==============================================================================
The Kalman Filter
==============================================================================

State-Space Representation
==============================================================================
Recall the basic state-space representation

.. math::

   \begin{align}
   \underset{(r \times 1)}{\boldsymbol{\xi}_{t+1}} & =
   \underset{(r \times r)}{F}\underset{(r \times 1)}{\boldsymbol{\xi}_{t}} +
   \underset{(r \times 1)}{\boldsymbol{v}_{t+1}} \\
   \underset{(n \times 1)}{\boldsymbol{Y}_{t}} & =
   \underset{(n \times k)}{A^{'}}\underset{(k \times 1)}{\boldsymbol{x}_{t}} +
   \underset{(n \times r)}{H^{'}}\underset{(r \times 1)}{\boldsymbol{\xi}_{t}} +
   \underset{(n \times 1)}{\boldsymbol{w}_{t}} \\
   E[\boldsymbol{v}_{t}\boldsymbol{v}_{\tau}^{'}] & = \begin{cases}
   \underset{(n \times n)}{Q} & t = \tau \\ 0 & \text{o/w} \end{cases} \\
   E[\boldsymbol{w}_{t}\boldsymbol{w}_{\tau}^{'}] & = \begin{cases}
   \underset{(n \times n)}{R} & t = \tau \\ 0 & \text{o/w} \end{cases} \\
   E[\boldsymbol{v}_{t}\boldsymbol{w}_{\tau}^{'}] & = 0 \hspace{4pt} \forall
   \hspace{4pt} t,\tau.
   \end{align}

Kalman Filter Overview
==============================================================================
Collect all known information at time :math:`\smash{t}` into a vector:

.. math::

   \smash{\underset{((n+k)t \times 1)}{ \boldsymbol{\mathcal{Y}_{t}}}
   =  (\boldsymbol{Y_{t}}^{'},
   \boldsymbol{Y_{t-1}}^{'},...,\boldsymbol{Y_{1}}^{'},
   \boldsymbol{x_{t}}^{'},\boldsymbol{x_{t-1}}^{'},...,\boldsymbol{x_{1}}^{'})^{'}}

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The Kalman Filter computes:

.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{t+1|t} & =
   \hat{E}[\boldsymbol{\xi}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
   \underset{(r \times r)}{P_{t+1|t}} & =
   E[(\boldsymbol{\xi}_{t+1}-\hat{\boldsymbol{\xi}}_{t+1|t})(\xi_{t+1} -
   \hat{\boldsymbol{\xi}}_{t+1|t})^{'}],
   \end{align}

where :math:`\smash{P_{t+1|t}}` is the MSE matrix for
:math:`\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}`.

Starting the Recursion
==============================================================================
We begin the recursion with

.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{1|0} & =
   E[\boldsymbol{\xi}_{1}|\mathcal{Y}_{0} = \emptyset] =
   E[\boldsymbol{\xi}_{1}] \\
   P_{1|0} & = E[(\boldsymbol{\xi}_{1} -
   E[\boldsymbol{\xi}_{1}])(\boldsymbol{\xi}_{1} -
   E[\boldsymbol{\xi}_{1}])^{'}].
   \end{align}

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According to the state equation, the unconditional expectation of
:math:`\smash{\boldsymbol{\xi}_{t}}` is:

.. math::

   \begin{align}
   E[\boldsymbol{\xi}_{t+1}] & = FE[\boldsymbol{\xi}_{t}] \\
   \implies E[\boldsymbol{\xi}_{t}] & = FE[\boldsymbol{\xi}_{t}] \\
   \implies (I_{r} - F)E[\boldsymbol{\xi}_{t}] & = 0 \\
   \implies E[\xi_{t}] & = 0.
   \end{align}

Starting the Recursion
==============================================================================
Further, the state equation also implies the unconditional variance of
:math:`\smash{\boldsymbol{\xi}_{t}}` is:

.. math::

   \begin{align}
   \underset{\Sigma}{\underbrace{E[\boldsymbol{\xi}_{t+1}\boldsymbol{\xi}_{t+1}^{'}]}}
   & = E[(F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1}) (F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1})^{'}] \\
   & =
   F\underset{\Sigma}{\underbrace{E[\boldsymbol{\xi}_{t}\boldsymbol{\xi}_{t}^{'}]}}F^{'} +
   F\underset{0}{\underbrace{E[\boldsymbol{\xi}_{t}\boldsymbol{v}_{t+1}^{'}]}} +
   \underset{0}{\underbrace{E[\boldsymbol{v}_{t+1}\boldsymbol{\xi}_{t}^{'}]}}F^{'} +
   \underset{Q}{\underbrace{E[\boldsymbol{v}_{t+1}\boldsymbol{v}_{t+1}^{'}]}} \\
   \implies \Sigma & = F \Sigma F^{'} + Q \\
   \implies Vec(P_{1|0}) & = Vec(\Sigma) = [I_{r^{2}} - (F \otimes
   F)]^{-1} Vec(Q).
   \end{align}

Forecasting :math:`\smash{Y_{t}}`
==============================================================================
Our objective will be to obtain
:math:`\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}` and
:math:`\smash{P_{t+1|t}}`, given values for
:math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}` and
:math:`\smash{P_{t|t-1}}`.

