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

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.

• under new US administration the temperature will be a stationary process. Dec 12, 2016 at 16:20

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.)

• Daily (or monthly) temperature will most often show cyclic behaviour over the year, so will not be stationary even when there is no long-term trend. Dec 12, 2016 at 4:13
• @kjetilbhalvorsen Thanks for the correction; my original line there was completely wrong. Dec 12, 2016 at 4:58
• Note that cyclostationarity is also amenable to modeling. Dec 12, 2016 at 7:17
• The interest rate is typically modeled as a stationary process. Really? I had a different opinion. Would you have a reference? (Of course, finding a good reference might not be easy for such a general statement.) Also, very good that you included the remark at the end. The terminology might be misleading, so the remark is really due there. Dec 14, 2016 at 20:09
• @RichardHardy I narrowed it to discuss only Vasicek and Cox-Ingersoll-Ross. Dec 14, 2016 at 21:55

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.

• The efficient market hypothesis does not imply excess returns should be zero in expectation. The efficient market hypothesis is that market prices reflect all available information. It is perfectly consistent with the efficient market hypothesis for expected returns to vary cross-sectionally across assets if the higher average returns are compensation for macro-economic risk. Eg. the risk premia for equities may be different than the risk premia for bonds etc... Dec 12, 2016 at 9:34
• Ah, you're right, I meant to say "stock returns". Thank you for the correction! Dec 12, 2016 at 16:07