Previsibilidade da taxa de câmbio: modelo de Markov-Switching multi-estado e tendência com suavidade controlada

The Hodrick-Prescott (HP) filter is a commonly used tool in macroeconomics used to extract a trend component from a time series. In this paper, we derive a new representation of the transformation of the data that is implied by the HP filter. This representation highlights that the HP filter is a symmetric weighted average plus a number of adjustments that are important near the beginning and end of the sample. The representation allows us to carry out a rigorous analysis of properties of the HP filter without using the ARMA-based approximation that has been used previously in the literature. Using this new representation, we characterize the large T behavior of the HP filter and find conditions under which it is asymptotically equivalent to a symmetric weighted average with weights independent of sample size. We also find that the cyclical component of the HP filter possesses weak dependence properties when the HP filter is applied to a stationary mixing process, a linear deterministic trend process, or a process with a unit root. This provides the first formal justification of the use of the HP filter as a tool to achieve weak dependence in a time series. In addition, a large smoothing parameter approximation to the HP filter is derived, and using this approximation, we find an alternative justification for the procedure given in Ravn and Uhlig (2002) for adjusting the smoothing parameter for the data frequency.

The Hodrick-Prescott (HP) filter is one of the most widely used econometric methods in applied macroeconomic research. The technique is nonparametric and seeks to decompose a time series into a trend and a cyclical component unaided by economic theory or prior trend specification. Like all nonparametric methods, the HP filter depends critically on a tuning parameter that controls the degree of smoothing. Yet in contrast to modern nonparametric methods and applied work with these procedures, empirical practice with the HP filter almost universally relies on standard settings for the tuning parameter that have been suggested largely by experimentation with macroeconomic data and heuristic reasoning about the form of economic cycles and trends. As recent research (Phillips and Jin, 2015) has shown, standard settings may not be adequate in removing trends, particularly stochastic trends, in economic data. This paper proposes an easy-to-implement practical procedure of iterating the HP smoother that is intended to make the filter a smarter smoothing device for trend estimation and trend elimination. We call this iterated HP technique the boosted HP filter in view of its connection to L_2-boosting in machine learning. The paper develops limit theory to show that the boosted HP (bHP) filter asymptotically recovers trend mechanisms that involve unit root processes, deterministic polynomial drifts, and polynomial drifts with structural breaks, thereby covering the most common trends that appear in macroeconomic data and current modeling methodology. In doing so, the boosted filter provides a new mechanism for consistently estimating multiple structural breaks even without knowledge of the number of such breaks. A stopping criterion is used to automate the iterative HP algorithm, making it a data-determined method that is ready for modern data-rich environments in economic research. The methodology is illustrated using three real data examples that highlight the differences between simple HP filtering, the data-determined boosted filter, and an alternative autoregressive approach. These examples show that the bHP filter is helpful in analyzing a large collection of heterogeneous macroeconomic time series that manifest various degrees of persistence, trend behavior, and volatility.

This article extends recent asymptotic theory developed for the Hodrick Prescott (HP) filter and boosted HP (bHP) filter to long-range dependent time series that have fractional Brownian motion (fBM) limit processes after suitable standardization. Under general conditions, it is shown that the asymptotic form of the HP filter is a smooth curve, analogous to the finding in Phillips and Jin for integrated time series and series with deterministic drifts. Boosting the filter using the iterative procedure suggested in Phillips and Shi leads under well-defined rate conditions to a consistent estimate of the fBM limit process or the fBM limit process with an accompanying deterministic drift when that is present. A stopping criterion is used to automate the boosting algorithm, giving a data-determined method for practical implementation. The theory is illustrated in simulations and two real data examples that highlight the differences between simple HP filtering and the use of boosting. The analysis is assisted by employing a uniformly and almost surely convergent trigonometric series representation of fBM.

