Real Time Estimation in Local Polynomial Regression, with Application to Trend-Cycle Analysis

Abstract

The paper focuses on the adaptation of local polynomial filters at the end of the sample period. We show that for real time estimation of signals (i.e., exactly at the boundary of the time support) we cannot rely on the automatic adaptation of the local polynomial smoothers, since the direct real time filter turns out to be strongly localized, and thereby yields extremely volatile estimates. As an alternative, we evaluate a general family of asymmetric filters that minimizes the mean square revision error subject to polynomial reproduction constraints; in the case of the Henderson filter it nests the well-known Musgrave's surrogate filters. The class of filters depends on unknown features of the series such as the slope and the curvature of the underlying signal, which can be estimated from the data. Several empirical examples illustrate the effectiveness of our proposal.

On the Spectral Properties of Matrices Associated with Trend Filters

“…the weight attached to the observation taken at the same time we are estimating the trend, as long as the span of the filter decreases. The leverage further tends to increase (up to unity) with high degrees of the fitting polynomial (for a formal proof, see Proietti and Luati, 2007). Here, we verify this phenomenon by choosing as smoothing matrix the circulant matrix with first and last rows replaced by any real time filter of the class introduced above.…”

“…that is obtained by partitioning the two-sided symmetric filter in two groups, w = [w ′ p , w ′ f ] ′ , where w p contains the weights attributed to the past and current observations and w f those attached to the future unavailable observations. The proof of (6) can be found in Proietti and Luati (2007), where detailed proofs of other results that will be used in this section, such as (8), are also available. Equation (6) represents the fundamental relationship which states how the asymmetric LPR filter weights are obtained from the symmetric ones.…”

“…As for the asymmetric filters, the reflecting have been chosen except for the case of the real time filter. In particular, the QL (Proietti and Luati, 2007) and Musgrave (1964) real time filters discussed in section 4.2 have been applied and compared. The smoothing matrix S is therefore equal to H except for the last row changed.…”

Growth Trends and Systematic Patterns of Booms and Busts‐Testing 200 Years of Business Cycle Dynamics

“…The local polynomial regression automatically corrects at the boundary if p − v is odd, as can be seen in Proietti and Luati (2008). Thus, to enhance the quality of the estimation at any boundary point, the total bandwidth used is kept the same as in the interior and a specific, asymmetric boundary kernel is applied to those points.…”

Data-driven local polynomial for the trend and its derivatives in economic time series

“…Local polynomial regression is also an automatic kernel carpentry and most well-known results on kernel regression can hence be adapted to this approach. Moreover, it is also widely applied to models with short-range (Opsomer, 1997;Opsomer et al, 2001;Francisco-Fernandez and Vilar-Fernandez, 2001;Proietti and Luati, 2008) and long-range (see e.g. Beran and Feng, 2002a) dependent errors.…”