V Time Varying Covariance and Correlation
|
|
- Herbert Strickland
- 6 years ago
- Views:
Transcription
1 V Time Varying Covariance and Correlaion DEFINITION OF CONDITIONAL CORRELATIONS. ARE THEY TIME VARYING? WHY DO WE NEED THEM? FACTOR MODELS DYNAMIC CONDITIONAL CORRELATIONS Are correlaions/covariances ime varying? Yes Correlaions are ime varying Derivaive prices of correlaion sensiive producs imply changes. Derivaives on correlaion now are raded. Time series esimaes change. There are many varieies. 1
2 How do we define ime varying correlaions? Recall ha for a single asse, we defined he condiional variance as he variance of he unpredicable par. Tha is, if y hen he condiional 2 variance is given by h E F 1. Hence if here is predicabiliy in he mean, we don include ha variaion in he condiional variance, we remove i firs. The same is rue for he condiional variance covariance marix. We define he condiional variance covariance marix for he par of y ha is no predicable. The condiional variance covariance marix is given by Eεε F 1 where ε y μ If here are no dynamics in he mean hen y 2
3 If mean reurns are no predicable hen y = and a wo dimensional covariance marix looks like: 2 r 1, 1, 1,2, Eyy F 1 E r1, r2, F 1 2 r 2, 1,2, 2, Condiional Covariance: E r r 1, 2, F 1 Condiional Variances CONDITIONAL CORRELATIONS Condiional correlaions are hen given by 1,2, 1,2, 1, 2, E 1 r1, r2, 2 2 1, 2, E r E r 1 1 3
4 ESTIMATION HISTORICAL CORRELATIONS i, j, s r r i, m j, m m1 s s 2 2 ri, mrj, m m1 m1 Exponenial smooher: q r r 1 q i, j, i, 1 j, 1 i, j, 1 Then i, j, q q i, j, q ii,, j, j, Use he same value for for each series 4
5 day hisorical correlaions beween AXP and GE C100_AXP_GE MULTIVARIATE MODELS 5
6 SOME MODELS ONE FACTOR MODEL ONE FACTOR MODEL WITH GARCH IDEOSYNCRATIC ERRORS MANY FACTOR MODEL MULTIVARIATE GARCH DYNAMIC CONDITIONAL CORRELATION ONE FACTOR ARCH One facor model such as CAPM There is one marke facor wih fixed beas and consan variance idiosyncraic errors independen of he facor. The marke has 2 some ype of ARCH wih variance m,. r r e i, i m, i, 2 2 ii,, i m, i If he marke has asymmeric volailiy, hen individual socks will oo. 6
7 CORRELATIONS Beween sock i and sock j assuming idiosyncracies are uncorrelaed. 2 i, j, i j m, i, j, 2 i j m, i im, j jm, Assuming beas are boh posiive, correlaions range from zero o one and increase wih marke volailiy. HOW TO ESTIMATE A ONE FACTOR MODEL FIT THE VOLATILITY OF THE MARKET ESTIMATE THE BETAS OF THE STOCKS AND THE VARIANCE OF THE IDIOSYNCRACIES 7
8 MARKET VOLATILITY Condiional Sandard Deviaion CALCULATE DYNAMIC CORRELATIONS h h 1 2h 2 8
9 AXP AND GE AGAIN C4_AXP_GE The mulivariae facor model Consider he case of n asses and le is he vecor of beas for he n socks. Then ' hm, Vn Where V n is he diagonal marix wih he idiosyncraic variances (v i s) on he diagonal. 9
10 The correlaions are given by: 1 ' R diag diag FACTOR DOUBLE ARCH Idiosyncraic errors also follow a GARCH models 2 ' hm, V, V ~ diagonal garch sd where h 0 V h 0 h n where i i iri1 hi 1 10
11 Mulifacor models Iniially, suppress he ime subscrip and consider our K facor model for asse i K ri i i 1F1i2F2... ikfk ei i ijfj ei j1 0 if i j Where Eee i j 2 I is convenien o wrie in marix noaion: r ' i i Bi F ei 1xK Kx1 vi if i=j If he facors are mean zero and he idiosyncraic errors are uncorrelaed wih he facors we ge ha he variance of asse i is: 2 Eri i EBF i eibf i ei EBF e F' B e i i i i E BFF B E BFe E e 2 ' 2 i i i i i BB v i i i EFFis he covariance marix of he facors. 0 11
12 The ideosyncraic errors are uncorrelaed across asses so ha he covariance of asse i wih asse j is E ri i rj i E BF i ei BjF e j E BF e F B e ' i i j j i ' j i j i j i j E BFF B B EFe E ef B E ee BB i j Now le r denoe a vecor of n asse reurns. r α B F e nx1 nx1 nxk Kx1 nx1 E E F F E F F EB FFB ee BEFFB EeeBB V rαrα B eb e B e Be Where I is a nxn ideniy marix and V is a nxn marix wih diagonal elemen i equal o v i. B is a marix wih i h row given by B i 12
