Lecture 6 Stephen G Hall MULTIVARIATE COINTEGRATION

IF WE WRITE OUT A MODEL IT WILL GENERALLY CONTAIN MANY COINTEGRATING VECTORS (MAYBE ONE FOR EACH EQUATION). THE SINGLE EQUATION ANALYSIS REALLY ASSUMES ONLY ONE COINTEGRATING VECTOR. YET FOR ECONOMICS TO BE INTERESTING WE REALLY NEED MANY COINTEGRATING VECTORS TO EXIST IN THE REAL WORLD. THE STRUCTURE AND CAUSALITY OF THE ECONOMY LIVES IN THE MANY COINTEGRATING VECTORS.

UNDERSTANDING HOW MULTIVARIATE COINTEGRATED SYSTEMS WORK, HOW TO ESTIMATE THEM AND HOW TO TEST HYPOTHESIS IS AT THE CORE OF UNDERSTANDING THE TIME SERIES BEHAVIOUR OF THE ECONOMY. IN THIS LECTURE WE EXPLORE TECHNIQUES FOR ESTIMATING AND TESTING MULTIPLE COINTEGRATING VECTORS AND THE UNDERLYING REASON WHY NON-STATIONARITY MAY BE GENERATED BY ECONOMIC SYSTEMS.

ECONOMICS AND COINTEGRATION (DAVIDSON AND HALL, EJ 1991,101,405, p ) THE CONDITIONAL ERROR CORRECTION SYSTEM LET Y BE A VECTOR OF ENDOGENOUS VARIABLES AND Z BE A VECTOR OF WEAKLY EXOGENOUS VARIABLES. THE ADL IS WHERE Z IS GENERATED BY A CLOSED VAR IF B(L) IS INVERTIBLE WE CAN SOLVE FOR THE FINAL FORM OF THE SYSTEM

NOW ASSUME THAT ECONOMIC THEORY SUGGESTS A SET OF EQUILIBRIUM RELATIONSHIPS. THESE ARE THE TARGET RELATIONSHIPS

NOTE HOWEVER THAT THIS CAN BE SEEN IF WE WRITE THE ADL IN ECM FORM WHERE C IS THE ERROR CORRECTION COEFFICIENT OR LOADING) MATRIX

note 1)THE TARGET RELATIONSHIPS REMAIN INVARIANT TO THERE LAG POSITION 2) THEY ARE ALSO INVARIANT TO STRUCTURAL OR REDUCED FORM SPECIFICATION

NON-STATIONARITY CAN ARISE IN ONE OF TWO BASIC WAYS. 1)THE SYSTEM MAY BE OF FULL RANK, SO B IS INVERTIBLE, AND Z IS NON-STATIONARY. THE Zs ARE EFFECTIVELY DRIVING THE SYSTEM. 2) THE UNSTABLE CASE. HERE B IS OF DEFICIENT RANK AND SO THE Y VARIABLES GENERATE UNIT ROOT BEHAVIOUR QUITE APART FROM THE BEHAVIOUR OF THE Zs.

THE UNSTABLE CASE TWO SUB CASES i) INSUFFICIENT TARGETS eg WAGES AND PRICES

ii) INCONSISTENT TARGETS eg WAGES AND PRICES

IMPLICATIONS UNDERSTANDING THE WORKINGS OF MULTIVARIATE SYSTEMS IS CRUCIAL WE NEED TO BE ABLE TO ESTIMATE NUMBERS OF COINTEGRATING VECTORS. WE ALSO NEED TO BE ABLE TO TEST HYPOTHESIS ABOUT HOW MANY COINTEGRATING VECTORS THERE ARE AND HOW THEY WORK IN THE SYSTEM.

SINGLE EQUATION TECHNIQUES CAN NEVER SATISFACTORILY ANSWER THESE POINTS. JOHANSEN PROVIDES THE ANSWER

ESTIMATING MANY COINTEGRATING VECTORS Johansen (1988) proposed a general framework for considering the possibility of multiple cointegrating vectors and this framework also allows questions of causality and general hypothesis tests to be carried out in a more satisfactory way. Begin by defining a VAR of a set of variables X as

Now we may express this system in vector error correction form(VECM) as,

Non-stationarity of X implies that will have deficient rank.

