時(shí)間序列分析及var模型(完整版)
【正文】 ionships 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。 Vector error correction model (VECM) Application: pairs trading Vector autoregression (VAR) 向量自回歸The classical linear regression model assumes strict exogeneity。 however, the (S)BIC prefers only one lag. I would prefer HQIC and try two lags first. If the second lag does not exhibit significant coefficient, we could try to reduce the lag length in line with (S)BIC.We run a VAR with two lags to explain current price changes in gold and silver. Table 2 provides the OLS estimates.Table 2: VAR model with two lagsWe see that silver prices (lag 2) affect current gold prices, and we can establish autocorrelation in both time series. To test whether gold Granger causes silver or vice versa, we run Granger causality tests reported in Table 3.Table 3: Granger causality testsHence, we confirm that past changes in silver prices can predict future gold price changes. This is very interesting, as it can be used to develop a trading strategy. Finally, we need to show that the VAR is stable (see Table 4).Table 4: Stability condition of the VARFinally, we can illustrate the impact of silver price changes on future gold price changes using an impulse response function. Figure 2 shows the impulse response function and confidence intervals derived from bootstrapping. If silver prices increase today by 1%, we should expect a significant decline in gold prices in two years by %.Figure 2: Impulse response function CointegrationWhen we explore Figure 1 a bit more carefully, we can see that silver and gold prices exhibit a certain degree of comovement. We could almost argue that they share a mon stochastic trend. The limitation of ARIMA and VAR models is that they can be only used if the time series are stationary. In our case, we had to firstdifference your time series to ensure stationarity. Firstdifferencing eliminates a lot of information in the time series. Is there no better way to analyse gold and silver prices.Long before the development of multiva
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