13 Now le s exend his o he ime varying case: E EFF E r α r α B B ee B B V Can be made ime varying as in: E 1 E 1FF E 1 r α r α B B ee B B V where E FF 1 If he facors are observable (like porfolios), we can apply a model for ime varying (co)variances o he facors. E FF 1 13
14 Principal Componens Can we idenify facors from he reurns daa raher han specifying hem? Recall ha he variance covariance marix of he K facor model reurns are given by: E ' r α r α B B V where E FF Wih he facors unknown, B is idenified only up o an orhogonal ransformaion. To see his, le G be ANY KxK marix such ha GG I hen B * =B G and F * =G F also yields he same variance covariance srucure since he marix *' * *' * E F F V GG GG B Β B' ' ' BV B' ΒV 14
15 This lack of idenificaion can be solved by imposing some condiion (if we don have names on he facors we don care which of he equivalen models we choose)! Typically, we impose ha he facors are uncorrelaed. This yields: B' DBV where D is a diagonal marix We can find a represenaion of he facors by finding he linear combinaion of he daa ha has he highes variance. In pracice, we solve Max xˆ x subjec o xx 1 x where ˆ is he esimaed variance covariance marix of he reurns. 15
16 The soluion is ha x 1 will be he eigenvecor associaed wih he larges eigenvalue of ˆ. Nex we simply normalize x 1 o sum o one so ha i is a valid vecor of porfolio weighs. The second principal componen solves he problem Max xˆx subjec o xx 1 and xˆx 0 x The new se of porfolio weighs correspond o he porfolio ha has he highes variance bu is uncorrelaed wih he firs porfolio. The soluion is ha x 2 will be he eigenvecor associaed wih he second larges eigenvalue of ˆ. Again, he weighs x 2 can be normalized o sum o 1. 16
17 This process can be repeaed K imes unil we have weighs w 1, w 2,, w K which are he normalized eigenvecors of K larges eigenvecors of he variance covariance marix of reurns. Once he weighs are known, we can consruc he porfolios and rea he facors as if hey are observed. How many facors? The larges eigenvalues should be associaed wih he common componens. The smaller eigenvalues will be associaed wih he ideosynraic variances saring wih he larges and hen becoming smaller. When n ges large, he ideosyncraic variances become an arbirarily small fracion of he overall variance. The facors do no go away. 17
18 Hence in large samples we can idenify he K facors up o an orhogonal ransformaion. Ofen we look a he scree plo of he eigenvalues from larges o smalles. Number of facors is deermined by he drop off. Looks like 4 facors here: Eigen value 18
19 Dynamic Condiional Correlaion o DCC is a new ype of mulivariae GARCH model ha is paricularly convenien for big sysems. See Engle(2002) or Engle(2005). o Jus las monh Engle s new ex came ou. DYNAMIC CONDITIONAL CORRELATION OR DCC 1. Esimae volailiies for each asse and compue he sandardized residuals or volailiy adjused reurns. 2. Esimae he ime varying covariances beween hese using a maximum likelihood crierion and one of several models for he correlaions. 3. Form he correlaion marix and covariance marix. They are guaraneed o be posiive definie. 19
20 HOW IT WORKS When wo asses move in he same direcion, he correlaion is increased slighly. This effec may be sronger in down markes (asymmery in correlaions). When hey move in he opposie direcion i is decreased. The correlaions ofen are assumed o only emporarily deviae from a long run mean CORRELATIONS UPDATE LIKE GARCH Approximaely, z z 1, 1 2, z z 1, 1 2,
21 An Asymmeric model allows correlaions o increase more when boh prices move down ogeher (like our asymmeric GARCH models. z z z z ( I )( I ) 1, 1 2, 1 1, 1 2, 1 z 0 z DCC Correlaions C9_AXP_GE 21
