The Kalman Filter¶
State-Space Representation¶
Recall the basic state-space representation
\[\begin{split}\begin{align}
\underset{(r \times 1)}{\boldsymbol{\xi}_{t+1}} & =
\underset{(r \times r)}{F}\underset{(r \times 1)}{\boldsymbol{\xi}_{t}} +
\underset{(r \times 1)}{\boldsymbol{v}_{t+1}} \\
\underset{(n \times 1)}{\boldsymbol{Y}_{t}} & =
\underset{(n \times k)}{A^{'}}\underset{(k \times 1)}{\boldsymbol{x}_{t}} +
\underset{(n \times r)}{H^{'}}\underset{(r \times 1)}{\boldsymbol{\xi}_{t}} +
\underset{(n \times 1)}{\boldsymbol{w}_{t}} \\
E[\boldsymbol{v}_{t}\boldsymbol{v}_{\tau}^{'}] & = \begin{cases}
\underset{(n \times n)}{Q} & t = \tau \\ 0 & \text{o/w} \end{cases} \\
E[\boldsymbol{w}_{t}\boldsymbol{w}_{\tau}^{'}] & = \begin{cases}
\underset{(n \times n)}{R} & t = \tau \\ 0 & \text{o/w} \end{cases} \\
E[\boldsymbol{v}_{t}\boldsymbol{w}_{\tau}^{'}] & = 0 \hspace{4pt} \forall
\hspace{4pt} t,\tau.
\end{align}\end{split}\]
Kalman Filter Overview¶
Collect all known information at time \(\smash{t}\) into a vector:
\[\smash{\underset{((n+k)t \times 1)}{ \boldsymbol{\mathcal{Y}_{t}}}
= (\boldsymbol{Y_{t}}^{'},
\boldsymbol{Y_{t-1}}^{'},...,\boldsymbol{Y_{1}}^{'},
\boldsymbol{x_{t}}^{'},\boldsymbol{x_{t-1}}^{'},...,\boldsymbol{x_{1}}^{'})^{'}}\]
The Kalman Filter computes:
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{t+1|t} & =
\hat{E}[\boldsymbol{\xi}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
\underset{(r \times r)}{P_{t+1|t}} & =
E[(\boldsymbol{\xi}_{t+1}-\hat{\boldsymbol{\xi}}_{t+1|t})(\xi_{t+1} -
\hat{\boldsymbol{\xi}}_{t+1|t})^{'}],
\end{align}\end{split}\]
where \(\smash{P_{t+1|t}}\) is the MSE matrix for \(\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}\).
Starting the Recursion¶
We begin the recursion with
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{1|0} & =
E[\boldsymbol{\xi}_{1}|\mathcal{Y}_{0} = \emptyset] =
E[\boldsymbol{\xi}_{1}] \\
P_{1|0} & = E[(\boldsymbol{\xi}_{1} -
E[\boldsymbol{\xi}_{1}])(\boldsymbol{\xi}_{1} -
E[\boldsymbol{\xi}_{1}])^{'}].
\end{align}\end{split}\]
According to the state equation, the unconditional expectation of \(\smash{\boldsymbol{\xi}_{t}}\) is:
\[\begin{split}\begin{align}
E[\boldsymbol{\xi}_{t+1}] & = FE[\boldsymbol{\xi}_{t}] \\
\implies E[\boldsymbol{\xi}_{t}] & = FE[\boldsymbol{\xi}_{t}] \\
\implies (I_{r} - F)E[\boldsymbol{\xi}_{t}] & = 0 \\
\implies E[\xi_{t}] & = 0.
