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

Abstract

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) January 2010 AbstractIn the context of linear state space models with Gaussian errors and known parameters, the Kalman filter generates best linear unbiased predictions of the underlying components together with their corresponding prediction mean squared errors (PMSE). However, in practice, the filter is run by substituting some parameters of the model by consistent estimates. In these circumstances, the PMSEs given by the Kalman filter do not take into account the parameter uncertainty and, consequently, they underestimate the true PMSEs. In this paper, we propose two new bootstrap procedures to obtain PMSE of the unobserved states based on obtaining replicates of the underlying states conditional on the information available at each moment of time. By conditioning on the available information, we simplify the procedure with respect to alternative bootstrap proposals previously available in the literature. Furthermore, we show that the new procedures proposed in this paper have better finite sample properties than the alternatives. We implement the proposed procedures 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 how taking into account the parameter uncertainty may change the prediction intervals constructed for those unobservable macroeconomic variables and, in particular, change the conclusions about the utility of the NAIRU as a macroeconomic indicator for expansions and recessions.

Review of Kalman Filter Employment in the NAIRU Estimation

“…This model of the NAIRU evolution is generally considered appropriate for the United States [18,20,31]. On the other hand, some authors [15,20] prefer a random walk with drift which is more suitable for European countries.…”

“…Another possibility is to start from the classic wage Phillips curve, although the price Phillips curve is preferred in practice over the wage curve. Some authors employ different models based on the forward-looking Phillips curve [20,21,31] or the New Keynesian Phillips curve [13,15,24,30], which are complemented by equations describing the relationship with unemployment, output or other variables. The Kalman filter can be appropriately used in such models to allow addressing the problem of unobservability of expected future inflation.…”

“…Such a model consists of the equation of GDP decomposition into trend and cycle, the forward-looking Phillips curve equation, the Okun's law equation, an investment equation, and other corresponding state equations describing evolution of the estimated variables over time. Other authors employ and develop this model even further [30,31]. Table 2 provides a summary of models used in the analyzed studies.…”

“…The model is transformed into a state-space form as mentioned, e.g., by [15,20,30,31] (or generally described by [37,38]), and then the Kalman filter is applied to it. The NAIRU and other unobservable variables are elements of the estimated state vector.…”

Bias-correction of Kalman filter estimators associated to a linear state space model with estimated parameters

“…Also, a bootstrap approach was adopted in the estimation of the mean squared prediction error of the best linear estimator of nonlinear functions of finitely many future observations in a stationary time series in Bandyopadhyay and Lahiri (2010). Rodríguez and Ruiz (2012) proposed two new bootstrap procedures to obtain MSE of the unobserved states which have better finite sample properties than both bootstraps alternatives and procedures based on the asymptotic approximation of the parameter distribution.…”