Linganisha mbinu

	Kipimo cha Uthabiti cha Ukozi wa Johansen	Kipimo cha Ukoanifu wa Johansen chenye Kuvunjika kwa Muundo
Nyanja	Ekonometriki	Ekonometriki
Familia	Regression model	Regression model
Mwaka wa asili	1988–2010	2000–2001
Mwanzilishi	Johansen (1988, 1991); robust extensions by Cavaliere, Rahbek, Taylor (2010) and others	Johansen (1988); structural-break extensions by Saikkonen & Lütkepohl (2000) and Lütkepohl, Müller & Saikkonen (2001)
Aina	Cointegration rank test (robust variant)	Cointegration test / VECM estimation
Chanzo asilia	Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica, 59(6), 1551–1580.	Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control, 12(2–3), 231–254.
Majina mbadala	outlier-robust Johansen test, robust trace test, robust maximum eigenvalue test, robust cointegration rank test	Johansen cointegration with breaks, break-robust Johansen test, cointegration test with regime shifts, structural change Johansen VECM
Zinazohusiana	5	5
Muhtasari	The Robust Johansen Cointegration test extends the classical Johansen (1988, 1991) likelihood-ratio framework for determining the cointegrating rank of a multivariate I(1) system to settings where standard Gaussian assumptions fail — in particular when the data exhibit outliers, fat-tailed innovations, or conditional heteroskedasticity. Robust modifications adjust residuals, re-weight observations, or bootstrap critical values so that rank inference remains valid under these violations.	The structural break Johansen cointegration test extends the standard maximum-likelihood Johansen procedure to settings where the multivariate time series exhibits level shifts or trend breaks. By incorporating dummy variables or shift regressors into the VECM, the test determines the cointegrating rank without confounding genuine long-run relationships with regime changes.