The Geometric Meaning of the Notion of Joint Unpredictability of a Bivariate VAR(1) Stochastic Process
|
|
- Primrose James
- 5 years ago
- Views:
Transcription
1 Econometrics 2013, 3, ; doi: /econometrics OPEN ACCESS econometrics ISSN Article The Geometric Meaning of the Notion of Joint Unpredictability of a Bivariate VAR(1) Stochastic Process Umberto Triacca Department of Computer Engineering, Computer Science Mathematics, University of L Aquila, Via Vetoio I Coppito, L Aquila, I-67100, Italy; umberto.triacca@ec.univaq.it; Tel.: , Fax.: Received: 21 August 2013; in revised form: 5 November 2013 / Accepted: 5 November 2013 / Published: 14 November 2013 Abstract: This paper investigates, in a particular parametric framework, the geometric meaning of joint unpredictability for a bivariate discrete process. In particular, the paper provides a characterization of the joint unpredictability in terms of distance between information sets in an Hilbert space. Keywords: Hilbert spaces; predictability; stochastic process JEL Classification: C18; C32 1. Introduction Let (Ω, F, P ) be a probability space {y t y 1,t, y 2,t ; t 0, 1,...} a bivariate stochastic process defined on (Ω, F, P ). We consider the differenced process { y t y 1,t, y 2,t ; t 1,...}, where is the first-difference operator. Following Caporale Pittis 1 Hassapis et al. 2, we say that the process { y t ; t 1,...} is jointly unpredictable if E( y t+1 σ(y t,..., y 0 )) 0 t (1) where σ(y t,..., y 0 ) is the σ-field generated by past vectors y i i 0,..., t. The goal of this paper is to show that the notion of joint unpredictability, in a particular parametric framework, can be characterized by a geometric condition. This characterization is given in terms of distance between information sets in an Hilbert space. In particular, we will show that the process { y t ; t 1,...} is jointly unpredictable if only if the information contained in its past is much
2 Econometrics 2013, distant from the information contained in its future. Even if our result is not as general as might seem desirable, we think that the intuition gained from this characterization makes the notion of joint unpredictability more clear. The rest of the paper is organized as follows. Section 2 presents the utilized mathematical framework. Sections 3 presents the geometric characterization. Section 4 concludes. 2. Preliminaries Definitions, notation, preliminary results from Hilbert space theory will be presented prior to establish the main result. An excellent overviews of the applications of Hilbert space methods to time series analysis can be found in Brockwell Davis 3. We use the following notations symbols. Let (Ω, F, P ) be a probability space. We consider the Hilbert space L 2 (Ω, F, P ) of all real square integrable rom variables on (Ω, F, P ). The inner product in L 2 (Ω, F, P ) is defined by z, w E(zw) for any z, w L 2 (Ω, F, P ). The space L 2 (Ω, F, P ) is a normed space the norm is given by w E(w 2 ) 1/2. The distance between z,w L 2 (Ω, F, P ) is d(z,w) z w. A sequence {z n } L 2 (Ω, F, P ) is said to converge to a limit point z L 2 (Ω, F, P ) if d(z n, z) 0 as n. A point z L 2 (Ω, F, P ) is a limit point of a set M (subset of L 2 (Ω, F, P )) if it is a limit point of a sequence from M. In particular, M is said to be closed if it contains all its limit points. If S is a arbitrary subset of L 2 (Ω, F, P ), then the set of all α 1 z α h z h (h 1, 2,...; α 1,..., α h arbitrary real numbers; z 1,..., z h arbitrary elements of S) is called a linear manifold spanned by S is symbolized by sp(s). If we add to sp(s) all its limit points we obtain a closed set that we call the closed linear manifold or subspace spanned by S, symbolized by sp(s). Two elements z, w L 2 are called orthogonal, we write z w, if z, w 0. If S is any subset of L 2 (Ω, F, P ), then we write x S if x s for all s S; similarly, the notation S T, for two subsets S T of L 2 (Ω, F, P ), indicates that all elements of S are orthogonal to all elements of T. For a given z L 2 (Ω, F, P ) a closed subspace M of L 2 (Ω, F, P ), we define the orthogonal projection of z on M, denoted by P (z M), as the unique element of M such that z P (z M) z w for any w M. We remember that if z M, then P (z M) 0. If M N are two arbitrary subsets of L 2 (Ω, F, P ), then the quantity d (M, N) inf { m n ; m M, n N} is called distance between M N. We close this section introducing some further definitions, concerning discrete stochastic processes in L 2 (Ω, F, P ). Let {x t } be a univariate stochastic process. We say that {x t } is integrated of order one (denoted x t I(1)) if the process { x t x t x t 1 } is stationary whereas {x t } is not stationary. We say that the bivariate stochastic process { y t y 1,t, y 2,t } is integrated of order one if y 1,t I(1) y 2,t I(1). A stochastic process {y t } Granger causes another stochastic process {x t }, with respect to a given information set I t that contains at least x t j, y t j, j > 0, if x t can be better predicted by using past values of y than by not doing so, all other information in I t (including the past of x) being
