E 4101/5101 Lecture 11: Cointegration, estimation …...Johansen test for rank(P) (other than 0 or 1) ML estimation of b for the case of’ rank(P) 2 No assumptions about weak exogeneity

Introduction Estimating a single cointegrating vector Testing r=0 against r=1

E 4101/5101Lecture 11: Cointegration, estimation and

testing: Part 1Ragnar Nymoen

Department of Economics, University of Oslo

6 april 2011

E 4101/5101 Lecture 11: Cointegration, estimation and testing: Part 1 Department of Economics, University of Oslo


Introduction IFor 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 + ε (2)

the “long-run” roots can be found as the roots of the characteristicequation associated with the matrix of levels coefficients Π.We write Π as the product of two matrices αn×r and β′r×n wherer ≡ rank(Π) :

Π = αβ′ (3)



Introduction II

We have: The stationary case:

rank(Π) = n =⇒ yt ∼ I(0)

The cointegrating case (reduced rank)

0 < rank(Π) < n =⇒ yt ∼ I(1)

and no cointegration (danger of spurious regression)

rank(Π) = 0 =⇒ yt ∼ I(1)



Introduction III

I We follow Davidson and MacKinnon, Ch 14, and discussestimation of known cointegrating vectors first, and after that,the testing of the hypothesis about absence of cointegration(more generally: reduced rank).

I This is a little “back to front” compared to Hamilton’s Ch 19,but the material covered is the same,and this organization hasthe advantage of bringing out the thin but consequential linebetween the spurious regression case and the cointegratingregression case.

The organization for this and the next lecture is:



Introduction IV1. rank(Π) is 1

Estimating a unique cointegrating vector by means of:The cointegration regressionThe ECM estimator

2. rank(Π) is 0 or 1Test rank(Π) = 0 against rank(Π) = 1,byEngle-Granger testsECM test

3. Test and ML estimation based on VARJohansen test for rank(Π) (other than 0 or 1)ML estimation of β for the case of’ rank(Π) ≥ 2 Noassumptions about weak exogeneity of variables with respectto β.



The cointegrating regression IWhen rank(Π) = 1, the cointegration vector is unique (subjectonly to normalization).Without loss of generality we set n = 1 and write yt = (yt , xt) asin a usual regression.The cointegration parameter β can be estimated by OLS on

yt = βxt + ut (4)

where ut ∼ I (0) by assumption.

(β− β) =∑T

t=1 xtut

∑Tt=1 x

2t

. (5)

Since xt ∼ I (1) we are in a the same situation as with the firstorder AR case with autoregressive parameter equal to one.



The cointegrating regression IIIn direct analogy we need to multiply (β− β) by T in order toobtain a non-degenerate asymptotic distribution:

T (β− β) =

1

T∑T

t=1 xtut

1

T 2 ∑Tt=1 x

2t

, (6)

=⇒ (β− β) converges to zero at rate T , instead of√T as in the

stationary case.

I This result is called the Engle-Granger super-consistencytheorem.

I Davidson and MacKinnon alos call β from OLS on (4) thelevels estimator.



The cointegrating regression III

I In the seminar, we saw that β is consistent also in the case ofa simultaneous equations system

I Remember that we assume r = 1 so the cointegration vectoris unique if it exists.

I More generally there is an identification issue, cf Lecture 10



The distribution of the Engle-Granger (levels) estimator I

The E-G estimator β has a non-standard distribution which willdepend on the details of the DGP.For the DGP

yt = βxt + ut , ut ∼ N(0, σ2u )

∆xt = εxt , εxt ∼ N(0, σ2ε )

E [utεxt ] = σuε

it can be shown, see Banerjee et al. (1983), ch 6.2.2



The distribution of the Engle-Granger (levels) estimator II

T (β− β)L−→

σue2

(Wε(1)2 + 1) + σε · h ·N(0,∫ 10 [Wε(r)]

2 dr)

σ2ε

∫ 10 [Wε(r)]

2 dr,

(7)

where h =√

σ2u − σ2

uε/σ2ε

I If σuε = 0 the non-centrality disappears from the numerator,but the bias is still a function of Brownian motions.

