Kim, C.-J. (1994): Dynamic Linear Models with Markov-Switching, Journal of Econometrics, 60, 122.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....variables. For instance, given observations up to time t, the distribution on the tth continuous state variable is a mixture of an exponentially growing (in t) number of Gaussians. Numerous approximate schemes have been developed to address the inference and learning problems [9] 10] 2] [11], 3] We address here a problem, to our knowledge, untreated in the literature on such hybrid models: Estimating the most likely sequence of hidden discrete and continuous variables given the observed data. This problem formulation is especially appropriate when the desired data interpretation ....

Kim C.-J. (1994), \Dynamic Linear Models with Markov-Switching," J. Econometrics 60, 1-22.

....the exact belief state at time t is a mixture of up to (2 Gaussians. We conclude that there is no hope of representing the exact belief state for anything but trivial SLDSs over a small number of time slices. There has been much work trying to deal with this problem, e.g. BSLK01, SS91, Kim94, GH00] Most of this work assumes that the number of discrete combination at every time slice is relatively small, and the difficulty comes mostly from the exponential dependence on t. In this section, we review some of the techniques to approximate the belief state under these conditions. In ....

C-J. Kim. Dynamic linear models with Markov-switching. Journal of Econometrics, 60:1--22, 1994.

....side of (3) causes the GARCH variable to be a function of the entire history of the state variable. If h were an ARCH(p) process, then h would depend only on the p most recent values of the state variable, as in Cai (1994) and Hamilton and Susmel (1994) Here I discuss how methods described in Kim (1994) can be applied to make estimation feasible for GARCH processes subject to markov switching. Clearly it is not practical to examine all of the possible sequences of past values of the state variable when evaluating the likelihood function for a sample of more than a thousand observations, as the ....

....subject to markov switching. Clearly it is not practical to examine all of the possible sequences of past values of the state variable when evaluating the likelihood function for a sample of more than a thousand observations, as the number of cases to consider exceeds 1000 by the time t = 10. Kim (1994) addresses this problem by introducing a collapsing procedure that greatly facilitates evaluation of the likelihood function at the cost of introducing a degree of approximation that does not appear to distort the calculated likelihood by much. The collapsing procedure, when applied to a GARCH ....

Kim, C.J. (1994), "Dynamic Linear Models with Markov Switching," Journal of Econometrics, January/February 1994, 1-22.

....averages of original parameters weighted by a best estimates of the switching states P (s t ) HMM variational paremeters log q t , on the other hand, measure the agreement of each individual LDS with the data. 3. 3 Approximate Generalized Pseudo Bayesian Inference The Generalized Psuedo Bayesian [1, 9] (GPB) approximation scheme is based on the general idea of collapsing a mixture of M t Gaussians onto a mixture of M r Gaussians, where r t(see [12] for a detailed review) While there are several variations on this idea, our focus is the GPB2 algorithm, which maintains a mixture of M 2 ....

C.-J. Kim, "Dynamic linear models with markov-switching," Journal of Econometrics, vol. 60, pp. 1--22, 1994.

....value of zero during recessions. Hamilton also provided an algorithm for estimating the probability of a recession at each time period based on a maximum likelihood approach. Since that time, several other authors have investigated modifications to the model specification (Lam, 1990; Hansen, 1992; Kim, 1994), computation of the recession probabilities (Albert and Chib, 1993) and the application of the models to various other data sources (Cecchetti et al., 1990; Hamilton and Lin, 1996) In this paper we propose a new model based on the structural time series model underlying exponential smoothing ....

....break in the growth rate. This idea is in line with the structural models including the SSM, which allows for structural breaks at each t through a change in the level l t . In another application, by estimating Hamilton generalised model, Lam (1990) in using maximum likelihood approach and Kim (1994) in using state space forms and Kalman filter approach failed to capture all seven recession periods mentioned by the NBER. The model captures only five recession periods and the low growth phases were shorter than those of NBER recessions. As a benchmark for the switching regime literature, ....

Kim C.-J. (1994): "Dynamic Linear Models with Markov Switching " Journal of Econometrics, V60, pp 1-22.

....nodes before their discrete ancestors clashes with our desire to eliminate all the nodes in slice t before we eliminate any in slice t 1. If we don t do strong triangulation, the number of mixture components becomes exponential in the length of the sequence. The standard approach (see e.g. [TSM85, BSL93, Kim94, WH97]) is to collapse the mixture into k components. If k = 1, this corresponds to computing the weak moments: p(i) X j p(i; j) i) X j (i; j)p(i; j) p(i) i) X j (i; j)p(i; j) p(i) X j (i; j) i) i; j) i) T p(i; j) p(i) These will ....

C-J. Kim. Dynamic linear models with Markov-switching. J. of Econometrics, 60:1-22, 1994.