22 C100_AXP_GE C4_AXP_GE C9_AXP_GE A simple approach o esimaing and forecasing large covariance marices Engle(2002), (2006) H DRD, D is diagonal marix of condiional sandard deviaions R is correlaion marix Based on sandardized reurns z 1 D r, V z R z 1 22
23 The general, mulivariae version of he DCC model looks like: ' R diag Q Q diag Q Q S z z Q where and are scalars. Hence he dynamics are he same for all series. S is he uncondiional correlaion marix. Esimaion T 1 1 L log H r ' H r 2T T 2 T T T 2 1 2log D r ' D r log R z ' R z z 1 2T 1 T 1 The log likelihood can be wrien as he sum of a par ha depends on he variances and a par ha depends on he correlaions. I is addiive and can be separaed ino wo esimaion problems. ' z 23
24 Bivariae Esimaion of correlaion parameers 1 z z 2* z z L 2 1 qi, j, 1 ri, jzi, 1 zj, 1 qi, j, 1 qi, j, i, j, q q T i, j, i, j, i, j, log(2 ) log 1 i, j, 2 1 i, j, ii,, ii,, o Engle proposes esimaing a model for all univariae pairs and hen averaging parameer esimaes (easy, bu no saisically sound). o An alernaive approach is o rea he pairs as a panel daase and simulaneously esimae DCC parameers (inference works here). Asymmeric Dynamic Correlaions of Global Equiy and Bond Reurns Lorenzo Capiello, Rober Engle and Kevin Sheppard Journal of Financial Economerics (2006) 24
25 Daa Weekly $ reurns Jan 1987 o Feb 2002 (785 observaions) 21 Counry Equiy Series from FTSE All-World Index 13 Daasream Benchmark Bond Indices wih 5 years average mauriy Europe AUSTRIA* BELGIUM* DENMARK* FRANCE* GERMANY* IRELAND* ITALY THE NETHERLANDS* SPAIN SWEDEN* SWITZERLAND* NORWAY UNITED KINGDOM* Ausralasia AUSTRALIA HONG KONG JAPAN* NEW ZEALAND SINGAPORE Americas CANADA* MEXICO UNITED STATES* *wih bond reurns 25
26 GARCH Models (asymmeric in orange) GARCH AVGARCH NGARCH EGARCH ZGARCH GJR-GARCH APARCH AGARCH NAGARCH 3EQ,8BOND 0 1BOND 6EQ,1BOND 8EQ,1BOND 3EQ,1BOND 0 1EQ,1BOND 0 26
27 AVERAGE EMU COUNTRY BOND RETURN CORRELATION 27
28 RESULTS Asymmeric Correlaions correlaions rise afer negaive reurns Shif in level of correlaions wih formaion of Euro Correlaions are rising no jus wihin EMU EMU Bond correlaions are especially high 28
Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1
Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien
More informationVolatility. Many economic series, and most financial series, display conditional volatility
Volailiy Many economic series, and mos financial series, display condiional volailiy The condiional variance changes over ime There are periods of high volailiy When large changes frequenly occur And periods
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationVectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1
Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies
More informationOBJECTIVES OF TIME SERIES ANALYSIS
OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging
More informationFinancial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS
Name SOLUTIONS Financial Economerics Jeffrey R. Russell Miderm Winer 009 SOLUTIONS You have 80 minues o complee he exam. Use can use a calculaor and noes. Try o fi all your work in he space provided. If
More informationTesting for a Single Factor Model in the Multivariate State Space Framework
esing for a Single Facor Model in he Mulivariae Sae Space Framework Chen C.-Y. M. Chiba and M. Kobayashi Inernaional Graduae School of Social Sciences Yokohama Naional Universiy Japan Faculy of Economics
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationGeneralized Least Squares
Generalized Leas Squares Augus 006 1 Modified Model Original assumpions: 1 Specificaion: y = Xβ + ε (1) Eε =0 3 EX 0 ε =0 4 Eεε 0 = σ I In his secion, we consider relaxing assumpion (4) Insead, assume
