{"id":36694,"date":"2021-10-29T07:14:46","date_gmt":"2021-10-29T07:14:46","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=36694"},"modified":"2026-03-11T15:56:07","modified_gmt":"2026-03-11T15:56:07","slug":"assumptions-underlying-linear-regression","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/assumptions-underlying-linear-regression\/","title":{"rendered":"Assumptions Underlying Linear Regression"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-1024x815.jpg\",\n  \"caption\": \"Correlated Error\",\n  \"width\": 1024,\n  \"height\": 815,\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  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-1024x849.jpg\",\n  \"caption\": \"Heteroskedasticity\",\n  \"width\": 1024,\n  \"height\": 849,\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  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1.jpg\",\n  \"caption\": \"Homoskedasticity\",\n  \"width\": 1590,\n  \"height\": 1341,\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  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation.jpg\",\n  \"caption\": \"Nonlinear Relation\",\n  \"width\": 1590,\n  \"height\": 1286,\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  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/6ouI_27iDVM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The classic normal linear regression model assumptions are as follows:<\/p>\n<p><strong>I.<\/strong> The relationship between the dependent variable, <em>Y<\/em>, and the independent variable, <em>X<\/em>, is <strong>linear<\/strong>.<\/p>\n<p>A linear relationship implies that the change in <em>Y<\/em> due to a one unit change in <em>X<\/em> is constant, regardless of the value taken by <em>X<\/em>. If the relationship between the two is not linear, the regression model will not capture the trend accurattely, a situation that will result in inaccurate predictions. The model will be biased and either underestimate or overestimate <em>Y<\/em> at various points. For example, the model \\(Y = \\beta_{0}+\\beta_{1}e^{\\beta_{1}x}\\) is non linear in \\(\u03b2_{1}\\), and therefore we should not attempt to fit a linear model between <em>X<\/em> and<em> Y.<\/em><\/p>\n<p>It also follows that&nbsp;the independent variable, <em>X<\/em>, must be non stochastic (must not be random). A random independent variable rules out a linear relationship between the dependent and independent variables.<\/p>\n<p>In addition, linearity means the residuals should not exhibit a discernible pattern when plotted against the independent variable but should instead be completely random. In the example below, we&#8217;re looking at a scenario where the residuals appear to show a pattern when plotted against the independent variable, <em>X<\/em>. This effectively serves as evidence of a non-linear relation.&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39327\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation.jpg\" alt=\"\" width=\"1590\" height=\"1286\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation-300x243.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation-1024x828.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation-768x621.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation-1536x1242.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Non-linear-Relation-400x324.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p><strong>II.<\/strong> The expectation of the error terms is <strong>zero<\/strong>.<\/p>\n<p>$$E(\\epsilon)=0$$<\/p>\n<p><strong>III. <\/strong>The error terms (residuals) must be normally distributed.<\/p>\n<p>A histogram of the residuals can be used to detect if the error term is normally distributed. A symmetric bell-shaped histogram indicates that the normality assumption is likely to be true.&nbsp;<\/p>\n<p><strong>IV. <\/strong>The variance of the error terms is <strong>constant<\/strong> across all observations.<\/p>\n<p>$$E(\\epsilon_i^2)=\\sigma_{\\epsilon}^2,\\ i=1,2, &#8230;, n$$<\/p>\n<p>This assumption is also known as the <strong>homoskedasticity<\/strong> assumption.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39330\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1.jpg\" alt=\"\" width=\"1590\" height=\"1341\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1-300x253.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1-1024x864.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1-768x648.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1-1536x1295.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/homoskedasticity-1-400x337.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p>In case residuals and the predicted values increase simultaneously, then such a situation is known as <strong>heteroscedasticity <\/strong>(or heteroskedasticity).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39331\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1.jpg\" alt=\"\" width=\"1590\" height=\"1319\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-300x249.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-1024x849.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-768x637.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-1536x1274.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Heteroskedasticity-1-400x332.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p>To test for heteroscedasticity, one ought to plot the least square residuals against the independent variable. If there is an evident pattern in the plot, then that is a manifestation of heteroskedasticity.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>V.<\/strong> The error terms, \\(\\epsilon\\), must be <strong>uncorrelated<\/strong> across all observations.<\/p>\n<p>$$E(\\epsilon_i\\epsilon_j)=0,\\ \\forall i\\neq j$$<\/p>\n<p>To verify this assumption, &nbsp;one should use a residual time series plot, which is a plot of residuals versus time. Fluctuating patterns around zero will indicate that the error term is dependent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39325\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error.jpg\" alt=\"\" width=\"1590\" height=\"1266\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-300x239.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-1024x815.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-768x612.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-1536x1223.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/Correlated-Error-400x318.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>","protected":false},"excerpt":{"rendered":"<p>The classic normal linear regression model assumptions are as follows: I. The relationship between the dependent variable, Y, and the independent variable, X, is linear. A linear relationship implies that the change in Y due to a one unit change&#8230;<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-36694","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Linear Regression Assumptions | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore key assumptions in linear regression, including non-random independent variables, zero-mean error terms, and homoscedasticity.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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