I Not normally distributed.I The same applies to the t−value based on β: It is not a normal



The distribution of the Engle-Granger (levels) estimator III

=⇒ Inference “in” the cointegration regression is generallyimpractical (because standard inference in not valid)

I This drawback is even more severe in DGPs with higher orderdynamics, because the disturbance of the cointegratingequation is autocorrelated also in the case of cointegration.

I σuε 6= 0, creates a “second order bias”: Correlation betweenregressor and disturbance in the cointegration regression doesnot damage the super-consistency property, but it creates afinite sample bias.

I Autocorrelation (due to any omitted higher order dynamics)adds to the second-order bias.



The distribution of the Engle-Granger (levels) estimator IV

I All the above (except the detailed expression for T (β− β) ofcourse) applies to more realistic cointegration regressions thatcontain an intercept and other constant terms.

Note:Hamilton’s book contains more general results for the cointegratingregression: Proposition 19.2, and Ch 19.3, p 601-608.



Modified Engle-Granger estimator IThere are important developments of the levels estimator that aimat “fixing” the second-order bias problem

I Phillips and Hansen fully modified estimator:Subtract an estimate of the bias from β (i.e. keep thecointegration regression simple).The modified estimator has an asymptotic normaldistribution, which allows inference on β.

I Saikonnen’s estimator,Is based on

yt = βxt + γ1∆xt+1 + γ2∆xt−1 + ut

or higher order lead/lags that “make” ut white-noise, seeDavidson and MacKinnon p 630.



ECM estimator I

The ECM represents a way of avoiding second order bias due todynamic mis-specification.This is because, under cointegration, the ECM is implied (therepresentation theorem)With n = 2, p = 1 and weak exogeneity of xt (= y2t) with respectto the cointegration parameter we have seen that the cointegratedVAR can be re-written as a conditional model and a marginalmodel (Lecture 10, slides)

∆yt = b∆xt + φ︸︷︷︸α11β11

yt−1 + γ︸︷︷︸α11β12

xt−1 + εt (8)

∆xt = εxt (9)



ECM estimator II

where b is the regression coefficient, and εt and εxt areuncorrelated normal variables (by regression).

∆yt = b∆x + φ(yt−1 +γ

φxt−1) + εt

= b∆x + φ(yt−1 +β12

β11xt−1) + εt

Normalization on yt−1 by setting β11 = −1, and defining β12 = β,for comparison with E-G estimaor, gives

∆yt = b∆x + φ(yt−1 − βxt−1) + εt



ECM estimator IIIThe ECM estimator βECM , is obtained from OLS on (8)

βECM = − γ

φ(10)

βECM is consistent if both γ and φ are consistent.OLS (by construction) chooses the γ and φ that give the bestpredictor yt−1 − βECMxt−1 for ∆yt .As T grows towards infinity, the true parameters γ, φ and β willtherefore be found.This is an example of the principle of canonical correlation, whichplays a central role in the Johansen method.



ECM estimator IV

Therefore, by direct reasoning:

γ −→T→∞

γ, φ −→T→∞

φ and βECM −→T→∞

β (11)

In fact:

I βECM is super-consistent

I βECM has better small sample properties than the E-G levelsestimator, since it is based on a well specified econometricmodel (avoids the second-order bias problem).

Inference:

I The distributions of γ and φ (under cointegration) can beshown to be so called “mixed normal” for large T .



ECM estimator VI Their variances are stochastic variables rather than parameters.

I However, the OLS based t-values of γ and φ areasymptotically N(0, 1).

I βECM is also “mixed normal”, but{γ

φ− β

}/√

Var(βECM) −→T−→∞

N(0, 1) (12)

where, despite the change in notation, it is clear thatVar(βECM) can be found by the formula as in Lecture 7.(Directly available from PcGive when the model is written inADL form).

I The generalization to n− 1 explanatory variables, interceptand dummies is also unproblematic.



ECM estimator VI

I Remember:The efficiency of the ECM estimator depends onthe assumed weak exogeneity of xt .



