# Random walk with drift

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## Random walk with drift

For a random walk with drift, the best forecast of tomorrow's price is today's price plus a drift term. One could think of the drift as measuring a trend in the price (perhaps reflecting long-term inflation). Given the drift is usually assumed to be constant. Related: Mean reversion.
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They showed evidence in favour of Taylor rule models and against the random walk with drift for Brazil and others countries.
In our exercise, we were able to outperform the random walk with drift, but not the random walk without drift.
A modified Theil inequality coefficient or U-coefficient greater than one indicates that the random walk benchmark or the random walk with drift has smaller absolute forecast errors than the competing methodologies.
In addition, error differential regression results for both the random walk benchmark and the random walk with drift benchmark are compared to the commercial and industrial property values for each of the four forecasting models.
Forecast results for the random walk and random walk with drift benchmarks are found in Tables 1.4 and 1.5.
Modified Theil Inequality Coefficients: Traditional Income Elasticity Model to Random Walk and Random Walk with Drift
The assumption of a random walk with drift process for the [K.sub.t]'s is not crucial for the analysis carried out in this article.
Under this assumption, the trend component is a random walk with drift:
Note that if [X.sub.t] had a constant non-zero mean, that is [X.sub.t] = [delta] + [[epsilon].sub.t], where [[epsilon].sub.t] is white noise, then [Y.sub.t] would be a random walk with drift: [Y.sub.t] = [Y.sub.t-1] + [delta] + [[epsilon].sub.t], E([DELTA][Y.sub.t]) = [delta].
For an economic time-series best characterized as a random walk with drift process, an appropriate forecasting model must account for the deterministic trend introduced by the drift term.
For example, the traditional DF tests may fail to distinguish between a covariance stationary series with a change in mean (i.e., structural break) and a random walk series, or between a random walk with drift series (a unit root series) with a one-time shock and a trend stationary series with a jump in the level.
That implies that the LTF RMSEs for the total number of water bills are statistically smaller than those obtained from the Random Walk with drift procedure for this variable [Diebold and Mariano, 1995].
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