{"id":13220,"date":"2021-04-11T04:01:17","date_gmt":"2021-04-11T04:01:17","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13220"},"modified":"2026-05-12T09:38:02","modified_gmt":"2026-05-12T09:38:02","slug":"autoregressive-models-and-multiperiod-forecasts","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/autoregressive-models-and-multiperiod-forecasts\/","title":{"rendered":"Autoregressive Models and Multiperiod Forecasts"},"content":{"rendered":"<p><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\": \"What is the two-step-ahead forecast for the AR(1) model?\",\r\n    \"text\": \"Consider an AR(1) model with the following prediction equation: x_t = 0.8 + 0.5 x_{t-1}. If the current value of x is 4.0, the two-step-ahead forecast is closest to: 0.64, 0.80, or 1.60.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is 0.64. First, the one-step-ahead forecast is x_{t+1} = 0.8 + 0.5 \u00d7 4.0 = 1.6. Then the two-step-ahead forecast is x_{t+2} = 0.8 + 0.5 \u00d7 1.6 = 0.64.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<h3 id=\"mce_22\" class=\"editor-rich-text__tinymce mce-content-body\" data-is-placeholder-visible=\"false\">\u00a0<\/h3>\r\n<p>The current-time values of a time series are related to the previous time values. This property is termed <strong>autoregressive.<\/strong> Autoregressive models are abbreviated (\\(AR_{p}\\)) models. \\(p\\) is known as the order of the model. It indicates the number of lagged values of the dependent variable used. More specifically, a time series that has been regressed on past values is an autoregressive model.<\/p>\r\n<p>In the autoregressive model, we <strong>abandon<\/strong> the <strong>dependent (y)<\/strong> and <strong>independent<\/strong> (x) notion and use\\(x_t\\) since there is no longer a difference.<\/p>\r\n<p>A <em>p<\/em>-order autoregressive model, \\(AR_{p}\\), is expressed as:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\text{b}_{2}\\text{x}_{\\text{t}-2}+&#8230;+\\text{b}_{\\text{p}}\\text{x}_{\\text{t}-\\text{p}}+\\epsilon_{\\text{t}}$$<\/p>\r\n<h3 style=\"text-align: justify;\">Example: Autoregressive Model<\/h3>\r\n<p>First-order autoregressive model: \\(\\text{AR}(1): \\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}\\)<\/p>\r\n<p>Second order autoregressive model: \\(\\text{AR}(2):\\text{x}_{t}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\text{b}_{2}\\text{x}_{\\text{t}-2}+\\epsilon_{\\text{t}}\\)<\/p>\r\n<p>Longer interval differences can be used to account for seasonality:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\text{b}_{2}\\text{x}_{\\text{t}-4}+\\epsilon_{\\text{t}}$$<\/p>\r\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 autoregressive models with our free trial. <\/a><\/div>\r\n<h2>Chain Rule of Forecasting<\/h2>\r\n<p>The chain rule of forecasting can be used to derive multiperiod forecasts using an \\(AR_{p}\\) model. It involves calculating a one-step-ahead forecast before a two-step ahead forecast as the independent variable is a lagged value of the dependent variable.<\/p>\r\n<h3>One-Step Ahead Forecast<\/h3>\r\n<p>The one-step-ahead forecast of an AR(1) model is given by:<\/p>\r\n<p>$$\\hat{\\text{x}}_{\\text{t}}=\\hat{\\text{b}}_{0}+\\widehat{\\text{b}}_{1}\\text{x}_{\\text{t}-1}$$<\/p>\r\n<h4>Example: Calculating the One-step Ahead Forecast<\/h4>\r\n<p>Given an AR(1) model where \\(\\hat{\\text{b}}_0=2\\) and \\(\\hat{\\text{b}}_1=1.8\\), the one-step-ahead forecast of \\(\\text{x}_{1}\\) when \\(\\text{x}_0=2\\) is <em>closest<\/em> to:\u00a0<\/p>\r\n<h5>Solution<\/h5>\r\n<p>$$\\hat{\\text{x}}_{\\text{t}}=\\hat{\\text{b}}_{0}+\\widehat{\\text{b}}_{1}\\text{x}_{\\text{t}-1}$$<\/p>\r\n<p>$$\\hat{\\text{x}}_{\\text{1}}=2+1.8\\times2=5.6$$<\/p>\r\n<h3>Two-Step Ahead Forecast<\/h3>\r\n<p>The two-step ahead forecast for an AR(1) model is determined as:<\/p>\r\n<p>$$\\hat{\\text{x}}_{\\text{t}+1}=\\hat{\\text{b}}_{0}+\\hat{\\text{b}}_{1}\\hat{\\text{x}}_{\\text{t}}$$<\/p>\r\n<h4>Example: Calculating the Two-step Ahead Forecast<\/h4>\r\n<p>Calculate the two-step ahead forecast of \\(\\text{x}_{2}\\)<\/p>\r\n<p>$$\\hat{\\text{x}}_{2}=2+1.8\\hat{\\text{x}}_1$$<\/p>\r\n<p>$$\\hat{\\text{x}}_{2}=2+1.8\\times5.6=12.08$$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Consider an AR(1) model with the following prediction equation:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=0.8+0.5\\text{x}_{\\text{t}-1}$$<\/p>\r\n<p>If the current value of \\(x\\) is 4.0, the two-step-ahead forecast is <em>closest to:<\/em><\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>0.64.<\/li>\r\n\t<li>0.80.<\/li>\r\n\t<li>1.60.<\/li>\r\n<\/ol>\r\n<h3>Solution<\/h3>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>One-step ahead forecast:<\/p>\r\n<p>If \\(\\text{x}_{\\text{t}}=4,\\) then \\(\\hat{\\text{x}}_{\\text{t}+1}=0.8+0.5\\times4=1.6\\)<\/p>\r\n<p>Two-step ahead forecast:\u00a0<\/p>\r\n<p>If \\(\\hat{\\text{x}}_{\\text{t}+1}=1.6,\\) then \\(\\hat{\\text{x}}_{\\text{t}+2}=0.8+0.5\\times1.6=0.64\\)<\/p>\r\n<\/blockquote>\r\n<p>Reading 5: Time Series Analysis<\/p>\r\n<p><em>LOS 5 (d) Describe the structure of an autoregressive (AR) model of order <\/em>p <em>and calculate one- and two period-ahead forecasts given the estimated coefficients.<\/em><\/p>\r\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>\r\n<p style=\"margin-top: 10px;\">Learn AR models, lagged variables, and time-series forecasting with clear study tools.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u00a0 The current-time values of a time series are related to the previous time values. This property is termed autoregressive. Autoregressive models are abbreviated (\\(AR_{p}\\)) models. \\(p\\) is known as the order of the model. It indicates the number of&#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":[290,216,230],"class_list":["post-13220","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-autoregressive-models-and-multiperiod-forecasts","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 v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Autoregressive Models and Forecasting | CFA Level II<\/title>\n<meta name=\"description\" content=\"Learn autoregressive models, AR(1) models, and multiperiod forecasting techniques using estimated coefficients in time series analysis.\" \/>\n<meta name=\"robots\" content=\"index, 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