Lecture 5 Stephen G. Hall COINTEGRATION

WE HAVE SEEN THE POTENTIAL PROBLEMS OF USING NON-STATIONARY DATA, BUT THERE ARE ALSO GREAT ADVANTAGES. CONSIDER FORMULATING A STRUCTURAL ECONOMIC MODEL. IT WILL CONTAIN MANY STRUCTURAL EQUILIBRIUM RELATIONSHIPS. BY EQUILIBRIUM WE MEAN RELATIONSHIPS WHICH WILL HOLD ON AVERAGE OVER A LONG PERIOD OF TIME (NOT NECESSARILY MARKET CLEARING)

BY ASSUMPTION DEVIATIONS FROM A VALID EQUILIBRIUM WILL BE A STATIONARY PROCESS SO WE CAN PARTITION THE DATA INTO THE LONG RUN EQUILIBRIUM AND THE REST. THIS CAN BE VERY USEFUL IN APPLIED WORK. THIS IS COINTEGRATION.

CONSIDER X,Y BOTH I(1) EVEN THOUGH BOTH ARE NON STATIONARY THERE IS A COMBINATION OF THE TWO WHICH IS CREATED BY THE FIRST EQUATION WHICH IS STATIONARY.

BOTH ARE DRIVEN BY THE COMMON STOCHASTIC TREND AND THE COINTEGRATING VECTOR IS

COINTEGRATION GENERAL DEFINITION; THE COMPONENTS OF THE VECTOR X ARE SAID TO BE COINTEGRATED OF ORDER d,b DENOTED, IF X IS I(d) AND THERE EXISTS A NON-ZERO VECTOR SUCH THAT THEN X IS COINTEGRATED AND IS THE COINTEGRATING VECTOR

IF X HAS n COMPONENTS THEN THERE MAY BE UP TO r COINTEGRATING VECTORS, r IS AT MOST n-1. THIS IMPLIES THE PRESENCE OF n-r COMMON STOCHASTIC TRENDS. WHEN n>1 AND r IS THE COINTEGRATING RANK OF THE SYSTEM.

THE GRANGER REPRESENTATION THEOREM THIS IMPORTANT THEOREM DEFINES SOME OF THE BASIC PROPERTIES OF COINTEGRATED SYSTEMS. LET X BE A VECTOR OF n I(1) COMPONENTS AND ASSUME THAT THERE EXISTS r COINTEGRATING COMBINATIONS OF X. THEN THERE EXISTS A VALID ECM REPRESENTATION FURTHER THERE ALSO EXISTS A MOVING AVERAGE REPRESENTATION WHERE C(1) HAS RANK n-r

IMPLICATIONS SO A VALID ECM REQUIRES THE PRESENCE OF A COINTEGRATING SET OF VARIABLES, OTHERWISE IT IS A CLASSIC SPURIOUS REGRESSION. THIS ALSO IMPLIES THAT THE TIME DATING OF THE LEVELS TERMS IS NOT IMPORTANT. THE EXISTENCE OF COINTEGRATION IMPLIES THAT GRANGER CAUSALITY MUST EXIST IN AT LEAST ONE DIRECTION BETWEEN THE VARIABLES OF THE SYSTEM. WE CAN START FROM THE MA REPRESENTATION AND DERIVE THE ECM - THESE TWO ARE OBSERVATIONALLY EQUIVALENT. HYLLEBERG AND MIZON(1989) EXTEND THIS TO ALSO INCLUDE THE EQUIVALENCE OF THE BEWLEY REPRESENTATION AND THE COMMON TRENDS REPRESENTATION.

ESTIMATING THE COINTEGRATING VECTORS THE CASE OF A UNIQUE COINTEGRATING VECTOR. THE ORIGINAL SUGGESTION MADE BY ENGLE AND GRANGER WAS SIMPLY TO EMPLOY A STATIC REGRESSION, eg IN THE BIVARIATE CASE WHERE WE ASSUME THAT X AND Y COINTEGRATE SO THAT e IS I(0)

AS e IS I(0) BY THE ASSUMPTION OF COINTEGRATION AND X IS I(1) SO THIS IS THE SUPER CONSISTENCY PROPERTY OF THE STATIC REGRESSION.

