ES1002 Econometrics of Time Series:...

Spurious regressionCo-integration

Error Correction ModelTesting for Cointegration

Vector Error Correction modelFinal ExamReferences

ES1002 Econometrics of Time Series: BasicsLecture 5: Cointegration and VECM

Hany Abdel-Latif

Egypt Scholars Economic Society

April 18, 2015Hany Abdel-Latif ES1002 Econometrics of Time Series .. Lecture 5




before we start

assignment 1comments on submissionsresults

assignment 2two excel files: dataset1.xls & dataset2.xlsdatasets assigned according to participants ID

dataset1.xls for ID end an odd numberdataset2.xls for ID end an even number

estimate a bivariate VAR modelwrite a 3-5 page report, including all diagnostic tests andinterpreting the resultsdue May 2 20:00 (Cairo Time)

Hany Abdel-Latif ES1002 Econometrics of Time Series .. Lecture 5




Outline

1 Spurious regression

2 Co-integration

3 Error Correction Model

4 Testing for Cointegration

5 Vector Error Correction model

6 Final Exam





The story short

classical econometric models assume that observed data comefrom a stationary process

constant mean and variance, over timemost economic and financial time series reveal the invalidity ofsuch assumption

economies evolve and grow over time

having a stationary time series would be an exception





Why stationarity matters

when data mean and variance are time variant

observations come from different distributions over time,posing difficult problems for empirical modellingif not accounted for, it can induce serious statistical mistakes

non-stationarity

can be due to evolution of the economy, legislative changes,technological change, and political turmoil inter aliacan be eliminated by transformations





What it means for regression

consider the ‘levels’ regression

yt = β1 + β2xt + ut

in the case where yt and xt are both non-stationary or I (1)variableswhat are the implications for the statistical properties of thedisturbance term

ut = yt − β1 − β2xt

ut will also be non-stationary or I (1)a linear combination of two (or more) I (1) variables is itselfand I (1) variable





Non-stationary errors

if ut is I (1)

ut is heteroscedastic because var(ut) increases with t;ut is not independent of uj for j 6= 0, the ACF of an I (1) seriesdoes not decay rapidly towards zero;cov(xt , ut) 6= 0, because the covariance between two anynon-stationary variables is, in general, non-zero

three of the assumptions of CLRM are violatedgiving the rise to the spurious regression problem





Co-integrated series

in certain cases, it may be possible to find a pair of constants,which we can denote π1 and π2, such that

vt = yt − π1 − π2xt

is stationary or I (0)

note that vt is just a linear function of yt and xt

if vt is stationary, then yt and xt are said to be cointegratedif two time series variables are cointegrated, they tend to’move together ’ over timethey are bound together by a long-run equilibrium relationship






even if two series are cointegrated, the residuals of the ’levels’regression yt = β1 + β2xt + ut may be serially correlatedtherefore we might originally have been thinking of fitting oneof the following specifications:

yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt

yt = β10 + β20xt + α1yt−1 + β21xt−1 + α2yt−2 + β22xt−2 + εt

or . . . however many lags are requiredthe lagged variables allow for the ’time-dependency ’including sufficient lagged terms to obtain a set of residualsthat are not serially correlated






yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt


specified in terms of I (1) variablesif yt and xt are cointegrated, both specifications can easily berearranged (reparameterised) so that they contain I(0)variables only






yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt

yt−yt−1 = β10 +β20(xt−xt−1)+(α1−1)yt−1 +(β21 +β20)xt−1 +εt






yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt

yt −yt−1 = β10+β20(xt−xt−1)+(α1 −1)yt−1 +(β21+β20)xt−1+εt






yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt

yt−yt−1 = β10+ β20 (xt −xt−1 )+(α1−1)yt−1+(β21+ β20)xt−1 +εt






yt = β10 + β20xt + α1yt−1 + β21xt−1 + εt

yt−yt−1 = β10 +β20(xt−xt−1)+(α1−1)yt−1 +(β21 +β20)xt−1 +εt

∆yt = δ20xt + ψ(yt−1 − π1 − π2xt−1) + εt

whereδ20 = β20 , ψ = α1 − 1,π1 = β10/(1− α1)

π2 = (β21 + β20)/(1− α1)





Error Correction Model

∆yt = δ20∆xt + ψ(yt−1 − π1 − π2xt−1) + εt

this is know as ’error correction model’the term ∆xt

represents the model’s short-run dynamicscontains information about the extent to which currentchanges in xt influence current changes in ythow ∆xt influences ∆yt






