e 4101/5101 lecture 12: cointegration, estimation and ... · i the johansen approach is covered in...

Introduction The likelihood approach to cointegration Canonical correlations Tests for Coint rank More general DGPs

E 4101/5101Lecture 12: Cointegration, estimation and

testing: Part 2Ragnar Nymoen

Department of Economics, University of Oslo

14 April 2011

E 4101/5101 Lecture 12: Cointegration, estimation and testing: Part 2 Department of Economics, University of Oslo


Introduction I

For the vector yt consisting of n× 1 variables, we have theGaussian VAR(p):

yt = Φ(L)yt−1 + εt (1)

We assume that if there are unit-roots in the associatedcharacteristic equation, they are located at the zero frequency.By using the transformed equation

∆yt = Φ∗(L)∆yt−1 + Πyt−1 + εt (2)

We write the levels coefficient matrix Π as the product of twomatrices αn×r and β′r×n where r ≡ rank(Π) :

Π = αβ′ (3)



Introduction II

We are interested in both the cointegrating case

0 < rank(Π) < n

and the case with no cointegration

rank(Π) = 0

I Since rank(Π) is given by the number of non-zero eigenvaluesof Π, one approach to testing is find the number ofeigenvalues that are significantly different from zero.

I Fortunately, this problem has a solution because theeigenvalues has an interpretation as a special kind of squaredcorrelation coefficients.



Introduction III

I This method has become known as the Johansen approach. Itis “likelihood based”, see Johansen (1995) and the underlyingassumption is a VAR with normal, or Gaussian, disturbances.

I Hamilton presentation of this approach is in Ch 20

I Davidson and MacKinnon in Ch 14.6

I The Johansen approach is covered in other textbooks as well,Juselius (2006) is perhaps the most comprehensive andaccessible.



Concentrated likelihood functions I

We first consider the n dimensional VAR(1):

∆yt = Πyt−1 + εt , t = 1, ..., T . (4)

εt∼N(0, Ω) (5)

The conditional log-likelihood function is

L(Π, Ω) = −Tn

2ln(2π)− T

2ln |Ω| − 1

2

T

∑t=1

ε′tΩ−1εt . (6)

which we write without any explicit notation for the conditioningon y0, i.e., as in Hamilton Ch. 11.1



Concentrated likelihood functions II

The ML-estimator for Ω is the usual one, namely:

Ω=1

T

T

∑t=1

εt ε′t (7)

where εt is the vector of OLS residualsGiven (7) we proceed to use the concentrated likelihood-function

Lc(Π) = −T

2ln

∣∣∣∣∣ 1

T

T

∑t=1

εt ε′t

∣∣∣∣∣ . (8)



Concentrated likelihood functions III

where we have dropped a constant, and maximize this function“under the restriction” that Π = αβ′, which gives

Lc(α,β) = −T

2ln

∣∣∣∣∣ 1

T

T

∑t=1

[(∆yt−1−αβ′yt−1)(∆yt−αβ′yt−1)′

∣∣∣∣∣(9)



Concentrated likelihood functions IV

The expresssion inside the determinant can be written as

1

T

T

∑t=1

∆yt∆y′t︸︷︷︸

S00

− αβ′1

T

T

∑t=1

yt−1y′t−1︸︷︷︸

S11

βα′−

− αβ′1

T

T

∑t=1

yt−1∆y′t−1︸︷︷︸

S10

− 1

T

T

∑t=1

∆yt−1y′t−1︸︷︷︸

S01

βα′

Lc(α,β) =− T

2ln∣∣S00 − αβ′S11 − αβ′S10 − S01βα′

∣∣E 4101/5101 Lecture 12: Cointegration, estimation and testing: Part 2 Department of Economics, University of Oslo


Concentrated likelihood functions VIf we first consider β as given, we obtain α(β) from

∂Lc(α,β)

∂α= 0⇒

α(β) = S01β(β′S11β)−1 (10)

Insertion in Lc gives a new concentrated likelihood:

Lcc(β) =− T

2ln∣∣∣S00−S01β(β′S11β)−1β′S10

∣∣∣If we define

Θ(β) =∣∣∣S00−S01β(β′S11β)−1β′S10

∣∣∣ (11)

we want to find the β that minimizes the function Λ(β).



