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Alan
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Need a proof (GARCH related)

November 10th, 2018, 5:45 pm

Let [$]Z[$] be a standard normal variate. I would like to see a proof that

(*) [$]E[\log (a Z^2 +b)] < 0[$] for all [$]\{(a,b): a > 0, b>0, a +b < 1\}[$].

In other words,

[$] \int_{-\infty}^{\infty} \log (a z^2 + b) \, e^{-z^2/2} \frac{dz}{\sqrt{2 \pi}} < 0[$] under the same conditions.

The connection with GARCH is that this condition is sufficient (according to others, and presumably also with [$]\omega > 0[$]) to establish that the GARCH(1,1) process

[$] \sigma^2_t = \omega + \sigma_{t-1}^2 (a Z^2_t + b)[$]

has a stationary distribution for [$]\sigma^2_t[$].

Numerically, I'm already convinced -- both in the validity of (*) and the existence of the stationary distribution under the given conditions. But, I would like to see a proof of the inequality (*).

Thanks!
 
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bearish
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Re: Need a proof (GARCH related)

November 10th, 2018, 6:35 pm

Let [$]Z[$] be a standard normal variate. I would like to see a proof that

(*) [$]E[\log (a Z^2 +b)] < 0[$] for all [$]\{(a,b): a > 0, b>0, a +b < 1\}[$].

In other words,

[$] \int_{-\infty}^{\infty} \log (a z^2 + b) \, e^{-z^2/2} \frac{dz}{\sqrt{2 \pi}} < 0[$] under the same conditions.

The connection with GARCH is that this condition is sufficient (according to others, and presumably also with [$]\omega > 0[$]) to establish that the GARCH(1,1) process

[$] \sigma^2_t = \omega + \sigma_{t-1}^2 (a Z^2_t + b)[$]

has a stationary distribution for [$]\sigma^2_t[$].

Numerically, I'm already convinced -- both in the validity of (*) and the existence of the stationary distribution under the given conditions. But, I would like to see a proof of the inequality (*).

Thanks!
Since [$]E[(a Z^2 +b)] =a+b[$] and [$]\log (a+b)<0 [$], it follows from Jensen's inequality and the concavity of the log function. 
 
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ppauper
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Re: Need a proof (GARCH related)

November 10th, 2018, 7:24 pm

[$] \int_{-\infty}^{\infty} \log (a z^2 + b) \, e^{-z^2/2} \frac{dz}{\sqrt{2 \pi}} < 0[$]
with a little playing around (writing [$]b/a=k^2[$]) maple seems to think that integral is
[$]\pi\,{\rm erfi}\left(\sqrt{\frac{b}{2a}}\right)+\ln\frac{a}{2}-\gamma-\frac{b}{a}\,{\rm hypergeom}\left([1, 1],[2, \frac{3}{2}],\frac{b}{2a}\right)[$]
with erfi the complex error function
 
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Alan
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Re: Need a proof (GARCH related)

November 10th, 2018, 7:44 pm

@bearish,  Thank you: short and sweet!

@ppauper, Thanks, also. Yes, I got this in Mathematica, too, and had plotted it earlier in [$]a+b<1[$], which was why I said that I was numerically convinced.

OK, problem solved -- Thanks again, guys!
 
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ppauper
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Re: Need a proof (GARCH related)

November 11th, 2018, 6:51 am

we have [$]0<a<1[$] and [$]0<b<1-a[$]
call [$]I(a,b)=\int_{-\infty}^{\infty} \log (a z^2 + b) \, e^{-z^2/2} \frac{dz}{\sqrt{2 \pi}}[$]
[$]I=0[$] when [$]a=0[$] and [$]b=1[$] as the integrand is zero
[$]\frac{\partial I}{\partial b}=\sqrt{\frac{\pi}{2ab}}\exp\left(\frac{b}{2a}\right)\,{\rm erfc}\left(\sqrt{\frac{b}{2a}}\right)>0[$]
for given [$]a[$], we will have [$]I(a,b)<I(a,1-a)=I_{m}(a)[$]
[$]\frac{\partial I_{m}}{\partial a}=\frac{1}{a}
-\frac{\pi^{1/2}}{2^{1/2}a^{3/2}\left(1-a\right)^{1/2}}\exp\left[\frac{1-a}{2a}\right]\,{\rm erfc}\left[\sqrt{\frac{1-a}{2a}}\right]<0[$]
which means that for [$]a[$] and [$]b[$] in your range, [$]I(a,b)<I_{m}(a)<I_{m}(0)=0[$]
 
