Thanks.

- Thread starter filipeferminiano
- Start date
- Tags auto-regressive process time series

Thanks.

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

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.

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.

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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.

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.