{"id":13431,"date":"2021-04-13T19:18:00","date_gmt":"2021-04-13T19:18:00","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13431"},"modified":"2026-06-01T17:28:18","modified_gmt":"2026-06-01T17:28:18","slug":"the-unit-root-test-for-nonstationary","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/the-unit-root-test-for-nonstationary\/","title":{"rendered":"The Unit Root Test for Nonstationary"},"content":{"rendered":"<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 does it mean if the null hypothesis cannot be rejected in a Dickey-Fuller test?\",\r\n    \"text\": \"If the null hypothesis H0: g1 = 0 under the Dickey-Fuller test cannot be rejected, what is the most accurate conclusion? A. The time series is covariance stationary. B. The time series does not have a unit root. C. The time series has a unit root.\",\r\n    \"answerCount\": 3,\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A. The time series is covariance stationary.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"B. The time series does not have a unit root.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"C. The time series has a unit root.\"\r\n      }\r\n    ],\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is C. In the Dickey-Fuller test, failing to reject the null hypothesis implies that b1 \u2212 1 = 0, or equivalently b1 = 1. This indicates that the time series contains a unit root. A time series with a unit root is not covariance stationary. If the null hypothesis is rejected, then the series does not have a unit root and may be covariance stationary.\"\r\n    },\r\n    \"author\": {\r\n      \"@type\": \"Organization\",\r\n      \"name\": \"AnalystPrep\"\r\n    }\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=dqlbUiTKbT5p_Abh\" 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\r\n<p>Unit root testing involves checking whether the time series is covariance stationary. We can either form an AR model and check for autocorrelations or perform a Dickey and Fuller test.<\/p>\r\n<p>A t-test is performed to examine the statistical significance of autocorrelations at various lags. The autocorrelations of a stationary process will be insignificantly different from zero at all lags or decrease rapidly to zero as the number of lags becomes large.<\/p>\r\n<p>Recall that an AR (1) model is said to be covariance stationary if the absolute value of \u00a0is less than 1. Additionally, the time series will have a unit root if the lag coefficient equals 1. However, <strong>random walks with drift<\/strong> are <strong>NOT<\/strong> covariance stationary. Therefore, we cannot use standard linear regression to test for covariance stationary\u00a0Dickey-Fuller unit root test is therefore used to test for unit roots.<\/p>\r\n<div style=\"margin: 18px 0;\"><a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">Strengthen your CFA Level II unit root test concepts with our Free Trial.<\/a><\/div>\r\n<h2>Dickey-Fuller Test<\/h2>\r\n<p>The Dickey-Fuller test uses the transformed AR(1) model by substracting \u00a0from both sides of the equation. i.e.,<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+b_{1}x_{t-1}+\\epsilon_{t}$$<\/p>\r\n<p>$$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}-\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>$$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{x}_{\\text{t}-1}(\\text{b}_{1}-1)+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>Let the transformed coefficient, \\(\\text{b}_{1}-1=\\text{g}\\). Then,<\/p>\r\n<p>$$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{g}_{1}\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>We then perform a t-test with the null hypothesis \\(\\text{H}_{0}:\\text{g}_{1}=0\\) (a test of \\(\\text{b}_1={1}\\), which implies the time-series has a unit root), against the alternative hypothesis \\(\\text{H}_{\\text{a}}:\\text{g}_{1}&lt;0\\) (a test of \\(\\text{b}_1={1}\\), which implies the time series has no unit root).<\/p>\r\n<p>If \\(\\text{g}_{1}\\) is not significantly different from 0, then \\(\\text{b}_{1}= 0\\), the series must have a unit root. The null hypothesis will be rejected if the time series does not have a unit root. On the other hand, failure to reject the null hypothesis means that the time series has a unit root.<\/p>\r\n<p>A time series with a unit root problem can be modeled by modeling the first differenced series with an autoregressive time series.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>If the null hypothesis \\(\\text{H}_{0}:\\text{g}_1=0\\) under the Dickey-Fuller test cannot be rejected; the <em>most accurate<\/em> conclusion is that:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>The time series is covariance stationary.<\/li>\r\n\t<li>The time series does not have a unit root.<\/li>\r\n\t<li>The time series has a unit root.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C. <\/strong><\/p>\r\n<p>If the null hypothesis cannot be rejected, then \\(\\text{b}_1-1=0\\) and \\(\\text{b}_1=1\\). This implies that the time series has a unit root. If the null hypothesis is rejected, the time series does not have a unit root.<\/p>\r\n<p><strong>A is incorrect.<\/strong>\u00a0Rejection of the null hypothesis implies that \\(\\text{b}_1-1=0\\) and so the time series has a unit root. However, a time series with a unit root is not covariance stationary.<\/p>\r\n<\/blockquote>\r\n<p>Reading 5:\u00a0Time Series Analysis<\/p>\r\n<p><em>LOS 5 (k) Describe the steps of the unit root test for nonstationary and explain the relation of the test to autoregressive time-series models.<\/em><\/p>\r\n\r\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a><p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Access CFA Level II quantitative methods study notes, practice questions, mock exams, and video lessons to strengthen your understanding of unit root tests and nonstationary time series analysis.<\/p><\/div>","protected":false},"excerpt":{"rendered":"<p>Unit root testing involves checking whether the time series is covariance stationary. We can either form an AR model and check for autocorrelations or perform a Dickey and Fuller test. A t-test is performed to examine the statistical significance 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":[216,230,297],"class_list":["post-13431","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-quantitative-method","tag-the-unit-root-test-for-nonstationary","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>Unit Root Test and Dickey-Fuller Test | CFA Level II<\/title>\n<meta name=\"description\" content=\"Learn unit root testing, the Dickey-Fuller test, and the augmented Dickey-Fuller test for identifying non-stationary time series data.\" \/>\n<meta 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