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# 统计代写|时间序列分析代写Time-Series Analysis代考|Regularization methods

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Regularization methods

Let $\mathbf{Z}t=\left[Z{1, t}, Z_{2, t}, \ldots, Z_{m, t}\right]^{\prime}, t=1,2, \ldots, n$, be a zero-mean $m$-dimensional time series with $n$ observations. It is well known that the least squares method can be used to fit the VAR $(p)$ model by minimizing
$$\sum_{t=1}^n\left|\mathbf{Z}t-\sum{k=1}^p \mathbf{\Phi}k \mathbf{Z}{t-k}\right|_2,$$
where ||$_2$ is Euclidean $\left(L^2\right)$ norm of a vector. In practice, more compactly, with data $\mathbf{Z}t=\left[Z{1, t}\right.$, $\left.Z_{2, t}, \ldots, Z_{m, t}\right]^{\prime}, t=1,2, \ldots, n$, we can present the VAR $(p)$ model in Eq. (10.2) in the matrix form,
$$\underset{n \times m}{\mathbf{Y}}=\underset{(n \times m p)}{\mathbf{X}} \underset{(m p \times m)}{\mathbf{D}}+\underset{(n \times m)}{\boldsymbol{\xi}},$$
where
$$\mathbf{Y}=\left[\begin{array}{c} \mathbf{Z}1^{\prime} \ \mathbf{Z}_2^{\prime} \ \vdots \ \mathbf{Z}_n^{\prime} \end{array}\right], \mathbf{X}=\left[\begin{array}{c} \mathbf{X}_1^{\prime} \ \mathbf{X}_2^{\prime} \ \vdots \ \mathbf{X}_n^{\prime} \end{array}\right], \boldsymbol{\Phi}=\left[\begin{array}{c} \boldsymbol{\Phi}_1^{\prime} \ \boldsymbol{\Phi}_2^{\prime} \ \vdots \ \boldsymbol{\Phi}_p^{\prime} \end{array}\right], \boldsymbol{\xi}=\left[\begin{array}{c} \mathbf{a}_1^{\prime} \ \mathbf{a}_2^{\prime} \ \vdots \ \mathbf{a}_n^{\prime} \end{array}\right],$$ and $$\mathbf{X}_t^{\prime}=\left[\mathbf{Z}{t-1}^{\prime}, \mathbf{Z}{t-2}^{\prime}, \ldots, \mathbf{Z}{t-p}^{\prime}\right]$$
So, minimizing Eq. (10.3) is equivalent to
$$\underset{\boldsymbol{\Phi}}{\operatorname{argmin}}|\mathbf{Y}-\mathbf{X \Phi}|_F$$
where ||$_F$ is the Frobenius norm of the matrix.
For a VAR model in high-dimensional setting, many regularization methods have been developed, which assume sparse structures on coefficient matrices $\boldsymbol{\Phi}_k$ and use regularization procedure to estimate parameters. These methods include the Lasso (Least Absolute Shrinkage and Selection Operator) method, the lag-weighted lasso method, and the hierarchical vector autoregression method, among others.

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

One of the most commonly used regularization methods is the lasso method proposed by Tibshirani (1996) and extended to the vector time series setting by Hsu et al. (2008). Formally, the estimation procedure for the VAR model is through
$$\underset{\boldsymbol{\Phi}}{\operatorname{argmin}}\left{|\mathbf{Y}-\mathbf{X} \boldsymbol{\Phi}|_F+\lambda|\operatorname{vec}(\boldsymbol{\Phi})|_1\right},$$
where the second term is the regularization through $L_1$ penalty with $\lambda$ being its control parameter. $\lambda$ can be determined by cross-validation. The lasso method does not impose any special assumption on the relationship of lag orders and tends to over select the lag order $p$ of the VAR model. This leads us to the development of some modified methods.
The lag-weighted lasso method
Song and Bickel (2011) proposed a method that incorporates the lag-weighted lasso (lasso and group lasso structures) approach for the high-dimensional VAR model. They placed group lasso penalties introduced by Yuan and Lin (2006) on the off-diagonal terms and lasso penalties on the diagonal terms. More specifically, if we denote $\boldsymbol{\Phi}(j,-j)$ as the vector composed of offdiagonal elements $\left{\phi_{j, i}\right}_{i \neq j}$, and $\boldsymbol{\Phi}k(j, j)$ as the $j$ thdiagonal element of $\boldsymbol{\Phi}_k$, then the regularization for $\boldsymbol{\Phi}_k$ is $$\sum{j=1}^m\left|\boldsymbol{\Phi}k(j,-j) \mathbf{W}(-j)\right|_2+\lambda \sum{j=1}^m w_j\left|\boldsymbol{\Phi}k(j, j)\right|,$$ where $\mathbf{W}(-j)=\operatorname{diag}\left(w_1, \ldots, w{j-1}, w_{j+1}, \ldots, w_m\right)$, an $(m-1) \times(m-1)$ diagonal matrix with $w_j$ being the positive real-valued weight associated with the $j$ th variable for $1 \leq j \leq m$, which is chosen to be the standard deviation of $Z_{j, t} . \lambda$ is the control parameter that controls the extent to which other lags are less informative than its own lags. The first term of Eq. (10.7) is the group lasso penalty, the second term is the lasso penalty, and they impose regularization on other lags and its own lags, respectively. Let $0<\alpha<1$ and $(k)^\alpha$ be the other control parameter for different regularization for different lags; the estimation procedure is based on
$$\underset{\boldsymbol{\Phi}1, \ldots, \boldsymbol{\Phi}_p}{\arg \min _k}\left{|\mathbf{Y}-\mathbf{X} \boldsymbol{\Phi}|_F+\sum{k=1}^p k^\alpha\left[\sum_{j=1}^m\left|\boldsymbol{\Phi}k(j,-j) \mathbf{W}(-j)\right|_2+\lambda \sum{j=1}^m w_j\left|\boldsymbol{\Phi}_k(j, j)\right|_1\right]\right}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Regularization methods