More informationRobert Kollmann. 6 September 2017
Appendix: Supplemenary maerial for Tracable Likelihood-Based Esimaion of Non- Linear DSGE Models Economics Leers (available online 6 Sepember 207) hp://dx.doi.org/0.06/j.econle.207.08.027 Rober Kollmann
More informationExponential Smoothing
Exponenial moohing Inroducion A simple mehod for forecasing. Does no require long series. Enables o decompose he series ino a rend and seasonal effecs. Paricularly useful mehod when here is a need o forecas
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationExponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits
DOI: 0.545/mjis.07.5009 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy,
More informationIntroduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
More informationState-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter
Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationDEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND
DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Asymmery and Leverage in Condiional Volailiy Models Michael McAleer WORKING PAPER
More informationComparing Means: t-tests for One Sample & Two Related Samples
Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion
More informationSummer Term Albert-Ludwigs-Universität Freiburg Empirische Forschung und Okonometrie. Time Series Analysis
Summer Term 2009 Alber-Ludwigs-Universiä Freiburg Empirische Forschung und Okonomerie Time Series Analysis Classical Time Series Models Time Series Analysis Dr. Sevap Kesel 2 Componens Hourly earnings:
More informationConcourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations
Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationAffine term structure models
Affine erm srucure models A. Inro o Gaussian affine erm srucure models B. Esimaion by minimum chi square (Hamilon and Wu) C. Esimaion by OLS (Adrian, Moench, and Crump) D. Dynamic Nelson-Siegel model (Chrisensen,
More informationI. Return Calculations (20 pts, 4 points each)
Universiy of Washingon Spring 015 Deparmen of Economics Eric Zivo Econ 44 Miderm Exam Soluions This is a closed book and closed noe exam. However, you are allowed one page of noes (8.5 by 11 or A4 double-sided)
More informationLinear Gaussian State Space Models
Linear Gaussian Sae Space Models Srucural Time Series Models Level and Trend Models Basic Srucural Model (BSM Dynamic Linear Models Sae Space Model Represenaion Level, Trend, and Seasonal Models Time Varying
More informationGMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
More informationLinear Combinations of Volatility Forecasts for the WIG20 and Polish Exchange Rates
Eliza Buszkowska Universiy of Poznań, Poland Linear Combinaions of Volailiy Forecass for he WIG0 and Polish Exchange Raes Absrak. As is known forecas combinaions may be beer forecass hen forecass obained
More informationTypes of Exponential Smoothing Methods. Simple Exponential Smoothing. Simple Exponential Smoothing
M Business Forecasing Mehods Exponenial moohing Mehods ecurer : Dr Iris Yeung Room No : P79 Tel No : 788 8 Types of Exponenial moohing Mehods imple Exponenial moohing Double Exponenial moohing Brown s
More informationSmoothing. Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T
Smoohing Consan process Separae signal & noise Smooh he daa: Backward smooher: A an give, replace he observaion b a combinaion of observaions a & before Simple smooher : replace he curren observaion wih
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationKriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationBias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé
Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070
More informationDEPARTMENT OF STATISTICS