\end{align}\end{split}\]
Starting the Recursion¶
Further, the state equation also implies the unconditional variance of \(\smash{\boldsymbol{\xi}_{t}}\) is:
\[\begin{split}\begin{align}
\underset{\Sigma}{\underbrace{E[\boldsymbol{\xi}_{t+1}\boldsymbol{\xi}_{t+1}^{'}]}}
& = E[(F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1}) (F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1})^{'}] \\
& =
F\underset{\Sigma}{\underbrace{E[\boldsymbol{\xi}_{t}\boldsymbol{\xi}_{t}^{'}]}}F^{'} +
F\underset{0}{\underbrace{E[\boldsymbol{\xi}_{t}\boldsymbol{v}_{t+1}^{'}]}} +
\underset{0}{\underbrace{E[\boldsymbol{v}_{t+1}\boldsymbol{\xi}_{t}^{'}]}}F^{'} +
\underset{Q}{\underbrace{E[\boldsymbol{v}_{t+1}\boldsymbol{v}_{t+1}^{'}]}} \\
\implies \Sigma & = F \Sigma F^{'} + Q \\
\implies Vec(P_{1|0}) & = Vec(\Sigma) = [I_{r^{2}} - (F \otimes
F)]^{-1} Vec(Q).
\end{align}\end{split}\]
Forecasting \(\smash{Y_{t}}\)¶
Our objective will be to obtain \(\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}\) and \(\smash{P_{t+1|t}}\), given values for \(\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}\) and \(\smash{P_{t|t-1}}\).
- \(\smash{\boldsymbol{x}_{t}}\) contains no information about \(\smash{\boldsymbol{\xi}_{t}}\) beyond what is contained in \(\smash{\boldsymbol{\mathcal{Y}}_{t-1}}\):
\[\smash{E[\boldsymbol{\xi}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}]
= E[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t-1}] =
\hat{\boldsymbol{\xi}}_{t|t-1}}.\]
According to the observation equation:
\[\begin{split}\begin{align}
\hat{\boldsymbol{Y}}_{t|t-1} & =
\hat{E}[\boldsymbol{Y}_{t}|
\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}] \\
& = A^{'}\boldsymbol{x}_{t} +
H^{'}E[\boldsymbol{\xi}_{t}|\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}] +
\underset{0}{\underbrace{E[\boldsymbol{w}_{t}| \boldsymbol{x}_{t},
\boldsymbol{\mathcal{Y}}_{t-1}]}} \\
& = A^{'}\boldsymbol{x}_{t} + H^{'} \hat{\boldsymbol{\xi}}_{t|t-1}.
\end{align}\end{split}\]
Forecast Error¶
The forecast error is:
\[\begin{split}\begin{align}
\boldsymbol{Y}_{t} - \hat{\boldsymbol{Y}}_{t|t-1} & =
A^{'}\boldsymbol{x}_{t} + H^{'}\boldsymbol{\xi}_{t} +
\boldsymbol{w}_{t} - A^{'}\boldsymbol{x}_{t} -
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1} \\
& = H^{'}(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1}) + \boldsymbol{w}_{t},
\end{align}\end{split}\]
which has the MSE matrix:
\[\begin{split}\begin{align}
E[(\boldsymbol{Y}_{t} -
\hat{\boldsymbol{Y}}_{t|t-1})(\boldsymbol{Y}_{t} & -
\hat{\boldsymbol{Y}}_{t|t-1})^{'}] \\
& = H^{'}\underset{P_{t|t-1}}{\underbrace{E[(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1})(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1})^{'}]}}H
+\underset{R}{\underbrace{E[\boldsymbol{w}_{t}\boldsymbol{w}_{t}^{'}]}} \\
& = H^{'}P_{t|t-1}H + R.