The Hodrick-Prescott (HP) filter is one of the most widely used econometric methods in applied macroeconomic research. The technique is nonparametric and seeks to decompose a time series into a trend and a cyclical component unaided by economic theory or prior trend specification. Like all nonparametric methods, the HP filter depends critically on a tuning parameter that controls the degree of smoothing. Yet in contrast to modern nonparametric methods and applied work with these procedures, empirical practice with the HP filter almost universally relies on standard settings for the tuning parameter that have been suggested largely by experimentation with macroeconomic data and heuristic reasoning about the form of economic cycles and trends. As recent research has shown, standard settings may not be adequate in removing trends, particularly stochastic trends, in economic data. This paper proposes an easy-to-implement practical procedure of iterating the HP smoother that is intended to make the filter a smarter smoothing device for trend estimation and trend elimination. We call this iterated HP technique the boosted HP filter in view of its connection to L_2-boosting in machine learning. The paper develops limit theory to show that the boosted HP filter asymptotically recovers trend mechanisms that involve unit root processes, deterministic polynomial drifts, and polynomial drifts with structural breaks – the most common trends that appear in macroeconomic data and current modeling methodology. In doing so, the boosted filter provides a new mechanism for consistently estimating multiple structural breaks. A stopping criterion is used to automate the iterative HP algorithm, making it a data-determined method that is ready for modern data-rich environments in economic research. The methodology is illustrated using three real data examples that highlight the differences between simple HP filtering, the data-determined boosted filter, and an alternative autoregressive approach. These examples show that the boosted HP filter is helpful in analyzing a large collection of heterogeneous macroeconomic time series that manifest various degrees of persistence, trend behavior, and volatility.

An important macro-economic issue of developed and developing countries is how best to decompose an economic time series into permanent (trend) and transitory (cycle) components. The issue is vital in empirical macroeconomics since, among other things, it relates to how one can estimate the output gap-the deviation of an economy's output from its potential or trend output. This paper considers how well the unobserved components model and the Hodrick Prescott (HP) filter decomposes real gross domestic product (GDP) in a small island developing state, the state of Barbados. The correlated unobserved components model for Barbados studied in the Agbeyegbe (2020) is modified to allow a second-order Markov trend. The effect of this modification is to make it possible to recover the HP trend as a particular case. The paper then compares several methods useful for trend decomposition of real GDP in Barbados. The competing methods are variants of two widely used trend-cycle decompositions of GDP that give markedly different estimates. Namely, methods based on the unobserved components model (UC) and the HP filter. The correlated unobserved components model produces smaller output gaps in amplitude, whereas the HP filter generates significant and persistent cycles. More specifically, the methods are: (i) the HP filter; (ii) an augmented HP filter (HP-AR), that allows for cyclical components to be serially correlated, introduced by Grant and Chan (2017b); (iii) the correlated unobserved components model (UCUR), without a break; (iv) the correlated unobserved components model (UCUR-t), with a break at time t; and (v) a correlated unobserved components model that allows for a second-order Markov trend process UCUR-2M. The result shows that for Barbados, with data covering the period 1967-2017, the correlated unobserved components model that allows for a break in trend fits the data better than the HP specification. These results are significant from a policy perspective. Knowing the correct duration of the business cycle is essential to providing appropriate recommendations; the result argues against the use of HP-filter in analyzing Barbados' business cycle. The result also finds that for Barbados, it is empirically important to correlate permanent and transitory shocks. By ignoring this correlation, researchers risk providing a misleading analysis of how the Barbadian economy works.

This paper analyzes the transmission characteristics of the Hodrick-Prescott (HP) filter and frequency characteristics of exponential series, defines 3dB bandwidths of both the HP filter and exponential time series. Formulas of the 3dB bandwidths of the HP filter and exponential time series are derived. Thus, a method to determine the smoothing parameters of the HP filter by the series length is proposed. The examples showed the proposed method can effectively control the magnitude of fluctuation components, provides a way for the HP filter to deal with different macroeconomic time series by selecting a suitable smoothing parameter.

