# Cointegration. Basic Ideas and Key results. Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board

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1 Cointegration Basic Ideas and Key results Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana September 14 19, 2009

2 Motivation Introduction Economic theory often suggests that certain pairs of economic or financial variables should be linked by a long-run economic relationship. Arbitrage arguments imply that the I(1) prices of certain financial time series are linked. The framework of cointegration deals with regression models with I(1) data.

3 Idea Behind Cointegration Many economic or financial time series appear to be I(1): I(1) variables tend to diverge as T, because their unconditional variances are proportional to T. Thus, it may seem that I(1) variables could never be expected to obey any sort of long-run equilibrium relationship. It is possible for two (or more) variables to be I(1), and yet a certain linear 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 an equilibrium relationship in the long-run, although they may diverge substantially from that equilibrium in the short run.

4 Some Examples The permanent income hypothesis (PIH) implies cointegration between 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 the nominal exchange rate and foreign and domestic prices. The Fisher equation implies cointegration between nominal interest rates and inflation. The expectations hypothesis of the term structure implies cointegration between nominal interest rates at different maturities.

5 Cointegration Consider two time series y 1,t and y 2,t, known to be I(1): Let y 1 and y 2 denote T -vectors: y 1 = (T 1) y 1,1 y 1,2. y 1,T y 2 = (T 1) y 2,1 y 2,2. y 2,T y 1,t and y 2,t will be cointegrated if there exists a vector η (1, η 2 ) such that, when y 1,t and y 2,t are in equilibrium: [y 1 y 2 ]η y 1 η 2 y 2 = 0 The η is called a cointegrating vector. It is clearly not unique!

6 Cointegration Realistically, y 1,t and y 2,t are likely to be changing over time systematically as well as stochastically. Cointegrating relationship: Y η = Xβ Y = [y 1 y 2 ] X = nonstochastic matrix (i.e., constant, trends)

7 Cointegration Relationship [y 1 y 2 ]η = 0 cannot be expected to hold exactly for all t. Equilibrium error: ν t = y tη x tβ In the general case when Y = [y 1 y 2... y m ], the m series y 1,t, y 2,t,..., y m,t are said to be cointegrated if there exists a vector η such that the equilibrium error ν t is I(0).

8 Consider the following two trend-stationary I(1) series: y 1,t = α 1 + β 1 t + u 1,t y 2,t = α 2 + β 2 t + u 2,t {u 1,t } t= and {u 2,t } t= are white noise processes. Linear combination of y 1,t and y 2,t with η = (1, η) : ν t = (α 1 ηα 2 ) + (β 1 ηβ 2 )t + u 1,t ηu 2,t ν t, in general, is still I(1). The only way the ν t series can be made I(0) is if η = β 1 /β 2. The effect of combining the two series is to remove the common linear trend.

9 Consider the following two independent random walk processes: y 1,t = w 1,t y 2,t = w 2,t w i,t = w i,t 1 + ɛ i,t ɛ i,t, i = 1, 2, are two independent white noise processes. Any linear combination of y 1,t and y 2,t must involve random walks w 1,t and w 2,t : y 1,t and y 2,t cannot be cointegrated unless w 1,t = w 2,t Once again, y 1,t and y 2,t must have a common trend.

10 In a bivariate case, the end result is that if y 1,t and y 2,t are cointegrated, then they must share exactly one common stochastic or deterministic trend. This observation, readily generalizes to multivariate cointegration: A set of m series that are cointegrated can be written as a covariance-stationary component plus a linear combination of a smaller set of common trends. The effect of cointegration is to purge these common trends from the resultant series. Obvious econometric questions: How to estimate the cointegrating vector η? How to test whether two or more variables are in fact cointegrated?

11 Cointegrating relationship between m series (y 1,t, y 2,t,..., y m,t ) can, in general, be written as ν t = y tη x tβ Rewrite the above equation as a linear regression and use OLS to estimate η: y 1 = Xβ + Y η + ν Coefficient of y 1 is arbitrarily normalized to unity. Y = [y 2 y 3... y m ]. Parameter vector η is equal to minus the (m 1) free elements of the parameter vector η.

