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2.6. LLNANDCLT
151
Ifg:R!Risdifferentiableatmandg
0
(m)6=0,then
p
nfg(
¯
X
n
g(m)g
d
!N(0,g
0
(m)
2
s
2
as n!¥
(2.27)
Thistheoremisusedfrequentlyinstatisticstoobtaintheasymptoticdistributionofestimators—
manyofwhichcanbeexpressedasfunctionsofsamplemeans
(Thesekindsofresultsareoftensaidtousethe“deltamethod”)
TheproofisbasedonaTaylorexpansionofgaroundthepointm
Takingtheresultasgiven,letthedistributionFofeachX
i
beuniformon[0,p/2]andletg(x)=
sin(x)
Derivetheasymptoticdistributionof
p
nfg(
¯
X
n
g(m)gandillustrateconvergenceinthesame
spiritastheprogramillustrate_clt.jldiscussedabove
Whathappenswhenyoureplace[0,p/2]with[0,p]?
Whatisthesourceoftheproblem?
Exercise2 Here’saresultthat’softenusedindevelopingstatisticaltests,andisconnectedtothe
multivariatecentrallimittheorem
Ifyoustudyeconometrictheory,youwillseethisresultusedagainandagain
AssumethesettingofthemultivariateCLTdiscussedabove,sothat
1. X
1
,...,X
n
isasequenceofIIDrandomvectors,eachtakingvaluesinR
k
2. m:=E[X
i
],andSisthevariance-covariancematrixofX
i
3. Theconvergence
p
n(
¯
X
n
m)
d
!N(0,S)
(2.28)
isvalid
Inastatisticalsetting,oneoftenwantstherighthandsidetobestandardnormal,sothatconfi-
denceintervalsareeasilycomputed
Thisnormalizationcanbeachievedonthebasisofthreeobservations
First,ifXisarandomvectorinR
k
andAisconstantandkk,then
Var[AX]=AVar[X]A
0
Second,bythecontinuousmappingtheorem,ifZ
n
d
!ZinRkandAisconstantandkk,then
AZ
n
d
!AZ
Third,ifSisakksymmetricpositivedefinitematrix,thenthereexistsasymmetricpositive
definitematrixQ,calledtheinversesquarerootofS,suchthat
QSQ
0
=I
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.7. LINEARSTATESPACEMODELS
152
HereIisthekkidentitymatrix
Puttingthesethingstogether,yourfirstexerciseistoshowthatifQistheinversesquarerootof
S,then
Z
n
:=
p
nQ(
¯
X
n
m)
d
!ZN(0,I)
Applyingthecontinuousmappingtheoremonemoretimetellsusthat
kZ
n
k
2
d
!kZk
2
GiventhedistributionofZ,weconcludethat
nkQ(
¯
X
n
m)k
2
d
!c
2
(k)
(2.29)
wherec
2
(k)isthechi-squareddistributionwithkdegreesoffreedom
(RecallthatkisthedimensionofX
i
,theunderlyingrandomvectors)
Yoursecondexerciseistoillustratetheconvergencein(2.29)withasimulation
Indoingso,let
X
i
:=
W
i
U
i
+W
i
where
• eachW
i
isanIIDdrawfromtheuniformdistributionon1,1]
• eachU
i
isanIIDdrawfromtheuniformdistributionon2,2]
• U
i
andW
i
areindependentofeachother
Hints:
1. sqrtm(A)computesthesquarerootofA.Youstillneedtoinvertit
2. YoushouldbeabletoworkoutSfromtheprocedinginformation
Solutions
Solutionnotebook
2.7 LinearStateSpaceModels
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.7. LINEARSTATESPACEMODELS
153
Contents
• LinearStateSpaceModels
– Overview
– TheLinearStateSpaceModel
– DistributionsandMoments
– StationarityandErgodicity
– NoisyObservations
– Prediction
– Code
– Exercises
– Solutions
“Wemayregardthepresentstateoftheuniverseastheeffectofitspastandthecause
ofitsfuture”–MarquisdeLaplace
Overview
Thislectureintroducesthelinearstatespacedynamicsystem
Easytouseandcarriesapowerfultheoryofprediction
Aworkhorsewithmanyapplications
• representingdynamicsofhigher-orderlinearsystems
• predictingthepositionofasystemjstepsintothefuture
• predictingageometricsumoffuturevaluesofavariablelike
– nonfinancialincome
– dividendsonastock
– themoneysupply
– agovernmentdeficitorsurplus
– etc.,etc.,...