A Tes for Mulivariae ARCH Effecs R. Sco Hacker and Abdulnasser Haemi-J 004: DEPARTMENT OF STATISTICS S-0 07 LUND SWEDEN A Tes for Mulivariae ARCH Effecs R. Sco Hacker Jönköping Inernaional Business School
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationTHE UNIVERSITY OF TEXAS AT AUSTIN McCombs School of Business
THE UNIVERITY OF TEXA AT AUTIN McCombs chool of Business TA 7.5 Tom hively CLAICAL EAONAL DECOMPOITION - MULTIPLICATIVE MODEL Examples of easonaliy 8000 Quarerly sales for Wal-Mar for quarers a l e s 6000
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationModeling and Forecasting Volatility Autoregressive Conditional Heteroskedasticity Models. Economic Forecasting Anthony Tay Slide 1
Modeling and Forecasing Volailiy Auoregressive Condiional Heeroskedasiciy Models Anhony Tay Slide 1 smpl @all line(m) sii dl_sii S TII D L _ S TII 4,000. 3,000.1.0,000 -.1 1,000 -. 0 86 88 90 9 94 96 98
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More informationChristos Papadimitriou & Luca Trevisan November 22, 2016
U.C. Bereley CS170: Algorihms Handou LN-11-22 Chrisos Papadimiriou & Luca Trevisan November 22, 2016 Sreaming algorihms In his lecure and he nex one we sudy memory-efficien algorihms ha process a sream
More informationDEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND
DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Dynamic Condiional Correlaions for Asymmeric Processes Manabu Asai and Michael McAleer
More informationL07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS. NA568 Mobile Robotics: Methods & Algorithms
L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS NA568 Mobile Roboics: Mehods & Algorihms Today s Topic Quick review on (Linear) Kalman Filer Kalman Filering for Non-Linear Sysems Exended Kalman Filer (EKF)
More informationECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates)
ECON 48 / WH Hong Time Series Daa Analysis. The Naure of Time Series Daa Example of ime series daa (inflaion and unemploymen raes) ECON 48 / WH Hong Time Series Daa Analysis The naure of ime series daa
More informationAsymmetry and Leverage in Conditional Volatility Models*
Asymmery and Leverage in Condiional Volailiy Models* Micael McAleer Deparmen of Quaniaive Finance Naional Tsing Hua Universiy Taiwan and Economeric Insiue Erasmus Scool of Economics Erasmus Universiy Roerdam
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationVector autoregression VAR. Case 1
Vecor auoregression VAR So far we have focused mosl on models where deends onl on as. More generall we migh wan o consider oin models ha involve more han one variable. There are wo reasons: Firs, we migh
More information13.3 Term structure models
13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)
More informationRegression with Time Series Data
Regression wih Time Series Daa y = β 0 + β 1 x 1 +...+ β k x k + u Serial Correlaion and Heeroskedasiciy Time Series - Serial Correlaion and Heeroskedasiciy 1 Serially Correlaed Errors: Consequences Wih
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationEE363 homework 1 solutions
EE363 Prof. S. Boyd EE363 homework 1 soluions 1. LQR for a riple accumulaor. We consider he sysem x +1 = Ax + Bu, y = Cx, wih 1 1 A = 1 1, B =, C = [ 1 ]. 1 1 This sysem has ransfer funcion H(z) = (z 1)
More informationUS AND LATIN AMERICAN STOCK MARKET LINKAGES
US AND LATIN AMERICAN STOCK MARKET LINKAGES Abdelmounaim Lahrech * School of Business and Adminisraion Al Akhawayn Universiy Kevin Sylweser Deparmen of Economics So. Illinois Universiy-Carbondale Absrac:
More informationForward guidance. Fed funds target during /15/2017
Forward guidance Fed funds arge during 2004 A. A wo-dimensional characerizaion of moneary shocks (Gürkynak, Sack, and Swanson, 2005) B. Odyssean versus Delphic foreign guidance (Campbell e al., 2012) C.
