Time Series, Stationarity!

[tharandeep]

Hi,

Im having difficulty on 2 question on time series:

(i) y(t) = 1.5 + 1.3y(t-1) - 0.3y(t-2) + u(t) where u(t) is white noise.

In this question I don't know what to do with the characteristic equation because there is the 1.5 element on its own. Would I use the characteristic equation to find out whether this is stationary, non stationary, or invertible?

Also, another question is

(ii) y(t) = -0.1y(t-1) + 0.42y(t-2) + u(t) + 1.2u(t-1)

Here I know this is an ARMA process and so am I correct in assuming I only need to deal with the AR part of it? If this is the case, then would the characteristic equation be:

1 + 0.1z - 0.42z^2 = 0 ???

Please help. I know this is probably very easy but I'm having some difficulty on Time Series at the moment.

Thanks!

 

[Hamilton]

emmm

in (i) the 1.5 part is going to cause it to be non-stationary , mind you it doesn't change the characteristic equation at all. just draw this and you'll see
Yt+1 = Yt + 1.5 where Y0 = 0 so Y1 will be 1.5 and Y2 will be 3 etc

in (ii) thats the correct characteristic equation and you only need to worry about the AR part to see if its stationary but if ya after auto covariances and auto correlations then the whole equation needs to be thought about.

 

[mugono]

Hi tharandeep,

To see if y(t) = 1.5 + 1.3y(t-1) - 0.3y(t-2) + u(t) where u(t) is white noise
is STATIONARY, you ALWAYS look at the characteristic equation of y(t)'s.

Don't be put off by the 1.5 on its own, this is just mu from the notes.

In the notes this is cleverly dealt with by (in the case of the equation above writing)

(1 - 1.3B + 0.3B^2) (Y(t) - 1.5) = u(t)
where B is the backshift operator. (Make sure you know why this is true

The 1.5 DOESN'T mean that the process in non-stationary. To answer that you need to evaluate the roots of the above characteristic equation!!!

Regarding part ii of your question:

YOU ARE RIGHT
The reason why you don't have to look at the white noise part is because it is always stationary regardless of the value of its parameters.

Hope this helps

Good luck

 

[Hamilton]

oops

dont know what I was thinking there , doh

 

[tharandeep]

Time series

Thank you both! I've finally got the hang of Time Series and it's all making sense.