According to the Basel III Framework, the gap between Credit to GDP Ratio and its long-run trend is the single best indicator for setting the Countercyclical Capital Buffer (CCB). The aim of setting the CCB is to increase the Capital Adequacy Requirement (CAR), in order to increase the resilience of the banking system in the case of upcoming financial difficulties. For calculating the long run trend of the Credit to GDP Ratio, the Basel Committee suggests to use the Hodrick-Prescott (HP) filter. In order to detrend the Credit to GDP Ratio, the HP filter only relies on the Credit to GDP Ratio itself and does not take into account other variables, which may be relevant to the risks to financial stability. Economic theory immediately suggests the Real Gross Domestic Product and the Real Estate Price Index as these relevant variables. During periods of negative Real Gross Domestic Product and Real Estate Price Index gaps, a high Credit to GDP Gap is less dangerous than is indicated by the HP filter. The reverse is true when gaps of these two variables are positive, that is a high Credit to GDP Gap is more dangerous for the financial system and the economy than is indicated by the HP filter. The present paper provides a theoretical and empirical justification of using Real GDP and Real Estate Price Index gaps in the process of detrending Credit to GDP ratio. Since the HP filter cannot work with different variables simultaneously, the paper introduces the Kalman filter as a solution. Comparing credit to GDP gaps calculated using different filters, the paper shows two cases when the Kalman filter outperformed the HP filter in Georgia between the years 2000 and 2016. The first case is the financial crisis of 2007-2008, during which the HP filter could only signal that a crisis was occurring after the fact, while the Kalman filter could work as an early warning indicator, informing about an upcoming crisis in the beginning of 2006. The second case is the first half of 2016, when the HP filter suggested to set the CCB while there was no financial crisis, which was correctly indicated by the Kalman filter.

In various fields of macroeconomic modelling, researchers often face the problem of decomposing time series into trend component and cycle fluctuations. While there are several potentially useful methods to perform the task in question, Hodrick-Prescott (HP) fi lter seems to have remained (despite some serious criticism) the most popular approach over the past decade. In this article I propose a straightforward and easy-to-implement bootstrap procedure for building pointwise and simultaneous confidence intervals around point estimates produced by HP filter. The principle of proposed method can be described as follows: first, we use maximum entropy bootstrap (Vinod, 2004, 2006) to approximate ensemble from which original time series is drawn and then apply the HP filter directly to each bootstrap replication. If necessary, the proposed method can be adapted to allow for uncertainty in the smoothing parameter. Practical usefulness of our approach is demonstrated with an application to the GDP data. Results are encouraging - obtained confi dence intervals for the trend and cyclical component are overall plausible thus supplying a researcher with some measure of uncertainty related to HP filtering. Finally, we demonstrate that a former approach to build confidence intervals for HP filter (Gallego and Johnson, 2005) leads to erratic inference for cycle due to the shape-destroying block bootstrap sampling.

In recent decades, in the research community of macroeconometric time series analysis, we have observed growing interest in the smoothing method known as the Hodrick–Prescott (HP) filter. This article examines the properties of an alternative smoothing method that looks like the HP filter, but is much less well known. We show that this is actually more like the exponential smoothing filter than the HP filter although it is obtainable through a slight modification of the HP filter. In addition, we also show that it is also like the low-frequency projection of Müller and Watson (2018, Econometrica 86, 775–804). We point out that these results derive from the fact that all three similar smoothing methods can be regarded as a type of graph spectral filter whose graph Fourier transform is discrete cosine transform. We then theoretically reveal the relationship between the similar smoothing methods and provide a way of specifying the smoothing parameter that is necessary for its application. An empirical examination illustrates the results.

