時(shí)間序列分析及var模型-閱讀頁(yè)
2025-07-11 18:30本頁(yè)面
【正文】 ut. 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。 thus, in the long run it tends to go back to its longterm equilibrium (mean). Based on the ratio, we could argue that gold seems to be overvalued pared to silver at the moment. Of course, taking the ratio suggests a very simple cointegrating vector – in fact we assume a onetoone relationship. Before we can use the Johansen procedure, we have to make sure that the time series have the same order of integration I(p). We already know that gold and silver prices are both I(1) time series. Table 5 shows the results of the Johansen test for cointegration. In line with the VAR model, we use two lags.Table 5: Johansen testThe null hypothesis that there is no cointegration (r=0) can be rejected if we use the trace statistic. However, the null hypothesis that we have one cointegrating vector (r=1) cannot be rejected. The problem is that the maxlambda statistic does not support cointegration. I also tried logprices instead, which is mon in analysing goldsilver ratios。 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。 Sons.15Dr Gerhard Kling, Quantitative Research Methods in Finance, University of Southampton
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