UNIVERSITY OF CALIFORNIA, SAN DIEGO DEPARTMENT OF ECONOMICS
|
|
- Isaac Daniel
- 5 years ago
- Views:
Transcription
1 2-7 UNIVERSITY OF LIFORNI, SN DIEGO DEPRTMENT OF EONOMIS THE JOHNSEN-GRNGER REPRESENTTION THEOREM: N EXPLIIT EXPRESSION FOR I() PROESSES Y PETER REINHRD HNSEN DISUSSION PPER 2-7 JULY 2
2 The Johansen-Granger Representation Theorem: n Explicit Expression for I() Processes Peter Reinhard Hansen y U San Diego and rown University reinharducsdedu Phone: (4) July, 2 bstract The Johansen-Granger representation theorem for the cointegrated vector autoregressive process is derived using the companion form This approach yields an explicit representation of all coe cients and initial values This result is useful for impulse response analysis, common feature analysis and asymptotic analysis of cointegrated processes y I thank Graham Elliott, James D Hamilton, Hans hristian Kongsted, nders Rahbek, and Halbert White for valuable comments ll errors remain my responsibility Financial support from the Danish Social Science Research ouncil and the Danish Research cademy is gratefully acknowledged
3 Introduction The Johansen-Granger representation theorem states that a vector autoregressive process (L)X t = " t, integrated of order one, has the representation X t = P t i= " i + (L)" t + ; where f(l)" t g is stationary if f" t g is stationary and where depends on initial values (X ;X ;:::); (see Johansen (99, 996)) Johansen s result gives explicit values of whereas the coe cients of the lag polynomial, (L); and the initial value, ; are given implicitly This representation of cointegrated processes is known as the Granger representation and is synonymous with the Wold representation for stationary processes ecause the representation divides X t into a random walk and a stationary process, it can be viewed as multivariate everidge- Nelson decomposition where the labels are permanent and transitory components, (see everidge and Nelson (98)) The Granger representation is valuable in the asymptotic analysis of cointegrated processes, where typically only an explicit expression for is needed Explicit values for the coe cients in (L) are useful in common feature analysis, (see Engle and Kozicki (993)), and in impulse response analysis, (see Lütkepohl and Reimers (992), Warne (993), and Lütkepohl and Saikkonen (997)), where the coe cients of (L) are interpreted as the transitory e ects of the shocks " t Similarly, in asymptotic analysis of the model with structural breaks, see Hansen (2), it is valuable to have an explicit value for : In this paper, explicit values of coe cients as well as initial values are found using the companion form, making use of the algebraic structure that characterizes this model From Johansen (996) we adopt the following de nitions: for an m n matrix a with full column rank n, we de ne ¹a = a(a a) and let the orthogonal complement of a; be the full rank m (m n) matrix a? that has a? a =: In Section 2 the explicit representation is derived In Section 3 we consider deterministic aspects of the representation Section 4 contains concluding remarks and the appendix contains relevant algebra The original Granger representation theorem, given by Engle and Granger (987), asserts the existence of an error correction representation of X t; under the assumptions that X t and X t have stationary and invertible VRM representations, for some matrix : The Johansen-Granger representation theorem, of Johansen (99, 996), makes assumptions on the autoregressive parameters, that precisely characterizes I() processes, and states results on the moving average representation of X t: 2
4 2 The Granger Representation for utoregressive Processes Integrated of Order One We consider the p-dimensional vector autoregressive process of order k X t = X t + 2 X t k X t k + D t + " t ; t =;:::;T; where the process deterministic terms are contained in D t and where " t, t =;:::;T is a sequence of independent identically distributed stochastic variables with mean zero 2 The process can be re-written in error correction form: k X X t = X t + i X t i + D t + " t ; t =;:::;T i= where = I + P k i= i and i = P k j=i+ j: The conditions that ensure that X t is integrated of order one, referred to as X t being I(), are stated in the following assumption: ssumption 2 The assumptions of the Johansen-Granger representation theorem are: (i) The roots of the characteristic polynomial det((z)) = det(i z 2 z 2 k z k ) are either outside the unit circle or equal to one (ii) The matrix has reduced rank r<p;and can therefore be expressed as the product = where and are p r matrices of full column rank r: (iii) The matrix?? has full rank, where =I P k i= i and where? and? are the orthogonal complements to and : The rst assumption ensures that the process is not explosive (roots in the unit circle) or seasonally cointegrated (roots on the boundary of the unit circle di erent from z = ), (see Hylleberg, Engle, Granger, and Yoo (99) or Johansen and Schaumburg (998)) The second ensures that there are at least p r unit roots and induces cointegration whenever r : The 2 The Granger representation is not relying on the assumptions on " t; since it is entirely an algebraic derivation However the iid assumption is important for some of the interpretations of the representation 3