• keyingredientofusefulmodels
– Friedman’spermanentincomemodelofconsumptionsmoothing
– Barro’smodelofsmoothingtotaltaxcollections
– RationalexpectationsversionofCagan’smodelofhyperinflation
– SargentandWallace’s“unpleasantmonetaristarithmetic”
– etc.,etc.,...
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.7. LINEARSTATESPACEMODELS
154
TheLinearStateSpaceModel
Objectsinplay
• Ann1vectorx
t
denotingthestateattimet=0,1,2,...
• Aniidsequenceofm1randomvectorsw
t
N(0,I)
• Ak1vectory
t
ofobservationsattimet=0,1,2,...
• AnnnmatrixAcalledthetransitionmatrix
• AnnmmatrixCcalledthevolatilitymatrix
• AknmatrixGsometimescalledtheoutputmatrix
Hereisthelinearstate-spacesystem
x
t+1
=Ax
t
+Cw
t+1
(2.30)
y
t
=Gx
t
x
0
N(m
0
,S
0
)
Primitives Theprimitivesofthemodelare
1. thematricesA,C,G
2. shockdistribution,whichwehavespecializedtoN(0,I)
3. thedistributionoftheinitialconditionx
0
,whichwehavesettoN(m
0
,S
0
)
GivenA,C,Ganddrawsofx
0
andw
1
,w
2
,...,themodel(2.30)pinsdownthevaluesofthese-
quencesfx
t
gandfy
t
g
Evenwithoutthesedraws,theprimitives1–3pindowntheprobabilitydistributionsoffx
t
gand
fy
t
g
Laterwe’llseehowtocomputethesedistributionsandtheirmoments
Martingaledifferenceshocks We’vemadethecommonassumptionthattheshocksareinde-
pendentstandardizednormalvectors
Butsomeofwhatwesaywillgothroughundertheassumptionthatfw
t+1
gisamartingaledif-
ferencesequence
Amartingaledifferencesequenceisasequencethatiszeromeanwhenconditionedonpastinfor-
mation
Inthepresentcase,sincefx
t
gisourstatesequence,thismeansthatitsatisfies
[w
t+1
jx
t
,x
1
,...]=0
Thisisaweakerconditionthanthatfw
t
gisiidwithw
t+1
N(0,I)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.7. LINEARSTATESPACEMODELS
155
Examples Byappropriatechoiceoftheprimitives,avarietyofdynamicscanberepresentedin
termsofthelinearstatespacemodel
Thefollowingexampleshelptohighlightthispoint
Theyalsoillustratethewisedictumfindingthestateisanart
Second-orderdifferenceequation Letfy
t
gbeadeterministicsequencethatsatifies
y
t+1
=f
0
+f
1
y
t
+f
2
y
1
s.t. y
0
,y
1
given
(2.31)
Tomap(2.31)intoourstatespacesystem(2.30),weset
x
t
=
2
4
1
y
t
y
1
3
5
A=
2
4
1
0
0
f
0
f
1
f
2
0
1
0
3
5
C=
2
4
0
0
0
3
5
G=
0 1 1 0
Youcanconfirmthatunderthesedefinitions,(2.30)and(2.31)agree
Thenextfigureshowsdynamicsofthisprocesswhenf
0
=1.1,f
1
=0.8,f
2
0.8,y
0
=y
1
=1
Lateryou’llbeaskedtorecreatethisfigure
UnivariateAutoregressiveProcesses Wecanuse(2.30)torepresentthemodel
y
t+1
=f
1
y
t
+f
2
y
1
+f
3
y
2
+f
4
y
3
+sw
t+1
(2.32)
wherefw
t
gisiidandstandardnormal
Toputthisinthelinearstatespaceformatwetakex
t
=
y
t
y
1
y
2
y
3
0
and
A=
2
6
6
4
f
1
f
2
f
3
f
4
1
0
0
0
0
1
0
0
0
0
1
0
3
7
7
5
C=
2
6
6
4
s
0
0
0
3
7
7
5
G=
1 0 0 0 0 0
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.7. LINEARSTATESPACEMODELS
156
ThematrixAhastheformofthecompanionmatrixtothevector
f
1
f
2
f
3
f
4
.
