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

#### Doug I. Jones

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

In previous sections, we have mainly reviewed methods that are based on the VAR model but with different model constraints and estimation procedures. However, there exist many other models for multivariate time series analysis, such as the transfer function model (Box et al. 2015), the state space model (Kalman 1960), and canonical correlation analysis (Box and Tiao 1977). More recently, Stock and Watson (2002a, b) introduced the factor model for dimension reduction and forecasting. The dynamic orthogonal component analysis was proposed by Matteson and Tsay (2011). In this review section, we will concentrate on the factor model.

The factor model is also called the diffusion index approach and can be written as
$$\mathbf{Z}t=\mathbf{L} \mathbf{F}_t+\varepsilon_t$$ where $\mathbf{F}_t=\left(F{1, t}, F_{2, t}, \ldots, F_{k, t}\right)^{\prime}$ is a $(k \times 1)$ vector of factors at time $t, \mathbf{L}=\left[\ell_{i, j}\right]$ is a $(m \times k)$ loading matrix, $\ell_{i, j}$ is the loading of the $i$ th variable on the $j$ th factor, $i=1,2, \ldots, m, j=1,2, \ldots, k$, and $\boldsymbol{\varepsilon}t=\left(\varepsilon{1, t}, \ldots, \varepsilon_{m, t}\right)^{\prime}$ is a $(m \times 1)$ vector of noises with $E\left(\varepsilon_t\right)=\mathbf{0}$, and $\operatorname{Cov}\left(\boldsymbol{\varepsilon}t\right)=\boldsymbol{\Sigma}$. Let $Z{i, t+\ell}$ be $i$ th component of $\mathbf{Z}{t+\ell}$, once values of factors are obtained, we can build a forecast equation for the $\ell$-step ahead forecast, such that $$Z{i, t+\ell}=\boldsymbol{\beta}^{\prime} \mathbf{F}t+\varepsilon{i, t+\ell},$$
where $\boldsymbol{\beta}=\left(\beta_1, \ldots, \beta_k\right)^{\prime}$ denotes the coefficient vector and $\varepsilon_{i, t+\ell}$ is a sequence of uncorrelated zero-mean random variables. Note that the Eq. (10.19) can be further extended to:
$$Z_{i, t+\ell}=\boldsymbol{\beta}^{\prime} \mathbf{F}t+\boldsymbol{\alpha}^{\prime} \mathbf{X}{i, t}+\varepsilon_{i, t+\ell},$$
where $\mathbf{X}{i, t}$ is a $m \times 1$ vector of lagged values of $Z{i, t+\ell}$ and/or other observed variables. We follow the approach proposed by Bai and $\mathrm{Ng}(2002)$ plus the penalty term $k[(\mathrm{~m}+n) / \mathrm{mn}] \log [\mathrm{mn} /$ $(m+n)$ ] to select the number of factors in our simulation studies and empirical examples. Other methods or penalties as described in Bai and $\mathrm{Ng}$ (2002) can also be used although this is beyond the scope of this chapter.

## 统计代写|时间序列分析代写Time-Series Analysis代考|The proposed method for high-dimension reduction

In many applications, a large number of individual time series may follow a similar pattern so that we can aggregate them together. By doing so, we can reduce the dimension of the multivariate time series to a manageable and meaningful size. Specifically, we will concentrate on the VAR model described in Section 10.2 and propose aggregation as our method of dimension reduction.

Given a vector time series, assume that after model identification, it follows the VAR (p) model,
$$\mathbf{Z}t=\sum{k=1}^p \mathbf{\Phi}k \mathbf{Z}{t-k}+\mathbf{a}t,$$ where $\mathbf{Z}_t$ is mean adjusted stationary $m$-dimensional original time series. Let $$\mathbf{Y}_t=\mathbf{A} \mathbf{Z}_t$$ where $\mathbf{A}$ is a $s \times m$ aggregation matrix with $s{1, t}, \ldots, Y_{s, t}\right]^{\prime}$. Presently, the elements in $\mathbf{A}$ are assumed to be binary, such that its $(i, j)$ element is 1 when $Z_{j, t}$ is included in the aggregate $Y_{i, t}$, and is 0 otherwise. In other words, the elements of row $i$ in $\mathbf{A}$ construct $Y_{i, t}$ as the sum of designated elements of $\mathbf{Z}_t$. We will call $\mathbf{Y}_t$ the aggregate series and $\mathbf{Z}_t$ the non-aggregate series. It can be shown that the aggregate series $\mathbf{Y}_t$ will also follow a $\operatorname{VAR}(p)$ model. However, in practice, we will normally use the same model identification procedure to fit a VAR $(P)$ model for some $P$ such that
$$\mathbf{Y}t=\sum{k=1}^P \boldsymbol{\Phi}k^{(a)} \mathbf{Y}{t-k}+\boldsymbol{\xi}_t,$$
where $\boldsymbol{\Phi}_k^{(a)}$ for $k=1, \ldots, P$ are $s \times s$ coefficient matrices, and $\boldsymbol{\xi}_t$ follows $s$-dimensional i.i.d. normal distribution with mean vector zero and covariance $\mathbf{\Sigma}^{(a)}$. The order $P$ can be selected by existing methods such as AIC, BIC, and sequential likelihood ratio test (a detailed review of order selection methods can be found in Lütkepohl, 2007).