ALSO THE ESTIMATE WILL BE ASYMPTOTICALLY INVARIANT TO NORMALIZATION AND TO MEASUREMENT ERROR AND SIMULTANEOUS EQUATION BIAS BUT THEY ARE SUBJECT TO A NON-STANDARD DISTRIBUTION AND SMALL SAMPLE BIAS.

Intuition stationary data x x x XXXXXX X x x x x X x x x

Intuition stationary data x x x XXXXXX X x x x x X x x x X x xx x x x x x X x x x x x x X x x x X x x x xx xxx

Intuition non-stationary data x x x X x x X x x x

Intuition non-stationary data x x x X x x X x x x

Intuition non-stationary data x x x X x x X x x x X x x x x X x x x x x x x x x X x x x x X x x x x x x

ENGLE GRANGER 2 STEP PROCEDURE GIVEN THE CONSISTENCY OF THE STATIC REGRESSION ENGLE AND GRANGER DEMONSTRATED THAT THE FULL ERROR CORRECTION MODEL COULD BE CONSISTENTLY ESTIMATED BY USING THE RESTRICTED PARAMETER ESTIMATES OF THE STATIC REGRESSION IN THE DYNAMIC MODEL. IN PRACTISE THIS CAN BE DONE BY USING THE LAGGED ERROR FROM THE STATIC REGRESSION. WHERE ALL THE ESTIMATED COEFFICIENTS IN THE SECOND STAGE HAVE STANDARD DISTRIBUTIONS.

THE SMALL SAMPLE BIAS IS A PROBLEM, IT WOULD ALSO BE DESIRABLE TO BE ABLE TO CONDUCT INFERENCE ON THE COINTEGRATING VECTOR. 3 POSSIBLE ALTERNATIVES. 1) ENGLE AND YOO(1989) 3 STEP ESTIMATOR. THEY SUGGEST A THIRD STEP WHICH GOES BACK TO THE STATIC REGRESSION AND CORRECTS THE SMALL SAMPLE BIAS, UNDER THE ASSUMPTION OF EXOGENEITY OF THE REGRESSORS. THIS IS A REGRESSION OF THE FORM THE CORRECTION IS THEN AND THE STANDARD ERRORS ON THE ADJUSTED COEFFICIENTS ARE GIVEN BY THE STANDARD ERRORS ON THE 3RD STAGE PARAMETERS

2) PHILLIPS FULLY MODIFIED ESTIMATOR. NOT ASSUMING THE EXOGENEITY OF THE REGRESSORS PHILLIPS HAS PROPOSED A NON-PARAMETRIC CORRECTION. LET THEN

3) DYNAMIC ESTIMATION AGAIN ASSUMING THE EXOGENEITY OF X PHILLIPS(1988) HAS SHOWN THAT THE SINGLE EQUATION DYNAMIC MODEL, GIVES ASYMPTOTIC ML ESTIMATES WITH STANDARD INFERENCE. THIS IS ASSUMING BOTH EXOGENEITY AND THE EXISTENCE OF A UNIQUE COINTEGRATING VECTOR. THE ASSUMPTION REGARDING THE EXISTENCE OF COINTEGRATION AND EXOGENEITY IS THEREFORE CRUCIAL.

EXOGENEITY AND COINTEGRATION. ENGLE AND YOO(1989) OUTLINE THE INTERACTION OF THESE TWO. EVEN THOUGH Y IS LAGGED IN X, X IS NOT WEAKLY EXOGENOUS WITH RESPECT TO Y. 4 CASES 1) NO RESTRICTIONS, NON-STANDARD INFERENCE PROBLEMS WITH SINGLE EQUATION ESTIMATION 2 ) X IS STRONGLY EXOGENOUS SINGLE EQUATION ESTIMATION IS FIML 3) X IS WEAKLY EXOGENOUS SINGLE EQUATION ESTIMATION IS FIML 4) X IS PREDETERMINED BUT NOT EXOGENOUS-OLS GIVES NON STANDARD DISTRIBUTIONS

