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Johansen test
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<h2 id="types">Types</h2> <p>There are two types of Johansen test, either with <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> or with <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a>, and the inferences might be a little bit different.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The null hypothesis for the trace test is that the number of cointegration vectors is <i>r</i> = <i>r</i>* < <i>k</i>, vs. the alternative that <i>r</i> = <i>k</i>. Testing proceeds sequentially for <i>r</i>* = 1,2, etc. and the first non-rejection of the null is taken as an estimate of <i>r</i>. The null hypothesis for the "maximum eigenvalue" test is as for the trace test but the alternative is <i>r</i> = <i>r</i>* + 1 and, again, testing proceeds sequentially for <i>r</i>* = 1,2,etc., with the first non-rejection used as an estimator for <i>r</i>. </p><p>Just like a <a href="/facts/Unit_root_test/y1Cb6Ipw">unit root test</a>, there can be a constant term, a trend term, both, or neither in the model. For a general <a href="/facts/Vector_autoregression/ELm2a06B">VAR</a>(<i>p</i>) model: </p> X t = μ + Φ D t + Π p X t − p + ⋯ + Π 1 X t − 1 + e t , t = 1 , … , T {\displaystyle X_{t}=\mu +\Phi D_{t}+\Pi _{p}X_{t-p}+\cdots +\Pi _{1}X_{t-1}+e_{t},\quad t=1,\dots ,T} <p>There are two possible specifications for error correction: that is, two vector <a href="/facts/Error_correction_model/SmLRkxzR">error correction models</a> (VECM): </p><p>1. The longrun VECM: </p> Δ X t = μ + Φ D t + Π X t − p + Γ p − 1 Δ X t − p + 1 + ⋯ + Γ 1 Δ X t − 1 + ε t , t = 1 , … , T {\displaystyle \Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-p}+\Gamma _{p-1}\Delta X_{t-p+1}+\cdots +\Gamma _{1}\Delta X_{t-1}+\varepsilon _{t},\quad t=1,\dots ,T} where Γ i = Π 1 + ⋯ + Π i − I , i = 1 , … , p − 1. {\displaystyle \Gamma _{i}=\Pi _{1}+\cdots +\Pi _{i}-I,\quad i=1,\dots ,p-1.\,} <p>2. The transitory VECM: </p> Δ X t = μ + Φ D t + Π X t − 1 − ∑ j = 1 p − 1 Γ j Δ X t − j + ε t , t = 1 , ⋯ , T {\displaystyle \Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-1}-\sum _{j=1}^{p-1}\Gamma _{j}\Delta X_{t-j}+\varepsilon _{t},\quad t=1,\cdots ,T} where Γ i = ( Π i + 1 + ⋯ + Π p ) , i = 1 , … , p − 1. {\displaystyle \Gamma _{i}=\left(\Pi _{i+1}+\cdots +\Pi _{p}\right),\quad i=1,\dots ,p-1.\,} <p>The two are the same. In both VECM, </p> Π = Π 1 + ⋯ + Π p − I . {\displaystyle \Pi =\Pi _{1}+\cdots +\Pi _{p}-I.\,} <p>Inferences are drawn on Π, and they will be the same, so is the explanatory power. </p> <h2 id="further-reading">Further reading</h2> <ul><li>Banerjee, Anindya; et al. (1993). <a href="https://archive.org/details/cointegrationerr00bane_133"><i>Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data</i></a>. New York: Oxford University Press. pp. <a href="https://archive.org/details/cointegrationerr00bane_133/page/n280">266</a>–268. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-828810-7.</li> <li>Favero, Carlo A. (2001). <a href="https://archive.org/details/appliedmacroecon00cafa"><i>Applied Macroeconometrics</i></a>. New York: Oxford University Press. pp. <a href="https://archive.org/details/appliedmacroecon00cafa/page/n56">56</a>–71. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-829685-1.</li> <li>Hatanaka, Michio (1996). <a href="https://books.google.com/books?id=ZQsaRNl5J60C&pg=PA219"><i>Time-Series-Based Econometrics: Unit Roots and Cointegration</i></a>. New York: Oxford University Press. pp. 219–246. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-877353-6.</li> <li><a href="/facts/G._S._Maddala/x3BUEmbJ">Maddala, G. S.</a>; Kim, In-Moo (1998). <a href="https://books.google.com/books?id=llXBvougICMC&pg=PA198"><i>Unit Roots, Cointegration, and Structural Change</i></a>. Cambridge University Press. pp. 198–248. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-58782-4.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica. 59 (6): 1551–1580. doi:10.2307/2938278. JSTOR 2938278. <a href="/wiki/Econometrica" target="_blank">/wiki/Econometrica</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>For the presence of I(2) variables see Ch. 9 of Johansen, Søren (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press. ISBN 978-0-19-877450-1. <a href="978-0-19-877450-1" target="_blank">978-0-19-877450-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Davidson, James (2000). Econometric Theory. Wiley. ISBN 0-631-21584-0. <a href="0-631-21584-0" target="_blank">0-631-21584-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hänninen, R. (2012). "The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach". Modern Time Series Analysis in Forest Products Markets. Springer. p. 66. ISBN 978-94-011-4772-9. <a href="978-94-011-4772-9" target="_blank">978-94-011-4772-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>