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Essays on Mathemati al Methods for E onomi sThesis defenseFranti²ek Brázdikfrantisek.brazdik� erge-ei. zCenter for E onomi Resear h and Graduate Edu ation of Charles UniversityCze h National Bank1August 31, 20091The views expressed here are my own and do not ne essarilyrepresent the views of the CNB.F. Brázdik Thesis defense 1 / 55
Outline1 Data Envelopment Analysis in Development E onomi s2 Models for Sto hasti Data Envelopment Analysis3 Announ ed Change of Monetary Regime
F. Brázdik Thesis defense 2 / 55
Methods of produ tivity analysisCompeting Methods for E� ien y Measurement:1 SFA approa hParametri methodSpe i� ation of produ tion fun tionyi = f (xi , β) + εi − vi ; where εi is error term and vi is positiveine� ien y term2 DEA approa hNon�parametri Properties of produ tion possibility setyi = f (xi )− ui ; where ui positive ine� ien y termF. Brázdik Thesis defense 3 / 55
E� ien y on ept OE=0R/0P, TE=0Q/0P, AE=0R/0Q
F. Brázdik Thesis defense 4 / 55
DEA Methodologyn homogenous DMUs: m inputs and s outputsT ⊂ Rm+s+ is general a produ tion possibility set, whereT = {(x , y) | using inputs x outputs y are produ ed}Properties of produ tion possibility set:ConvexityIne� ien y property � free disposalMinimum extrapolationNo free lun hE� ien y dominan e: DMU is dominated when there exist aDMU that an produ e the same levels of outputs with lessintensive use of inputsF. Brázdik Thesis defense 5 / 55
DEA Input Oriented ModelInput oriented model:minλj ,θj ,ej ,sj θjs.t. θjxij − ixλj − eij = 0, i = 1, . . . ,m;ryλj − srj = yrj , r = 1, . . . , k;
ϕ(1Tλj) = ϕ;
λj , ej , sj ≥ 0,θ proportional redu tion of inputsej , sj non-proportional sla ksλ intensity variableϕ = 1 variable returns to s aleϕ = 0 onstant returns to s aleF. Brázdik Thesis defense 6 / 55
Data Envelopment Analysis in Development E onomi sPresentation outline1 Data Envelopment Analysis in Development E onomi s2 Models for Sto hasti Data Envelopment Analysis3 Announ ed Change of Monetary Regime
F. Brázdik Thesis defense 7 / 55
Data Envelopment Analysis in Development E onomi sMotivationMotivation:Unique data setSu ess of �Green Revolution�Growth of Indonesian ri e produ tion over 1950�1980 periodGoals:Test farm size�produ tivity relationTownsend, Kirsten and Vink (1998), Helfand and Levine (2004):farm size�produ tivity relationship re onsiderationEvaluate impa t of intensi� ation program and other fa tors onfarm's e� ien yFarm spe i� fa tors: labor, fertilizers, et .E onomi fa tors: pri es of inputsEnvironmental fa tors: lo ation, wet-dry period, et .F. Brázdik Thesis defense 8 / 55
Data Envelopment Analysis in Development E onomi sMethodologyStage 1: DEAPri e distortions: Input oriented modelTime invariant produ tion frontierTime varying produ tion frontierStage 2: TobitE� ien y s ores � ensored variableE� ien y model estimationRandom e�e t modelMundlak's orre tion: Handling Correlation of individual hara teristi s and unobserved heterogeneityF. Brázdik Thesis defense 9 / 55
Data Envelopment Analysis in Development E onomi sE� ien y s oresStage 1:High orrelation of average DEA s ore ranking with SFArankings: 0.7127 � 0.8214Average te hni al e� ien y s ores range from 0.60 to 0.77High average s ale e� ien y 0.90Approximately 70% of farms are lo ated in DRS region ofprodu tion possibility setE� ien y s ores are onsistent a ross modelsNo signi� ant te hnologi al hange over onsidered period �Malmquist indexProdu tion growth was mainly driven by expansion of area usedfor produ tionHigh degree of heterogeneity in s oresF. Brázdik Thesis defense 10 / 55
