What is a stationary process?

Question

How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?

Answer 1

A stationary process is one where the mean and variance don't change over time. This is technically "second order stationarity" or "weak stationarity", but it is also commonly the meaning when seen in literature.

In first order stationarity, the distribution of $(X_{t+1}, ..., X_{t+k})$ is the same as $(X_{1}, ..., X_{k})$ for all values of $(t, k)$.

You can see whether a series is stationary through it's autocorrelation function (ACF): $\rho_k = Corr(X_t, X_{t-k})$. When the ACF of the time series is slowly decreasing, this is an indication that the mean is not stationary; conversely, a stationary series should converge on zero quickly.

For instance, white noise is stationary, while a random walk is not. We can simulate these distributions easily in R (from a prior answer of mine):

op <- par(mfrow = c(2,2), mar = .5 + c(0,0,0,0))

N <- 500
# Simulate a Gaussian noise process
y1 <- rnorm(N)
# Turn it into integrated noise (a random walk)
y2 <- cumsum(y1)

plot(ts(y1), xlab="", ylab="", main="", axes=F); box()
plot(ts(y2), xlab="", ylab="", main="", axes=F); box()
acf(y1, xlab="", ylab="", main="", axes=F); box()
acf(y2, xlab="", ylab="", main="", axes=F); box()

par(op)

Which ends up looking somewhat like this:

If a time series varies over time, it is possible to make it stationary through a number of different techniques.

Answer 2

A process is defined here and is simply a collection of random variables indexed (in general) by time.

Otherwise I know the concept stated by Shane under the name of "weak stationarity", strong stationary processes are those that have probability laws that do not evolve through time.

More formally let $X_t$ be a given process, then let's call $P_X$ the probability law of the process, then for any given finite set of times $(t_1,...,t_n)$, and some borelian sets $(A_1,...,A_n)$ and any real $h>0$, $X_t$ is said (strongly) stationary iff :

$P_X(X_{t_1}\in A_1,...,X_{t_n}\in A_n)=P_X(X_{t_1+h}\in A_1,...,X_{t_n+h}\in A_n)$

For Gaussian process weak stationarity is equivalent to stationarity.

Regards

Answer 3

A very excellent discussion of stationarity as it relates to trading can be found in Sherry's (Sherrys'?) Mathematics of Technical Analysis (poorly organized, but very useful book). As he puts it, if the price changes of a stock, etc., are stationary over a time period, the underlying rules generating the price changes are effectively unchanged. The hypothesis is that a trading algorithm has little chance of working on a non-stationary series of price changes.

Below are some charts I've generated using the technique he describes in the book. Basically, you break your data set into two parts, histogram the prices changes, and construct cumulative distribution functions (cdfs). Then, do a Pearson's $\chi^2$ test to see if one cdf is significantly different than the other. You can also just look at the cdfs by eye. The case below is a borderline case. You can see the cdfs look a little bit different, but $\chi^2$ needs to be better than 14.07 to reach the 0.05 significance level in Pearson's test.