The heart of the Johansen procedure is simply to decompose Into 2 matricies and both of which are Nxr such that and so the rows of β may be defined as the r distinct cointegrating vectors and the rows of show how these cointegrating vectors are loaded into each equation in the system.

eg suppose a three variable system where r=1

and if r=2

ML ESTIMATION THE STANDARD LIKELIHOOD FUNCTION FOR THE SYSTEM IS NOW REWRITE THE SYSTEM AS

NOW WE CAN PARTIAL OUT THE EFFECTS OF THE DYNAMICS BY REGRESSING EACH PART ON THE DYNAMICS SO THAT THE MODEL BECOMES

THE LIKELIHOOD FUNCTION IS THEN AND CONCENTRATING WE GET

SO THAT AND THIS WILL BE MAXIMISED WHEN WE MINIMISE

THE SOLUTION TO THIS PROBLEM IS GIVEN AS A GENERALISED EIGENVALUE PROBLEM AND LET E BE THE CORRESPONDING MATRIX OF EIGENVECTORS AND D IS A DIAGONAL MATRIX CONSISTING OF THE ORDERED EIGENVALUES

THEN A VALID COINTEGRATING VECTOR WILL PRODUCE A SIGNIFICANTLY NON-ZERO EIGENVALUE AND THE ESTIMATE OF THE COINTEGRATING VECTOR WILL BE GIVEN BY THE CORRESPONDING EIGENVECTOR.

THE COINTEGRATING SPACE THE MATRIX OF EIGENVECTORS IS SAID TO SPAN THE COINTEGRATING SPACE THIS IS BECAUSE THEY ARE NOT THE SAME THING AS THE TARGET RELATIONSHIPS. THE COINTEGRATING VECTORS MAY BE ANY LINEAR COMBINATION OF THE UNDERLYING TARGET RELATIONS. WE HAVE IN EFFECT ESTIMATED AN UNIDENTIFIED SYSTEM AND ALL WE REALLY KNOW ABOUT IS THE BOUNDARIES OF THE SPACE THAT THE TARGET RELATIONSHIPS MUST LIE WITHIN.

SO IN THE ABSENCE OF IDENTIFYING RESTRICTIONS WE HAVE TO BE VERY CAREFUL IN INTERPRETING THE COINTEGRATING VECTORS WHEN THERE IS MORE THAN ONE. THERE SHOULD BE AS MUCH INTEREST IN THE LONG RUN MATRIX AND THE LOADING MATRIX AS IN THE DIRECT ESTIMATES OF THE COINTEGRATING VECTORS.

INFERENCE ON THE COINTEGRATING SPACE THE FIRST THING WE NEED TO KNOW IS THE NUMBER OF COINTEGRATING VECTORS IN THE SYSTEM. WE CAN CONSTRUCT A LIKELIHOOD RATIO TEST (THE TRACE STATISTIC) THIS TESTS THE HYPOTHESIS THAT THE NO. OF CVs IS AT MOST r AGAINST THE ALTERNATIVE THAT IT IS GREATER

AND THE MAXIMAL-EIGENVALUE TEST (lambda-max) THIS TESTS THE HYPOTHESIS THAT NO OF CVs=r AGAINST THE ALTERNATIVE THAT IT IS r+1 THESE TESTS HAVE NON-STANDARD DISTRIBUTIONS WHICH DEPEND ON NUISANCE PARAMETERS

Constants: where should they go No constant implies no drift terms but also no scaling in the CVs Restricted constant, scaling in the CVs but no drift if loading weights are zero Unrestricted constant scaling in CVs if they are there, but if no CVs then the constant implies a drift term in the first difference model. Hence a trend in the model

5% CRITICAL VALUES FOR TRACE TEST N-rNo CUNR CREST C

5% CRITICAL VALUES FOR LAMBDA MAX TEST N-rNO CUNRS CREST C

MAXIMUM LIKELIHOOD ESTIMATION OF THE UK WAGE MODEL (HALL 1989) EIGENVECTORS EIGENVA LUES REAL WAGES PRODUC TIVITY UNEMPHOURS

The Lambda-Max test rTestC.V

So then just normalise the first vector to compare with the OLS results Real wages Produnemphours ML results OLS results

HYPOTHESIS TESTING ONE OF THE KEY RESULTS IN JOHANSEN IS THAT NON STANDARD DISTRIBUTIONS ARISE WHEN THE NULL AND THE ALTERNATIVE ASSUME A DIFFERENT COINTEGRATING RANK. SO ASSUMING WE HAVE DETERMINED THE NUMBER OF CVs FROM ABOVE, CONDITIONAL ON THAT ASSUMPTION WE CAN CONDUCT STANDARD LIKELIHOOD BASED TESTS.