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- :math:`\smash{\boldsymbol{x}_{t}}` contains no information about
  :math:`\smash{\boldsymbol{\xi}_{t}}` beyond what is contained in
  :math:`\smash{\boldsymbol{\mathcal{Y}}_{t-1}}`:

.. math::

   \smash{E[\boldsymbol{\xi}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}]
   = E[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t-1}] =
   \hat{\boldsymbol{\xi}}_{t|t-1}}.

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According to the observation equation:

.. math::

   \begin{align}
   \hat{\boldsymbol{Y}}_{t|t-1} & =
   \hat{E}[\boldsymbol{Y}_{t}|
   \boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}] \\
   & = A^{'}\boldsymbol{x}_{t} +
   H^{'}E[\boldsymbol{\xi}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}] +
   \underset{0}{\underbrace{E[\boldsymbol{w}_{t}| \boldsymbol{x}_{t},
   \boldsymbol{\mathcal{Y}}_{t-1}]}} \\
   & = A^{'}\boldsymbol{x}_{t} + H^{'} \hat{\boldsymbol{\xi}}_{t|t-1}.
   \end{align}

Forecast Error
==============================================================================
The forecast error is:

.. math::

   \begin{align}
   \boldsymbol{Y}_{t} - \hat{\boldsymbol{Y}}_{t|t-1} & =
   A^{'}\boldsymbol{x}_{t} + H^{'}\boldsymbol{\xi}_{t} +
   \boldsymbol{w}_{t} - A^{'}\boldsymbol{x}_{t} -
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1} \\
   & = H^{'}(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1}) + \boldsymbol{w}_{t},
   \end{align}

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which has the MSE matrix:

.. math::

   \begin{align}
   E[(\boldsymbol{Y}_{t} -
   \hat{\boldsymbol{Y}}_{t|t-1})(\boldsymbol{Y}_{t} & -
   \hat{\boldsymbol{Y}}_{t|t-1})^{'}] \\
   & = H^{'}\underset{P_{t|t-1}}{\underbrace{E[(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1})(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1})^{'}]}}H
   +\underset{R}{\underbrace{E[\boldsymbol{w}_{t}\boldsymbol{w}_{t}^{'}]}} \\
   & = H^{'}P_{t|t-1}H + R.
   \end{align}

Forecast MSE
==============================================================================
We have used the fact that:

.. math::

   \smash{E[\boldsymbol{w}_{t}(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1})] = 0}

because :math:`\smash{E[\boldsymbol{w}_{t}\boldsymbol{\xi}_{t}^{'}] =
0}` and because

.. math::

   \smash{E[\boldsymbol{w}_{t}\hat{\boldsymbol{\xi}}_{t|t-1}^{'}] =
   E[\boldsymbol{w}_{t}(F\xi_{t-1})^{'}] =
   E[\boldsymbol{w}_{t}\boldsymbol{\xi}_{t-1}^{'}]F^{'} = 0}.

Update the forecast of :math:`\smash{\xi_{t}}`
==============================================================================
After we observe :math:`\smash{\boldsymbol{Y}_{t}}`, we can obtain a
new forecast of :math:`\smash{\boldsymbol{\xi}_{t}}`:

.. math::

   \smash{\hat{\boldsymbol{\xi}}_{t|t} =
   E[\boldsymbol{\xi}_{t}|\boldsymbol{Y}_{t},\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}]
   = E[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}]}.

Update the forecast of :math:`\smash{\xi_{t}}`
==============================================================================
The formula for updating a linear projection in this fashion is:

.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{t|t} & = \hat{\boldsymbol{\xi}}_{t|t-1} +
   E[({\boldsymbol{\xi}}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1})
   (\boldsymbol{Y}_{t}-\hat{\boldsymbol{Y}}_{t|t-1})^{'}] \\
   & \hspace{2in} \times E[(\boldsymbol{Y}_{t} -
   \hat{\boldsymbol{Y}}_{t|t-1})(\boldsymbol{Y}_{t} -
   \hat{\boldsymbol{Y}}_{t|t-1})^{'}]^{-1} \hspace{3pt}
   (\boldsymbol{Y}_{t} - \hat{\boldsymbol{Y}}_{t|t-1}) \\
   & = \hat{\boldsymbol{\xi}}_{t|t-1} + E[(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t-1})(\boldsymbol{w}_{t}^{'} +
   (\boldsymbol{\xi}_{t} - \hat{\boldsymbol{\xi}}_{t|t-1})^{'}H)] \\
   & \hspace{2in} \times (H^{'}P_{t|t-1}H +
   R)^{-1}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} -
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
   & = \hat{\boldsymbol{\xi}}_{t|t-1} +
   \overset{K_{t}}{\overbrace{P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}}}
   \hspace{3pt} (\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} -
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}).
   \end{align}

