Financial Risk Manager Part 1

Explanation

A time series with a unit-root is indeed a random walk time series with a drift, described using an AR(1) model where the lag coefficient is 1. In time series analysis, a unit-root test checks whether a time series variable is non-stationary and possesses a unit-root. The presence of a unit-root indicates that the time series does not have a tendency to return to a long-run mean and has a high level of persistence.

Key Points:

• The AR(1) model, or first-order autoregressive model, is a common approach for modeling time series data
• If the lag coefficient is 1, it indicates that the series is a random walk, meaning that the current value is equal to the previous value plus a random error
• The 'drift' term in a random walk model represents a constant trend over time
• Therefore, a unit-root time series can be accurately described as a random walk with a drift, modeled using an AR(1) model with a lag coefficient of 1

Why other options are incorrect:

• Choice B: A random walk time series with a drift described using an AR(1) model whose lag coefficient is 0 does not characterize a unit-root time series. The lag coefficient of 0 would imply that the value at any given point in time does not depend on its previous value, which contradicts the concept of a unit-root where there is persistence in values over time.
• Choice C: Time series with unit roots are not covariance stationary. Covariance stationarity requires that mean and variance are constant over time and covariances for different intervals depend only on the length of the interval, but not on where they are in time. However, a unit root process violates these conditions as it has a changing mean and variance over time.
• Choice D: A unit-root time series is not stationary and does not revert to its mean over time. Instead, it follows a random walk, which means it tends to wander away from its initial value and does not exhibit mean reversion. This is contrary to the properties of stationary time series, which do revert to their mean over time.