# Deriving the common LIML estimator from first principles

David Hendry (1976) comments that deriving the LIML estimator is hard. I tend to agree. Guido Imbens has a nice expression here which reads \begin{eqnarray} \hat{\beta}_{LIML} = (X'(I - \lambda M_Z) X)^{-1} (X'(I-\lambda M_Z) Y), \end{eqnarray} which comes from a model \begin{eqnarray} y = X \beta + \epsilon \\ X = Z \Pi + V \end{eqnarray} Now, sadly he doesn't give a detailed derivation. I got as far as having a matrix \begin{eqnarray} \begin{pmatrix} y' P_Z z & z' P_Z X \\ X' P_Z Y & X' P_Z Z \end{pmatrix} \end{eqnarray} whose choice of eigenvector $b = \begin{pmatrix} 1 \\ -\beta \end{pmatrix}$ that corresponds to the smallest eigenvalue of said matrix should correspond to the LIML estimate that Imbens proposes.

Unfortunately, I am stuck in seeing the connection between the two and have trouble arriving at the above expression.

Can someone point me to a step-by-step derivation or suggest a way in which I can derive an estimator of $\beta$ that equals the LIML estimator above?