Data Envelopment Analysis in Development E onomi sProdu tion Fa torsStage 2:HYV employment and share ropping positively related withe� ien y s oreNo signi� ant e� ien y bene�t from intensi� ation programparti ipationNo signi� ant e�e t of wet period: ine� ient irrigation systemsPositive e�e t of family labor share: quality of laborSize�e� ien y relation:�U� shaped relation � quadrati Threshold � apx. 1.41 ha and apx. 1.9 (using Mundlak's orre tion)Threshold oin ides with farm size on other islandsF. Brázdik Thesis defense 11 / 55
Data Envelopment Analysis in Development E onomi sCon lusionsAdopt �best�pra ti e� produ tion mixes: 23%�42% proportionalredu tion of all inputsPositive returns of swit h to HYVAdjust farm size: Pooling plotsReform of subsidies system to avoid overuse of inputs: Pesti idespri esPersonalization of intensi� ation program
F. Brázdik Thesis defense 12 / 55
Models for Sto hasti Data Envelopment AnalysisPresentation outline1 Data Envelopment Analysis in Development E onomi s2 Models for Sto hasti Data Envelopment Analysis3 Announ ed Change of Monetary Regime
F. Brázdik Thesis defense 13 / 55
Models for Sto hasti Data Envelopment AnalysisIntrodu tionGoals of Produ tivity analysis:Estimate produ tion fun tionMeasure distan e between observation and produ tion possibilityfrontierEvaluate e� ien y of observed produ tion pointsGoals:Develop the SDEA oriented modelsCompare e� ien y rankingsF. Brázdik Thesis defense 14 / 55
Models for Sto hasti Data Envelopment AnalysisMotivationProblem:DEA - extreme point methodInvalid e� ien y evaluationRobustness of resultsSolution:SDEA - inputs and outputs are random variables
F. Brázdik Thesis defense 15 / 55
Models for Sto hasti Data Envelopment AnalysisSDEA approa h
1 2 3 4 5 6Input
1
2
3
4
OutputStochastic Frontier
DMU1
DMU2
DMU3
DMU4
DMU5
DMU5 IneffiecntDMUs
DMU1 EfficientDMUs
EstimatedProductionfrontier
F. Brázdik Thesis defense 16 / 55
Models for Sto hasti Data Envelopment AnalysisMethodologyTheoreti al Work:Oriented modelsModels with variable returns to s aleLinearized modelsAppli ations:Solver: fast; large size problems; solutions with low number ofzero elementsStudy: Indonesian ri e farms e� ien yComparing results with parametri methodsF. Brázdik Thesis defense 17 / 55
Models for Sto hasti Data Envelopment AnalysisSDEA MethodologySho k stru ture: xij = xij + aijεyij = yij + bijεModel:maxλ
Prob(eT (Xλ− xj) + eT (yj − Y λ) < 0)− αs.t. Prob(i xλ < xij) ≥ 1− ǫ, i = 1, . . . ,m;Prob(r yλ > yrj) ≥ 1− ǫ, r = 1, . . . , s;λ ≥ 0,F. Brázdik Thesis defense 18 / 55
Models for Sto hasti Data Envelopment AnalysisSDEA problemInput oriented model:minλj ,θj θj − ǫ(Prob(1T (Xλj − θj xj) + 1T (yj − Y λj) < 0)− α)s.t. Prob(i xλj < θj xij) ≥ 1− ǫ, i = 1, . . . ,m;Prob(r yλj > yrj) ≥ 1− ǫ, r = 1, . . . , s;
ϕ(1Tλj) = ϕ;
λj ≥ 0.θ proportional redu tion of inputsej , sj non-proportional sla ksλ intensity variableϕ = 1 variable returns to s aleϕ = 0 onstant returns to s aleF. Brázdik Thesis defense 19 / 55
Models for Sto hasti Data Envelopment AnalysisE� ien y s ores distribution0
12
34
5D
ensi
ty
0 .2 .4 .6 .8 1Efficiency score
CCRnorm