A trick, and a result from multivariate analysis IThe determinant of a 2× 2 matrix can be written as∣∣∣∣ a b

c d

∣∣∣∣ = ad − bc =

d(a− bd−1c)a(d − ba−1c)

The same holds for partitioned matrices.With suitable definitions of a, b, c and d we can formulate adeterminant∣∣∣∣ S00 S01β

β′S10 β

′S11β

∣∣∣∣ = Θ(β) ·

∣∣∣β′S11β∣∣∣

|S00|∣∣∣β′S11β−β

′S10S−100 fi

′S01β

∣∣∣ (12)

We have a case of c = b′, meaning that

S01βS−100 β′S10 = β

′S10S−100 S01β



A trick, and a result from multivariate analysis II

and the two ways of writing the determinant in (12) give

Θ(β)

|S00|=∣∣∣β′S11β

∣∣∣−1 ∣∣∣β′S11β−β′S10S−100 S01β

∣∣∣ (13)

Since S00 does not depend on β, minimizing Θ(β) is equivalent tominimize: ∣∣∣β′S11β−β

′S10S−100 S01β

∣∣∣∣∣∣β′S11β∣∣∣ (14)

This problem has a standard solution in multivariate analysis,Anderson (1951):



A trick, and a result from multivariate analysis IIIThe columns of β are estimated by the eigenvectors correspondingto the r largest solutions of the generalized eigenvalue problem:

ρS11−S10S−100 S01 = 0 (15)

The idea is now to determine the r largest eigenvaluesρ1 ≥ ρ2 ≥ . . . ρr . . . ≥ ρn ≥ 0.The columns of β are found as the corresponding r eigenvectors.The full set of eigenvectors is[

ρiS11−S10S−100 S01

]vi = 0, i = 1, 2, . . . , n (16)

Below, we will show that the matrix V, with the vi vectors ascolumns satisfies the normalization

V′S11V=In×n (17)



A trick, and a result from multivariate analysis IV

by virtue of being so called canonical variates.

We finally define a selection matrix P′=[Ir×r ,0

′]

. This gives

β = VP (18)

and

β′S11β=Ir×r (19)

while the ML estimator of α is obtained by substitution in (10). Inthe light of (19), this gives simply

α = S01β. (20)



Identification of the cointegration spaceI A common to say that the r vectors β given by (18) span the

(cointegrating) space containing the true β asymptotically.I In that special sense they are consistent estimates of β. In

view of (19):

β′ 1

T

T

∑t=1

yt−1y′t−1β=Ir×r

1

T

T

∑t=1

β′yt−1y

′t−1β=Ir×r

meaning that the I (0) disequilibrium terms have unit variance andare orthogonal.

I However this an arbitrary normalizationI As we learned in Lecture 10, the cointegration vectors are

unidentified in general (if r = 1, up to a constant)E 4101/5101 Lecture 12: Cointegration, estimation and testing: Part 2 Department of Economics, University of Oslo


Canonical variates and correlations I

We have T observations of ∆yt and yt−1. Collect in a 2n× Tmatrix X:

X2n×T =

[X0

X1

]. (21)

Assume that we want to reduce the dimentionality of the problemby finding the two linear combinations of ∆yt and yt−1 that havethe highest correlation. We define two new variables

u=a′X0 (22)

and

z=b′X1 (23)



Canonical variates and correlations II

where a and b is 1× n. From the definitions of variance andcovariance of linear combinations:

Cov(u,z) = a′S01b = Cov(u,z) = b

′S10a, (24)

where the Sij matrices are defined above:

Var(u) = E(u′u) = a

′S00a (25)

and

Var(z) = E(z′z) = b

′S11b (26)

We define the first pair of canonical variates as the pair u1, z1 withvariance equal to 1 and which maximizes Corr(u,z).



Canonical variates and correlations III

Formally, choose the a and b that maximize

L = a′S01b− λ1(a

′S00a− 1)− λ2(b

′S11b− 1). (27)

1oc:

S01b−2λ1S00a = 0 (28)

S10a−2λ2S11b = 0 (29)

Pre-multiply in (28) by a′, and in (29) by b

′:

a′S01b=2λ1 (30)

b′S10a = 2λ2. (31)



Canonical variates and correlations IVBut note that

a′S01b = b

′S10a = Corr(u,z) ≡ R (32)

since the variances are 1. Hence we have

a′S01b = R = 2λ1 = 2λ2. (33)

and (28) and (29) can be re-expressed as

S01b = RS00a (34)

S10a = RS11b. (35)

From (34):

a =1

RS−100 S01b (36)



Canonical variates and correlations Vsubstitution in (35) gives[

S10S−100 S01 − R2S11

]b = 0. (37)

Similarly:

b =1

RS−111 S10a (38)

and [S01S−111 S10 − R2S00

]a = 0. (39)

R2 is found from∣∣S10S−100 S01 − R2S11

∣∣ = ∣∣S11S10S−100 S01 − R2Inxn∣∣ = 0 (40)

meaning that R2 is the eigenvalue to the matrix S11S10S−100 S01 (orS−100 S01S−111 S10).



