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# 金融代写|金融计量经济学代考Financial Econometrics代考|MTH9891 Multiple Time Series Modeling

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## 金融代写|金融计量经济学代考Financial Econometrics代考|Multiple Time Series Modeling

Multiple time series analysis is concerned with modeling and estimation of dynamic relationships among ‘ $m$ ‘ related time series $y_{1 t}, \ldots, y_{m t}$, based on observations over $T$ equally spaced time points $t=1, \ldots, T$, and also between these series and potential exogenous time series variables $x_{1 t}, \ldots, x_{n t}$, observed over the same time period. We shall explore the use of these techniques in leveraging statistical arbitrage in multiple markets in Chapters $5 \& 6$. We first introduce a general model for multiple time series modeling, but will specialize to vector autoregressive (VAR) models for more detailed investigation. As some concepts are similar to those of the univariate models discussed in Chapter 2 , the presentation will be brief.

Let $Y_{t}=\left(y_{1 t}, \ldots, y_{m t}\right)^{\prime}$ be an $m \times 1$ multiple time series vector of response variables and let $X_{t}=\left(x_{1 t}, \ldots, x_{n t}\right)^{\prime}$ be an $n \times 1$ vector of input or predictor time series variables. Let $\epsilon_{t}$ denote an $m \times 1$ white noise vector of errors, independently distributed over time with $\mathrm{E}\left(\epsilon_{t}\right)=0$ and $\operatorname{Cov}\left(\epsilon_{t}\right)=\Sigma_{\epsilon \epsilon}$, a positive-definite matrix. We consider the multivariate time series model
$$Y_{t}=\sum_{s=0}^{p} C_{s} X_{t-s}+\epsilon_{t},$$
where the $C_{s}$ are $m \times n$ matrices of unknown parameters. In the general setting, the ‘input’ vectors $X_{t}$ could include past (lagged) values of the response series $Y_{t}$. In the context of trading $Y_{t}$ could denote the prices of related assets, $X_{t-s}$ could represent the past values of $Y_{t}$ and the values of volume, volatility, market, industry factors of all related assets, etc. Note that the model (3.21) can be written in the form of a multivariate regression model. An important issue that arises in the multiple series modeling is as follows: Even for moderate values of the dimensions $m$ and $n$, the number of parameters that must be estimated in model (3.21), when no constraints are imposed on the matrices $C_{s}$ can become quite large. Because of the potential complexity of these models, it is often useful to consider dimension reduction procedures such as reduced-rank regression methods described in the last section and more in detail in Reinsel and Velu (1998) [289]. This may also lead to more efficient, interpretable models due to the reduction in the number of unknown parameters. The key quantities that play a role are cross-covariance matrices that relate $X_{t-s}$ to $Y_{t}$, for appropriate values of ‘ $s$ ‘.

## 金融代写|金融计量经济学代考Financial Econometrics代考|Co-Integration, Co-Movement and Commonality in Multiple Time Series

The financial time series generally exhibit non-stationary behavior. For the stationary series, the dynamic relationships among components of the vector time series, $Y_{t}$, has been studied in various ways. While the estimation and inference aspects are similar to the univariate series discussed in Chapter 2 , because of the sheer dimension of $Y_{t}$, there are more challenges in the precise estimation of the model coefficients but there are also more opportunities to study the linkages among the components of $Y_{t}$ that may be insightful. For example, this provides an opportunity to investigate the behavior of a portfolio of stocks. The number of unknown elements in (3.22) is $m^{2}(p+1)$ which can be quite large even for modest values of ‘ $m$ ‘ and ‘ $p$ ‘; therefore considerations of the dimension reduction aspects in modeling (3.22) have been given by Box and Tiao (1977) [54] and Brillinger (1981) [59] among others. The traditional tool used in multivariate analysis for dimension reduction, principal components method is based on the contemporaneous covariance matrix, and thus it ignores the time dependence. While Result 4, stated in the previous section, indicates that the linear combination can exhibit time dependence, we should expect the optimal linear combination to exhibit other desirable features, which are described here.

Identifying the common features among the series have been focused on by several authors; for example, Engle and Kozicki (1993) [127], Vahid and Engle (1993) [318] among others. The series if they are stationary, are studied for their co-movement and if they are non-stationary, they are studied for their co-integration. A method that can capture both aspects can be stated as a reduced-rank autoregressive modeling procedure which we briefly describe below. The basic $\operatorname{VAR}(p)$ model in (3.22) without the constant term can be written as
$$Y_{t}=C Y_{t-1}^{*}+\epsilon_{t},$$

## 金融代写|金融计量经济学代考FINANCIAL ECONOMETRICS代 考|MULTIPLE TIME SERIES MODELING

$$Y_{t}=\sum_{s=0}^{p} C_{s} X_{t-s}+\epsilon_{i},$$

289
. 由于末知参数数量的减少，这也可能导致更有效、可解释的模型。发挥莋用的关键量是相关的交叉协方差矩阵 $X_{t-s}$ 至 $Y_{t}$, 对于适当的值’ $s$ !

## 金融代写|金融计量经济学代考FINANCIAL ECONOMETRICS代 考|CO-INTEGRATION, CO-MOVEMENT AND COMMONALITY IN MULTIPLE TIME SERIES

54

59

127
, 瓦䆖德和恩格尔 1993
318

$$Y_{t}=C Y_{t-1}^{*}+\epsilon_{t}$$

## Matlab代写

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