13A.6 Johansen's method: reduced-rank regression

13A.6.1Concentrating out the short run#

Take the VECM (13A.9)–(13A.10) with Gaussian errors $\varepsilon_t\sim N(0,\Omega)$. Regress $\Delta y_t$ and $y_{t-1}$ separately on the short-run regressors $(1, \Delta y_{t-1},\dots,\Delta y_{t-p+1})$ and call the residuals $R_{0t}$ and $R_{1t}$. By the Frisch–Waugh–Lovell theorem the likelihood, maximised over $\nu$ and the $\Gamma_i$, depends only on the concentrated model

Define moment matrices $S_{ij} = T^{-1}\sum_t R_{it}R_{jt}'$ for $i,j\in\{0,1\}$.

13A.6.2Maximising over $\alpha$ and $\Omega$, then $\beta$#

For fixed $\beta$, (13A.16) is a multivariate regression of $R_0$ on the $r$ regressors $\beta'R_1$, so

and the concentrated log-likelihood is $-\tfrac T2\ln|\hat\Omega(\beta)| + \text{const}$.

Hence

Proof

By (13A.17) we must minimise the ratio $|\beta'(S_{11}-S_{10}S_{00}^{-1}S_{01})\beta|/|\beta'S_{11}\beta|$ over $n\times r$ matrices $\beta$. The ratio is unchanged by $\beta\mapsto\beta M$ for invertible $M$, so normalise $\beta'S_{11}\beta = I_r$; the problem becomes: minimise $|I_r - \beta'S_{10}S_{00}^{-1}S_{01}\beta|$ subject to $\beta'S_{11}\beta=I_r$. Write $\beta = S_{11}^{-1/2}U$ with $U'U = I_r$ and $B = S_{11}^{-1/2}S_{10}S_{00}^{-1}S_{01}S_{11}^{-1/2}$ (symmetric, eigenvalues $\hat\lambda_i\in[0,1)$, the same as those of (13A.18)). We minimise $|I_r - U'BU|$. By the Poincaré separation theorem the eigenvalues $\mu_1\ge\dots\ge\mu_r$ of $U'BU$ satisfy $\mu_i\le\hat\lambda_i$, so $|I_r - U'BU| = \prod(1-\mu_i) \ge \prod_{i\le r}(1-\hat\lambda_i)$, with equality when $U$ spans the top $r$ eigenvectors of $B$. Mapping back gives $\beta = (v_1,\dots,v_r)$, and substituting into $-\tfrac T2\ln|\hat\Omega|$ gives (13A.19).∎

The $\hat\lambda_i$ are the squared canonical correlations between $R_0$ and $R_1$: how much of the change in prices is predictable from each direction of the lagged levels.

13A.6.3The two rank tests#

From (13A.19), the likelihood-ratio statistic for $H(r)$: rank $\le r$, against $H(n)$: unrestricted, is

and for $H(r)$ against $H(r+1)$,

Test sequentially: $r = 0$ first; if rejected, $r = 1$; stop at the first non-rejection. The limiting distributions are functionals of an $(n-r)$-dimensional Brownian motion and depend on how the constant and trend enter (inside or outside the cointegrating relation; Johansen 1995 distinguishes five cases). For a pair with an unrestricted constant (coint_johansen(det_order=0) in statsmodels), the 5% critical values reported there are 15.49 ($\lambda_{\text{trace}}(0)$), 14.26 ($\lambda_{\max}(0)$) and 3.84 (both tests at $r=1$).