3 Econometrics 2013, used in either case. More formally, we say that {y t } is Granger causal for {x t } with respect to H xy (t) sp {x t, y t, x t 1, y t 1,...} if x t+1 P (x t+1 H xy (t)) 2 < x t+1 P (x t+1 H x (t)) 2 where H x (t) sp {x t, x t 1,...}. Two stochastic processes, {x t }, {y t }, both of which are individually I(1), are said to be cointegrated if there exists a non-zero constant β such that {z t x t βy t } is a stationary (I(0)) process. It is important to note that cointegration between two variables implies the existence of causality (in the Granger sense) between them in at least one direction (see Granger 4). 3. A Geometric Characterization In this section we assume that { y t y 1,t, y 2,t ; t 0, 1,... } be a bivariate stochastic process defined on (Ω, F, P ), integrated of order one, with y 1,0 y 2,0 0, that has a VAR(1) representation y t Ay t 1 + u t (2) where A a 11 a 12 a 21 a 22 is a fixed (2 2) coefficient matrix u t, is i.i.d. with E (u t ) 0 E(u t u t) Σ σ1 2 0 for all t E(u 0 σ2 2 t u s) 0 for s t. In this framework we have that {y 2,t } does not Granger cause {y 1,t } if only if a Similarly, {y 1,t } does not Granger cause {y 2,t } if only if a We observe that the VAR residuals are usually correlated hence the covariance matrix Σ is seldom a diagonal matrix. However, because the main aim of this study is pedagogical, we assume that Σ is diagonal for analytical convenience. We consider the following information sets: I y1 (t+) { y 1,t+1, y 1,t+2,...}, I y2 (t+) { y 2,t+1, y 2,t+2,...}, H y1 (t) sp { y 1,t, y 1,t 1,...} H y2 (t) sp { y 2,t, y 2,t 1,...}. Theorem 3.1. Let y t be a VAR(1) process defined as in (2). The differenced process { y t ; t 1,...} is jointly unpredictable if only if d (I y1 (t+), H y2 (t)) σ y1 d (I y2 (t+), H y1 (t)) σ y2 Theorem 1 provides a geometric characterization of the notion of joint unpredictability of a bivariate process in term of distance between information sets. It is important to note that d (I y1 (t+), H y2 (t)) σ y1 d (I y2 (t+), H y1 (t)) σ y2 Thus we have that the process { y t y 1,t, y 2,t ; t 1,...} is jointly unpredictable if only if the distances d (I y1 (t+), H y2 (t)) d (I y2 (t+), H y1 (t)) achieve their maximum value, respectively.
4 Econometrics 2013, It is intuitive to think that if these distances achieve their maximum value, then σ(y t,..., y 0 ) does not contain any valuable information about the future of the differenced series, y t y 1,t, y 2,t hence these are jointly unpredictable with respect to the information set σ(y t,..., y 0 ), that is E( y t+1 σ(y t,..., y 0 )) 0. We recall that Theorem 1 holds only in a bivariate setting Lemmas In order to prove Theorem 1, we need the following lemmas. Lemma 3.2. Let V be a closed subspace of L 2 (Ω, F, P ) G a subset of L 2 (Ω, F, P ) such that g η R, g G. G V if only if d (G, V ) η. Proof. Focker Triacca (5, p. 767). Lemma 1 establishes a relationship between the orthogonality of sets/spaces in the Hilbert space L 2 (Ω, F, P) their distance. We note that the orthogonality between G V holds if only if the distance d (G, V ) achieves the maximum value. In fact, d (G, V ) can not be greater than η since 0 V. Lemma 3.3. The processes {y 1,t } {y 2,t } are not cointegrated if only if A I. Proof. By (2) we have y 1,t y 1,t a 11 1 a 12 a 21 a 22 1 y 1,t 1 y 2,t 1 + These equations must be balanced, that is the order of integration of (a 11 1)y 1,t 1 + a 12 y 2,t 1 a 21 y 1,t 1 + (a 22 1)y 2,t 1 must be zero. ( ) If A I, since (a 11 1)y 1,t 1 + a 12 y 2,t 1 I(0) a 21 y 1,t 1 + (a 22 1)y 2,t 1 I(0), we can have three cases. Case (1) A a ij, with a ij 0 i, j 1, 2, i j a ii 1 i 1, 2. Case (2) a 11 a 12 A 0 1 with a 11 1 a Case (3) with a 21 0 a A a 21 a 22 In all three cases, there exists at least a not trivial linear combination of the processes {y 1,t } {y 2,t } that is stationary. Thus we can conclude that {y 1,t } {y 2,t } are cointegrated. ( ) If A I, then a 12 a 21 0 so {y 1,t } does not Granger cause {y 2,t } {y 2,t } does not Granger cause {y 1,t }. It follows that {y 1,t } {y 2,t } are not cointegrated.
