Consider a time series of the inflation rate \\((\\text{y}_{\\text{t}})\\)<\/em> regressed on a time series of interest rates \\((\\text{x}_{\\text{t}})\\):<\/p>\r\n $$\\text{y}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}+\\epsilon_{\\text{t}}$$<\/p>\r\n In this case, we have two different time series, \\(\\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 The results of a Dickey-Fuller test can either be:<\/p>\r\n 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 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 This has a similar implication to outcome two above.<\/p>\r\n 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 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 We can test whether the two time series are cointegrated through the following steps:<\/p>\r\n 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 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 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 most accurate<\/em> conclusion is that:<\/p>\r\n The correct answer is B.<\/strong><\/p>\r\n 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 Reading 5: Time Series Analysis<\/p>\r\n 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","protected":false},"excerpt":{"rendered":" [vsw id=”-SilFtkpBK8″ source=”youtube” width=”611″ height=”344″ autoplay=”no”] 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…<\/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":"\nOutcomes of the Dickey-Fuller test<\/h2>\r\n
\u00a0 \u00a0 \u00a0 \u00a0a. Both the time series are covariance stationary.<\/h5>\r\n
\u00a0 \u00a0 \u00a0 \u00a0b. Only the dependent variable time series is covariance stationary.<\/strong><\/h5>\r\n
\u00a0 \u00a0 \u00a0 \u00a0 c. Only the independent variable time series is covariance stationary.<\/h5>\r\n
\u00a0 \u00a0 \u00a0 \u00a0 d. None of the time series is covariance stationary<\/h5>\r\n
Testing for Cointegration<\/h2>\r\n
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Question<\/h2>\r\n
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Solution<\/h4>\r\n