==============================================================================
ML Estimation of State-Space Models
==============================================================================

Kalman Filter Forecasts
==============================================================================
The Kalman filter forecasts :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}` and
:math:`\smash{\hat{\boldsymbol{Y}}_{t|t-1}}` are linear projections of
:math:`\smash{\boldsymbol{\xi}_{t}}` and :math:`\smash{\boldsymbol{Y}_{t}}` on
:math:`\smash{(\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1})}`.

.. raw:: <!--pause-->

- They are optimal among all forecasts that are linear functions of
  :math:`\smash{(\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1})}`.

.. raw:: <!--pause-->

- If :math:`\smash{\boldsymbol{\xi}_{1}}` and
  :math:`\smash{\{\boldsymbol{w}_{t},\boldsymbol{v}_{t}\}_{t=1}^{T}}` are multivariate
  Gaussian, :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}` and
  :math:`\smash{\hat{\boldsymbol{Y}}_{t|t-1}}` are optimal among *all*
  forecasts that are functions of
  :math:`\smash{(\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1})}` (linear and
  non-linear).

Conditional Distribution of :math:`\smash{\boldsymbol{Y}_t}`
==============================================================================
The distribution of
:math:`\smash{\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}}` is also
multivariate Gaussian, of the form:

.. math::

   \smash{\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1} \sim
   MVN(A^{'}\boldsymbol{x}_{t} + H^{'}\hat{\boldsymbol{\xi}}_{t|t-1},H^{'}P_{t|t-1}H
   +R)}

Conditional Distribution of :math:`\smash{\boldsymbol{Y}_t}`
==============================================================================
Thus, the density function is

.. math::

   \begin{align}
   f_{\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},
   \boldsymbol{\mathcal{Y}}_{t-1}} & (\boldsymbol{Y}_{t}|
   \boldsymbol{x}_{t},
   \boldsymbol{\mathcal{Y}}_{t-1},\boldsymbol{\theta}) \\
   & = (2\pi)^{-n/2} | H^{'}P_{t|t-1}H + R|^{-1/2} \\
   & \hspace{1in} \times
   \exp\bigg\{-\frac{1}{2}\left(\boldsymbol{Y}_{t} - A^{'}\boldsymbol{x}_{t} -
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}\right) \\
   & \hspace{2.5in} \times \left(H^{'}P_{t|t-1}H +
   R\right)^{-1} \\
   & \hspace{3.5in} \times \left(\boldsymbol{Y}_{t} - A^{'}\boldsymbol{x}_{t} -
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}\right)^{'}\bigg\}
   \end{align}

where :math:`\smash{\boldsymbol{\theta}}` aggregates all known parameters in
:math:`\smash{F, A, H, Q,}` and :math:`\smash{R}`.

Log-likelihood
==============================================================================
The log-likelihood is the joint density

.. math::

   \smash{\ell(\boldsymbol{\theta}) =  \sum_{t=1}^{T}
   \log\left(f_{\boldsymbol{Y}_{t}|\boldsymbol{X}_{t},
   \boldsymbol{\mathcal{Y}}_{t-1}}
   (\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},
   \boldsymbol{\mathcal{Y}}_{t-1},\boldsymbol{\theta})\right)}

.. raw:: <!--pause-->

- The log-likelihood can be maximized numerically with respect to
  :math:`\smash{F(\boldsymbol{\theta}), A(\boldsymbol{\theta}), H(\boldsymbol{\theta}),
  Q(\boldsymbol{\theta})}`, and :math:`\smash{R(\boldsymbol{\theta})}`.

.. raw:: <!--pause-->

- This is an exact log likelihood and yields exact MLEs.

.. raw:: <!--pause-->

- Maximum likelihood estimation for :math:`\smash{MA}` and
  :math:`\smash{ARMA}` can be performed in this manner.

Basic Prescription
==============================================================================
1. Guess :math:`\smash{\boldsymbol{\theta}^{(0)}}`

.. raw:: <!--pause-->

2. Given :math:`\smash{\boldsymbol{\theta}^{(s)}}`, compute
   :math:`\smash{F(\boldsymbol{\theta}^{(s)}), A(\boldsymbol{\theta}^{(s)}),
   H(\boldsymbol{\theta}^{(s)}), Q(\boldsymbol{\theta}^{(s)}),}` and
   :math:`\smash{R(\boldsymbol{\theta}^{(s)})}`.

.. raw:: <!--pause-->

3. Use the Kalman Filter to iteratively compute
   :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}` and
   :math:`\smash{P_{t|t-1}}`, :math:`\smash{t =1,\ldots,T}`.

.. raw:: <!--pause-->

4. Compute the log-likelihood using
   :math:`\smash{H(\boldsymbol{\theta}^{(s)}), A(\boldsymbol{\theta}^{(s)}),
   R(\boldsymbol{\theta}^{(s)})}`, and
   :math:`\smash{\{\hat{\boldsymbol{\xi}}_{t|t-1},P_{t|t-1}\}_{t=1}^{T}}`.

.. raw:: <!--pause-->

5. Use a numerical method to update
   :math:`\smash{\boldsymbol{\theta}^{(s+1)}}`.

.. raw:: <!--pause-->

6. If :math:`\smash{||\boldsymbol{\theta}^{(s+1)} - \boldsymbol{\theta}^{(s)}|| <
   \tau}`, stop.  Otherwise, set :math:`\smash{i = i +1}` and return
   to step 2.

Basic Prescription
==============================================================================
Updating :math:`\smash{\boldsymbol{\theta}^{(i)} \rightarrow
\boldsymbol{\theta}^{(i+1)}}` may involve numerical or analytical
derivatives.

.. raw:: <!--pause-->

- Analytical derivatives of the log likelihood with respect to each
  :math:`\smash{\theta_{i}}` will involve

.. math::

   \smash{ \frac{\partial \hat{\boldsymbol{\xi}}_{t|t-1}
   (\boldsymbol{\theta})}{\partial \theta_{i}} \hspace{4pt} \text{
   and } \hspace{4pt} \frac{\partial P_{t|t-1}}{\partial
   \theta_{i}}}.

.. raw:: <!--pause-->

- These derivatives can be updated recursively similar to
  :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}` and :math:`\smash{P_{t|t-1}}`.