5 third assumption restricts the process from being I(2); because (iii) together with (ii) ensures that the number of unit roots is exactly p r: Under these assumptions, Johansen (99) showed that X t has the representation X t = P t i= (" i + D i )+(L)(" t + D t )+; where =? (??)?: y using the companion form of the process, it is possible to obtain explicit values for the coe cients of the lag polynomial (L) = + L + 2 L 2 + ; and the initial values contained in ; as I show below The following lemma will be useful Lemma 22 Let a and b be m n matrices, m n with full column rank n; and let a? and b? be their orthogonal complements, respectively The following ve statements are equivalent (i) The matrix (I + b a) does not have as an eigenvalue (ii) Let v be a vector in R m : Then (b a)v =implies v =: (iii) The matrix b a has full rank (iv) The m m matrix (b; a? ) has full rank (v) The matrix b? a? has full rank Proof The equivalence of (i), (ii) and (iii) is straightforward, and the identity j(a; a? )jj(b; a? )j = j(a; a? ) (b; a? )j = j a b j = ja bjja?a? j a? b a? a? proves that (iii) holds if and only if (iv) holds Finally, the identity j(b; b? )jj(b; a? )j = j(b; b? ) (b; a? )j = j b b j = jb bjjb? a?j b? b b? a? completes the proof 2 The ompanion Form We transform the process into the companion form, by de ning X t = Xt ;X t ;:::;X t k+ 4
6 so that with suitable de nitions X t = X t + t + " t = X t + t + " t ; which converts the process to a vector autoregressive process of order one The needed de nitions are = + 2 k k 2 k I I I I I ; = k I I ; = I I I I I ; " t = " t ; t = D t : It is easily veri ed that the orthogonal complements of and are given by? =?? k? ;? =?? : Lemma 23 Let ; ; and be de ned as above, and assume that ssumption 2 holds Then the eigenvalues of the matrix (I + ) are all less than one in absolute value 5
7 Proof y ssumption 2 (iii); the identity?? =?(I k )? shows that?? has full rank, and by Lemma 22, we have that is not an eigenvalue of (I + ) However we need to show that the eigenvalues are smaller than one in absolute value Therefore consider an eigenvector v =(v ;:::;v k ) 6=of (I + ); eg (I + )v = v: The upper r + p rows of (I + )v yields v + ( v + v k v k ) = v ( v + v k v k ) = v 2 which implies v 2 =( )v ; and the remaining part implies v 2 = v 3 = = k 2 v k : The case =clearly ful lls j j < so assume 6= : Multiply the second set of equations by ( )= k and substitute z == to obtain [I( z) z ( z)z k ( z)z k ]v k =: This is equivalent to I( z) z k X i ( z)z i =; i= and since ssumption 2 has jzj > we conclude that j j < : The result has the implication that under ssumption 2 the sum P i= ( + ) i is convergent with limit ( ), such that a process de ned by Y t = P i= ( + ) i u t i is stationary whenever u t is stationary Lemma 24 With the de nitions above we have the identities: (I ) = (I ) ¹ I = (I ) ¹ + ( I)+ ¹?? : Proof Since I = ( ) +?(??)? = ¹ +?¹?; the rst identity follows from (I ) = (I )( ¹ +?¹? ) 6
8 = (I ) ¹ +?¹?? (??)?? ¹? = (I ) ¹ ; and the second follows by applying the rst identity and that = ¹?? : We are now ready to formulate the main result Theorem 25 (The Johansen-Granger representation theorem) Let a process be given by the equation k X X t = X t + i X t i + D t + " t ; i= and assume that ssumption 2 holds Then the process has the representation tx X t = (" i + D i )+(L)(" t + D t )+(X X k X k+ ) i= where =? (??)? and where the coe cients of (L) are given by i = GQ i E ;2 where G = ((I ); ;:::; k ) Q = I + k 2 k k 2 k I I E ;2 = (I p ;I p ; ; ; ) : Proof Under ssumption 2 the pk pk matrix ( ;?) has full rank We can therefore obtain the Granger representation for X t by nding the moving average representation for the processes X t and? X t individually and then stacking them and multiplying by ( ;? ) : First, consider the process X t =(I + ) X t + (" t + t ): 7