5 Econometrics 2013, Lemma 3.4. If {y 1,t } {y 2,t } are cointegrated, then a 11 + a 22 1 < 1. Proof. We subtract y 1,t 1, y 2,t 1 from both sides of Equation (2) by obtaining y 1,t y 2,t y 2,t a 11 1 a 12 a 21 a 22 1 α 2 y 1,t 1 y 2,t 1 If {y 1,t } {y 2,t } are cointegrated, we have y 1,t α 1 y 1,t 1 β 1 β 2 + β 1 ϑ 1 ϑ 2 ϑ 1 ϑ 2 α 1 α 2 y 2,t 1 1 β 2 /β 1 y 1,t 1 1 β y 1,t 1 y 2,t 1 (y 1,t 1 βy 2,t 1 ) + y 2,t 1 where β (β 2 /β 1 ) is the cointegration coefficient ϑ 1 β 1 α 1 ϑ 2 β 1 α 2 are the speed of adjustment coefficients. We observe that y 1,t β y 2,t ϑ 1 y 1,t 1 βϑ 2 y 1,t 1 βϑ 1 y 2,t 1 + β 2 ϑ 2 y 2,t 1 + β (3) By rearranging Equation (3) we obtain an AR(1) model for y 1,t βy 2,t : y 1,t βy 2,t δ(y 1,t 1 βy 2,t 1 ) + β where δ 1 + ϑ 1 βϑ 2 a 11 + a Since {y 1,t } {y 2,t } are cointegrated, {y 1,t βy 2,t } is a stationary process so a 11 + a 22 1 < 1 Lemma 3.5. The process { y t y 1,t, y 2,t ; t 0, 1,... } is jointly unpredictable if only if A I Proof. ( ) process { y t y 1,t, y 2,t ; t 0, 1,... } is jointly unpredictable, then E(y t σ(y t 1,..., y 1 )) y t 1 On the other h, since y t Ay t 1 + u t with E (u t ) 0 E(u t u t) Σ for all t E(u t u s) 0 for s t, we have that E(y t σ(y t 1,..., y 1 )) Ay t 1
6 Econometrics 2013, Hence we have so Ay t 1 y t 1 A I ( ) If A I then y t y t 1 + u t with E (u t ) 0 E(u t u t) Σ for all t E(u t u s) 0 for s t, hence we have E(y t σ(y t 1,..., y 0 )) y t 1 Thus we can conclude that the process { y t y 1,t, y 2,t ; t 1,... } is jointly unpredictable. Before to conclude this subsection we observe that Equation (2) can be written in lag operator notation. The lag operator L is defined such that Ly t y t 1. We have that (I AL)y t u t or 1 a 11 L a 12 L a 21 L 1 a 22 L y 1,t y 2,t 3.2. Proof of Theorem 1 Sufficiency. If then, by Lemma 1, we have d (I y1 (t+), H y2 (t)) σ y1 d (I y2 (t+), H y1 (t)) σ y2 I y1 (t+) H y2 (t) I y2 (t+) H y1 (t) Now we assume that a 12 a 21 are not both equal to zero. We can have three cases. Case (1) a 12 0 a This implies that r 1,t (a 11 1)y 1,t + a 12 y 2,t + Thus y 2,t < y 1,t+1, y 2,t > E( y 1,t+1 y 2,t ) (a 11 1)E(y 1,t ) + a 12 E(y 2,t ) + E(+1 ) t (a 11 1)E(y 1,t ) + a 12 E( u 2,s ) (a 11 1)E(y 1,t ) + a 12 σ 2 2 s1
7 Econometrics 2013, Now, we note that Thus but this is absurd since E(y 1,t ) a t 11E(y 1,0 ) 0 < y 1,t+1, y 2,t > a 12 σ I y1 (t+) H y2 (t) Case (2) a 12 0 a In this case we have < y 2,t+1, y 1,t > a 21 σ Again this is absurd since I y2 (t+) H y1 (t) Case (3) a 12 0 a We note that y 1,t (1 a 22 L)γ(L) a 12 Lγ(L) a 21 Lγ(L) (1 a 11 L)γ(L) y 2,t where γ(l) 1 L (1 a 11 L)(1 a 22 L) a 12 a 21 L 2 By Lemma 2, we have that {y 1t } {y 2t } are cointegrated hence the matrix a 11 1 a 12 A I a 21 a 22 1 has rank 1. It follows that a 12 a 21 (1 a 11 )(1 a 22 ) Thus γ(l) 1 L (1 a 11 L)(1 a 22 L) (1 a 11 )(1 a 22 )L 2 1 L (1 L)(1 + L) (1 L)(a 11 + a 22 )L L (a 11 + a 22 )L 1 1 (a 11 + a 22 1)L 1 1 δl where δ a 11 + a Since {y 1t } {y 2t } are cointegrated, by Lemma 3 we have that δ < 1 hence Now, we can have two cases. γ(l) 1 + δl + δ 2 L
8 Econometrics 2013, Case (a) δ 0. In this case we have y 1,t a a 12 1 y 2,t a a 11 1 Thus but this is absurd since < y 1,t+1, y 2,t > a 12 σ2 2 0 < y 2,t+1, y 1,t > a 21 σ1 2 0 I y1 (t+) H y2 (t) I y2 (t+) H y1 (t) Case (b) δ 0. In this case we have Thus y 1,t + (a 11 1) 1 + δ(a 11 1) a a (a 11 1) δ i 1 i + a 1 i 1 i i0 y 2,t + (a 22 1) 1 + δ(a 22 1) a a 21 δ (a 22 1) δ i 1 i + a 21 δ i 1 i i0 < y 1,t+1, y 2,t > (a 11 1)σ1 1 δ a δ (a 22 1) σ 1 δ 2a < y 2,t+1, y 1,t > (a 22 1)σ2 1 δ a δ (a 11 1) σ 1 δ 1a Now, we consider the system 1 + (a22 1) δ 1 δ σ (a 11 1)σ1 1 δ 2 (a 22 1)σ2 1 δ (a11 1) δ 1 δ σ i0 i0 a 12 a The determinant of the matrix 1 + (a22 1) δ 1 δ σ (a 11 1)σ1 1 δ 2 (a 22 1)σ2 1 δ (a11 1) δ 1 δ σ is σ 2 1σ 2 2 ( 1 δ ) 1 + δ
