Model Uncertainty, State Uncertainty, and State-Space Models

Abstract

State-space models have been increasingly used to study macroeconomic and financial problems. A state-space representation consists of two equations, a measurement equation which links the observed variables to unobserved state variables and a transition equation describing the dynamics of the state variables. In this paper, we show that a classic linear-quadratic macroeconomic framework which incorporates two new assumptions can be analytically solved and explicitly mapped to a state-space representation. The two assumptions we consider are the model uncertainty due to concerns for model misspecification (robustness) and the state uncertainty due to limited information constraints (rational inattention). We show that the state-space representation of the observable and unobservable can be used to quantify the key parameters on the degree of model uncertainty. We provide examples on how this framework can be used to study a range of interesting questions in macroeconomics and international economics.

Robust RLS Wiener Signal Estimators for Discrete-Time Stochastic Systems with Uncertain Parameters

“…Hence,z(k) has the different value from the nominal signal z(k) of the state-space model (2), not including the uncertain quantities. In (1), as the sum of the degraded signalz(k) and the observation noise v(k),y(k) is measured. The state-space model without including the uncertain quantities ∆H(k) and ∆Φ(k) in (1) is expressed as…”

“…In (2), z(k) represents the signal to be estimated. H is an m by n observation matrix, x(k) is the state vector and v(k) is the white observation noise with the auto-covariance function in (1). Also, the auto-covariance function of the input noise w(k) is given in (1).…”

“…H is an m by n observation matrix, x(k) is the state vector and v(k) is the white observation noise with the auto-covariance function in (1). Also, the auto-covariance function of the input noise w(k) is given in (1). It is assumed that the signal and the observation noise are mutually independent and have zero means.…”

“…Under the linear least-squares estimation problem of the signal z(k) in section 2, Theorem 1 presents the robust RLS Wiener fixed-point smoothing and filtering algorithms. Theorem 1 Let the state equation and the observation equation, including the uncertain quantities ∆Φ and ∆H respectively, be given by (1). Let Φ and H represent the system and observation matrices respectively for the signal process of z(k).…”

“…The state-space model in macroeconomic and financial problems includes parameter uncertainties [1]. An improved robust Kalman filter is applied to precise point positioning/inertial navigation system (PPP/INS) [2].…”