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Cuchulainn
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Re: Need a proof (GARCH related)

November 12th, 2018, 8:43 am

Cool.
 
jackwalter
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Re: Need a proof (GARCH related)

November 13th, 2018, 2:23 pm

Thank you so much for delivering this information! The created thread is indeed useful. I'll keep it under my control.
 
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Cuchulainn
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Re: Need a proof (GARCH related)

November 13th, 2018, 4:21 pm

Thank you so much for delivering this information! The created thread is indeed useful. I'll keep it under my control.
I'm a bit pushed for time. Can you help me with my thesis on Kalman filters?

http://custom-written-papers.org/blog/c ... rch-papers

Custom-written-papers.org services are carried our by a qualified team of writers. Our goal is to save your time and provide you with work of high quality. All of the custom written research papers, custom written essays and other assignments we deliver are edited by experienced proofreader. While writing custom papers or custom essays, our specialists use diverse kinds of sources including foreign literature, monographs and primary research. Custom written term papers are formatted in accordance to APA, MLA, Chicago, Turabian or other specific style requested by you. Our clients may order paper writing help of any complexity and be sure that our specialists follow the instructions. Custom-written-papers.org specialists do not set aside any of your teacher’s requirements. We treat our clients with understanding and you get a custom written essay without the jitters.

Impressive.
 
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Paul
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Re: Need a proof (GARCH related)

November 13th, 2018, 5:01 pm

Their blurb needs proof reading.
 
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Cuchulainn
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Re: Need a proof (GARCH related)

November 13th, 2018, 7:31 pm

Their blurb needs proof reading.
I always check the authenticity by asking for the C++ code. It's rampant. It makes me mad.

I would be a good domain expert to interview for a fraud detection system using Markov (HMM). There are good and bad state machines.

It our day Paul I suppose kids were more idealistic. People did maths for fun.
 
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ppauper
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Re: Need a proof (GARCH related)

November 13th, 2018, 8:56 pm

the internet has made cheating global.
When I was in school. people said that one of the frats had a filing cabinet full of old essays from past members that current members could reuse.
Now you just go on the internet
 
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Cuchulainn
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Re: Need a proof (GARCH related)

November 14th, 2018, 5:20 pm

the internet has made cheating global.
When I was in school. people said that one of the frats had a filing cabinet full of old essays from past members that current members could reuse.
Now you just go on the internet
Any skeletons?
 
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bearish
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Re: Need a proof (GARCH related)

November 15th, 2018, 3:19 am

Since nobody else has brought it up, there is one claim in the advertisement that was so rudely dumped into this nice thread that seems just wrong: "Custom-written-papers.org specialists do not set aside any of your teacher’s requirements." I am pretty sure that most teacher's requirements include a do-it-yourself clause. In fact, it's probably the most important one. 
 
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ppauper
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Re: Need a proof (GARCH related)

November 15th, 2018, 7:44 am

the internet has made cheating global.
When I was in school. people said that one of the frats had a filing cabinet full of old essays from past members that current members could reuse.
Now you just go on the internet
Any skeletons?
I was never a brother, just went to a few of their parties which were open to anyone prepared to pay a couple of bucks to get in.
It's a national frat so no doubt the essay bank is a national policy, and according to wiki, a number of notable people were (are? do people join for life?) members.
Including a number of nobel laureates, so I'm sure there's a good collection of essays in various filing cabinets across the land. No doubt everything is online now.
On the flip side, a number congressmen/senators/cabinet secretaries from both parties were/are members, as is at least one president from many many many years ago.