9 Econometrics 2013, Since σ 2 1σ 2 2 > 0 δ 1+δ 1, we have that σ 2 1σ 2 2 ( 1 δ ) δ Thus a 12 0, a 21 0 implies that < y 1,t+1, y 2,t > 0 or < y 2,t+1, y 1,t > 0, but this is absurd since I y1 (t+) H y2 (t) I y2 (t+) H y1 (t) In all Cases (1 3) we obtain an absurd conclusion, thus we can state that Now, we prove that a 11 a We have that a 12 0, a 21 0 y i,t (a ii 1)y i,t + u it i 1, 2 Since the error term u t, is stationary these equations must be balanced, that is the order of integration of y i,t (a ii 1)y i,t must be the same. By the hypothesis that y i,t I(1), it follows that y it I(0) (i.e., stationary) (a ii 1)y i,t is I(1), hence y i,t (a ii 1)y i,t + u i,t i 1, 2 implies that a 11 a Thus A I hence, by Lemma 4, it follows that the process { y t ; t 1,...} is jointly unpredictable. Necessity. If the process { y t ; t 1,...} is is jointly unpredictable, then by Lemma 4 it follows that A I hence y 1,t y 2,t t. This implies that P ( y 1,t+h H y2 (t)) 0 P ( y 2,t+h H y1 (t)) 0 h > 0. Therefore we have that y 1,t+h H y2 (t) y 2,t+h H y1 (t) h > 0. Thus, by Lemma 1, it follows that Theorem 1 is proved. d (I y1 (t+), H y2 (t)) σ y1 d (I y2 (t+), H y1 (t)) σ y2 4. Conclusions In this paper we have considered the following geometric condition concerning the distance between information sets d (I y1 (t+), H y2 (t)) σ y1 d (I y2 (t+), H y1 (t)) σ y2 (4) It says that the distances d (I y1 (t+), H y2 (t)) d (I y2 (t+), H y1 (t)) achieve their maximum value, respectively. Theorem 1 tells us that, under the hypothesis that the process y t follows a bivariate VAR(1) model, the condition Equation (4) represents a geometric characterization of the notion of joint unpredictability. If this condition holds, the processes y 1 y 2 are jointly unpredictable since the past of the bivariate process y t does not contain any valuable information about the future of the differenced series. The information in the past is too far from the future information. Even if the bivariate VAR(1) assumption is far from general, we think that this geometric characterization is useful in order to throw light on the concept of joint unpredictability of a stochastic process.
10 Econometrics 2013, Acknowledgments We thank two anonymous referees for their helpful constructive comments, Fulvia Focker for helpful comments. Conflicts of Interest The author declares no conict of interest. References 1. Caporale, G.M.; Pittis, N. Cointegration predictability of asset prices. J. Int. Money Financ. 1998, 17, Hassapis C.; Kalyvitis, S.; Pittis, N. Cointegration joint efficiency of international commodity markets. Q. Rev. Econ. Financ. 1999, 39, Brockwell, P.J.; Davis, R.A. Time Series: Theory Methods, 2nd ed.; Springer-Verlag: New York, NY, USA, Granger, C.W.J. Developments in the study of cointegrated economic variables. Oxf. Bull. Econ. Stat. 1986, 48, Focker, F.; Triacca, U. Interpreting the concept of joint unpredictability of asset returns: A distance approach. Physica A 2006, 2, c 2013 by the author; licensee MDPI, Basel, Switzerl. This article is an open access article distributed under the terms conditions of the Creative Commons Attribution license (
Econ 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationUnit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series
econometrics Article Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series Nunzio Cappuccio 1 and Diego Lubian 2, * 1 Department of Economics and Management Marco Fanno,
More informationMeasuring the Distance between Sets of ARMA Models
Article Measuring the Distance between Sets of ARMA Models Umberto Triacca Department of Computer Engineering, Computer Science and Mathematics, University of L Aquila, Via Vetoio Coppito, L Aquila I-67010,
More informationIntroduction to Algorithmic Trading Strategies Lecture 3
Introduction to Algorithmic Trading Strategies Lecture 3 Pairs Trading by Cointegration Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Distance method Cointegration Stationarity
More informationISSN Article
Econometrics 2014, 2, 203-216; doi:10.3390/econometrics2040203 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Article Testing for A Set of Linear Restrictions in VARMA Models
More informationThe PPP Hypothesis Revisited
1288 Discussion Papers Deutsches Institut für Wirtschaftsforschung 2013 The PPP Hypothesis Revisited Evidence Using a Multivariate Long-Memory Model Guglielmo Maria Caporale, Luis A.Gil-Alana and Yuliya
More informationA Non-Parametric Approach of Heteroskedasticity Robust Estimation of Vector-Autoregressive (VAR) Models
Journal of Finance and Investment Analysis, vol.1, no.1, 2012, 55-67 ISSN: 2241-0988 (print version), 2241-0996 (online) International Scientific Press, 2012 A Non-Parametric Approach of Heteroskedasticity
