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ion 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. Nevertheless, we can use the shortterm dynamics captured in the VAR to do shortterm trading. Pairs trading APPLICATION The idea of pairs trading is that we trade two similar shares that are driven by similar macroeconomic factors. In our example, we focus on the US steel industry and try to identify a trading strategy based on United States Steel Corporation and Titan International.First we need to modify the time dimension, as NASDAQ reports the latest share prices first. We selected 5years of daily closing prices for our analysis.*Time dimension needs to be modifiedgen t=_nreplace t=1264ttsset t*Line chart twoway (line us_steel t) (line titan t)Let’s have a quick look at a line chart bining both share prices. Obviously both share prices are nonstationary, which we should confirm first using DickeyFuller tests. We run DickeyFuller tests based on share prices and firstdifferenced time series. The tests confirm that both time series are I(1). Hence, we can try to find a cointegration relation following the Johansen procedure. Before we do that, I suggest that we explore a VAR model and determine the optimal lag structure.*Returnsgen r_us=ln(us_steel)ln()gen r_titan=ln(titan)ln()*DickeyFullerdfuller us_steeldfuller titandfuller r_usdfuller r_titanWe cannot establish any VAR lag structure, which shows that there is hardly any serialcorrelation of returns. This is great news for the Efficient Market Hypothesis – but bad news for us. When you explore AC and PAC function, you will also discover that univariate analysis won’t go anywhere, as autocorrelation in returns is not present. Luckily, we can find cointegration with Titan adjusting back to longterm equilibrium conditions. Based on the VECM, we can predict the levels (share prices) of both stocks. These are onestep ahead forecast.When should you buy and sell? This also depends on transaction costs. In pairs trading, you go long in one stock and short in the second stock to hedge y