Engle-Granger test

I The easiest approach is to use an ADF regression to the testnull-hypothesis of a unit-root in the residuals ut from thecointegrating regression (4).

I The motivation for the ∆ut−j is as before: to whiten theresiduals of the ADF regression

I The DF critical values are shifted to the left as deterministicterms, and/or more I (1) variables in the regression are added.

I See Figure 14.4 in Davidson and Mackinnon





The ECM test

I As we have seen, r = 0 corresponds to φ = 0 in the ECMmodel in (8):

∆yt = b∆xt + φyt−1 + γxt−1 + εt

I It also comes as no surprise that the t-value tφ have typicalDF-like distributions under H0 : φ = 0.

I See Figure 14.5 in Davidson and Mackinnon,

I and Ericsson and MacKinnon (2002) for critical values.





Why use ECM test instead of the Engle-Granger test? IThe size of the test (the probability of type 1 error) is more or lessthe same for the two tests.However, the power of the ECM test is generally larger than forthe E-G test.If tECMφ is the ECM test based on (8), it can be shown that

tECMφ∼=

σeσε

tEGτ , (13)

where tEG is the E-G test using

∆ut = τut−1 + et (14)

The “t-values”, and therefore the power, will be equal whenσe = σε .



Why use ECM test instead of the Engle-Granger test? IIWe can say something about when this will happen: Start with theECM and bring it on ADL form:

yt = bxt + (1 + φ)yt−1 + (γ− b)xt−1 + εt

(1− (1 + φ)L)yt = (b+ (γ− b)L)xt + εt

Assume next that the following Common factor restriction holds:

(b+ (γ− b)L)

(1− (1 + φ)L)= β (15)

so that

b = β

(γ− b) = −β(1 + φ)



Why use ECM test instead of the Engle-Granger test? IIIyt = βxt + (1 + φ)yt−1 − β(1 + φ)xt−1 + εt (16)

∆yt − β∆xt = φ(yt−1 − βxt−1) + εt

If we replace β by β, we haveThe ECM model (8) implies the Dickey-Fuller regression

∆yt − β∆xt︸︷︷︸∆ut

= φ(yt−1 − βxt−1)︸︷︷︸ut−1

+ εt (17)

when the Common factor restriction in (15) is true.

I If the Common factor restriction is invalid, the E-G test isbased on a mis-specified model.

I As a consequence σe > σε , and there is a loss of powerrelative to ECM test.



Why use ECM test instead of the Engle-Granger test? IV

I Common factor restrictions are easy to test in PcGive, also forhigher order multivariate ADL

I You can read more about Common factor restrictions inDavidson and MacKinnon p 294-298

I There the focus is that if we have an ADL(p, p) model withwhite-noise errors and m Common factor restrictions hold, wecan obtain a more parsimonious ADL(p −m, p −m).

I The disturbance will then be autocorrelated but thisautocorrelation is a “convenient simplification, not anuiscance”.

I Note that Common factors restrict the dynamic multipliers ina quite marked way.



Why use ECM test instead of the Engle-Granger test? V

I For example if the restriction in (15) holds, y in (16) has thesame impact and long-run multiplier with respect to x .

I We can see these implications in our case:

yt =(b+ (γ− b)L)

(1− (1 + φ)L)xt +

1

1− (1 + φ)Lεt

which gives

yt = βxt + ut ,

ut = (1 + φ)ut−1 + εt

Same impact and long-run multiplier, and AR(1) disturbance.



References

Banerjee, A., J. Dolado, J. W. Galbraith and D. F. Hendry (1993):Co-integration, Error-Correction and the Econometric Analysis ofNon-Stationary Data, Oxford University PressEricsson, N.E. and J.G. MacKinnon (2002): Distributions oferror-correction tests for cointegration, Econometrics Journal, 5,285–318


Documents

E 4101/5101 Lecture 11: Cointegration, estimation …...Johansen test for rank(P) (other than 0 or 1) ML estimation of b for the case of’ rank(P) 2 No assumptions about weak exogeneity