More informationOn the Error Correction Model for Functional Time Series with Unit Roots
On the Error Correction Model for Functional Time Series with Unit Roots Yoosoon Chang Department of Economics Indiana University Bo Hu Department of Economics Indiana University Joon Y. Park Department
More informationQuestions and Answers on Unit Roots, Cointegration, VARs and VECMs
Questions and Answers on Unit Roots, Cointegration, VARs and VECMs L. Magee Winter, 2012 1. Let ɛ t, t = 1,..., T be a series of independent draws from a N[0,1] distribution. Let w t, t = 1,..., T, be
More informationECON 7335 INFORMATION, LEARNING AND EXPECTATIONS IN MACRO LECTURE 1: BASICS. 1. Bayes Rule. p(b j A)p(A) p(b)
ECON 7335 INFORMATION, LEARNING AND EXPECTATIONS IN MACRO LECTURE : BASICS KRISTOFFER P. NIMARK. Bayes Rule De nition. Bayes Rule. The probability of event A occurring conditional on the event B having
More informationLecture Notes 1: Vector spaces
Optimization-based data analysis Fall 2017 Lecture Notes 1: Vector spaces In this chapter we review certain basic concepts of linear algebra, highlighting their application to signal processing. 1 Vector
More informationLecture 2: Univariate Time Series
Lecture 2: Univariate Time Series Analysis: Conditional and Unconditional Densities, Stationarity, ARMA Processes Prof. Massimo Guidolin 20192 Financial Econometrics Spring/Winter 2017 Overview Motivation:
More informationMultivariate forecasting with VAR models
Multivariate forecasting with VAR models Franz Eigner University of Vienna UK Econometric Forecasting Prof. Robert Kunst 16th June 2009 Overview Vector autoregressive model univariate forecasting multivariate
More informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 IX. Vector Time Series Models VARMA Models A. 1. Motivation: The vector
More informationSTAT 151A: Lab 1. 1 Logistics. 2 Reference. 3 Playing with R: graphics and lm() 4 Random vectors. Billy Fang. 2 September 2017
STAT 151A: Lab 1 Billy Fang 2 September 2017 1 Logistics Billy Fang (blfang@berkeley.edu) Office hours: Monday 9am-11am, Wednesday 10am-12pm, Evans 428 (room changes will be written on the chalkboard)
More informationEcon 424 Time Series Concepts
Econ 424 Time Series Concepts Eric Zivot January 20 2015 Time Series Processes Stochastic (Random) Process { 1 2 +1 } = { } = sequence of random variables indexed by time Observed time series of length
More informationA TIME SERIES PARADOX: UNIT ROOT TESTS PERFORM POORLY WHEN DATA ARE COINTEGRATED
A TIME SERIES PARADOX: UNIT ROOT TESTS PERFORM POORLY WHEN DATA ARE COINTEGRATED by W. Robert Reed Department of Economics and Finance University of Canterbury, New Zealand Email: bob.reed@canterbury.ac.nz
More informationLecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem
Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Defining cointegration Vector
More informationStable Process. 2. Multivariate Stable Distributions. July, 2006
Stable Process 2. Multivariate Stable Distributions July, 2006 1. Stable random vectors. 2. Characteristic functions. 3. Strictly stable and symmetric stable random vectors. 4. Sub-Gaussian random vectors.
More informationConvolution Based Unit Root Processes: a Simulation Approach
International Journal of Statistics and Probability; Vol., No. 6; November 26 ISSN 927-732 E-ISSN 927-74 Published by Canadian Center of Science and Education Convolution Based Unit Root Processes: a Simulation
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationGeneralized Impulse Response Analysis: General or Extreme?
MPRA Munich Personal RePEc Archive Generalized Impulse Response Analysis: General or Extreme? Kim Hyeongwoo Auburn University April 2009 Online at http://mpra.ub.uni-muenchen.de/17014/ MPRA Paper No. 17014,
More information7. Integrated Processes
7. Integrated Processes Up to now: Analysis of stationary processes (stationary ARMA(p, q) processes) Problem: Many economic time series exhibit non-stationary patterns over time 226 Example: We consider
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More informationSTA441: Spring Multiple Regression. This slide show is a free open source document. See the last slide for copyright information.
