【正文】 however, we can use a more relaxed concept, namely weak exogeneity. As we use lagged values of both dependent variables, we can argue that these lagged values are known to us, as we observed them in the previous period. We call these variables predetermined. Predetermined (lagged) variables fulfil weak exogeneity in the sense that they have to be uncorrelated with the contemporaneous error term in t. We can still use OLS to estimate the following system of equations, which is called a VAR in reduced form.(3) yt=α1+i=1pβ11iyti+i=1pβ12ixti+ε1t(4) xt=α2+i=1pβ21iyti+i=1pβ22ixti+ε2tThe beauty of this model is that we don’t need to predefine whether x or y are endogenous (the dependent variable). In fact, we can test whether x (y) is endogenous or exogenous using Granger causality tests. The idea of Granger causality is that past observations (lagged dependent variables) can influence current observations – but not vice versa. So the idea is rather simple: the past affects the present, and the present does not affect the past. STATA provides Granger causality tests after conducting a VAR analysis, which is based on testing the joint hypothesis that past realisations do not Granger cause the present realisation of the dependent variable.In many applications, VAR models make a lot of sense, as a clear direction of causality cannot be predefined. For instance, there is a substantial literature on the benefits of internationalisation (. entering foreign market through crossborder Mamp。金融時(shí)間序列及面板數(shù)據(jù)往往都有很強(qiáng)的自相關(guān)性，假定解釋變量影響因變量。 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 mo