9 Since all the eigenvalues of (I + ); according to Lemma 23, are smaller than one in absolute value, the process has the stationary representation X t = (L)(" t + t ) where i =(I+ ) i, and where by stationary we mean that X t E X t is stationary Next consider the random walk? X t =? X t +? (" t + t ) tx =?X + i=?(" i + i ): representationforx t is now obtained as X t =( ;?) (L)(" t + t ) P t i=? (" i + i )+? X : Theentirematrix( ;? ) is given in the ppendix, for our purposes we only need its upper p rows that de ne the equation for X t Theserowsaregivenby (I ) ¹ ; s ;:::; s k ;¹? with the de nition s i = i + + k : For simplicity, we de ne F = (I ) ¹ ; s ;:::; s k and obtain the representation for X t : X t = (F; ¹? ) (L)(" t + t ) P t i=? (" i + i )+? X = F (L)(" t + t )+ ¹? tx = (L)(" t + D t )+ i=?(" i + i )+ ¹??X tx (" i + D i )+; i= 8
10 where the initial value is explicitly given by = ¹??X = (X X k X (k ) ); and the coe cients of the polynomial (L) are given by i = F (I + ) i E = FD i E ;2 ; with the additional de nitions D = (I + ) = I ;E = I p and E ;2 = I p I p : ecause (I + ) = (I + ) we have that D = Q where Q is as given in the theorem Thus, the coe cients can be written as i = FD i E ;2 = F Q i E = GQ i E ;2 where G = F = (I ); s ;:::; s k ; where we applied the identity (I ) ¹ =(I ) of Lemma 24 This completes the proof orollary 26 The coe cients of (L) can be obtained recursively from the formula ix i = i + ( + j ) i j ; i =; 2;:::; j= 9
11 where = I and = I and where we set j =for j k: Proof From the proof of the Johansen-Granger representation theorem we have that i = GQ i E ;2 So by de ning i = ;i 2;i k;i = Q i = Q i E ;2 ; tedious algebra (given in the ppendix) leads to the relation i = i + 2;i ; = ; i =; 2;::: and the structure of Q yields the equation ix 2;i = ( + j ) 2;i j ; 2; = I i =; 2;:::: j= y inserting 2;i j = i j i j we nd the equation of the corollary one s a special case we formulate the representation for the vector autoregressive process of order orollary 27 Let X t = X t + " t be a process ful lling ssumption 2 Then we have the representation tx X X t = " i +( ) i I + "t i + X i= i= where =? (?? )? : The result of orollary 27 is derived directly in Johansen (996) by dividing the process into its stationary and non-stationary part with the identity I = +? (??)?: The proof of Theorem 25 made use of the more general identity I =(I ) ¹ + ( I)+ ¹?? of Lemma 24, which simpli es to the identity in Johansen (996) when =I; as is the case for a VR() process 3 Deterministic Terms In this section we study the stationary polynomial s role for the deterministic term The deterministic part plays an important role for the asymptotic analysis of this model, because the limits
12 of some test statistics depend on the deterministic term The literature has developed a notation for models with di erent deterministic terms which we shall adopt First we analyze the model H : This model contains only a constant D t = ¹ ; which in general will give rise to a linear trend in the process X t Next, we also analyze its sub-model H ; which has the deterministic term D t = ½ This is equivalent to the restriction on the constant ¹ =; which is precisely what is needed for X t not to have a linear trend We also analyze the models H and H : Model H has a linear deterministic trend D t = ¹ + ¹ t; which gives rise to a quadratic trend in the process X t, and the sub-model H, has the deterministic trend restricted to D t = ¹ + ½ t, which prevents the X t from having a quadratic trend 3 The Models H and H When the deterministic term is simply a constant ¹ = D t ; the Granger representation is given by tx X t = " t i + (L)" t + ()¹ + ¹ t + : i= So unless ¹ =; the constant ¹ leads to a deterministic linear trend in the process X t The matrix () is calculated in the appendix and is found to be () = (I ) ¹ ¹ (I ) = ª; Ã k! X i i i= where =(I ) ¹ ; =¹ ( I) and ª= P k i= i i = P k P k i= j=i j: This result encompasses two ndings from Hansen and Johansen (998) The rst is that E( X t )= ()¹ =¹ ( I)¹ ; and the second is that in H ; where ¹ = ½ ; the linear trend vanishes while the constant in the processisgivenby()¹ = (I ) ¹ ½:
13 32 Models H and H When the deterministic term contains a linear trend, D t = ¹ + ¹ t; the deterministic part of the Granger representation is given by 2 ¹ t 2 + ¹ + 2 ¹ t + (L)(¹ + ¹ t) ; (see Hansen and Johansen (998)) This can be re-written as 2 ¹ t 2 + ¹ + Ã! 2 + () X ¹ t + ()¹ i i ¹ : (3) So unless? ¹ =the linear trend ¹ leads to a quadratic deterministic trend in the process X t : The only term of (3) not derived previously, is P i= i i: This term is derived in the appendix and is given by i= + ª + ª ª ªª k(k ) ª: 2 In model H where the linear trend is restricted to ¹ = ½ ; (3) reduces to ¹ (I ) ¹ ½ t + ()¹ + ( I) ¹ () ½ which encompasses a result from Johansen (996, equation 52), because the expression for ; in Johansen (996, equation 52), equals ¹ (I ) ¹ ½ 4 onclusion We gave an explicit expression of the moving average representation for processes integrated of order one using the companion form for the process The explicit expression is useful to have in studies of impulse response functions and in common features analysis s a side bene t the approach gives a new proof of the Johansen-Granger representation theorem, a proof that some might nd more intuitive and easy to follow than previous proofs 2
