Linganisha mbinu

	Kipimo cha Uthabiti cha Ukozi wa Johansen	Kielelezo cha Usahihishaji wa Hitilafu wa Kielekezi (VECM)
Nyanja	Ekonometriki	Ekonometriki
Familia	Regression model	Regression model
Mwaka wa asili	1988–2010	1987
Mwanzilishi	Johansen (1988, 1991); robust extensions by Cavaliere, Rahbek, Taylor (2010) and others	Robert F. Engle and Clive W. J. Granger
Aina	Cointegration rank test (robust variant)	Multivariate time-series model
Chanzo asilia	Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica, 59(6), 1551–1580.	Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), 251–276.
Majina mbadala	outlier-robust Johansen test, robust trace test, robust maximum eigenvalue test, robust cointegration rank test	VECM, error correction VAR, cointegrated VAR, vector equilibrium correction model
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 Vector Error Correction Model extends the Vector Autoregression (VAR) framework to a system of variables that share one or more long-run equilibrium relationships. It jointly models short-run dynamics and the speed at which each variable corrects back toward equilibrium after a shock, making it the standard tool for analysing cointegrated multivariate time series.