Lecture 30 – Inference in the VECM Recall the VEC representation of a n-dimensional cointegrated system with cointegrating rank h – ∆yt = D + C1∆yt-1 +…+ Cp-1∆yt-p+1 + C0yt-1 + εt where εt ~ w.n. (Ω) C0 = -BA’, A is an nxh matrix, h < n, which spans the CI space of y (i.e., A has rank h and A’yt ~ I(0)), B is an nxh “factor-loading” matrix Last time we looked at Johansen’s algorithm for producing the QMLE of the system’s parameters for given h (and p).

Testing the Cointegrating Rank, h Recall that part of the output of the Johansen algorthim is a set of n eigenvalues that are derived from the residual var-cov matrices of the auxilliary regressions: nλλ ˆ...1 >> If yt ~ CI(h), then λh+1=λh+2=…=λn=0.

1. The λ-trace test Consider H0: h = h0 vs. HA: h > h0 (h0 ε 0,1,…,n-1) The LR statistic:

2*(LHA-LH0) = = λ∑+

−−n

hiT

10

)ˆ1log( λ trace

Johansen showed that under H0 this statistic converges in distribution to a random variable, which depends only on n-h0 and whether/how an intercept is included in the VECM. (“trace statistic”? – the asymptotic null distribution of this statistic is the distribution of the trace of a random matrix)

2. The λ-max test Consider H0: h = h0 vs. HA: h = h0+1 (h0 ε 0,1,…,n-1) The LR statistic: 2*(LHA-LH0) = = λ)ˆ1log( 10 +

−− hT λ max

Johansen showed that under H0 this statistic converges in distribution to a random variable, which depends only on n-h0 and whether/how an intercept is included in the VECM. (“max statistic”? – the asymptotic null distribution of this statistic is the

distribution of the maximum eigenvalue of a random matrix)

Note – Case 2 vs. Case 3 An intercept would typically be included in the VECM and auxilliary regressions to allow for a nonzero intercept in the cointegrating relationships and, in some cases, deterministic trends in the components of y. Allowing for a nonzero intercept in the cointegrating relationships but not allowing for deterministic trends imposes restrictions on the VECM (and auxilliary regressions) intercept term that should be accounted for in estimating that model. (Hamilton, 643-645.) In testing restrictions on h, the model is estimated the same way (unrestricted intercepts in the auxilliary regressions) regardless of whether there are deterministic trends or not. However, the limiting distribution of the test statistics depend on whether there are no deterministic trends (Case 2) or there are deterministic trends (Case 3).

Testing for the presence of deterministic trends in y: Under the null hypothesis that there are no deterministic trends in yt , 2(LU-LR) ~ X2(n-h) where LU is the log-likelihood value when the VECM is fit without restricting the intercept and LR is the log-likelihood value when the VECM is fit subject to the null restriction.

Testing linear restrictions on the cointegrating space – The cointegrating space is an h-dimensional subspace of Rn spanned by the columns of A. Suppose we want to test the restriction that A lies in a particular m-dimension subspace of Rn, h < s < n. That is, consider H0: A = Q*α where Q is an nxs matrix of known constants α is an unknown sxh parameter matrix. For example – Suppose h = 1 and the null hypothesis is that the first n-1 elements of y are cointegrated; i.e., the coefficient on yn,t is zero for all CI vectors of y. What are Q and α? Under H0,

))(()]ˆ1log()ˆ1[log()(2 2

1,0,0 snhTLL

h

DAiiA −Χ→−−−=−− ∑ λλ Causality tests in CI systems? See Phillips and Toda (1993,1994)