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## Consistency of the maximum likelihood estimate for Non-homogeneous Markov-switching models (2013)

Citations: | 1 - 1 self |

### Citations

1934 | A New Approach to Economic Analysis of Nonstationary Time Series
- Hamilton
- 1989
(Show Context)
Citation Context ...nd direction Introduction Recent decades have seen extensive interest in time series models with regime switchings. One of the most influential paper in this field is the one by Hamilton in 1989 (see =-=[18]-=-) where Markov-Switching AutoRegressive (MS-AR) models were introduced. It became one of the most popular nonlinear time series model. MS-AR models combine several autoregressive models to describe th... |

957 |
Markov chains and stochastic stability
- Meyn, Tweedie
(Show Context)
Citation Context ... we consider the pdf (w.r.t. µ0) β given by β(x, y) := For every x0, x−1 ∈ E and every y 0 −s ∈ K s+1 , we have γ(x, y) α . qθ(x0, y0|x−1, y −1 −s) ≥ αβ(x0, y0). x ′ ,y −1 −s Due to classical results =-=[29]-=-, this implies the ψ-irreducibility, the strong aperiodicity (the whole space is νs-small and νs+1-small, with νs and νs+1 equivalent to µ), the Harris recurrence (since we can decompose the whole set... |

689 |
Time Series: a Dynamical Systems Approach
- Tong
- 1990
(Show Context)
Citation Context ... xk−1 and is parametrized using indicator functions, we obtain the Threshold AutoRegressive (TAR) models which is an other important family of models with regime switching in the literature (see e.g. =-=[31]-=-). HMMs, MS-AR and TAR models have been used in many fields of applications and their theoretical properties have been extensively studied (see e.g. [31], [11] and [6]). Models with non-homogeneous Ma... |

434 |
The Statistics of Directional Data.
- Mardia
- 1972
(Show Context)
Citation Context ...es but in this work we have chosen to focus on the von Mises process initially introduced in [5]. It is based on the von Mises distribution which is a natural distribution for circular variables (see =-=[28]-=-) admitting a pdf fγ (with respect to the Lebesgue measure on T) given by ∀y ∈ T, fγ(y) = 1 exp (κ cos(y − φ)) = 2πI0(κ) 1 2πI0(κ) ∣ ∣e γe−iy∣ ∣ ∣ , (29) for some complex parameter γ := κeiφ (with κ ≥... |

249 | Nonlinear Time Series: Nonparametric and Parametric Methods. - Fan, Yao - 2003 |

194 |
Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis,
- Krolzig
- 1997
(Show Context)
Citation Context ...R model has been fitted to this time series. In practice, we have used the EM algorithm to compute the MLE. The recursions of this algorithm are relatively similar to the ones of the MS-AR model (see =-=[24]-=-, [8]). To facilitate the comparison with (26), we have also considered AR models of order s = 2 and a lag r = 2 for the transition probabilities. The fitted model is the following ⎧ ⎪⎨ 0.54 +1.11 Yk−... |

147 |
Regime Switching with TimeVarying Transitions Probabilities,
- Diebold, Lee, et al.
- 1994
(Show Context)
Citation Context ... adapt the standard numerical estimation procedure which are available for the homogeneous models, such as the forward-backward recursions or the EM algorithm, to the non-homogeneous models (see e.g. =-=[8]-=-, [22], [20]). However, we could not find any theoretical results on the asymptotic properties of the MLE for these models and this paper aims at filling this gap. The paper is organized as follows. I... |

115 | Maximum-likelihood estimation for hidden Markov models
- Leroux
- 1992
(Show Context)
Citation Context ...Ω+, F+) associated to the invariant measure ¯νθ and by Ēθ the corresponding expectation. The question of consistency of the MLE has been studied by many authors in the context of usual HMMs (see e.g. =-=[26, 25, 9]-=-) and MS-AR models (see [10] and references therein). The aim of this section is to state consistency results of MLE for general NHMS-AR. The proof of the following theorem is a direct but careful ada... |