STA441: Spring 2018 Multiple Regression This slide show is a free open source document. See the last slide for copyright information. 1 Least Squares Plane 2 Statistical MODEL There are p-1 explanatory
More information7. Integrated Processes
7. Integrated Processes Up to now: Analysis of stationary processes (stationary ARMA(p, q) processes) Problem: Many economic time series exhibit non-stationary patterns over time 226 Example: We consider
More information[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More informationOrthogonal Projection and Least Squares Prof. Philip Pennance 1 -Version: December 12, 2016
Orthogonal Projection and Least Squares Prof. Philip Pennance 1 -Version: December 12, 2016 1. Let V be a vector space. A linear transformation P : V V is called a projection if it is idempotent. That
More informationECON 4160, Lecture 11 and 12
ECON 4160, 2016. Lecture 11 and 12 Co-integration Ragnar Nymoen Department of Economics 9 November 2017 1 / 43 Introduction I So far we have considered: Stationary VAR ( no unit roots ) Standard inference
More information1 Regression with Time Series Variables
1 Regression with Time Series Variables With time series regression, Y might not only depend on X, but also lags of Y and lags of X Autoregressive Distributed lag (or ADL(p; q)) model has these features:
More informationMultivariate Time Series
Multivariate Time Series Notation: I do not use boldface (or anything else) to distinguish vectors from scalars. Tsay (and many other writers) do. I denote a multivariate stochastic process in the form
More information10) Time series econometrics
30C00200 Econometrics 10) Time series econometrics Timo Kuosmanen Professor, Ph.D. 1 Topics today Static vs. dynamic time series model Suprious regression Stationary and nonstationary time series Unit
More informationStudies in Nonlinear Dynamics & Econometrics
Studies in Nonlinear Dynamics & Econometrics Volume 9, Issue 2 2005 Article 4 A Note on the Hiemstra-Jones Test for Granger Non-causality Cees Diks Valentyn Panchenko University of Amsterdam, C.G.H.Diks@uva.nl
More informationBootstrapping the Grainger Causality Test With Integrated Data
Bootstrapping the Grainger Causality Test With Integrated Data Richard Ti n University of Reading July 26, 2006 Abstract A Monte-carlo experiment is conducted to investigate the small sample performance
More informationLakehead University ECON 4117/5111 Mathematical Economics Fall 2003
Test 1 September 26, 2003 1. Construct a truth table to prove each of the following tautologies (p, q, r are statements and c is a contradiction): (a) [p (q r)] [(p q) r] (b) (p q) [(p q) c] 2. Answer
More informationGeneralized Impulse Response Analysis: General or Extreme?
Auburn University Department of Economics Working Paper Series Generalized Impulse Response Analysis: General or Extreme? Hyeongwoo Kim Auburn University AUWP 2012-04 This paper can be downloaded without
More informationA Simple Computational Approach to the Fundamental Theorem of Asset Pricing
Applied Mathematical Sciences, Vol. 6, 2012, no. 72, 3555-3562 A Simple Computational Approach to the Fundamental Theorem of Asset Pricing Cherng-tiao Perng Department of Mathematics Norfolk State University
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationMore Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order Restriction
Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 4, pp. 700-716 c 2007, Indian Statistical Institute More Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order
More informationCointegration, Stationarity and Error Correction Models.
Cointegration, Stationarity and Error Correction Models. STATIONARITY Wold s decomposition theorem states that a stationary time series process with no deterministic components has an infinite moving average
More informationDepartamento de Estadfstica y Econometrfa Statistics and Econometrics Series 27. Universidad Carlos III de Madrid December 1993 Calle Madrid, 126
-------_._-- Working Paper 93-45 Departamento de Estadfstica y Econometrfa Statistics and Econometrics Series 27 Universidad Carlos III de Madrid December 1993 Calle Madrid, 126 28903 Getafe (Spain) Fax
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationVector spaces. DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis.
Vector spaces DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Vector space Consists of: A set V A scalar
More informationEconometrics. Week 11. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 11 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 30 Recommended Reading For the today Advanced Time Series Topics Selected topics
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationComment On The Interpretation of Instrumental Variables in the Presence of Specification Errors: A Causal Comment
econometrics Comment On The Interpretation of Instrumental Variables in the Presence of Specification Errors: A Causal Comment Burkhard Raunig Economic Studies Division, Central Bank of Austria, Otto-Wagner-Plat
More informationConsistent Bivariate Distribution
A Characterization of the Normal Conditional Distributions MATSUNO 79 Therefore, the function ( ) = G( : a/(1 b2)) = N(0, a/(1 b2)) is a solu- tion for the integral equation (10). The constant times of
More informationThe distribution of a sum of dependent risks: a geometric-combinatorial approach
The distribution of a sum of dependent risks: a geometric-combinatorial approach Marcello Galeotti 1, Emanuele Vannucci 2 1 University of Florence, marcello.galeotti@dmd.unifi.it 2 University of Pisa,
More informationMultivariate Time Series: VAR(p) Processes and Models
Multivariate Time Series: VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t 1 + + Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with
More information9) Time series econometrics
30C00200 Econometrics 9) Time series econometrics Timo Kuosmanen Professor Management Science http://nomepre.net/index.php/timokuosmanen 1 Macroeconomic data: GDP Inflation rate Examples of time series
More informationWhen Do Wold Orderings and Long-Run Recursive Identifying Restrictions Yield Identical Results?
Preliminary and incomplete When Do Wold Orderings and Long-Run Recursive Identifying Restrictions Yield Identical Results? John W Keating * University of Kansas Department of Economics 334 Snow Hall Lawrence,
More informationUnit Roots in White Noise?!
