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3.9. ESTIMATIONOFSPECTRA
381
The most t common estimator of the spectral l density of f this process s is the periodogram of
X
0
,...,X
1
,whichisdefinedas
I(w):=
1
n
1
å
t=0
X
t
e
itw
2
,
w2R
(3.99)
(Recallthatjzjdenotesthemodulusofcomplexnumberz)
Alternatively,I(w)canbeexpressedas
I(w)=
1
n
8
<
:
"
1
å
t=0
X
t
cos(wt)
#
2
+
"
1
å
t=0
X
t
sin(wt)
#
2
9
=
;
ItisstraightforwardtoshowthatthefunctionIisevenand2p-periodic(i.e.,I(w)=Iw)and
I(w+2p)=I(w)forallw2R)
Fromthesetworesults,youwillbeabletoverifythatthevaluesofIon[0,p]determinethevalues
ofIonallofR
Thenextsectionhelpstoexplaintheconnectionbetweentheperiodogramandthespectraldensity
Interpretation Tointerprettheperiodogram,itisconvenienttofocusonitsvaluesattheFourier
frequencies
w
j
:=
2pj
n
j=0,...,1
InwhatsenseisI(w
j
)anestimateoff(w
j
)?
Theanswerisstraightforward,althoughitdoesinvolvesomealgebra
Withabitofeffortonecanshowthat,foranyintegerj>0,
1
å
t=0
e
itw
j
=
1
å
t=0
exp
i2pj
t
n
=0
Letting
¯
Xdenotethesamplemeann
1
å
1
t=0
X
t
,wethenhave
nI(w
j
)=
1
å
t=0
(X
t
¯
X)e
itw
j
2
=
1
å
t=0
(X
t
¯
X)e
itw
j
1
å
r=0
(X
r
¯
X)e
irw
j
Bycarefullyworkingthroughthesums,onecantransformthisto
nI(w
j
)=
1
å
t=0
(X
t
¯
X)
2
+2
1
å
k=1
1
å
t=k
(X
t
¯
X)(X
t k
¯
X)cos(w
j
k)
Nowlet
ˆ
g(k):=
1
n
1
å
t=k
(X
t
¯
X)(X
t k
¯
X),
k=0,1,...,1
Thisisthesampleautocovariancefunction,thenatural“plug-inestimator”oftheautocovariance
functiong
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.9. ESTIMATIONOFSPECTRA
382
(“Plug-inestimator”isaninformaltermforanestimatorfoundbyreplacingexpectationswith
samplemeans)
Withthisnotation,wecannowwrite
I(w
j
)= ˆg(0)+2
1
å
k=1
ˆ
g(k)cos(w
j
k)
Recallingourexpressionfor fgivenabove,weseethatI(w
j
)isjustasampleanalogoff(w
j
)
Calculation Let’snowconsiderhowtocomputetheperiodogramasdefinedin(3.99)
Therearealreadyfunctionsavailablethatwilldothisforus—anexampleisperiodograminthe
DSP.jlpackage
However,itisverysimpletoreplicatetheirresults,andthiswillgiveusaplatformtomakeuseful
extensions
ThemostcommonwaytocalculatetheperiodogramisviathediscreteFouriertransform,which
inturnisimplementedthroughthefastFouriertransformalgorithm
Ingeneral,givenasequencea
0
,...,a
1
,thediscreteFouriertransformcomputesthesequence
A
j
:=
1
å
t=0
a
t
exp
i2p
tj
n
,
j=0,...,1
Witha
0
,...,a
1
storedinJuliaarraya,thefunctioncallfft(a)returnsthevaluesA
0
,...,A
1
asaJuliaarray
Itfollowsthat,whenthedataX
0
,...,X
1
isstoredinarrayX,thevaluesI(w
j
)attheFourier
frequencies,whicharegivenby
1
n
1
å
t=0
X
t
exp
i2p
tj
n
2
,
j=0,...,1
canbecomputedbyabs(fft(X)).^2 / length(X)
Note:TheJuliafunctionabsactselementwise,andcorrectlyhandlescomplexnumbers(bycom-
putingtheirmodulus,whichisexactlywhatweneed)
Here’safunctionthatputsallthistogether
function periodogram(x::Array):
length(x)
I_w abs(fft(x)).^n
2pi [0:n-1./ n
# Fourier frequencies
w, I_w w[1:int(n/2)], I_w[1:int(n/2)] # Truncate to interval [0, pi]
return w, I_w
end
Let’sgeneratesomedataforthisfunctionusingtheARMAtypefromQuantEcon
(Seethelectureonlinearprocessesfordetailsonthisclass)
