Theoretical moment question

[ac_1]

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

[TomDoan]

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).

[ac_1]

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).

[TomDoan]

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.

[TomDoan]

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.

[ac_1]

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).

[TomDoan]

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).