AR(1) Model - Jorion CD Question

[notjusttp]

10. Suppose X follows an AR(1) model: X(t) = 0.1 + 0.8*X(t-1) + e(t), where, E(e(t)) = 0. What is the long term mean of X?

B is correct. For a AR(1) model of the form: X(t) = alpha + beta X(t-1) + e(t), where E[e(t)] = 0, the long term mean of X is alpha / (1-beta).
For this problem, the long term mean of X is 0.5000 = 0.1 / (1.0 - 0.8).


My Doubt.

1) What is this AR(1) model?
2) It is covered in which topic?

Thanks and rgds
Amit

 

[David Harper CFA FRM]

Hi Amit,

It refers to Linda Allen p70: "Next period’s expectations are a weighted sum of today’s value, Xt, and the long run mean a/(1 - b). Here b is the key parameter, often termed “the speed of reversion” parameter. If b = 1 then the process is a random walk – a nonstationary process with an undefined (infinite) long run mean, and, therefore, next period’s expected value is equal to today’s value. If b < 1 then the process is mean reverting."

However, current testability is low/non existent.

Hull is now more relevant, so more relevant would be to find the long-run mean variance of a GARCH (1,1) process; i.e., LR variance = omega/(1-alpha-beta)

David

 

[notjusttp]

Hi David,

thanks a lot for the detailed explanation...cheers..Amit