Spectral Analysis

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Lecture 3

• Spectral Analysis

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Frequency domain approach

• Examines contributions of different frequencies
  in explaining the variance.
• Analysis based on the estimated spectral density
  function.
• Provides the information on the properties of the
  time series data.
• Applied to econometric problems.

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Example

• Monthly growth rate of IP (p. 169), T 513
• peaks at k 18, 44, 89, 128, 171, 210
• cycle vk k/T 18/513, 44/513, .., 210/513
• period Tk 1/vk 28.5, 12, months
• (2.5 yrs business cycle, 12, .. seasonality,, )
• frequency wk 2?vk 2?(218/513), ..
• (per unit time in radian)

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Use of the spectral density function

• S(wk) of X, where wk 2?vk
• Total area under the curve from 0 to ?
• .5 Var(X)
• (Symmetric from ? to 2?)
• We examine if low or high frequency dominates.
• Examples (using PEST program)
• unit root process (low)
• white noise (horizontal line)
• Stationary MA(1), AR(1) process (high)

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Background

• Fourier transformation
• Xt ? over k0 to T/2 Xt(vk)
• ? akcos(wkt) bksin(wkt)
• where ak and bk are orthogonal Fourier
  coefficients.
• Xt(vj) and Xt(vk) are orthogonal.
• Variance decomposition
• Var(Xt) ? Var(Xt(vk)) ? ?k2
• The variance is decomposed over different
  frequencies.

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• Another form of (Discrete) Fourier transformation
• X(k) T-1 ? Xt exp(-iwkt) Xc(k) - iXs(k)
• Inverse Fourier transformation
• Xt sum over k-(T/2) to (T/2) X(k)exp(iwkt)
• Periodogram
• I(wk) 2TXc(k)2 Xs(k)2
• .. Not-consistent estimator for the spectral
  density

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Spectral density function

• Spectral density function
• Sx(wk) (1/2?)? over j - to?j exp(-iwj)
• (1/2?)?0 2? over j0 to ?j cos(wj)
• fx(wk) Sx(wk)/ ?0
• .. Normalized spectral density
• Inversion
• ?0 integral from -? to? Sx(wk) dw

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• Smoothed spectral density
• Sx(wk) (1/2?)? over j - to?j exp(-iwj)
• (1/2?)?0 2? over k1 toM wn(k)?j
  cos(wj)
• where wn(k) is a lag window (kernel)
• M is a bandwidth.
• Note Automatic bandwidth by Andrews(1991)

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Applications to Econometrics

• Spectral density at frequency zero
• Sx(0) (1/2?)? over j - to ?j
• longrun variance 2? Sx(0)
• ?2 ?0 2? over k1 toM wn(k)?j
• captures unknown error structure
• (non-parametric estimation)

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• Autocorrelation-heteroskedasticity consistent
  standard error in regression
• Recall
• Whites Heteroskedasticity consistent standard
  error
• Extension to allow for autocorrelation as well.
• Example

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• Hannans efficient estimator
• yt Xt? ut with unknown autocorrelation
• Transform yt Xt in frequency domain, then
• do OLS on the transformed variables, say yt
  Xt.
• Transformation is based on the cross spectral
  density of
• yt ut (also, Xt ut), then inverse
  transformation

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• Goodness-of-fit test
• .. Testing for a white noise process (or any
  ARMA)
• Based on the cumulative peridogram
• Max difference follows Kolmogorov-Smirnov
  statistics.

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Cross, coherence phase spectra

• Cross Spectrum
• Using cross covariance, ?XY(j)
• Coherence Spectrum
• like correlation coefficient
• Phase spectrum
• lead lag analysis (like Causality)

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• Bi-spectrum
• Bi-varaite joint density
• S(w1, w2)
• Testing for linearity