Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem

Transcription

1 Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem Prof. Massimo Guidolin Financial Econometrics Winter/Spring 2018

2 Overview Defining cointegration Vector Error Correction VAR models Testing for cointegration and Johansen s method Lecture 5: Unit Roots, Cointegration and Error Correction Models Prof. Guidolin 2

3 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 stationary o Situations in which the choice to simultaneously difference all nonstationary 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 them which is integrated of order d-b 22

4 Getting Intuition Through One Realistic Case For concreteness, let s consider a N = 2 case in which the variables are log dividends (ld t ) and log stock prices (lp t ) We represent their dynamic process as a restricted VAR(1): This can be re-written in compact form as: This is why the VAR is restricted Consider the realistic case, when our variables are non-stationary, which is obtained simply by setting b 1 = 1: a 2 > 0 This way, ld t becomes a random walk with drift; because lp t is a linear function of a random walk, it becomes itself a random walk 23

5 Getting Intuition Through One Realistic Case Cointegration can be identified from the need for models to be balanced in terms of LHS vs. RHS orders of integration To understand the essence of cointegration, consider reparameterizing 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 lp t-1-1 ld t-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) = a 0 + I(0) + I(0) o Of course the model for ld t is I(0) 24

6 Getting Intuition Through One Realistic Case Cointegrating vector = coefficients that balance the model This model is balanced if and only if lp t-1-1 ld t-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, lp t-1 and ld t-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 1 ld t-1 as the long-run equilibrium for the log-stock price, lp* t-1 = 1 ld t-1, you understand meaning of the correction part 25

7 Getting Intuition Through One Realistic Case If a 1 < 1, then a 1 1 < 0, and when lp t-1 < 1 ld t-1 implies that lp t > 0, i.e., when prices are below their long-run equilibrium defined by dividends, then prices will increase When lp t-1 > 1 ld t-1, lp t < 0, i.e., when prices are above their longrun 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 Small alpha High alpha 26

8 Testing Bivariate Cointegrating Relationships 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 stationary o Suppose P t and F t are both I(1) and estimate the long-run equilibrium relationship: o If the variables are cointegrated, an OLS regression yields a superconsistent estimator of the cointegrating parameters κ 0 and κ 1 o OLS estimator converges faster (at a rate T) than in OLS models using stationary variables, where the convergence rate is traditionally T 1/2 o Test consists of no-intercept ADF tests applied to 27

9 Multivariate Cointegrating Relationships 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 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 There will exist a sort of indeterminacy as to which relationship holds 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 Their advantage that all the r N 1 cointegrating relationships will be tested and estimated Consider the multivariate generalization of the single-equation dynamic model derived in Appendix B 28

10 Johansen s Method 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 interpreted as a long-run coefficient matrix, because in equilibrium, all the y t i are zero, and setting u t to their expectation of zero yields * (*) * 29

11 Johansen s Method 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 0 o The eigenvalues, λ i s, are put in ascending order λ 1 λ 2... λ g 0 o By construction, they must be less than 1 in absolute value and positive, and λ 1 will be the largest, while λ g will be the smallest If the variables are not cointegrated, the rank of will not be significantly different from zero, so λ i 0 i o If rank( ) = 1, then ln(1 λ 1 ) will be negative and ln(1 λ i ) = 0 i > 1 o If the eigenvalue i is non-zero, then ln(1 λ i ) < 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: 30

12 Johansen s Method 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 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 N 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 (N) since this would correspond to the original y t being stationary 31

13 Johansen s Method If has zero rank, then y t depends only on y t j and not on y t 1, so that there is no long-run relationship between the elements of y t 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 the VECM, also known as the adjustment speed coefficients o For example, suppose that N = 4, then o If r = 1, so that there is one cointegrating vector, then α and β will be See posted example (optional) on cointegration among US real stock prices, dividends, and earnings no rejection DCF for earnings 32

14 Appendix A: An Example of I(2) Process 33

15 Appendix B: Deriving the ADF Test Regression 34

16 Appendix C: Cointegration = Sharing Common Trends 35