Coefficient Instability

Just as in regression models, the coefficients in time-series models are often unstable across different sample periods. Regression coefficient estimates derived from an earlier sample period can differ from those approximated using a later period. Similarly, the estimated regression coefficients vary depending on the length of the sample periods.

In selecting a sample period for estimating a time-series model, we should seek to assure ourselves that the time series was stationary in the sample period.

Random Walk Process

A random walk is a time series in which the value of the series in one period is the value of the series in the previous period plus an unpredictable random error: x_t = x_t-1 + ε_t

• b₀ = 0, and b₁ = 1
• E(ε_t) = 0: The expected value of each error term is 0.
• E(ε_t²) = σ²: The variance of the error term is constant.
• E(ε_i ε_j) = 0 if i != j: There is no serial correlation in the error terms.

A random walk with drift is a random walk with a nonzero intercept term. It is expressed as: x_t = b₀ + b₁ x_t-1 + ε_t, where b₀ = constant drift, and b₁ = 1.

A random walk has an undefined mean reversion level. The mean-reverting level = b₀/(1 - b₁), with a b₁ = 1 ....

A random walk is not covariance stationary. The variance of a random walk process does not have an upper bound. As t increases, the variance grows with no upper bound. However, the covariance stationary property suggests that the mean and variance terms of a time series must remain constant over time.