CCRlog
FE
FEsp
SDEA−CCR vs. SFADensity estimates
01
23
45
Den
sity
0 .2 .4 .6 .8 1Efficiency score
BCCnorm
BCClog
FE
FEsp
SDEA−BCC vs. SFADensity estimates
F. Brázdik Thesis defense 20 / 55
Models for Sto hasti Data Envelopment AnalysisRanking onsisten eSpearman orrelation oe� ientSDEACCRN BCCN CCRLN BCCLNSFAFE 0.2534∗∗ 0.2448∗∗ -0.0224 -0.0292FEsp 0.2115∗∗ 0.2399 -0.0835∗∗ -0.0762∗Note: ∗∗ and ∗ means signi� an e at 1%, 5% respe tivelyLow values of ranking orrelation oe� ientsData with high degree of variationF. Brázdik Thesis defense 21 / 55
Announ ed Change of Monetary RegimePresentation outline1 Data Envelopment Analysis in Development E onomi s2 Models for Sto hasti Data Envelopment Analysis3 Announ ed Change of Monetary Regime
F. Brázdik Thesis defense 22 / 55
Announ ed Change of Monetary RegimeMotivationCze h Republi is onsidering monetary union entryMa roe onomi stability in small open e onomy environment:Collard & Dellas (2002)varian e of seriesevolution of varian eCurren y peg regime an support ma roe onomi stability:Cu he-Curti et al. (2008): rigidity in the goods marketDellas and Tavlas (2003): presen e of nominal rigiditiesSmall open e onomy: Behavior of after the announ ement ofswit h toward the ex hange rate stability rule (unilateral peg)F. Brázdik Thesis defense 23 / 55
Announ ed Change of Monetary RegimeModels of regime swit hQuestions:How will the response to sho ks of interest rates hange overthe transition period?What monetary regime is optimal for transition?Are business y les getting syn hronized over the transitionperiod?Goal:Modeling a monetary regime swit h in DSGE modelIntrodu e new theoreti al framework for regime swit h modelingFarmer, Waggoner and Zha (2007): Re ent works rely onMarkov swit hing pro essesF. Brázdik Thesis defense 24 / 55
Announ ed Change of Monetary RegimeModel IJustiniano and Preston (2004) framework:Two ountries:Home � small e onomyOptimizing agents: households and �rmsForeign � large e onomy (monetary union)Exogenous pro essesDomesti agents:Households: habit formationFirms: domesti produ ers, importers, and �nal good produ erF. Brázdik Thesis defense 25 / 55
Announ ed Change of Monetary RegimeModel IIModel features:No apitalAll goods are tradableComplete markets: Symmetri equilibriumZero in�ation steady stateNominal rigidities: Monopolisti ompetitionMonopolisti ompetition: Intermediate goodIn�ation indexation of good pri esImporters: Law of one pri e gapFinal good aggregation: Dixit-Stiglitz formF. Brázdik Thesis defense 26 / 55
Announ ed Change of Monetary RegimeModel IIIDomesti monetary poli y rules:Pre-transition:Targeting of in�ation, output gap or hange in nominalex hange rateTransition:Follow pre-transition rule with knowledge of regime swit hPost-transition:Rule of o�setting foreseen hanges in the nominal ex hange rate
F. Brázdik Thesis defense 27 / 55
Announ ed Change of Monetary RegimeMonetary poli y rulesGeneralization of monetary regimes:Pre-transition regime (independent monetary poli y):i It = ρi it−1 + (1− ρi)(ρππCPIt + ρyyt + ρe∆et)Post-transition regime (stability of ex hange rate):iUt = ρe ∑∞j=t (12)t−j∆Et [ej ]Transition regime:iTt = regimet i It + (1− regimet) iUt , where regimet ∈ {0, 1}where 0 ≤ ρi < 1, ρπ > 1, ρy > 0 and ρe ≥ 0and ρe = 2.0F. Brázdik Thesis defense 28 / 55