Canonical variates and correlations VI

I The eigenvector b is then determined from (37).

I The matrix S11S10S−100 S01 has n eigenvalues, but R = a′S01b

is the number we wish to maximize, therefore the solution isto choose the largest eigenvalue, R2

1 , and the associatedeigenvectors b1 and a1.

I The two first canonical variates must therefore be

u1 = a′1X0 (41)

z1 = b′1X1, (42)

Corr(u1,z1) ≡ R1

I R1 is called the (first) canonical correlation coefficient.



Canonical variates and correlations VII

I The next n− 1 pairs of canonical variates are defined in thesame way as the first (unit variance in particular)

I In addition it is required that all pairs are uncorrelated.

I The pairs are ordered by the size of the associated R2i .



Canonical correlation and ML I

I The eigenvalue problem (40) is the same problem as the MLapproach was leading to, compare (15).

I Hence, there are several important relationships betweenJohansen’s method and canonical correlations and variates:

ρ1 = maxa,b

Corr(u,z) = R21

ρi = R2i , i = 1, 2, . . . , n

β = [b1, b2, . . . , br ] = [b1, b2, . . . , br , . . . , bn]P ≡ VP



Canonical correlation and ML II

I Finally

V′S11V= In×n and β

′S11β=Ir×r

see (17) and (19), hold by definition for canonical variates,variance equal to 1 and uncorrelated.



Canonical correlation for open systems

I Canonical correlation analysis is not limited to cases withequal number of variables in the two groups X0 and X1.

I If there are p variables in X0 and q variables in X1, themaximum number of eigenvalues becomes minp, q.

I Assume for example the we include non-modelled(“exogenous”) I (1) variables in the X1 matrix, so that p < q=⇒ maximum number of cointegrating vectors is p

I And the rank of the relevant Π matrix is full.



Canonical correlations and single equation OLS I

I Canonical correlation coefficients are interpretable asgeneralizations of usual correlations coefficients

I Consider the case of p = 1 and q > 1. Linear regression givescanonical variates X0 and b

′X1and

maxb

Corr(X0 , b′X1)=R

implies thatR2 = ρ1

From the formulae above, for p = 1 and q > 1

R =1

s0

√S01S−111 S10 (43)



Canonical correlations and single equation OLS IIwhere s0 = S00, is consistent for σ2

0 , the variance of ∆yt . S01

is a 1× q vector

Assuming normality, the conditional variance of ∆yt given yt−1 er

σ21 = σ2

0 − S01S−111 S10

showing that R2 from (43) is an estimate of the relativeunexplained variation:

σ20 − σ2

1

σ20

= R2

σ20 = σ2

1 ⇐⇒ R2 = 0

σ21 = 0⇐⇒ R2 = 1



Canonical correlations and single equation OLS III

I When p = 1, both canonical correlation and the Johansenmethod are equivalent to usual regression analysis

I When p > 1, R21 is larger than any R-squares from individual

regressions.

I The interpretation is that more than one variable equilibriumcorrects,

I This information is not used when we do single equationregression, which is therefore inefficient when weak exogeneitydoes not hold for the variables in the cointegrating vector.



Testing the hypotheses about reduced rank I

From (13), and neglecting additive constants, the maximizedlikelihood is

L∗ = −T

2ln∣∣∣β′(S11−S10S−100 S01)β

∣∣∣ (44)

From

β′S11β = Ir×r

and

β(S10S−100 S01)β = ρrxr



Testing the hypotheses about reduced rank II

where ρrxr is the diagonal matrix with the eigenvalues ofS10S−100 S01, we obtain

L∗ = −T

2ln |Ir×r − ρrxr rxr | (45)

= −T

2

r

∑i=1

ln(1− ρi ).

When the Π is estimated freely, we have

L∗∗=− T

2

n

∑i=1

ln(1− ρi ) (46)



Testing the hypotheses about reduced rank IIIThe Likelihood-ratio test of the hypothesis that there are at most rcointegrating vectors 0 ≤ r < n, and n− r unit-roots:

ηr = 2L∗(β)−L∗(V) (47)

= −Tn

∑i=r+1

ln(1− ρi ), r = 0, 1, 2, . . . , n− 1

is called the trace-test. It’s distribution is tabulated in HamiltonTable B.10, “Case 1”.

