{"id":13128,"date":"2021-04-09T20:08:24","date_gmt":"2021-04-09T20:08:24","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13128"},"modified":"2026-01-07T20:17:34","modified_gmt":"2026-01-07T20:17:34","slug":"predicted-trend-value-of-a-time-series","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/predicted-trend-value-of-a-time-series\/","title":{"rendered":"Predicted Trend Value of a Time Series"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"VideoObject\",\r\n  \"@id\": \"https:\/\/analystprep.com\/videos\/times-series-analysis-2025-level-ii-cfa-quantitative-methods-module-5\",\r\n  \"name\": \"Time-series Analysis (2025 Level II CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 5)\",\r\n  \"description\": \"In this comprehensive lesson, Professor James Forjan, PhD, CFA, teaches Time-Series Analysis for the CFA Level II Quantitative Methods topic. This reading builds directly on regression analysis concepts from Level I and introduces time-based data modeling for financial forecasting and trend analysis. You will learn how to identify patterns in financial data, test for stationarity, model relationships between past and current values, and forecast future outcomes using autoregressive and moving-average processes.\",\r\n  \"uploadDate\": \"2021-12-24\",\r\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/-SilFtkpBK8\/0.jpg\",\r\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=-SilFtkpBK8\",\r\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8\",\r\n  \"duration\": \"PT55M01S\",\r\n  \"publisher\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Log-linear trend model prediction\",\r\n    \"text\": \"Consider a log-linear trend model with an intercept term of 0.5 and a slope coefficient of 0.10. The predicted value of the trend after six periods is closest to:\\n\\nA. 1.1.\\n\\nB. 1.6.\\n\\nC. 3.0.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"C. 3.0.\\n\\nIn a log-linear trend model, the predicted value is calculated using the formula:\\n\\ny_t = e^(b0 + b1 * t)\\n\\nSubstituting the given values:\\n\\ny_t = e^(0.5 + (0.10 \u00d7 6)) = e^(3.0) \u2248 3.0.\\n\\nTherefore, the predicted value after six periods is 3.0.\"\r\n    }\r\n  }\r\n}\r\n<\/script>\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/log-linear-trends\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3.jpg\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3.jpg\",\r\n  \"caption\": \"Log-Linear Trends\",\r\n  \"width\": 1590,\r\n  \"height\": 1217,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/euro-usd-historical-trends\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2.jpg\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2.jpg\",\r\n  \"caption\": \"Euro\/USD Historical Trends\",\r\n  \"width\": 1590,\r\n  \"height\": 1148,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/example-of-time-series\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1.jpg\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1.jpg\",\r\n  \"caption\": \"Example of Time Series\",\r\n  \"width\": 1590,\r\n  \"height\": 1212,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8?si=DAD8tgXNSn509CuT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\r\n<p>A time series shows data on a variable&#8217;s outcome in different periods\u2014for example, a time series showing EURUSD exchange rates between a given time interval.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14865\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1.jpg\" alt=\"Example of Time Series\" width=\"1590\" height=\"1212\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1-300x229.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1-1024x781.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1-768x585.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1-1536x1171.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_1-400x305.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Time series models explain the past and predict the future of a time series using a trend, which is a long-term pattern in a specific direction.<\/p>\r\n<div class=\"cta-button\">\r\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\r\n     class=\"button small-button\"\r\n     target=\"_blank\"\r\n     rel=\"noopener noreferrer\">\r\n    Want to master time-series trend forecasts with exam-style questions? Start a free trial.\r\n  <\/a>\r\n<\/div>\r\n\r\n<h2>Types of Trends<\/h2>\r\n<h3>Linear Trend Models<\/h3>\r\n<p>In a linear trend, the dependent variable changes at a constant amount with time. Consider a time series with a linear trend. The value of the time series at time \\(t\\) can be expressed as:<\/p>\r\n<p>$$\\text{y}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{t}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>Where:<\/p>\r\n<p>\\(\\text{y}_{\\text{t}}\\) = Value of the time series at time \\(t\\).<\/p>\r\n<p>\\(\\text{b}_{0}\\) = Intercept term.<\/p>\r\n<p>\\(\\text{b}_{1}\\) = Trend coefficient.<\/p>\r\n<p>\\(\\text{t}\\) = Time, the independent variable. \\(t=1, 2, &#8230;, \\text{T}\\).<\/p>\r\n<p>\\(\\epsilon_{\\text{t}}\\) = Random error term.