Thenextfigureshowsdynamicsofthisprocesswhen
f
1
=0.5,f
2
0.2,f
3
=0,f
4
=0.5,s=0.2,y
0
=y
1
=y
2
=y
3
=1
VectorAutoregressions Nowsupposethat
• y
t
isak1vector
• f
j
isakkmatrixand
• w
t
isk1
Then(2.32)istermedavectorautoregression
Tomapthisinto(2.30),weset
x
t
=
2
6
6
4
y
t
y
1
y
2
y
3
3
7
7
5
A=
2
6
6
4
f
1
f
2
f
3
f
4
I
0
0
0
0
I
0
0
0
0
I
0
3
7
7
5
C=
2
6
6
4
s
0
0
0
3
7
7
5
G=
0 0 0 0
whereIisthekkidentitymatrixandsisakkmatrix
Seasonals Wecanuse(2.30)torepresent
1. thedeterministicseasonaly
t
=y
4
2. theindeterministicseasonaly
t
=f
4
y
4
+w
t
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.7. LINEARSTATESPACEMODELS
157
Infactbotharespecialcasesof(2.32)
Withthedeterministicseasonal,thetransitionmatrixbecomes
A=
2
6
6
4
0 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 1 0
3
7
7
5
ItiseasytocheckthatA
=
I,whichimpliesthatx
t
isstrictlyperiodicwithperiod4:
1
x
t+4
=x
t
Suchanx
t
processcanbeusedtomodeldeterministicseasonalsinquarterlytimeseries.
Theindeterministicseasonalproducesrecurrent,butaperiodic,seasonalfluctuations.
TimeTrends Themodely
t
=at+bisknownasalineartimetrend
Wecanrepresentthismodelinthelinearstatespaceformbytaking
A=
1 1
0 1
C=
0
0
G=
a b
(2.33)
andstartingatinitialconditionx
0
=
0 1
0
Infactit’spossibletousethestate-spacesystemtorepresentpolynomialtrendsofanyorder
Forinstance,let
x
0
=
2
4
0
0
1
3
5
A=
2
4
1 1 1 0
0 1 1 1
0 0 0 1
3
5
C=
2
4
0
0
0
3
5
Itfollowsthat
A
t
=
2
4
t t(1)/2
0 1
t
0 0
1
3
5
Thenx
0
t
=
t(1)/2 1
,sothatx
t
containslinearandquadratictimetrends
Asavariationonthelineartimetrendmodel,consider:math:y_t=at+b
Tomodify(2.33)accordingly,weset
A=
1 1
0 1
C=
0
0
G=
a b
(2.34)
1
TheeigenvaluesofAare(1, 1,ii).
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.7. LINEARSTATESPACEMODELS
158
Moving Average e Representations A nonrecursive e expression n for x
t
as a a function of
x
0
,w
1
,w
2
,...,w
t
canbefoundbyusing(2.30)repeatedlytoobtain
x
t
=Ax
1
+Cw
t
(2.35)
=A
2
x
2
+ACw
1
+Cw
t
.
.
.
=
1
å
j=0
A
j
Cw
t j
+A
t
x
0
Representation(2.35)isamovingaveragerepresentation
Itexpressesfx
t
gasalinearfunctionof
1. currentandpastvaluesoftheprocessfw
t
gand
2. theinitialconditionx
0
Asanexampleofamovingaveragerepresentation,letthemodelbe
A=
1 1
0 1
C=
1
0
YouwillbeabletoshowthatA
=
t
0 1
andA
j
C=
1 0
0
Substitutingintothemovingaveragerepresentation(2.35),weobtain
x
1t
=
1
å
j=0
w
t j
+
t
x
0
wherex
1t
isthefirstentryofx
t
Thefirsttermontherightisacumulatedsumofmartingaledifferences,andisthereforeamartin-
gale
Thesecondtermisatranslatedlinearfunctionoftime
Forthisreason,x
1t
iscalledamartingalewithdrift
DistributionsandMoments
UnconditionalMoments Using(2.30), it’seasytoobtainexpressionsforthe(unconditional)
meansofx
t
andy
t
We’llexplainwhatunconditionalandconditionalmeansoon
Lettingm
t
:=
[x
t
]andusinglinearityofexpectations,wefindthat
m
t+1
=Am
t
with m
0
given
(2.36)
Herem
0
isaprimitivegivenin(2.30)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.7. LINEARSTATESPACEMODELS
159
Thevariance-covariancematrixofx
t
isS
t
:=
[(x
t
m
t
)(x
t
m
t
)
0
]
Usingx
t+1
m
t+1
=A(x
t
m
t
)+Cw
t+1
,wecandeterminethismatrixrecursivelyvia
S
t+1
=AS
t
A
0
+CC
0
