時間序列分析及var模型-wenkub
2023-07-11 18:30:47 本頁面
【正文】 next step is to allow that lagged values of xt can affect yt and vice versa. This means that we obtain a system of equations for two dependent variables (yt and xt). Both dependent variables are influenced by past realisations of yt and xt. By doing that, we violate strict exogeneity (see Lecture 2)。 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 multivariate time series econometrics, people realised that gold and silver seem to have a mon movement around a longterm equilibrium (goldsilver price ratio). Moreover, the idea of equilibrium conditions in economics and the availability of macroeconomic time series led to the development of cointegration analysis. The idea is very simple. Even if two (or more) time series are nonstationary and hence have stochastic trends, they might be still driven by the same underlying factors that lead to their stochastic behaviour. Therefore, we analyse the time series in levels and see whether we can find a longterm equilibrium – a socalled cointegrating vector. Before we explore the Johansen procedure, let’s look at the goldsilver ratio over time shown in Figure 3.Figure 3: The goldsilver ratio, 19002010The ratio looks like a meanreverting process。 the cocalled errorcorrection ponent that captures the deviat
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