cointegration - basic ideas and key resultsthe permanent income hypothesis (pih) implies...

IntroductionCommon Trends

Estimating Cointegrating VectorsTesting For Cointegration

Error-Correction Representation

CointegrationBasic Ideas and Key results

Egon ZakrajsekDivision of Monetary Affairs

Federal Reserve Board

Summer School in Financial MathematicsFaculty of Mathematics & Physics

University of LjubljanaSeptember 14–19, 2009




Motivation

Economic theory often suggests that certain pairs of economic orfinancial variables should be linked by a long-run economicrelationship.

Arbitrage arguments imply that the I(1) prices of certain financialtime series are linked.

The framework of cointegration deals with regression models withI(1) data.




Idea Behind Cointegration

Many economic or financial time series appear to be I(1):

I(1) variables tend to diverge as T →∞, because theirunconditional variances are proportional to T .

Thus, it may seem that I(1) variables could never be expected toobey any sort of long-run equilibrium relationship.

It is possible for two (or more) variables to be I(1), and yet a certainlinear combination of those variables to be I(0)!

If that is the case, the I(1) variables are said to be cointegrated:

If two or more I(1) variables are cointegrated, they must obey anequilibrium relationship in the long-run, although they maydiverge substantially from that equilibrium in the short run.




Some Examples

The permanent income hypothesis (PIH) implies cointegrationbetween consumption and income.

Money demand models imply cointegration between money,nominal income, prices, and interest rates.

Growth theory models imply cointegration between income,consumption, and investment.

Purchasing power parity (PPP) implies cointegration between thenominal exchange rate and foreign and domestic prices.

The Fisher equation implies cointegration between nominalinterest rates and inflation.

The expectations hypothesis of the term structure impliescointegration between nominal interest rates at differentmaturities.




Cointegration

Consider two time series y1,t and y2,t, known to be I(1):

Let y1 and y2 denote T -vectors:

y1(T×1)

=

y1,1

y1,2...

y1,T

y2(T×1)

=

y2,1

y2,2...

y2,T

y1,t and y2,t will be cointegrated if there exists a vectorη ≡ (1, η2)′ such that, when y1,t and y2,t are in equilibrium:

[y1 y2]η ≡ y1 − η2y2 = 0

The η is called a cointegrating vector.

It is clearly not unique!




Cointegration

Realistically, y1,t and y2,t are likely to be changing over timesystematically as well as stochastically.

Cointegrating relationship:

Y η = Xβ

Y = [y1 y2]X = nonstochastic matrix (i.e., constant, trends)




Cointegration

Relationship [y1 y2]η = 0 cannot be expected to hold exactly for all t.

Equilibrium error:νt = y′tη − x′tβ

In the general case when Y = [y1 y2 . . . ym], the m seriesy1,t, y2,t, . . . , ym,t are said to be cointegrated if there exists a vector ηsuch that the equilibrium error νt is I(0).




Common Trends

Consider the following two trend-stationary I(1) series:

y1,t = α1 + β1t+ u1,t

y2,t = α2 + β2t+ u2,t

{u1,t}∞t=−∞ and {u2,t}∞t=−∞ are white noise processes.

Linear combination of y1,t and y2,t with η = (1,−η)′:

νt = (α1 − ηα2) + (β1 − ηβ2)t+ u1,t − ηu2,t

νt, in general, is still I(1).The only way the νt series can be made I(0) is if η = β1/β2.The effect of combining the two series is to remove the commonlinear trend.




Common Trends

Consider the following two independent random walk processes:

y1,t = w1,t

y2,t = w2,t

wi,t = wi,t−1 + εi,t

εi,t, i = 1, 2, are two independent white noise processes.

Any linear combination of y1,t and y2,t must involve random walksw1,t and w2,t:

y1,t and y2,t cannot be cointegrated unless w1,t = w2,t

Once again, y1,t and y2,t must have a common trend.




Common Trends

In a bivariate case, the end result is that if y1,t and y2,t arecointegrated, then they must share exactly one common stochastic ordeterministic trend.

This observation, readily generalizes to multivariate cointegration:A set of m series that are cointegrated can be written as acovariance-stationary component plus a linear combination of asmaller set of common trends.The effect of cointegration is to purge these common trends fromthe resultant series.

Obvious econometric questions:How to estimate the cointegrating vector η?How to test whether two or more variables are in factcointegrated?




Estimating Cointegrating Vectors

Cointegrating relationship between m series (y1,t, y2,t, . . . , ym,t) can,in general, be written as

νt = y′tη − x′tβ

Rewrite the above equation as a linear regression and use OLS toestimate η:

y1 = Xβ + Y ∗η∗ + ν

Coefficient of y1 is arbitrarily normalized to unity.

Y ∗ = [y2 y3 . . . ym].Parameter vector η∗ is equal to minus the (m− 1) free elementsof the parameter vector η.