\end{align}\end{split}\]
Forecast MSE¶
We have used the fact that:
\[\smash{E[\boldsymbol{w}_{t}(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1})] = 0}\]
because \(\smash{E[\boldsymbol{w}_{t}\boldsymbol{\xi}_{t}^{'}] = 0}\) and because
\[\smash{E[\boldsymbol{w}_{t}\hat{\boldsymbol{\xi}}_{t|t-1}^{'}] =
E[\boldsymbol{w}_{t}(F\xi_{t-1})^{'}] =
E[\boldsymbol{w}_{t}\boldsymbol{\xi}_{t-1}^{'}]F^{'} = 0}.\]
Update the forecast of \(\smash{\xi_{t}}\)¶
After we observe \(\smash{\boldsymbol{Y}_{t}}\), we can obtain a new forecast of \(\smash{\boldsymbol{\xi}_{t}}\):
\[\smash{\hat{\boldsymbol{\xi}}_{t|t} =
E[\boldsymbol{\xi}_{t}|\boldsymbol{Y}_{t},\boldsymbol{x}_{t},\boldsymbol{\mathcal{Y}}_{t-1}]
= E[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}]}.\]
Update the forecast of \(\smash{\xi_{t}}\)¶
The formula for updating a linear projection in this fashion is:
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{t|t} & = \hat{\boldsymbol{\xi}}_{t|t-1} +
E[({\boldsymbol{\xi}}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1})
(\boldsymbol{Y}_{t}-\hat{\boldsymbol{Y}}_{t|t-1})^{'}] \\
& \hspace{2in} \times E[(\boldsymbol{Y}_{t} -
\hat{\boldsymbol{Y}}_{t|t-1})(\boldsymbol{Y}_{t} -
\hat{\boldsymbol{Y}}_{t|t-1})^{'}]^{-1} \hspace{3pt}
(\boldsymbol{Y}_{t} - \hat{\boldsymbol{Y}}_{t|t-1}) \\
& = \hat{\boldsymbol{\xi}}_{t|t-1} + E[(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t-1})(\boldsymbol{w}_{t}^{'} +
(\boldsymbol{\xi}_{t} - \hat{\boldsymbol{\xi}}_{t|t-1})^{'}H)] \\
& \hspace{2in} \times (H^{'}P_{t|t-1}H +
R)^{-1}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} -
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
& = \hat{\boldsymbol{\xi}}_{t|t-1} +
\overset{K_{t}}{\overbrace{P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}}}
\hspace{3pt} (\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} -
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}).
\end{align}\end{split}\]
Update the forecast of \(\smash{\xi_{t}}\)¶
The associated MSE is:
\[\begin{split}\begin{align}
P_{t|t} & = E[(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t})(\boldsymbol{\xi}_{t} -
\hat{\boldsymbol{\xi}}_{t|t})^{'}] \\
& = P_{t|t-1} -
P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}H^{'}P_{t|t-1}.
\end{align}\end{split}\]
Forecasting \(\smash{\boldsymbol{\xi}_{t+1}}\)¶
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{t+1|t} & =
\hat{E}[\boldsymbol{\xi}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
& = F\hat{E}[\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}] +
E[\boldsymbol{v}_{t+1}|\boldsymbol{\mathcal{Y}}_{t}] \\
& = F\hat{\boldsymbol{\xi}}_{t|t} \\
& = F\hat{\boldsymbol{\xi}}_{t|t-1} + F
K_{t}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}).
\end{align}\end{split}\]
\[\begin{split}\begin{align}
P_{t+1|t} & = E[(\boldsymbol{\xi}_{t+1} -
\hat{\boldsymbol{\xi}}_{t+1|t})(\boldsymbol{\xi}_{t+1} -
\hat{\boldsymbol{\xi}}_{t+1|t})^{'}] \\
& = E[( F\boldsymbol{\xi}_{t} + \boldsymbol{v}_{t+1} -
F\hat{\boldsymbol{\xi}}_{t|t})(F\boldsymbol{\xi}_{t} +
\boldsymbol{v}_{t+1} - F\hat{\boldsymbol{\xi}}_{t|t})^{'}] \\
& = F P_{t|t} F^{'} + Q \\
& = F(P_{t|t-1} - P_{t|t-1} H(H^{'}P_{t|t-1}H + R)^{-1} \hspace{3pt}
H^{'}P_{t|t-1})F^{'} + Q.
