Explanation

A random walk process is a special case of an AR(1) model with an intercept equal to 0 and a slope equal to 1.

Mathematical Definition:

• Random walk: $Y_t = Y_{t-1} + \epsilon_t$
• AR(1) model: $Y_t = b_0 + b_1Y_{t-1} + \epsilon_t$
• When $b_0 = 0$ and $b_1 = 1$, the AR(1) model becomes a random walk

Why other options are incorrect:

A. Is a covariance-stationary time series - FALSE

• Random walks are non-stationary
• Variance increases over time: $Var(Y_t) = t\sigma^2$
• Mean may not be constant

B. Has a well-defined mean-reverting level - FALSE

• Mean-reverting level for AR(1) is $\frac{b_0}{1-b_1}$
• For random walk ($b_1 = 1$), this becomes undefined (division by zero)
• Random walks do not revert to any mean level

Key Properties of Random Walks:

• Unit root process ($b_1 = 1$)
• Non-stationary
• No tendency to revert to mean
• Variance grows with time
• Commonly used to model stock prices and other financial variables

Therefore, option C correctly describes the relationship between random walks and AR(1) models.