{"id":95838,"date":"2021-07-22T11:06:07","date_gmt":"2021-07-22T15:06:07","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=95838"},"modified":"2022-11-21T09:47:46","modified_gmt":"2022-11-21T14:47:46","slug":"grokking-linear-regression-analysis-in-finance-part-iii","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/grokking-linear-regression-analysis-in-finance-part-iii\/","title":{"rendered":"Grokking Linear Regression Analysis in Finance \u2013 Part III"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>See&nbsp;<a href=\"\/campus\/ibkr-quant-news\/grokking-linear-regression-analysis-in-finance\/\">Part I<\/a>&nbsp;and <a href=\"\/campus\/ibkr-quant-news\/grokking-linear-regression-analysis-in-finance-part-ii\/\">Part II<\/a> for an overview of the linear models and the concept of \u201cregression\u201d.<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-multiple-linear-regression\">Multiple linear regression<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let\u2019s now say we believe there are multiple factors that tell us something about&nbsp;<em>KO<\/em>&#8216;s returns. They could be&nbsp;<em>SPY<\/em>&#8216;s returns, its competitor PepsiCo\u2019s (<em>NASDAQ : PEP<\/em>) returns, and the US Dollar index (<em>ICE : DX<\/em>) returns. We denote these variables with the letter&nbsp;XX&nbsp;and add subscripts for each of them. We use the notation&nbsp;X<sub>i,1<\/sub>,X<sub>i,2<\/sub>&nbsp;and&nbsp;X<sub>i,3<\/sub>&nbsp;to refer to the&nbsp;i<sup>th<\/sup>&nbsp;observation of&nbsp;<em>SPY, PEP<\/em>&nbsp;and&nbsp;<em>DX<\/em>&nbsp;returns respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Like before, let\u2019s put them all in an equation format to make things explicit.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"744\" height=\"79\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti.png\" alt=\"\" class=\"wp-image-95883 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti.png 744w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti-700x74.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti-300x32.png 300w\" data-sizes=\"(max-width: 744px) 100vw, 744px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 744px; aspect-ratio: 744\/79;\" \/><\/figure>\n\n\n\n<p>\u03b2<sub>0<\/sub>,\u03b2<sub>1<\/sub>,\u03b2<sub>2<\/sub> and \u03b2<sub>3<\/sub> are the model parameters in equation&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/linear-regression\/#mjx-eqn-ref2\">2<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here, we have a multiple linear regression model to describe the relation between&nbsp;YY&nbsp;(the returns on&nbsp;<em>KO<\/em>) and&nbsp;X<sub>i<\/sub>;i = 1, 2, 3&nbsp;(the returns on&nbsp;<em>SPY, PEP,<\/em>&nbsp;and&nbsp;<em>DX<\/em>&nbsp;respectively).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We call it multiple, since there is more than one explanatory variable (three, in this case); and we call it linear, since the coefficients are linear.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When we go from one to two explanatory variables, we can visualize it as a 2-D plane (which is the generalization of a line) in three dimensions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For ex.&nbsp;Y = 3 \u2212 2X<sub>1<\/sub> + 4X<sub>2<\/sub> can be plotted as shown below.<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"445\" height=\"463\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti-2.png\" alt=\"\" class=\"wp-image-95887 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti-2.png 445w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/07\/regression-quantinsti-2-300x312.png 300w\" data-sizes=\"(max-width: 445px) 100vw, 445px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 445px; aspect-ratio: 445\/463;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">As we add more features, we move to n-dimensional planes (called hyperplanes) in&nbsp;(n + 1)&nbsp;dimensions which are much harder to visualize (anything above three dimensions is). Nevertheless, they would still be&nbsp;<em>linear in their coefficients<\/em>&nbsp;and hence the name.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The objective of multiple linear regression is to find the \u201cbest\u201d possible values for&nbsp;\u03b2<sub>0<\/sub>,\u03b2<sub>1<\/sub>,\u03b2<sub>2<\/sub> and \u03b2<sub>3<\/sub> such that the formula can \u201caccurately\u201d calculate the value of&nbsp;Y<sub>i<\/sub>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In our example here, we have three&nbsp;X\u2032s.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiple regression allows for any number of&nbsp;X\u2032s&nbsp;(as long as they are less than the number of observations).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Stay tuned for the next installment in which Vivek will discuss linear regression of a non-linear relationship.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Visit QuantInsti for additional insight on this topic:&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/linear-regression\/\">https:\/\/blog.quantinsti.com\/linear-regression\/<\/a>.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Any trading symbols displayed are for illustrative purposes only and are not intended to portray recommendations.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Vivek Krishnamoorthy demonstrates a multiple linear regression model, used for econometrics purposes.<\/p>\n","protected":false},"author":646,"featured_media":62531,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,343,349,338,350,341,344],"tags":[4922,1006,4404,10009,10008,10010],"contributors-categories":[13654],"class_list":["post-95838","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-regions","tag-econometrics","tag-fintech","tag-linear-regression","tag-multiple-linear-regression-model","tag-regression-coefficient","tag-simple-linear-regression-model","contributors-categories-quantinsti"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- 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