Announ ed Change of Monetary RegimeInformation bu�er IFuture information is added to the state spa eAgents foresee the future hanges of monetary regimeRegime indi ator:regimet = inft,1inft,1 = inft−1,2 + νt,1inft,2 = inft−1,3 + νt,2...inft,N−1 = inft−1,N + νt,N−1inft,N = νt,N , (1)F. Brázdik Thesis defense 29 / 55
Announ ed Change of Monetary RegimeInformation bu�er IIinft,i , i ∈ 1, . . . ,N are new endogenous variables,νt,i , i ∈ 1, . . . ,N are information sho ks in the period t.Announ ement is modeled as a series of information sho ksrealization
νk,i =
{ 1, i ≤ T ;0, i > T ,νl ,i = 0, ∀i and in the all subsequent periods l , l > kνl ,i is zero mean and zero varian e random variableF. Brázdik Thesis defense 30 / 55
Announ ed Change of Monetary RegimeSolutionThree models:Model of independent poli y: linearTransition period model: quadrati Final period model: linear1 Solve model:Easy for independent a �nal period modelTransition period: Se ond order approximation of the monetarypoli y ruleDynare++: fast solver for large problems2 Estimate modelDynare: Bayesian estimation3 De�ne s enarios:Evaluate information sho ksSimulate the linear modelF. Brázdik Thesis defense 31 / 55
Announ ed Change of Monetary RegimeEstimation resultsHigh value of the openness parameter: 0.35Inverse elasti ity of labor supply: 1.08Monetary poli y rule: high interest rate smoothing, in�ationstability is almost 3 times more preferred than output stabilitySlightly more rigidity in domesti good se tor than in importedgoodIn�ation indexation: 0.56
F. Brázdik Thesis defense 32 / 55
Announ ed Change of Monetary RegimeIrfsHow will the response to sho ks of interest rates hange over thetransition period?Compare responses:Examine the e�e t of the transition period lengthExamine the e�e ts of hoi e of the transition period regimeChoi e of weights in the monetary poli y rule to re�e t standardregimes
F. Brázdik Thesis defense 33 / 55
Announ ed Change of Monetary RegimeIrf (Transition length): Te hnology sho k1 3 5 7 9 11 13 15 17
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Dev
iatio
n
Periods
CPI inflation
1 3 5 7 9 11 13 15 17
0.1
0.2
0.3
0.4
0.5
Dev
iatio
n
Periods
Output
1 3 5 7 9 11 13 15 17
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Dev
iatio
n
Periods
∆ e
1 3 5 7 9 11 13 15 17
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Dev
iatio
n
Periods
Nominal int. rate
1 3 5 7 9 11 13 15 17
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
Dev
iatio
n
Periods
Imported inflation
1 3 5 7 9 11 13 15 17
−1
−0.8
−0.6
−0.4
−0.2
0
Dev
iatio
n
Periods
Domestic inflation
F. Brázdik Thesis defense 34 / 55
Announ ed Change of Monetary RegimeIrf (Transition length): Preferen e sho k1 3 5 7 9 11 13 15 17
0
0.05
0.1
0.15
0.2
0.25
0.3
Dev
iatio
n
Periods
CPI inflation
1 3 5 7 9 11 13 15 17
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
Dev
iatio
n
Periods
Output
1 3 5 7 9 11 13 15 17
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
Dev
iatio
n
Periods
∆ e
1 3 5 7 9 11 13 15 17
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
Dev
iatio
n
Periods
Nominal int. rate
1 3 5 7 9 11 13 15 17
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Dev
iatio
n
Periods
Imported inflation
1 3 5 7 9 11 13 15 17
0
0.1
0.2
0.3
0.4
0.5
0.6
Dev
iatio
n
Periods
Domestic inflation
F. Brázdik Thesis defense 35 / 55
