{"id":11501,"date":"2021-02-24T14:05:59","date_gmt":"2021-02-24T14:05:59","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11501"},"modified":"2026-05-01T12:25:01","modified_gmt":"2026-05-01T12:25:01","slug":"estimating-the-parameters-of-simple-linear-regression","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/estimating-the-parameters-of-simple-linear-regression\/","title":{"rendered":"Estimating the Parameters of a Simple Linear Regression"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Predicted ROE based on a linear regression model\",\n    \"text\": \"Jane estimates a model that regresses her company\u2019s return on equity (ROE) against its growth opportunities (GO). She comes up with the following linear regression:\\n\\nROE_i = 3 + 1.5 * GO_i + \u03f5_i. The predicted value of the company\u2019s ROE if its GO is 10% is closest to:\\n\\nA. 18%\\n\\nB. 1.5%\\n\\nC. 15%\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. 18%.\\n\\nUsing the linear regression equation, ROE_i = 3 + 1.5 * 10 = 18%. Therefore, if the growth opportunity (GO) is 10%, the predicted ROE is 18%.\\n\\nB is incorrect because 1.5% is the slope of the regression, not the predicted ROE.\\n\\nC is incorrect because 15% results from multiplying the slope (1.5) by the independent variable (GO), but the full regression equation must be used to calculate the predicted ROE.\"\n    }\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/r3Hs3NHW-GE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>While conducting a regression analysis, we start with the dependent variable whose variation we want to explain and the independent variable that explains the changes in the dependent variable. The <strong>least-square criterion<\/strong> is used to measure the accuracy of a straight line by minimizing the squared deviations from the line.<\/p>\n<p style=\"text-align: left;\">$$Y=b_0+b_1X_1+\\epsilon$$<\/p>\n<p>Where:<\/p>\n<p>\\(Y\\) = Dependent Variable<\/p>\n<p>\\(b_0\\) = Intercept<\/p>\n<p>\\(b_1\\) = Slope Coefficient<\/p>\n<p>\\(X\\) = Independent Variable<\/p>\n<p>\\(\\epsilon\\) = Error Term (Noise)<\/p>\n<p>\\(b_0\\) and \\(b_1\\) are known as <strong>Regression Coefficients.<\/strong><\/p>\n<p>Since the population parameters values \\(\\beta_0\\) and \\(\\beta_1\\) cannot be observed in a regression model we use only \u00a0and \u00a0 which are estimates, and testing is based on estimated values in relation to the hypothesized population values. When conducting a simple linear regression, the estimated slope \\(\\hat\\beta_0\\) and the intercept \\(\\hat\\beta_1\\) are such that the sum of the squared vertical distance from the observations to the fitted line is minimized. The sum of squares error (SSE) is:<\/p>\n<p>$$\\sum_{i-1}^{n}(Y_{i}-\\hat{\\beta}_{0}-\\hat{\\beta}_{1}X_{i})^2$$<\/p>\n<p>$$\\sum_{i=1}^{n} e_{i}^{2}$$<\/p>\n<p>With the help of linear squares regression, we can estimate the values of the population parameters of\u00a0\\(\\hat\\beta_0\\) and \\(\\hat\\beta_1\\) A line can fit through observations of X and Y, explaining the value need by Y for any value of X. The residual and the dependent variable have the same units of measurement.<\/p>\n<p>To calculate the intercept \\(\\hat\\beta_0\\) and the slope \\(\\hat\\beta_1\\) for a given sample of (Y, X) pairs of observations, we divide the covariance of Y and X by the variance of X.<\/p>\n<p>$$\\hat{\\beta}_{1}=\\frac{\\text{Cov(X,Y)}}{\\text{Var(X)}}$$<\/p>\n<p>The intercept is calculated by using the mean of Y and the mean of X.<\/p>\n<p>$$\\hat{\\beta}_{0}=\\hat{Y}-\\hat{\\beta}_{1}\\hat{X}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(\\hat{Y}\\) = Mean of Y<\/li>\n<li data-tadv-p=\"keep\">\\(\\hat{X}\\) = Mean of X<\/li>\n<\/ul>\n<div style=\"text-align: center; margin: 25px 0;\"><a style=\"display: inline-block; padding: 12px 20px; border: 2px solid #2f5cff; border-radius: 999px; color: #2f5cff; text-decoration: none; background: #f7f9fc; white-space: nowrap;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Understand regression parameters with our free trial. <\/a><\/div>\n<h4><strong>Example: \u00a0Estimating Slope and Intercept for the ROA model<\/strong><\/h4>\n<p>$$\\small{\\begin{array}{l|l|l|l|l|l}\\textbf{Company}&amp;{\\textbf{ROA}\\\\ (\\textbf{Y}_{\\textbf{i}})}&amp;{\\textbf{CAPEX}\\\\ (\\textbf{X}_{\\textbf{i}})}&amp;{\\\\(\\textbf{Y}_{\\textbf{i}}-\\bar{\\textbf{Y}})^{2}}&amp;{\\\\ (\\textbf{X}_{\\textbf{i}}-\\bar{\\textbf{X}_{\\textbf{i}}})^{2}}&amp;{\\\\(\\textbf{Y}_{\\textbf{i}}-\\bar{\\textbf{Y}})^{2}}<br \/>\n\\\\\\hline\\text{1} &amp; 7.0 &amp; 0.9 &amp; 31.36 &amp; 23.43 &amp; 27.10 \\\\ \\hline \\text{2} &amp; 5.0 &amp; 0.6 &amp; 57.76 &amp; 26.42 &amp; 39.06 \\\\ \\hline \\text{3} &amp; 17.0 &amp; 5.2 &amp; 19.36 &amp; 0.29 &amp; -2.38 \\\\ \\hline\\text{4} &amp; 22.0 &amp; 12 &amp; 88.36 &amp; 39.19 &amp; 58.84 \\\\ \\hline \\text{5} &amp;12.0 &amp; 10 &amp;0.36 &amp; 18.15 &amp; -2.56\\\\ \\hline \\textbf{Sum} &amp; 63.0 &amp; 28.7 &amp; 197.2 &amp; 107.48 &amp; 120.06 \\\\ \\hline\\text{Mean} &amp; 12.6 &amp; 5.74 &amp; &amp; &amp;\\\\ \\end{array}}$$<\/p>\n<p>Slope coefficient: $$\\hat{\\beta}_{1}=\\frac{\\text{120.06}}{\\text{107.48}}=\u00a01.12$$\u00a0<\/p>\n<p>Intercept:$$\\hat{\\beta}_{0}=\\text{12.6}-\\text{(1.12}\\times\\text{5.74)}= 6.171$$\u00a0<\/p>\n<p>ROA regression model: $$Y=6.171+1.12X_1+\\epsilon$$<\/p>\n<p>&nbsp;<\/p>\n<h2><strong>Interpreting the regression coefficients<\/strong><\/h2>\n<p>The intercept is the expected mean value of the dependent variable when the value of the independent variable is zero. However, in some contexts, this is not always true. The slope is the one-unit change in the independent variable that will cause a change in the dependent variable. A positive slope indicates that the dependent and independent variables are moving in the same direction. On the other hand, a negative slope indicates that the dependent and independent variables are in opposite directions.<\/p>\n<p>From the previous example, we would interpret the coefficients as follows:<\/p>\n<ol>\n<li>The return on assets for a company is 6.171% if the company has no capital expenditure.<\/li>\n<li>If Capex increases by one unit, the ROA will increase by 1.12%.<\/li>\n<\/ol>\n<p>The regression line&#8217;s least squares fitting is that the residual term&#8217;s expected value is zero. The focus, however, is on minimizing the sum of the squared residual terms.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<h3><strong>Cross-Sectional vs. Time-series Regressions.<\/strong><\/h3>\n<p>A cross-sectional regression involves several observations of Y and X for the same period.\u00a0 The observations for this type of regression can be drawn from different sources.<\/p>\n<p>In a time-series regression, the data is drawn from observation of the same source but in different periods.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Jane estimates a model that regresses her company\u2019s return on equity (ROE) against its growth opportunities (GO). She comes up with the following linear regression:<\/p>\n<p>$$ROE_i=3+1.5GO_i+\\epsilon_i$$. The predicted value of the company\u2019s ROE if its GO is 10% is <em>closest<\/em> to:<\/p>\n<ol type=\"A\">\n<li>18%.<\/li>\n<li>1.5%.<\/li>\n<li>15%.<\/li>\n<\/ol>\n<h3>Solution<\/h3>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p>$$ROE_i =3+1.5\\times10 =18\\% $$<\/p>\n<p><strong>B is incorrect. <\/strong>1.5% is the slope and not the ROE.<\/p>\n<p><strong>C is incorrect. <\/strong>15% results from the multiplying slope by the independent variable.<\/p>\n<\/blockquote>\n<p>Reading 0: Introduction to Linear Regression<\/p>\n<p><em>LOS 0 (b) Describe the least squares criterion, how it is used to estimate regression coefficients, and their interpretation<\/em><\/p>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 20px; border-radius: 999px; background-color: #1a73e8; color: #ffffff; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a><\/p>\n<p style=\"font-size: 15px; margin-top: 12px; color: #555;\">Learn how to estimate regression coefficients, interpret slope and intercept, and apply least squares methods to CFA Level II quantitative exam questions.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>While conducting a regression analysis, we start with the dependent variable whose variation we want to explain and the independent variable that explains the changes in the dependent variable. The least-square criterion is used to measure the accuracy of a&#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":[232,216,230],"class_list":["post-11501","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-assumptions-underlying-linear-regression","tag-cfa-level-2","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>Estimating Parameters in Linear Regression<\/title>\n<meta name=\"description\" content=\"Learn how to estimate parameters in a simple linear regression model, including slope and intercept, and how they are used in model fitting.\" \/>\n<meta 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