時(shí)間序列分析及var模型-wenkub.com
2025-06-23 18:30 本頁面
【正文】 hence, both ratios don’t seem to be stationary. Vector errorcorrection model (VECM)The VECM bines VAR and cointegration into one framework. The VAR is extended by including deviations from the longterm equilibrium defined by the cointegration vector. The coefficient of the deviation from the longterm equilibrium indicates the speed of adjustment back into equilibrium.The VECM capture the longterm relationship and the shortterm dynamics of two or more time series. Let’s see how it works in the case of gold and silver prices. Table 6 reports the VECM specification, which resembles the VAR with two lags. It also contains the CE ponent。 hence, we can assess the dynamic impact of a marginal change of one variable on another. The standard OLS regression provides coefficients, and coefficients refer to the partial impact of an explanatory variable on the dependent variable. In the case of VAR models, the relationship bees dynamic, as a change of one variable (say x) in t can affect x and y in t+1. The impact on x and y in t+1 in turn affects x and y in t+2 and so on until the impact dies out. Impulse response functions are very useful in illustrating the shortterm dynamics in a model.Let’s look at an example to see how VAR modelling works. In Lecture 5, we tried very hard to understand gold prices. We extend our univariate model by exploring the relationships between gold and silver prices. Linking two (similar) assets or securities is a very mon trading strategy, which is called pairstrading.Before we do any sophisticated modelling, it is always beneficial to look at some line charts. Figure 1 shows the indexed time series of nominal gold and silver prices from 1900 to 2010.Figure 1: Nominal gold and silver prices, indexed, 19002010We can see that there is a certain degree of comovement, which we might be able to exploit for our trading strategy. Before we can use VAR, we need to ensure that both time series are stationary. It is obvious from Figure 1 that gold and silver prices are not stationary. However, after taking a firstdifference we can show that price changes are stationary. So both time series are I(1).The next step is to determine the optimal lag length using information criteria. Table 1 shows different specifications using the varsoc mand.Table 1: Determining the optimal lag length using information criteriaBased on the AIC and HQIC, two lags are optimal。We now relax both assumptions using a VAR model. VAR models can be regarded as a generalisation of AR(p) processes by adding additional time series. Hence, we enter the field of multivariate time series analysis. VAR模型可以當(dāng)作是在一般的自回歸過程中加入時(shí)間序列。 Vector error correction model (VECM)Page | 15Lecture 66. Time series analysis: Multivariate models Learning outes Application: pairs trading Vector autoregression (VAR) 向量自回歸The classical linear regression model assumes strict exogeneity。Let’s look at a standard AR(p) process for two variables (yt and xt).(1) yt=α1+i=1pβ1iyti+ε1t(2) xt=α2+i=1pβ2ixti+ε2tThe
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