Unit Roots in White Noise?! A.Onatski and H. Uhlig September 26, 2008 Abstract We show that the empirical distribution of the roots of the vector auto-regression of order n fitted to T observations of
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More information13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process. Strict Exogeneity
Outline: Further Issues in Using OLS with Time Series Data 13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process I. Stationary and Weakly Dependent Time Series III. Highly Persistent
More informationCointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Financial Econometrics / 56
Cointegrated VAR s Eduardo Rossi University of Pavia November 2013 Rossi Cointegrated VAR s Financial Econometrics - 2013 1 / 56 VAR y t = (y 1t,..., y nt ) is (n 1) vector. y t VAR(p): Φ(L)y t = ɛ t The
More informationIt is easily seen that in general a linear combination of y t and x t is I(1). However, in particular cases, it can be I(0), i.e. stationary.
6. COINTEGRATION 1 1 Cointegration 1.1 Definitions I(1) variables. z t = (y t x t ) is I(1) (integrated of order 1) if it is not stationary but its first difference z t is stationary. It is easily seen
More information1 Linear Difference Equations
ARMA Handout Jialin Yu 1 Linear Difference Equations First order systems Let {ε t } t=1 denote an input sequence and {y t} t=1 sequence generated by denote an output y t = φy t 1 + ε t t = 1, 2,... with
More informationStock Prices, News, and Economic Fluctuations: Comment
Stock Prices, News, and Economic Fluctuations: Comment André Kurmann Federal Reserve Board Elmar Mertens Federal Reserve Board Online Appendix November 7, 213 Abstract This web appendix provides some more
More informationInner Product, Length, and Orthogonality
Inner Product, Length, and Orthogonality Linear Algebra MATH 2076 Linear Algebra,, Chapter 6, Section 1 1 / 13 Algebraic Definition for Dot Product u 1 v 1 u 2 Let u =., v = v 2. be vectors in Rn. The
More informationUniversity of Pretoria Department of Economics Working Paper Series
University of Pretoria Department of Economics Working Paper Series Predicting Stock Returns and Volatility Using Consumption-Aggregate Wealth Ratios: A Nonlinear Approach Stelios Bekiros IPAG Business
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2
MA 575 Linear Models: Cedric E Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2 1 Revision: Probability Theory 11 Random Variables A real-valued random variable is
More informationCointegration between Trends and Their Estimators in State Space Models and Cointegrated Vector Autoregressive Models
econometrics Article Cointegration between Trends and Their Estimators in State Space Models and Cointegrated Vector Autoregressive Models Søren Johansen * and Morten Nyboe Tabor Department of Economics,
More informationNotes on Time Series Modeling
Notes on Time Series Modeling Garey Ramey University of California, San Diego January 17 1 Stationary processes De nition A stochastic process is any set of random variables y t indexed by t T : fy t g
More informationMultivariate Least Weighted Squares (MLWS)
() Stochastic Modelling in Economics and Finance 2 Supervisor : Prof. RNDr. Jan Ámos Víšek, CSc. Petr Jonáš 23 rd February 2015 Contents 1 2 3 4 5 6 7 1 1 Introduction 2 Algorithm 3 Proof of consistency
More informationINVARIANT SUBSPACES FOR CERTAIN FINITE-RANK PERTURBATIONS OF DIAGONAL OPERATORS. Quanlei Fang and Jingbo Xia
INVARIANT SUBSPACES FOR CERTAIN FINITE-RANK PERTURBATIONS OF DIAGONAL OPERATORS Quanlei Fang and Jingbo Xia Abstract. Suppose that {e k } is an orthonormal basis for a separable, infinite-dimensional Hilbert
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationHilbert Spaces: Infinite-Dimensional Vector Spaces
Hilbert Spaces: Infinite-Dimensional Vector Spaces PHYS 500 - Southern Illinois University October 27, 2016 PHYS 500 - Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October
More information1 Quantitative Techniques in Practice
1 Quantitative Techniques in Practice 1.1 Lecture 2: Stationarity, spurious regression, etc. 1.1.1 Overview In the rst part we shall look at some issues in time series economics. In the second part we
More informationExogeneity and Causality
Università di Pavia Exogeneity and Causality Eduardo Rossi University of Pavia Factorization of the density DGP: D t (x t χ t 1, d t ; Ψ) x t represent all the variables in the economy. The econometric
More informationECON 4160, Spring term Lecture 12
ECON 4160, Spring term 2013. Lecture 12 Non-stationarity and co-integration 2/2 Ragnar Nymoen Department of Economics 13 Nov 2013 1 / 53 Introduction I So far we have considered: Stationary VAR, with deterministic
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More informationProblem set 1 - Solutions
EMPIRICAL FINANCE AND FINANCIAL ECONOMETRICS - MODULE (8448) Problem set 1 - Solutions Exercise 1 -Solutions 1. The correct answer is (a). In fact, the process generating daily prices is usually assumed
More informationCointegrated VARIMA models: specification and. simulation
Cointegrated VARIMA models: specification and simulation José L. Gallego and Carlos Díaz Universidad de Cantabria. Abstract In this note we show how specify cointegrated vector autoregressive-moving average
More informationAPPLIED TIME SERIES ECONOMETRICS
APPLIED TIME SERIES ECONOMETRICS Edited by HELMUT LÜTKEPOHL European University Institute, Florence MARKUS KRÄTZIG Humboldt University, Berlin CAMBRIDGE UNIVERSITY PRESS Contents Preface Notation and Abbreviations
More informationThe autocorrelation and autocovariance functions - helpful tools in the modelling problem
The autocorrelation and autocovariance functions - helpful tools in the modelling problem J. Nowicka-Zagrajek A. Wy lomańska Institute of Mathematics and Computer Science Wroc law University of Technology,
More informationFinancial Econometrics
Financial Econometrics Multivariate Time Series Analysis: VAR Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) VAR 01/13 1 / 25 Structural equations Suppose have simultaneous system for supply
More information(I AL BL 2 )z t = (I CL)ζ t, where
ECO 513 Fall 2011 MIDTERM EXAM The exam lasts 90 minutes. Answer all three questions. (1 Consider this model: x t = 1.2x t 1.62x t 2 +.2y t 1.23y t 2 + ε t.7ε t 1.9ν t 1 (1 [ εt y t = 1.4y t 1.62y t 2
More informationProf. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis
Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More information7 Introduction to Time Series Time Series vs. Cross-Sectional Data Detrending Time Series... 15
Econ 495 - Econometric Review 1 Contents 7 Introduction to Time Series 3 7.1 Time Series vs. Cross-Sectional Data............ 3 7.2 Detrending Time Series................... 15 7.3 Types of Stochastic
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More information9. Multivariate Linear Time Series (II). MA6622, Ernesto Mordecki, CityU, HK, 2006.