Maravall and del Rio (2001), analized the time aggregation properties of the Hodrick-Prescott (HP) filter, which decomposes a time series into trend and cycle, for the case of annual, quarterly, and monthly data, and showed that aggregation of the disaggregate component cannot be obtained as the exact result from direct application of an HP filter to the aggregate series. The present paper shows how, using several criteria, one can find HP decompositions for different levels of aggregation that provide similar results. We use as the main criterion for aggregation the preservation of the period associated with the frequency for which the filter gain is ½; this criterion is intuitive and easy to apply. It is shown that the Ravn and Uhlig (2002) empirical rule turns out to be a first-order approximation to our criterion, and that alternative — more complex — criteria yield similar results. Moreover, the values of the parameter \lambda of the HP filter, that provide results that are approximately consistent under aggregation, are considerably robust with respect to the ARIMA model of the series. Aggregation is seen to work better for the case of temporal aggregation than for systematic sampling. Still a word of caution is made concerning the desirability of exact aggregation consistency. The paper concludes with a clarification having to do with the questionable spuriousness of the cycles obtained with HP filter.

The use of the Hodrick-Prescott (HP) filter is presented as an alternative to the traditional digital filtering and spline smoothing methods currently used in biomechanics. In econometrics, HP filtering is a standard tool used to decompose a macroeconomic time series into a nonstationary trend component and a stationary residual component. The use of the HP filter in the present work is based on reasonable assumptions about the jerk and noise components of the raw displacement signal. Its applicability was tested on 4 kinematic signals with different characteristics. Two are well known signals taken from the literature on biomechanical signal filtering, and the other two were acquired with our own motion capture system. The criterion for the selection of cutoff frequency was based on the power spectral density of the raw displacement signals. The results showed the technique to be well suited to filtering biomechanical displacement signals in order to obtain accurate higher derivatives in a simple and systematic way. Namely, the HP filter and the generalized cross-validated quintic spline (GCVSPL) produce similar RMS errors on the first (0.1063 vs. 0.1024 m/s2) and second (23.76 vs. 23.24 rad/s2) signals. The HP filter performs slightly better than GCVSPL on the third (0.209 vs. 0.236 m/s2) and fourth (1.596 vs. 2.315 m/s2) signals.

The Whittaker–Henderson (WH) graduation is a smoothing method for equally spaced one-dimensional data such as time series. It includes the Bohlmann filter, the Hodrick–Prescott (HP) filter, and the Whittaker graduation as special cases. Among them, the HP filter is the most prominent trend-cycle decomposition method for macroeconomic time series such as real gross domestic product. Recently, a modification of the HP filter, the boosted HP (bHP) filter, has been developed, and several studies have been conducted. The basic idea of the modification is to achieve more desirable smoothing by extracting long-term fluctuations remaining in the smoothing residuals. Inspired by the modification, this paper develops the boosted version of the WH graduation, which includes the bHP filter as a special case. Then, we establish its properties that are fundamental for applied work. To investigate the properties, we use a spectral decomposition of the penalty matrix of the WH graduation

In this paper, we estimate the time-varying COVID-19 contact rate of a Susceptible-Infected-Recovered (SIR) model. Our measurement of the contact rate is constructed using data on actively infected, recovered and deceased cases. We propose a new trend filtering method that is a variant of the Hodrick-Prescott (HP) filter, constrained by the number of possible kinks. We term it the sparse HP filter and apply it to daily data from five countries: Canada, China, South Korea, the UK and the US. Our new method yields the kinks that are well aligned with actual events in each country. We find that the sparse HP filter provides a fewer kinks than the l1 trend filter, while both methods fitting data equally well. Theo-retically, we establish risk consistency of both the sparse HP and l1 trend filters. Ultimately, we propose to use time-varying contact growth rates to document and monitor outbreaks of COVID-19.

Sparse HP filter: Finding kinks in the COVID-19 contact rate

Filtering economic time series: On the cyclical properties of Hamilton's regression filter and the Hodrick-Prescott filter