2018年-IMF国际货币组织全球_A_Closed_Form_Multivariate_Linear_Filter_24页_790kb

Summary of "A Closed Form Multivariate Linear Filter" by Francis Vitek

Core Content

This paper introduces a new class of closed-form multivariate linear filters to jointly decompose time series variables into cyclical and trend components, incorporating stochastic linear restrictions. The filters are designed to address limitations of traditional univariate methods, such as the Hodrick-Prescott (HP) filter and Lucas filter, which are criticized for being atheoretic and not capturing dynamic interrelationships among variables.

Main Points

• Objective: The paper aims to provide a closed-form solution for decomposing multiple time series variables into cyclical and trend components, while considering both static and dynamic stochastic linear restrictions.
• Two Filters Introduced:
  1. Ordinary Multivariate Linear Filter (OMLF): Features homogeneous penalty term difference orders and static restrictions.
  2. Generalized Multivariate Linear Filter (GMLF): Features heterogeneous penalty term difference orders and dynamic restrictions.
• Key Features:
  • The filters operate over the entire sample in one step, avoiding sensitivity to initial conditions.
  • They inherit properties from both univariate closed-form filters and linear unobserved components models.
  • The OMLF nests the HP and Lucas filters under specific parameter settings.
  • The GMLF allows for more flexible modeling of dynamic relationships between cyclical and trend components.
• Application: The GMLF is applied to estimate potential output, the natural rate of unemployment, and the natural rate of interest for the United States, conditional on equilibrium conditions derived from a calibrated New Keynesian model.
• Statistical Properties:
  • The filters balance the minimization of cyclical and trend component variances.
  • The penalty parameters (λ, γ, ψ) control the smoothness of the trend components and the adherence to restrictions.
• Economic Significance:
  • The estimates show economically meaningful deviations from univariate filters.
  • Sensitivity analysis reveals the sources of these deviations.

Key Information

I. Introduction

• Unobserved variables (like potential output, natural unemployment rate, and natural interest rate) are essential in monetary economics.
• Traditional univariate filters are criticized for not capturing dynamic relationships and for being atheoretic.
• The paper proposes multivariate filters that address these issues by incorporating both static and dynamic restrictions.

II. Multivariate Filters

A. Ordinary Multivariate Linear Filter (OMLF)

• Minimizes the objective function that includes:
  • Squared cyclical components.
  • Squared differences of trend components of order $d$.
  • Quadratic penalties for deviations of static combinations of cyclical and trend components from zero.
• The unique global minimum is derived using matrix notation and satisfies the equation:
  $$
  \operatorname{Vec}(\overline{Y}{|T}) = \left[ I{NT} + \left(I_N \otimes (\lambda^2 (\Delta^d)^T \Delta^d)\right) + \left(\left(\gamma^2 \Phi \Phi^T + \psi^2 \Theta \Theta^T\right) \otimes I_T\right]^{-1} \left[ I_{NT} + \left(\gamma^2 \Phi \Phi^T \otimes I_T\right)\right] \operatorname{Vec}(Y)
  $$
• It is a generalization of the HP and Lucas filters.

B. Generalized Multivariate Linear Filter (GMLF)

• Extends the OMLF to allow for heterogeneous penalty term difference orders and dynamic restrictions.
• The objective function includes:
  • Squared cyclical components.
  • Squared differences of trend components of variable orders $d_i$.
  • Dynamic combinations of cyclical and trend components with lag and lead orders $P_1, P_2$ and $Q_1, Q_2$.
• The solution is given by:
  $$
  \operatorname{Vec}(\overline{Y}T) = \left[ I{NT} + \sum_{d=1}^D \left(\Lambda_d \Lambda_d^T \otimes (\Delta^d)^T \Delta^d\right) + \sum_{p=-P_1}^{P_2} \left((\Phi_p \Gamma)(\Phi_p \Gamma)^T \otimes (L_P^p)^T L_P^p\right) + \sum_{q=-Q_1}^{Q_2} \left((\Theta_q \Psi)(\Theta_q \Psi)^T \otimes (L_Q^q)^T L_Q^q\right)\right]^{-1} \left[ I_{NT} + \sum_{p=-P_1}^{P_2} \left((\Phi_p \Gamma)(\Phi_p \Gamma)^T \otimes (L_P^p)^T L_P^p\right)\right] \operatorname{Vec}(Y)
  $$

III. New Keynesian Model

• The model is an extension of the closed economy New Keynesian model, incorporating rational expectations and nominal price rigidity.
• Households:
  • Maximize intertemporal utility subject to dynamic budget constraints.
  • The utility function includes external habit formation and labor supply decisions.
  • The model yields necessary first-order conditions for consumption, labor, and bond/stock holdings.
• Firms:
  • Intermediate firms produce differentiated goods and sell shares.
  • The model includes a partial indexation rule for prices, where a fraction $1 - \omega$ of firms adjust prices optimally, and the rest follow a rule based on past inflation.
  • The final output good is produced from intermediate goods, with constant returns to scale.

IV. Estimation of Potential Output and Natural Rates

• The GMLF is used to estimate potential output, natural rate of unemployment, and natural rate of interest, conditional on equilibrium conditions from a calibrated New Keynesian model.
• The estimates differ from univariate filters, and the sensitivity analysis reveals the impact of parameter perturbations on the results.
• Tables show the sensitivity of estimation results to variations in smoothing, weight, and structural parameters.

V. Conclusion

• The paper presents a closed-form multivariate linear filter that addresses the limitations of univariate filters by incorporating dynamic and static restrictions.
• The filters are computationally efficient and do not rely on initial conditions.
• The application to the U.S. economy shows economically significant results, supporting the use of these filters in monetary policy analysis.