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Two cointegrated series contain a single unit root. Each series can be formulated as the sum of a common unit root plus a stationary component. Most textbooks covering cointegration will cover such formulations - see Hamilton's (1994) discussion of Phillips' "triangular representation" of a cointegrated vector, for example. Simulating is likely to be easy ...


You are asking an interesting question. Firstly, a Submartingale has increasing or equal expectation (not decreasing). Secondly, the process $dX_t=X_tdW_t$ is a true martingale (not strictly local), since its solution (by Ito): $$X_t=X_0e^{W_t-\frac{t}{2}}$$ has $E(X_t)=X_0$ constant expectation ($e^{-\frac{t}{2}}E(e^{W_t})=1, W_t\sim N(0,t)$). The ...

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