9. Multivariate Linear Time Series (II). MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Introduction to Time Series and Forecasting. P.J. Brockwell and R. A. Davis, Springer Texts
More informationLecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem
Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Stochastic vs. deterministic
More informationVolume 30, Issue 1. The relationship between the F-test and the Schwarz criterion: Implications for Granger-causality tests
Volume 30, Issue 1 The relationship between the F-test and the Schwarz criterion: Implications for Granger-causality tests Erdal Atukeren ETH Zurich - KOF Swiss Economic Institute Abstract In applied research,
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationFinal Exam November 24, Problem-1: Consider random walk with drift plus a linear time trend: ( t
Problem-1: Consider random walk with drift plus a linear time trend: y t = c + y t 1 + δ t + ϵ t, (1) where {ϵ t } is white noise with E[ϵ 2 t ] = σ 2 >, and y is a non-stochastic initial value. (a) Show
More informationThe regression model with one stochastic regressor (part II)
The regression model with one stochastic regressor (part II) 3150/4150 Lecture 7 Ragnar Nymoen 6 Feb 2012 We will finish Lecture topic 4: The regression model with stochastic regressor We will first look
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationSubset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models
econometrics Article Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models Richard A Ashley 1 and Xiaojin Sun 2, * 1 Department of Economics, Virginia Tech, Blacksburg, VA 24060, USA;
More informationACE 562 Fall Lecture 2: Probability, Random Variables and Distributions. by Professor Scott H. Irwin
ACE 562 Fall 2005 Lecture 2: Probability, Random Variables and Distributions Required Readings: by Professor Scott H. Irwin Griffiths, Hill and Judge. Some Basic Ideas: Statistical Concepts for Economists,
More informationTitle. Description. var intro Introduction to vector autoregressive models
Title var intro Introduction to vector autoregressive models Description Stata has a suite of commands for fitting, forecasting, interpreting, and performing inference on vector autoregressive (VAR) models
More informationGeometric interpretation of signals: background
Geometric interpretation of signals: background David G. Messerschmitt Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-006-9 http://www.eecs.berkeley.edu/pubs/techrpts/006/eecs-006-9.html
More informationLecture 3: Multiple Regression
Lecture 3: Multiple Regression R.G. Pierse 1 The General Linear Model Suppose that we have k explanatory variables Y i = β 1 + β X i + β 3 X 3i + + β k X ki + u i, i = 1,, n (1.1) or Y i = β j X ji + u
More informationSimultaneous Equation Models Learning Objectives Introduction Introduction (2) Introduction (3) Solving the Model structural equations
Simultaneous Equation Models. Introduction: basic definitions 2. Consequences of ignoring simultaneity 3. The identification problem 4. Estimation of simultaneous equation models 5. Example: IS LM model
More informationCh. 15 Forecasting. 1.1 Forecasts Based on Conditional Expectations
Ch 15 Forecasting Having considered in Chapter 14 some of the properties of ARMA models, we now show how they may be used to forecast future values of an observed time series For the present we proceed
More informationOn Least Squares Linear Regression Without Second Moment
On Least Squares Linear Regression Without Second Moment BY RAJESHWARI MAJUMDAR University of Connecticut If \ and ] are real valued random variables such that the first moments of \, ], and \] exist and
More informationDEPARTMENT OF ECONOMICS
ISSN 0819-64 ISBN 0 7340 616 1 THE UNIVERSITY OF MELBOURNE DEPARTMENT OF ECONOMICS RESEARCH PAPER NUMBER 959 FEBRUARY 006 TESTING FOR RATE-DEPENDENCE AND ASYMMETRY IN INFLATION UNCERTAINTY: EVIDENCE FROM
More informationThe Game of Normal Numbers
The Game of Normal Numbers Ehud Lehrer September 4, 2003 Abstract We introduce a two-player game where at each period one player, say, Player 2, chooses a distribution and the other player, Player 1, a
More information