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[IrisToolbox] for Macroeconomic Modeling

Intro to Kalman filter

jaromir.benes@iris-toolbox.com

\[ \newcommand{\E}[1]{\mathrm E_{#1}\!} \newcommand{\C}[1]{{\color{darkred}#1}} \]

Overview

  • Major Kalman filter advances appeared in the 1950s–1960s primarily for spacecraft control (dynamic positioning)

  • Recursive form of a conditioning problem

    • Estimate directly unobserved quantities
    • Reduce computational demands
    • Run in real time

  • In macro, we could do stacked time

  • A number of useful characteristics from recursive

    • Prediction errors
    • Likelihood
    • Conditional MSEs

Reverse logic

Simulation: "causality"

Initial conditions + shocks \(\to\) variables

Kalman filtering: "diagnostics"

Observations of variables \(\to\) the most likely combination of initial conditions and shocks

State space system

Run for \(t = 1, \dots, T\):

Transition equations: Dynamics
\[ x_t = T \, x_{t-1} + k + R \, e_t \]


Measurement equations: Static transformation
\[ y_t = Z \, x_t + d + H \, u_t \]

Initializing the filter

To start the Kalman filter, we need to describe the distribution of the initial condition

\[ x_{0|0} \equiv \E{0} \left[ x_0 \right] \]
\[ P_{0|0} \equiv \mathrm{MSE}_0\! \left[x_0\right] \equiv \E{0} \left[ \left(x_{0|0} - x_0\right) \left(x_{0|0}-x_0\right)' \right] \]

The Kalman filter algorithm itself does not specify how to come up with the initial condition

  • Use the asymptotic (steady-state) distribution

  • Use the result of a preceding run of a filter

  • Arbitrary set up initial condition, stochastic or fixed

Prediction step


Prediction step for transition variables
\[ x_{1|0} = T \, x_{0|0} + k + R \, 0 \]
\[ P_{1|0} = T \, P_{0|0} \, T' + R \, \Omega \, R' \]


Prediction step for measurement variables
\[ y_{1|0} = Z \, x_{1|0} + d + H \, 0= Z \left( T\, \C{x_{0|0}} + k + R \, \C{0}\right) + d + H\,\C{0} \]
\[ F_1 = MSE_0\!\left[y_1\right] = Z \, P_{1|0} \, Z' + \Sigma_{x,t} = Z \, \left( T \, \C{P_{0|0}} \, T' + \C{\Sigma_{x,t}} \right) \, Z' + \C{\Sigma_{y,t}} \]


where \(\quad \Sigma_x = R \, \mathrm E\!\left[ e \, e' \right] \, R', \quad \Sigma_y = H \, \mathrm E\!\left[ u \, u' \right] \, H'\)

Observe data

  • Observation of \(y_1\) becomes available

  • In general, our prediction will be wrong by some margin, \(y_{1|0}\ne y_1\)

  • Prediction error \(\nu_1 \equiv y_{1|0} - y_1\)

Updating step

  • Reconcile your prediction with the observation by changing whatever can be changed that shaped our prediction, i.e. whatever is stochastic and not fixed
  • State in previous period (initial condition) \(x_{0|1} \ne x_{0|0}\)
  • Transition shocks in current period \(e_{1|1} \ne 0\)
  • Measurement errors in current period \(u_{1|1} \ne 0\)
  • Infinitely many combinations \(\to\) find the likelihood maximizing combination

Logic of the prediction and updating steps


![[kalman-filter.png]]

Conditioning problem in updating step

  • Equivalent to the conditional normal distribution formulas

Minimize

\[ \begin{multline} \left(x_{0|1}-x_{0|0}\right)' P_{0|0}^{-1} \left(x_{0|1}-x_{0|0}\right) \ \cdots \\[0pt] + \left(e_{1|1}-0\right)' \, \Omega_e{}^{-1} \, \left(e_{1|1}-0\right) \ \cdots \\[0pt] + \left(u_{1|1}-0\right)' \, \Omega_u{}^{-1} \, \left(u_{1|1}-0\right) \end{multline} \]

subject to

\[ Z\left(T\,x_{0|1} + k + R \, e_{1|1}\right) + d + H \, u_{1|1} = y_1 \]

The result of an updating step

  • New set of "inputs" reflecting the observation of \(y_{1}\): \(x_{0|1}\), \(e_{1|1}\), \(u_{1|1}\)

  • Accordingly, the MSE matrices will also change (shrink)

  • Recompute \(x_{1|1}\), \(P_{1|1}\)

Wash, rinse, repeat

  • Using \(x_{1|1}\), \(P_{1|1}\) as a new starting point, calculate new predictions for \(t=2\)

  • Repeat the entire procedure

Backward smoothing

  • Arriving at the final time \(T\), we use \(y_T\) to update \(x_{T-1|T}\), \(e_{T|T}\), \(u_{T|T}\)

  • But wait... we're changing \(x_{T-1|T-1}\) into \(x_{T-1|T}\)... that makes things inconsistent with \(y_{T-1}\)...

  • We now also need to calculate \(e_{T-1|T}\), \(u_{T-1|T}\) and \(x_{T-2|T}\), etc...

  • ...all the way back to \(e_{1|T}\), \(u_{1|T}\), \(x_{0|T}\)

Three sets of values

\[ t = 1, \dots, T \]

  • Prediction step \(x_{t|t-1}\)

  • Updating step \(x_{t|t}\)

  • Smoothing step \(x_{t|T}\)

Likelihood function

  • Prediction error decomposition approach to likelihood function evaluation
\[ \log \mathcal L \propto - \sum_{t=1}^{T}\ \nu_t{}' \, F_t{}^{-1} \, \nu_t \]
  • Allows to break the overall likelihood into the contributions of individual periods