<\/p>\r\n<p>The following shows a linear trend of the EURUSD between 05 January 2021 to 06 February 2021.<\/p>\r\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14867\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2.jpg\" alt=\"Euro\/USD Historical Data\" width=\"1590\" height=\"1148\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2-300x217.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2-1024x739.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2-768x555.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2-1536x1109.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_2-400x289.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Predicted Value of a Linear Trend<\/h4>\r\n<p>The predicted value of a time series with a linear trend in period five is expressed as:<\/p>\r\n<p>$$\\widehat{y_{5}}=\\widehat{b_{0}}+\\widehat{b_{1}}(5)$$<\/p>\r\n<h4>Out of Sample Prediction<\/h4>\r\n<p>The predicted value of a time series for period \\(T+1\\) is given by:<\/p>\r\n<p>$$\\widehat{\\text{y}}_{T+1}=\\widehat{b}_{0}+\\widehat{b_{1}}(T+1)$$<\/p>\r\n<h4>Example: Predicted Value of a Linear trend<\/h4>\r\n<p>Given that \\(b_{0}=2\\) and \\(b_{1}=1.6\\), the predicted value of y after four periods is closest to:<\/p>\r\n<p>$$y_{t}=b_{0}+b_{1}t+\\epsilon_{t}$$<\/p>\r\n<p>$$y_{4}=2+1.6\\times4=8.4$$<\/p>\r\n<h2>Log-Linear Trend Models<\/h2>\r\n<p>A log-linear trend analysis relies on an independent variable that changes exponentially over time or grows at a constant rate. It often works better with financial time series.<\/p>\r\n<p>A time series with exponential growth can be modeled as:<\/p>\r\n<p>$$y_{t}=e^{b_{0}+b_{1}(t)}$$<\/p>\r\n<p>Where:<\/p>\r\n<p>\\(y_{t}\\) = The value of the dependent variable at time \\(t\\).<\/p>\r\n<p>\\(b_{0}\\) = Intercept term.<\/p>\r\n<p>\\(b_{1}\\) = Constant growth rate.<\/p>\r\n<p>\\(t\\) = time = \\(1,2,3,&#8230;,T\\).<\/p>\r\n<p>Taking the natural log on both sides of the above equation generates a log-linear model expressed as:<\/p>\r\n<p>$$\\text{lny}_{\\text{t}}=\\text{ln ln}(e^{b_{0}+b_{1}t})$$<\/p>\r\n<p>$$\\text{ln ln}\\ \\text{y}_{\\text{t}}=\\ \\text{b}_{0}+\\text{b}_{1}\\text{t}$$<\/p>\r\n<p>The following figure shows different forms of log-linear trends.<\/p>\r\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14871\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3.jpg\" alt=\"Log-linear Trends\" width=\"1590\" height=\"1297\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3-300x245.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3-1024x835.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3-768x626.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3-1536x1253.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_3-400x326.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Example: Predicted Value of a Log-Linear Trend<\/h4>\r\n<p>If \\(b_{0}=3\\) and \\(b_{1}=1.5\\),\u00a0\u00a0the predicted value of y after four periods is:<\/p>\r\n<p>$$\\text{ln y}_{t}=b_{0}+b_{1}t+\\epsilon_{t}$$<\/p>\r\n<p>$$\\text{ln y}_{4}= 3+1.5\\times4$$<\/p>\r\n<p>$$\\text{y}_{4}=e^{9}=8103.08$$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Consider a log-linear trend model with an intercept term of 0.5 and a slope coefficient of 0.10. The predicted value of the trend after six periods is <em>closest to:<\/em><\/p>\r\n<p>\u00a0 \u00a0 \u00a0A. 1.1.<\/p>\r\n<p>\u00a0 \u00a0 \u00a0B. 1.6.<\/p>\r\n<p>\u00a0 \u00a0 \u00a0C. 3.0.<\/p>\r\n<h3>Solution<\/h3>\r\n<p>The correct answer is <strong>C.<\/strong><\/p>\r\n<p>$$y_{t}=e^{b_{0}+b_{1}(t)}$$<\/p>\r\n<p>$$y_{t}=e^{0.5+(0.10\\times6)}=3.0$$<\/p>\r\n<\/blockquote>\r\n<p>Reading 5: Time Series Analysis<\/p>\r\n<p><em>LOS 5 (a)\u00a0Calculate and evaluate the predicted trend value for a time series, modeled as either a linear trend or a log-linear trend, given the estimated trend coefficients.<\/em><\/p>\r\n<div class=\"cta-button\">\r\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\r\n     class=\"button large-button\"\r\n     target=\"_blank\"\r\n     rel=\"noopener noreferrer\">\r\n    Start Free Trial \u2192<br>\r\n    <span style=\"font-weight: normal;\">\r\n      Practice trend models and time-series forecasting with full solutions\r\n    <\/span>\r\n  <\/a>\r\n<\/div>\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>A time series shows data on a variable&#8217;s outcome in different periods\u2014for example, a time series showing EURUSD exchange rates between a given time interval. Time series models explain the past and predict the future of a time series using&#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":[286],"class_list":["post-13128","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-predicted-trend-value-of-a-time-series","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>Predicted Trend Value of a Time Series | CFA Level II<\/title>\n<meta name=\"description\" content=\"Explains how to estimate the predicted trend value in a time series using regression, trend components, and time-based data examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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