Announ ed Change of Monetary RegimeTransition period: Welfare evaluation IWhat monetary regime is optimal for the transition?Assumptions:Pre-transition period: estimated regimeTransition period: Optimal regimeWelfare evaluation:Santa reu (2005):Lt = τVar(πt) + (1− τ)Var(yt) + τ4 (∆it),where τ ∈< 0, 1 >F. Brázdik Thesis defense 36 / 55
Announ ed Change of Monetary RegimeLoss fun tion evaluation
19
1725
330 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
20
40
60
80
100
120
τ
Loss function
Transition length
Loss
F. Brázdik Thesis defense 37 / 55
Announ ed Change of Monetary RegimeOptimal fun tion for the transition: ρi
1
9
17
25
33
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
τTransition length
ρi
F. Brázdik Thesis defense 38 / 55
Announ ed Change of Monetary RegimeOptimal fun tion for the transition: ρπ
1
9
17
25
33
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
τTransition length
ρπ
F. Brázdik Thesis defense 39 / 55
Announ ed Change of Monetary RegimeOptimal fun tion for the transition: ρy
1
9
17
25
33
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Transition lengthτ
ρy
F. Brázdik Thesis defense 40 / 55
Announ ed Change of Monetary RegimeOptimal fun tion for the transition: ρe
1
9
17
25
33
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
0
1
2
3
4
5
6
7
Transition lengthτ
ρe
F. Brázdik Thesis defense 41 / 55
Announ ed Change of Monetary RegimeBusiness y les orrelationsAre business y les getting syn hronized over the transition period?Ex hange rate stabilization vs. lost of monetary poli y in�uen eon in�ationInterest rate gets more orrelated with the hanges in theex hange rate over the transition period
F. Brázdik Thesis defense 42 / 55
Announ ed Change of Monetary RegimeCorrelation: Foreign interest rate5 9 13
0.4
0.5
0.6
0.7
0.8
0.9
1
Cor
rela
tion
Periods
Nominal int. rate
5 9 13−0.1
−0.05
0
0.05
0.1
0.15
Cor
rela
tion
Periods
CPI inflation
5 9 130.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
Cor
rela
tion
Periods
Output
5 9 13−0.4
−0.2
0
0.2
0.4
0.6
0.8
1C
orre
latio
n
Periods
∆ e
F. Brázdik Thesis defense 43 / 55
Announ ed Change of Monetary RegimeCorrelation: Domesti interest rate5 9 13
0
0.5
1
1.5
2
Cor
rela
tion
Periods
Nominal int. rate
5 9 13−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Cor
rela
tion
Periods
CPI inflation
5 9 13−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Cor
rela
tion
Periods
Output
5 9 13−0.2
0
0.2
0.4
0.6
0.8
1
1.2C
orre
latio
n
Periods
∆ e
F. Brázdik Thesis defense 44 / 55
Announ ed Change of Monetary RegimePoli y impli ationsThe in�ation-interest rate orrelation drops mainly in the initialand late phase of the transition.Consistently with the experiment design the interest rate -ex hange rate orrelation in reasesIn�uen e of monetary poli y on in�ation and output is alteredfrom anti- y li al to pro- y li alInitial loss: In rease in interest rate signals to depre iation underthe post-transition regime
F. Brázdik Thesis defense 45 / 55
ResponseResponse I1 Chapter 1:Link between onvexity type and returns to s ale is drawnThe relations between results of optimization problem take formof theoremThe use of IPM method is reasoned by readiness of IPM solver odeRelative measure statement was orre ted
F. Brázdik Thesis defense 46 / 55
ResponseResponse II2 Chapter 2:Input orientation is used while strong distortions to inputs pri esare presentFarms were delivering lose to self-su� ien y levels while someinputs were wastedFor the robustness he k di�erent types of e� ien y measureare usedχj = (