Estimating Cointegrating Vectors

Two potentially serious problems:

Endogeneity: If (y1,t, y2,t, . . . , ym,t) are cointegrated, they aresurely determined jointly. The error term νt will almost certainlybe correlated with the regressors!

Spurious regression: We are regressing an I(1) variable on oneor more other I(1) variables!

Nevertheless, OLS may be used to obtain a consistent estimate of anormalized cointegrating vector η.




Properties of the OLS Estimator of η∗

Important caveats (see Stock (1987) and Phillips (1991)):

T (η∗ − η∗) converges in distribution to a nonnormal RV notnecessarily centered at zero.

The OLS estimator η∗ is consistent and converges to η∗ at rate Tinstead of the usual rate

√T . That is, η∗ is super consistent.

Asymptotically, there is no simultaneity bias.

In general, the asymptotic distribution of T (η∗ − η∗) isasymptotically biased and nonnormal. The usual OLS standarderrors are not correct.

Even though the asymptotic bias goes to zero as T →∞, η∗ maybe substantially biased in finite samples.

The OLS estimator η∗ is also not efficient.




Dynamic OLS (DOLS)

Asymptotically more efficient estimates of η∗ may be obtained byDOLS (see Stock & Watson 1993):

y1 = Xβ + Y ∗η∗ +p∑

j=−p

∆Y ∗−jγj + ν

DOLS specification simply adds p leads and p lags of the firstdifference of Y ∗ to the standard cointegrating regression.

The addition of leads and lags removes the deleterious effectsthat short-run dynamics of the equilibrium process νt have on theestimate of the cointegrating vector η∗.




Properties of DOLS Estimator of η∗

The DOLS estimator η∗ is consistent, asymptotically normallydistributed, and efficient.

Asymptotically valid standard errors for the individual elementsof η∗ are given by their corresponding HAC (e.g., Newey-West)standard errors.

If T is not large relative to p(m− 1), there may be so manyadditional regressors that finite-sample properties of the DOLSestimator η∗ will actually be quite poor.




Testing for Cointegration

Basic idea (Engle & Granger 1987):If (y1,t, y2,t, . . . , ym,t) are cointegrated, the true equilibriumerror process νt must be I(0).

If they are not cointegrated, then νt must be I(1).

Test the null hypothesis of no cointegration against thealternative of cointegration by performing a unit root test on theequilibrium error process νt.




Residual-Based Cointegration Tests

Engle-Granger (EG) 2-step procedure:Choose the normalization (i.e., η1 = 1).Form the cointegrating residual νt = Y ′t η

∗.Test whether or not νt has a unit root—that is, is an I(1) process.Rejection of the null hypothesis at a pre-specified significancelevel implies that the m series are cointegrated.

Two cases to consider:The proposed cointegrating vector η is pre-specified.The proposed cointegrating vector η is estimated from the dataand an estimate of the cointegrating residual is formed:

νt = Y ′t η

Tests using the pre-specified cointegrating vector are much morepowerful!




Testing for Cointegration when η∗ is Unknown

ADF test:

∆νt = (α− 1)νt +p∑

j=1

θj∆νt−j + et

νt = y1t −X ′β − Y ′2tν

η = OLS or DOLS estimate of the cointegrating vector η




Testing for Cointegration when η∗ is Unknown

Key results:

Phillips and Ouliaris (1990) show that residual-based unit roottests applied to the estimated cointegrating residuals do not havethe usual Dickey-Fuller distributions under the null hypothesis ofno-cointegration.Because of the spurious regression phenomenon under the nullhypothesis, the distribution of these tests have asymptoticdistributions that depend on:

Deterministic terms in the regression used to estimate η.(m− 1) = number of varibles Y2t.

These distributions are known as Phillips-Ouliaris distributionsand critical values have been tabulated.




Error-Correction Model

Consider:

Yt = [y1t y2t]′ = bivariate I(1) process

Yt is cointegrated with η = [1 − η2]′

Error-Correction Model (ECM) (Engle & Granger (1987)):

∆y1t = α1 + λ1[y1t−1 − ηy2t−1]

+∑

j

β1j∆y1t−j +∑

j

γ1j∆y2t−j + e1t

∆y2t = α2 + λ2[y1t−1 − ηy2t−1]

+∑

j

β2j∆y1t−j +∑

j

γ2j∆y2t−j + e2t




Error Correction Model

ECM links the long-run equilibrium relationship between y1t

and y2t implied by cointegration with the short-run dynamicadjustment mechanism that describes how the two series reactwhen they move out of long-run equilibrium.

Parameters λ1 and λ2 measure the speed of adjustment of y1t

and y2t to the long-run equilibrium, respectively.

cointegration - basic ideas and key resultsthe permanent income hypothesis (pih) implies...

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