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thank you for helping out.
here i restate the problem, sorry i misspelled it in the
e-mail.
math problem:
the equation ...
Yt = a + b*Yt-1 + c*Xt-1
is a recursive difference equation.
need to rewrite the Yt part as a function of Xt-n by method of
recursive substitution so that the whole equation is a function
of Xt-n, a,b,c only.
see also my comments below.
> Hi Bilo,
>
> since the X-term is to remain unchanged we may consider this problem as an
> inhomogeneous linear recurrence in Y(t), which is usually solvable. However,
> different coefficients require different methods, and sometimes complex
> numbers will be involved.
i just need to rewrite the equation and don't require a full solution.
this is a garch model and garch literature refers to the fact that
to be able to derive a log likelihood function the equation needs to
be rewritten as a function of only past innovations Xt-n
and the recursions Yt-1 taken out of the equation.
>
> The closed form would contain t, though, so that we would get something like
> f(a,b,c,t,X(t-1)).
yes.
>
> If you can specify the coefficients a, b, c, and also indicate the initial
> condition for Y(0) or Y(1), I will try to work it out for you. No
> guarantees, but I can give it a try.
a,b,c are parameters that will have to be estimated.
the constrains for those are:
b > 0, c > 0, a > 0 b + c < 1 ( ie standard garch constraints )
Y(0) = 0
Y(1) = a, not sure here, but a + b*0 + c*0
i was able to find this reference of the transformation of
the garch model by recursive substitution, here is
the excerpt:
"...garch model is Vt^2 = a + b*Vt-1^2 + c*Et-1^2
if we recursively substitute for the lagged variance on the right
hand side, we can express the conditional variance as a
weighted average of all the lagged squared residuals Et only
removing the reference to Vt-1
Vt^2 = a / (1-b) + c*Sum[b^n-1 * Et-n^2 ], over n.
we see that garch(1,1) variance specification is analogous to
the sample variance, but that it down-weights more distant lagged squared errors..."
if this transformation is correct, this means that by analogy
Yt = a + b*Yt-1 + c*Xt-1 could be transformed by recursive substitution ( and
possibly approximation ) into ( i took ^2 out, don't know if this is ok or not )
Yt = a / (1 - b) + c*Sum[b^n-1 * Xt-n ], over n
so, i would like to know if this transformation is correct and how
they got to it.
this might save you some time since it might be the end result of the transformation.
i however fail to see how they got to that especially the term a / ( 1-b) and need help.
i would appreciate if you or someone could help in any way.
bilo.
ps. Yt = a + b*Yt-1 + c*Xt-1 equation could also be looked at as a recursive
filter, so if anyone knows filters help out please.
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