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We consider cointegration rank estimation for a p-dimensional Fractional Vector ErrorCorrection Model. We propose a new two-step procedure which allows testing for furtherlong-run equilibrium relations with possibly di¤erent persistence levels. The rst stepconsists in estimating the parameters of the model under the null hypothesis of thecointegration rank r = 1; 2,... p - 1: This step provides consistent estimates of theorder of fractional cointegration, the cointegration vectors, the speed of adjustment tothe equilibrium parameters and the common trends. In the second step we carry out asup-likelihood ratio test of no-cointegration on the estimated p - r common trends thatare not cointegrated under the null. The order of fractional cointegration is re-estimatedin the second step to allow for new cointegration relationships with di¤erent memory. Weaugment the error correction model in the second step to adapt to the representation ofthe common trends estimated in the rst step. The critical values of the proposed testsdepend only on the number of common trends under the null, p��r; and on the interval ofthe orders of fractional cointegration b allowed in the estimation, but not on the order offractional cointegration of already identi ed relationships. Hence this reduces the set ofsimulations required to approximate the critical values, making this procedure convenientfor practical purposes. In a Monte Carlo study we analyze the nite sample properties ofour procedure and compare with alternative methods. We nally apply these methodsto study the term structure of interest rates.
error correction model; gaussian var model; likelihood ratio tests; maximum likelihood estimation