# 时间序列分析代考

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

$$\mathbf{Z}t=\mathbf{L} \mathbf{F}t+\varepsilon_t$$其中$\mathbf{F}_t=\left(F{1, t}, F{2, t}, \ldots, F_{k, t}\right)^{\prime}$是$(k \times 1)$的因子向量$t, \mathbf{L}=\left[\ell_{i, j}\right]$是$(m \times k)$的加载矩阵$\ell_{i, j}$是$i$变量在$j$因子上的加载，$i=1,2, \ldots, m, j=1,2, \ldots, k$$\boldsymbol{\varepsilon}t=\left(\varepsilon{1, t}, \ldots, \varepsilon_{m, t}\right)^{\prime}是(m \times 1)的噪声向量E\left(\varepsilon_t\right)=\mathbf{0}和\operatorname{Cov}\left(\boldsymbol{\varepsilon}t\right)=\boldsymbol{\Sigma}。设Z{i, t+\ell}为\mathbf{Z}{t+\ell}的i分量，一旦得到各因子的值，我们就可以为\ell -步预测建立预测方程，如$$ Z{i, t+\ell}=\boldsymbol{\beta}^{\prime} \mathbf{F}t+\varepsilon{i, t+\ell}, $$其中\boldsymbol{\beta}=\left(\beta_1, \ldots, \beta_k\right)^{\prime}为系数向量，\varepsilon_{i, t+\ell}为不相关的零均值随机变量序列。注意，式(10.19)可以进一步扩展为:$$ Z_{i, t+\ell}=\boldsymbol{\beta}^{\prime} \mathbf{F}t+\boldsymbol{\alpha}^{\prime} \mathbf{X}{i, t}+\varepsilon_{i, t+\ell}, $$其中\mathbf{X}{i, t}是Z{i, t+\ell}和/或其他观察变量滞后值的m \times 1向量。我们遵循Bai和\mathrm{Ng}(2002)加上惩罚项k[(\mathrm{~m}+n) / \mathrm{mn}] \log [\mathrm{mn} /$$(m+n)$]提出的方法，在我们的模拟研究和实证例子中选择因素的数量。如Bai和$\mathrm{Ng}$(2002)所述的其他方法或处罚也可以使用，尽管这超出了本章的范围。

## 统计代写|时间序列分析代写Time-Series Analysis代考|The proposed method for high-dimension reduction

$$\mathbf{Z}t=\sum{k=1}^p \mathbf{\Phi}k \mathbf{Z}{t-k}+\mathbf{a}t,$$其中$\mathbf{Z}t$为平均调整后平稳$m$维原始时间序列。设$$\mathbf{Y}_t=\mathbf{A} \mathbf{Z}_t$$，其中$\mathbf{A}$是一个$s \times m$聚合矩阵，$s{1, t}, \ldots, Y{s, t}\right]^{\prime}$。目前，假设$\mathbf{A}$中的元素是二进制的，因此当$Z_{j, t}$包含在聚合$Y_{i, t}$中时，其$(i, j)$元素为1，否则为0。换句话说，$\mathbf{A}$中$i$行的元素将$Y_{i, t}$构造为$\mathbf{Z}_t$的指定元素之和。我们将称$\mathbf{Y}_t$为聚合系列，称$\mathbf{Z}_t$为非聚合系列。可以看出，聚合序列$\mathbf{Y}_t$也将遵循$\operatorname{VAR}(p)$模型。然而，在实践中，我们通常会使用相同的模型识别程序来拟合一些$P$的VAR $(P)$模型
$$\mathbf{Y}t=\sum{k=1}^P \boldsymbol{\Phi}k^{(a)} \mathbf{Y}{t-k}+\boldsymbol{\xi}_t,$$

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