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
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3.9. ESTIMATIONOFSPECTRA
383
Here’sacodesnippetthat,oncetheprecedingcodehasbeenrun,generatesdatafromtheprocess
X
t
=0.5X
1
+e
t
0.8e
2
(3.100)
wherefe
t
giswhitenoisewithunitvariance,andcomparestheperiodogramtotheactualspectral
density
import PyPlot: plt
import QuantEcon: ARMA
40
# Data size
phi, theta 0.5, [0-0.8]
# AR and MA parameters
lp ARMA(phi, , theta)
simulation(lp, ts_length=n)
fig, ax plt.subplots()
x, y periodogram(X)
ax[:plot](x, y, "b-", lw=2, alpha=0.5, label="periodogram")
x_sd, y_sd spectral_density(lp, two_pi=False, resolution=120)
ax[:plot](x_sd, y_sd, "r-", lw=2, alpha=0.8, label="spectral density")
ax[:legend]()
plt.show()
Runningthisshouldproduceafiguresimilartothisone
Thisestimatelooksratherdisappointing,butthedatasizeisonly40,soperhapsit’snotsurprising
thattheestimateispoor
However,ifwetryagainwithn = 1200theoutcomeisnotmuchbetter
Theperiodogramisfartooirregularrelativetotheunderlyingspectraldensity
Thisbringsustoournexttopic
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ARGENTAND
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OHN
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TACHURSKI
April20,2016
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3.9. ESTIMATIONOFSPECTRA
384
Smoothing
Therearetworelatedissueshere
Oneisthat,giventhewaythefastFouriertransformisimplemented,thenumberofpointswat
whichI(w)isestimatedincreasesinlinewiththeamountofdata
Inotherwords,althoughwehavemoredata,wearealsousingittoestimatemorevalues
Asecondissueisthatdensitiesofalltypesarefundamentallyhardtoestimatewithoutparame-
tericassumptions
Typically,nonparametricestimationofdensitiesrequiressomedegreeofsmoothing
Thestandardwaythatsmoothingisappliedtoperiodogramsisbytakinglocalaverages
Inotherwords,thevalueI(w
j
)isreplacedwithaweightedaverageoftheadjacentvalues
I(w
j p
),I(w
j p+1
),...,I(w
j
),...,I(w
j+p
)
Thisweightedaveragecanbewrittenas
I
S
(w
j
):=
p
å
‘= p
w(‘)I(w
j+‘
)
(3.101)
wheretheweightswp),...,w(p)areasequenceof2p+1nonnegativevaluessummingtoone
Ingenerally,largervaluesofpindicatemoresmoothing—moreonthisbelow
Thenextfigureshowsthekindofsequencetypicallyused
Notethesmallerweightstowardstheedgesandlargerweightsinthecenter,sothatmoredistant
valuesfromI(w
j
)havelessweightthancloseronesinthesum(3.101)
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
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3.9. ESTIMATIONOFSPECTRA
385
EstimationwithSmoothing Ournextstepistoprovidecodethatwillnotonlyestimatethe
periodogrambutalsoprovidesmoothingasrequired
Suchfunctionshavebeenwritteninestspec.jlandareavailableviaQuantEcon.jl
Thefileestspec.jlareprintedbelow
#=
Functions for working with periodograms of f scalar r data.
@author : Spencer Lyon <spencer.lyon@nyu.edu>
@date : : 2014-08-21
References
----------
http://quant-econ.net/jl/estspec.html
=#
import DSP
"""
Smooth the data in x using convolution with a window of requested size and type.
