What is a stationary time series? What are some examples?

Question

In my econometrics class, my teacher defined a stationary time series thus: "Loosely speaking, a time series is stationary if its stochasitc properties and its temporal dependence structure do not change over time." I am confused as to what some examples would be. Would temperature throughout the years be stationary, assuming that there isn't any trend? Does stationarity mean that the only movement in the data is attributed to random, white noise? What are some examples? I am at a loss for examples.

Answer 1

Perhaps a simple example from finance might help intuition. Let $R_t$ be the interest rate for period $t$ (note this is a random variable).

Numerous interest rate models (eg. Vasicek or Cox-Ingersoll-Ross) imply the rate is stationary process. If you earn the interest rate $R_t$ each period and start with $V_0$ dollars, then the quantity of dollars you have at time $t$ is given by:

$$V_t = V_0 \prod_{\tau=1}^t \left(1 + R_\tau \right)$$

The process $\left\{ V_t \right\}$ is NOT stationary. There's no unconditional mean or variance.

Other examples from econ and finance:

• Let $Y_t$ be aggregate output (i.e. GDP) of the economy at time $t$.

  • $y_t = \log(Y_t)$ is almost certainly not a stationary process.
  • The growth in log output (i.e. $y_t - y_{t-1}$) is typically treated as a stationary process

• Let $S_t$ be the price of overall market portfolio.

  • $s_t = \log(S_t)$ is almost certainly not a stationary process.
  • The log return $r_t = s_t - s_{t-1}$ of the market portfolio is typically treated as a stationary process.

A random walk or a Wiener process (the continuous time analogue to a random walk) are canonical examples of non-stationary processes. On the other hand, increments of a random walk or a Wiener process are stationary processes.

Temperature

As @kjetil points out, temperature is not a stationary process. For example, the distribution over temperatures in January is not the same as the distribution over temperatures in June. The joint distribution changes when shifted in time.

On the other hand, let $\mathbf{y}_t$ be a 12 by 1 vector for year $t$ where each entry of the vector denotes the average temperature for a month. You might be able to argue that $\mathbf{y}_t$ is a stationary process.

-- Update As @bright-star points out in the comments, this is the basic idea behind cyclostationarity. The temperature on a specific day as $t$ varies across years may be a stationary process.

Sunspots

One of the first time-series models was developed by Yule and Walker to model the 11-year sunspot cycle.

Let $y_t$ be the number of sunspots in year $t$. They modeled the number of sunspots in a year as a stationary process using the AR(2) model:

$$ y_t = a + b y_{t-1} + c y_{t-2} + \epsilon_t $$

A stationary process can have patterns, cycles, etc...

Be aware of the two common definitions of stationarity.

Somewhat loosely:

• A process is strictly stationary if the joint distribution is time invariant.
• A process is covariance stationary if the unconditional expectation and the autocovariance exist and do not vary over time.

(Perhaps an obscure, technical remark, but strict stationarity does not imply covariance stationarity and covariance stationarity does not imply strict stationarity.)

Answer 2

A stationary process' distribution does not change over time. An intuitive example: you flip a coin. 50% heads, regardless of whether you flip it today or tomorrow or next year.

A more complex example: by the efficient market hypothesis, excess stock returns should always fluctuate around zero. There is no trend; as soon as they can predict returns, traders exploit the trend until it vanishes. So no matter when you observed excess returns, it would still be distributed WN(0,$\sigma$).

As you said, it would randomly vary according to a white noise process.