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lengths, that could not be captured with univariate linear filters. Exam­ ples of research in both directions can be found in Sims (1977), Lahiri and Moore (1991), Stock and Watson (1993), and Hamilton (1994) and (1989). Although the first approach is known to present serious limitations,the new and more sophisticated methods developed in the second approach (most notably, multivariate and nonlinear extensions) are at an early stage, and have proved still unreliable, displaying poor behavior when moving away from the sample period . Despite the fact that business cycle estimation is basic to the conduct of macroeconomic policy and to monitoring of the economy, many decades of attention have shown that formal modeling of economic cycles is a frustrating issue. As Baxter and King (1999) point out, we still face at present the same basic question "as did Burns and Mitchell fifty years ago: how should one isolate the cyclical component of an eco­ nomic time series? In particular, how should one separate business-cycle elements from slowly evolving secular trends, and rapidly varying seasonal or irregular components?" Be that as it may, it is a fact that measuring (in some way) the busi­ ness cycle is an actual pressing need of economists, in particular of those related to the functioning of policy-making agencies and institutions, and of applied macroeconomic research.

Inhaltsverzeichnis

Frontmatter

1. Introduction and Brief Summary

Abstract
This monograph addresses the problem of measuring economic cycles (also called business cycles) in macroeconomic time series. In the decade that followed the Great Depression, economists developed an interest in the possible existence of (more or less systematic) cycles in the economy; see, for example, Haberler (1944) or Shumpeter (1939). It became apparent that in order to identify economic cycles, one had to remove from the series seasonal fluctuations, associated with short-term behavior, and the long-term secular trend, associated mostly with technological progress. Burns and Mitchell (1946) provided perhaps the first main reference point for much of the posterior research. Statistical measurement of the cycle was broadly seen as capturing the variation of the series within a range of frequencies, after the series has been seasonally adjusted and detrended. (Burns and Mitchell suggested a range of frequencies associated with cycles with a period between, roughly, two and eight years.)
Regina Kaiser, Agustín Maravall

2. A Brief Review of Applied Time Series Analysis

Abstract
In the introduction it was mentioned that the present standard technique used in applied work to estimate business cycles consists of applying a moving average (MA) filter, most often the HP filter, to a seasonally adjusted (SA) series, most often adjusted with the X11 filter. It was also mentioned that the procedure presents several drawbacks which, as shown, can be seriously reduced by incorporating some time series analysis tools, such as ARIMA models and signal extraction techniques. Before proceeding further, it will prove convenient to review some concepts and tools of applied time series analysis.
Regina Kaiser, Agustín Maravall

3. ARIMA Models and Signal Extraction

Abstract
Back to the Wold representation (2.18) of a stationary process, z t = Ψ(B)a t , the representation is of no help from the point of view of fitting a model because, in general, the polynomial Ψ(B) will contain an infinite number of parameters. Therefore we use a rational approximation of the type
$$ \Psi \left( B \right) \doteq \frac{{\theta \left( B \right)}} {{\varphi \left( B \right)}}, $$
where θ(B) and φ(B) are finite polynomials in B of order q and p, respectively. Then we can write
$$ z_t = \frac{{\theta \left( B \right)}} {{\varphi \left( B \right)}}a_t , or $$
$$ \hat p_t $$
(3.1)
Regina Kaiser, Agustín Maravall

4. Detrending and the Hodrick-Prescott Filter

Abstract
A well-known family of ad hoc niters, designed to capture a band of frequencies in the low frequency range, is the Butterworth family of filters (see, e.g., Otnes and Enochson (1978)). The filters are typically expressed by means of the sine or tangent functions. For the sine-type case, the filter is represented, in the frequency domain, by its gain function which, for the two-sided filter, is given by
$$ G(\omega ) = \left[ {1 + \left( {\frac{{\sin \left( {{\omega \mathord{\left/ {\vphantom {\omega 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}} {{\sin \left( {{{\omega _0 } \mathord{\left/ {\vphantom {{\omega _0 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} \right)^{2d} } \right]^{ - 1} ; 0 \leqslant \omega \leqslant \pi . $$
(4.1)
Regina Kaiser, Agustín Maravall

5. Some Basic Limitations of the Hodrick-Prescott Filter

Abstract
In Section 4.1 we presented the HP filter as a symmetric two-sided filter. Given that the concurrent estimator is a projection on a subset of the set of information that provides the final estimator, the latter cannot be less efficient. Besides, concurrent estimators, obtained with a one-sided filter, induce phase effects that distort the timing of events and harm early detection of turning points.
Regina Kaiser, Agustín Maravall

6. Improving the Hodrick-Prescott Filter

Abstract
In Chapter 5 we saw that the filter implies large revisions for recent periods (roughly, for the last two years). The imprecision in the cycle estimator for the last quarters implies, in turn, a poor performance in early detection of turning points. Furthermore, as was just mentioned, direct inspection of Figure 5.3 shows another limitation of the HP filter: the cyclical signal it provides seems rather uninformative. Seasonal variation has been removed, but a large amount of noise remains in the signal, making its reading and the dating of turning points difficult. In the next two sections, we proceed to show how these two shortcomings can be reduced with some relatively simple modifications.
Regina Kaiser, Agustín Maravall

7. Hodrick-Prescott Filtering Within a Model-Based Approach

Abstract
What we have suggested in the previous section is to estimate the cycle in steps. First, the AMB method is used to obtain the trend-cycle estimator \( \hat p_t \) (i.e., the noise-free SA series). In a second step, the HP filter is applied to \( \hat p_t \).
Regina Kaiser, Agustín Maravall

Backmatter

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