More informationGranger Causality Among Pre-Crisis East Asian Exchange Rates. (Running Title: Granger Causality Among Pre-Crisis East Asian Exchange Rates)
Granger Causaliy Among PreCrisis Eas Asian Exchange Raes (Running Tile: Granger Causaliy Among PreCrisis Eas Asian Exchange Raes) Joseph D. ALBA and Donghyun PARK *, School of Humaniies and Social Sciences
More informationDistribution of Estimates
Disribuion of Esimaes From Economerics (40) Linear Regression Model Assume (y,x ) is iid and E(x e )0 Esimaion Consisency y α + βx + he esimaes approach he rue values as he sample size increases Esimaion
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationEstimation Uncertainty
Esimaion Uncerainy The sample mean is an esimae of β = E(y +h ) The esimaion error is = + = T h y T b ( ) = = + = + = = = T T h T h e T y T y T b β β β Esimaion Variance Under classical condiions, where
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationProblem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
More informationLicenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
More informationDistribution of Least Squares
Disribuion of Leas Squares In classic regression, if he errors are iid normal, and independen of he regressors, hen he leas squares esimaes have an exac normal disribuion, no jus asympoic his is no rue
More informationACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H.
ACE 56 Fall 5 Lecure 8: The Simple Linear Regression Model: R, Reporing he Resuls and Predicion by Professor Sco H. Irwin Required Readings: Griffihs, Hill and Judge. "Explaining Variaion in he Dependen
More informationMaximum Likelihood Estimation of Time-Varying Loadings in High-Dimensional Factor Models. Jakob Guldbæk Mikkelsen, Eric Hillebrand and Giovanni Urga
Maximum Likelihood Esimaion of Time-Varying Loadings in High-Dimensional Facor Models Jakob Guldbæk Mikkelsen, Eric Hillebrand and Giovanni Urga CREATES Research Paper 2015-61 Deparmen of Economics and
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationModel Reduction for Dynamical Systems Lecture 6
Oo-von-Guericke Universiä Magdeburg Faculy of Mahemaics Summer erm 07 Model Reducion for Dynamical Sysems ecure 6 v eer enner and ihong Feng Max lanck Insiue for Dynamics of Complex echnical Sysems Compuaional
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationReferences are appeared in the last slide. Last update: (1393/08/19)
SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be
More informationExponentially Weighted Moving Average (EWMA) Chart Based on Six Delta Initiatives
hps://doi.org/0.545/mjis.08.600 Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives KALPESH S. TAILOR Deparmen of Saisics, M. K. Bhavnagar Universiy, Bhavnagar-36400 E-mail: kalpesh_lr@yahoo.co.in
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationOutline. lse-logo. Outline. Outline. 1 Wald Test. 2 The Likelihood Ratio Test. 3 Lagrange Multiplier Tests
Ouline Ouline Hypohesis Tes wihin he Maximum Likelihood Framework There are hree main frequenis approaches o inference wihin he Maximum Likelihood framework: he Wald es, he Likelihood Raio es and he Lagrange
More informationGeorey E. Hinton. University oftoronto. Technical Report CRG-TR February 22, Abstract
Parameer Esimaion for Linear Dynamical Sysems Zoubin Ghahramani Georey E. Hinon Deparmen of Compuer Science Universiy oftorono 6 King's College Road Torono, Canada M5S A4 Email: zoubin@cs.orono.edu Technical
More information(a) Set up the least squares estimation procedure for this problem, which will consist in minimizing the sum of squared residuals. 2 t.
Insrucions: The goal of he problem se is o undersand wha you are doing raher han jus geing he correc resul. Please show your work clearly and nealy. No credi will be given o lae homework, regardless of
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationAdvanced time-series analysis (University of Lund, Economic History Department)
Advanced ime-series analysis (Universiy of Lund, Economic Hisory Deparmen) 30 Jan-3 February and 6-30 March 01 Lecure 9 Vecor Auoregression (VAR) echniques: moivaion and applicaions. Esimaion procedure.
