Unobserved State Variables Research Articles

Inference in Dynamic Discrete Choice Models With Serially Correlated Unobserved State Variables

This paper develops a method for inference in dynamic discrete choice models with serially correlated unobserved state variables. Estimation of these models involves computing high-dimensional integrals that are present in the solution to the dynamic program and in the likelihood function. First, the paper proposes a Bayesian Markov chain Monte Carlo estimation procedure that can handle the problem of multidimensional integration in the likelihood function. Second, the paper presents an efficient algorithm for solving the dynamic program suitable for use in conjunction with the proposed estimation procedure.

Parameter Estimation of Electricity Spot Models from Futures Prices

We consider a slight perturbation of the Schwartz-Smith model for the electricity futures prices and the resulting modified spot model. Using the martingale property of the modified price under the risk neutral measure, we derive the arbitrage free model for the spot and futures prices. We estimate the parameters of the model by the method of maximum likelihood using the Kalman filter's estimate of the unobservable state variables, coupled with the usual statistical techniques. The main advantage of the new model is that it avoids the inclusion of artificial noise to the observation equation for the implementation of Kalman filter. The extra noise is build in within the model in an arbitrage free setting.

B02 温度調節器の非線形モデル予測制御(OS6 非線形制御理論とその応用1)

In this paper, we describe a thennal control of a thermoelectric cooler / heater with Peltier device which is operated by the Peltier Effect. Generally, it is said that the thermal control with Peltier device is difficult because of its nonlineariry. In order to deal with the nonlinearity, we use nonlinear model predictive control for the thermal control of the device. A nonlinear model of the thermal system with Peltier device is derived. The validity of the model is evaluated by experimental results. Nonlinear model predictive control system is designed for the model. Recursive Kalman filter is used to estimate the unobservable state variables. The experimental results show the accurate control performance. The robustness against themal load is also verified by the experiments.

Brain Network Analysis of Seizure Evolution

The human brain is one of the most complex biological systems. Neuroscientists seek to understand the brain function through detailed analysis of neuronal excitability and synaptic transmission. In this study, we propose a network analysis framework to study the evolution of epileptic seizures. We apply a signal processing approach, derived from information theory, to investigate the synchronization of neuronal activities, which can be captured by electroencephalogram (EEG) recordings. Two networktheoretic approaches are proposed to globally model the synchronization of the brain network. We observe some unique patterns related to the development of epileptic seizures, which can be used to illuminate the brain function governed by the epileptogenic process during the period before a seizure. The proposed framework can provide a global structural patterns in the brain network and may be used in the simulation study of dynamical systems (e.g. the brain) to predict oncoming events (e.g. seizures). To analyze long-term EEG recordings in the future, we discuss how the Markov-Chain Monte Carlo (MCMC) methodology can be applied to estimate the clique parameters. This MCMC framework fits very well with this work as the epileptic evolution can be considered to be a system with unobservable state variables and nonlinearities.

Canonical Term‐Structure Models with Observable Factors and the Dynamics of Bond Risk Premia

We derive a canonical representation for the no‐arbitrage discrete‐time term structure models with both observable and unobservable state variables, popularized by Ang and Piazzesi (2003). We conduct a specification analysis based on this canonical representation and we analyze how alternative parameterizations affect estimated risk premia, impulse response functions, and variance decompositions. We find a trade‐off between the need to obtain parsimonious parameterizations and the ability of the models to match observed patterns of variation in risk premia. We also find that more richly parameterized models uncover a greater influence of macroeconomic fundamentals on the long‐end of the yield curve.

Canonical Term-Structure Models with Observable Factors and the Dynamics of Bond Risk Premiums

We derive a canonical representation for the no-arbitrage discrete-time term structure models with both observable and unobservable state variables, popularized by Ang and Piazzesi (2003). We conduct a specification analysis based on this canonical representation and we analyze how alternative parametrizations affect estimated risk premia, impulse response functions and variance decompositions. We find a trade-off between the need to obtain parsimonious parametrizations and the ability of the models to match observed patterns of variation in risk premia. We also find that more richly parametrized models uncover a greater influence of macroeconomic fundamentals on the long-end of the yield curve.

경사각을 갖는 관성항법시스템 초기 정밀정렬의 오차 분석

In this paper, performance of the initial alignment for INS whose attitude is not leveled is investigated. Observability of the initial alignment filter is analyzed and estimation errors of the estimated state variables are derived. First, the observability is analyzed using the rank test of observability matrix and the normalized error covariance of the Kalman filter based on the 10-state model. In result, it can be seen that the accelerometer biases on horizontal axes are unobservable. Second, the steady-state estimation errors of the state variables are derived using the observability equation. It is verified that the estimates of the state variables have errors due to the unobservable state variables and the non-leveling tilt angles of a vehicle containing the INS. Especially, this paper shows that the larger the tilt angles of the vehicle are, the larger the estimation errors corresponding to the sensor biases are. Finally, it is shown that the performance of the 8-state model excepting the accelerometer biases on horizontal axes is better than that of the 10-state model in the initial alignment by simulation.

Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models

We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330-336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and lambda) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330-336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.

Analysis of commodity prices with the particle filter

The behavior of commodities prices is fundamental to real-asset investment decisions, hedging, and pricing financial derivatives. Schwartz and Smith [Schwartz, E.S., Smith, J.E. (2000). Short term-variations and long-term dynamics in commodity prices. Management Science, 46, 893-911.] proposed a two-factor model for describing the stochastic processes of commodity prices, in which the two factors are short-term variations and equilibrium prices. These are both unobserved state variables that are estimated using the Kalman filter. The estimation is based on the observation of future prices for different maturities. The authors have carried out this process without incorporating jumps in the short-term variation of prices. Here we aim to demonstrate that the inclusion of jumps better explains the behavior of oil prices, and in fact creates difficulties in the estimation of state variables. This is because the variables become non-Gaussian so the Kalman filter is not recommended. Another methodology, called the particle filter, is more suitable in this case, and we describe its application in this article.

Learning about Investment Risk: The Effects of Structural Uncertainty on Dynamic Investment and Consumption

This paper examines the effects of incomplete information on dynamic investment and consumption in a general equilibrium model where shocks to capital are unobservable and there is structural or parameter uncertainty regarding the volatility of these shocks; i.e., the investment risk. In this model higher investment reduces the estimation error on the unobservable state variables, and thereby also reduces the estimation error on the unknown parameter. Investment policy therefore affects the speed of learning and is itself subsequently affected as estimation errors fall over time. We quantify the interaction of learning with investment by numerically computing equilibrium investment and consumption in a version of the model calibrated by aggregate US data. We find that imperfect observability (of productivity shocks) by itself does not significantly affect equilibrium investment and consumption. The introduction of structural or parameter uncertainty has significant effects on equilibrium investment and consumption, however. Moreover, investment and consumption levels in the presence of structural uncertainty are more quantitatively consistent with the data when compared to versions of the model that assume either complete information or incomplete information without structural uncertainty. The presence of structural uncertainty also appears to make consumption growth more volatile compared to the benchmark cases. Finally, and consistent with the theoretical analysis, the marginal propensity to invest is not monotone increasing with time even though the estimation errors are being reduced due to learning.

Term Structure Forecasts of Long-Term Consumption Growth

AbstractRelying on a simple general equilibrium model of the term structure, we show that both nominal yields and real consumption growth rates can be affine in the unobservable state variables. We can then express real consumption growth rates in terms of nominal yields rather than the unobservable state variables with the coefficients of the resultant forecasting relation being endogenously determined by the term structure model. Using term structure data over the 1985 to 2000 sample period, the empirical evidence is consistent with our model more accurately predicting real consumption growth rates than a regression model based on the term spread.

A Reinforcement Learning Scheme for a Partially-Observable Multi-Agent Game

We formulate an automatic strategy acquisition problem for the multi-agent card game as a reinforcement learning problem. The problem can approximately be dealt with in the framework of a partially observable Markov decision process (POMDP) for a single-agent system. Hearts is an example of imperfect information games, which are more difficult to deal with than perfect information games. A POMDP is a decision problem that includes a process for estimating unobservable state variables. By regarding missing information as unobservable state variables, an imperfect information game can be formulated as a POMDP. However, the game of Hearts is a realistic problem that has a huge number of possible states, even when it is approximated as a single-agent system. Therefore, further approximation is necessary to make the strategy acquisition problem tractable. This article presents an approximation method based on estimating unobservable state variables and predicting the actions of the other agents. Simulation results show that our reinforcement learning method is applicable to such a difficult multi-agent problem.

Model-based reinforcement learning: a computational model and an fMRI study

In this paper, we discuss an optimal decision-making problem in an unknown environment on the bases of both machine learning and brain learning. We present a model-based reinforcement learning (RL) in which the environment is directly estimated. Our RL performs action selection according to the detection of environmental changes and the current value function. In a partially-observable situation, in which the environment includes unobservable state variables, our RL incorporates estimation of unobservable variables. We propose a possible functional model of our RL, focusing on the prefrontal cortex and the anterior cingulate cortex. To test the model, we conducted a human imaging study during a sequential learning task, and found significant activations in the dorsolateral prefrontal cortex and the anterior cingulate cortex during RL. From a comparison of the mean activations in the earlier and later learning phases, we suggest that the dorsolateral prefrontal cortex maintains and manipulates the environmental model, while the anterior cingulate cortex is related to the uncertainty of action selection. These experimental results are consistent with our model.

