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or clue that the relationship
between Y and X is spurious.
It indicates that Y and X do notmove
together in the long-run.Warwick Business School 6
Spurious regression: the cause
The OLS estimator is:If Y and X are stationary and cov(X,ε)=0 (X is exogenous) then the OLS estimator is consistent:However with Y and X~I(1) and β=0 then ε~I(1). In that case the OLS estimator is inconsistent–it does notconverge on β=0 ⇒Y and X appear to be related (even as T→∞).The reason is that the stochastictrends in X and ε(both are I(1)) causes the sample covariance between X and εto diverge(in probability) ⇒it does nottend to cov(X,ε)=0. ()ΣΣΣΣΣΣ−−−−+=+==2112112ˆiiiiiiiiiixTxTxTxxTxyxεβεββSample covariancebetween X and εSample variance of X()ββεεppiiXxT→⇒→Σ−ˆ,cov1yprobabilitin econvergenc''≡→pWarwick Business School 7
CointegrationIn generallinear combinations of I(1) variables form spurious relationships:We get the case just analyzed when:Therefore, in OLS estimation with I(1) variables, typically: 1.Point estimators are inconsistent because the error term/z is I(1)2.The t-stats follow non –normal distributions (see slide 4).An important exceptionto 1. is where there are values of the β’s such that z~I(0):Σ=+=njjtjtIXz10)1(~ββ.0 ),1(~=−−=ββμIXYztttΣ=+=njjtjtIXz10)0(~ββz is a linear combination of I(1) variables which is I(0).This combination of the variables is called a: COINTEGRATING RELATIONSHIPIn this case z is CI(1,1) (‘cointegratedof order one-one’) In general if z is a linear combination of I(d) variables which is I(d-b) (b>0) then z is CI(d,b).The relationship is spuriousbecause there is no tendency for the series to move togetherin the long-run (z is I(1)).Warwick Business School 8
Cointegration
The intuition behind cointegrationis that the I(1) variables share the same fundamentals/long-run components:
⇒The variables share a common stochastic trend
Individuallythe variables vary widely in the long-run:
⇒Their variance is infinite.
⇒Their spectra are infinite at frequency zero(⇒∞long-run variation).
But in combinationthe variables move togetherin the long-run:
⇒The variance of the combination is finite.
⇒The spectrum of the combination at frequency zero is finite.
In effect the dominant long-run components of the individual variables ‘cancel out’in the cointegratingrelationship.Warwick Business School 9
Cointegration
Cointegrationis a veryimportant concept in empirical finance because it means that variables which:
•Have no equilibriumtendency individually(because they are I(1))
•Are nonetheless bound together in equilibrium as a group(because a linear combination of the variables is I(0))
Cointegrationanalysis is therefore very important in analyzing the long-run/equilibrium properties of a system of non-stationary variables.
Clive Granger shared the Nobel Prize in Economics in 2003 (with Robert Engle who got it for ARCH):
“for methods of analyzing economic time series with common trends (cointegration)”.Warwick Business School 10
Cointegration: properties of OLS estimator
The cointegratingrelationship can be written as a linear regression equation:The fact that ε~I(0) makes a bigdifference to the OLS estimator compared to the spurious regressions analysis:–Not only are the OLS estimators consistent they are SUPER-CONSISTENT.–This means the estimator conv