lecture 5 cointegration

7/27/2019 Lecture 5 Cointegration

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Lecture 5Stephen G. Hall

COINTEGRATION


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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)


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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.


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CONSIDER X,Y BOTH I(1)

e=Y

v+Y=X

tt

ttt

EVEN THOUGH BOTH ARE NON STATIONARY THERE IS A COMBINATION OF THE

TWO WHICH IS CREATED BY THE FIRST EQUATION WHICH IS STATIONARY.


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BOTH ARE DRIVEN BY THE COMMON STOCHASTIC TREND

eit

0=i

AND THE COINTEGRATING VECTOR IS

)(1,-


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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

b)CI(d,X~

0>bdb),-I(dXt ,~

THEN X IS COINTEGRATED AND

IS THE COINTEGRATING VECTOR


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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

nxrIS

AND r IS THE COINTEGRATING RANK OF THE SYSTEM.


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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 EXISTSr COINTEGRATING COMBINATIONS OF X. THEN THERE EXISTS A VALID ECM

REPRESENTATION

rRANKHASWHERE

++X-=XL)-(L)(1 tk-tt

FURTHER THERE ALSO EXISTS A MOVING AVERAGE REPRESENTATION

)+((L)C+)+C(1)(=

)+C(L)(=XL)-(1

t*

t

tt

WHERE C(1) HAS RANK n-r


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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 THESYSTEM.

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.


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ESTIMATING THE COINTEGRATING VECTORS

THE CASE OF A UNIQUE COINTEGRATING VECTOR.

THE ORIGINAL SUGGESTION MADE BY ENGLE AND GRANGER WAS SIMPLY TOEMPLOY A STATIC REGRESSION, eg IN THE BIVARIATE CASE

e+X=Y ttt WHERE WE ASSUME THAT X AND Y COINTEGRATE SO THAT e IS I(0)

)X(eX+=

)X(YX=

1-2t

t

1=t

tt

T

1=t

1-2t

T

1=t

tt

T

1=t


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AS e IS I(0) BY THE ASSUMPTION OF COINTEGRATION AND X IS I(1)

(1)OeXT

OXT

ptt

T

1=t

1-

p2t

T

1=t

1-

(T)

~

~

SO

)T(O)-(

(1)O)-T(

1-p

p

~

~

THIS IS THE SUPER CONSISTENCY PROPERTY OF THE STATIC REGRESSION.


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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.


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Intuition stationary data

x

x

x

X

X

X

X x

x x x

X x x x


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Intuition stationary data

x

x

x

X

X

X

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

X x x x

X x x x

xx xxx


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Intuition non-stationary data

x

x

x

X x x

X x x x


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x

x

x

X x x

X x x x

X x x x

X x x x


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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

X x x x

x

X x x

x x x x

ENGLE GRANGER 2 STEP PROCEDURE


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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. INPRACTISE THIS CAN BE DONE BY USING THE LAGGED ERROR FROM THE

STATIC REGRESSION.

t1-ttt +e+X(L)=Y(L)

WHERE

X-Y=e ttt

ALL THE ESTIMATED COEFFICIENTS IN THE SECOND

STAGE HAVE STANDARD DISTRIBUTIONS.

THE SMALL SAMPLE BIAS IS A PROBLEM IT WOULD ALSO BE DESIRABLE TO


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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

ttt +)X(-=

THE CORRECTION IS THEN

+=3

AND THE STANDARD ERRORS ON THE ADJUSTED COEFFICIENTS ARE GIVEN

BY THE STANDARD ERRORS ON THE 3RD STAGE PARAMETERS

2) PHILLIPS FULLY MODIFIED ESTIMATOR


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2) PHILLIPS FULLY MODIFIED ESTIMATOR.

NOT ASSUMING THE EXOGENEITY OF THE REGRESSORS PHILLIPS HAS

PROPOSED A NON-PARAMETRIC CORRECTION.

LET

u=X

u+X=Y

2tt

1ttt

THEN

)uuE(=

-

1=

X-Y=Y

T-XYX=

k20

0=k

211-22

+

t

1-

2212t

+

t

+

t+t

T

1=t

2t

T

1=t

-1+

3) DYNAMIC ESTIMATION


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3) DYNAMIC ESTIMATION

AGAIN ASSUMING THE EXOGENEITY OF X PHILLIPS(1988) HAS SHOWN THAT

THE SINGLE EQUATION DYNAMIC MODEL,

tt2tt1t +X(L)d+)X-Y(d=Y 11

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


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EXOGENEITY AND COINTEGRATION.