Update the forecast of :math:`\smash{\xi_{t}}`
==============================================================================
The associated MSE is:

.. math::

   \begin{align}
   P_{t|t} & = E[(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t})(\boldsymbol{\xi}_{t} -
   \hat{\boldsymbol{\xi}}_{t|t})^{'}] \\
   & = P_{t|t-1} -
   P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}H^{'}P_{t|t-1}.
   \end{align}

Forecasting :math:`\smash{\boldsymbol{\xi}_{t+1}}`
==============================================================================
.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{t+1|t} & =
   \hat{E}[\boldsymbol{\xi}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
   & = F\hat{E}[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}] +
   E[\boldsymbol{v}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
   & = F\hat{\boldsymbol{\xi}}_{t|t} \\
   & = F\hat{\boldsymbol{\xi}}_{t|t-1} + F
   K_{t}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}).
   \end{align}

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.. math::

   \begin{align}
   P_{t+1|t} & = E[(\boldsymbol{\xi}_{t+1} -
   \hat{\boldsymbol{\xi}}_{t+1|t})(\boldsymbol{\xi}_{t+1} -
   \hat{\boldsymbol{\xi}}_{t+1|t})^{'}] \\
   & = E[( F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1} -
   F\hat{\boldsymbol{\xi}}_{t|t})(F\boldsymbol{\xi}_{t} +
   \boldsymbol{v}_{t+1} - F\hat{\boldsymbol{\xi}}_{t|t})^{'}] \\
   & = F P_{t|t} F^{'} + Q \\
   & = F(P_{t|t-1} - P_{t|t-1} H(H^{'}P_{t|t-1}H + R)^{-1} \hspace{3pt}
   H^{'}P_{t|t-1})F^{'} + Q.
   \end{align}

Forecasting :math:`\smash{\boldsymbol{Y}_{t+1}}`
==============================================================================
.. math::

   \smash{\hat{\boldsymbol{Y}}_{t+1|t} =
   E[\boldsymbol{Y}_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{\mathcal{Y}}_{t}]=
   A^{'}\boldsymbol{x}_{t+1} + H^{'} \hat{\boldsymbol{\xi}}_{t+1|t}},

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which has associated MSE:

.. math::

   \smash{E[(\boldsymbol{Y}_{t+1} - \hat{\boldsymbol{Y}}_{t+1|t})(\boldsymbol{Y}_{t+1} -
   \hat{\boldsymbol{Y}}_{t+1|t})^{'}] = H^{'}P_{t+1|t}H + R}.

Forecasting :math:`\smash{\boldsymbol{Y}_{t+s}}`
==============================================================================
Iterating on the state equation:

.. math::

   \begin{gather}
   \boldsymbol{\xi}_{t+s} = F^{s}\boldsymbol{\xi}_{t} +
   F^{s-1}\boldsymbol{v}_{t+1} + F^{s-2}\boldsymbol{v}_{t+2} +
   \ldots + F\boldsymbol{v}_{t+s-1} + \boldsymbol{v}_{t+s} \\
   \implies
   E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\xi}_{t},\boldsymbol{\mathcal{Y}}_{t}]
   = F^{s}\boldsymbol{\xi}_{t}.
   \end{gather}

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By the law of iterated expectations:

.. math::

   \smash{\hat{\boldsymbol{\xi}}_{t+s|t} =
   E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\mathcal{Y}}_{t}] =
   E[E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\xi}_{t},\boldsymbol{\mathcal{Y}}_{t}]|\boldsymbol{\mathcal{Y}}_{t}]
   = E[F^{s}\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}] =
   F^{s}\hat{\boldsymbol{\xi}}_{t|t}}.

Forecasting :math:`\smash{\boldsymbol{Y}_{t+s}}`
==============================================================================
The forecast error is:

.. math::

   \smash{\boldsymbol{\xi}_{t+s} - \hat{\boldsymbol{\xi}}_{t+s|t} =
   F^{s}(\boldsymbol{\xi}_{t} - \hat{\boldsymbol{\xi}}_{t|t}) +
   F^{s-1}\boldsymbol{v}_{t+1} + \ldots + F\boldsymbol{v}_{t+s-1} +
   \boldsymbol{v}_{t+s}},

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with MSE:

.. math::

   \smash{P_{t+s|t} = F^{s}P_{t|t}(F^{'})^{s} +
   F^{s-1}Q(F^{'})^{s-1} + \ldots + FQF^{'} + Q }.

