{"id":36750,"date":"2021-10-29T18:15:49","date_gmt":"2021-10-29T18:15:49","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=36750"},"modified":"2026-04-22T17:44:06","modified_gmt":"2026-04-22T17:44:06","slug":"functional-forms-for-simple-linear-regression","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/functional-forms-for-simple-linear-regression\/","title":{"rendered":"Functional Forms for Simple Linear Regression"},"content":{"rendered":"<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/6ouI_27iDVM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For non-linear relationships, we can use several functional forms to potentially transform the data to be used in linear regression. The following are the three often-used log transformation functional forms:<\/p>\n<ol>\n<li><strong>Log-lin model<\/strong>: In this log transformation, the dependent variable is logarithmic while the independent variable is linear. It is represented as \\(lnY_i = b_0 + b_1 X_i\\). The slope coefficient in the log-lin model is the relative change in the dependent variable for an absolute change in the independent variable. That is, a one-unit increment in \\(X\\) triggers a \\(100b_1\\) percent change in \\(Y\\).<\/li>\n<li><strong>Lin-log model<\/strong>: In this case, the dependent variable is linear while the independent variable is logarithmic. It is represented as \\(Y_i = b_0 + b_1 lnX_i\\). The slope coefficient in the lin-log model is responsible for the absolute change in the dependent variable for a relative change in the independent variable. Put another way, if X increases by 1%, \\(Y\\) will change by \\(b_1\\) per 100 units.<\/li>\n<li><strong>Log-log model<\/strong>: In this log transformation, both the dependent and independent variables are logarithmic. It is represented as \\(lnY_i = b_0 + b_1 lnX_i\\). The slope coefficient in the log-log model is the relative change in the dependent variable for a relative change in the independent variable. In other words, if \\(X\\) increases by 1%, \\(Y\\) will change by \\(b_1\\)<\/li>\n<\/ol>\n<h2>Selecting the Correct Functional Form<\/h2>\n<p>To settle on the correct functional form, consider the following goodness of fit measures:<\/p>\n<ol style=\"list-style-type: upper-roman;\">\n<li>Coefficient of determination (R<sup>2<\/sup>). High value of is better.<\/li>\n<li>F-statistic. High value of F-statistic is better.<\/li>\n<li>Standard error of the estimate (S<sub>e<\/sub>). A low value of S<sub>e<\/sub> is better.<\/li>\n<\/ol>\n<p>Aside from the factors cited above, the patterns in residuals can also be analyzed when evaluating a \u00a0model. \u00a0Residuals are random and uncorrelated in a good model.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For non-linear relationships, we can use several functional forms to potentially transform the data to be used in linear regression. The following are the three often-used log transformation functional forms: Log-lin model: In this log transformation, the dependent variable is&#8230;<\/p>\n","protected":false},"author":15,"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-36750","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Functional Forms in Linear Regression | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore different functional forms in simple linear regression and their impact on model accuracy and interpretation.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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