lecture 4: cointegration and error correction...

Lecture 4: Cointegration and Error Correction Models

Prof. Massimo Guidolin

20192– Financial Econometrics Spring 2015

Overview

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General notion

The case of log dividends and log stock prices

Error correction models

Multivariate cointegrating relationships

Johansen’s procedure

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

General concept

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In finance and macroeconomics, most popular series contain a unit root, i.e., they are I(1) series (random walks) o For instance, the natural log of UK aggregate dividends and stock

prices in the lecture notes As we shall see, however there exists a linear combination (their

difference) of them that becomes stationary o When the difference makes them stationary, we say that the two

series are cointegrated with a cointegrating vector (1, -1)’ In general, we say that two non-stationary series integrated of

order d are cointegrated of order b, if there exists a linear combination of them which is integrated of order d-b

For concreteness and also to help you follow the lecture notes (in case you work on them), let’s consider a N = 2 case in which the variables are log dividends (ldt) and log stock prices (lpt)

Linear combinations of I(1) series mat exist that make the result stationary; in this case the series are cointegrated


One important case of cointegrated variables

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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


This is why the VAR is restricted


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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 + x 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 (ECM) If we interpret 1ldt-1 as the long-run equilibrium level for the log-

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

Cointegrating vector = coefficients that balance the model An error correction model represents all variables as I(0)

showing the adjustment mechanism that drives them back towards the cointegrating relationship



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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 defined by 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. Guidolin Small alpha High alpha

Matters are a bit more complicated in the multivariate case In general, among N non-stationary series we may have up to N-1

cointegrating vectors o The single equation dynamic modeling we have used in the bivariate

example may cause serious troubles when there are multiple cointegrating vectors

o See the lecture notes for one such example o The solution of this identification problem requires a framework to

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

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

Consider the multivariate generalization of the single-equation dynamic model discussed above

Multivariate Cointegrating Relationships

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Among N non-stationary series we may have up to N-1 cointegrating vectors


Suppose that a set of g variables (g ≥ 2) are under consideration that are I(1) and which are thought may be cointegrated and for which we estimate an unrestricted VAR(k):

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

Johansen test centers around an examination of the matrix that can be interpreted as a long-run coefficient matrix, because in equilibrium, all the yt−i will be zero, and setting ut , to their expected value of zero will yield

Johansen’s method

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Johansen’s methods is based on an analysis of the rank of the matrix in the VECM representation


The test for cointegration is calculated by examining the rank of the matrix via its eigenvalues

The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from 0 o The eigenvalues, denoted λi are put in ascending order λ1 ≥ λ2 ≥ . . . ≥ λg o 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 ∀i o If rank() = 1, then ln(1 − λ1) will be negative and ln(1 − λi) = 0 ∀i > 1 o 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

Johansen’s method

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The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from 0


There are two test statistics for cointegration under the Johansen approach:

where r is the number of cointegrating vectors under the null hypothesis and λ-hat are the estimated values for the eigenvalues o Intuitively, the larger is an eigenvalue, the larger and more negative

will be ln(1 − λi-hat) and hence the larger will be the test statistic Each eigenvalue will have associated with it a different

cointegrating 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 an unspecified or general alternative that they are more than r

Johansen’s method

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Two tests are used within the method, trace and max eigenvalue


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

values depend on g − r , the number of non-stationary components and whether constants are included in each of the equations

o If the test statistic is greater than the critical value, we reject the null hypothesis 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 (g) since this would

correspond to the original yt being stationary If has zero rank, then yt depends only on yt−j and not on yt−1, so

that there is no long-run relationship between the elements of yt−1: Hence there is no cointegration

Johansen’s method

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Both the λtrace and λmax tests have non-standard distributions and their null hypotheses are slightly different


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

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

amount of each cointegrating vector entering each equation of the VECM, also known as the adjustment speed coefficients For example, suppose that g = 4, then

If r = 1, so that there is one cointegrating

vector, then α and β will be

Johansen’s method

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For 1 < rank() < g, there are r cointegrating vectors = αβ, where β gives the cointegrating vectors, while α

contains the adjustment speed coefficients


Carefully read these Lecture Slides + class notes

Possibly read BROOKS, chapter 8. Lecture Notes are available on Prof. Guidolin’s personal web page Campbell, J. Y. and R. Shiller (1987) "Cointegration and Present Value Models",

Journal of Political Economy, 95, 1062-1088.

Reading List/How to prepare the exam

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

lecture 4: cointegration and error correction...

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