  In this paper, we investigate the consistency and asymptotic efficiency of an
estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are
not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$
are in the right half space (i.e., eigenvalues with positive real parts). In
this case the process grows exponentially fast. (2) The eigenvalues of $F$ are
on the left half space (i.e., the eigenvalues with negative or zero real
parts). The process where all eigenvalues of $F$ have negative real parts is
called a stable process and has a unique invariant (i.e., stationary)
distribution. In this case the process does not grow. When the eigenvalues of
$F$ have zero real parts (i.e., the case of zero eigenvalues and purely
imaginary eigenvalues) the process grows polynomially fast. Considering (1) and
(2) separately, we first show that an estimator, $\hat{F}$, of $F$ is
consistent. We then combine them to present results for the general
Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic
efficiency of the estimator.