85 |
Inference in hidden Markov models
- Cappé, Moulines, et al.
- 2005
(Show Context)
Citation Context ...ing in the literature (see e.g. [31]). HMMs, MS-AR and TAR models have been used in many fields of applications and their theoretical properties have been extensively studied (see e.g. [31], [11] and =-=[6]-=-). Models with non-homogeneous Markov switchings have also been considered in the literature. In particular, they have been used to describe breaks associated with events such as financial crises or a... |

85 | A non-homogeneous hidden Markov model for precipitation occurrence
- Hughes, Guttorp, et al.
- 1999
(Show Context)
Citation Context ...th events such as financial crises or abrupt changes in government policy in econometric time series (see [22] and references therein). They are also popular for meteorological applications (see e.g. =-=[20]-=-, [4], [33]) with the regimes describing the so-called ”weather types”. In most cases it is assumed that the evolution of {Xk} depends not only on lagged values of the process of interest but also on ... |

82 |
Identifiability of finite mixtures.
- Teicher
- 1963
(Show Context)
Citation Context ...nce p̄θ1(y k k−s) > 0 (the invariant pdf h1 satisfies h1 > 0 and the transition pdf qθ satisfies qθ > 0 by construction), this last equality also holds for Lebesgue almost every y k k−s. According to =-=[23]-=-, finite mixtures of Gaussian distribution are identifiable. Due to (7), this implies in particular that if 2∑ x=1 π(1)x N (y; a (1) x , σ (1) x ) = M∑ x=1 π(2)x N (y; a (2) x , σ (2) x ) for − a.e. y... |

65 | Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime,
- Douc, Moulines, et al.
- 2004
(Show Context)
Citation Context ... measure ¯νθ and by Ēθ the corresponding expectation. The question of consistency of the MLE has been studied by many authors in the context of usual HMMs (see e.g. [26, 25, 9]) and MS-AR models (see =-=[10]-=- and references therein). The aim of this section is to state consistency results of MLE for general NHMS-AR. The proof of the following theorem is a direct but careful adaptation of the proof of [10,... |

63 |
The Analysis of Directional Time Series: Applications to Wind Speed and Direction.
- BRECKLING
- 1989
(Show Context)
Citation Context ... very few models for time series of wind direction which is an important meteorological parameter for many applications. Some models have been proposed in the literature for circular time series (see =-=[5]-=-, [13],[17], [21]) and some of them have been applied to time series of wind direction. However they are not able to catch the complex features of the time series of wind direction considered in this ... |

56 | Exponential forgetting and geometric ergodicity in hidden Markov models
- Gland, Mevel
- 2000
(Show Context)
Citation Context ...Ω+, F+) associated to the invariant measure ¯νθ and by Ēθ the corresponding expectation. The question of consistency of the MLE has been studied by many authors in the context of usual HMMs (see e.g. =-=[26, 25, 9]-=-) and MS-AR models (see [10] and references therein). The aim of this section is to state consistency results of MLE for general NHMS-AR. The proof of the following theorem is a direct but careful ada... |

51 | Asymptotics of the maximum likelihood estimator for general hidden markov models
- Douc, Matias
(Show Context)
Citation Context ...Ω+, F+) associated to the invariant measure ¯νθ and by Ēθ the corresponding expectation. The question of consistency of the MLE has been studied by many authors in the context of usual HMMs (see e.g. =-=[26, 25, 9]-=-) and MS-AR models (see [10] and references therein). The aim of this section is to state consistency results of MLE for general NHMS-AR. The proof of the following theorem is a direct but careful ada... |

46 |
Hidden Markov Models for Time Series: An Introduction Using R.
- Zucchini, MacDonald
- 2009
(Show Context)
Citation Context ...with the 44 years of data). hal-00831448, version 1 - 7 Jun 2013 (see [2] for more recent references on this topic). HMMs have also been proposed for modeling time series of wind direction (see [17], =-=[37]-=-). However HMMs assume that successive observations are conditionally independent given the latent weather type and fail in reproducing the strong relation which exists between the wind conditions at ... |

