Autocovariance function of $MA(q)$
I'm trying to compute the autocovariance function $gamma_{X}$, where: $$X_t = Z_t + sum_{k=1}^q theta_k Z_{t-k} =: (F_{{alpha}_{theta}}Z)_t, ; forall tin mathbb{Z},text{ with :} $$
$$ Zsim mathit{W.N}(0,sigma^2)text{ and }; forall jin mathbb{Z} ,; alpha_{theta_j} = mathbf{1}_{{j = 0}} + theta_j cdot mathbf{1}_{{1leq j leq q}} .$$
Now, the A.C function of a filtered white noise is : $gamma_{X}(h) = sigma^2 sum_{jinmathbb{Z}}alpha_j alpha_{j+h}, ; hin mathbb{Z}.$
I compute : $$alpha_j alpha_{j+h} = mathbf{1}_{{j = h =0}} + theta_{|h|} cdot mathbf{1}_{{1leq |h| leq q,; j=-h}} + theta_jtheta_{j+h} cdot mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}}.$$
I'm ashamed to say that the last indicator is giving me some trouble to find the correct bounds on $j$ and $h$.
I find that $$theta_jtheta_{j+h} mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}} = theta_jtheta_{j+|h|} mathbf{1}_{{|h|leq q-1,; 1leq j leq q-h,; hgeq0}}+theta_jtheta_{j-|h|} mathbf{1}_{{|h|leq q-1,; 1-hleq j leq q,; hleq0}}, $$ wheras it should be:
$theta_jtheta_{j+|h|} mathbf{1}_{{mathbf{1leq}|h|leq q-1,; 1leq j leq mathbf{q-|h|}}}$.
inequality time-series
asked Nov 26 at 10:52
deque
413111
add a comment |
I'm trying to compute the autocovariance function $gamma_{X}$, where: $$X_t = Z_t + sum_{k=1}^q theta_k Z_{t-k} =: (F_{{alpha}_{theta}}Z)_t, ; forall tin mathbb{Z},text{ with :} $$
$$ Zsim mathit{W.N}(0,sigma^2)text{ and }; forall jin mathbb{Z} ,; alpha_{theta_j} = mathbf{1}_{{j = 0}} + theta_j cdot mathbf{1}_{{1leq j leq q}} .$$
Now, the A.C function of a filtered white noise is : $gamma_{X}(h) = sigma^2 sum_{jinmathbb{Z}}alpha_j alpha_{j+h}, ; hin mathbb{Z}.$
I compute : $$alpha_j alpha_{j+h} = mathbf{1}_{{j = h =0}} + theta_{|h|} cdot mathbf{1}_{{1leq |h| leq q,; j=-h}} + theta_jtheta_{j+h} cdot mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}}.$$
I'm ashamed to say that the last indicator is giving me some trouble to find the correct bounds on $j$ and $h$.
I find that $$theta_jtheta_{j+h} mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}} = theta_jtheta_{j+|h|} mathbf{1}_{{|h|leq q-1,; 1leq j leq q-h,; hgeq0}}+theta_jtheta_{j-|h|} mathbf{1}_{{|h|leq q-1,; 1-hleq j leq q,; hleq0}}, $$ wheras it should be:
$theta_jtheta_{j+|h|} mathbf{1}_{{mathbf{1leq}|h|leq q-1,; 1leq j leq mathbf{q-|h|}}}$.
inequality time-series
asked Nov 26 at 10:52
deque
413111
add a comment |
I'm trying to compute the autocovariance function $gamma_{X}$, where: $$X_t = Z_t + sum_{k=1}^q theta_k Z_{t-k} =: (F_{{alpha}_{theta}}Z)_t, ; forall tin mathbb{Z},text{ with :} $$
$$ Zsim mathit{W.N}(0,sigma^2)text{ and }; forall jin mathbb{Z} ,; alpha_{theta_j} = mathbf{1}_{{j = 0}} + theta_j cdot mathbf{1}_{{1leq j leq q}} .$$
Now, the A.C function of a filtered white noise is : $gamma_{X}(h) = sigma^2 sum_{jinmathbb{Z}}alpha_j alpha_{j+h}, ; hin mathbb{Z}.$
I compute : $$alpha_j alpha_{j+h} = mathbf{1}_{{j = h =0}} + theta_{|h|} cdot mathbf{1}_{{1leq |h| leq q,; j=-h}} + theta_jtheta_{j+h} cdot mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}}.$$
I'm ashamed to say that the last indicator is giving me some trouble to find the correct bounds on $j$ and $h$.
I find that $$theta_jtheta_{j+h} mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}} = theta_jtheta_{j+|h|} mathbf{1}_{{|h|leq q-1,; 1leq j leq q-h,; hgeq0}}+theta_jtheta_{j-|h|} mathbf{1}_{{|h|leq q-1,; 1-hleq j leq q,; hleq0}}, $$ wheras it should be:
$theta_jtheta_{j+|h|} mathbf{1}_{{mathbf{1leq}|h|leq q-1,; 1leq j leq mathbf{q-|h|}}}$.
inequality time-series
asked Nov 26 at 10:52
deque
413111
I'm trying to compute the autocovariance function $gamma_{X}$, where: $$X_t = Z_t + sum_{k=1}^q theta_k Z_{t-k} =: (F_{{alpha}_{theta}}Z)_t, ; forall tin mathbb{Z},text{ with :} $$
$$ Zsim mathit{W.N}(0,sigma^2)text{ and }; forall jin mathbb{Z} ,; alpha_{theta_j} = mathbf{1}_{{j = 0}} + theta_j cdot mathbf{1}_{{1leq j leq q}} .$$
Now, the A.C function of a filtered white noise is : $gamma_{X}(h) = sigma^2 sum_{jinmathbb{Z}}alpha_j alpha_{j+h}, ; hin mathbb{Z}.$
I compute : $$alpha_j alpha_{j+h} = mathbf{1}_{{j = h =0}} + theta_{|h|} cdot mathbf{1}_{{1leq |h| leq q,; j=-h}} + theta_jtheta_{j+h} cdot mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}}.$$
I'm ashamed to say that the last indicator is giving me some trouble to find the correct bounds on $j$ and $h$.
I find that $$theta_jtheta_{j+h} mathbf{1}_{{1leq j leq q,; 1leq j+h leq q}} = theta_jtheta_{j+|h|} mathbf{1}_{{|h|leq q-1,; 1leq j leq q-h,; hgeq0}}+theta_jtheta_{j-|h|} mathbf{1}_{{|h|leq q-1,; 1-hleq j leq q,; hleq0}}, $$ wheras it should be:
$theta_jtheta_{j+|h|} mathbf{1}_{{mathbf{1leq}|h|leq q-1,; 1leq j leq mathbf{q-|h|}}}$.
inequality time-series
inequality time-series
asked Nov 26 at 10:52
deque
413111
asked Nov 26 at 10:52
deque
413111
asked Nov 26 at 10:52
deque
413111
asked Nov 26 at 10:52
deque
413111
asked Nov 26 at 10:52
deque
413111
413111
add a comment |
add a comment |
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%2f3014181%2fautocovariance-function-of-maq%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3014181%2fautocovariance-function-of-maq%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
