Ch. 24 Johansen's mle for Cointegration We have so far considered only single-equation estimation and testing for cointe- gration. While the estimation of single equation is convenient and often consis- tent, for some purpose only estimation of a system provides sufficient information This is true, for example, when we consider the estimation of multiple cointe- grating vectors, and inference about the number of such vectors. This chapter examines methods of finding the cointegrating rank and derive the asymptotic distributions. To develop these results, we first begin with a discussion of canon- ical correlation analys 1 Canonical Correlation 1.1 Population Canonical Correlations Let the(n1 x 1) vector yt and the(n2 x 1)vector xt denote stationary ran- dom vector that are measured as deviation from their population means, so that e(yty represent the variance-covariance matrix of yt. In general, there might be complicated correlations among the element of yt and xt, i.e yt yu E(yty E(yx E(xty E(xtx' -[ If the two set are very large, the investigator may wish to study only a few of linear combination of y and xt which yield most highly correlated. He may find that the interrelation is completely described by the correlation between the first few canonical variate We now define two new(n x 1)random vectors, n, and $t, where n the smaller of ni and n2. These vectors are linear combinations of yt and xt, respectively Here, K and A' are(n xni) and(n n2)matrices, respectively. The matrices K and A' are chosen such that the following conditions holdsCh. 24 Johansen’s MLE for Cointegration We have so far considered only single-equation estimation and testing for cointe￾gration. While the estimation of single equation is convenient and often consis￾tent, for some purpose only estimation of a system provides sufficient information. This is true, for example, when we consider the estimation of multiple cointe￾grating vectors, and inference about the number of such vectors. This chapter examines methods of finding the cointegrating rank and derive the asymptotic distributions. To develop these results, we first begin with a discussion of canon￾ical correlation analysis. 1 Canonical Correlation 1.1 Population Canonical Correlations Let the (n1 × 1) vector yt and the (n2 × 1) vector xt denote stationary ran￾dom vector that are measured as deviation from their population means, so that E(yty ′ t ) represent the variance-covariance matrix of yt . In general, there might be complicated correlations among the element of yt and xt , i.e. E  yt xt   yt xt ′ =  E(yty ′ t ) E(ytx ′ t ) E(xty ′ t ) E(xtx ′ t )  =  Σyy Σyx Σxy Σxx  . If the two set are very large, the investigator may wish to study only a few of linear combination of yt and xt which yield most highly correlated. He may find that the interrelation is completely described by the correlation between the first few canonical variate. We now define two new (n×1) random vectors, ηt and ξt , where n the smaller of n1 and n2. These vectors are linear combinations of yt and xt , respectively: ηt ≡ K′yt , ξt ≡ A′xt . Here, K′ and A′ are (n×n1) and (n×n2) matrices, respectively. The matrices K′ and A′ are chosen such that the following conditions holds. 1