44 |
The geometric ergodicity of nonlinear autoregressive models,
- An, Huang
- 1996
(Show Context)
Citation Context ...odel used in [22] or a Gaussian kernel (see (18)) could also be considered. 2.1.2 Properties of this Markov chain Various authors have studied the ergodicity of MS-AR ([35], [34], [14]) and TAR ([7], =-=[3]-=-) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition. Here we will use a strict drift condition. Let ‖ · ‖ be some norm on R s ... |

35 |
A hidden Markov model for downscaling synoptic atmospheric patterns to precipitation amounts’.
- Bellone, Hughes, et al.
- 2000
(Show Context)
Citation Context ...nts such as financial crises or abrupt changes in government policy in econometric time series (see [22] and references therein). They are also popular for meteorological applications (see e.g. [20], =-=[4]-=-, [33]) with the regimes describing the so-called ”weather types”. In most cases it is assumed that the evolution of {Xk} depends not only on lagged values of the process of interest but also on stric... |

35 | Estimation of Markov regime-switching regression models with endogenous switching’.
- Kim, Piger, et al.
- 2008
(Show Context)
Citation Context ...sidered in the literature. In particular, they have been used to describe breaks associated with events such as financial crises or abrupt changes in government policy in econometric time series (see =-=[22]-=- and references therein). They are also popular for meteorological applications (see e.g. [20], [4], [33]) with the regimes describing the so-called ”weather types”. In most cases it is assumed that t... |

26 |
A hidden Markov model for space-time precipitation
- Zucchini, Guttorp
- 1991
(Show Context)
Citation Context ...eature of meteorological time series. A classical approach for modeling these meteorological regimes (or ”weather types”) consists in introducing a hidden (or latent) variable. This idea goes back to =-=[36]-=- where HMMs were proposed for modelling the space-time evolution of daily rainfall 16E S Wind direction S W N E W E 0 5 10 15 20 25 30 Time (days) N Figure 3: Wind direction for the month of January ... |

25 |
Stochastic downscaling of precipitation: From dry events to heavy rainfalls
- Vrac, Naveau
- 2007
(Show Context)
Citation Context ...uch as financial crises or abrupt changes in government policy in econometric time series (see [22] and references therein). They are also popular for meteorological applications (see e.g. [20], [4], =-=[33]-=-) with the regimes describing the so-called ”weather types”. In most cases it is assumed that the evolution of {Xk} depends not only on lagged values of the process of interest but also on strictly ex... |

18 |
A correlation coefficient for circular data.
- Fisher, Lee
- 1983
(Show Context)
Citation Context ...f the dynamics of the process although it remains some significant discrepancies. In particular, the fitted NHMS-AR model slightly underestimates the circular autocorrelation function defined as (see =-=[12]-=-) ρ(k) = E[cos(Y0) cos(Yk)] + E[sin(Y0) sin(Yk)] − E[sin(Y0) cos(Yk)] − E[cos(Y0) sin(Yk)] E[cos(Y0) 2 ]E[sin(Y0) 2 ] − E[sin(Y0) cos(Y0)] 2 for lags between 2 and 5 days and some coefficients of the ... |

16 |
On the ergodicity of TAR(1) processes.
- Chen, Tsay
- 1991
(Show Context)
Citation Context ...bit model used in [22] or a Gaussian kernel (see (18)) could also be considered. 2.1.2 Properties of this Markov chain Various authors have studied the ergodicity of MS-AR ([35], [34], [14]) and TAR (=-=[7]-=-, [3]) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition. Here we will use a strict drift condition. Let ‖ · ‖ be some norm on... |

