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
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
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, random walks with drift<\/strong> are 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 The Dickey-Fuller test uses the transformed AR(1) model by substracting \u00a0from both sides of the equation. i.e.,<\/p>\r\n $$\\text{x}_{\\text{t}}=\\text{b}_{0}+b_{1}x_{t-1}+\\epsilon_{t}$$<\/p>\r\n $$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}-\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n $$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{x}_{\\text{t}-1}(\\text{b}_{1}-1)+\\epsilon_{\\text{t}}$$<\/p>\r\n Let the transformed coefficient, \\(\\text{b}_{1}-1=\\text{g}\\). Then,<\/p>\r\n $$\\text{x}_{\\text{t}}-\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{g}_{1}\\text{x}_{\\text{t}-1}+\\epsilon_{\\text{t}}$$<\/p>\r\n 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}<0\\) (a test of \\(\\text{b}_1={1}\\), which implies the time series has no unit root).<\/p>\r\n 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 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 If the null hypothesis \\(\\text{H}_{0}:\\text{g}_1=0\\) under the Dickey-Fuller test cannot be rejected; the most accurate<\/em> conclusion is that:<\/p>\r\n The correct answer is C. <\/strong><\/p>\r\n 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 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 Reading 5:\u00a0Time Series Analysis<\/p>\r\n 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","protected":false},"excerpt":{"rendered":" [vsw id=”-SilFtkpBK8″ source=”youtube” width=”611″ height=”344″ autoplay=”no”] 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…<\/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":"\nDickey-Fuller Test<\/h2>\r\n
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Question<\/h2>\r\n
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Solution<\/h4>\r\n