Stationarity

Hi there,

I know this is quite basic but I'm wondering if someone could explain stationarity to me? Is it not just saying that the variables are identical?

Many thanks

Scotty

 

Not so much that they are identical, but identical in joint distribution.

For a process, Say {Xt : t>=0} to be stationary the joint distribution of X_t1,X_t2,X_t3,X_t4....,X_tn has to be identical to the joint distribution of X_(t1+k),X_(t2+k),...,X_(tn+k). This has to hold for all values of t1,t2,t3,...,tn, n and k (Ie. all possible lengths of sequence and all possible lags of that sequence). It's saying the statistical properties have to remain strictly constant over time. Notice how rigorous this requirement is.. Saying the distributions are equal for those joint random variables means all moments have to be equal. This includes skewness, kurtosis, 5th order moments etc. This is almost impossible to demonstrate in practice, usually it's sufficient to just have the first two moments remaining constant over time (Weak stationarity).

 

Could you perhaps give me an example of when there is stationarity, yet the random variables are not identical?

Thanks

Scotty

 

As i said it's extremely difficult to show that a process is strictly stationary, this definition is almost never demonstrable due to the volume of algebra that would be required. Usually we just show the 1st 2 moments are constant over time. For example a series of independent and identically distributed random variables Zt for t=1,2,3,4.... Where Zt~N(0,s^2) for all t is weak stationary, Since E[Zt]=0 for all t (Mean is constant) and Cov(Zi,Zj) = s^2 iff i=j. (Covariance depends only on the lag between i and j, since if i=!j then Cov(Zi,Zj)=0), so the 2nd moment is constant over time) Infact in this case since a normal random variable is entirely characterised by it's 1st 2 moments weak stationary implies strict stationary but in general this is not true

 

I appreciate the replies but I'm still not really getting the difference between random variables being identical and being stationary.

Is it just two different ways of describing the same thing:
Identical: When comparing 2 or more random variables
Stationary: When comparing 1 random variable changing over time

???

Many thanks

Scotty

 

A stochastic process is by definition a sequence of random variables. A normal distribution for instance is one random variable, say X~N(0,1). If we index a sequence of these random variables with time, lets use discrete indexing, say let t=1,2,3,4,...n. Then at each time we have a normal random variable X1,X2,X3,...,Xn. We would like to say something about how the statistical properties of this process vary over time (or if they are fixed, ie. stationary.) PDFs (or CDF's if you like) are what characterise random variables. For the process to be strictly stationary all possible joint distributions (PDF's) of all possible sequences of the process have to be identical at all possible lags. This would prove that the processes statistical properties have no tendency to change over time. This is extremely difficult to prove in most cases, so this definition of stationary is not usually used. A weaker condition exists called weak stationarity. This only requires that the first two moments of the process X remain constant over time, and assumes nothing about the joint distributions of the process. This is in general very easy to demonstrate, simply by showing the mean of the process is not time dependent, and the covariance of the process is also not time dependent (it only depends on the lag, not the times involved..

 

OK I think I understand stationarity and identical random variables separately.
I just think that they imply each other (only strict stationarityy): if the statistical properties do not change over time, then surely for every time step the random variables are identical...

Weak stationarity I can see a difference.

Scotty

 

Hello Scotty,

You can have two randon variables with different moments and defined by different probablity distributions (i.e. they are different) and yet both can be stationary.

Why do you think stationarity implies identical?

 

In the notes doesn't it say that all statistical properties do not change with time so for example the expectation and variance are constant.

Therefore as moments, hence MGF's are identical then surely the distributions are identical?...Right?...

Scotty