Stationarity and differencing

What stationary means

A time series is weakly stationary when its mean and variance do not change with time and the covariance between $x_t$ and $x_{t+h}$ depends only on the lag $h$. Such a series keeps coming back to the same level and spreads out by the same amount, so what you estimate from one stretch of it is valid for the next. Daily returns are close to stationary. Prices are not: they wander, and the longer you watch them the farther they can be from where they started.

The standard model of a non-stationary price is the random walk $x_t=x_{t-1}+\varepsilon_t$: each step is a fresh shock, and the shocks add up, so the variance after $t$ steps is $t\sigma^{2}$ and grows without limit. Differencing turns it into the stationary series $\Delta x_t=\varepsilon_t$. A series that becomes stationary after one difference is called integrated of order one, written $I(1)$. If a series is instead a deterministic trend plus stationary noise, the right fix is to subtract the fitted trend, and differencing would only add noise. Do not difference more often than needed: a series that was already stationary becomes over-differenced, with a negative autocorrelation that was not there before.

• Weak stationarity: constant mean and variance, and an autocovariance that depends only on the lag.
• Differencing removes a stochastic trend; detrending removes a deterministic one.
• Spurious regression: two unrelated random walks regressed on each other often show a high $R^2$ and a significant slope.

Lab · what's graded

• The mean of the differences15 pts
• A log return15 pts
• How far a random walk wanders15 pts
• Spot the stationary series15 pts
• Match the vocabulary15 pts
• Checking a series before modelling10 pts
• Why a regression can mislead15 pts