1. Spectral Analysis

2. 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*.

3. 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)

4. 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)

5. 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.

6. 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

7. 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

8. 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)

9. 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)

10. 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

11. 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

12. 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.

13. Cross, coherence & phase spectra
Cross Spectrum
Using cross covariance, XY(j)
Coherence Spectrum
like correlation coefficient
Phase spectrum
lead & lag analysis (like Causality)

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

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

23. z=e^(iw)

24. Fourier Transform

25. Fourier Transform

26. Laplace Transform