....In signal processing, finite impulse response (FIR) filters are applied usually. Their AR filter approximation is rather poor, while we will show that MA filtering can model the filter characteristics sufficiently well. An earlier proposal of algorithms for state space models with Markov switching [12] might be seen as a more general approach containing ARMA models as a special case. However, in the ARMA case, that algorithm is not as efficient as the one proposed in this paper. e.g. its algorithmic complexity as well as the necessary amount of auxiliary variables is of order O(q 3 ) for ....

C.-J. Kim, "Dynamic linear models with Markov-switching," J. Econometrics, vol. 60, pp. 1--22, 1994.

....from the Bayesian network literature can be applied to dynamical systems. In particular, it has been shown that estimation in LDSs and inference in HMMs are special cases of inference in DBNs. The focus of this paper is on a subclass of DBN models called Switching Linear Dynamic Systems [2, 26, 17, 9, 22]. Intuitively, these models attempt to describe a complex nonlinear dynamic system with a succession of linear models that are indexed by a switching variable. While other approaches such as learning weighted combinations of linear models are possible, the switching approach has an appealing ....

....the above equations represent a generalization of the parameter update equations of classical (non switching) LDS models [8] 3 Previous Work SLDS models and their equivalents have been studied in statistics, time series modeling, and target tracking since early 1970 s. Bar Shalom [2] and Kim [17] have developed a number of approximate pseudo Bayesian inference techniques based on mixture component truncation or collapsing is SLDSs. They did not address the issue of learning system parameters. Shumway and Stoffer [26] presented a systematic view of inference and learning in SLDS while ....

C.-J. Kim. Dynamic linear models with markov-switching. Journal of Econometrics, 60:1-- 22, 1994.

....low variational state transition matrix ( whose determinant is indicated by in Figure 3.3) Through further iterations the variational inference algorithm converges to the true switching state sequence. 3. 3 Approximate Generalized Pseudo Bayesian Inference The Generalized Psuedo Bayesian [2, 11] (GPB) approximation scheme is based on the general idea of collapsing , i.e. representing a mixture of Gaussians with a mixture of Gaussians, where J (see [13] for a detailed review) While there are several variations on this idea, our focus is the GPB2 algorithm, which maintains a ....

C.-J. Kim. Dynamic linear models with markov-switching. Journal of Econometrics, 60:1--22, 1994. 7

....and learning techniques from the BN literature can be applied to dynamical systems. In particular, it has been shown that estimation in LDSs and inference in HMMs are special cases of inference in DBNs. The focus of this paper is on a subclass of DBN models called Switching Linear Systems [2, 22, 14, 8, 19]. Intuitively, these models attempt to describe a complex nonlinear dynamic system with a succession of linear models that are indexed by a switching variable. While other approaches such as learning weighted combinations of linear models are possible, the switching approach has an appealing ....

....of our learned models over simple smoothness priors. We believe these methods provide a useful alternative to detailed biomechanical modeling. SLDS models and their equivalents have been studied in statistics, time series modeling, and target tracking since early 1970 s. Bar Shalom [2] and Kim [14] have developed a number of approximate pseudo Bayesian inference techniques based on mixture component truncation or collapsing is SLDSs. They did not address the issue of learning system parameters. Shumway and Stoffer [22] presented a systematic view of inference and learning in SLDS while ....

C.-J. Kim, "Dynamic linear models with markovswitching, " Journal of Econometrics, vol. 60, pp. 1--22, 1994.

....p 11 p 12 Delta Delta Delta p 1k p 21 p 22 Delta Delta Delta p 2k . p k1 p k2 Delta Delta Delta p kk 3 7 7 7 7 5 (10) Thus, we have specified the dynamic structure of the Markov switching model. The state inference is adopted from Hamilton (1994, Chap. 22) and Kim (1994) 6 . It has to be calculated iteratively. First, set the starting value for the state inference equal to the model s parameter ae, i.e. the (k Theta 1) vector of probabilities that the system is governed by a particular regime in period one given no information from observations. Then, the ....

Kim, C.-J. (1994): "Dynamic linear models with Markov switching," Journal of Econometrics, 60, 1--22.

....above. Model 1 is our benchmark model with no further restrictions. Model 2 is a version that does not allow for the plucking type asymmetry, that is 0 = i y t t . Model 3 does not allow for switches in the growth rate of the stochastic trend, that is 0 1 = All models are estimated via Kim s (1993a, 1993b, 1994) approximate maximum likelihood algorithm. Table 2 contains the estimated parameters and standard errors for Models 1 3. 13 Our discussion will focus on model 1, the benchmark model. The other models will be of primary interest in performing hypothesis tests regarding the presence of asymmetry. ....

Kim, C.-J., 1994, Dynamic linear models with Markov-switching, Journal of Econometrics 60, 1-22.

