Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters

Prediction intervals in State Space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, where the true parameters are substituted by consistent estimates. This approach has two limitations. First, it does not incorporate the uncertainty due to parameter estimation. Second, the Gaussianity assumption of future innovations may be inaccurate. To overcome these drawbacks, Wall and Stoffer (2002) propose to obtain prediction intervals by using a bootstrap procedure that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. The bootstrap procedure proposed by Wall and Stoffer (2002) is further complicated by fact that the intervals are obtained for the prediction errors instead of for the observations. In this paper, we propose a bootstrap procedure for constructing prediction intervals in State Space models that does not need the backward representation of the model and is based on obtaining the intervals directly for the observations. Therefore, its application is much simpler, without loosing the good behavior of bootstrap prediction intervals. We study its finite sample properties and compare them with those of the standard and the Wall and Stoffer (2002) procedures for the Local Level Model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series

Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters

Prediction intervals in State Space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, where the true parameters are substituted by consistent estimates. This approach has two limitations. First, it does not incorporate the uncertainty due to parameter estimation. Second, the Gaussianity assumption of future innovations may be inaccurate. To overcome these drawbacks, Wall and Stoffer (2002) propose to obtain prediction intervals by using a bootstrap procedure that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. The bootstrap procedure proposed by Wall and Stoffer (2002) is further complicated by fact that the intervals are obtained for the prediction errors instead of for the observations. In this paper, we propose a bootstrap procedure for constructing prediction intervals in State Space models that does not need the backward representation of the model and is based on obtaining the intervals directly for the observations. Therefore, its application is much simpler, without loosing the good behavior of bootstrap prediction intervals. We study its finite sample properties and compare them with those of the standard and the Wall and Stoffer (2002) procedures for the Local Level Model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series.NAIRU, Output gap, Parameter uncertainty, Prediction Intervals, State Space Models

Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters

In the context of linear state space models with known parameters, the Kalman filter (KF) generates best linear unbiased predictions of the underlying states together with their corresponding Prediction Mean Square Errors (PMSE). However, in practice, when the filter is run with the parameters substituted by consistent estimates, the corresponding PMSE do not take into account the parameter uncertainty. Consequently, they underestimate their true counterparts. In this paper, we propose two new bootstrap procedures to obtain PMSE of the unobserved states designed to incorporate this latter uncertainty. We show that the new bootstrap procedures have better finite sample properties than bootstrap alternatives and than procedures based on the asymptotic approximation of the parameter distribution. The proposed procedures are implemented for estimating the PMSE of several key unobservable US macroeconomic variables as the output gap, the Non-accelerating Inflation Rate of Unemployment (NAIRU), the long-run investment rate and the core inflation. We show that taking into account the parameter uncertainty may change their prediction intervals and, consequently, the conclusions about the utility of the NAIRU as a macroeconomic indicator for expansions and recessions.The second author gratefully acknowledges the financial support from Project ECO2009-08100 by the Spanish GovernmentPublicad

Unemployment and Hysteresis: A Nonlinear Unobserved Components A Nonlinear Unobserved Components A Nonlinear Unobserved Components A Nonlinear Unobserved Components A Nonlinear Unobserved Components Approach

A new test for hysteresis based on a nonlinear unobserved components model is proposed. Observed unemployment rates are decomposed into a natural rate component and a cyclical component. Threshold type nonlinearities are introduced by allowing past cyclical unemployment to have a different impact on the natural rate depending onthe regime of the economy. The impact of lagged cyclical shocks on thecurrent natural component is the measure of hysteresis. To derive anappropriate p-value for a test for hysteresis two alternative bootstrapalgorithms are proposed: the first is valid under homoskedastic errorsand the second allows for heteroskedasticity of unknown form. A MonteCarlo simulation study shows the good performance of both bootstrapalgorithms. The bootstrap testing procedure is applied to data fromItaly, France and the United States. We find evidence of hysteresis forall countries under study.Hysteresis, Unobserved Components Model, Threshold Autoregressive Models, Nuisance parameters, Bootstrap

UNEMPLOYMENT AND HYSTERESIS: A NONLINEAR UNOBSERVED COMPONENTS APPROACH

The aim of this paper is to find a possible hysteresis effect on unemployment rate series from Italy, France and the United States. We propose a definition of hysteresis taken from Physics which allows for nonlinearities. To test for the presence of hysteresis we use a nonlinear unobserved components model for unemployment series. The estimation methodology used can be assimilated into a threshold autoregressive representation in the framework of a Kalman filter. To derive an appropriate p-value for a test for hysteresis we propose two alternative bootstrap procedures: the first is valid under homoskedastic errors and the second allows for general heteroskedasticity. We investigate the performance of both bootstrap procedures using Monte Carlo simulation.Hysteresis; Unobserved Components Model; Threshold Autoregressive Models; Nuisance parameters; Bootstrap