∆yt = δ20∆xt + ψ(yt−1 − π1 − π2xt−1) + εt

this is know as ’error correction model’the term ψ(yt−1 − π1 − π2xt−1)

can be also written ψvt−1known as the error correction mechanism

recall yt−1 = β1 + β2xt−1 represents the long-run equilibriumrelationship between xt and yt , accordingly:

if vt−1 = yt−1 − π1 − π2xt−1 > 0, yt−1 was above itsequilibrium value at t − 1if vt−1 = yt−1 − π1 − π2xt−1 < 0, yt−1 was below itsequilibrium value at t − 1






∆yt = δ20∆xt + ψ(yt−1 − π1 − π2xt−1) + εt

we should expect that ψ < 0, so thatvt−1 > 0 (yt−1 too high) ⇒ ψvt−1 < 0 ⇒ tendency ∆yt < 0(yt falling)

vt−1 < 0 (yt−1 too low) ⇒ ψvt−1 > 0 ⇒ tendency ∆yt > 0(yt rising)





Including more lags


the error correction representation

∆yt = δ20∆xt +δ11∆yt−1 +δ21∆xt−1 +ψ(yt−1−π1−π2xt−1) + εt

whereδ20, δ11, δ21, ψ, π1, π2 are transformations of β10, β20, α1,β21, α2, β22

here we included additional lagged difference terms ∆yt−1,∆xt−1 among the short-run dynamics





Engel-Granger test for cointegration

Engle and Granger (1987) developed a two-step residuals-basedprocedure for testing for cointegration, and incorporating acointegrating relationship into an estimated modelit is necessary to test each of the variables individually forstationarity or non-stationarity, , using a unit root testthe Engle-Granger procedure is applicable only if both variablesare non-stationary and I(1)if either or both of xt and yt are non-stationary and I (2), thenthe procedure could be followed using the first-differences ofthe I (2) variable






step 1

estimate the model yt = β1 + β2xt + vt using OLSsave the residuals v̂t = yt − β̂1 + β̂2xt , test v̂t for stationaritytest H0: ρ = 0 against ρ < 0 in one of:

∆v̂t = ρv̂t−1 + εt ,∆v̂t = ρv̂t−1 + δ1∆v̂t−1 + εt ,∆v̂t = ρv̂t−1 + δ1∆v̂t−1 + δ2∆v̂t−2 + εt ,or . . . however many lags are required

decisionaccept H0 ⇒ v̂t is non-stationary ⇒ STOP yt , xt are NOTcointegratedreject H0 ⇒ v̂t is stationary ⇒ PROCEED to step 2 yt , xtcointegrated






notelag length

AIC or SIC can be used to select the lag-length for the ADFautoregressionselecting the correct lag-length is important, because the resultof the cointegration test is sensitive to the lag-length chosen

constant and trendno constant term or trend is required in the ADFautoregression, because the sample mean of v̂t is zero and v̂t isuntrendedcritical values

a separate set of critical values (produced by Engle andGranger) is required to determine acceptance or rejection ofH0

in the multivariate case, the critical values are dependent onthe number of xjt ’s included on the RHS of the cointegratingregressionHany Abdel-Latif ES1002 Econometrics of Time Series .. Lecture 5





step 2

obtain the estimated error correction model, by estimating oneof the following using OLS:

∆yt = δ20∆xt + ψv̂t + εt

or∆yt = δ20∆xt + δ11∆yt−1 + δ21∆xt−1 + ψv̂t + εt

or

. . . however many lags are required






note

AIC or SIC can be used to select the lag-length for the lagged∆yt ’s and ∆xt ’s in the error correction modelit is not possible to perform hypothesis tests on π1 and π2, aserious limitation of the Engle-Granger procedure;but hypothesis tests on δ20, δ11, δ21 and ψ can be performed





Johansen’s approach

Johansen (1988) demonstrated that cointegration can also bemodelled within a modified VAR frameworkin order to keep notation as simple as possible, we will considerthe bivariate caseconsider the following bivariate VAR(1) model

Yt = β0 + β1Yt−1 + ut

or (y1ty2t

)=

(β10β20

)+

(β11 α11α21 β21

)(y1t−1y2t−1

)+

(u1tu2t

)Hany Abdel-Latif ES1002 Econometrics of Time Series .. Lecture 5




Vector Error Correction Model

(y1ty2t

)=

(β10β20

)+

(β11 α11α21 β21

)(y1t−1y2t−1

)+

(u1tu2t

)suppose y1t and y2t are both non-stationary or I (1)

but a linear combination of y1t and y2t exists which isstationary or I (0)

therefore y1t and y2t are cointegrated






(y1ty2t

)=

(β10β20

)+

(β11 α11α21 β21

)(y1t−1y2t−1

)+

(u1tu2t

)in this case, the bivariate VAR(1) model can bereparameterised so that it is expressed in terms of I (0)variables only, as follows:

