Restriction in Cointegration Analysis
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Restrictions allows us to include economic theory into non-structural VECM models. They are useful in making coherency between two opposite school of econometrics: traditional structural simultaneous equation approach and the vector autoregression (VAR) models based on reduced form. The paper regards economic interpretation in the case of restricting particular matrices of VECM representation. Restrictions on short-term dependency matrix are connected with the rank of lags in the VECM model choice and the short-run exogeneity. The reduced rank condition with respect to the long-run multiplier matrix imply a lack of the system joint stationarity. Restrictions concern not only the vector error correction model parameter, but the matrices of its solution may be restricted too. In the case of the model with jointly stationary variables I(0) the vector moving average (VMA) representation is such solution, in the case of I(1) variables model - common I(1) trends model. The row restrictions imposed on orthogonal complements of cointegration matrix may be useful in the stationarity analysis of system variables and hence may be compared with the standard unit root tests results. On the other hand, restriction on adjustment matrix orthogonal complements are helpful in impulse - response analysis. Almost all economically interpreted restrictions may be included in the alternative (dual) form - the row restrictions on the original matrix may be performed as the column conditions on the orthogonal complement of such matrix. The likelihood ratio test allows us to verify such overidentifying restrictions. The table in the and of the paper clarifies the type of restrictions classification. The restrictions interpretation is rather simple in the case of I(1) model. The l(2) variables inclusion imply serious complications.
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