{"id":29248,"date":"2022-12-23T17:38:53","date_gmt":"2022-12-23T17:38:53","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=29248"},"modified":"2025-12-31T08:49:25","modified_gmt":"2025-12-31T08:49:25","slug":"formulate-and-interpret-a-multiple-regression-model-that-includes-qualitative-independent-variables","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/formulate-and-interpret-a-multiple-regression-model-that-includes-qualitative-independent-variables\/","title":{"rendered":"Formulate and Interpret a Multiple Regression Model That Includes Qualitative Independent Variables"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"How many dummy variables are needed when the first quarter is used as the reference category?\",\n    \"text\": \"Ninel Khan, a technical trader, is analyzing the seasonality of the price of the GBP\/USD. Khan believes the price is significantly different in the first quarter compared to the other three quarters. She uses the first quarter as the reference point in the regression. The number of dummy variables that Khan will need in her regression equation is:\\n\\nA. 2.\\nB. 3.\\nC. 4.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"3.\\n\\nWhen a categorical variable has n categories, a regression model requires n \u2212 1 dummy variables to avoid multicollinearity. In this case, there are four quarters, and the first quarter is used as the reference category. Therefore, three dummy variables are needed to represent the remaining quarters.\",\n      \"dateCreated\": \"2025-12-31\"\n    }\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/Uqmkw12EQl4\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>Dummy variables<\/strong> are binary variables used to quantify the effect of qualitative independent variables. A dummy variable is assigned a value of 1 if a particular condition is met and, otherwise, a value of 0. The number of dummy variables for \\(n\\) different classes must equal \\(n-1\\).<\/p>\n<p>The intercept term measures the average value of the dependent variable of the omitted class. On the other hand, the estimated coefficient on each dummy variable measures the average incremental effect of that dummy variable on the dependent variable.<\/p>\n<h4>Example: Incorporating Dummy Variables in a Multiple Regression Model<\/h4>\n<p>Adil Suleman, CFA, wishes to identify possible drivers of a company&#8217;s percentage return on capital (ROC). He identifies performance measures, including margin (%), sales and debt ratios, and demographic measures, such as the region and the economic sector, as possible drivers of ROC.<\/p>\n<p>The dummy variable \u201cregion\u201d is coded 1 when a company is located in the northern region and 0 if it&#8217;s in the southern region. On the other hand, the dummy variable \u201ceconomic sector\u201d is coded 1 when a company belongs to the banking sector and 0 when it belongs to the technology sector.<\/p>\n<p>Suleman regresses ROC against sales, debt ratio, profit margin, region, and sector to obtain the following regression output.<\/p>\n<p>$$ \\begin{array}{l|c} \\text{Regression Statistics} &amp; \\\\ \\hline \\text{Multiple R} &amp; 0.8851 \\\\ \\hline \\text{R Square} &amp; 0.7833 \\\\ \\hline \\text{Adjusted R Square} &amp; 0.7263 \\\\ \\hline \\text{Standard Error} &amp; 0.9561 \\\\ \\hline \\text{Observations} &amp; 25 \\\\ \\end{array} $$<\/p>\n<p>$$ \\textbf{ANOVA} \\\\ \\begin{array}{c|c|c|c|c} &amp; \\text{Df} &amp; \\text{SS} &amp; \\text{MS} &amp; \\text F &amp; \\text{Significance F} \\\\ \\hline \\text{Regression} &amp; 5 &amp; 62.7924 &amp; 12.5585 &amp; 13.7389 &amp; 0.0000 \\\\ \\hline \\text{Residual} &amp; 19 &amp; 17.3676 &amp; 0.9141 &amp; &amp; \\\\ \\hline \\text{Total} &amp; 24 &amp; 80.1600 &amp; &amp; &amp; \\\\ \\end{array} $$<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} &amp; \\text{Coefficients} &amp; \\text{Standard} &amp; \\text{t Stat} &amp; \\text{P-value} \\\\ &amp; &amp; \\text{Error} &amp; &amp; \\\\ \\hline \\text{Intercept} &amp; 10.1241 &amp; 0.8503 &amp; 11.9069 &amp; 0.0000 \\\\ \\hline \\text{Sales}&amp; 0.0010 &amp; 0.0004 &amp; 2.4003 &amp; 0.0268 \\\\ \\hline \\text{Debt ratio} &amp; 0.0166 &amp; 0.0138 &amp; 1.2017 &amp; 0.2443 \\\\ \\hline \\text{Profit Margin} &amp; 0.1807 &amp; 0.0552 &amp; 3.2713 &amp; 0.0040 \\\\ \\hline \\text{Region} &amp; 2.1755 &amp; 0.6061 &amp; 3.5896 &amp; 0.0020 \\\\ \\hline \\text{Sector} &amp; \u22120.8703 &amp; 0.4202 &amp; \u22122.0709 &amp; 0.0522 \\\\ \\end{array} $$<\/p>\n<h4><strong>Interpreting the Results<\/strong><\/h4>\n<p>From the above results, the multiple regression equation can be expressed as follows:<\/p>\n<p>$$ \\begin{align*} ROC &amp; = 10.1241 + 0.001SAL + 0.0166DR \\\\ &amp; + 0.1807PM + 2.1755REG \u2212 0.8703SEC \\end{align*} $$<\/p>\n<h3>Adjusted R<sup>2<\/sup><\/h3>\n<p>72.63% of the variation in the return on capital is explained by three quantitative regressors (sales, debt ratio, and profit margin) and two qualitative regressors (region and sector).<\/p>\n<h3>Testing the Significance of the Overall Model<\/h3>\n<p>\\(H_0 : b_1 = b_2 = b_3 = b_4 = b_5 = 0\\) versus at \\(H_a\\) : At least one \\(b_j \\neq 0\\)<\/p>\n<p>From the ANOVA table, column \u201cSignificance F,\u201d notice that the p-value is less than 5%. Thus, reject \\(H_0\\) in favor of \\(H_a\\). Conclude that the model is statistically significant.<\/p>\n<h3>Testing the Significance of Dummy Variables<\/h3>\n<p>The coefficient of the region in this regression model is positive and statistically significant at the 0.05 level since the p-value is less than 0.05. We can, therefore, conclude that the northern region is significantly different from the southern region at the 5% significance level.<\/p>\n<p>On the contrary, the banking sector is not significantly different from the technology sector since the p-value is greater than 0.05. Additionally, the sign of this coefficient is negative.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Ninel Khan, a technical trader, is analyzing the seasonality of the price of the GBP\/USD. Khan believes the price is significantly different in the first quarter compared to the other three quarters. She uses the first quarter as the reference point in the regression. The number of dummy variables that Khan will need in her regression equation is:<\/p>\n<ol type=\"A\">\n<li>2.<\/li>\n<li>3.<\/li>\n<li>4.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>If we need to differentiate among \\(n\\) categories, the regression should include \\(n \u2212 1\\) dummy variables. In this case, we have four quarters. Thus, three dummy variables are needed.<\/p>\n<\/blockquote>\n<p>Reading 4: Extensions of Multiple Regression<\/p>\n<p><em>Los 4 (b) Formulate and interpret a multiple regression model that includes qualitative independent variables<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dummy variables are binary variables used to quantify the effect of qualitative independent variables. A dummy variable is assigned a value of 1 if a particular condition is met and, otherwise, a value of 0. The number of dummy variables&#8230;<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[],"class_list":["post-29248","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-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>Formulate and Interpret a Multiple Regression Model That Includes Qualitative Independent Variables - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Dummy variables are binary variables used to quantify the effect of qualitative independent variables.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/formulate-and-interpret-a-multiple-regression-model-that-includes-qualitative-independent-variables\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formulate and Interpret a Multiple Regression Model That Includes Qualitative Independent Variables - 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