1. The Kalman Filter Lecture to MSc Time Series Econometrics, Spring 2015

2. KF and state space systems State equation Measurement equation KF is a recursive procedure for optimally estimating the true, unobserved state from the observed, presumed noisy signal.

3. -Example of a state-space system; -Task: uncover true, unobserved [green] from observed [blue] -Using insight that true is a persistent process. -Persistence, measurement noise and state shocks all to be estimated. Calibration: Rho=0.95 Sig_u=1 Sig_v=3 X(1)=0

4. Remarks on the KF Kalman(1960). His PhD thesis! Ubiquitous in economics, finance, physics, biology… Literature on nonlinear, non-Gaussian filters General treatment of multivariate case in textbooks by Hamilton, Harvey, Luktepohl. Simple univariate discussion in notes by Nimark. Wide range of uses in time series and macro…

5. Uses of the KF in time series and macro Forecasting an unobserved series Computing the likelihood of a DSGE model Estimating time-varying-parameter time series models Almost duality with some optimal control problems in macro [and engineering…] – see Lunqvistd and Sargent. KF closely related to models of least squares learning, and recursive least squares econometrics

6. History RE Kalman hired into US research centre as part of cold war effort to accelerate ballistics and space research. K Filter an outgrowth of Maths of optimal control under continuous time: Ricatti (1720) Emerged in parrallel from work by xxx starting from formula for recursive least squares estimation.

7. Kalman Filter and Apollo K Filter used in Apollo navigation computers. This transcript from the Armstrong moon expedition: “Roger. We saw you up stirring around, and we thought that you were probably eating your breakfast there. Just in general, we'll probably start coming up with the uplink of the state vectors and the target loads and what have you at about 190 50, somewhere in that area, and get you started to work.” [Thanks to @ZacGross/@BirkbeckEFS]

8. Preamble: optimal weighting of 2 noisy signals Measurement equation Optimal estimate given one signal z_1 Variance of our optimal estimate given z_1 signal.

9. Weighting 2 noisy signals: Second signal arrives Expression for optimal estimate xhat given our 2 signals Weight a set to minimise variance of estimate about true value of x Optimal value for a. Variance of estimate at optimum.

10. Scalar Kalman Filter example Same logic as our ‘two signal’ example Only repeated many times over as new data come in. 2 signals comprise i) what we know before the data come in, and ii) what we know after

11. KF and state space systems State equation Measurement equation KF is a recursive procedure for optimally estimating the true, unobserved state from the observed, presumed noisy signal.

12. Scalar Kalman recursion from the beginning Want optimal xhat conditional on history of signals Start with prior over initial condition for state, which has some variance. Use state equation to propagate forwards an estimate for x_1 This propagated estimate for x_1 has a variance.

13. Scalar KF recursion, ctd… Updated estimate after arrival of signal z_1 in period 1 Optimal Kalman gain Variance of x_1/1 Equivalent expression for the variance, which you will prove in an exercise.

14. Scalar KF recursion, ctd… Propagate forwards one more period to get variance of period 2 estimate Substituting in for p_1/1 By induction, general expression for t/t-1 variance Period t Kalman gain

15. The KF recursion recapitulated in words Initial period priors about the state, and prior variance Loop to recursively calculate all subsequent posteriors Involving updates of state estimates, variances of state estimates, and the time-varying gain. Let’s restate it in algebra/pseudo-code, clearly enough that you could program it.

16. Kalman Filter algorithm in algebra Form priors about initial state, and variance of this estimate: propagate forwards to period 1 estimates using state equation. For every t: Compute Kalman Gain Compute x_t/t from gain and x_t/t-1 Let t=t+1 Compute x_t/t-1 from the state equation Update p_t/t-1 Stop when t=T.

17. Multivariate state-space system and the KF Multivariate state-space system Our Kfiltered estimate for state X

18. Studies using the Kalman Filter Laubach and Williams – Estimates US trend growth, natural rate of interest – Non-technical summary herehere

19. Studies using the Kalman Filter Greenslade et al – Estimates ‘NAIRU’, a natural rate of unemployment like concept for the UK – Semi-structural Phillips Curve is the transition equation. – Observables are inflation, unemployment.