Multivariate trend-cycle extraction with the Hodrick-Prescott filter

TL;DR

This paper extends the Hodrick-Prescott filter to multivariate time series using a seemingly unrelated approach, enabling efficient estimation of trend and cycle components in macroeconomic data.

Contribution

It introduces a multivariate generalization of the Hodrick-Prescott filter based on the seemingly unrelated time series approach and employs the META estimation method for simplicity and efficiency.

Findings

The proposed method accurately extracts trend-cycle components in multivariate macroeconomic data.

META estimation simplifies the parameter estimation process, making it faster and more stable.

Empirical application demonstrates the method's effectiveness on European industrial production data.

Abstract

The Hodrick-Prescott filter represents one of the most popular method for trend-cycle extraction in macroeconomic time series. In this paper we provide a multivariate generalization of the Hodrick-Prescott filter, based on the seemingly unrelated time series approach. We first derive closed-form expressions linking the signal-noise matrix ratio to the parameters of the VARMA representation of the model. We then show that the parameters can be estimated using a recently introduced method, called "Moment Estimation Through Aggregation (META)". This method replaces the traditional multivariate likelihood estimation with a procedure that requires estimating univariate processes only. This makes the estimation simpler, faster and better-behaved numerically. We prove that our estimation method is consistent and asymptotically normal distributed for the proposed framework. Finally, we present an empirical application focusing on the industrial production of several European countries.