Bootstrapping unobserved component models

En esta tesis, proponemos el uso de técnicas bootstrap para incorporar la incertidumbre de la estimación de los parámetros en modelos de componentes inobservados expresado en un contexto de modelos de espacio de los estados. A lo largo de los capítulos usamos simulaciones Monte Carlo y datos reales para mostrar los resultados de los procedimientos propuesto

Bootstrap Approximation to Prediction MSE for State-Space Models with Estimated Parameters

We propose a simple but general bootstrap method for estimating the Prediction Mean Square Error (PMSE) of the state vector predictors when the unknown model parameters are estimated from the observed series. As is well known, substituting the model parameters by the sample estimates in the theoretical PMSE expression that assumes known parameter values results in under-estimation of the true PMSE. Methods proposed in the literature to deal with this problem in state-space modelling are inadequate and may not even be operational when fitting complex models, or when some of the parameters are close to their boundary values. The proposed method consists of generating a large number of series from the model fitted to the original observations, re-estimating the model parameters using the same method as used for the observed series and then estimating separately the component of PMSE resulting from filter uncertainty and the component resulting from parameter uncertainty. Application of the method to a model fitted to sample estimates of employment ratios in the U.S.A. that contains eighteen unknown parameters estimated by a three-step procedure yields accurate results. The procedure is applicable to mixed linear models that can be cast into state-space form. (Updated 6th October 2004

Monetary Policy and Data Uncertainty: A Case Study of Distribution, Hotels and Catering Growth

This paper is a case study of the real world monetary policy data uncertainty problem. The initial and the latest release for growth rates of the distribution, hotels and catering sector are combined with official data on household income and two surveys in a state-space model. Though important to the UK economy, the distribution, hotels and catering sector is apparently difficult to measure. One finding is that the initial release data is not important in predicting the latest release. It could be that the statistical office develop the initial release as a building block towards the final release rather than an estimate of it. Indeed, there is multicollinearity between the initial release and the retail sales survey, which would then contain the same early available information. A second finding is that the estimate of the later release is sensitive to the estimate of the average historical growth rate. This means that establishing priors for this parameter and testing for shift structural breaks should be very important.Data Uncertainty; Distribution Sector; Kalman Filter; Monetary Policy

Solving, Estimating and Selecting Nonlinear Dynamic Economic Models without the Curse of Dimensionality

A welfare analysis of a risky policy is impossible within a linear or linearized model and its certainty equivalence property. The presented algorithms are designed as a toolbox for a general model class. The computational challenges are considerable and I concentrate on the numerics and statistics for a simple model of dynamic consumption and labor choice. I calculate the optimal policy and estimate the posterior density of structural parameters and the marginal likelihood within a nonlinear state space model. My approach is even in an interpreted language twenty time faster than the only alternative compiled approach. The model is estimated on simulated data in order to test the routines against known true parameters. The policy function is approximated by Smolyak Chebyshev polynomials and the rational expectation integral by Smolyak Gaussian quadrature. The Smolyak operator is used to extend univariate approximation and integration operators to many dimensions. It reduces the curse of dimensionality from exponential to polynomial growth. The likelihood integrals are evaluated by a Gaussian quadrature and Gaussian quadrature particle filter. The bootstrap or sequential importance resampling particle filter is used as an accuracy benchmark. The posterior is estimated by the Gaussian filter and a Metropolis- Hastings algorithm. I propose a genetic extension of the standard Metropolis-Hastings algorithm by parallel random walk sequences. This improves the robustness of start values and the global maximization properties. Moreover it simplifies a cluster implementation and the random walk variances decision is reduced to only two parameters so that almost no trial sequences are needed. Finally the marginal likelihood is calculated as a criterion for nonnested and quasi-true models in order to select between the nonlinear estimates and a first order perturbation solution combined with the Kalman filter.stochastic dynamic general equilibrium model, Chebyshev polynomials, Smolyak operator, nonlinear state space filter, Curse of Dimensionality, posterior of structural parameters, marginal likelihood

Multivariate state space methods for official statistics and climate modelling

This thesis explores how state space models, which are a type of econometric models designed to analyse time series data, can be employed to achieve more accurate and realistic estimates of official statistics, and to model and forecast regional concentrations of air pollutants. Specifically, a novel approach is presented, which incorporates survey-based, claimant counts and Google Trends data in order to provide more timely and accurate estimates of Dutch unemployment, over only employing survey-based data. A new method is proposed to model the relationship between the latter and claimant counts data as time-varying, which allows us to promptly tackle changes in such relationship and therefore achieve more realistic real-time estimates of Dutch unemployment. Time-varying relationships can potentially be modelled with other, already existing, econometric techniques, than the one proposed in this thesis, and the reasons why they have not been considered further are here documented. Finally, a novel spatial type of state space model is employed in order to model regional concentrations of nitrogen dioxide (NO2) in the Netherlands. The (time-varying) effects on this air pollutant of meteorological conditions, traffic intensity and geographical location of the Dutch regions, are accounted for in the model. The latter is further used to forecast regional NO2 concentrations for different scenarios of traffic intensity, and can therefore be potentially employed for evaluation of pollution-reduction policies