A Random Walk Process

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A random walk process A simple random walk model A random walk is defined as a process where the current value of a variable is composed of the past value plus an error term defined as a white noise (a normal variable with zero mean and variance one). Algebraically a random walk is represented as follows: yt = yt 1 + t − The implication of a process of this type is that the best prediction of y for next period is the current value, or in other words the process does not allow to predict the change (yt yt 1). That is, the change of y is absolutely random. − − It can be shown that the mean of a random walk process is constant but its variance is not. Therefore a random walk process is nonstationary, and its variance increases with t. In practice, the presence of a random walk process makes the forecast process very simple since all the future values of yt+s for s > 0, is simply yt. A random walk model with drift A drift acts like a trend, and the process has the following form: yt = yt 1 + a + t − For a > 0 the process will show an upward trend. The third graph presented in the previous page was generated assuming a = 1. This process shows both a deterministic trend and a stochastic trend. Using the general solution for the previous process, n y = y + at + t 0 X t t=1 n where y0 + at is the deterministic trend and t=1 t is the stochastic trend. The relevance of the random walk model isP that many economic time series follow a pattern that resembles a trend model. Furthermore, if two time series are independent random walk processes then the relationship between the two does not have an economic meaning. If one still estimates a regression model between the two the following results are expected: (a) High R2 (b) Low Durbin-Watson statistic (or high ρ) (c) High t ratio for the slope coefficient This indicates that the results from the regression are spurious. A regression in terms of the changes can provide evidence against previous spurious results. If the coefficient of a regression ∆yt = a0 + β1∆xt + νt is not significant, then this is an indication that the relationship between y and x is spurious, and one should proceed by selecting other explanatory variable. If the Durbin-Watson test is passed then the two series are conintegrated, and a regression between them is appropriate. { 1 {.

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