## Random Walk Model

The random walk model is widely used in the area of finance. The stock prices or exchange rates (Asset prices) follow a random walk. A common and serious departure from random behavior is called a random walk (non-stationary), since today’s stock price is equal to yesterday stock price plus a random shock.

There are two types of random walks

- Random walk without drift (no constant or intercept)
- Random walk with drift (with a constant term)

**Definition**

A time series said to follow a random walk if the first differences (difference from one observation to the next observation) are random.

Note that in a random walk model, the time series itself is not random, however, the first differences of time series are random (the differences changes from one period to the next).

A random walk model for a time series $X_t$ can be written as

\[X_t=X_{t-1}+e_t\, \, ,\]

where $X_t$ is the value in time period $t$, $X_{t-1}$ is the value in time period $t-1$ plus a random shock $e_t$ (value of error term in time period $t$).

Since the random walk is defined in terms of first differences, therefore, it is easier to see the model as

\[X_t-X_{t-1}=e_t\, \, ,\]

where the original time series is changed to a first difference time series, that is the time series is transformed.

The transformed time series:

- Forecast the future trends to aid in decision making
- If time series follows random walk, the original series offers little or no insights
- May need to analyze first differenced time series

Consider a real-world example of daily US-dollar-to-Euro exchange rate. A plot of entire history (of daily US-dollar-to-Euro exchange rate) from January 1, 1999, to December 5, 2014 looks like

The historical pattern from above plot looks quite interesting, with many peaks and valleys. The plot of the daily*changes*(first difference) would look like

The volatility (variance) has not been constant over time, but the day-to-day changes are almost completely random.

Remember that, random walk patterns are also widely found elsewhere in nature, for example, in the phenomenon of Brownian Motion that was first explained by Einstein.