14 ppendix : Proofs The Inverse of ( ;? ) In the proof of the Johansen-Granger representation theorem we need an explicit expression for the rst p rows of ( ;? ) : The entire matrix is given by ( ;?) = (I ) ¹ s s k ¹? (I ) ¹ s I s k ¹? (I ) ¹ s s k ¹? (I ) ¹ s s k ¹? (I ) ¹ s s k I ¹? ; which is veri ed by multiplying it by ( ;? ) and using the identity (I ) ¹ =(I ) from Lemma 24 2 The Expression for i In orollary 26 we asserted the relation i = i + 2;i ;i=; 2;:::: This relation is proved as follows First, notice from the equation for i given by i = ;i 2;i k;i = I + k 2 k k 2 k I I i ; = E ;2 that k;i = 2;i k+2 k 2; and that ;i = ;i + 2;i,why ;i = P i j= 2;j: So that ix 2;i = ( + j ) 2;i j ; 2; = I i =; 2;:::; j= and note that 2;i = 2;i + + k k;i : Next consider i = G i =(I ) ;i s 2;i s k k;i 3
15 = (I ) ( 2;i + ;i )+( I) 2;i s 2 3;i s k k;i = (I ) 2;i +(I ) ;i s 2 3;i s k k;i = (I ) 2;i +(I ) ;i ( s ) 2;i ( s k 2 k 2) k ;i ( k k ) k;i = G i + 2;i = i + 2;i : Which completes the proof 3 n Expression for () In the analysis of the deterministic terms we need to calculate X () = F I + i E = F E : i= The inverse of is given by = 2 k 2 k I 2 k 2 k I I I I I ; = ¹ (I ) ¹ ¹ (I ) s ¹ (I ) s k (I ) ¹ s I s k (I ) ¹ s I s k I : So 4
16 = ¹ (I ) ¹ (I ) s ¹ (I ) s k s + I s k s s k + I and therefore we nd E = ¹ (I ) ; () and nally that () = ¹ (I ) (I ) ¹ ; s ;:::; s k = (I ) ¹ ¹ k X Xk ( I) j : = ª i= j=i where =(I ) ¹ ; =¹ ( I) and ª= P k P k i= j=i j = P k i= i i: 4 n Expression for P i= i i Inthecasewherethedeterministictermisgivenby D t = ¹ + ¹ t we make use of X i= I + i i = ³ 2 + : 5
17 Thesecondtermiscalculatedinthecasewith D t = ¹, and the term we need to add is given by 2 E = ª + ª + + ª +(k ) thus F 2 E = ª ª ªª k(k ) ª 2 so that X ³ i i = F + 2 i= E = + ª + ª ª ªª k(k ) ª: 2 References everidge, S, and R Nelson (98): New pproach to Decompositions of Time Series Into Permanent and Transitory omponents with Particular ttentions to Measurement of the usiness ycle, Journal of Monetary Economics, 7, 5 74 Engle, R F, and W J Granger (987): o-integration and Error orrection: Representation, Estimation and Testing, Econometrica, 55, Engle, R F, and S Kozicki (993): Testing for ommon Features, Journal of usiness and Economic Statistics,, Hansen, P, and S Johansen (998): Workbook on ointegration Oxford University Press, Oxford Hansen, P R (2): Structural hanges in ointegrated Processes, PhD thesis, University of alifornia at San Diego Hylleberg, S, R F Engle, W J Granger, and S Yoo (99): Seasonal Integration and ointegration, Journal of Econometrics, 44, Johansen, S (99): Estimation and Hypothesis Testing of ointegration Vectors in Gaussian Vector utoregressive Models, Econometrica, 59, (996): Likelihood ased Inference in ointegrated Vector utoregressive Models Oxford University Press, Oxford, 2nd edn 6
18 Johansen, S, and E Schaumburg (998): Likelihood nalysis of Seasonal ointegration, Journal of Econometrics, 88, Lütkepohl, H, and H-E Reimers (992): Impulse Response nalysis of ointegrated Systems, Journal of Enonomic Dynamics and ontrol, 6, Lütkepohl, H, and P Saikkonen (997): Impulse Response nalysis in In nite Order ointegrated Vector utoregressive Processes, Journal of Econometrics, 8, Warne, (993): ommon Trends Model: Identi cation, Estimation and Inference, Seminar Paper No 555, IIES, Stockholm University 7
Representation of cointegrated autoregressive processes 1 Representation of cointegrated autoregressive processes with application
Representation of cointegrated autoregressive processes Representation of cointegrated autoregressive processes with application to fractional processes Søren Johansen Institute of Mathematical Sciences
More informationCointegration Lecture I: Introduction
1 Cointegration Lecture I: Introduction Julia Giese Nuffield College julia.giese@economics.ox.ac.uk Hilary Term 2008 2 Outline Introduction Estimation of unrestricted VAR Non-stationarity Deterministic
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 information(Y jz) t (XjZ) 0 t = S yx S yz S 1. S yx:z = T 1. etc. 2. Next solve the eigenvalue problem. js xx:z S xy:z S 1
Abstract Reduced Rank Regression The reduced rank regression model is a multivariate regression model with a coe cient matrix with reduced rank. The reduced rank regression algorithm is an estimation procedure,
