時間序列分析及var模型(已修改)
2025-07-08 18:30 本頁面
【正文】 Page | 15Lecture 66. Time series analysis: Multivariate models Learning outes Vector autoregression (VAR) Cointegration Vector error correction model (VECM) Application: pairs trading Vector autoregression (VAR) 向量自回歸The classical linear regression model assumes strict exogeneity。 hence, there is no serial correlation between error terms and any realisation of any independent variable (lead or lag). As we discovered, serial correlation (or autocorrelation) is very mon in financial time series and panel data. Furthermore, we assumed a predefined relation of causality: explanatory variable affect the dependent variable. 傳統(tǒng)的線性回歸模型假設嚴格的外生性，誤差項與可實現(xiàn)的獨立變量之間沒有序列相關性。金融時間序列及面板數(shù)據(jù)往往都有很強的自相關性，假定解釋變量影響因變量。We now relax both assumptions using a VAR model. VAR models can be regarded as a generalisation of AR(p) processes by adding additional time series. Hence, we enter the field of multivariate time series analysis. VAR模型可以當作是在一般的自回歸過程中加入時間序列。Let’s look at a standard AR(p) process for two variables (yt and xt).(1) yt=α1+i=1pβ1iyti+ε1t(2) xt=α2+i=1pβ2ixti+ε2tThe 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, 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。A). There is evidence that multinationals outperform local peers due to the benefits of operating in many countries. At the same time, we know that
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