Pricing Power Derivatives: A Two-Factor Jump-Diffusion Approach

We propose a two-factor jump-diffusion model with seasonality for the valuation of electricity future contracts. The model we propose is an extension of Schwartz and Smith (Management Science, 2000) long-term / short-term model. One of the main contributions of the paper is the inclusion of a jump component, with a non-constant intensity process (probability of occurrence of jumps), in the short-term factor. We model the stochastic behaviour of the underlying (unobservable) state variables by Affine Diffusions (AD) and Affine Jump Diffusions (AJD). We obtain closed form formulas for the price of futures contracts using the results by Duffie, Pan and Singleton (Econometrica, 2000). We provide empirical evidence on the observed seasonality in risk premium, that has been documented in the PJM market. This paper also complements the results provided by the equilibrium model of Bessembinder and Lemmon (Journal of Finance, 2002), and provides an easy methodology to extract risk-neutral parameters from forward data.

Iterative and Recursive Estimation in Structural Nonadaptive Models

An inference method, called latent backfitting, is proposed. This method appears well suited for econometric models where the structural relationships of interest define the observed endogenous variables as a known function of unobserved state variables and unknown parameters. This nonlinear state-space specification paves the way for iterative or recursive EM-like strategies. In the E steps, the state variables are forecasted given the observations and a value of the parameters. In the M steps, these forecasts are used to deduce estimators of the unknown parameters from the statistical model of latent variables. The proposed iterative/recursive estimation is particularly useful for latent regression models and for dynamic equilibrium models involving latent state variables. Practical implementation issues are discussed through the example of term structure models of interest rates.

A Computationally Practical Simulation Estimation Algorithm for Dynamic Panel Data Models with Unobserved Endogenous State Variables

This paper develops a simulation estimation algorithm that is particularly useful for estimating dynamic panel data models with unobserved endogenous state variables. The new approach can easily deal with the commonly encountered and widely discussed initial conditions problem, as well as the more general problem of missing state variables during the sample period. Repeated sampling experiments on dynamic probit models with serially correlated errors indicate that the estimator has good small sample properties. We apply the estimator to a model of married women's labor force participation decisions. The results show that the rarely used Polya model, which is very difficult to estimate given missing data problems, fits the data substantially better than the popular Markov model. The Polya model implies far less state dependence in employment status than the Markov model. It also implies that observed heterogeneity in education, young children and husband income are much more important determinants of participation, while race is much less important.

Dynamic Portfolio Choice: A Simulation Approach

We present a simulation-based method for solving realistic portfolio choice problems that potentially involve non-standard preferences and a large number of assets with arbitrary return distribution. Specifically, the return distribution can be time-varying as a function of many observable or unobservable state variables and can even be path-dependent. Furthermore, the method is flexible enough to accommodate intermediate consumption, parameter and model uncertainty, and portfolio constraints. We first establish the properties of the method for the choice between a stock index and cash when the stock returns are either iid or predictable by the dividend yield. We then explore the optimal asset allocation across ten industry portfolios that exhibit momentum through its empirical pattern of own- and cross-serial correlations of returns.

MCMC Analysis of Diffusion Models With Application to Finance

This article proposes a new method for estimation of parameters in diffusion processes from discrete observations. The method is based on Markov-chain Monte Carlo methodology and applies to a wide class of models including systems with unobservable state variables and nonlinearities. The method is applied to the estimation of parameters in one-factor (CEV) interest-rate models and a two-factor model with a latent stochastic volatility component (SV). The CEV model is found to do a poor job in capturing the time-varying volatility interest-rate data. The SV model provides vastly superior fit to that of the CEV model. The article also presents a simulation study that demonstrates that the method provides accurate parameter estimates of the SV model at moderate sample sizes.

Gaussian Estimation of a Two‐factor Continuous Time Model of the Short‐term Interest Rate

This paper considers the econometric estimation of a two‐factor model of the short‐term interest rate. We develop a procedure for the time series estimation of its parameters, based on recently developed Gaussian estimation methods which are extended to handle unobservable state variables. The main methodological contribution is the derivation of an exact discrete model and the exact Gaussian likelihood function in terms of the discrete observations and structural parameters of the two‐factor model. The model is estimated on one month euro‐currency interest rate data for seven currencies – the Belgium franc, Canadian dollar, Dutch guilder, French franc, German mark, Italian lira, and the US dollar over the period February 1981 to December 1995 – and our results indicate that the method works well in practice. The empirical estimates reported in this study can be used to compare estimates from the calibration of an arbitrage‐free analogue of the model to market prices of interest rate caps and swaptions for use in the financial markets. (J.E.L.: C13, E43).

Business cycle durations

While the development of Markov switching extensions to time series modeling has provided a useful way of characterizing business cycle dynamics, these models are not without their weaknesses. One problem is posed by the fact that since the state space for the unobserved state variables grows with the sample size, sampling distributions for maximum-likelihood estimates are difficult to establish. A second problem is that since the transition probabilities are constant, the conditional expected duration of phase is constant. This paper extends the model so that the information contained in leading indicator data can be used to forecast transition probabilities. These transition probabilities can then be used to calculate expected durations. The model is applied to US data to evaluate its ability to explain observed business cycle durations. The technical problems encountered with classical techniques are avoided by using Bayesian methods. Gibbs sampling techniques are used to calculate expected posterior durations.