The RATS Software Forum

Hi Tom,

For the following process:

y(t) = mu + phi1*y(t-1) + beta*trend + eps(t)

y = series
mu = constant
phi1 = coefficient
beta = coefficient
trend = time-trend
eps = white noise process

I can calculate the E[y(t)] and Var(y(t)) for phi1=1, using recursive back-substitution, but having problems regarding E[y(t)] and Var[y(t)] for |phi1|<1, I get

E[y(t)] = mu*(1 + phi1 + phi1^2 + ... + phi1^(n-1)) +
(phi1^(n))*E[y(0)] +
E[beta*(n) + phi1*beta*(n-1) + phi1^2*beta*(n-2) + ... + phi1^(n-1)*beta*(n)]

As E[eps(t)]=0 by definition, and as |phi1|<1 with n->infinity, the middle-term cancels to zero.

So the first term is

E[y(t)] = mu/(1-phi1) via infinite power series result: from i=0 ∑ to i=(n-1) phi^i = 1/(1-phi1) as per 'textbook(s)'.

Same reason for the beta term and therefore,

E[y(t)] = (mu + beta*(t))/(1-phi1)

Correct?

What about Var[y(t)]?

Regards,
Amarjit

mu and beta*t are known and so don't affect the variance. So you just have the AR part, which is well-documented. sigma^2/(1-phi^2).

TomDoan wrote: ↑Mon Aug 26, 2024 7:17 am mu and beta*t are known and so don't affect the variance. So you just have the AR part, which is well-documented. sigma^2/(1-phi^2).

Thanks, please can you provide textbook reference(s).

For the variance of an AR(1) process? Effectively any time series book you could think of. The deterministic terms drop out in computing the variance.

I'm not sure what it is that you're trying to do, but the calculations of the mean are much more complicated than that---you need the sum of (phi^n)(t-n) where you're forgetting the interaction between the phi^n and n. The mean of that process isn't a pleasant expression in the underlying parameters, but it's a linear time trend. Which is why people who need to work with something like that are more likely to write the process as

tau(t)=a+bt
z(t)=phi*z(t-1)+eps(t)
y(t)=tau(t)+z(t)

i.e. linear trend plus AR(1) noise. The two are equivalent.

The moments of the model

y(t) = mu + phi1*y(t-1) + beta*trend + eps(t)

are not in any textbook I have seen.

Yes, agreed they are for AR(p), MA(q).

And (for the third time), the deterministic parts drop out of the variance, so the variance is just that for a garden-variety AR(1).

Calculating the (time-varying) mean of the process out of this form requires being able to work with power series. Which is why you probably can't find it. And again, there is a simpler equivalent form of a trend with noise. (Which, by the way, is the form that is used by BOXJENK).