TESTING COINTEGRATION FOR THE ABOVE ESTIMATION PROCEDURES TO BE VALID WE NEED TO ESTABLISH THAT THE VARIABLES DO COINTEGRATE. IN A SINGLE EQUATION CONTEXT THIS AMOUNTS TO CHECKING THAT THE RESIDUALS OF THE FOLLOWING REGRESSION ARE I(0) THIS IS SIMPLY A MATTER OF CHECKING A SERIES FOR STATIONARITY. BUT THE ERROR PROCESS IS A CONSTRUCTED SERIES FROM ESTIMATED PARAMETERS SO THE TESTS HAVE DIFFERENT DISTRIBUTIONS. MAIN TESTS ARE THE COINTEGRATING REGRESSION DURBIN WATSON (CRDW), THE (AUGMENTED) DICKEY-FULLER TESTS AND THE PHILLIPS NON- PARAMETRIC TESTS.

THE CRDW TEST THE DICKEY-FULLER TEST THE PHILLIPS(1987) AND PHILLIPS AND PERON(1988) TESTS SAME DISTRIBUTION AS THE ADF. THE DISTRIBUTION OF THESE TESTS VARIES WITH T AND n, THE NUMBER OF X VARIABLES.

5% CRITICAL VALUES FOR COINTEGRATION TESTS T=100 nCRDWADF(1)ADF(4) PHILLIPS t TEST AGAIN HAS AN ADF DISTRIBUTION AND CAN BE CONSTRUCTED USING EITHER THE ESTIMATED PARAMETER OR ITS VALUE UNDER THE NULL (ie. 0). OTHER SINGLE EQUATION TESTS SOMETIMES REPORTED ARE THE SINGLE VARIABLE VERSION OF MULTIVARIATE STATISTICS (eg. JOHANSEN). SEE NEXT LECTURE.

MacKINNON(1991) GIVES THE MOST COMPREHENSIVE SET OF CRITICAL VALUES USING RESPONSE SURFACES. THESE ALLOW A RANGE OF DIFFERENT SAMPLE SIZES TO BE HANDLED. A FORMULA IS ESTIMATED OF THE FORM, AND A TABLE GIVES THE PARAMETERS. SO FOR 200 OBSERVATIONS WHEN n=6 THE 5% CRITICAL VALUE IS

Example There were a number of early papers which discussed and proposed the notion of cointegration. But the main literature really starts with a special issue on cointegration of the oxford bulletin of economics and statistics in 1986, vol 48 no 3. This contained a number of theory pieces and the first application for the Granger and Engle procedure which provides a good blueprint for the overall procedure Hall(1986) ‘An Application of the Granger and Engle two-step Estimation Procedure to UK aggregate Wage data’ A subsequent paper also provides the first application of the Johansen Procedure to the same data set.

Table 1 the time series properties of the variables VariableDFADF LW LP LPROD LAVH UPC DLW DLPC DLPROD DLAVH DUPC LW-LP D(LW-LP)

The basic Sargan model LW= LP + 1.1LPROD CRDW = 0.24 DF =-1.7 ADF =-2.6 R 2 = RCO:

The combined Sargan-Phillips model LW= LP LPROD UPC CRDW = 0.28 DF = ADF = -3.0 R 2 = RCO:

The combined Sargan-Phillips model with average hours LW= LP LPROD UPC – 1.79LAVH CRDW = 0.74 DF =- 4.7 ADF = R 2 = RCO:

Testing the exclusion of three of the variables LPLPRODSUPC CRDW DF ADF R2R RCO

Changing the dependent variable Dep variable ConstantLPLPRO D UPCLAVHR2R2 LW LP UPC LAVH LPROD

Final Dynamic Equation DLW = EDP – 1.18 DDUPC(-1) DLAVH DLW(-2) –0.26Z(-1) (1.4) (6.0) (1.4) (8.6) (2.9) (3.3) DW=1.99 BP(16)=23.7 SEE=0.012 CHiSQ(12)=2.3 CHOW(64,12)=0.22 Z is the error correction term