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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 = 2vk = 2(18/513), .. (per unit time in radian)
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Use of the spectral density function
S(wk) of X, where wk = 2vk 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 Xt= {over k=0 to T/2} Xt(vk)
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)) = k2 … The variance is decomposed over different frequencies.
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Background 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
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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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Spectral density function
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)
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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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Applications to Econometrics
Autocorrelation-heteroskedasticity consistent standard error in regression Recall: White’s Heteroskedasticity consistent standard error Extension to allow for autocorrelation as well. Example
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Applications to Econometrics
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
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Applications to Econometrics
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
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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
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Fourier Transform
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Laplace Transform
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