{"id":11814,"date":"2021-03-02T06:28:19","date_gmt":"2021-03-02T06:28:19","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11814"},"modified":"2026-07-07T20:12:39","modified_gmt":"2026-07-07T20:12:39","slug":"predicted-value-dependent-variable","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/predicted-value-dependent-variable\/","title":{"rendered":"The  Predicted Value of a Dependent Variable"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Multiple Regression (2022 Level II CFA\u00ae Exam \u2013 Reading 2)\",\n  \"description\": \"In this lesson, Professor James Forjan explains multiple regression for the CFA Level II Quantitative Methods curriculum. The video covers regression modeling, interpretation of coefficients, statistical significance, and how regression is applied in financial analysis for the CFA exam.\",\n  \"uploadDate\": \"2022-04-05\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/8E2AbtAb0a8\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=8E2AbtAb0a8\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8\",\n  \"duration\": \"PT43M31S\",\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/wp-content\/uploads\/2020\/01\/AnalystPrep-Logo.png\"\n    }\n  }\n}\n<\/script><br \/>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Given the regression results, what is the predicted value of return on capital (ROC)?\",\n    \"text\": \"Consider the regression model ROC = 8.6531 + 0.0009(Sales) + 0.0229(Debt Ratio) + 0.2996(Profit Margin). If sales = 1000, debt ratio = 20, and profit margin = 20%, what is the predicted value of ROC?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. Substituting the values into the regression equation: ROC = 8.6531 + (0.0009 \u00d7 1000) + (0.0229 \u00d7 20) + (0.2996 \u00d7 20) = 8.6531 + 0.9 + 0.458 + 5.992 = 16.03%.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"7.38%.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"8.29%.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"16.03%.\"\n      }\n    ]\n  }\n}\n<\/script><br \/>\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8?si=A6zjdcc_MaFFJNJS\" 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<h2>The Predicted Value of the Dependent Variable<\/h2>\n<p>The following steps are followed to predict the value of a dependent variable in a multiple regression model.<\/p>\n<ol type=\"i\">\n<li>Determine the regression coefficient estimates.<\/li>\n<li>Obtain the forecasted values of the independent variables.<\/li>\n<li>Calculate the predicted value of the dependent variable using the equation:<\/li>\n<\/ol>\n<p>$$\\hat{Y_{i}}=\\widehat{b_{0}}+\\widehat{b_{1}}\\hat{X_{1i}}+\\widehat{b_{2}}\\hat{X_{2i}}+&#8230;+\\widehat{b_{k}}\\hat{X_{ki}}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(\\hat{Y}\\) = the predicted value of the dependent variable.<\/li>\n<li data-tadv-p=\"keep\">\\(\\widehat{b_{j}}\\) = the estimated slope coefficient for the \\(j^{th}\\) independent variable.<\/li>\n<li data-tadv-p=\"keep\">\\(\\hat{X_{ji}}\\) = the forecasted value of the \\(j^{th}\\) independent variable, \\(j = 1, 2, \u2026, k\\).<\/li>\n<\/ul>\n<div style=\"margin: 18px 0;\"><a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff ; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8 ;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">Master CFA Level 2 predicted value concepts with AnalystPrep\u2019s Free Trial. <\/a><\/div>\n<h3>Example: Calculating the Predicted Value of a Dependent Variable<\/h3>\n<p>Consider the following regression equation of the price of USDX on inflation rates and real interest rates:<\/p>\n<p>$$P=b_{0}+b_1INF+b_2IR+\\epsilon_{t}$$<\/p>\n<p>The following table gives the regression results:<\/p>\n<p>$$\\small{\\begin{array}{l|c}\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.8264\\\\ \\hline\\text{R Square} &amp; 0.6830\\\\ \\hline\\text{Adjusted R Square} &amp; 0.5924\\\\ \\hline\\text{Standard Error} &amp; 5.3537\\\\ \\hline\\text{Observations} &amp; 10\\\\ \\end{array}}$$<\/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>Use the estimated regression equation above to calculate the predicted price of the US dollar index (USDX), assuming the inflation rate is 3.5% and the real rate of interest is 4%.<\/p>\n<h4>Solution<\/h4>\n<p>$$\\hat{Y_{i}}=\\widehat{b_{0}}+\\widehat{b_{1}}\\hat{X_{1i}}+\\widehat{b_{2}}\\hat{X_{2i}}+&#8230;+\\widehat{b_{k}}\\hat{X_{ki}}$$<\/p>\n<p>$$\\begin{align*}\\hat{Y_{i}}&amp;=81+(-276\\times0.035)+(902\\times0.04)\\\\&amp;=$107.42\\end{align*}$$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Consider the following multiple regression results of the return on capital (ROC) on performance measures (profit margin (%), sales and debt ratio).<\/p>\n<p>$$\\small{\\begin{array}{l|c}\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.7906\\\\ \\hline\\text{R Square} &amp; 0.6251\\\\ \\hline\\text{Adjusted R Square} &amp; 0.5715\\\\ \\hline\\text{Standard Error} &amp; 1.1963\\\\ \\hline\\text{Observations} &amp; 25\\\\ \\end{array}}$$<\/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>Given that sales = 1000, debt ratio = 20, and profit margin = 20%, the predicted value of the return on capital (ROC) according to the regression model is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>7.38%.<\/li>\n<li>8.29%.<\/li>\n<li>16.03%.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p><strong>The correct answer is C.<\/strong><\/p>\n<p>The regression equation is expressed as:<\/p>\n<p>$$\\text{ROC} = 8.653+0.0009S+0.0229DR+0.2996PM$$<\/p>\n<p>$$\\begin{align*}\\text{ROC} &amp;= 8.6531+ (0.0009\\times 1000)+(0.0229\\times 20) + (0.2996\\times 20)\\\\&amp;=16.03\\%\\end{align*}$$<\/p>\n<\/blockquote>\n<p>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (e) Calculate and interpret\u00a0 a predicted value for the dependent variable, given an estimated regression model and assumed values for the independent variables.<\/em><\/p>\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a> <\/p>\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Strengthen your understanding of predicted values and regression analysis in CFA Level 2 Quantitative Methods with study notes, practice questions, and mock exams.\n<\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Predicted Value of the Dependent Variable The following steps are followed to predict the value of a dependent variable in a multiple regression model. Determine the regression coefficient estimates. Obtain the forecasted values of the independent variables. Calculate the&#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,"rs_blank_template":"","rs_page_bg_color":"","slide_template_v7":"","footnotes":""},"categories":[102,229],"tags":[216,242,230],"class_list":["post-11814","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-interval-of-a-regression-coefficient-and-the-predicted-value-of-a-dependent-variable","tag-quantitative-method","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v28.8 (Yoast SEO v28.8) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Predicted Value of a Dependent Variable | CFA Level II<\/title>\n<meta name=\"description\" content=\"Learn how to calculate the predicted value of a dependent 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