{"id":11746,"date":"2021-03-01T10:26:30","date_gmt":"2021-03-01T10:26:30","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11746"},"modified":"2026-01-08T14:01:57","modified_gmt":"2026-01-08T14:01:57","slug":"multiple-regression-equation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/multiple-regression-equation\/","title":{"rendered":"Multiple Regression Equation"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Stuart Sellers, CFA, wants to identify possible drivers of a company's percentage return on capital (ROC). Stuart identifies performance measures including the profit margin (PM), the sales, and debt ratios as possible drivers of ROC. Which of the following is the most appropriate regression equation given the data?\",\n    \"text\": \"Stuart Sellers, CFA, wants to identify possible drivers of a company's percentage return on capital (ROC). Stuart identifies performance measures including the profit margin (PM), the sales, and debt ratios as possible drivers of ROC. Using the regression output provided in Exhibit 1, which of the following is the most appropriate regression equation? A. ROC = 0.5831 - 0.00365 Sales + 0.01252 DebtRatio - 0.00654 PM B. ROC = 0.4139 + 0.00055 Sales - 0.01265 DebtRatio + 0.00546 PM C. ROC = 0.4972 + 0.00479 Sales + 0.00187 DebtRatio - 0.00654 PM\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. A multiple regression model is expressed as: Y = b0 + b1X1 + b2X2 + ... + bkXk + \u03b5. Using the coefficients from Exhibit 1\u2014Sales, Debt Ratio, and Profit Margin\u2014the most appropriate regression equation is: ROC = 0.4139 + 0.00055 Sales - 0.01265 DebtRatio + 0.00546 PM. This correctly applies each coefficient to the respective independent variable.\"\n    }\n  }\n}\n<\/script><br \/>\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8?si=UzHr37wRwXMy-3bN\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11.jpg\",\n  \"caption\": \"Multiple Regression\",\n  \"width\": 1590,\n  \"height\": 1112,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script><\/p>\n<p>Multiple regression allows us to evaluate the effect of two or more independent variables on a given dependent variable. Multiple regression with two explanatory variables and one intercept term can be represented in the following 3D diagram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14852\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11.jpg\" alt=\"Multiple Regression\" width=\"1590\" height=\"1112\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11-300x210.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11-1024x716.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11-768x537.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11-1536x1074.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/Img_11-400x280.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Mathematically, a multiple regression model wityh two or more explanatory variables takes the following form:<\/p>\n<p>$$Y_{i}=b_{0}+b_{1}X_{1,i}+b_{2}X_{2,i}+&#8230;+b_{k}X_{k,i}+\\epsilon_{i}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li>\\(Y_i\\) = Dependent variable.<\/li>\n<li>\\(b_0\\) = Intercept term.<\/li>\n<li>\\(b_1, b_2,&#8230;,b_k\\) = Slope coefficients.<\/li>\n<li>\\(X_{1,i}, X_{2,i}, &#8230;,X_{k,i}\\) = Independent variables.<\/li>\n<li>\\(\\epsilon_{i}\\) = Error term.<\/li>\n<li>\\(n\\) = Number of observations.<\/li>\n<\/ul>\n<p>A regression equation has \\(k\\) slope coefficients and \\(k+1\\) regression coefficients.<\/p>\n<p>The intercept term is defined as the value of the dependent variable when the independent variables are zero. On the other hand, the slope coefficient is defined as the estimated change in the dependent variable given a one-unit change in the value of the independent variable, keeping the other independent variables constant.<\/p>\n<div style=\"margin: 0 0 20px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    style=\"\n      display: inline-block;\n      border: 2px solid #1e63ff;\n      color: #1e63ff;\n      background: #ffffff;\n      padding: 10px 14px;\n      border-radius: 10px;\n      font-weight: 500;\n      line-height: 1.35;\n      text-decoration: none;\n    \"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n  ><br \/>\n    Want to practice multiple regression equations and coefficient interpretation with CFA\u00ae-style questions? Try AnalystPrep\u2019s free trial now.<br \/>\n  <\/a>\n<\/div>\n<h4>Example: Multiple Regression in the Investment World<\/h4>\n<p>James Chase, an investment analyst, wants to determine the impact of inflation rates and real rates of interest on the price of the US Dollar index (USDX). Chase uses the multiple regression model below:<\/p>\n<p>$$P=b_0+b_1INF+b_2IR+\\epsilon_t$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(P\\) = Price of USDX.