On the least squares estimation of multiple-regime threshold autoregressive models

Liu, Dong; Ling, Shiqing

doi:10.1016/j.jeconom.2011.11.006

Cited by 85 publications

(57 citation statements)

References 41 publications

(33 reference statements)

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“…The finite fouth moment in Theorem 4.1 is inherited from Hansen (2000). When there is a threshold effect in variance in model (2.1) as Chan (1993), Li and Ling (2012) discussed the limiting distribution of the estimated threshold. In this case, the approximation in Theorem 4.1 should be asymmetric, see a further discussion in Yu (2012) and Yu (2015).…”

Section: Approximating Distributions Of Thementioning

confidence: 99%

“…It has been extensively applied in many areas, including economics, finance, biological and environmental sciences, among others, see Chan and Kutoyants (2010) and Tong (2011) for nice reviews. The asymptotic theory of the least squares estimator (LSE) of two-regime TAR models was established by Chan (1993) and Chan and Tsay (1998), and was further developed by Li and Ling (2012) and Li et al (2013) for the multiple-regime TAR model and the TMA model, respectively. Hansen (2000) studied the LSE for two-regime threshold AR/regression models when the threshold effect is vanishingly small.…”

Section: Introductionmentioning

confidence: 99%

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Statistical Inference for Structurally Changed Threshold Autoregressive Models

Gao¹,

Ling²

2019

STAT SINICA

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In this paper, we study the theory and methodology for statistical inferences of threshold and change-point in the threshold autoregressive models. It is shown that the least squares estimators (LSE) of the threshold and change-point are n-consistent, and converge weakly to the minimizer of a compound Poisson process and the location of minima of a two-sided random walk, respectively.When the magnitude of change in the parameters of the state regimes or the time horizon is small, it is further shown that these limiting distributions can be approximated by a class of known distributions. The LSE of slope parameters are √ n-consistent and asymptotically normal. Furthermore, a likelihood-ratio based confidence set is given for the threshold and change-point, respectively. Simulation study is carried out to assess the performance of our procedure. The proposed theory and methodology are further illustrated by a tree-ring dataset.

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Section: Approximating Distributions Of Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Statistical Inference for Structurally Changed Threshold Autoregressive Models

Gao¹,

Ling²

2019

STAT SINICA

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“…For multi-threshold stochastic regression models (e.g., Ertel and Fowlkes (1976), Liu, Wu, and Zidek (1997), Gonzalo and Pitarakis (2002), Li and Ling (2012)), we can use the NeSS algorithm to obtain a sequential estimate of multiple thresholds, one at a time, by the NeSS algorithm. However, it is known that the limiting distribution of such a sequential estimate is different from that of a joint estimate.…”

Section: Discussionmentioning

confidence: 99%

“…For the former, see, e.g., Bacon and Watts (1971), Goldfeld and Quandt (1972), Maddala (1977), Quandt (1983), and others. For the latter, see, e.g., Chan (1993), who first showed that the least squares estimator (LSE) of the threshold parameter is super-consistent and obtained its limiting distribution theoretically; Hansen (1997Hansen ( , 2000, who presented an alternative approximation to the limiting distribution of the estimated threshold when the threshold effect diminishes as the sample size increases; Gonzalo and Pitarakis (2002), who developed a sequential estimation approach that makes the estimation of multiple threshold models computationally feasible and formally discussed the large sample properties; Li and Ling (2012), who established the asymptotic theory of LSE in multiple threshold models and proposed a resampling method for implementing the limiting distribution of the estimated threshold directly when the threshold effect is fixed. Other significant results related to threshold models include Tsay (1989Tsay ( , 1998, Hansen (1996), Caner and Hansen (2001), Gonzalo and Wolf (2005), Seo and Linton (2007), and Yu (2012), among others.…”

Section: Introductionmentioning

confidence: 99%

“…However, when the threshold is unknown, the ordinary least squares method for linear regression cannot be applied immediately since the threshold parameter lies in an indicator function. This issue has been commonly tackled by using the single grid search (SGS) algorithm over a feasible threshold space; see Tong and Lim (1980), Chan (1993), Hansen (1997Hansen ( , 2000, Gonzalo and Pitarakis (2002), Li and Ling (2012), Yu (2012), and others. The SGS algorithm requires least squares operations of order O(n) for single threshold models, where n is the sample size.…”

Section: Introductionmentioning

confidence: 99%

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Nested sub-sample search algorithm for estimation of threshold models

Li¹,

Tong²

2016

STAT SINICA

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NESTED SUB-SAMPLE SEARCH ALGORITHM FOR ESTIMATION OF THRESHOLD MODELS Dong Li and Howell Tong Tsinghua University and London School of Economics & Political ScienceAbstract: Threshold models have been popular for modelling nonlinear phenomena in diverse areas, in part due to their simple fitting and often clear model interpretation. A commonly used approach to fit a threshold model is the (conditional) least squares method, for which the standard grid search typically requires O(n) operations for a sample of size n; this is substantial for large n, especially in the context of panel time series. This paper proposes a novel method, the nested subsample search algorithm, which reduces the number of least squares operations drastically to O(log n) for large sample size. We demonstrate its speed and reliability via Monte Carlo simulation studies with finite samples. Possible extension to maximum likelihood estimation is indicated.

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Quasi‐maximum exponential likelihood estimation for double‐threshold GARCH models

2021

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We consider the nonparametric inference for the double‐threshold generalized autoregressive conditional heteroscedastic models. The quasi‐maximum exponential likelihood estimators (QMELEs) of the model parameters are obtained, and their asymptotic properties are established. Simulation studies imply that the estimators are asymptotically normally distributed. An empirical investigation of stock returns illustrates our findings. Both the simulations and the example indicate that the QMELE is feasible, reliable and appropriate to fit the financial time series data of the Hang Seng Index.

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On the least squares estimation of multiple-regime threshold autoregressive models

Cited by 85 publications

References 41 publications

Statistical Inference for Structurally Changed Threshold Autoregressive Models

Statistical Inference for Structurally Changed Threshold Autoregressive Models

Nested sub-sample search algorithm for estimation of threshold models

Quasi‐maximum exponential likelihood estimation for double‐threshold GARCH models

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