I The testing is sequential; η0,η1, . . . , ηn−1.

I Not that if the largest squared correlation coefficient ρ0 issmall, the whole sequence η0,η1, . . . , ηn−1 will be small valuesas a result of low multivariate correlation between the I (0)variables in ∆yt and the I(1) variables in yt−1.



Testing the hypotheses about reduced rank IVI The number of cointegrating vector is r + 1 if the last

significant test is ηr (the H0 of n− r unit-roots is rejected.

I Since yt is a multivariate I(1) process for the whole sequenceof H0 s,

I Therefore ηr will be a function of Brownian motions (and nota Chi-square).

I An alternative test hypothesis formulation is H0: r = r ∗,against H1: r = r ∗ + 1 This leads to the maximal-eigenvaluetest:

ζr = −T ln(1− ρr+1), r = 0, 1, 2, . . . , n− 1. (48)



Testing the hypotheses about reduced rank V

I An interesting special case is:H0: r = r ∗ = n− 1 against H1:r = r ∗ = n.

I This becomes in effect a test of I (1), with a single commontrend under H0.

I In this case the two tests coincide, and it has the asymptoticdistribution of the square of the Dickey-Fuller t-statistic.



VAR(p)

I In the usual way, the VAR(p) can be written in terms ofdifferences and lagged levels.

I With reference to FLW theorem: regress out the effect of thep − 1 differenced variables from ∆yt and yt−1 and proceed toanalyse these OLS residuals

I Can be interpreted as yet another “layer” of likelihoodconcentration.

I Based on the residuals the derivation of the ML-estimators isas above.

I Note the similarity to the ADF test



Deterministic terms

I Lecture 10: It matters a great deal whether the constant isrestricted to be in the cointegrating space or not

I It also affects the distributions of the tests

I µ free in DGP and in the model: Table B.10 Case 3.

µ in the model but µ = αµ0 in DGP: Table B.10, Case 2.µ = αµ0 in the model and in DGP: Not tabulated in Hamilton.

I Restricted linear trends seem relevant for economic data,again the distributions for the tests are affected, the book byJuselius is a good reference but MacKinnon, Haug andMichelis even more comprehensive (and with programs).



Similarity in rank testing

I Since the distributions are different for different forms ofdeterministic non-stationarity, there is a premium on the testprocedure that gives similarity

I The advise for data with visible drift in levels:

I include an deterministic trend as restricted together with anunrestricted constant.

I After rank determination, can test significance of the restrictedtrend with standard inference

I Shift in levels

I Include restricted step dummy and a free impulse dummy.

I Exogenous variables, see table and program by MacKinnon,Haug and Michelis (1999).



I(0) variables in the VAR?

A misunderstanding that sometimes occurs is that “there can beno stationary variables in he cointegrating relationships”.Consider for example:

−y1t + β12y2t + β13y3t + β14y4t = ecm1t (49)

β21y1t − y2t + β23y3t + β24y4t = ecm2t (50)

If y1 is the log of real-wages, y2 productivity, y3 relative importprices, and y4 the rate of unemployment, then the first relationshipmay be a bargaining based wage equation which can be identifiedif there are restrictions in the second (price) equation.y4t ∼ I (0), most sensibly, but we want to estimate and test thetheory β14 = 0.Hence: specify the VAR with y4t included.



From I(1) to I(0)I When the rank has been determined, we are back in the

stationary-case.I The distribution of the identified cointegration coefficents are

“mixed normal” so that conventional asymptotic inference canbe performed on this β.

I The determination of rank actually let’s us move from theI (1) VAR, to the cointegrated VAR that contains only I (0)variables

I Another name for this I (0) model is the vector equilibriumcorrection model, VECM.

I The VECM can usually be analysed further, using the tools ofthe stationary model.

I Hence, co-integration analysis is an important step in theanalysis, but just one step.



References

Anderson, T. W. (1951) Estimating Linear Restrictions onRegressions Coeffcicients for Multivariate Normal Distributions,Annals of Mathematical Statistics, 22, 327-51Johansen, S. (1995), Likelihood-Based Inference in CointegratedVector Auto-Regressive Models, Oxford University PressJuselius, K (2004) The Cointegrated VAR Model, Methodologyand Applications, Oxford University PressMacKinnon, J., A. A. Haug and L. Michelis (1999) NumericalDistributions Functions of Likelihood Ratio Tests for Cointegration,with programs.


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