13 |
Time series analysis of circular data.
- Fisher, Lee
- 1994
(Show Context)
Citation Context ... few models for time series of wind direction which is an important meteorological parameter for many applications. Some models have been proposed in the literature for circular time series (see [5], =-=[13]-=-,[17], [21]) and some of them have been applied to time series of wind direction. However they are not able to catch the complex features of the time series of wind direction considered in this work. ... |

13 |
Markov models for circular and linear-circular time series.
- Holzmann, Munk, et al.
- 2006
(Show Context)
Citation Context ...models for time series of wind direction which is an important meteorological parameter for many applications. Some models have been proposed in the literature for circular time series (see [5], [13],=-=[17]-=-, [21]) and some of them have been applied to time series of wind direction. However they are not able to catch the complex features of the time series of wind direction considered in this work. We us... |

13 |
On stability of nonlinear AR processes with Markov switching
- Yao, Attali
- 2000
(Show Context)
Citation Context ...k functions such as the probit model used in [22] or a Gaussian kernel (see (18)) could also be considered. 2.1.2 Properties of this Markov chain Various authors have studied the ergodicity of MS-AR (=-=[35]-=-, [34], [14]) and TAR ([7], [3]) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition. Here we will use a strict drift condition.... |

13 |
Consistent estimation of linear and non-linear autoregressive models with markov regime. Journal of time series analysis
- Krishnamurthy, Ryden
- 1998
(Show Context)
Citation Context ...∈ER f(y) > 0, and • E = {1, ...,M} with M ≥ 2 and (8) replaced by any transition kernel p1,θ satisfying (1). 8 2.3 Consistency of MLE The results given in this section generalize the results given in =-=[12, 17]-=- for homogeneous MS-AR models with linear Gaussian autoregressive models. Corollary 9. Assume that Hypotheses 4 and 6 hold true for every θ. Let Θ be a compact subset of Θ̃. Then, for all θ ∈ Θ there ... |

12 |
Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum-likelihood estimator
- Francq, Roussignol
- 1998
(Show Context)
Citation Context ...such as the probit model used in [22] or a Gaussian kernel (see (18)) could also be considered. 2.1.2 Properties of this Markov chain Various authors have studied the ergodicity of MS-AR ([35], [34], =-=[14]-=-) and TAR ([7], [3]) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition. Here we will use a strict drift condition. Let ‖ · ‖ b... |

11 | Markov-switching autoregressive models for wind time series. Environ
- Ailliot, Monbet
- 2012
(Show Context)
Citation Context ... systematic validation is performed on a longer time series in the next section. 3.2 Wind direction Various approaches have been proposed in the literature for modeling time series of wind speed (see =-=[1]-=- and references therein). In comparison, there exist only very few models for time series of wind direction which is an important meteorological parameter for many applications. Some models have been ... |

11 | The Nagaev-Guivarc'h method via the Keller-Liverani theorem - Hervé, Pène - 2010 |

9 |
On square-integrability of an AR process with Markov switching
- Yao
- 2001
(Show Context)
Citation Context ...tions such as the probit model used in [22] or a Gaussian kernel (see (18)) could also be considered. 2.1.2 Properties of this Markov chain Various authors have studied the ergodicity of MS-AR ([35], =-=[34]-=-, [14]) and TAR ([7], [3]) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition. Here we will use a strict drift condition. Let ‖... |

6 |
A Markov Process for Circular Data
- Kato
- 2010
(Show Context)
Citation Context ... for time series of wind direction which is an important meteorological parameter for many applications. Some models have been proposed in the literature for circular time series (see [5], [13],[17], =-=[21]-=-) and some of them have been applied to time series of wind direction. However they are not able to catch the complex features of the time series of wind direction considered in this work. We use data... |

