Lecture 29 – Cointegration IV Johansen’s MLE of the CI SpaceAssume that yt is an n-dimensional I(1) process with VEC form:Δyt = C1Δyt-1 +…+ Cp-1Δyt-p+1 + C0yt-1 + εtwhere 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” matrixNotes –1. The nxn matrix C0 has rank h2. This existence of this representation of a CI process is an implication of the “Granger Representation Theorem” (Engle and Granger, Ecta, 1987.)3. Consider the special case with n = 2, p =1, h = 1.Δy1tb1 (a1y1t-1 + a2y2t-1 ) + ε1t = Δy2t b2 (a1y1t-1 + a2y2t-1 ) + ε2t where a1y1t + a2y2t is a zero-mean I(0) process.How to estimate the VECM? OLS applied to each equation? The problem is that the OLS estimator does not constrain the matrix C0 to be rank h. Quasi-MLE: Act under the assumption that the ε’s are normally distributed, i.e., εt ~ i.i.d. N(0,Ω) and then maximize the log-likelihood function with respect to the elements of C1,…,Cp-1, A, B, and Ω. (How to select h?)The QMLE approach appears problematic because of the apparent need to estimate A and B(and, possibly, the other parameters) numerically.Johansen, in a series of papers, provided:- A simple procedure based on the principles of “reduced-rank” regressions to compute the MLE of the VECM for a given h.- The asymptotic distributions of likelihood ratio statistics for testing the size of h (including H0: h = 0)- The asymptotic distributions of the MLEs of A and B. (The distributions of the C-hatsare standard.)Johansen’s MLE for Estimating the VECM – Assume h and p are known.The log-likelihood function for the sample y1,…,yT, conditional on the initial observations y0,…,yp-1 and the assumption that the ε’s are normally distributed is:L(D,C1,…,Cp-1, B, A, Ω) =TttTTn11'21detlog2)2log(2whereεt = Δyt –D-C1Δyt-1 - … - Cp-1Δyt-p+1+BA’yt-1(Note that we’ve added an intercept here, for generality. More on its interpretation, restrictions, etc. later.)1. Fit Auxilliary RegressionsFit a p-1-order VAR to Δyt (applying OLS equation-by-equation):tptpttuyFyFFyˆˆ...ˆˆ11110Fit a regression (OLS equation by equation) of yt on 1, Δyt-1,…,Δyt-p+1:tptpttvyGyGGyˆˆ...ˆˆ111102. Compute the (squared) sample canonical coefficients for the u-hats and v-hats.That is, compute the eigenvalues nˆ,...,ˆ1 ofuvuuvuvvˆˆˆˆ11 where   T T TuvttvuttuuttvvuvTuuTvvT1 1 1''''ˆˆˆ1ˆ,ˆˆ1ˆ,ˆˆ1ˆ and, WLOG,nˆ...ˆ1.3. The MLE of the cointegrating space is:]ˆ...ˆ[ˆ1 haaA where iaˆ is the eigenvector of uvuuvuvvˆˆˆˆ11 associated with iˆ.4. The MLE of the remaining parameters are:ABuvˆˆˆ00ˆ'ˆˆˆˆGABFD 1,...,1ˆ'ˆˆˆˆ piGABFCiii)''ˆˆˆ()'ˆˆˆ(1ˆ1ttTttvABuvABuTNotes –1. Given p and h, the maximum value of the likelihood function is:)ˆ1(2ˆdetlog22)2log(2*1hiuuTTTnTnL This statistic is the basis for testing the size of h and p. Note that its calculation only requires steps 1 and 2 above.2. In the VECM, we left the intercept term, D, unrestricted. This turns out to mean thati. Each of the h CI relationships can have an interceptii. The n-h variables that are not CI withone another can have drifts.If we want to allow (i) but rule out (ii) this imposes an additional constraint on the likelihood function and requires a modification of the algorithm. (See Hamilton, pp.