Question

1. Explain whether time series that follows a random walk with drift is weakly dependent.

Answer 1

Ans) Random walks is basically an auto regressive process where we can consider p = 1. Thus, a time series generated by this process cannot be assumed to be weakly dependent.

Function for this is, yt = yt - 1 + e1 =y0 +et + et +1 +.....+e1

et is an I.i.d sequence with mean 0 and variance _{t^{2 , assume y0 is known.}}

The expected value of yt is always y0, but does not depend on t,

Var yt = Var (et +et-1+....+et) =t , so it increases with t.

Thus we can say that a random walk is highly persistent.

Now on the other hand a random walk with drift is an example of a highly persistent series that is also trending

yt = a0 + yt -1 + et = a0+et+ et - 1+....+et + y0.

Hence it can also have a unit root process with drift if et is weakly dependent.

Thus moving ahead with that point, a random walk with drift would signal a linear time dependent component that changes with time.