More informationTime Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley
Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the
More informationGMM-based inference in the AR(1) panel data model for parameter values where local identi cation fails
GMM-based inference in the AR() panel data model for parameter values where local identi cation fails Edith Madsen entre for Applied Microeconometrics (AM) Department of Economics, University of openhagen,
More informationby Søren Johansen Department of Economics, University of Copenhagen and CREATES, Aarhus University
1 THE COINTEGRATED VECTOR AUTOREGRESSIVE MODEL WITH AN APPLICATION TO THE ANALYSIS OF SEA LEVEL AND TEMPERATURE by Søren Johansen Department of Economics, University of Copenhagen and CREATES, Aarhus University
More informationCointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Fin. Econometrics / 31
Cointegrated VAR s Eduardo Rossi University of Pavia November 2014 Rossi Cointegrated VAR s Fin. Econometrics - 2014 1 / 31 B-N decomposition Give a scalar polynomial α(z) = α 0 + α 1 z +... + α p z p
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 informationFractional integration and cointegration: The representation theory suggested by S. Johansen
: The representation theory suggested by S. Johansen hhu@ssb.no people.ssb.no/hhu www.hungnes.net January 31, 2007 Univariate fractional process Consider the following univariate process d x t = ε t, (1)
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 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 informationEquivalence of several methods for decomposing time series into permananent and transitory components
Equivalence of several methods for decomposing time series into permananent and transitory components Don Harding Department of Economics and Finance LaTrobe University, Bundoora Victoria 3086 and Centre
More information1 First-order di erence equation
References Hamilton, J. D., 1994. Time Series Analysis. Princeton University Press. (Chapter 1,2) The task facing the modern time-series econometrician is to develop reasonably simple and intuitive models
More informationTests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables
Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables Josep Lluís Carrion-i-Silvestre University of Barcelona Dukpa Kim y Korea University May 4, 28 Abstract We consider
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationStochastic Trends & Economic Fluctuations
Stochastic Trends & Economic Fluctuations King, Plosser, Stock & Watson (AER, 1991) Cesar E. Tamayo Econ508 - Economics - Rutgers November 14, 2011 Cesar E. Tamayo Stochastic Trends & Economic Fluctuations
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 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 informationMarkov-Switching Models with Endogenous Explanatory Variables. Chang-Jin Kim 1
Markov-Switching Models with Endogenous Explanatory Variables by Chang-Jin Kim 1 Dept. of Economics, Korea University and Dept. of Economics, University of Washington First draft: August, 2002 This version:
More informationLECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT
MARCH 29, 26 LECTURE 2 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT (Davidson (2), Chapter 4; Phillips Lectures on Unit Roots, Cointegration and Nonstationarity; White (999), Chapter 7) Unit root processes
More informationPanel cointegration rank testing with cross-section dependence
Panel cointegration rank testing with cross-section dependence Josep Lluís Carrion-i-Silvestre y Laura Surdeanu z September 2007 Abstract In this paper we propose a test statistic to determine the cointegration
More informationChapter 2. GMM: Estimating Rational Expectations Models
Chapter 2. GMM: Estimating Rational Expectations Models Contents 1 Introduction 1 2 Step 1: Solve the model and obtain Euler equations 2 3 Step 2: Formulate moment restrictions 3 4 Step 3: Estimation and
More informationY t = ΦD t + Π 1 Y t Π p Y t p + ε t, D t = deterministic terms
VAR Models and Cointegration The Granger representation theorem links cointegration to error correction models. In a series of important papers and in a marvelous textbook, Soren Johansen firmly roots
More informationTesting for Regime Switching: A Comment
Testing for Regime Switching: A Comment Andrew V. Carter Department of Statistics University of California, Santa Barbara Douglas G. Steigerwald Department of Economics University of California Santa Barbara
More informationChapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models
Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example........................... 2.2 Discussion...................................
