{"id":13567,"date":"2021-04-14T10:11:02","date_gmt":"2021-04-14T10:11:02","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13567"},"modified":"2026-06-01T17:14:18","modified_gmt":"2026-06-01T17:14:18","slug":"cointegration","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/cointegration\/","title":{"rendered":"Cointegration"},"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 conclusion is reached if the Engle-Granger Dickey-Fuller test fails to reject the null hypothesis?\",\r\n    \"text\": \"An analyst tests the residuals from two time series for a unit root using the Dickey-Fuller test and determines the critical values using the Engle-Granger procedure. The test fails to reject the null hypothesis. What is the most accurate conclusion? A. The two series are cointegrated. B. The two series are not cointegrated. C. The error terms are covariance stationary.\",\r\n    \"answerCount\": 3,\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A. The two series are cointegrated.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"B. The two series are not cointegrated.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"C. The error terms are covariance stationary.\"\r\n      }\r\n    ],\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is B. If the Engle-Granger Dickey-Fuller test fails to reject the null hypothesis of a unit root in the residuals, the residuals are not covariance stationary. As a result, the two time series are not cointegrated. Without cointegration, a linear regression between the two non-stationary series may produce spurious results and should not be relied upon for inference.\"\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>Consider a time series of the inflation rate <em>\\((\\text{y}_{\\text{t}})\\)<\/em> regressed on a time series of interest rates \\((\\text{x}_{\\text{t}})\\):<\/p>\r\n<p>$$\\text{y}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}+\\epsilon_{\\text{t}}$$<\/p>\r\n<p>In this case, we have two different time series, <em>\\(\\text{y}_{\\text{t}}\\)<\/em> and \\(\\text{x}_{\\text{t}}\\). Either one of the time series is subject to non-stationarity. Recall that we test for non-stationarity using the Dickey-fuller test for each time series to check for unit roots.<\/p>\r\n\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 cointegration concepts with our Free Trial.<\/a><\/div>\r\n\r\n<h2>Outcomes of the Dickey-Fuller test<\/h2>\r\n<p>The results of a Dickey-Fuller test can either be:<\/p>\r\n<h5>\u00a0 \u00a0 \u00a0 \u00a0a. Both the time series are covariance stationary.<\/h5>\r\n<p>This means that the time series do not have a unit root. In this case, we use linear regression to test the relationship between the two series. Besides, the regression coefficients are statistically reliable.<\/p>\r\n<h5>\u00a0 \u00a0 \u00a0 \u00a0b.<strong> Only the dependent variable time series is covariance stationary.<\/strong><\/h5>\r\n<p>This means that some of the linear regression assumptions are violated. Therefore, the estimated regression coefficients and the standard error would be inconsistent.<\/p>\r\n<h5>\u00a0 \u00a0 \u00a0 \u00a0 c. Only the independent variable time series is covariance stationary.<\/h5>\r\n<p>This has a similar implication to outcome two above.<\/p>\r\n<h5>\u00a0 \u00a0 \u00a0 \u00a0 d. None of the time series is covariance stationary<\/h5>\r\n<p>If none of the time series is covariance stationary, check whether or not they are cointegrated. Two time-series are said to be cointegrated if an economic or financial relationship exists between them, preventing them from diverging without bound in the long run.<\/p>\r\n<p>We can test for cointegration using either the Engle-Granger or Dickey-Fuller test. The error term will be stationary, and the hypothesis tests will be valid if both the time series are covariance stationary and cointegrated.<\/p>\r\n<h2>Testing for Cointegration<\/h2>\r\n<p>We can test whether the two time series are cointegrated through the following steps:<\/p>\r\n<ul>\r\n\t<li>Regress one variable on the other variable using the model: $$\\text{y}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}+\\epsilon_{\\text{t}}$$<\/li>\r\n\t<li>Test the errors for a unit root using the Dickey-Fuller test. The critical t-values are determined using the Engle and Granger test.<\/li>\r\n<\/ul>\r\n<p>Rejection of the null hypothesis implies that the error terms are covariance stationary, and both series are cointegrated. Recall that the null hypothesis is that the error term has a unit root.<\/p>\r\n<ul>\r\n\t<li>Use regression to model the relationship between the two-time series if they are cointegrated.<\/li>\r\n<\/ul>\r\n<p>If the (Engler-Granger) Dickey-Fuller test fails to reject the null hypothesis, we conclude that the error term is not covariance stationary. This means that the two series are not cointegrated. Consequently, the linear regression is invalid.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>An analyst tests the two time series errors for a unit root using the Dickey-Fuller test and determines the critical values using the Engle and Granger test. The test fails to reject the null hypothesis. The <em>most accurate<\/em> conclusion is that:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>The two series are cointegrated.<\/li>\r\n\t<li>The two series are not cointegrated.<\/li>\r\n\t<li>The error terms are 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>If the (Engler-Granger) Dickey-Fuller test fails to reject the null hypothesis, we conclude that the error terms are not covariance stationary. This means that the two series are not cointegrated. Consequently, a linear regression cannot be used.<\/p>\r\n<\/blockquote>\r\n<p>Reading 5: Time Series Analysis<\/p>\r\n<p><em>LOS 5 (n) Explain how time-series variables should be analyzed for nonstationary and\/or cointegration before use in linear regression.<\/em><\/p>\r\n\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 cointegration and long-term relationships in time series analysis.<\/p><\/div>\r\n","protected":false},"excerpt":{"rendered":"<p>Consider a time series of the inflation rate \\((\\text{y}_{\\text{t}})\\) regressed on a time series of interest rates \\((\\text{x}_{\\text{t}})\\): $$\\text{y}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}+\\epsilon_{\\text{t}}$$ In this case, we have two different time series, \\(\\text{y}_{\\text{t}}\\) and \\(\\text{x}_{\\text{t}}\\). Either one of the time series is subject to&#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":[216,300,230],"class_list":["post-13567","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-cointegration","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>Cointegration - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Understand the outcomes of the Dickey-Fuller test and Engle-Granger test in determining covariance stationarity and cointegration.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/cointegration\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cointegration - 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