More informationTime series Decomposition method
Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,
More informationModeling the Volatility of Shanghai Composite Index
Modeling he Volailiy of Shanghai Composie Index wih GARCH Family Models Auhor: Yuchen Du Supervisor: Changli He Essay in Saisics, Advanced Level Dalarna Universiy Sweden Modeling he volailiy of Shanghai
More informationExplaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationDynamic models for largedimensional. Yields on U.S. Treasury securities (3 months to 10 years) y t
Dynamic models for largedimensional vecor sysems A. Principal componens analysis Suppose we have a large number of variables observed a dae Goal: can we summarize mos of he feaures of he daa using jus
More informationLecture 3: Exponential Smoothing
NATCOR: Forecasing & Predicive Analyics Lecure 3: Exponenial Smoohing John Boylan Lancaser Cenre for Forecasing Deparmen of Managemen Science Mehods and Models Forecasing Mehod A (numerical) procedure
More informationLecture #31, 32: The Ornstein-Uhlenbeck Process as a Model of Volatility
Saisics 441 (Fall 214) November 19, 21, 214 Prof Michael Kozdron Lecure #31, 32: The Ornsein-Uhlenbeck Process as a Model of Volailiy The Ornsein-Uhlenbeck process is a di usion process ha was inroduced
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationChapter 15. Time Series: Descriptive Analyses, Models, and Forecasting
Chaper 15 Time Series: Descripive Analyses, Models, and Forecasing Descripive Analysis: Index Numbers Index Number a number ha measures he change in a variable over ime relaive o he value of he variable
More informationTime series model fitting via Kalman smoothing and EM estimation in TimeModels.jl
Time series model fiing via Kalman smoohing and EM esimaion in TimeModels.jl Gord Sephen Las updaed: January 206 Conens Inroducion 2. Moivaion and Acknowledgemens....................... 2.2 Noaion......................................
More informationWhy is Chinese Provincial Output Diverging? Joakim Westerlund, University of Gothenburg David Edgerton, Lund University Sonja Opper, Lund University
Why is Chinese Provincial Oupu Diverging? Joakim Weserlund, Universiy of Gohenburg David Edgeron, Lund Universiy Sonja Opper, Lund Universiy Purpose of his paper. We re-examine he resul of Pedroni and
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More information0.1 MAXIMUM LIKELIHOOD ESTIMATION EXPLAINED
0.1 MAXIMUM LIKELIHOOD ESTIMATIO EXPLAIED Maximum likelihood esimaion is a bes-fi saisical mehod for he esimaion of he values of he parameers of a sysem, based on a se of observaions of a random variable
More informationRichard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen
Mulivariae Regular Variaion wih Applicaion o Financial Time Series Models Richard A. Davis Colorado Sae Universiy Bojan Basrak Eurandom Thomas Mikosch Universiy of Groningen Ouline + Characerisics of some
More informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More informationThe Real Exchange Rate, Real Interest Rates, and the Risk Premium. Charles Engel University of Wisconsin
The Real Exchange Rae, Real Ineres Raes, and he Risk Premium Charles Engel Universiy of Wisconsin How does exchange rae respond o ineres rae changes? In sandard open economy New Keynesian model, increase
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationØkonomisk Kandidateksamen 2005(II) Econometrics 2. Solution
Økonomisk Kandidaeksamen 2005(II) Economerics 2 Soluion his is he proposed soluion for he exam in Economerics 2. For compleeness he soluion gives formal answers o mos of he quesions alhough his is no always
More informationInstitute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
More informationTourism forecasting using conditional volatility models
Tourism forecasing using condiional volailiy models ABSTRACT Condiional volailiy models are used in ourism demand sudies o model he effecs of shocks on demand volailiy, which arise from changes in poliical,
More information