### Abstract

For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

Original language | English (US) |
---|---|

Pages (from-to) | 1951-1966 |

Number of pages | 16 |

Journal | Journal of Economic Dynamics and Control |

Volume | 34 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2010 |

### Fingerprint

### Keywords

- Certainty equivalence
- Entropy
- Hidden Markov models
- Kalman filter
- Misspecification
- Robustness

### ASJC Scopus subject areas

- Economics and Econometrics
- Applied Mathematics
- Control and Optimization

### Cite this

*Journal of Economic Dynamics and Control*,

*34*(10), 1951-1966. https://doi.org/10.1016/j.jedc.2010.05.004

**Robust hidden Markov LQG problems.** / Hansen, Lars Peter; Mayer, Ricardo; Sargent, Thomas.

Research output: Contribution to journal › Article

*Journal of Economic Dynamics and Control*, vol. 34, no. 10, pp. 1951-1966. https://doi.org/10.1016/j.jedc.2010.05.004

}

TY - JOUR

T1 - Robust hidden Markov LQG problems

AU - Hansen, Lars Peter

AU - Mayer, Ricardo

AU - Sargent, Thomas

PY - 2010/10

Y1 - 2010/10

N2 - For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

AB - For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

KW - Certainty equivalence

KW - Entropy

KW - Hidden Markov models

KW - Kalman filter

KW - Misspecification

KW - Robustness

UR - http://www.scopus.com/inward/record.url?scp=77956884459&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956884459&partnerID=8YFLogxK

U2 - 10.1016/j.jedc.2010.05.004

DO - 10.1016/j.jedc.2010.05.004

M3 - Article

VL - 34

SP - 1951

EP - 1966

JO - Journal of Economic Dynamics and Control

JF - Journal of Economic Dynamics and Control

SN - 0165-1889

IS - 10

ER -