##### Arguments
- x::Array: An n array y containing the data to smooth
- window_len::Int(7): An odd integer giving g the e length of the window
- window::AbstractString("hanning"): A string giving the window type. Possible values
are flat, hanning, hamming, bartlett, or blackman
##### Returns
- out::Array: The array of smoothed data
"""
function smooth(x::Array, window_len::Int=7, window::AbstractString="hanning")
if length(x) window_len
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ARGENTAND
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TACHURSKI
April20,2016
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3.9. ESTIMATIONOFSPECTRA
386
throw(ArgumentError("Input vector r length h must be >= window length"))
end
if window_len 3
throw(ArgumentError("Window length h must t be at least 3."))
end
if iseven(window_len)
window_len += 1
println("Window length must be odd, reset to $window_len")
end
windows Dict("hanning" => DSP.hanning,
"hamming" => DSP.hamming,
"bartlett" => DSP.bartlett,
"blackman" => DSP.blackman,
"flat" => DSP.rect # moving average
)
# Reflect x x around d x[0] and x[-1] prior to convolution
round(Int, window_len 2)
xb x[1:k]
# First k elements
xt x[end-k+1:end# Last k elements
[reverse(xb); x; reverse(xt)]
# === Select window values === #
if !haskey(windows, window)
msg "Unrecognized window type '$window'"
print(msg " Defaulting to hanning")
window "hanning"
end
windows[window](window_len)
return conv(w ./ sum(w), s)[window_len+1:end-window_len]
end
"Version of smooth where window_len and window are keyword arguments"
function smooth(x::Array; window_len::Int=7, window::AbstractString="hanning")
smooth(x, window_len, window)
end
function periodogram(x::Vector)
length(x)
I_w abs(fft(x)).^./ n
2pi (0:n-1./ # Fourier frequencies
# int rounds to nearest integer. We want to round up or take 1/2 2 + + 1 to
# make sure e we e get the whole interval l from m [0, pi]
ind iseven(n) round(Int, n 1) : ceil(Int, n 2)
w, I_w w[1:ind], I_w[1:ind]
return w, I_w
end
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3.9. ESTIMATIONOFSPECTRA
387
function periodogram(x::Vector, window::AbstractString, window_len::Int=7)
w, I_w periodogram(x)
I_w smooth(I_w, window_len=window_len, window=window)
return w, I_w
end
"""
Computes the periodogram
I(w) = (1 / / n) ) | sum_{t=0}^{n-1} x_t e^{itw} |^2
at the Fourier r frequences s w_j := 2 pi j / / n, , j = 0, ..., n - 1, using g the e fast
Fourier transform. . Only y the frequences s w_j j in [0, pi] and corresponding values
I(w_j) are returned. . If f a window type is s given n then smoothing is performed.
##### Arguments
- x::Array: An n array y containing the data to smooth
- window_len::Int(7): An odd integer giving g the e length of the window
- window::AbstractString("hanning"): A string giving the window type. Possible values
are flat, hanning, hamming, bartlett, or blackman
##### Returns
- w::Array{Float64}: Fourier frequencies at t which h the periodogram is evaluated
- I_w::Array{Float64}: The periodogram at frequences w
"""
periodogram
"""
Compute periodogram from data x, using prewhitening, smoothing and recoloring.
The data is fitted to an AR(1) model for r prewhitening, , and the residuals are
used to compute a first-pass periodogram m with h smoothing. . The e fitted
coefficients are then used for recoloring.
