Archives

  • 2018-07
  • 2018-10
  • 2018-11
  • 2019-04
  • 2019-05
  • 2019-06
  • 2019-07
  • 2019-08
  • 2019-09
  • 2019-10
  • 2019-11
  • 2019-12
  • 2020-01
  • 2020-02
  • 2020-03
  • 2020-04
  • 2020-05
  • 2020-06
  • 2020-07
  • 2020-08
  • 2020-09
  • 2020-10
  • 2020-11
  • 2020-12
  • 2021-01
  • 2021-02
  • 2021-03
  • 2021-04
  • 2021-05
  • 2021-06
  • 2021-07
  • 2021-08
  • 2021-09
  • 2021-10
  • 2021-11
  • 2021-12
  • 2022-01
  • 2022-02
  • 2022-03
  • 2022-04
  • 2022-05
  • 2022-06
  • 2022-07
  • 2022-08
  • 2022-09
  • 2022-10
  • 2022-11
  • 2022-12
  • 2023-01
  • 2023-02
  • 2023-03
  • 2023-04
  • 2023-05
  • 2023-07
  • 2023-08
  • 2023-09
  • 2023-10
  • 2023-11
  • 2023-12
  • 2024-01
  • 2024-02
  • 2024-03
  • 2024-04
  • 2024-05

    • Combining Eq with an observation

    2018-10-30

    Combining Eq. (20) with an observation equation (Eq. (15)), we have a state space model with Markov switching:where and Markov states and are independent with transition matrices and , respectively. Therefore, the solution method proposed by Farmer et al. (2008) allows rewriting the MS-DSGE model as a fixed-parameter model in an extended state vector, as shown in Eq. (17). The MSV solution not only satisfies the definition of the solution to the models, but also can be rewritten as an MS-VAR. Combining this solution to an observation equation, we have a state space model with Markov switching, expressed by Eq. (21). In what follows, the estimation of this model is discussed. The state space model with Markov switching contains unobserved states and also unobserved Markov states. The presence of these two sets of unobservable variables implies that the standard Kalman filter cannot be applied, as it will not be possible to make inference and to calculate transition probabilities at the same time. However, with unobserved Markov states, the inference can be conditioned on the current and past values of and . As pointed out by Kim and Nelson (1999), each iteration of the filter implies that the number of cases increases in M, where M stands for the number of regimes. This makes the problem with finding the solution to the model computationally intensive. That being said, Kim and Nelson (1999) proposed an approximation to make the filter more operational. This approximation causes a limited number of states to be taken along iterations in each melanin and to be “collapsed” at the end of each iteration. To apply the approximation, a new state variable is defined, , which indexes both and and whose transition matrix is given by , where represents the Kronecker product. According to Kim and Nelson (1999), , and can be traced out, which implies the existence of possible paths for the state variables in each time period. Intuitively, Kim and Nelson\'s (1999) algorithm runs the Kalman filter for each one of the paths and, thereafter, a weighted average is obtained using the weights given by the probabilities of each path. The Bayesian approach is used to estimate the model following two steps. The first one combines the likelihood function, obtained from Kim and Nelson\'s (1999) algorithm, with prior distribution for the parameters. A combination of numerical maximizers is used to calculate the approximate posterior mode. In this case, the initial values are refined by the simplex algorithm and employed in the CSMINWEL optimization routine, proposed by Christopher Sims. The posterior mode is used as initial value for the Metropolis Hastings algorithm with 100,000 iterations. The second step consists in utilizing the mean and variance of the last 1000 iterations from the first step to run the main Metropolis Hastings algorithm. This step consists of 200,000 iterations. The models to be estimated have the following specifications: The prior distribution of the model\'s parameters was determined based on studies available in the Brazilian literature, such as in Kanczuk (2002), Araújo et al. (2006), Silveira (2008), Furlani et al. (2010), Palma and Portugal (2011) and Castro (2011). Table 1 shows the distribution for each parameter, in addition to the mean, standard deviation, and bounds. The economic openness parameter () was set at 0.23 according to the volume of exports/imports vis-à-vis the domestic output in the analyzed period. Given that the discount factor is calibrated in such a way that the value assumed by the average real long-term interest rate is equal to , with an average interest rate of 10.8% p.a., the intertemporal discount rate () was calibrated at 0.975. Consonant with Palma and Portugal (2011), the inverse of the elasticity of labor supply () is assumed to follow a gamma distribution with a mean of 2 and a standard deviation of 0.35. Other parameters defined according to these authors were: the fractions of firms that do not adjust their prices ( and ); the elasticity of substitution between domestic goods (); the degree of habit persistence; and indexation rates ( and ). It was assumed that the inverse of the intertemporal elasticity () follows a gamma distribution with a mean of 1.2 and a standard deviation of 0.2.