{"id":13350,"date":"2021-04-13T12:49:39","date_gmt":"2021-04-13T12:49:39","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13350"},"modified":"2026-05-01T15:46:30","modified_gmt":"2026-05-01T15:46:30","slug":"random-walk-process","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/random-walk-process\/","title":{"rendered":"Random Walk Process"},"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 most accurate statement about a random walk?\",\r\n    \"text\": \"Which of the following statements is most accurate about a random walk? A. It has a finite mean-reverting level. B. It has an undefined mean-reverting level. C. It is covariance stationary.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"A random walk has an undefined mean-reverting level. As a result, it does not revert to a long-run mean and is not covariance stationary.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<h3 id=\"mce_22\" class=\"editor-rich-text__tinymce mce-content-body\" data-is-placeholder-visible=\"false\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8\" width=\"611\" height=\"344\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h3>\r\n<p>A time series is said to follow a <strong>random walk process<\/strong> if the predicted value of the series in one period is equivalent to the <strong>value<\/strong> of the series in the <strong>previous period<\/strong> <strong>plus<\/strong> a<strong> random error<\/strong>.<\/p>\r\n<p>A simple random walk process can be expressed as follows:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>Where:<\/p>\r\n<ul>\r\n\t<li>\\(\\text{x}_{\\text{t}}\\) = Best prediction tomorrow.<\/li>\r\n\t<li>\\(\\text{x}_{\\text{t}-1}\\) = Best value today.<\/li>\r\n\t<li>\\(\\epsilon_{\\text{t}}\\) = Random error term.<\/li>\r\n<\/ul>\r\n<div style=\"background-color: #f5f7fa; padding: 20px; text-align: center; margin: 30px 0; border-radius: 8px;\">\r\n<div style=\"max-width: 620px; margin: 0 auto;\"><a style=\"display: flex; align-items: center; justify-content: center; width: 100%; padding: 10px 20px; border: 2px solid #1a73e8; border-radius: 999px; text-decoration: none; color: #1a73e8; font-size: 15px; font-weight: 600; line-height: 1.4;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Practice random walk models with our Free Trial <\/a><\/div>\r\n<\/div>\r\n<h2>Characteristics of Random Walk Time Series<\/h2>\r\n<ul>\r\n\t<li>An AR(1) time series with \\(\\beta_{0}=0\\) and \\(\\beta_{1}=1\\) is a random walk. This is because the best prediction for tomorrow is the best value today plus a random error term.<\/li>\r\n\t<li>The expected value of the <strong>error term<\/strong> \\(\\epsilon_{\\text{t}}\\) is equal to zero.<\/li>\r\n\t<li>The variance of the residuals is constant.\u00a0<\/li>\r\n\t<li><strong>Random walk with drift<\/strong>: A time series follows a random walk with drift if it has a non-zero constant intercept term. It is expressed as:<\/li>\r\n<\/ul>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>Where:<\/p>\r\n<ul>\r\n\t<li>\\(\\text{b}_{0}\\) = Constant drift<\/li>\r\n\t<li>\\(\\text{b}_{1}\\) = 1<\/li>\r\n<\/ul>\r\n<p>Note that a random walk is expressed as:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n<ul>\r\n\t<li>\\(\\text{b}_{0}=0\\) for a random walk without drift;<\/li>\r\n\t<li>\\(\\text{b}_{0}\\neq 0\\) for a random walk with drift; and<\/li>\r\n\t<li>\\(\\text{b}_{1}=1\\) for a random walk with or without drift.<\/li>\r\n<\/ul>\r\n<p>A random walk has an <strong>undefined mean reversion level<\/strong>. If has a mean-reverting level, i.e., \\(\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}},\\) then \\(\\text{x}_{\\text{t}}=\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}\\). However, in a random walk, \\(\\text{b}_{0}=0\\) and \\(\\text{b}_{1}=1\\), so, \\(\\text{x}_{\\text{t}}=\\frac{0}{1-1}=0\\).<\/p>\r\n<p>A random walk is <strong>not covariance stationary.<\/strong> The covariance stationary property suggests that the mean and variance terms of a time series remain constant over time. However, the variance of a random walk process does not have an upper bound. As \\(t\\) increases, the variance grows with no upper bound. This implies that we cannot use standard regression analysis on a time series that appears to be a random walk.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The <em>most accurate<\/em> statement about a random walk is that it:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>Has a finite mean-reverting level.<\/li>\r\n\t<li>Has an undefined mean-reverting level.<\/li>\r\n\t<li>Is covariance stationary.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is B.<\/strong><\/p>\r\n<p>A random walk process has an undefined mean-reverting level, and thus it is not covariance stationary.<\/p>\r\n<\/blockquote>\r\n<p>Reading 5: Time Series Analysis<\/p>\r\n<p><em>LOS 5 (i) Describe characteristics of random walk processes and contrast them to covariance stationary processes.<\/em><\/p>\r\n<p>&nbsp;<\/p>\r\n<div style=\"background-color: #f5f7fa; padding: 32px 20px; text-align: center; margin: 40px 0; border-radius: 8px;\"><a style=\"display: inline-block; background-color: #1a73e8; color: #ffffff; text-decoration: none; font-size: 16px; font-weight: 600; padding: 12px 28px; border-radius: 999px; line-height: 1.4;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"margin: 14px auto 0; max-width: 620px; font-size: 15px; line-height: 1.6; color: #444444;\">Build CFA Level II readiness with structured study materials, guided learning, and focused practice on time series models and forecasting concepts.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>A time series is said to follow a random walk process if the predicted value of the series in one period is equivalent to the value of the series in the previous period plus a random error. A simple random&#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":[216,230,295],"class_list":["post-13350","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-quantitative-method","tag-random-walk-process","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>Random Walk Process Explained<\/title>\n<meta name=\"description\" content=\"Learn random walk processes in time series analysis, including random walk with drift and how they differ from stationary processes.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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