$$\sum_{t=1}^n\left|\mathbf{Z}t-\sum{k=1}^p \mathbf{\Phi}k \mathbf{Z}{t-k}\right|2,$$ 其中|| $_2$为向量的欧几里得$\left(L^2\right)$范数。在实践中，更简洁地说，对于数据$\mathbf{Z}t=\left[Z{1, t}\right.$, $\left.Z{2, t}, \ldots, Z_{m, t}\right]^{\prime}, t=1,2, \ldots, n$，我们可以将Eq.(10.2)中的VAR $(p)$模型以矩阵形式表示，
$$\underset{n \times m}{\mathbf{Y}}=\underset{(n \times m p)}{\mathbf{X}} \underset{(m p \times m)}{\mathbf{D}}+\underset{(n \times m)}{\boldsymbol{\xi}},$$

$$\mathbf{Y}=\left[\begin{array}{c} \mathbf{Z}1^{\prime} \ \mathbf{Z}_2^{\prime} \ \vdots \ \mathbf{Z}_n^{\prime} \end{array}\right], \mathbf{X}=\left[\begin{array}{c} \mathbf{X}_1^{\prime} \ \mathbf{X}_2^{\prime} \ \vdots \ \mathbf{X}_n^{\prime} \end{array}\right], \boldsymbol{\Phi}=\left[\begin{array}{c} \boldsymbol{\Phi}_1^{\prime} \ \boldsymbol{\Phi}_2^{\prime} \ \vdots \ \boldsymbol{\Phi}_p^{\prime} \end{array}\right], \boldsymbol{\xi}=\left[\begin{array}{c} \mathbf{a}_1^{\prime} \ \mathbf{a}_2^{\prime} \ \vdots \ \mathbf{a}_n^{\prime} \end{array}\right],$$和$$\mathbf{X}_t^{\prime}=\left[\mathbf{Z}{t-1}^{\prime}, \mathbf{Z}{t-2}^{\prime}, \ldots, \mathbf{Z}{t-p}^{\prime}\right]$$

$$\underset{\boldsymbol{\Phi}}{\operatorname{argmin}}|\mathbf{Y}-\mathbf{X \Phi}|_F$$

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

$$\underset{\boldsymbol{\Phi}}{\operatorname{argmin}}\left{|\mathbf{Y}-\mathbf{X} \boldsymbol{\Phi}|F+\lambda|\operatorname{vec}(\boldsymbol{\Phi})|_1\right},$$ 其中第二项是通过$L_1$惩罚进行正则化，$\lambda$是其控制参数。$\lambda$可以通过交叉验证来确定。lasso方法没有对滞后阶数之间的关系作任何特殊的假设，并倾向于过度选择VAR模型的滞后阶数$p$。这导致我们发展了一些改进的方法。 滞后加权套索法 Song和Bickel(2011)提出了一种将滞后加权套索(套索和组套索结构)方法纳入高维VAR模型的方法。他们将Yuan和Lin(2006)引入的套索罚放在非对角线项上，套索罚放在对角线项上。更具体地说，如果我们将$\boldsymbol{\Phi}(j,-j)$表示为由非对角元素$\left{\phi{j, i}\right}{i \neq j}$组成的向量，将$\boldsymbol{\Phi}k(j, j)$表示为$\boldsymbol{\Phi}_k$的$j$的th对角元素，则$\boldsymbol{\Phi}_k$的正则化为$$\sum{j=1}^m\left|\boldsymbol{\Phi}k(j,-j) \mathbf{W}(-j)\right|_2+\lambda \sum{j=1}^m w_j\left|\boldsymbol{\Phi}k(j, j)\right|,$$，其中$\mathbf{W}(-j)=\operatorname{diag}\left(w_1, \ldots, w{j-1}, w{j+1}, \ldots, w_m\right)$是$(m-1) \times(m-1)$对角矩阵，$w_j$是$1 \leq j \leq m$的变量与$j$相关联的正实值权，选择为$Z_{j, t} . \lambda$的标准差是控制参数，它控制其他滞后比自身滞后信息量少的程度。Eq.(10.7)的第一项是group lasso penalty，第二项是group lasso penalty，它们分别对其他lag和自身lag进行正则化。设$0<\alpha<1$和$(k)^\alpha$为针对不同滞后的不同正则化的另一个控制参数;估计过程是基于
$$\underset{\boldsymbol{\Phi}1, \ldots, \boldsymbol{\Phi}p}{\arg \min _k}\left{|\mathbf{Y}-\mathbf{X} \boldsymbol{\Phi}|_F+\sum{k=1}^p k^\alpha\left[\sum{j=1}^m\left|\boldsymbol{\Phi}k(j,-j) \mathbf{W}(-j)\right|_2+\lambda \sum{j=1}^m w_j\left|\boldsymbol{\Phi}_k(j, j)\right|_1\right]\right}$$

## MATLAB代写

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