1. Stationary processes
2. Stationarity
To obtain parsimony in a time series model you often assume some form of distributional invariance over time, or stationarity.
For an observed time series: Fluctuations appear random. However, the same type of random behavior often holds from one time period to the next.
For example, returns on stocks or changes in interest rates each have very different behavior from the previous year.
But their mean, standard deviation, and other statistical properties are often similar from one year to the next.
3. Weak stationarity - I
A process is weakly stationary if its mean, variance, and covariance are unchanged by time shifts. That is, there is a common constant mean mu and a variance sigma-squared for all times t.
4. Weak stationarity - II
And, the covariance between Y at time T and at time S only depends on how close T and S are, not the time indices T and S themselves.
For example, if Y is a stationary process, then the covariance between Y at times 2 and 5 is the same as that between times 7 and 10, since they are both three time units apart.
5. Stationarity: why?
Why focus on stationary models? A stationary process can be modeled with many fewer parameters.
You do not need a different mean for each Y_t; rather they all have a common mean mu.
You can estimate mu accurately by y-bar, the sample mean.
6. Stationarity: when?
When a time series is observed, a natural question is: Is it stationary? Specifically, is a stationary process model appropriate for this time series data?
Many financial time series do not exhibit stationarity. However:
The changes in the series are often approximately stationary.
Perhaps after applying a log() transformation.
A stationary series should show random oscillation around some fixed level, a phenomenon called mean-reversion.
7. Stationarity example
As an example, Inflation rates are shown in the top figure. They do not naturally return to a specific fixed level, as they are heavily influenced by the current federal monetary policy. If you instead consider the changes in the rate series as shown on the bottom, you now see very quick reversion to a common mean of zero, and there are no clear patterns over time in the change series.
8. Let's practice!
Great! Now, let's finish this chapter with some exercises!