Set out the features of a random walk and how it differs from a covariance stationary process
The idea
A random walk is a series whose value each period equals the previous value plus an unpredictable error: x(t) = x(t − 1) + ε. With drift, a constant is added each period, so the expected value k periods ahead is the current value plus k times the drift. A random walk has a slope coefficient b1 of exactly 1, so it has no finite mean-reverting level, and its variance grows without limit over time. It is therefore not covariance stationary, unlike an AR(1) with b1 below 1 in absolute value, which has a constant mean and variance. Its best forecast without drift is simply today's value. Exchange rates and asset prices often behave this way.
In the app: 2 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
Take the free diagnostic →More in Time-Series Analysis
- Work out a trend forecast for a time series, whether fitted as a linear or a log-linear trend, from given trend coefficients, and judge the result
- Explain what decides the choice between a linear and a log-linear trend for a given series, and assess where trend models fall short
- Explain why a time series needs to be covariance stationary and what it means when a series lacks stationarity
- Set out how an order-p autoregressive (AR) model is built and, from its fitted coefficients, work out forecasts one and two periods ahead
- Explain how residual autocorrelations let you check whether an autoregressive model suits a time series
- Say what mean reversion is and work out the level a series tends to revert to
- Distinguish forecasts made within the sample from those made outside it, and rank time-series models by forecast accuracy using root mean squared error
- Explain why time-series model coefficients can be unstable
- Explain what unit roots mean for time-series work, when they tend to arise and how to detect them, and show how to transform a series with a unit root so that an AR model can be used on it
- Set out the stages of testing a series for a unit root (nonstationarity) and explain how the test connects to autoregressive models
- Explain how to detect and adjust for seasonal effects in a time-series model, and work out and explain a forecast from an AR model that includes a seasonal lag
- Say what autoregressive conditional heteroskedasticity (ARCH) is and show how ARCH models help forecast a time series' variance
- Explain how to check time-series variables for cointegration and/or nonstationarity before putting them into a linear regression
- Choose a suitable time-series model for a given investment problem and give reasons for the choice