Saya perlu mendapatkan ekspresi analitik untuk fungsi autocovariance dari proses ARMA (2,1) yang dilambangkan oleh:
Jadi, saya tahu itu:
jadi saya bisa menulis:
kemudian, untuk mendapatkan versi analitik dari fungsi autocovariance, saya perlu mengganti nilai - 0, 1, 2 ... sampai saya mendapatkan rekursi yang valid untuk semua lebih besar dari beberapa integer.
Karenanya, saya mengganti dan menyelesaikannya untuk mendapatkan:
sekarang saya dapat menyederhanakan dua pertama istilah ini, dan kemudian menggantikan seperti sebelumnya:
maka saya gandakan delapan istilah, yaitu:
So, I am left needing to resolve the four remaining terms. I want to use the same logic for lines 1, 2, 5 and 6 as I used on lines 4 and 7 - for example for line 1:
because .
Similarly for lines 2, 5 and 6. But I have a model solution that suggests the expression for simplifies to:
This suggests my simplification as described above would miss the term with the coefficient - which under my logic should be 0. Is my logic at fault, or is the model solution I found incorrect?
The worked solution also suggest that "analogously" can be found as:
and for :
I hope the question is clear. Any assistance will be much appreciated. Thank you in advance.
This is a question related to my research, and is not in preparation for any exam or coursework.
sumber
OK. So the process of writing the post actually pointed me to the solution.
Consider the Expectation terms 1, 2, 5 and 6 from above that I thought should be 0.
Immediately for terms 5 -E[ϵtyt−1] - and 6 - E[ϵtyt−2] : these terms are definitely zero, because yt−1 and yt−2 are independent of ϵt and E[ϵt]=0 .
However, terms 1 and 2 look as though the Expectation is of two correlated variables. So, consider the expressions foryt−1 and yt−2 thus:
And recall term 1 -ϕ1θ1E[ϵt−1yt−1] . If we multiply both sides of the expression for yt−1 by ϵt−1 and then take Expectations, it is clear that all terms on the right hand side except the last become zero (because the values of yt−2 , yt−3 , and ϵt−2 are independent of ϵt−1 and E[ϵt−1]=0 ) to give:
So term 1 becomes+ϕ1θ1σ2ϵ . For term 2, it should be clear that, by the same logic, all terms are zero.
Hence the original model answer was correct.
However, if anyone can suggest an alternative way to obtain a general (even if messy) solution, I would be very pleased to hear it!
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