Estimation of restricted models and structural forms

Efficient estimation of a general structural form model such as (32.7) with restrictions on the parameter matrices is more complicated. Of course, identifying restrictions are necessary for consistent estimation. In practice, various over­identifying restrictions are usually available, typically in the form of zero restrictions on Г * (j = 0,…, p – 1), C* and B*. In addition, there may be a rank restriction for П* given by the number of cointegrating relations. Alternatively,

П* may be replaced by the product a*P*’, if identifying restrictions are available for the cointegrating relations and/or the loading matrix a*. Restrictions for a* are typically zero constraints, meaning that some cointegrating relations are excluded from some of the equations of the system. In some cases it is possible to estimate P* in a first stage, for example, using a reduced form procedure which ignores some or all of the structural restrictions on the short-term parameters. Let the estimator be S*. Because the estimators of the cointegrating parameters converge at a better rate than the estimators of the short-term parameters they may be treated as fixed in a second-stage procedure for the structural form. In other words, a systems estimation procedure may be applied to

Г *0 A yt = a*S*’yt-1 + Г *Ayt_1 + … + Г **_1Ayt_v+1 + C*Dt + B*zt + Ht. (32.9)

If only exclusion restrictions are imposed on the parameter matrices in this form, standard three-stage LS or similar methods may be applied which result in estimators of the short-term parameters with the usual asymptotic properties. Important results on estimating models with integrated variables are due to Phillips and his co-workers (e. g. Phillips, 1987, 1991).

If deterministic variables are to be included in the cointegration relations this requires a suitable reparameterization of the model. Such reparameterizations for intercepts and linear trend terms are presented in Section 4.2, where tests for the cointegrating rank are discussed. In that context a proper treatment of determin­istic terms is of particular importance. Therefore, a more detailed discussion is deferred to Section 4.2. In a subsequent analysis of the model the parameters of the deterministic terms are often of minor interest and therefore the properties of the corresponding estimators are not treated in detail here (see, however, Sims et al., 1990).

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