{-1} \boldsymbol{\varepsilon}t\right} \ & =-\frac{1}{2} \sum{t=1}^T\left{m \log (2 \pi)+\log \left|\boldsymbol{\Gamma}_t\right|+\log \left|\hat{\mathbf{U}} \hat{\boldsymbol{\Lambda}} \hat{\mathbf{U}}^{\prime}\right|+\boldsymbol{\varepsilon}_t^{\prime} \hat{\mathbf{U}} \hat{\boldsymbol{\Lambda}}^{-1 / 2} \mathbf{V}^{\prime} \boldsymbol{\Gamma}_t^{-1} \mathbf{V} \hat{\boldsymbol{\Lambda}}^{-1 / 2} \hat{\mathbf{U}}^{\prime} \boldsymbol{\varepsilon}_t\right}, \end{aligned}
where $\log \left|\hat{\boldsymbol{\Omega}} \boldsymbol{\Gamma}_t \hat{\boldsymbol{\Omega}}^{\prime}\right|=\log \left|\boldsymbol{\Gamma}_t\right|+\log \left|\hat{\mathbf{\Omega}} \hat{\mathbf{\Omega}}^{\prime}\right|, \hat{\mathbf{\Omega}} \hat{\mathbf{\Omega}}^{\prime}=\hat{\mathbf{U}} \hat{\boldsymbol{\Lambda}} \hat{\mathbf{U}}^{\prime}$, and $\boldsymbol{\theta}=\left(\boldsymbol{\theta}_1^{\prime}, \boldsymbol{\theta}_2^{\prime}\right)$ with $\boldsymbol{\theta}_1$ being a vector of dimension $m(m-1) / 2$ characterizing the $m \times m$ orthogonal matrix $\mathbf{V}, \boldsymbol{\theta}_2$ being the GARCH parameter vector of dimension $2 m$ for $\boldsymbol{\Gamma}_t$. We note here that by using unconditional information first, van der Weide (2002) showed that the number of parameters to be estimated for $\mathbf{V}$ is $m(m-1) / 2$ instead of $m^2$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|The weighted scatter estimation method

It is well known that sample covariance is not the most efficient method to estimate the population covariance. The estimation of the linkage matrix using PCA based on sample covariance is prone to outliers, and outliers are quite common in economic and business data. In this section, we propose a new weighted scatter estimation method (WSE) to estimate the linkage matrix, which inherits many nice properties of robust estimation.

For this method, after singular value decomposition of the linkage matrix, in the first step, $\mathbf{U}$ and $\boldsymbol{\Lambda}$ are still consistently estimated through PCA of the unconditional sample covariance in Eq. (6.35). However, in the second step, $\mathbf{V}$ is estimated based on weighted multivariate scatter estimators of $\mathbf{s}_{\mathbf{t}}=\mathbf{V}^{\prime} \mathbf{r}_t$, and the univariate GARCH parameters in Eq. (6.32) are estimated separately in the third step. The proposed estimation of the linkage matrix does not require the complicated distribution form and any optimization of an objective function; thus, it is free of computational and convergence problems, even when the dimension is high. This property makes the new method numerically attractive and easy to apply.

Under the GO-GARCH specification Eqs. (6.30)-(6.34), we see that $\operatorname{Var}\left(\mathbf{s}t\right)=\operatorname{Var}\left(\mathbf{r}_t\right)=\mathbf{I}_m . \mathbf{V}$ is unidentifiable through PCA on unconditional variance, as for any orthogonal matrix $\mathbf{Q}$, Var $\left(\mathbf{Q} \mathbf{s}_t\right)=\operatorname{Var}\left(\mathbf{Q V}^{\prime} \mathbf{r}_t\right)=\mathbf{I}_m$. The key idea of the proposed estimation method is that $\mathbf{V}$ can be identified through PCA on weighted multivariate scatter estimators, denoted as $\hat{\mathbf{H}}_w$, that assign weights to each observation based on predefined measures with respect to the distribution hyper-contour and higher moments. There are many ways to define weighted multivariate scatter estimation. For example, we can apply the weighting scheme of M-estimation, which is used in the robust statistics criterion. The concept can be easily extended to other weighting schemes. Define $\hat{\mathbf{H}}_w$ as the solution of the equation: $$\frac{1}{T} \sum{t=1}^T w\left(g_t\right) \mathbf{s}t \mathbf{s}{t-1}^{\prime}=\mathbf{H},$$
where $g_t=\mathbf{s}t^{\prime} \mathbf{H}^{-1} \mathbf{s}_t \geq 0$, the squared Mahalanobis distance, $\mathbf{s}_t=\boldsymbol{\Lambda}^{-1 / 2} \mathbf{U}^{\prime} \boldsymbol{\varepsilon}_t=\mathbf{V r { t }}$, and $w(g), g \geq 0$, is a weighting function with conditions given in the following Theorem 6.1. We define analogously the functional form of multivariate scatter at a distribution $F_{\mathrm{s}}$ in $R^m$, denoted as $\mathbf{H}w\left(F{\mathrm{s}}\right)$, to be the solution of the equation:
$$E\left[w(g) \mathbf{s s}^{\prime}\right]=\mathbf{H},$$
where s, without the time subscript, denotes a $m \times 1$ random vector following distribution $F_{\mathbf{s}}$, $g=\mathbf{s}^{\prime} \mathbf{H}^{-1} \mathbf{s} \geq 0$, and $E$ is the expectation operator over $\mathbf{s}$. If $\mathbf{s}1, \ldots, \mathbf{s}_T$ is a sample series of length $T$, and $F{\mathrm{s}}$ is the corresponding empirical distribution with $P\left(\mathbf{s}t\right)=T^{-1}, t=1, \ldots, T$, Eq. (6.40) becomes Eq. (6.39), and $\mathbf{H}_w\left(F{\mathrm{s}}\right)$ becomes $\hat{\mathbf{H}}_w$. We have the following results.

## 统计代写|时间序列分析代写Time-Series Analysis代考|The two-step estimation method

$$\hat{\mathbf{\Sigma}}=T^{-1 / 2} \Sigma_{t=1}^T \varepsilon_t \varepsilon_t^{\prime}=\hat{\mathbf{U}} \hat{\mathbf{\Lambda}} \hat{\mathbf{U}}^{\prime}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The weighted scatter estimation method

$$E\left[w(g) \mathbf{s s}^{\prime}\right]=\mathbf{H},$$

## MATLAB代写

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