6 |
Confidence intervals for hidden markov model parameters. British journal of mathematical and statistical psychology
- Visser, Raijmakers, et al.
(Show Context)
Citation Context ...k−2))−1 (Xk = 2) (2.25,178) (-64.1,-1.12) (10) 5 where the italic values in parenthesis below the parameter values correspond to 95% confidence intervals computed using parametric bootstrap (see e.g. =-=[25]-=-). These values reflect the finite sample properties of the estimates. The estimate of π (x) − and π (x) + are not given because they are very close to 0. It means that these technical parameters have... |

5 |
Identifiability of Finite Mixtures of von Mises Distributions. Annals of Statistics 9:1130–1131
- FRASER, HSU, et al.
- 1981
(Show Context)
Citation Context ...,(1) +∑ s ℓ=1 γ(x) ℓ,(1) eiy k−ℓ (yk) = with fγ defined by (29) where m denotes the Lebesgue measure on T. M∑ x=1 ¯Pθ2(Xk = x|y k−1 k−s )f γ (x) 0,(2) +∑ s ℓ=1 γ(x) ℓ,(2) eiy k−ℓ (yk) 24According to =-=[16]-=-, finite mixtures of von Mises distribution are identifiable. This implies in particular that if M∑ π (x) 1 fγ (x) M∑ (y) = π (x) 2 fγ (x)(y) for m − a.e. y x=1 with γ (x) 1 = γ (x′) 1 for x = x ′ a... |

4 |
Lectures on the coupling method, Corrected reprint of the 1992 original
- Lindvall
- 2002
(Show Context)
Citation Context ... E pθ(Y n ) dm⊗(n−k+1) E (˜x n k ) pθ(Y n k |Xk = xk, Y k−1 k |Xk = ˜xk, Y k−1 k−s , k−s ) . (35) ) dmE(˜xk) From this last inequality (since 0 < p1,− < p1,+ < ∞), we directly get the following (from =-=[27]-=-). 19Corollary 22. (as [10, Cor. 1]) For all m ≤ k ≤ n and every probability measures m1 and m2 on E, we have, ¯ Pθ − a.s. ∫ ¯Pθ(Xk ∥ ∈ ·|Xm = xm, Y n ∫ m−s+1) dm1(xm) − ¯Pθ(Xk ∈ ·|Xm = xm, Y n m−s+1... |

1 |
Identifiability of finite mixtures. Annals of mathematical statistics
- Teicher
- 1963
(Show Context)
Citation Context ...x ′ ),(2). C Identifiability for the gaussian model: proof of Proposition 9 Using similar arguments as in Appendix B, existing results on the identifiability of mixture of Gaussian distributions (see =-=[30]-=-) we obtain that if ¯ PY θ1 = ¯ PY , then for all x ∈ {1, 2} and y ∈ R θ2 ( β (x) ) ( 0,(1) , β(x) 1,(1) , ..., β(x) r,(1) , σ(x) (1) = β (x) ) 0,(2) , β(x) 1,(2) , ..., β(x) r,(2) , σ(x) (2) and p1,θ... |

1 |
Maartje EJ Raijmakers, and Peter Molenaar. Confidence intervals for hidden markov model parameters. British journal of mathematical and statistical psychology
- Visser
(Show Context)
Citation Context ...yk−2)) −1 (Xk = 2) (2.25,178) (-64.1,-1.12) (28) where the italic values in parenthesis below the parameter values correspond to 95% confidence intervals computed using parametric bootstrap (see e.g. =-=[32]-=-). The estimate of π (x) − and π (x) + are not given because they are very close to 0. It means that these technical parameters have no practical importance and can be fixed equal to an arbitrary smal... |

1 |
Some theoretical results on markov-switching autoregressive models with gamma innovations. Comptes Rendus de l’Acadmie des sciences
- Ailliot
(Show Context)
Citation Context ...df which may not be bounded close to the origin depending on the values of the parameters. The results given in [9] do not apply directly to this model whereas we will show that (3) applies (see also =-=[1]-=-). Third, to prove the result in the stationary case, we replace Harris recurrence by (5) which is equivalent to each one of the two following properties • for any initial measure ν on E ×Ks, we have ... |