Ich muss analytische Ausdrücke für die Autokovarianzfunktion eines ARMA (2,1) -Prozesses ableiten , die bezeichnet werden durch:
Also, ich weiß das:
damit ich schreiben kann:
Um dann die analytische Version der Autokovarianzfunktion abzuleiten, muss ich Werte von - 0, 1, 2 ... einsetzen, bis ich eine Rekursion erhalte, die für alle größer als eine ganze Zahl gültig ist .
Daher setze ich und arbeite dies durch, um zu erhalten:
Jetzt kann ich die ersten beiden Begriffe vereinfachen und dann wie bisher durch ersetzen :
dann multipliziere ich die acht Terme, die sind:
Daher muss ich die vier verbleibenden Bedingungen noch klären. Ich möchte für die Zeilen 1, 2, 5 und 6 dieselbe Logik verwenden wie für die Zeilen 4 und 7 - zum Beispiel für Zeile 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.
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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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