{"id":181969,"date":"2022-11-07T16:12:00","date_gmt":"2022-11-07T21:12:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/traders-insight\/autocorrelation-and-autocovariance-calculation-examples-and-more-part-i\/"},"modified":"2023-02-13T17:03:04","modified_gmt":"2023-02-13T22:03:04","slug":"autocorrelation-and-autocovariance-calculation-examples-and-more-part-i","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/autocorrelation-and-autocovariance-calculation-examples-and-more-part-i\/","title":{"rendered":"Autocorrelation and Autocovariance: Calculation, Examples, and More &#8211; Part I"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Autocorrelation and autocovariance are one of the most critical metrics in financial time series econometrics. Both functions are based on covariance and correlation metrics. You will learn more about them. This easy-to-learn essential guide will help you understand better about ARMA models.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>What is autocovariance?<\/li>\n\n\n\n<li>What is autocorrelation?<\/li>\n\n\n\n<li>What are the autocovariance and autocorrelation at lag zero?<\/li>\n\n\n\n<li>Calculation of autocovariance with an example<\/li>\n\n\n\n<li>Calculation of autocorrelation with an example<\/li>\n\n\n\n<li>Computation of autocovariance and autocorrelation in Python<\/li>\n\n\n\n<li>Plot the autocorrelation function in Python<\/li>\n\n\n\n<li>Computation of autocovariance and autocorrelation in R<\/li>\n\n\n\n<li>Plot the autocorrelation functions in R<\/li>\n\n\n\n<li>What is partial autocorrelation?<\/li>\n\n\n\n<li>Computation of partial autocorrelation in Python and R<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">You might have encountered yourself trying to learn the Autoregressive Moving Average (ARMA) model. You then started to see a lot of use of covariances and correlations, but strangely enough, you see those two words with the prefix &#8220;<em>auto&#8221;&nbsp;<\/em>and you get frightened!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Don\u2019t worry, this article will help you understand their details. Just keep the focus on the article and everything will be ok!<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"what-is-autocovariance\">What is autocovariance?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">First, you need to understand what&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/covariance-correlation\/\">covariance and correlation<\/a>&nbsp;are. Remember that covariance is applied to 2 assets. The autocovariance is the same as the covariance.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The only difference is that the autocovariance is applied to the same asset, i.e., you compute the covariance of the asset price return X with the same asset price return X, but from a previous period.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>How\u2019s that possible?<\/em>&nbsp;Simple, check it out:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-1.png\" alt=\" class=\" class=\"wp-image-165244 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where X and Y can be the returns of asset X and Y, respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, the autocovariance function can be defined as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-2.png\" alt=\" class=\" class=\"wp-image-165248 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-9.png\" alt=\" class=\" class=\"wp-image-165251 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-default\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"what-is-autocorrelation\">What is autocorrelation?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In simple terms,&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/autoregression\/\">autocorrelation<\/a>&nbsp;is the same as the correlation function! To be specific, autocorrelation, as the autocovariance, is applied to the same asset.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Check the difference between correlation and autocorrelation (also called serial correlation) below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-3.png\" alt=\" class=\" class=\"wp-image-165259 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-10.png\" alt=\" class=\" class=\"wp-image-165262 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Now let\u2019s check the autocorrelation:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-12.png\" alt=\" class=\" class=\"wp-image-165283 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-11.png\" alt=\" class=\" class=\"wp-image-165271 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You might ask us:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Why variance and not the multiplication of the standard deviation of the returns at the different lags?<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Well, you must remember that an ARMA model is applied to stationary time series. This topic belongs to&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/time-series-analysis\/\">time series analysis<\/a>. Consequently, it is assumed that the price returns, if stationary, have the same variance for any lag, i.e.:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-5.png\" alt=\" class=\" class=\"wp-image-165267 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"what-are-the-autocovariance-and-autocorrelation-at-lag-zero\">What are the autocovariance and autocorrelation at lag zero?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Interesting question and simple to answer! Let\u2019s see first for the former:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-6.png\" alt=\" class=\" class=\"wp-image-165276 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Can you guess what the last part resembles?<\/em><br>It is the variance of the price returns!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consequently, the autocovariance of the returns at lag 0 is the variance of the returns.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Can you guess now what the autocorrelation of the returns would be at lag 0?<\/em>Let\u2019s use the formulas to find out:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-7.png\" alt=\" class=\" class=\"wp-image-165300 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Since we know, from the above algebraic calculation, that the covariance of the same variable is its variance, we have the following:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/autocorrelation-quantinsti-8.png\" alt=\" class=\" class=\"wp-image-165298 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Consequently, the autocorrelation function for any asset price return at lag 0 is always 1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Stay tuned for the next installment in this series to learn about calculation of the autocovariance.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Visit QuantInsti for additional insight on this topic: <a href=\"https:\/\/blog.quantinsti.com\/autocorrelation-autocovariance\/\">https:\/\/blog.quantinsti.com\/autocorrelation-autocovariance\/<\/a>.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Autocorrelation and autocovariance are one of the most critical metrics in financial time series econometrics.<\/p>\n","protected":false},"author":825,"featured_media":181981,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,343,349,338,350,341,351,352,344,2197],"tags":[14058,9374,14059,4922,595,494],"contributors-categories":[13654],"class_list":["post-181969","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-programing-languages","category-python-development","category-ibkr-quant-news","category-quant-asia-pacific","category-quant-development","category-quant-europe","category-quant-north-america","category-quant-regions","category-quant-south-america","tag-arma-models","tag-autocorrelation","tag-autocovariance","tag-econometrics","tag-python","tag-quant","contributors-categories-quantinsti"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.8) - 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