\end{align}\end{split}\]
Forecasting \(\smash{\boldsymbol{Y}_{t+1}}\)¶
\[\smash{\hat{\boldsymbol{Y}}_{t+1|t} =
E[\boldsymbol{Y}_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{\mathcal{Y}}_{t}]=
A^{'}\boldsymbol{x}_{t+1} + H^{'} \hat{\boldsymbol{\xi}}_{t+1|t}},\]
which has associated MSE:
\[\smash{E[(\boldsymbol{Y}_{t+1} - \hat{\boldsymbol{Y}}_{t+1|t})(\boldsymbol{Y}_{t+1} -
\hat{\boldsymbol{Y}}_{t+1|t})^{'}] = H^{'}P_{t+1|t}H + R}.\]
Forecasting \(\smash{\boldsymbol{Y}_{t+s}}\)¶
Iterating on the state equation:
\[\begin{split}\begin{gather}
\boldsymbol{\xi}_{t+s} = F^{s}\boldsymbol{\xi}_{t} +
F^{s-1}\boldsymbol{v}_{t+1} + F^{s-2}\boldsymbol{v}_{t+2} +
\ldots + F\boldsymbol{v}_{t+s-1} + \boldsymbol{v}_{t+s} \\
\implies
E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\xi}_{t},\boldsymbol{\mathcal{Y}}_{t}]
= F^{s}\boldsymbol{\xi}_{t}.
\end{gather}\end{split}\]
By the law of iterated expectations:
\[\smash{\hat{\boldsymbol{\xi}}_{t+s|t} =
E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\mathcal{Y}}_{t}] =
E[E[\boldsymbol{\xi}_{t+s}|\boldsymbol{\xi}_{t},\boldsymbol{\mathcal{Y}}_{t}]|\boldsymbol{\mathcal{Y}}_{t}]
= E[F^{s}\boldsymbol{\xi}_{t}|\boldsymbol{\mathcal{Y}}_{t}] =
F^{s}\hat{\boldsymbol{\xi}}_{t|t}}.\]
Forecasting \(\smash{\boldsymbol{Y}_{t+s}}\)¶
The forecast error is:
\[\smash{\boldsymbol{\xi}_{t+s} - \hat{\boldsymbol{\xi}}_{t+s|t} =
F^{s}(\boldsymbol{\xi}_{t} - \hat{\boldsymbol{\xi}}_{t|t}) +
F^{s-1}\boldsymbol{v}_{t+1} + \ldots + F\boldsymbol{v}_{t+s-1} +
\boldsymbol{v}_{t+s}},\]
with MSE:
\[\smash{P_{t+s|t} = F^{s}P_{t|t}(F^{'})^{s} +
F^{s-1}Q(F^{'})^{s-1} + \ldots + FQF^{'} + Q }.\]
Forecasting \(\smash{\boldsymbol{Y}_{t+s}}\)¶
Rewrite the observation equation:
\[\smash{\boldsymbol{Y}_{t+s} = A^{'}\boldsymbol{x}_{t+s} +
H^{'}\boldsymbol{\xi}_{t+s} + \boldsymbol{w}_{t+s}}.\]
Thus,
\[\smash{\hat{\boldsymbol{Y}}_{t+s|t} =
E[\boldsymbol{Y}_{t+s}|\boldsymbol{x}_{t+s},\boldsymbol{\mathcal{Y}}_{t}]
= A^{'}\boldsymbol{x}_{t+s} + H^{'}\hat{\boldsymbol{\xi}}_{t+s|t}}.\]
Forecasting \(\smash{\boldsymbol{Y}_{t+s}}\)¶
The forecast error is:
\[\begin{split}\begin{align}
\boldsymbol{Y}_{t+s} - \hat{\boldsymbol{Y}}_{t+s|t} & =
A^{'}\boldsymbol{x}_{t+s} + H^{'}\boldsymbol{\xi}_{t+s} +