##### Arguments
- x::Array: An n array y containing the data to smooth
- window_len::Int(7): An odd integer giving g the e length of the window
- window::AbstractString("hanning"): A string giving the window type. Possible values
are flat, hanning, hamming, bartlett, or blackman
##### Returns
- w::Array{Float64}: Fourier frequencies at t which h the periodogram is evaluated
- I_w::Array{Float64}: The periodogram at frequences w
"""
function ar_periodogram(x::Array, window::AbstractString="hanning", window_len::Int=7)
# run regression
x_current, x_lagged x[2:end], x[1:end-1# x_t t and d x_{t-1}
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TACHURSKI
April20,2016
3.9. ESTIMATIONOFSPECTRA
388
coefs linreg(x_lagged, x_current)
# get estimated values and compute residual
est [ones(x_lagged) x_lagged] coefs
e_hat x_current est
phi coefs[2]
# compute periodogram on residuals
w, I_w periodogram(e_hat, window, window_len)
# recolor and return
I_w I_w ./ abs(phi .* exp(im.*w)).^2
return w, I_w
end
Thelistingdisplaysthreefunctions,smooth(),periodogram(),ar_periodogram(). Wewilldis-
cussthefirsttwohereandthethirdonebelow
Theperiodogram()functionreturnsaperiodogram,optionallysmoothedviathesmooth()func-
tion
Regardingthesmooth()function,sincesmoothingaddsanontrivialamountofcomputation,we
haveappliedafairlytersearray-centricmethodbasedaroundconv
Readersarelefttoeitherexploreorsimplyusethiscodeaccordingtotheirinterests
Thenextthreefigureseachshowsmoothedandunsmoothedperiodograms,aswellasthetrue
spectraldensity
(Themodelisthesameasbefore—seeequation(3.100)—andthereare400observations)
Fromtopfiguretobottom,thewindowlengthisvariedfromsmalltolarge
Inlookingatthefigure,wecanseethatforthismodelanddatasize,thewindowlengthchosenin
themiddlefigureprovidesthebestfit
Relativetothisvalue,thefirstwindowlengthprovidesinsufficientsmoothing,whilethethird
givestoomuchsmoothing
Ofcourseinrealestimationproblemsthetruespectraldensityisnot visibleandthechoiceof
appropriatesmoothingwillhavetobemadebasedonjudgement/priorsorsomeothertheory
Pre-FilteringandSmoothing Inthecodelistingaboveweshowedthreefunctionsfromthefile
estspec.jl
The third function inthefile(ar_periodogram())adds apre-processingstep p to o periodogram
smoothing
Firstwedescribethebasicidea,andafterthatwegivethecode
Theessentialideaisto
1. Transformthedatainordertomakeestimationofthespectraldensitymoreefficient
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ARGENTAND
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TACHURSKI
April20,2016
3.9. ESTIMATIONOFSPECTRA
389
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3.9. ESTIMATIONOFSPECTRA
390
2. Computetheperiodogramassociatedwiththetransformeddata
3. Reversetheeffectofthetransformationontheperiodogram,sothatitnowestimatesthe
spectraldensityoftheoriginalprocess
Step1iscalledpre-filteringorpre-whitening,whilestep3iscalledrecoloring
Thefirststepiscalledpre-whiteningbecausethetransformationisusuallydesignedtoturnthe
dataintosomethingclosertowhitenoise
Whywouldthisbedesirableintermsofspectraldensityestimation?
Thereasonisthatwearesmoothingourestimatedperiodogrambasedonestimatedvaluesat
nearbypoints—recall(3.101)
Theunderlyingassumptionthatmakesthisagoodideaisthatthetruespectraldensityisrela-
tivelyregular—thevalueofI(w)isclosetothatofI(w
0)
whenwisclosetow
0
Thiswillnotbetrueinallcases,butitiscertainlytrueforwhitenoise
Forwhitenoise,Iisasregularaspossible—itisaconstantfunction
Inthiscase,valuesofI(w
0
)atpointsw
0
neartoprovidedthemaximumpossibleamountof
informationaboutthevalueI(w)
Anotherwaytoputthisisthat if Iisrelativelyconstant, , thenwecanusealargeamountof
smoothingwithoutintroducingtoomuchbias
TheAR(1)Setting Let’sexaminethisideamorecarefullyinaparticularsetting—wherethe
dataisassumedtobeAR(1)
(MoregeneralARMAsettingscanbehandledusingsimilartechniquestothosedescribedbelow)
SupposeinpartcularthatfX
t
giscovariancestationaryandAR(1),with
X
t+1
=m+fX
t
+e
t+1
(3.102)
wheremandf2( 1,1)areunknownparametersandfe
t
giswhitenoise
ItfollowsthatifweregressX
t+1
onX
t
andanintercept, theresidualswillapproximatewhite
noise
Let
• gbethespectraldensityoffe
t
g—aconstantfunction,asdiscussedabove
• I
0
betheperiodogramestimatedfromtheresiduals—anestimateofg
• fbethespectraldensityoffX
t
g—theobjectwearetryingtoestimate
InviewofanearlierresultweobtainedwhilediscussingARMAprocesses,fandgarerelatedby
f(w)=
1
feiw
2
g(w)
(3.103)
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
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