Stationarity Flashcards
What are the conditions for stationarity?
Expectation and variance of time series is independent of t and the autocovariance function just depend on the time difference r-s
Is white noise stationary?
Yes by definition - if white noise are normally distributed its also strictly stationary
Whats the difference between strict and weak stationarity
Strict stationarity is too strong in practice for most applications as imposes conditions on the distributions. Weak stationarity imposes conditions only on the first two moments of a series
What does strict stationarity mean
A time series {Xt} is strict stationary is the joint distributions of (Xt1, Xt2,… Xtk) and (Xt1+n, Xt2+n,… Xtk+n are the same for all ti integers. This Means that the probabilistic behaviour of all realisations of the time series is identical to the time shifted set.)
Is a random walk stationary?
No - expectation depends on t
How can we relate strict stationarity to stationarity
If a time series {Xt} is strictly stationary and the expectation of Xt squared is finite then {Xt} is also stationary. The converse is not true without further conditions!
Is there any case where stationarity implies strict stationarity?
Yes if the time series is Gasussian (all finite distributions are gaussian) - exception
What is a good quality of the sample mean estimator?
It is an unbiased estimator for the mean function (constahnt in case of stationary time series)
how do we usually conduct time series analysis with data?
Use sampled data , maybe only one realisation alot fo the time and we assume stationarity of the sampel data and use averages over the single realisation to estimate population means and variances etc
What do significant peaks in the sample autocorrelation function ellude to?
If they are outside of the interval of 1.96 * 1/root n for large n then might provide evidence of a significant autocorrelation at lag h.
