Stationarity

Describing Stationarity

• An assumption of most time-series models, such as AR, MA, ARMA, and ARCH, is they are stationary

• A model is stationary if it satisfies the following criteria:

  • Mean($s_{t}$) is constant
  • Var($s_{t}$) is constant with time (i.e. no heteroskedasticity)
  • There's no seasonality in the data
• The criteria for stationarity is more general compared to the criteria for white noise models
• Therefore, if a model is a white noise model, then it is also stationary
• However, just because a model is stationary, doesn't necessarily imply the model is a white noise model

Testing for Stationarity

1. We can graph the residuals to visually detect stationarity

2. We can perform global and local checks of the criteria across the time series data

   • Specifically, we would perform checks to ensure the mean is constant, variance is constant, and no seasonality
   • Global checks imply checking these criteria across the entire time series dataset
   • Local checks imply creating slices of the time series data and testing the criteria on these slices
   • A method of performing local checks could include running a rolling window across the data set and testing the criteria on each iteration
3. Perform a true statistical test, such as the augmented Dickey-Fuller test (i.e. ADF test)

Ensuring Stationarity

• We can find a transformation of some non-stationary time-series model, where the transformation maps a non-stationary time-series model to a stationary time-series model
• Then, we can work with our transformed, stationary time-series model to satisfy the time-series models' assumption of stationarity
• For example, a linear time-series model is a non-stationary model represented as the following:

$$y_{t} = \beta_{0} + \beta_{1}t + \epsilon_{t}$$

• However, we can transform the linear time-series model to a stationary model:

$$z_{t} = y_{t} - y_{t-1}$$

References

• Stationarity Video