More informationNUMERICAL DISTRIBUTION FUNCTIONS OF LIKELIHOOD RATIO TESTS FOR COINTEGRATION
JOURNAL OF APPLIED ECONOMETRICS J. Appl. Econ. 14: 563±577 (1999) NUMERICAL DISTRIBUTION FUNCTIONS OF LIKELIHOOD RATIO TESTS FOR COINTEGRATION JAMES G. MACKINNON, a * ALFRED A. HAUG b AND LEO MICHELIS
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
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 informationMultivariate Time Series: Part 4
Multivariate Time Series: Part 4 Cointegration Gerald P. Dwyer Clemson University March 2016 Outline 1 Multivariate Time Series: Part 4 Cointegration Engle-Granger Test for Cointegration Johansen Test
More informationThe Role of "Leads" in the Dynamic Title of Cointegrating Regression Models. Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji
he Role of "Leads" in the Dynamic itle of Cointegrating Regression Models Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji Citation Issue 2006-12 Date ype echnical Report ext Version publisher URL http://hdl.handle.net/10086/13599
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 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 informationOn Econometric Analysis of Structural Systems with Permanent and Transitory Shocks and Exogenous Variables
On Econometric Analysis of Structural Systems with Permanent and Transitory Shocks and Exogenous Variables Adrian Pagan School of Economics and Finance, Queensland University of Technology M. Hashem Pesaran
More informationReferences. Press, New York
References Banerjee, Dolado J, Galbraith JW, Hendry DF (1993) Co-integration, errorcorrection and the econometric analysis of non stationary data. Oxford University Press, New York Bellman R (1970) Introduction
More informationA New Approach to Robust Inference in Cointegration
A New Approach to Robust Inference in Cointegration Sainan Jin Guanghua School of Management, Peking University Peter C. B. Phillips Cowles Foundation, Yale University, University of Auckland & University
More informationLinear Algebra. James Je Heon Kim
Linear lgebra James Je Heon Kim (jjk9columbia.edu) If you are unfamiliar with linear or matrix algebra, you will nd that it is very di erent from basic algebra or calculus. For the duration of this session,
More informationARIMA Models. Richard G. Pierse
ARIMA Models Richard G. Pierse 1 Introduction Time Series Analysis looks at the properties of time series from a purely statistical point of view. No attempt is made to relate variables using a priori
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 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 informationNew Introduction to Multiple Time Series Analysis
Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer Contents 1 Introduction 1 1.1 Objectives of Analyzing Multiple Time Series 1 1.2 Some Basics 2
More informationLabor-Supply Shifts and Economic Fluctuations. Technical Appendix
Labor-Supply Shifts and Economic Fluctuations Technical Appendix Yongsung Chang Department of Economics University of Pennsylvania Frank Schorfheide Department of Economics University of Pennsylvania January
More informationTesting the null of cointegration with structural breaks
Testing the null of cointegration with structural breaks Josep Lluís Carrion-i-Silvestre Anàlisi Quantitativa Regional Parc Cientí c de Barcelona Andreu Sansó Departament of Economics Universitat de les
More informationModelling of Economic Time Series and the Method of Cointegration
AUSTRIAN JOURNAL OF STATISTICS Volume 35 (2006), Number 2&3, 307 313 Modelling of Economic Time Series and the Method of Cointegration Jiri Neubauer University of Defence, Brno, Czech Republic Abstract:
More informationEstimation and Testing for Common Cycles
Estimation and esting for Common Cycles Anders Warne February 27, 2008 Abstract: his note discusses estimation and testing for the presence of common cycles in cointegrated vector autoregressions A simple
More informationCointegration Tests Using Instrumental Variables Estimation and the Demand for Money in England
Cointegration Tests Using Instrumental Variables Estimation and the Demand for Money in England Kyung So Im Junsoo Lee Walter Enders June 12, 2005 Abstract In this paper, we propose new cointegration tests
More informationTest for Structural Change in Vector Error Correction Models 1
中京大学経済学論叢 25 号 24 年 3 月 Test for Structural Change in Vector Error Correction Models Junya MASUDA 2 Abstract In this paper we suggest a test for cointegration rank when change point is known and model
More informationChapter 1. GMM: Basic Concepts
Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating
More informationMatrix Polynomial Conditions for the Existence of Rational Expectations.
Matrix Polynomial Conditions for the Existence of Rational Expectations. John Hunter Department of Economics and Finance Section, School of Social Science, Brunel University, Uxbridge, Middlesex, UB8 PH,
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 informationModeling Expectations with Noncausal Autoregressions
Modeling Expectations with Noncausal Autoregressions Markku Lanne * Pentti Saikkonen ** Department of Economics Department of Mathematics and Statistics University of Helsinki University of Helsinki Abstract
More informationStructural Breaks in the Cointegrated Vector. Autoregressive Model
Structural Breaks in the ointegrated Vector Autoregressive Model Peter Reinhard Hansen y Department of Economics University of alifornia, San Diego November, 999 Abstract We generalize the cointegrated
More informationA Test of Cointegration Rank Based Title Component Analysis.
