Indeed, there is assumption of no serial correlation, but errors can be correlated between responses. Yet, this correlation is to be estimated (after computing ) so it cannot be used before.

CCE is not ML estimator, Sunberg99:

The classical estimator is not ML (unless q=p). Heuristically this is because only p-vector part of the q-vector is used for estimation of the unknown x, and the remaining q-p components contain some (little) information about

Brown87: Confidence and Conflict in Multivariate Calibration, Philip J. Brown; Rolf Sundberg; Journal of the Royal Statistical Society. Series B (Methodological), Vol. 49, No. 1. (1987), pp. 46-57.

I may not understand the initial setup, but equation system (1) appears to be just a system of q equations with the same exogenous variables in each equation, a “reduced form” system in the terminology of econometrics. The estimator of alpha and B in (3a) and (3b) is like an OLS. I am not clear here why the vectors of ones is not just placed in the “X” and standard notation used, but that is not my real question.

My problem comes from the G and the maximum likelihood estimator. Is G diagonal? If so, then I will trust you on the algebra for the rest. But if G is not diagonal, i.e. not known to be diagonal apriori, then (3a) isn’t the ML estimator of B.

Let me restate my question in the time dimension and equation dimension. As I understand, we are assuming no serial correlation, and thus the covariance matrix for each equation is diagonal. But what about between equations? If there is correlation of the error terms between equations — known in econometrics as a system of “Seemingly Unrelated Regression,” or SUR, equations — then that information would be in the likelihood function (the covariance of the i_th error term in equation k with the i_th error term in equation k’).

Further, while I am decades out of date with the research, I don’t believe the finite sample distribution of SUR estimators for non-orthogonal regressors has been reduced to a tractable form and incorporated into the standard statistical software. Now it may be that the inverse estimation problem is more tractable although that is not likely, and thus, I don’t think that those F-statistics can be anything more than an approximation to the critical regions.

Since the proxies are the original dependents, I can’t say as to whether or not one can hypothesize zero cross-equation correlation, but with temperatures themselves the cross-equation (geographical correlation) is not zero (although a SUR, with ARMA, estimation of temperature doesn’t offer much more than a standard, one-equation ARMA estimation).

1) http://www.climateaudit.org/pdf/statistics/brown.1982.jrss.pdf

2,3) I have only paper copies of those

I’ am Walid. I’m working on multivariate calibration and intervals calibration. I m needing copies of the following 3 articles:

1- Brown 82: Multivariate Calibration, Journal of the Royal Statistical Society. Ser B. Vol. 44, No. 3, pp. 287-321

2- Williams 69: Regression methods in calibration problems. Bull. ISI., 43, 17-28

3 -Krutchkoff 67: Classical and inverse regression methods of calibration. Technometrics, 9, 425-439..

Please contact me on: walidgani@yahoo.fr

Done.

The second equation corresponds to one new observation Y’ (row vector), which doesn’t belong to calibration data. X’ is transposed in the original Brown’s text (xi in 2.2), I guess the reason is to
have unknown vector X’ ( to be estimated) in column vector form.