{"id":13414,"date":"2021-04-13T14:04:05","date_gmt":"2021-04-13T14:04:05","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13414"},"modified":"2026-05-01T15:57:22","modified_gmt":"2026-05-01T15:57:22","slug":"unit-roots-for-time-series-analysis","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/unit-roots-for-time-series-analysis\/","title":{"rendered":"Unit Roots for Time-Series Analysis"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the most appropriate way to transform a time series with a unit root?\",\n    \"text\": \"Which of the following is the most appropriate approach to transforming a time series with a unit root problem?\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The most appropriate approach is to model the first differences of the time series. A time series with a unit root is non-stationary in levels but can be transformed into a covariance-stationary series by first differencing. This involves subtracting the previous period\u2019s value from the current value and then modeling the differenced series.\",\n      \"dateCreated\": \"2026-01-21\"\n    }\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\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8\" width=\"611\" height=\"344\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h3>\n<h2>The Unit Root Problem<\/h2>\n<p>An AR(1) series is said to be covariance stationary if the absolute value of the lag coefficient \\(\\text{b}_{1}\\) is less than 1. If the absolute value of \\(\\text{b}_{1}=1\\), the time series is said to have a unit root. All random walks have a unit root since they have \\(\\text{b}_{1}=1\\). This implies that they are not covariance stationary; hence we cannot apply the standard linear regression to test for \\(\\text{b}_{1}=1\\).<\/p>\n<p>A Dickey-Fuller test can be used to establish if the time series has a unit root. A time series with unit roots should be transformed by first-differencing it to a covariance stationary time series, which can be effectively analyzed using regression analysis.<\/p>\n<p>First-differencing is a technique that involves subtracting the dependent variable in the immediately preceding period from the current value of the time series to define a new dependent variable, \\(y\\). Thus, we model the change in the value of the dependent variable.<\/p>\n<p>$$\\text{y}_{\\text{t}}=\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\epsilon_{\\text{t}}$$<\/p>\n<p>The first-differenced time series can then be modeled as an autoregressive time series. A properly differenced random walk time series is covariance stationary with a mean reversion level of 0.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Which of the following is the <em>most appropriate<\/em> approach to transforming a time series with a unit root problem?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>Performing a Dickey-Fuller test.<\/li>\n<li>Modeling the first differences of the time series.<\/li>\n<li>Performing a log-linear transformation.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p><strong>The correct answer is B.<\/strong><\/p>\n<p>A time series with a unit root can be first differenced to transform it into one that is covariance stationary. This is followed by estimating an autoregressive model for the first-differenced series.<\/p>\n<p>First differencing involves subtracting the value of the time series in the immediately preceding period from the current value of the series to define a new dependent variable, \\(y\\).<\/p>\n<p><strong>A is incorrect. <\/strong>The Dickey fuller test is used to determine if the time series has a unit root.<\/p>\n<p><strong>C is incorrect.<\/strong>\u00a0A log-linear transformation is appropriate when data grows at a constant rate.<\/p>\n<\/blockquote>\n<p><em>Reading 5: Time Series Analysis <\/em><\/p>\n<p><em>LOS 5 (j) Describe implications of unit roots for time-series analysis, explain when unit-roots are likely to occur and how to test for them, and demonstrate how a time series with a unit root can be transformed so it can be analyzed with an AR model.<\/em><\/p>\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><\/p>\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 statistical testing techniques.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Unit Root Problem An AR(1) series is said to be covariance stationary if the absolute value of the lag coefficient \\(\\text{b}_{1}\\) is less than 1. If the absolute value of \\(\\text{b}_{1}=1\\), the time series is said to have a&#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":[216,230,296],"class_list":["post-13414","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-quantitative-method","tag-unit-roots-for-time-series-analysis","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>Unit Roots in Time Series | AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn what a unit root is in time series analysis and how it affects stationarity, model validity, and forecasting accuracy.\" \/>\n<meta name=\"robots\" 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