Outline what a series must satisfy in order to be covariance stationary.
The idea
A series is covariance stationary when its first two moments do not depend on time. Three conditions must hold. First, the mean is constant, with E(yt) = μ for every date t. Second, the variance is constant and finite. Third, the covariance between yt and yt−k depends only on the displacement k, never on the date t. Nothing is said about the shape of the distribution, so normality is not required, and the conditional variance may still change as long as the unconditional one is constant. Typical violators are a random walk (variance grows with t), a series with a deterministic trend (mean moves) and an explosive process with an autoregressive coefficient above one in absolute value. Covariance stationarity is what lets sample averages estimate fixed parameters, so exams ask which of the three conditions a given series breaks.
A practice question
Which set of conditions must a time series satisfy to be covariance stationary?
- Normal distribution at every date and independence between observationsWhy not: Describes independent Gaussian noise. Covariance stationarity restricts only the first two moments and allows dependence between observations.
- Constant mean, constant finite variance, and autocovariance that depends only on the lag - correct
- Zero mean, unit variance, and zero autocovariance at every nonzero lagWhy not: Describes standardised white noise, a special case. The general requirement is a constant mean and variance with lag-dependent autocovariance.
- Constant mean, a variance that may drift, and a lag-one autocovariance that is constantWhy not: Lets the variance change over time. The variance must be constant and finite as well as the autocovariances depending only on the lag.
A covariance-stationary series has a time-invariant mean, a constant finite variance and autocovariances gamma(k) that depend only on the lag k. No distributional shape is required, and dependence over time is allowed.
A harder one
A risk manager estimates the lag-1 autocovariance of a monthly credit spread series separately for 2005 to 2012 and 2013 to 2020. She obtains 0.80 in the first sub-period and 0.20 in the second, while the mean and variance are the same in both. Which covariance-stationarity requirement does this evidence call into question?
- Autocovariance depending only on the lag, not on calendar time - correct
- Constant meanWhy not: The mean is stated to be the same in both sub-periods, so this condition is not contradicted by the evidence.
- Finite varianceWhy not: The variance is the same and finite in both sub-periods; only the dependence structure changes.
- Normality of the distributionWhy not: Normality is not a requirement of covariance stationarity at all, so it cannot be the condition in question.
The lag-1 autocovariance should be the same whichever window is used. A shift from 0.80 to 0.20 shows that gamma(1) depends on the date, violating the third condition, while mean and variance are unchanged.
The hardest tier
Consider four processes, each with e_t white noise: (i) y_t = 0.5 y_(t-1) + e_t; (ii) y_t = y_(t-1) + e_t; (iii) y_t = 0.3 + e_t + 0.9 e_(t-1); (iv) y_t = 1.1 y_(t-1) + e_t. Which processes are covariance stationary?
- (i), (iii) and (iv)Why not: Treats the explosive process (iv), with phi = 1.1, as stationary; stationarity needs the absolute AR coefficient below one.
- (i), (ii) and (iii)Why not: Treats the unit-root random walk (ii) as stationary; its variance grows with t.
- (i) and (iii) - correct
- (i) onlyWhy not: Rejects the MA(1) in (iii) wrongly; a finite-order MA is always covariance stationary, whatever the size of theta.
AR(1) is stationary when |phi| < 1, so (i) qualifies, (ii) has a unit root and (iv) is explosive. Any finite MA, such as (iii) with theta = 0.9, is stationary because it is a finite sum of white noise.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Stationary Time Series
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