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    • Finally monetary policy is assumed to

    2018-10-25

    Finally, monetary policy is assumed to be expressed by a Taylor rule. Through the interest rate, the central bank reacts to movements in inflation rate and in output. Additionally, the central bank can react to nominal exchange rate depreciation. Thus, the interest rate rule is given by: Eq. (11) demonstrates that the nominal interest rate reacts to the past inflation rate, to current inflation, to output, to nominal exchange rate movement, and to an interest rate shock or, in general, to a monetary policy shock . Parameter represents the degree of interest rate smoothing, and , and are the inflation reaction, output, and exchange rate movement coefficients, respectively. Shock can be interpreted as a non-systematic component of the monetary policy. Therefore, the model contains 22 variables including four terms of expectation. The exogenous processes of model are: preference shock ; technology shock ; cost-push shock ; risk premium shock ; monetary policy shock ; foreign output shock ; foreign inflation shock ; and foreign interest rate shock . All disturbances are assumed to be independent AR(1) processes, except for which follows an i.i.d. process. Hence, we have:
    MS-DSGE model solution and estimation procedures Farmer et al. (2008) proposed a method to solve the model represented in Eq. (16). The option for this method served a dual purpose: computational efficiency, i.e., in general, the algorithm converges quickly; and presence of necessary conditions for the existence of a solution. The solution procedure proposed allows rewriting the MS-DSGE model as a fixed-parameter model in an extended state vector:where The matrices of parameters , , and are functions of the parameters and of the transition probabilities. The extended state vector is defined as: Shocks are:where , is the ith column of the identity matrix. According to Farmer et al. (2008), Eq. (19) contains two types of shocks. The first block, represented by the first our website of , includes the Markov-switching shocks. The second block, second element of , contains the shocks to structural equations. Moreover, the authors mentioned above showed that both types of shocks have zero mean. The solution to the Markov switching model, defined by Farmer et al. (2008), is represented as a stochastic process such that: These conditions can be satisfied by a broad set of fundamental and non-fundamental solutions (also known as sunspot or bubble). Nevertheless, Farmer et al. (2008) focus on fundamental solutions or, following the nomenclature used by the authors, minimal state variable (MSV) solutions which, unlike fundamental solutions, have a finite number and rely only on state variables (Cho and Moreno, 2011). In addition, the authors proved that the MSV solution to the system represented in Eq. (17) solves the model from Eq. (16), i.e., the MSV solution of the fixed-parameter model with an extended state vector is also the solution to the MS-DSGE model. In this case, matrix plays a crucial role in the determination of the MSV solution. To ensure the stochastic process is bounded, the solution should be in a linear subspace. For that to occur, Farmer et al. (2008) define a matrix such that . This way, matrix guarantees that constraints are imposed on in the case of regime 1, whereas the definition of Eq. (17) makes sure that in the case of regime 1. The definition of matrix for imposition of the constraints occurs as follows: first, an initial value is set for and matrices and are calculated, where the superscript refers to the iteration steps; second, the decomposition QZ of and is computed; third, matrices and are ordered such that the number of elements on the diagonal is rearranged in increasing order, in this case, for and when , where is a whole number; fourth, if is formed by the last rows of , then which is reiterated until Reversion mutation converges. If the iteration process converges, it means that the solution to Eq. (17) is the solution to the original model – Eq. (16). Gensys is applied to verify the existence and unicity and also to compute the solution to Eq. (16). If a single solution exists, this solution can be rewritten in the form of a Markov switching VAR (MS-VAR):