{"id":13178,"date":"2021-04-11T01:37:30","date_gmt":"2021-04-11T01:37:30","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13178"},"modified":"2026-05-01T14:53:52","modified_gmt":"2026-05-01T14:53:52","slug":"linear-or-log-linear-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/linear-or-log-linear-model\/","title":{"rendered":"Linear or Log-Linear 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\": \"Which model is most appropriate for predicting a stock price that has increased over the last decade at a relatively constant growth rate?\",\n    \"text\": \"Neeth Shinu, a portfolio manager, has determined that the price of a certain stock has increased over the last decade, but its growth rate has remained relatively constant. The model that is most appropriate for predicting the future prices of the stock is:\\n\\nA. Pt = b0 + b1(t) + \u03b5t\\nB. ln(pt) = b0 + b1(pt\u22121) + \u03b5t\\nC. ln(pt) = b0 + b1(t) + \u03b5t\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"C. ln(pt) = b0 + b1(t) + \u03b5t\"\n    }\n  }\n}\n<\/script> <script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/linear-log-linear-model-img-11\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-768x459.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-768x459.jpg\",\n  \"caption\": \"Example of Linear or Log-Linear Model\",\n  \"width\": 768,\n  \"height\": 459,\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  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/linear-or-log-linear-model\/\"\n  }\n}\n<\/script><\/p>\n<h3 id=\"mce_22\" class=\"editor-rich-text__tinymce mce-content-body\" data-is-placeholder-visible=\"false\"><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/-SilFtkpBK8?autoplay=0&amp;loop=0&amp;rel=0\" width=\"611\" height=\"344\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h3>\n<p>A linear trend model should model a time series that increases over time by a constant amount. On the other hand, a time series that grows at a constant rate should be modeled by a log-linear model.<\/p>\n<p>To decide between linear and log-linear trend models, one should plot the data. If the data points are equally distributed above and below the regression line, use a linear trend model.<\/p>\n<p>On the other hand, if the data points are persistently above or below the trend line, the residuals are serially correlated. This is a violation of one of the key regression model assumptions that residuals are uncorrelated across all observations. For instance, stock prices and sales data are best modeled using log-linear models.<\/p>\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 time-series models with our free trial. <\/a><\/div>\n<h2><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14876\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11.jpg\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-300x179.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-1024x612.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-768x459.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-1536x919.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_11-400x239.jpg 400w\" alt=\"Linear vs Log-linear Trend Models\" width=\"1590\" height=\"951\" \/>Limitations of Trend Models<\/h2>\n<p>The error terms may be correlated. This violates the linear regression assumption that residuals are uncorrelated with each other, referred to as serial correlation.<\/p>\n<p>Serial correlation makes the linear regression estimates inconsistent and invalid. Serial correlation can be detected by performing a Durbin-Watson test. If the Durbin-Watson statistic is significantly different from 2, then the residual terms are serially correlated.<\/p>\n<p>We can correct serial correlation by transforming the data or using other regression models such as an autoregressive model, which will be discussed later in this reading.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Neeth Shinu, a portfolio manager, has determined that the price of a certain stock has increased over the last decade, but its growth rate has remained relatively constant. The model that is <em>most appropriate<\/em> for predicting the future prices of the stock is:<\/p>\n<ol type=\"A\">\n<li>\\(\\text{P}_{\\text{t}}+\\text{b}_{0}+\\text{b}_{1}(\\text{t})+\\epsilon_{\\text{t}}\\)<\/li>\n<li>\\(\\text{ln}(\\text{p}_{\\text{t}})=\\text{b}_{0}+\\text{b}_{1}(\\text{p}_{\\text{t}-1})+\\epsilon_{\\text{t}}\\)<\/li>\n<li>\\(\\text{ln}(\\text{p}_{\\text{t}})=\\text{b}_{0}+\\text{b}_{1}(\\text{t})+\\epsilon_{\\text{t}}\\)<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p><strong>The correct answer is C.<\/strong><\/p>\n<p>A log-linear model is most suitable for a time series that grows at a constant growth rate.<\/p>\n<p><strong>A is incorrect.<\/strong>\u00a0It is a linear trend model which is suitable for a time series that increases over time by a constant amount.<\/p>\n<p><strong>\u00a0<\/strong><strong>B is incorrect.<\/strong>\u00a0This is an autoregressive model, which is appropriate in the presence of serial correlation.<\/p>\n<\/blockquote>\n<p>Reading 5: Time Series Analysis<\/p>\n<p><em>LOS 5 (b) Describe factors that determine whether a linear or a log-linear trend should be used with a particular time series and evaluate limitations of trend models.<\/em><\/p>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 14px 26px; background: #4a76d1; color: #fff; border-radius: 999px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a><\/p>\n<p style=\"margin-top: 10px;\">Learn linear vs log-linear models, trend analysis, and forecasting techniques with clear study tools.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A linear trend model should model a time series that increases over time by a constant amount. On the other hand, a time series that grows at a constant rate should be modeled by a log-linear model. To decide between&#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":[216,287,288],"class_list":["post-13178","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-linear-or-log-linear-model","tag-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>Linear vs Log-Linear Models | AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn when to use linear and log-linear models in time series analysis and how each model captures trends and growth rates.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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