\boldsymbol{w}_{t+s} - A^{'}\boldsymbol{x}_{t+s} -
H^{'}\hat{\boldsymbol{\xi}}_{t+s|t} \\
& = H^{'}(\boldsymbol{\xi}_{t+s} -
\hat{\boldsymbol{\xi}}_{t+s|t}) + \boldsymbol{w}_{t+s},
\end{align}\end{split}\]
with MSE:
\[\smash{E[(\boldsymbol{Y}_{t+s} -
\hat{\boldsymbol{Y}}_{t+s|t})(\boldsymbol{Y}_{t+s} -
\hat{\boldsymbol{Y}}_{t+s|t})^{'}] = H^{'}P_{t+s|t}H + R}.\]
Summary of Kalman Filter Steps¶
- Start with forecast \(\smash{\hat{\boldsymbol{\xi}}_{1|0}}\) and associated MSE matrix \(\smash{P_{1|0}}\)
- Given forecast \(\smash{\hat{\boldsymbol{\xi}}_{t|t-1}}\) and MSE \(\smash{P_{t|t-1}}\), compute
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{t|t} & = \hat{\boldsymbol{\xi}}_{t|t-1} +
\overset{K_{t}}{\overbrace{P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}}}
\hspace{3pt} (\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
P_{t|t} & = P_{t|t-1} -
P_{t|t-1}H(H^{'}P_{t|t-1}H + R)^{-1}H^{'}P_{t|t-1}.
\end{align}\end{split}\]
Summary of Kalman Filter Steps¶
- Given \(\smash{\hat{\boldsymbol{\xi}}_{t|t}}\) and MSE \(\smash{P_{t|t}}\), compute
\[\begin{split}\begin{align}
\hat{\boldsymbol{\xi}}_{t+1|t} & =
F\hat{\boldsymbol{\xi}}_{t|t-1} + F
K_{t}(\boldsymbol{Y}_{t}-A^{'}\boldsymbol{x}_{t} +
H^{'}\hat{\boldsymbol{\xi}}_{t|t-1}) \\
P_{t+1|t} & = F(P_{t|t-1} - P_{t|t-1} H(H^{'}P_{t|t-1}H + R)^{-1}
\hspace{3pt} H^{'}P_{t|t-1})F^{'} + Q.
\end{align}\end{split}\]
- Given \(\smash{\hat{\boldsymbol{\xi}}_{t+1|t}}\) and MSE \(\smash{P_{t+1|t}}\), compute
\[\begin{split}\begin{align}
\hat{\boldsymbol{Y}}_{t+1|t} & =
A^{'}\boldsymbol{x}_{t+1} + H^{'} \hat{\boldsymbol{\xi}}_{t+1|t}
\\
E[(\boldsymbol{Y}_{t+1} -
\hat{\boldsymbol{Y}}_{t+1|t})(\boldsymbol{Y}_{t+1} -
\hat{\boldsymbol{Y}}_{t+1|t})^{'}] & = H^{'}P_{t+1|t}H + R.
\end{align}\end{split}\]
Example: Long-Run Risks¶
\[\begin{split}\begin{gather}
x_{t+1} = \rho x_{t} + \varphi_{e}\sigma e_{t+1} \\
g_{t+1} = \log\left(C_{t+1}/C_{t}\right) = \mu + x_{t} + \sigma
\eta_{t+1} \\
g_{d,t+1} = \log\left(D_{t+1}/D_{t}\right) = \mu_{d} + \phi
x_{t} + \varphi_{d}\sigma u_{t+1} \\
\varphi_{t+1},u_{t+1},\eta_{t+1} \overset{i.i.d.}{\sim} N(0,1) \\
\end{gather}\end{split}\]
where \(\smash{C_{t}}\) and \(\smash{D_{t}}\) are aggregate consumption and aggregate dividends.