Time series problem - variance of w mean in an AR(1) process

#1
Hi, I'm learning about Time Series, and I'm with trouble in one exercise (attached). I don't even know how to start it. It's about proofing for a auto-regressive process, AR(1) that the variance of w mean (w is the difference of order d that transform an nonstationary process into a stationary one) is dependent on the sample variance and from the first estimated autocorrelation. The file attached is better explained. Someone can help me?
Thanks.
 

BGM

TS Contributor
#2
The right hand side of the equation do look like the estimated variance of the average of an AR(1) process.

Is it the full question? it will be much more helpful give you can give a more detailed and precise definition of \( \bar{W} \) in the left hand side.

There are other TS users like vinux who is very good in time series may help too :p
 
#3
Yes, this is the entire question. Sorry for this, I explain a little more in this post.
Suppose a time serie [TEX] X_{t} [/TEX], if its plot indicates that this is not a stationary time serie, so we get the first order difference, that is [TEX] X_{t} - X_{t - 1} = W_{t}[/TEX].Other form we can denote this is [TEX] W_{t} = \Delta^{1}X_{t} [/TEX], that is the first difference, and generally for [TEX] d [/TEX] differences [TEX] W_{t} = \Delta^{d}X_{t} [/TEX]. Now, we have a serie with [TEX] T - 1 = n [/TEX] cases and a new sample mean called [TEX] \bar{W} [/TEX].
The estimated autocorrelation, [TEX] r_{j} [/TEX] can be calculated as [TEX] \frac{c_{j}}{c_{0}} [/TEX] and
[TEX] c_{j} = \frac{1}{T}\sum_{t = 1}^{T - j}[(X_{t} - \bar{X})(X_{t}_{j} - \bar{X})], j = 0,1,..., T - 1 [/TEX]
OBS: In this part of the last equation [TEX] (X_{t}_{j} - \bar{X}) [/TEX], it's actually t + j, but for some reason latex doesn't compile it.
 
Last edited:

vinux

Dark Knight
#4
Following may be a hint to question:

Under the assumption, [tex] W_t [/tex] is (stationary) AR(1) process you could arrive at Var[tex] (\bar W) [/tex]

\( Var(\bar W) = Var( \frac{X_1+ \cdots+X_T}{T}) \)
\( = \frac{1}{T^2} \sum_{i=1}^{T} \sum_{j=1}^{T} Cov(X_i,X_j)\)

you know the covariance of every lag of AR(1) process. Simplify the above expression and substitute AR(1) covariance. You may need to take as limit on T for simplified expression.

EDIT: Thanks BGM to mentioning my name.
 

BGM

TS Contributor
#5
Another issue:

Not sure why OP emphasized on the differenced series:

\( W_t^{(d)} = \sum_{j=0}^d \binom {d} {j} (-1)^j X_{t-j} \)

Anyway if the sample mean is taken on this differenced series, you may use it, combining the hints given by vinux to compute the variance. (Please check! :))

Another point is that on the right hand side which sample autocorrelation are you using? with what lag? And why there are plug-in estimators (the sample variance and sample autocorrelation) while there are also the true population paramter \( \phi \) in the denominator? This is quite confusing.