Manufacturing Earnings and Cycles: New Evidence

A Appendix

A.l Estimation of the Spectrum

A.1.1 Autoregressive Spectra

To estimate the spectra, we fit autoregressive models in the time domain, and
calculate the spectra of the estimated models. This method is based on the
seminal work by Burg (1967), who shows that the resulting spectrum is for-
mally identical to a spectrum derived on the Maximum Entropy Principle. This
is seen to be a more reasonable approach then the normally used periodogram
estimator. The periodogram employs the assumption that all the covariances
outside the sample period are zero. Given that economic time series are notori-
ously short, this seems to be a problematic assumption¹⁵ Consider a univariate
AR model of order p, with residual variance σ². The spectrum is given by

Equation (Al) is the analogue to the univariate spectrum in equation (1).
With a VAR model of order p, the spectral density matrix is given by

F(ω) = —A(ω) ¹ΣA(ω) *; ω ∈ [—π,π].
2π

Σ is the error variance-covariance matrix of the model, and A(ω) is the Fourier
transform of the matrix lag polynomial A(T) = I — A_xT — ∙ ∙ ∙ — A_pL^p.¹⁶ The
diagonal elements of this matrix are the analogue to the univariate spectrum
in equation (1), and the off-diagonal elements are the cross-spectra defined in
footnote 7.

A.2 Modified Baxter-King Filter

Baxter and King (1999) construct a bandpass Elter of finite order K which
is optimal in the sense that it is an approximate bandpass filter with trend-
reducing properties and Symmetricweights, which ensure that there is no phase

¹⁵See the discussion in (Priestley, 1981, p. 432, 604-607). A recent applications to eco-
nomic time series is A’Hearn and Woitek (2001).

¹⁶Z is the backshift operator; the superscript denotes the complex conjugate transpose.

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