lecture 5: unit roots, cointegration and error...

Lecture 5: Unit Roots, Cointegration and Error Correction Models – The

Spurious Regression Problem

Prof. Massimo Guidolin

20192– Financial Econometrics

Winter/Spring 2018

Overview

2Lecture 5: Unit Roots, Cointegration and Error Correction Models – Prof. Guidolin

Defining cointegration

Vector Error Correction VAR models

Testing for cointegration and Johansen’s method

Cointegration: General Concept and Definition In finance and macroeconomics, most popular series contain a unit

root, i.e., they are I(1) series (random walks)o For instance, the US aggregate dividends and stock prices

As we shall see, however there may exist a linear combination (e.g.,their weighted difference) of them that becomes stationaryo Situations in which the choice to simultaneously difference all nonsta-

tionary series may be a mistake, as it will imply a loss of information,possibly invalid inference, and suboptimal predictive performance

o Economic theories often useful, e.g., no-arbitrage relationships We say that two non-stationary series integrated of order d are

cointegrated of order b, if there exists a linear combination of themwhich is integrated of order d-b

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Getting Intuition Through One Realistic Case

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For concreteness, let’s consider a N = 2 case in which the variables are log dividends (ldt) and log stock prices (lpt)

We represent their dynamic process as a restricted VAR(1):

This can be re-written in compact form as:

Consider the realistic case, when our variables are non-stationary, which is obtained simply by setting b1 = 1:

This way, ldt becomes a random walk with drift; because lpt is a linear function of a random walk, it becomes itself a random walk

Lecture 4: Cointegration and Error Correction Models – Prof. Guidolin

This is why the VARis restricted

a2 > 0

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To understand the essence of cointegration, consider re-parameterizing the model in the following way:

The model for changes in log-prices, i.e., for log-stock returns, is “balanced” if and only if lpt-1 - 1ldt-1 is I(0)

For a model to be balanced it means that it must involve variables of the same level of integration, i.e., I(0) = a0 + • I(0) + I(0)o Of course the model for ldt is I(0)

Cointegration can be identified from the need for models to be balanced in terms of LHS vs. RHS orders of integration



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This model is balanced if and only if lpt-1 - 1ldt-1 is I(0) However, this implies that a 1 must exist such that a weighted sum

(difference) of two I(1) variables, must be stationary Hence, lpt-1 and ldt-1 are cointegrated, with cointegrating vector

equal to (1 - 1)’ The model written as

in its first equation is called an error correction model (VECM) An error correction model represents all variables as I(0) showing

the adjustment mechanism that drives them back towards the cointegrating relationship

Dropping error correction and just estimate a VAR, model is invalid If we interpret 1ldt-1 as the long-run equilibrium for the log-stock

price, lp*t-1 = 1ldt-1, you understand meaning of the correction part

Cointegrating vector = coefficients that balance the model



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If a1 < 1, then a1 – 1 < 0, and when lpt-1 < 1ldt-1 implies that lpt> 0, i.e., when prices are below their long-run equilibrium definedby dividends, then prices will increase When lpt-1 > 1ldt-1, lpt < 0, i.e., when prices are above their long-

run equilibrium defined by dividends, prices will decrease The parameter in the ECM specification determines the speed of

adjustment in the presence of disequilibrium The system defined by the ECM based on a cointegrating

relationship is self-equilibrating

Lecture 4: Cointegration and Error Correction Models – Prof. GuidolinSmall alpha High alpha


Cointegrated I(d) variables are such because they share at least one common stochastic trend, see Appendix C for an example

Two alternative and fundamental ways to test for cointegration:① Univariate, regression-based tests (Engle and Granger’s, 1987) that exploit the idea that a regression can be used to find at least one (the mean-squared error minimal) cointegrating relationship ② Multivariate, VECM-based multi-cointegration tests, Johansen’s