<\/li>\n<li data-tadv-p=\"keep\">\\(INF\\) = Inflation rate.<\/li>\n<li data-tadv-p=\"keep\">\\(IR\\) = Real rate of interest.<\/li>\n<li data-tadv-p=\"keep\">\\(\\epsilon_{t}\\) = Error term.<\/li>\n<\/ul>\n<p>The regression of the price of USDX on inflation and real interest rates generates the following results:<\/p>\n<p>$$\\small{\\begin{array}{l|c|c|c|c}{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t Stat} &amp; \\textbf{P-value}\\\\ \\hline\\text{Intercept} &amp; 81 &amp; 7.9659 &amp; 10.1296 &amp; 0.0000\\\\ \\hline\\text{Inflation rates} &amp; -276 &amp; 233.0748 &amp; -1.1833 &amp; 0.2753\\\\ \\hline\\text{Real interest Rates} &amp; 902 &amp; 279.6949 &amp; 3.2266 &amp; 0.0145\\\\ \\end{array}}$$<\/p>\n<p>The multiple regression equation can be expressed as:<\/p>\n<p>$$P=81-276INF+902IR$$<\/p>\n<p>The regression coefficient estimate of the inflation rate is negative. This indicates that an increase in the inflation rates causes a decrease in the price of the US Dollar index (USDX).<\/p>\n<p>Furthermore, the positive real rate of interest coefficient implies that an increase in the real interest rate is accompanied by an increase in the price of USDX.<\/p>\n<p>Additionally, the t-statistic indicates that only the real interest rate variable is significant at the 5% significance level.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Adil Suleman, CFA, wishes to identify possible drivers of a company\u2019s percentage return on capital (ROC). Suleman identifies performance measures, including the profit margin (%), sales, and debt ratio, as possible drivers of ROC.<\/p>\n<p>He obtains the following results from the regression of ROC on profit margin (%), sales, and the debt ratio.<\/p>\n<p style=\"text-align: left;\">$$ \\textbf{Exhibit 1: Summary Output} $$<\/p>\n<p>$$\\small{\\begin{array}{l|c|c|c|c|c}{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t Stat} &amp; \\textbf{P-value} \\\\ \\hline\\text{Intercept} &amp; 8.6531 &amp; 0.9174 &amp; 9.4323 &amp; 0.0000 \\\\ \\hline \\text{Sales} &amp; 0.0009 &amp; 0.0005 &amp; 1.7644 &amp; 0.0922\\\\ \\hline\\text{Debt ratio} &amp; 0.0229 &amp; 0.0165 &amp; 1.3880 &amp; 0.1797 \\\\ \\hline\\text{Profit Margin%} &amp; 0.2996 &amp; 0.0564 &amp; 5.3146 &amp; 0.0000\\\\ \\end{array}}$$<\/p>\n<p>Use the notations S = Sales, DR=Debt ratio, and PM = Profit margin. The <em>most appropriate<\/em> expression of the multiple regression equation that can be used to test the effects of the changes in the values of sales, debt ratio, and profit margin (%) on ROC is:<\/p>\n<p>\u00a0 \u00a0 \u00a0 A. ROC = 8.6531 + 0.0005S + 0.0165DR + 0.0564PM<\/p>\n<p>\u00a0 \u00a0 \u00a0 B. ROC = 8.653 + 0.0009S + 0.0229DR + 0.2996PM<\/p>\n<p>\u00a0 \u00a0 \u00a0 C. ROC = 0.9174 + 0.0005S + 0.0165DR + 0.0564PM<\/p>\n<h3>Solution<strong>\u00a0<\/strong><\/h3>\n<p><strong>The correct answer is B.<\/strong><\/p>\n<p>A multiple regression model is expressed as:<\/p>\n<p>$$Y_{i}=b_{0}+b_{1}X_{1,i}+b_{2}X_{2,i}+&#8230;+b_{k}X_{k,i}+\\epsilon_{i}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(Y_i\\) = Dependent variable.<\/li>\n<li data-tadv-p=\"keep\">\\(b_0\\) = Intercept term.<\/li>\n<li data-tadv-p=\"keep\">\\(b_1, b_2,&#8230;,b_k\\) = slope coefficients.<\/li>\n<li data-tadv-p=\"keep\">\\(X_{1,i}, X_{2,i}, &#8230;,X_{k,i}\\) = Independent variables.<\/li>\n<li data-tadv-p=\"keep\">\\(\\epsilon_{i}\\) = Error term.<\/li>\n<\/ul>\n<\/blockquote>\n<p>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (a) Formulate a multiple regression equation to describe the relationship between a dependent variable and several independent variables and determine the statistical significance of each independent variable.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Multiple regression allows us to evaluate the effect of two or more independent variables on a given dependent variable. Multiple regression with two explanatory variables and one intercept term can be represented in the following 3D diagram: Mathematically, a multiple&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[216,239,230],"class_list":["post-11746","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-multiple-regression-equation","tag-quantitative-method","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Multiple Regression Equation | CFA Level II Quant Notes<\/title>\n<meta name=\"description\" content=\"Explains the multiple regression formula, including independent variables, 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