with S
0
given
(2.37)
Aswithm
0
,thematrixS
0
isaprimitivegivenin(2.30)
Asamatterofterminology,wewillsometimescall
• m
t
theunconditionalmeanofx
t
• S
t
theunconditionalvariance-convariancematrixofx
t
Thisistodistinguishm
t
andS
t
fromrelatedobjectsthatuseconditioninginformation,tobede-
finedbelow
However, youshouldbeawarethatthese“unconditional”momentsdodependon theinitial
distributionN(m
0
,S
0
)
MomentsoftheObservations Usinglinearityofexpectationsagainwehave
[y
t
]=
[Gx
t
]=Gm
t
(2.38)
Thevariance-covariancematrixofy
t
iseasilyshowntobe
Var[y
t
]=Var[Gx
t
]=GS
t
G
0
(2.39)
Distributions Ingeneral,knowingthemeanandvariance-covariancematrixofarandomvector
isnotquiteasgoodasknowingthefulldistribution
However,therearesomesituationswherethesemomentsalonetellusallweneedtoknow
OnesuchsituationiswhenthevectorinquestionisGaussian(i.e.,normallydistributed)
Thisisthecasehere,given
1. ourGaussianassumptionsontheprimitives
2. thefactthatnormalityispreservedunderlinearoperations
Infact,it’swell-knownthat
uN(u¯,Sand v=a+Bu =) vN(a+B¯u,BSB
0
)
(2.40)
Inparticular,givenourGaussianassumptionsontheprimitivesandthelinearityof(2.30)wecan
seeimmediatelythatbothx
t
andy
t
areGaussianforallt0
2
Sincex
t
isGaussian, tofindthedistribution, allweneedtodoisfinditsmeanandvariance-
covariancematrix
Butinfactwe’vealreadydonethis,in(2.36)and(2.37)
2
Thecorrectwaytoarguethisisbyinduction.Supposethatx
t
isGaussian.Then(2.30)and(2.40)implythatx
t+1
isGaussian.Sincex
0
isassumedtobeGaussian,itfollowsthateveryx
t
isGaussian.Evidentlythisimpliesthateachy
t
isGaussian.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.7. LINEARSTATESPACEMODELS
160
Lettingm
t
andS
t
beasdefinedbytheseequations,wehave
x
t
N(m
t
,S
t
)
(2.41)
Bysimilarreasoningcombinedwith(2.38)and(2.39),
y
t
N(Gm
t
,GS
t
G
0
)
(2.42)
EnsembleInterpretations Howshouldweinterpretthedistributionsdefinedby(2.41)–(2.42)?
Intuitively,theprobabilitiesinadistributioncorrespondtorelativefrequenciesinalargepopula-
tiondrawnfromthatdistribution
Let’sapplythisideatooursetting,focusingonthedistributionofy
T
forfixedT
Wecangenerateindependentdrawsofy
T
byrepeatedlysimulatingtheevolutionofthesystem
uptotimeT,usinganindependentsetofshockseachtime
Thenextfigureshows20simulations,producing20timeseriesforfy
t
g,andhence20drawsofy
T
Thesysteminquestionistheunivariateautoregressivemodel(2.32)
Thevaluesofy
T
arerepresentedbyblackdotsintheleft-handfigure
Intheright-handfigure,thesevaluesareconvertedintoarotatedhistogramthatshowsrelative
frequenciesfromoursampleof20y
T
‘s
(The parameters s and source code for the figures s can n be e found d in file lin-
ear_models/paths_and_hist.pyfromtheapplicationsrepository)
Hereisanotherfigure,thistimewith100observations
Let’snowtrywith500,000observations,showingonlythehistogram(withoutrotation)
Theblacklineisthedensityofy
T
calculatedanalytically,using(2.42)
Thehistogramandanalyticaldistributionareclose,asexpected
Bylookingatthefiguresandexperimentingwithparameters,youwillgainafeelforhowthe
distributiondependsonthemodelprimitiveslistedabove
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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