A Test of Cointegration Rank Based Title Component Analysis Author(s) Chigira, Hiroaki Citation Issue 2006-01 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/13683 Right
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 informationResiduals-based Tests for Cointegration with GLS Detrended Data
Residuals-based Tests for Cointegration with GLS Detrended Data Pierre Perron y Boston University Gabriel Rodríguez z Ponti cia Universidad Católica del Perú Revised: October 19, 015 Abstract We provide
More informationECONOMETRIC METHODS II TA session 2 Estimating VAR(p) processes
ECONOMETRIC METHODS II TA session 2 Estimating VAR(p) processes Fernando Pérez Forero May 3rd, 2012 1 Introduction In this second session we will cover the estimation of Vector Autoregressive (VAR) processes
More informationIdenti cation and Frequency Domain QML Estimation of Linearized DSGE Models
Identi cation and Frequency Domain QML Estimation of Linearized DSGE Models hongjun Qu y Boston University Denis Tkachenko z Boston University August, ; revised: December 6, Abstract This paper considers
More informationVector error correction model, VECM Cointegrated VAR
1 / 58 Vector error correction model, VECM Cointegrated VAR Chapter 4 Financial Econometrics Michael Hauser WS17/18 2 / 58 Content Motivation: plausible economic relations Model with I(1) variables: spurious
More informationTESTS FOR STOCHASTIC SEASONALITY APPLIED TO DAILY FINANCIAL TIME SERIES*
The Manchester School Vol 67 No. 1 January 1999 1463^6786 39^59 TESTS FOR STOCHASTIC SEASONALITY APPLIED TO DAILY FINANCIAL TIME SERIES* by I. C. ANDRADE University of Southampton A. D. CLARE ISMA Centre,
More informationBayesian Inference in the Time Varying Cointegration Model
Bayesian Inference in the Time Varying Cointegration Model Gary Koop University of Strathclyde Roberto Leon-Gonzalez National Graduate Institute for Policy Studies Rodney W. Strachan y The University of
More informationEstimating the Number of Common Factors in Serially Dependent Approximate Factor Models
Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models Ryan Greenaway-McGrevy y Bureau of Economic Analysis Chirok Han Korea University February 7, 202 Donggyu Sul University
More informationDifference equations. Definitions: A difference equation takes the general form. x t f x t 1,,x t m.
Difference equations Definitions: A difference equation takes the general form x t fx t 1,x t 2, defining the current value of a variable x as a function of previously generated values. A finite order
More informationThe Predictive Space or If x predicts y; what does y tell us about x?
The Predictive Space or If x predicts y; what does y tell us about x? Donald Robertson and Stephen Wright y October 28, 202 Abstract A predictive regression for y t and a time series representation of
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 informationOn Standard Inference for GMM with Seeming Local Identi cation Failure
On Standard Inference for GMM with Seeming Local Identi cation Failure Ji Hyung Lee y Zhipeng Liao z First Version: April 4; This Version: December, 4 Abstract This paper studies the GMM estimation and
More informationNumerically Stable Cointegration Analysis
Numerically Stable Cointegration Analysis Jurgen A. Doornik Nuffield College, University of Oxford, Oxford OX1 1NF R.J. O Brien Department of Economics University of Southampton November 3, 2001 Abstract
More informationModeling Expectations with Noncausal Autoregressions
MPRA Munich Personal RePEc Archive Modeling Expectations with Noncausal Autoregressions Markku Lanne and Pentti Saikkonen Department of Economics, University of Helsinki, Department of Mathematics and
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 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 informationSUPPLEMENTARY TABLES FOR: THE LIKELIHOOD RATIO TEST FOR COINTEGRATION RANKS IN THE I(2) MODEL
SUPPLEMENTARY TABLES FOR: THE LIKELIHOOD RATIO TEST FOR COINTEGRATION RANKS IN THE I(2) MODEL Heino Bohn Nielsen and Anders Rahbek Department of Economics University of Copenhagen. heino.bohn.nielsen@econ.ku.dk
More informationThe Kuhn-Tucker Problem
Natalia Lazzati Mathematics for Economics (Part I) Note 8: Nonlinear Programming - The Kuhn-Tucker Problem Note 8 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19). The Kuhn-Tucker
More informationAn Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis
An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis M. Hashem Pesaran Trinity College, Cambridge, England Yongcheol Shin Department of Applied Economics, University of Cambridge,
More informationReduced rank regression in cointegrated models
Journal of Econometrics 06 (2002) 203 26 www.elsevier.com/locate/econbase Reduced rank regression in cointegrated models.w. Anderson Department of Statistics, Stanford University, Stanford, CA 94305-4065,
More informationCovariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
More informationModeling Expectations with Noncausal Autoregressions
MPRA Munich Personal RePEc Archive Modeling Expectations with Noncausal Autoregressions Markku Lanne and Pentti Saikkonen Department of Economics, University of Helsinki, Department of Mathematics and
More informationParametric Inference on Strong Dependence
Parametric Inference on Strong Dependence Peter M. Robinson London School of Economics Based on joint work with Javier Hualde: Javier Hualde and Peter M. Robinson: Gaussian Pseudo-Maximum Likelihood Estimation
