Stationarity of a time series
Section of Dependence along a series, part of Time series
Stationarity of a time series
| Type | Section |
| Taught in | Dependence along a series |
| Part | Time series |
| Questions using it | 2 |
Without some repetition, a series of length T does not identify a different mean at every time and a different covariance for every pair. Strict stationarity says that any block of observations has the same joint distribution as the same block shifted by \tau. The path changes. The distribution does not. Weak stationarity asks only for two moments: a mean \mu that does not depend on time, and a covariance that depends on the two times only through their difference, \gamma_{t-s}=\mathrm{Cov}(X_s,X_t). The lag-zero value \gamma_0=\mathrm{Var}(X_t) is then constant as well. The autocorrelation is \gamma_h/\gamma_0, which lies in [-1,1]. For a Gaussian process, weak stationarity and strict stationarity are the same statement. Otherwise weak is weaker.
The variogram looks at the change rather than the level: \mathrm{Var}(X_s-X_t)=2(\gamma_0-\gamma_{t-s}) when the process is weakly stationary. People often care about how much tomorrow moves from today, not about the variance of today’s temperature in isolation. A variogram can exist when the covariance function does not, because the difference X_s-X_t can be well behaved when X_t itself is not.
Theorem 1 If \{X_t\} is weakly stationary with mean zero, then X_t=\sum_{j=0}^{\infty} a_j\varepsilon_{t-j}+\nu_t, where the \varepsilon_t are uncorrelated with mean zero and variance \sigma^2, the coefficients a_j are constants, \nu_t is linearly determined by its own past, and the innovations are uncorrelated with \nu_t. If the mean is \mu, add \mu to the right-hand side.
Every weakly stationary series is a moving average of infinite order, plus a deterministic piece. In practice the mean is subtracted first and put back at the end. The modelling choices are then which of those pieces we keep: is the dependence Markov, or seasonal, or both; is the season only in the mean; are the lags added or multiplied? A multiplicative series is often logged, and then the forecast is on the log scale. Jensen’s inequality says \mathrm{E}(\log X)\neq\log\mathrm{E}(X). Exponentiating a forecast of the log is not a forecast of the mean on the original scale.
White noise is the building block inside that sum: uncorrelated innovations with mean zero and constant variance. The next chapter is a forecasting application rather than a further derivation of white noise, AR(p), and MA(q).
An AR(1) with coefficient 1 is a random walk. Say what happens to the variance of X_t as t grows, and why that process cannot be weakly stationary.
For a weakly stationary series, show \mathrm{Var}(X_s-X_t)=2(\gamma_0-\gamma_{|t-s|}) from the definition of \gamma.
Why does a time series of length T, with a different mean at each time and a free covariance for every pair, run out of degrees of freedom? What does weak stationarity put in place of those free parameters?
Used in
- Taught in Dependence along a series
- Questions that use it (2) — C47 (answer) · C49 (answer)
- All 7 questions on Dependence along a series
See also
Stationary distribution · Stationarity and detailed balance · Limiting distribution · Strict stationarity · Time series · Unit root · Weak stationarity · Wold decomposition