ENGLE AND YOO(1989) OUTLINE THE INTERACTION OF THESE TWO.

v+)X-Y(+Y=X

u+X+X=Y

t1-t1-t1-tt

tttt

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) 0=== X IS STRONGLY EXOGENOUS SINGLE EQUATIONESTIMATION IS FIML

0= 3) X IS WEAKLY EXOGENOUS SINGLE EQUATION ESTIMATION ISFIML

0==4) X IS PREDETERMINED BUT NOT EXOGENOUS-OLSGIVESNON STANDARD DISTRIBUTIONS

TESTING COINTEGRATION


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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 THEFOLLOWING REGRESSION ARE I(0)

X-Y=u ttt

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.


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5% CRITICAL VALUES FOR COINTEGRATION TESTS T=100

n CRDW ADF(1) ADF(4)

2 0.38 -3.38 -3.17

3 0.48 -3.76 -3.62

4 0.58 -4.12 -4.02

5 068 -4.48 -4.36

PHILLIPS t TEST AGAIN HAS AN ADF DISTRIBUTION AND CAN BECONSTRUCTED 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


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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,

T+T+=CV -22-11

AND A TABLE GIVES THE PARAMETERS.

SO FOR 200 OBSERVATIONS WHEN n=6 THE 5% CRITICAL VALUE IS

4.7906=0.00028-0.0856-4.7048=

)11.17(200-)17.12(200-4.7048=CV-2-1

Example


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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 no3. 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 EstimationProcedure 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


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Table 1 the time series properties of the variables

Variable DF ADF

LW 10.9 2.6

LP 14.5 1.9

LPROD 3.8 3.3

LAVH -0.3 -0.5

UPC 5.2 1.8

DLW -3.5 -1.4

DLPC -1.4 -0.9

DLPROD -8.0 -2.4

DLAVH -11.3 -4.6

DUPC -2.4 -2.5LW-LP 2.6 2.2

D(LW-LP) -8.5 -3.6

The basic Sargan model


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The basic Sargan model

LW= -5.49+ 0.99LP + 1.1LPROD

CRDW = 0.24 DF =-1.7 ADF =-2.6 R2 = 0.9972

RCO: 0.86 0.72 0.52 0.35 0.18 0.04 0.08 -0.20 -0.27 -0.29 -0.32 -0.34

The combined Sargan-Phillips model


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The combined Sargan Phillips model

LW= -5.6 + 1.03LP + 1.07LPROD - 0.72UPC

CRDW = 0.28 DF =- 2.12 ADF = -3.0 R2 = 0.9974

RCO: 0.85 0.7 0.49 0.29 0.1 -0.06 -0.18 -0.31 -0.37 -0.39 -0.41 -0.43

The combined Sargan-Phillips model with average hours


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The combined Sargan Phillips model with average hours

LW= 2.88 + 1.02LP + 0.93LPROD - 0.61UPC 1.79LAVH

CRDW = 0.74 DF =- 4.7 ADF = -2.88 R2 = 0.999374

RCO: 0.63 0.39 0.09 -0.1 -0.03 -0.06 -0.05 -0.04 -0.06 -0.05 -0.06 -0.02

Testing the exclusion of three of the variables


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Testing the exclusion of three of the variables

LP LPRODS UPCCRDW 0.05 0.339 0.64

DF -0.68 -2.648 -3.66

ADF -1.43 -1.37 -2.14

R2 0.95 0.99 0.999

RCO1 0.96 0.82 0.68

2 0.92 0.73 0.47

3 0.86 0.64 0.22

4 0.78 0.55 0.06

5 0.65 0.57 0.14

6 0.58 0.52 0.13

7 0.5 0.46 0.14

Changing the dependent variable


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Changing the dependent variable

Dep

variable

Constant LP LPRO

D

UPC LAVH R2

LW 2.88 1.02 0.93 -0.61 -1.79 0.9993

LP 2.79 1.03 0.88 -0.73 -1.78 0.0088

UPC 1.74 1.2 0.85 -3.52 -1.65 0.8508

LAVH 6.89 1.01 0.86 -0.57 -2.64 0.8096

LPROD 2.28 0.966 1.21 -0.56 -1.66 0.9746

Final Dynamic Equation


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a y a c quat o

DLW = -0.007 + 1.04 EDP 1.18 DDUPC(-1) - 0.98 DLAVH + 0.22 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

lecture 5 cointegration

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