Show that sum of these two Random Variables is conditionally normal distributed (from IGARCH model)
$begingroup$
According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed.
$σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(epsilon_{t-1}^2 - 1)$
$r_t = σ_t * epsilon_t$
$epsilon_t ∼ N(0,1)$
According to the book $r_{t+1} + r_{t+2}$ conditional on $epsilon_{t}$ & $σ_t$ is normally distributed. The $r_{t+1}$ term makes sense to be conditional normally distributed given $epsilon_{t}$ & $σ_t$ since then the $σ_{t+1}$ term in $r_{t+1} = σ_{t+1}* epsilon_{t+1}$ is known, and therefore $r_{t+1}$ is just a normal random variable.
But for $r_{t+2}$, the $σ_{t+2}^2 =σ_{t+1}^2 + (1 − α)σ_{t+1}^2(epsilon_{t+1}^2 - 1)$ is not known and is still a random variable. So I don't see how $r_{t+2}$ is conditionally normally distributed.
Any help would be greatly appreciated. Thanks!
(I posted a similar question Conditional distribution at time t+1 given information at time t is normally distributed, showing that conditional distribution of sum is also normal
before but got no responses, so I shortened the question down as much as I could)
normal-distribution conditional-probability time-series
asked Jan 18 at 3:58
SladeSlade
3110
$endgroup$
add a comment |
$begingroup$
According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed.
$σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(epsilon_{t-1}^2 - 1)$
$r_t = σ_t * epsilon_t$
$epsilon_t ∼ N(0,1)$
According to the book $r_{t+1} + r_{t+2}$ conditional on $epsilon_{t}$ & $σ_t$ is normally distributed. The $r_{t+1}$ term makes sense to be conditional normally distributed given $epsilon_{t}$ & $σ_t$ since then the $σ_{t+1}$ term in $r_{t+1} = σ_{t+1}* epsilon_{t+1}$ is known, and therefore $r_{t+1}$ is just a normal random variable.
But for $r_{t+2}$, the $σ_{t+2}^2 =σ_{t+1}^2 + (1 − α)σ_{t+1}^2(epsilon_{t+1}^2 - 1)$ is not known and is still a random variable. So I don't see how $r_{t+2}$ is conditionally normally distributed.
Any help would be greatly appreciated. Thanks!
(I posted a similar question Conditional distribution at time t+1 given information at time t is normally distributed, showing that conditional distribution of sum is also normal
before but got no responses, so I shortened the question down as much as I could)
normal-distribution conditional-probability time-series
asked Jan 18 at 3:58
SladeSlade
3110
$endgroup$
add a comment |
$begingroup$
According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed.
$σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(epsilon_{t-1}^2 - 1)$
$r_t = σ_t * epsilon_t$
$epsilon_t ∼ N(0,1)$
According to the book $r_{t+1} + r_{t+2}$ conditional on $epsilon_{t}$ & $σ_t$ is normally distributed. The $r_{t+1}$ term makes sense to be conditional normally distributed given $epsilon_{t}$ & $σ_t$ since then the $σ_{t+1}$ term in $r_{t+1} = σ_{t+1}* epsilon_{t+1}$ is known, and therefore $r_{t+1}$ is just a normal random variable.
But for $r_{t+2}$, the $σ_{t+2}^2 =σ_{t+1}^2 + (1 − α)σ_{t+1}^2(epsilon_{t+1}^2 - 1)$ is not known and is still a random variable. So I don't see how $r_{t+2}$ is conditionally normally distributed.
Any help would be greatly appreciated. Thanks!
(I posted a similar question Conditional distribution at time t+1 given information at time t is normally distributed, showing that conditional distribution of sum is also normal
before but got no responses, so I shortened the question down as much as I could)
normal-distribution conditional-probability time-series
asked Jan 18 at 3:58
SladeSlade
3110
$endgroup$
According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed.
$σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(epsilon_{t-1}^2 - 1)$
$r_t = σ_t * epsilon_t$
$epsilon_t ∼ N(0,1)$
According to the book $r_{t+1} + r_{t+2}$ conditional on $epsilon_{t}$ & $σ_t$ is normally distributed. The $r_{t+1}$ term makes sense to be conditional normally distributed given $epsilon_{t}$ & $σ_t$ since then the $σ_{t+1}$ term in $r_{t+1} = σ_{t+1}* epsilon_{t+1}$ is known, and therefore $r_{t+1}$ is just a normal random variable.
But for $r_{t+2}$, the $σ_{t+2}^2 =σ_{t+1}^2 + (1 − α)σ_{t+1}^2(epsilon_{t+1}^2 - 1)$ is not known and is still a random variable. So I don't see how $r_{t+2}$ is conditionally normally distributed.
Any help would be greatly appreciated. Thanks!
(I posted a similar question Conditional distribution at time t+1 given information at time t is normally distributed, showing that conditional distribution of sum is also normal
before but got no responses, so I shortened the question down as much as I could)
normal-distribution conditional-probability time-series
normal-distribution conditional-probability time-series
asked Jan 18 at 3:58
SladeSlade
3110
asked Jan 18 at 3:58
SladeSlade
3110
asked Jan 18 at 3:58
SladeSlade
3110
asked Jan 18 at 3:58
SladeSlade
3110
asked Jan 18 at 3:58
SladeSlade
3110
3110
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077817%2fshow-that-sum-of-these-two-random-variables-is-conditionally-normal-distributed%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077817%2fshow-that-sum-of-these-two-random-variables-is-conditionally-normal-distributed%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
