DATA-DRIVEN NONPARAMETRIC SPECTRAL DENSITY ESTIMATORS FOR ECONOMIC TIME SERIES: A MONTE CARLO STUDY

ABSTRACT

Spectral analysis at frequencies other than zero plays an increasingly important role in econometrics. A number of alternative automated data-driven procedures for nonparametric spectral density estimation have been suggested in the literature, but little is known about their finite-sample accuracy. We compare five such procedures in terms of their mean-squared percentage error across frequencies. Our data generating processes (DGP) include autoregressive-moving average (ARMA) models, fractionally integrated ARMA models and nonparametric models based on 16 commonly used macroeconomic time series. We find that for both quarterly and monthly data the autoregressive sieve estimator is the most reliable method overall.

Keywords:

• Business cycle measurement
• Model identification
• Periodogram smoothing
• Autocovariance smoothing
• Autoregressive sieve
• Bandwidth selection

ACKNOWLEDGMENTS

We thank Jeremy Berkowitz, Frank Diebold, Phil Howrey, Miles Kimball, Jan Kmenta, Andrew Levin, Essie Maasoumi, Shinichi Sakata, Jonathan Wright, two anonymous refereees, and the associate editor for helpful comments. The opinions expressed in this paper are those of the authors and do not necessarily reflect views of the European Central Bank.

Notes

Frequency domain analysis has a long tradition in econometrics, dating back to the 1960s. For a survey of early applications of spectral analysis in econometrics see Granger and Engle.[15]

A survey of these techniques can be found in Den Haan and Levin.25-26

Previous studies of spectral density estimators tended to focus on the integrated mean-squared error of the spectral density estimator (see e.g. 21-24. Beamish and Priestley[30] also consider other criteria.

If the frequency of a cycle is T, the period of the cycle is 2π /T time units. For example, for T = π, the corresponding cycle will be of length 2 quarters; for T = π /2, it will be of length 4 quarters, etc.

The spectral estimates in Fig. 1 have been computed using the autoregressive sieve method, which performs best among the methods analyzed in this paper.

Politis and Romano[35] suggest replacing the Bartlett window by a window with trapezoidal shape. They show that use of the latter window in the case of ARMA-DGPs results in a faster rate of convergence of the spectral estimator. Preliminary simulation evidence, however, showed that the estimator recommended by Politis and Romano did not have systematically lower mean-squared percentage errors for our DGPs than the estimator based on the Bartlett window.

This condition is somewhat stronger than that required for the consistency of estimators of slope parameters in stationary autoregressive processes: p(T) → ∞ as T → ∞ subject to [pbar] _T  = o(T ^1/2) (see 44-45.

All data are from the DRI Economics Database. The raw data codes are PRNEW, IP, FYGM3, FM1, EXRJAN, and LPMHU.

All data are from the DRI Economics Database. For a precise definition of the private output variable and the corresponding deflator see King and Watson.[1] The other variables are listed in footnote 7.

In the case of the real interest rate the AIC selected a pure AR representation. Instead, we selected the secondary minimum of the AIC, corresponding to an ARMA(1, 2) process.