More informationEconomics 241B Review of Limit Theorems for Sequences of Random Variables
Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de nitions of convergence focus on the outcome sequences of a random variable. Convergence
More informationIntroduction to the Mathematical and Statistical Foundations of Econometrics Herman J. Bierens Pennsylvania State University
Introduction to the Mathematical and Statistical Foundations of Econometrics 1 Herman J. Bierens Pennsylvania State University November 13, 2003 Revised: March 15, 2004 2 Contents Preface Chapter 1: Probability
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 informationNonsense Regressions due to Neglected Time-varying Means
Nonsense Regressions due to Neglected Time-varying Means Uwe Hassler Free University of Berlin Institute of Statistics and Econometrics Boltzmannstr. 20 D-14195 Berlin Germany email: uwe@wiwiss.fu-berlin.de
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 information7. MULTIVARATE STATIONARY PROCESSES
7. MULTIVARATE STATIONARY PROCESSES 1 1 Some Preliminary Definitions and Concepts Random Vector: A vector X = (X 1,..., X n ) whose components are scalar-valued random variables on the same probability
More informationNotes on Generalized Method of Moments Estimation
Notes on Generalized Method of Moments Estimation c Bronwyn H. Hall March 1996 (revised February 1999) 1. Introduction These notes are a non-technical introduction to the method of estimation popularized
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 informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models School of Economics and Management - University of Geneva Christophe Hurlin, Université of Orléans University of Orléans April 2018 C. Hurlin (University of Orléans)
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 informationTesting for non-stationarity
20 November, 2009 Overview The tests for investigating the non-stationary of a time series falls into four types: 1 Check the null that there is a unit root against stationarity. Within these, there are
More informationADVANCED ECONOMETRICS I (INTRODUCTION TO TIME SERIES ECONOMETRICS) Ph.D. FAll 2014
ADVANCED ECONOMETRICS I (INTRODUCTION TO TIME SERIES ECONOMETRICS) Ph.D. FAll 2014 Professor: Jesús Gonzalo Office: 15.1.15 (http://www.eco.uc3m.es/jgonzalo) Description Advanced Econometrics I (Introduction
More informationIntroduction to Linear Algebra. Tyrone L. Vincent
Introduction to Linear Algebra Tyrone L. Vincent Engineering Division, Colorado School of Mines, Golden, CO E-mail address: tvincent@mines.edu URL: http://egweb.mines.edu/~tvincent Contents Chapter. Revew
More informationTests of the Co-integration Rank in VAR Models in the Presence of a Possible Break in Trend at an Unknown Point
Tests of the Co-integration Rank in VAR Models in the Presence of a Possible Break in Trend at an Unknown Point David Harris, Steve Leybourne, Robert Taylor Monash U., U. of Nottingam, U. of Essex Economics
More informationVAR Models and Cointegration 1
VAR Models and Cointegration 1 Sebastian Fossati University of Alberta 1 These slides are based on Eric Zivot s time series notes available at: http://faculty.washington.edu/ezivot The Cointegrated VAR
More informationGMM estimation of spatial panels
MRA Munich ersonal ReEc Archive GMM estimation of spatial panels Francesco Moscone and Elisa Tosetti Brunel University 7. April 009 Online at http://mpra.ub.uni-muenchen.de/637/ MRA aper No. 637, posted
More informationEstimation and Inference with Weak Identi cation
Estimation and Inference with Weak Identi cation Donald W. K. Andrews Cowles Foundation Yale University Xu Cheng Department of Economics University of Pennsylvania First Draft: August, 2007 Revised: March
More informationVector autoregressions, VAR
1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS17/18 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,
More informationFINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 2015
FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 205 Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by
More informationSimple Estimators for Semiparametric Multinomial Choice Models
Simple Estimators for Semiparametric Multinomial Choice Models James L. Powell and Paul A. Ruud University of California, Berkeley March 2008 Preliminary and Incomplete Comments Welcome Abstract This paper
More informationTrend-Cycle Decompositions
Trend-Cycle Decompositions Eric Zivot April 22, 2005 1 Introduction A convenient way of representing an economic time series y t is through the so-called trend-cycle decomposition y t = TD t + Z t (1)
More informationLecture 11: Eigenvalues and Eigenvectors
Lecture : Eigenvalues and Eigenvectors De nition.. Let A be a square matrix (or linear transformation). A number λ is called an eigenvalue of A if there exists a non-zero vector u such that A u λ u. ()
More informationSIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER. Donald W. K. Andrews. August 2011
SIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER By Donald W. K. Andrews August 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1815 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
More informationIdenti cation and Frequency Domain QML Estimation of Linearized DSGE Models
Identi cation and Frequency Domain QML Estimation of Linearized DSGE Models hongjun Qu y Boston University Denis Tkachenko z Boston University August, Abstract This paper considers issues related to identi
More information