....the parameters as follows: T t t t r r r f l 1 2 1 , ln( max q q , A.10) where q b s s d e e = 0 1 0 1 q p P . 11 In addition to making the above inferences, we also obtain the smoothed probability , 1 Pr[ 1 = T T t r r S by employing Kim s (1994) smoothing algorithm. Specifically, given the filtered probability , Pr[ 1 = t t r r j S , which can be found by collapsing across states for (A.7) and the conditional probability , Pr[ 2 1 = t t t r r j S , given in equation (A.2) we iterate backwards through the following ....

Kim, C.-J., 1994, "Dynamic Linear Models with Markov-Switching," Journal of Econometrics 60, 1-22.

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Kim, Chang-Jin, 1994, \Dynamic Linear Models with Markov-Switching," Journal of Econometrics, 60, 1-22.

....above. Model 1 is our benchmark model with no further restrictions. Model 2 is a version that does not allow for the plucking type asymmetry, that is 0 = i y t t . Model 3 does not allow for switches in the growth rate of the stochastic trend, that is 0 1 = All models are estimated via Kim s (1993a, 1993b, 1994) approximate maximum likelihood algorithm. Table 2 contains the estimated parameters and standard errors for Models 1 3. 14 Our discussion will focus on model 1, the benchmark model. The other models will be of primary interest in performing hypothesis tests regarding the presence of asymmetry. ....

Kim, C.-J. (1994), `Dynamic linear models with Markov-switching', Journal of Econometrics 60, 1-22.

....the parameters as follows: T t t t r r r f l 1 2 1 , ln( max q q , A.10) where q b s s d e e = 0 1 0 1 q p P . 10 In addition to making the above inferences, we also obtain the smoothed probability , 1 Pr[ 1 = T T t r r S by employing Kim s (1994) smoothing algorithm. Specifically, given the filtered probability , Pr[ 1 = t t r r j S , which can be found by collapsing across states for (A.7) and the conditional probability , Pr[ 2 1 = t t t r r j S , given in equation (A.2) we iterate backwards through the following ....

Kim, C.-J., 1994, "Dynamic Linear Models with Markov-Switching," Journal of Econometrics 60, 1-22.

....2 ) 1 ( e e e s s s = ss ee1 2 0 2 . 7) Pr[ SS q tt = 00 1 and Pr[ SS p tt = 11 1 . The model allows us to investigate how the behavior of U.S. stock prices has changed throughout the past 70 years. The model is estimated via approximate maximum likelihood as discussed in Kim (1994). Readers are referred to the Appendix for details. Empirical results for the above model can be summarized as follows. First, both filtered and smoothed inferences about 2 t e s , t=1,2, T, suggest that most of the high volatility occurred in the 1930 s, although there is some evidence of a ....

....1 2 y p hh As a byproduct of running the above filter for given appropriate initial values , we get the following log likelihood function: A.13) T t t t y f l 1 1 ) ln( ln t y q . For more details on state space models with Markov switching and their inferences, refer to Kim (1994) and Kim and Nelson (1999) ....

Kim, C.-J., 1994, "Dynamic Linear Models with Markov-Switching," Journal of Econometrics 60, 1-22.

....Model 9 Since the state variables, t t S S 2 1 and , are unobserved, our model is nonlinear, and calculation of the exact Gaussian likelihood function is not possible. To estimate the parameters, as well as the unobserved components, we cast our model into its state space representation and use Kim s (1994) approximate maximum likelihood estimation algorithm. Section 1 of the Appendix presents the state space representation of our model. Section 2 a presents a detailed description of the estimation algorithm. Section 3 demonstrates how we construct t C from t c D . 4. Permanent and Transitory ....

Kim, C.-J., " Dynamic Linear Models with Markov Switching," Journal of Econometrics 60 (1994), 1-22.

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Kim, C.-J. (1994): Dynamic Linear Models with Markov-Switching, Journal of Econometrics, 60, 122.

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C-J. Kim. Dynamic linear models with Markov-switching. J. of Econometrics, 60:1--22, 1994.

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Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1--22. Lauritzen, S. L. and Spiegelhalter, D. J. (1988). Local computations with probabilities on graphical structures and their application to expert systems. J. Royal Statistical Society B, pages 157--224.

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C-J. Kim, "Dynamic linear Models with Markov Switching," J. of Econometrics, Vol. 60, pp1-22, 1994

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C.-J. Kim. Dynamic linear models with Markov switching. Journal of Econometrics, 60:1--22, 1994.

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C.-J. Kim, "Dynamic linear models with markov-switching," J. Econometrics, vol. 60, 1994.

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C.-J. Kim, "Dynamic linear models with Markov-switching," Journal of Econometrics, vol. 60, pp. 1--22, 1994. 351

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