IN A SYSTEM HOWEVER NOT ALL RESTRICTIONS ARE VALID CAN WE TEST 0.5=0. NO! SO THIS DOES NOT FORM A REAL RESTRICTION ON THE MODEL

WE FORMULATE TESTS WITHIN THIS FRAMEWORK BY SETTING A RESTRICTION MATRIX H WHICH EFFECTIVELY REDUCES THE NUMBER OF FREE PARAMETERS TO BE ESTIMATED WHERE H IS AN nxs MATRIX AND s<r IS THE NUMBER OF RESTRICTIONS SOME CASES; UNRESTRICTED (2 CVs )

1 RESTRICTION 2 RESTRICTIONS

WE THEN SIMPLY REPEAT THE ML ESTIMATION PROCEDURE SUBJECT TO THESE RESTRICTIONS. THIS WILL PRODUCE A NEW SET OF EIGENVECTORS BASED ON THE RESTRICTION AND THE LR TEST IS

THE LOADING MATRIX ATTENTION OFTEN FOCUSES ON THE CVs RATHER THAN THE LOADING MATRIX BUT THIS IS ALSO OF INTEREST (SEE BELOW). WE CAN CONSTRUCT SIMILAR TESTS OF RESTRICTIONS ON THE LOADING MATRIX AS FOLLOWS WHERE A IS AN nxs MATRIX AND s<r IS THE NUMBER OF RESTRICTIONS SOME CASES; UNRESTRICTED (2 CVs)

1 RESTRICTION 2 RESTRICTIONS

WE THEN AGAIN SIMPLY REPEAT THE ML ESTIMATION PROCEDURE SUBJECT TO THESE RESTRICTIONS. THIS WILL PRODUCE A NEW SET OF EIGENVECTORS BASED ON THE RESTRICTION AND THE LR TEST IS

Conclusion The Johansen Multivariate procedure is a powerful way of analysing data It allows a complex interaction of causality and structure which allows us to understand systems in a much deeper way The remaining challenge is the identification problem

Example The Long Run Determination of the UK Monetary Aggregates Hall, Henry and Wilcox Bank of England Study

dfadfdfadfdfadf lm lnm lm lm lm lgdp lpgd lqce lcpi I(0) I(1) I(2)

dfadfdfadfdfadf ltfe lptfe rtb bdr cons cc cda cap I(0) I(1) I(2)

dfadfdfadfdfadf bssr lfw Ltw snd lsm I(0) I(1) I(2)

Log of Real M0

Log of Real Consumption

Log of real M0 minus the log of real Consumption

Log of real M0 minus 0.5*log of real Consumption

eigvallm0lcpiLqce rLR test5%CV

bssr lcc lcda lcap crdw df adf R2R Cointegrating vectors on lm0-lcpi-lqce

Similar results A similar set of results showing that log versions of the financial innovation variables also failed to cointegrate

So where now? Practitioners explain the result by financial innovation. What drives this? Interest rates! But summed over time because of learning

bssr Time Crdw Df Adf R2R2.99 Tests of cointegration on lm0-lcpi-lqce with cumulated interest rates

Eigenvm-p-q rLR test5%CV Johansen test of financial innovation

Seems to work! Cumulated interest rates seem to pick up the trend movement in velocity

Dynamic Model So we can go on and estimate a standard ECM based on this long run relationship

SE=0.01 R 2 =0.65 DW=2.3 ARCH(1)=1.8 RESET(4)=7.0 BJ(2)=1.7 LM(1)=1.8 LM(2)=3.9 LM(4)=4.6 LM(8)=9.3 LB(1)=1.5 LB(2)-3,7 LB(4)=4.4 LB(8)=12.9 CHSQ(8)=4.9 CHSQ(12)=4.8 CHSQ(24=16.7

Sample of recursive parameter estimates

conclusion A parsimonious and very stable dynamic model Cointegration has made an important missing effect very obvious Fixing this improves things enormously