12 Two potentially serious problems: Endogeneity: If (y 1,t, y 2,t,..., y m,t ) are cointegrated, they are surely determined jointly. The error term ν t will almost certainly be correlated with the regressors! Spurious regression: We are regressing an I(1) variable on one or more other I(1) variables! Nevertheless, OLS may be used to obtain a consistent estimate of a normalized cointegrating vector η.

13 Properties of the OLS Estimator of η Important caveats (see Stock (1987) and Phillips (1991)): T (ˆη η ) converges in distribution to a nonnormal RV not necessarily centered at zero. The OLS estimator ˆη is consistent and converges to η at rate T instead of the usual rate T. That is, ˆη is super consistent. Asymptotically, there is no simultaneity bias. In general, the asymptotic distribution of T (ˆη η ) is asymptotically biased and nonnormal. The usual OLS standard errors are not correct. Even though the asymptotic bias goes to zero as T, ˆη may be substantially biased in finite samples. The OLS estimator ˆη is also not efficient.

14 Dynamic OLS (DOLS) Asymptotically more efficient estimates of η may be obtained by DOLS (see Stock & Watson 1993): y 1 = Xβ + Y η + p j= p Y jγ j + ν DOLS specification simply adds p leads and p lags of the first difference of Y to the standard cointegrating regression. The addition of leads and lags removes the deleterious effects that short-run dynamics of the equilibrium process ν t have on the estimate of the cointegrating vector η.

15 Properties of DOLS Estimator of η The DOLS estimator ˆη is consistent, asymptotically normally distributed, and efficient. Asymptotically valid standard errors for the individual elements of ˆη 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 many additional regressors that finite-sample properties of the DOLS estimator ˆη will actually be quite poor.

16 Testing for Cointegration Basic idea (Engle & Granger 1987): If (y 1,t, y 2,t,..., y m,t ) are cointegrated, the true equilibrium error process ν t must be I(0). If they are not cointegrated, then ν t must be I(1). Test the null hypothesis of no cointegration against the alternative of cointegration by performing a unit root test on the equilibrium error process ν t.

17 Residual-Based Cointegration Tests Engle-Granger (EG) 2-step procedure: Choose the normalization (i.e., η 1 = 1). Form the cointegrating residual ν t = Y t η. Test whether or not ν t has a unit root that is, is an I(1) process. Rejection of the null hypothesis at a pre-specified significance level 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 data and an estimate of the cointegrating residual is formed: ˆν t = Y t ˆη Tests using the pre-specified cointegrating vector are much more powerful!

18 Testing for Cointegration when η is Unknown ADF test: ˆν t = (α 1)ˆν t + p θ j ˆν t j + e t j=1 ˆν t = y 1t X ˆβ Y 2tˆν ˆη = OLS or DOLS estimate of the cointegrating vector η

19 Testing for Cointegration when η is Unknown Key results: Phillips and Ouliaris (1990) show that residual-based unit root tests applied to the estimated cointegrating residuals do not have the usual Dickey-Fuller distributions under the null hypothesis of no-cointegration. Because of the spurious regression phenomenon under the null hypothesis, the distribution of these tests have asymptotic distributions that depend on: Deterministic terms in the regression used to estimate η. (m 1) = number of varibles Y 2t. These distributions are known as Phillips-Ouliaris distributions and critical values have been tabulated.

20 Error-Correction Model Consider: Y t = [y 1t y 2t ] = bivariate I(1) process Y t is cointegrated with η = [1 η 2 ] Error-Correction Model (ECM) (Engle & Granger (1987)): y 1t = α 1 + λ 1 [y 1t 1 ηy 2t 1 ] + j β 1j y 1t j + j γ 1j y 2t j + e 1t y 2t = α 2 + λ 2 [y 1t 1 ηy 2t 1 ] + j β 2j y 1t j + j γ 2j y 2t j + e 2t

21 Error Correction Model ECM links the long-run equilibrium relationship between y 1t and y 2t implied by cointegration with the short-run dynamic adjustment mechanism that describes how the two series react when they move out of long-run equilibrium. Parameters λ 1 and λ 2 measure the speed of adjustment of y 1t and y 2t to the long-run equilibrium, respectively.

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