1. The random walk (RW) model
2. Random walk
A Random Walk or R-W is a simple example of an unstable or non-stationary process.
It has no specified mean or variance.
It exhibits very strong dependence over time, with each observation closely related to its immediate neighbors.
However, its changes or increments follow a white noise process, which is stable and stationary.
3. Random walk
Let's look at some plots of Random Walk time series.
In figure A you note that there is strong persistence in the series. Neighboring observations are similar, and short upward and downward trends appear at random. It crosses the dashed horizontal zero line a few times. Figure B exhibits many of the same features. Figure C begins near zero, but then drifts downward, whereas figure D begins near zero, but then drifts upward. Each of these are Random Walk time series.
4. Random walk
Now let's look at the random walk model. It is defined recursively. Today's or the current observation is equal to yesterday's or the previous observation plus noise or error. This error is specifically mean zero white noise.
In practice, simulating a Random Walk time series requires an initial point, Y_zero. Given Y_zero and the first white noise term you generate the first Random Walk observation, and then you proceed to time index two. Often, the value zero is used as the initial point Y_zero, for simplicity. The random walk model has only one parameter, that is the variance of the white noise.
5. Random walk - I
The terms in a random walk can be rearranged as a first difference series. Here, you see that the difference between the current observations Y_t and the previous observation Y_{t-1} is equal to the noise term. That is, the difference or change series is white noise.
6. Random walk - II
If you start with a Random Walk time series, as appears on the left, and apply the diff() function in R to compute the difference series, you can see the result of this simple transformation on the right. The differences, or changes, or increments, of a Random Walk series are a White Noise series.
7. Random walk with drift - I
The random walk model can be extended to include an intercept or drift coefficient. This adds a constant, C, to the recursive formula. The Random Walk with a Drift has two parameters, the drift constant C and the White Noise variance sigma-squared.
What is the first difference series of a Random Walk with a Drift? It is simply a constant plus noise, which is a White Noise process with mean C.
8. Random walk with drift - II
Let's look at some plots of Random Walk with a Drift time series.
Figure A has a drift constant of C equal to zero, that is there is no drift. Figure B has a positive drift coefficient, and you can see a clear upward trend over time. Figure C has a negative drift coefficient, and you can see a clear downward trend over time. Figure D is similar to figure B, but is has a larger positive drift coefficient, and the upward trend over time is steeper.
9. Let's practice!
Great! Now let's practice simulating and estimating Random Walk models!