Stationary processes

Time Series Analysis in R

David S. Matteson

Associate Professor at Cornell University

Stationarity

  • Stationary models are parsimonious.
  • Stationary processes have distributional stability over time.

Observed time series:

  • Fluctuate randomly.
  • But behave similarly from one time period to the next.
Time Series Analysis in R

Weak stationarity - I

Weak stationary: mean, variance, covariance constant over time.

$Y_1, Y_2$, ...is a weakly stationary process if:

  • Mean $ \mu $ of $ Y_t$ is same (constant) for all $t$.
  • Variance $ \sigma ^2$ of $ Y_t$ is same (constant) for all $t$.
  • And….
Time Series Analysis in R

Weak stationarity - II

Covariance of $ Y_t$ and $ Y_s$ is same (constant) for all $ \vert t - s \vert = h$, for all$ h$.

$$ Cov(Y_2, Y_5) = Cov(Y_7, Y_{10})$$

since each pair is separated by three units of time.

Time Series Analysis in R

Stationarity: why?

A stationary process can be modeled with fewer parameters.

For example, we do not need a different expectation for each $ Y_t$; rather they all have a common expectation, $ \mu$.

  • Estimate $ \mu$ accurately by $ \bar y $.
Time Series Analysis in R

Stationarity: when?

Many financial time series do not exhibit stationarity, however:

  • The changes in the series are often approximately stationary.
  • A stationary series should show random oscillation around some fixed level; a phenomenon called mean-reversion.
Time Series Analysis in R

Stationarity example

Inflation rates and changes in inflation rates:

Time Series Analysis in R

Let's practice!

Time Series Analysis in R

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