Econometrics II

For weakly stationary processes we defined the autcovariance and autocorrelation functions that codify part of the dependence inherent in the process. Given this such a process is termed regular iff it is comprised of random variables that are asymptotically uncorrelated. Furthermore,  a weakly stationary process is termed short memory iff its autocovariance function is absolutely summable, a condition that implies asymptotic uncorrelateness with sufficiently fast rate, and thereby implies regularity.

We examined the definition of a white noise process as a canonical example of weak stationarity. By examples we showed that such processes exist.

We occupied ourselves with comparisons, via examples in the framework of independence, of the notions of stationarity and weak stationarity. Hence homogeneity and enough moment existence conditions imply that both properties hold, heterogeneity and enough moment conditions imply that the first does not, yet the second holds, homogeneity with insufficient moment existence conditions imply that the first holds while the second does not, and heterogeneity with insufficient moment existence conditions imply that neither holds. Later in the course we will examine more complex examples involving dependence.

We also occupied ourselves with the question of invariance of strict and weak stationarity w.r.t. transformations. We saw that the former is invariant w.r.t. to measurable pointwise transformations while the latter is generally not since for example those might destroy moment existence conditions. Both are preserved by transformations w.r.t. powers of the lag operator.

We begun to examine the first method-in this course-of the construction of a process via some transformation of a given one. We defined the concept of a causal linear process w.r.t. to a white noise process and a sequence of absolutely summable coefficients. After some preparation involving series of such sequences we have proven that this process is weakly stationary, given some hidden to us results that imply that this process is well defined and in this framework the expectation with the series operator commute.

Notes for the above can be found here.

Exercise: Prove or disprove-If the process is Gaussian, then it is strictly stationary iff it is weakly stationary.