{"id":11996,"date":"2021-03-03T20:38:49","date_gmt":"2021-03-03T20:38:49","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11996"},"modified":"2026-06-25T09:24:06","modified_gmt":"2026-06-25T09:24:06","slug":"multiple-regression-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/multiple-regression-model\/","title":{"rendered":"Multiple Regression Model"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Adil Suleman, CFA, wishes to establish the possible drivers of a company\u2019s percentage return on capital (ROC). He regresses ROC on profit margin (%), sales, and debt ratio and obtains the following regression statistics: Multiple R = 0.79, R Square = 0.62, Adjusted R Square = 0.57, Standard Error = 1.20, and 25 observations. The most accurate interpretation of the multiple R-squared for the model is:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The correlation between the predicted and actual values of the dependent variable is 0.89. Since the multiple R-squared is 0.79, the correlation between the predicted and actual values is the square root of 0.79, which is approximately 0.89.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"Explained variation in the dependent variable is 21% of the total variation.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The correlation between predicted and actual values of the dependent variable is 0.89.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The correlation between predicted and actual values of the dependent variable is 0.79.\"\n      }\n    ]\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8\" width=\"611\" height=\"344\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Consider the multiple regression of the price of the US Dollar index on inflation rates and real interest rates. The estimated regression model is expressed as:<\/p>\n<p>$$P=81-276INF+902IR$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li>P = Price of USDX.<\/li>\n<li>INF = Inflation rate.<\/li>\n<li>IR = Real interest rate.<\/li>\n<\/ul>\n<p>The estimated regression model is commonly interpreted using the slope coefficients. The slope coefficients show the expected change in the price of the USDX for a one-unit change in inflation rates or real interest rates.<\/p>\n<p>From the above model, we can deduce the following:<\/p>\n<p>\u00a0 \u00a0 \u00a0 i. The price of the USDX is $81 when both inflation and real interest rates are 0%.<\/p>\n<p>\u00a0 \u00a0 \u00a0ii. A 1% increase in the inflation rate (keeping the real interest rate constant) decreases the price of the USDX by $276.<\/p>\n<p>\u00a0 \u00a0 \u00a0iii. A 1% increase in the real rate of interest (keeping the inflation rate constant) increases the price of the USDX by $902.<\/p>\n<div style=\"text-align: center; margin: 28px 0;\"><a style=\"display: inline-block; background: #1a73e8; color: #ffffff; padding: 12px 26px; border-radius: 40px; font-size: 16px; font-weight: 500; text-decoration: none; line-height: 1.4;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Learn Multiple Regression Models with our Free Trial <\/a><\/div>\n<h2>Cheatsheet for Evaluating a Multiple Regression Model<\/h2>\n<p>The following is a summarized guideline for assessing a multiple regression model.<\/p>\n<p>1. Check whether the model is correctly specified.<\/p>\n<ul>\n<li>Investigate functional form misspecification, time series model misspecification, and nonstationarity.<\/li>\n<\/ul>\n<p>2. Check whether the individual coefficients are statistically significant.<\/p>\n<ul>\n<li>By performing t-tests on individual coefficients.<\/li>\n<li>For a two-tailed test of the regression coefficient, reject the null hypothesis if the t-statistic is greater than the upper critical t-value or lower than the lower critical t-value. The conclusion is that the regression coefficient is statistically significantly different from the null hypothesis value at the given significance level.<\/li>\n<\/ul>\n<p>3. Check the validity of the model (statistical significance)<\/p>\n<ul>\n<li><strong>Perform an F-test:<\/strong> It is a one-tailed test that checks for the overall fit of the model.<\/li>\n<li><strong>Use R<sup>2<\/sup>:<\/strong> It is the percentage of variation in the dependent variable that is explained by the independent variables. Adjusted R<sup>2<\/sup> is preferred as it adjusts for the number of independent variables.<\/li>\n<\/ul>\n<p>4. Assess the presence of heteroskedasticity.<\/p>\n<ul>\n<li>Check for conditional heteroskedasticity by carrying out a Breusch-pagan chi-square test.<\/li>\n<li>If present, use white-corrected standard errors to correct the standard errors of the estimated regression coefficients.<\/li>\n<\/ul>\n<p>5. Examine if there is autocorrelation (serial correlation).<\/p>\n<ul>\n<li>Perform the Durbin-Watson test.<\/li>\n<li>If present, adjust coefficient standard errors using the Hansen method.<\/li>\n<\/ul>\n<p>6. Check for multicollinearity<\/p>\n<ul>\n<li>Compare T-test, F-test, and R<sup>2<\/sup> to check for confliction.<\/li>\n<li>Check for high correlation among independent variables.<\/li>\n<li>If present, use stepwise regression to systematically eliminate variables from the regression until multicollinearity is minimized.<\/li>\n<\/ul>\n<blockquote>\n<h2>Question<\/h2>\n<p>Adil Suleman, CFA, wishes to establish the possible drivers of a company\u2019s percentage return on capital (ROC). Suleman identifies performance measures such as 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>\u00a0$$\\small{\\begin{array}{lc}\\hline{}&amp;\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.79\\\\ \\text{R Square} &amp; 0.62\\\\ \\text{Adjusted R Square} &amp; 0.57\\\\ \\text{Standard Error} &amp; 1.20\\\\ \\text{Observations} &amp; 25\\\\ \\hline\\end{array}}$$<\/p>\n<p>The <em>most accurate <\/em>interpretation of the multiple R-squared for the above model is that:<\/p>\n<p>\u00a0 \u00a0 \u00a0 A. Explained variation in the dependent variable is 21% of the total variation.<\/p>\n<p>\u00a0 \u00a0 \u00a0 B. The correlation between predicted and actual values of the dependent variable is 0.89.<\/p>\n<p>\u00a0 \u00a0 \u00a0 C. The correlation between predicted and actual values of the dependent variable is 0.79.<\/p>\n<h3>Solution<\/h3>\n<p><strong>The correct answer is B.<\/strong><\/p>\n<p>The multiple R-squared for the regression is 0.79; thus, the model explains 79% of the variation in the dependent variable. The correlation between the predicted and actual values of the dependent variable is the square root of the multiple R-squared:<\/p>\n<p>$$\\text{Correlation}=\\sqrt{0.79}\u22480.89$$<\/p>\n<\/blockquote>\n<p>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (o) Evaluate and interpret a multiple regression model and its results.<\/em><\/p>\n<div style=\"background: #f5f7fb; padding: 24px 18px; border-radius: 12px; text-align: center; margin: 36px 0 18px;\"><a style=\"display: inline-block; background: #1a73e8; color: #ffffff; padding: 10px 24px; border-radius: 40px; font-size: 16px; font-weight: bold; text-decoration: none; margin-bottom: 16px;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a><\/p>\n<div style=\"font-size: 14px; color: #333333; max-width: 650px; margin: 0 auto; line-height: 1.6;\">Master multiple regression models, coefficient interpretation, model evaluation, and regression analysis techniques with CFA Level II exam-style practice.<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Consider the multiple regression of the price of the US Dollar index on inflation rates and real interest rates. The estimated regression model is expressed as: $$P=81-276INF+902IR$$ Where: P = Price of USDX. INF = Inflation rate. IR = Real&#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":[216,252,230],"class_list":["post-11996","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-multiple-regression-model","tag-quantitative-method","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Multiple Regression Analysis | CFA Level II<\/title>\n<meta name=\"description\" content=\"Learn multiple regression analysis in CFA Level II, including assumptions, F-tests, and adjusted R\u00b2 for evaluating model fit and statistical significance.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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