Forecasting :math:`\smash{\boldsymbol{Y}_{t+s}}`
==============================================================================
Rewrite the observation equation:

.. math::

   \smash{\boldsymbol{Y}_{t+s} = A^{'}\boldsymbol{x}_{t+s} +
   H^{'}\boldsymbol{\xi}_{t+s} + \boldsymbol{w}_{t+s}}.

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

Thus,

.. math::

   \smash{\hat{\boldsymbol{Y}}_{t+s|t}  =
   E[\boldsymbol{Y}_{t+s}|\boldsymbol{x}_{t+s},\boldsymbol{\mathcal{Y}}_{t}]
   = A^{'}\boldsymbol{x}_{t+s} + H^{'}\hat{\boldsymbol{\xi}}_{t+s|t}}.

Forecasting :math:`\smash{\boldsymbol{Y}_{t+s}}`
==============================================================================
The forecast error is:

.. math::

   \begin{align}
   \boldsymbol{Y}_{t+s} - \hat{\boldsymbol{Y}}_{t+s|t} & =
   A^{'}\boldsymbol{x}_{t+s} + H^{'}\boldsymbol{\xi}_{t+s} +
   \boldsymbol{w}_{t+s} - A^{'}\boldsymbol{x}_{t+s}  -
   H^{'}\hat{\boldsymbol{\xi}}_{t+s|t} \\
   & = H^{'}(\boldsymbol{\xi}_{t+s} -
   \hat{\boldsymbol{\xi}}_{t+s|t}) + \boldsymbol{w}_{t+s},
   \end{align}

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

with MSE:

.. math::

   \smash{E[(\boldsymbol{Y}_{t+s} -
   \hat{\boldsymbol{Y}}_{t+s|t})(\boldsymbol{Y}_{t+s} -
   \hat{\boldsymbol{Y}}_{t+s|t})^{'}] = H^{'}P_{t+s|t}H + R}.

Summary of Kalman Filter Steps
==============================================================================
1. Start with forecast :math:`\smash{\hat{\boldsymbol{\xi}}_{1|0}}`
   and associated MSE matrix :math:`\smash{P_{1|0}}`

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

2. Given forecast :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}`
   and MSE :math:`\smash{P_{t|t-1}}`, compute

.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{t|t} & = \hat{\boldsymbol{\xi}}_{t|t-1} +
   \overset{K_{t}}{\overbrace{P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}}}
   \hspace{3pt} (\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
   P_{t|t} & = P_{t|t-1} -
   P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}H^{'}P_{t|t-1}.
   \end{align}

Summary of Kalman Filter Steps
==============================================================================
3. Given :math:`\smash{\hat{\boldsymbol{\xi}}_{t|t}}` and MSE
   :math:`\smash{P_{t|t}}`, compute

.. math::

   \begin{align}
   \hat{\boldsymbol{\xi}}_{t+1|t} & =
   F\hat{\boldsymbol{\xi}}_{t|t-1} + F
   K_{t}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
   H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
   P_{t+1|t} & = F(P_{t|t-1} - P_{t|t-1} H(H^{'}P_{t|t-1}H + R)^{-1}
   \hspace{3pt} H^{'}P_{t|t-1})F^{'} + Q.
   \end{align}

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4. Given :math:`\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}` and MSE
   :math:`\smash{P_{t+1|t}}`, compute

.. math::

   \begin{align}
   \hat{\boldsymbol{Y}}_{t+1|t} & =
   A^{'}\boldsymbol{x}_{t+1} + H^{'} \hat{\boldsymbol{\xi}}_{t+1|t}
   \\
   E[(\boldsymbol{Y}_{t+1} -
   \hat{\boldsymbol{Y}}_{t+1|t})(\boldsymbol{Y}_{t+1} -
   \hat{\boldsymbol{Y}}_{t+1|t})^{'}] & = H^{'}P_{t+1|t}H + R.
   \end{align}

Example: Long-Run Risks
==============================================================================
.. math::

   \begin{gather}
   x_{t+1} = \rho x_{t} + \varphi_{e}\sigma e_{t+1} \\
   g_{t+1} = \log\left(C_{t+1}/C_{t}\right) = \mu + x_{t} + \sigma
   \eta_{t+1} \\
   g_{d,t+1} = \log\left(D_{t+1}/D_{t}\right) = \mu_{d} + \phi
   x_{t} + \varphi_{d}\sigma u_{t+1} \\
   \varphi_{t+1},u_{t+1},\eta_{t+1} \overset{i.i.d.}{\sim} N(0,1) \\
   \end{gather}

where :math:`\smash{C_{t}}` and :math:`\smash{D_{t}}` are aggregate
consumption and aggregate dividends.