Regina Kaiser

Measuring business cycles in economic time series

1 Introduction and Brief Summary.- 2 A Brief Review of Applied Time Series Analysis.- 2.1 Some Basic Concepts.- 2.2 Stochastic Processes and Stationarity.- 2.3 Differencing.- 2.4 Linear Stationary Process, Wold Representation. and Auto-correlation Function.- 2.5 The Spectrum.- 2.6 Linear Filters and Their Squared Gain.- 3 ARIMA Models and Signal Extraction.- 3.1 ARIMA Models.- 3.2 Modeling Strategy, Diagnostics and Inference.- 3.2.1 Identification.- 3.2.2 Estimation and Diagnostics.- 3.2.3 Inference.- 3.2.4 A Particular Class of Models.- 3.3 Preadjustment.- 3.4 Unobserved Components and Signal Extraction.- 3.5 ARIMA-Model-Based Decomposition of a Time Series.- 3.6 Short-Term and Long-Term Trends.- 4 Detrending and the Hodrick-Prescott Filter.- 4.1 The Hodrick-Prescott Filter: Equivalent Representations.- 4.2 Basic Characteristics of the Hodrick-Prescott Filter.- 4.3 Some Criticisms and Discussion of the Hodrick-Prescott Filter.- 4.4 The Hodrick-Prescott Filter as a Wiener-Kolmogorov Filter.- 4.4.1 An Alternative Representation.- 4.4.2 Derivation of the Filter.- 4.4.3 The Algorithm.- 4.4.4 A Note on Computation.- 5 Some Basic Limitations of the Hodrick-Prescott Filter.- 5.1 Endpoint Estimation and Revisions.- 5.1.1 Preliminary Estimation and Revisions.- 5.1.2 An Example.- 5.2 Spurious Results.- 5.2.1 Spurious Crosscorrelation.- 5.2.2 Spurious Autocorrelation Calibration.- 5.2.3 Spurious Periodic Cycle.- 5.3 Noisy Cyclical Signal.- 6 Improving the Hodrick-Prescott Filter.- 6.1 Reducing Revisions.- 6.2 Improving the Cyclical Signal.- 7 Hodrick-Prescott Filtering Within a Model-Based Approach.- 7.1 A Simple Model-Based Algorithm.- 7.2 A Complete Model-Based Method Spuriousness Reconsidered.- 7.3 Some Comments on Model-Based Diagnostics and Inference.- 7.4 MMSE Estimation of the Cycle: A Paradox.- References.- Author Index.

Time Series Segmentation Procedures to Detect, Locate and Estimate Change-Points

This article deals with the problem of detecting, locating, and estimating the change-points in a time series process. We are interested in finding changes in the mean and the autoregressive coefficients in piecewise autoregressive processes, as well as changes in the variance of the innovations. With this objective, we propose an approach based on the Bayesian information criterion (BIC) and binary segmentation. The proposed procedure is compared with several others available in the literature which are based on cusum methods (Inclan and Tiao, J Am Stat Assoc 89(427):913–923, 1994), minimum description length principle (Davis et al., J Am Stat Assoc 101(473):229–239, 2006), and the time varying spectrum (Ombao et al., Ann Inst Stat Math 54(1):171–200, 2002). We computed the empirical size and power properties of the available procedures in several Monte Carlo scenarios and also compared their performance in a speech recognition dataset.

Some Basic Limitations of the Hodrick-Prescott Filter

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.

Hodrick-Prescott Filtering Within a Model-Based Approach

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 \( \hat p_t \).

Introduction and Brief Summary

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.)

A Brief Review of Applied Time Series Analysis

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.

Improving the Hodrick-Prescott Filter

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.

ARIMA Models and Signal Extraction

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

Estimation of the business cycle: A modified Hodrick-Prescott filter

\Psi \left( B \right) \doteq \frac{{\theta \left( B \right)}} {{\varphi \left( B \right)}},