Engle and Granger’s methodology seeks to determine whether the residuals of an estimated equilibrium relationship are stationaryo Suppose Pt and Ft are both I(1) and estimate the long-run equilibrium

relationship:o If the variables are cointegrated, an OLS regression yields a super-

consistent estimator of the cointegrating parameters κ0 and κ1,o OLS estimator converges faster (at a rate T) than in OLS models using

stationary variables, where the convergence rate is traditionally T1/2

o Test consists of no-intercept ADF tests applied to

Testing Bivariate Cointegrating Relationships

27Lecture 4: Cointegration and Error Correction Models – Prof. Guidolin

Matters are a bit more complicated in the multivariate case In general, among N non-stationary series of the same integration

order, we may have up to N-1 cointegrating vectorso The single equation dynamic modeling we have used in the bivariate

example may cause serious troubles when there are multiplecointegrating vectors

o There will exist a sort of indeterminacy as to which relationship holdso The solution of this identification problem requires a framework to

allow the researcher to find the number of cointegrating vectorsamong a set of variables and to identify them

The procedure proposed by Johansen (1988, 1992) within a VARframework achieves both results

Their advantage that all the r ≤ N – 1 cointegrating relationshipswill be tested and estimated

Consider the multivariate generalization of the single-equationdynamic model derived in Appendix B

Multivariate Cointegrating Relationships


Suppose that a N ≥ 2 variables are I(1) and follow:

To use Johansen’s test, the VAR needs to be turned into a V-ECM:

Johansen test centers around the matrix that can be interpretedas a long-run coefficient matrix, because in equilibrium, all the yt−iare zero, and setting ut to their expectation of zero yields

Johansen’s Method

(*)

*

*

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Formal tests based on the rank of the matrix via its eigenvalues, to determine the number of cointegrating relationships/vectors

The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from 0o The eigenvalues, λis, are put in ascending order λ1 ≥ λ2 ≥ . . . ≥ λg ≥ 0o By construction, they must be less than 1 in absolute value and

positive, and λ1 will be the largest, while λg will be the smallest If the variables are not cointegrated, the rank of will not be

significantly different from zero, so λi ≃ 0 ∀io If rank() = 1, then ln(1 − λ1) will be negative and ln(1 − λi) = 0 ∀i > 1o If the eigenvalue i is non-zero, then ln(1 − λi) < 0 ∀i > 1, for to have a

rank of 1, the largest eigenvalue must be significantly non-zero, while others will not be significantly different from 0

Two test statistics for cointegration under the Johansen approach:

Johansen’s Method


Each eigenvalue will have associated with it a differentcointegrating vector, which will be the corresponding eigenvector

A significant eigenvalue indicates a significant cointegrating vector λtrace is a joint test where the null is that the number of

cointegrating vectors is less than or equal to r against anunspecified or general alternative that they are more than r

λmax conducts separate tests on each eigenvalue, and has as its nullhypothesis that the number of cointegrating vectors is r against analternative of r + 1o The distribution of the test statistics is non-standard: the critical values

depend on N − r , the number of non-stationary components and whetherconstants are included in each of the equations

o If the test statistic is greater than the critical value, we reject the nullhypothesis of r cointegrating vectors in favor of the alternative

o The testing is conducted in a sequence and under the null r is the rank of : it cannot be of full rank (N) since this would

correspond to the original yt being stationary

Johansen’s Method


If has zero rank, then yt depends only on yt−j and not on yt−1, sothat there is no long-run relationship between the elements of yt−1:Hence there is no cointegration

For 1 < rank() < N, there are r cointegrating vectors You can show that can be defined as the product of two matrices,

say,m α and β of dimension (N × r) and (N × g), respectively, The matrix β gives the cointegrating vectors, while α gives the

amount of each cointegrating vector entering each equation of theVECM, also known as the adjustment speed coefficientso For example, suppose that N = 4, theno If r = 1, so that there is one

cointegrating vector, thenα and β will be

See posted example (optional) on cointegration among US real stockprices, dividends, and earnings ⟹ no rejection DCF for earnings

Johansen’s Method


Appendix C: Cointegration = Sharing Common Trends

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lecture 5: unit roots, cointegration and error...

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