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Mr. Francis Vitek
This paper considers the problem of jointly decomposing a set of time series variables into cyclical and trend components, subject to sets of stochastic linear restrictions among these cyclical and trend components. We derive a closed form solution to an ordinary problem featuring homogeneous penalty term difference orders and static restrictions, as well as to a generalized problem featuring heterogeneous penalty term difference orders and dynamic restrictions. We use our Generalized Multivariate Linear Filter to jointly estimate potential output, the natural rate of unemployment and the natural rate of interest, conditional on selected equilibrium conditions from a calibrated New Keynesian model.
Mr. Francis Vitek and Ulric Eriksson von Allmen

include Clark (1989) , Apel and Jansson (1999) , and Laubach and Williams (2003) . But full information maximum likelihood estimation of linear unobserved components models, facilitated by the recursive multivariate linear filter due to Kalman (1960) , can be computationally complex, yielding cyclical and trend component estimates that can be sensitive to model specification choices and initial conditions. This paper considers the problem of jointly decomposing a set of time series variables into cyclical and trend components, subject to sets of stochastic linear

Mr. Francis Vitek

Front Matter Page Monetary and Capital Markets Department Contents I. Introduction II. The Multivariate Filters A. The Ordinary Multivariate Linear Filter B. The Generalized Multivariate Linear Filter III. The New Keynesian Model A. Households B. Firms C. Policy D. Equilibrium IV. Estimation of Potential Output and Natural Rates A. Restrictions B. Results Figure 1. Estimated Trend Components from Multivariate versus Univariate Filters Figure 2. Estimated Cyclical Components from Multivariate versus Univariate

Mr. Francis Vitek

Appendix A. Proof of Proposition 1 The Ordinary Multivariate Linear Filter (OMLF) solves the following minimization problem, where y i , t = y ^ i , t + y ¯ i , t , while G ≤ N and H ≤ N : min ⁡ { { y ¯ i , t } i = 1 N } t = 1 T S ( { { y ¯ i , t } i = 1 N } t = 1 T ) = ∑ ​ i = 1 N

Michal Andrle and Mr. Benjamin L Hunt

be expressed as a (multivariate) linear filter representation, x t | T = Σ i = t 0 T w 1 , t y 1 , t − i + … + Σ i = t 0 T w n , t y n , t − i = Σ k = 1 n ξ k , ( 7 ) where a particular unobservable variable x t | T —the output gap in this case–is expressed as a weighted average of an

Michal Andrle
This paper discusses several popular methods to estimate the ‘output gap’. It provides a unified, natural concept for the analysis, and demonstrates how to decompose the output gap into contributions of observed data on output, inflation, unemployment, and other variables. A simple bar-chart of contributing factors, in the case of multi-variable methods, sharpens the intuition behind the estimates and ultimately shows ‘what is in your output gap.’ The paper demonstrates how to interpret effects of data revisions and new data releases for output gap estimates (news effects) and how to obtain more insight into real-time properties of estimators.