Find out about spactral analysis

1. 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 , 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
(28.5/12=2.3 yrs business cycle, 12, ..
seasonality,, )
frequency wk = 2vk = 2(18/513), ..
(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 k=0 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)) =  k
2
… 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) = 2T[Xc(k)2 + Xs(k)2]
.. Not-consistent estimator for the spectral density
Background

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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 j=0 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 k=1 toM} wn(k)j cos(wj)]
where wn(k) is a lag window (kernel)
M is a bandwidth.
Note: Automatic bandwidth by Andrews(1991)
Spectral density function

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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 k=1 toM} wn(k)j
… captures “unknown” error structure
(non-parametric estimation)

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 Autocorrelation-heteroskedasticity
consistent standard error in regression
Recall:
White’s Heteroskedasticity consistent standard
error
Extension to allow for autocorrelation as well.
Example
Applications to Econometrics

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Hannan’s 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
Applications to Econometrics

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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.
Applications to Econometrics

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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

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This Side: Long Time Period
j small, wj small, & T large
Short Time Period

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z=e^(iw)

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Fourier Transform
F(a) is the Fourier transform of f(x)
f x
F
F a

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Fourier Transform

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Laplace Transform
F(t) Laplace Transform
f s
L
F t