(y1t − y1t−1y2t − y2t−1

)=

(β10β20

)+

(β11 − 1 α11α21 β21 − 1

)(y1t−1y2t−1

)+

(u1tu2t

)






(y1ty2t

)=

(β10β20

)+

(β11 α11α21 β21

)(y1t−1y2t−1

)+

(u1tu2t


(y1t −y1t−1

y2t − y2t−1

)=

(β10β20

)+

(β11 −1 α11α21 β21 − 1

)(y1t−1y2t−1

)+

(u1tu2t

)






(y1ty2t

)=

(β10β20

)+

(β11 α11α21 β21

)(y1t−1y2t−1

)+

(u1tu2t


(y1t − y1t−1y2t −y2t−1

)=

(β10β20

)+

(β11 − 1 α11

α21 β21 −1

)(y1t−1y2t−1

)+

(u1tu2t

)






(y1t − y1t−1y2t − y2t−1

)=

(β10β20

)+

(β11 − 1 α11α21 β21 − 1

)(y1t−1y2t−1

)+

(u1tu2t

)(

∆y1t∆y2t

)=

(β10β20

)+

(π11 π12π21 π22

)(y1t−1y2t−1

)+

(u1tu2t

)where π11 = β11 − 1, π12 = α11, π21 = α21, π22 = β21 − 1or

∆Yt = β0 + πYt−1 + ut

which is a Vector Error Correction Model (VECM) representationHany Abdel-Latif ES1002 Econometrics of Time Series .. Lecture 5




Johansen’s approach

note

in the Engle-Granger formulation,there is a presumption that yt is partly determined by xt,accordingly ∆yt depends on ∆xt , as well as ∆yt−1 and ∆xt−1

in the Johansen formulation,y1t and y2t are treated symmetrically, and no causation isassumed between the current values in either direction∆y1t depends only on the lagged values of ∆y1t−1, ∆y2t−1,and higher-order lags if applicablesimilarly, ∆y2t depends only on the lagged values ∆y1t−1,∆y2t−1 etc





Conditions for cointegration

recall

the rank of any matrix is the number of linearly independentrows or columnsloosely speaking, two rows are linearly independent if it is notpossible to express one of the rows as a multiple of another






recall

suppose that

A =

(3 47 9

), B =

(3 62 4

), C =

(0 00 0

)rank(A) = 2, because row 2 cannot be expressed as a multipleof row 1 (or vice versa), therefore A has two linearlyindependent rowsrank(B) = 1, because row 1 is 1.5 times row 2, therefore rows1 and 2 are not linearly independent: B has only one linearlyindependent rowrank(C) = 0, because a square matrix containing zeros onlyhas no independent rows






Johansen showed that the condition fora stationary or I(0) linear combination of y1t and y2t to existin other words, the condition for y1t and y2t to be cointegrated

depends on the rank of the matrix π =

(π11 π12π21 π22

)in the

VECM






π =

(π11 π12π21 π22

)

if y1t and y2t are both stationary or I(0)

rank(π) = 2in this case, any linear combination of y1t and y2t is stationarythe matrix π contains 2 cointegrating vectorsin one sense, y1t and y2t are trivially cointegratedhowever, it would not be common practice to refer to y1t andy2t as ’cointegrated’ in this case






π =

(π11 π12π21 π22

)

if y1t and y2t are both non-stationary or I(1), but a linearcombination of y1t and y2t exists which is stationary or I(0)

rank(π) = 1y1t and y2t are cointegratedthe matrix π contains 1 cointegrating vector






π =

(π11 π12π21 π22

)

if y1t and y2t are both non-stationary or I(1), and NOstationary linear combination of y1t and y2t exists

rank(π) = 0y1t and y2t are NOT cointegratedthe matrix π contains NO cointegrating vectors





Testing for cointegration

in Johansen’s VECM formulation, to test for cointegration weneed to test hypotheses concerning the rank of the matrix πlet r denote rank(π)

Johansen developed two test statistics, known as the tracestatistic and the maximal eigenvalue statisticusing Johansen’s notation, these are denoted λtrace and λmax

the formulation of the null hypothesis differs very slightlybetween the two proceduresin both cases, acceptance or rejection of the null is decided bycomparing the test statistic with special critical valuescompiled by Johansen





Testing for cointegration





Final Exam

when: choose the dayMay 16 OR May 30

structuremultiple choice questionscovers all lectures

length2 hours

inquiriescourse page (Piazza)by email





References

Engle, R. F. and Granger, C. W. (1987), ‘Co-integration and errorcorrection: representation, estimation, and testing’,Econometrica: journal of the Econometric Society pp. 251–276.


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