1 An algorithm to estimate time varying parameter SURE models under different type of restrictions Susan Orbe 1 , Eva Ferreira 1 and Juan Rodr´ ıguez-P´ oo 2 1 Dpto de Econometr´ ıa y Estad´ ıstica, Universidad del Pa´ ıs Vasco, Spain 2 Dpto de Econom´ ıa, Universidad de Cantabria, Spain Keywords. Nonparametric methods, constrained estimators, seasonal re- strictions, cross restrictions 1 Introduction A system of seemingly unrelated regression equations (SURE) is considered where constraints on the coefficients are allowed. These possible constraints can be of two different types, either single or cross equation restrictions. An example of the first type are seasonal restrictions (see Hylleberg, 1986). The second type appears on empirical specifications of production and con- sumption systems. In production theory, for example, the estimation of the structural parameters is subject to restrictions implied by homogeneity, prod- uct exhaustion, symmetry and monotonicity (see Jorgenson, 2000). Also in consumption, when demand functions are specified, the assumption of linear budget constraints lead to adding up and homogeneity restrictions. Further- more, additional restrictions are sometimes suggested by the theory, as the Slutsky symmetry condition and the Engle condition (see Deaton and Muell- bauer, 1980). Motivated by the previous analysis, a nonparametric method is proposed to estimate time varying coefficients subject to these types of restrictions in a SURE framework. A closed form for the time varying coefficient estimators is provided so the coefficients can be estimated without the need of iterative procedures. 2 A general time-varying parameter SURE model Let us consider the m-th equation in the SURE model given by y mt = β m1t x m1t + β m2t x m2t + ... + β mpmt x mpmt + u mt (1) for m =1,...,M t =1,...,T ; where the endogenous variable in the m-th equation, y m , is explained by a set of p m explanatory variables, x m1 ,x m2 ,..., x mpm . Thus, the total number of explanatory variables is P = ∑ M m=1 p m . The vector of unknown coefficients needed to be estimated is β mt =(β m1t ,β m2t ,... ...,β mpmt ) ′ . For the error terms, u mt , it is assumed that: E (u mt ) = 0, E (u mt u m ′ t )= σ mm ′ t and E (u mt u m ′ t ′ )=0, ∀m,t. This covariance struc- ture allows for heteroskedasticity and also for a time varying contemporary