【正文】 dels 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。 Vector error correction model (VECM) 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。Page | 15Lecture 66. Time series analysis: Multivariate models Learning outes the cocalled errorcorrection ponent that captures the deviation from the longterm equilibrium in the previous period. So the CE is a lagged and hence predetermined variable, as required by OLS and the VAR framework.Table 6: VECM based on gold and silver pricesThe speed of adjustment is not significant, which undermines the idea of a longrun equilibrium in gold and silver prices as suggested by the literature.We can explore structural changes, when we plot the predicted longterm equilibrium over time.Figure 4: Predicted longterm equilibrium based on VECMIt is very obvious that the longterm equilibrium undergoes structural breaks. We could split the time period into a stable and unstable period. Yet the main issue with structural breaks is that they appear to be obvious ex post – but nearly impossible to predict ex ante.For instance, if we focus on 19001980, we obtain a very strong result that underlines cointegration and adjustments into the longterm equilibrium. Hence, we conclude that the goldsilver ratio is no longer a reliable phenomenon that we could rely on.