θ∗j − 1T e∗j1T xj ) 1T yj1TYλ∗j (page 53)Harvest ost is fra tion of rop re eived by workers: Output ompositionLabor is measured as man hoursF. Brázdik Thesis defense 47 / 55
ResponseResponse III3 Chapter 3:Mark-up sho ks may take form of demand sho ksThe extension by wage mark-up sho k may help to explain thevarian e in employmentFixing may be bene� ial for ountries in stress or small ountriesIn the future work on this topi higher order approximation ofunderlying model is going to be onsidered
F. Brázdik Thesis defense 48 / 55
Referen esReferen es IR.F. Towsend and J. Kirsten and N. VinkFarm size, Produ tivity and Returns to S ale in Agri ultureRevisited: A Case Study of Wine Produ ers in South Afri aAgri ultural E onomi s,1998,19S. M. Helfand and E. S. LevineFarm Size and the Determinants of Produ tive E� ien y in theBrazilian Center-WestAgri ultural E onomi s,2004,31
F. Brázdik Thesis defense 49 / 55
Referen esReferen es IIRoger E.A. Farmer and Daniel F. Waggoner and Tao ZhaUnderstanding the New-Keynesian Model when Monetary Poli ySwit hes RegimesNational Bureau of E onomi Resear h, Working Paper Series,12965,Harris Dellas and G. S. TavlasWage rigidity and monetary unionCEPR Dis ussion Papers, Jan, 2003Ni olas A. Cu he-Curti and Harris Dellas and Jean-Mar NatalIn�ation Targeting in a Small Open E onomyInternational Finan e, 11, 2008F. Brázdik Thesis defense 50 / 55
Referen esReferen es IIIMundell, R.A.,A theory of optimum urren y areasThe Ameri an E onomi Review 51 (4), 1961
F. Brázdik Thesis defense 51 / 55
AppendixChapter 2: Se ond stageTobit:χ∗ij = βT x + νi + ǫij , where χit is ensored variablerandom e�e ts, νi , are iid N(0, σ2ν) and ǫit are iid N(0, σ2ǫ )independently of νiUnobserved heterogeneity modeling:Mundlak (1978): unobserved heterogeneity an be modeled as afun tion of means of in luded regressorsνi = βxi + αi
αi is a part of farm's unobserved heterogeneity and un orrelatedwith regressorsxi is ve tor of farm i means for individual regressors xi over theobserved periodF. Brázdik Thesis defense 52 / 55
AppendixChapter 1: Linearized ModelLinearized input oriented model:minλj ,qkr ,hki ,θj θj + ǫ[1T (Xλj − θj xj) + 1T (yj − Y λj)++δ(1T (Aλj − θjaj) + 1T (bj − Bλj))σεΦ
−1(α)]++ǫ(
∑sr=1(q1r + q2r ) +∑mi=1(h1i + h2i))s.t. i xλj ≤ θj xij + (h1i + h2i)σεΦ−1(ǫ),iaλj − θjaij = h1i − h2i , i = 1, . . . ,m,yjλj ≤ r y + (q1r + q2r )σεΦ−1(ǫ),brj − rbλj = q1r − q2r , r = 1, . . . , s,
ϕ(1Tλj) = ϕ,
λj ≥ 0, qkr ≥ 0, hki ≥ 0, k = 1, 2F. Brázdik Thesis defense 53 / 55
AppendixChapter 3: Correlation with foreign in�ation rate5 9 13
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Cor
rela
tion
Periods
Nominal int. rate
5 9 130.1
0.15
0.2
0.25
0.3
Cor
rela
tion
Periods
CPI inflation
5 9 130.15
0.155
0.16
0.165
0.17
0.175
0.18
0.185
0.19
0.195
Cor
rela
tion
Periods
Output
5 9 13−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05C
orre
latio
n
Periods
∆ e
F. Brázdik Thesis defense 54 / 55
AppendixChapter 3: Correlation with foreign output5 9 13
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Cor
rela
tion
Periods
Nominal int. rate
5 9 130.12
0.14
0.16
0.18
0.2
0.22
0.24
Cor
rela
tion
Periods
CPI inflation
5 9 130.48
0.49
0.5
0.51
0.52
0.53
Cor
rela
tion
Periods
Output
5 9 13−0.2
−0.1
0
0.1
0.2
0.3C
orre
latio
n
Periods
∆ e
F. Brázdik Thesis defense 55 / 55