This is a playful introduction to the technique of Kalman Filtering which is used in a variety of applications ranging from control systems for rockets to economic forecasting. It is intended to communicate intuitions to beginners. It concludes with a brief description of how the New York Fed uses it for nowcasting GDP.

1. Where in the World is
Carmen Sandiego?
How Kalman can help find Carmen

2. Carmen’s True Path
Tim
T=1
T=3
T=4
T=5

3. Sightings
Tim
True positions
are in 2-d but
shown in 1-d
here

4. Estimation: Where is Carmen?
At each time point estimate her current
position (distribution)
Combine info from current sighting
with past ones
Why are past sightings important?
Carmen’s moves have only a
limited speed
See slide “Dynamics: How Carmen
Moves”
Y
X more likely at
T=2 than Y X

5. Key aspects of estimation
All sightings compete to contribute to estimates
Contribution weights determined by recency
Greater concordance of sightings improves accuracy

6. Putting some FLESH
ON THE BONE

7. Sightings II
Tim
Density falls
off as we
move away
from the
middle
Sighting
distribution
centered at
true position
Sighting
distributions
are normal
2-d positions

8. DynaMICS: How Carmen MOves
In the present case we model it as a random walk
Pt = (xt, yt)
Pt+1 = Pt + 𝝐
𝝐 ~ 𝒩(0, σ2 I)

9. estimation IS RECURSIVE
Prior At time t, we have prob dist Pt
Evidence sightings Sgtt
Posterior Pt+1= Bayes update of Pt by Sgtt
Calculations easier if measurement error is normal
Pt+1 is our estimate of Carmen’s position at time t+1

10. Predict-Measure-Correct
Post mean estimate of position at time t
Predict Use model of Carmen’s movement to predict position
Predt+1
Measure observed position (sighting) Sgtt+1
Errort+1=Sgtt+1- Predt+1
Correct Post+1 = Predt+1 + Kt+1 Errort+1

11. GEneralizing to a
framework:
STATE SPACE
REPRESENTATION

12. STATES
Hidden state Carmen’s position -- unknown except through noisy
measurements (sightings)
State can be a vector
Classical particle: kinematic state
Fair values for stocks in a basket

13. DYNAMICS
State Update Equations Stochastic and Linear
St+1 = G St + 𝝐
𝝐 ~ 𝒩(𝛍, 𝛀)
In Carmen’s case
G=I
𝛍=0, 𝛀=𝞂2I

14. Measurement
Measurements are generally noisy
Example: Carmen’s sightings = true position + Gaussian
noise
Sometimes we can only measure a function of the state
Example: iPhone accelerometer measures only
acceleration part of state
In general, we have Observation Equations

15. Predict-Measure-Correct II
Ss@t estimate of time-s state @time=t
Predict Use dynamics to predict (at time t) mean state at time t+1
St+1@t = G St@t
Measure Ot+1
Errort+1=Ot+1- St+1@t
Correct St+1@t+1= St+1@t + Kt+1 Errort+1

16. Where Can we APplY THis
A bit different from other types of signal processing filters (eg Fourier filters, PCA
filtering)
We need a model of hidden state dynamics
Linear state updates
Normally distributed updating noise
We need a model of measurement and errors
Observations = Linear function of state
Normally distributed measurement noise
Need to spec
parameters
State update eqns
Observation equations

17. For a deeper dive
There are many excellent articles. As usual, Wikipedia is a pretty good source.
Here is one of my favorites:
Richard J. Meinhold and Nozer D. Singpurwalla Understanding the Kalman Filter.
The American Statistician, Vol. 37, No. 2. (May, 1983), pp. 123-127.
I have written a Jupyter notebook that lets you play around with the filter for a simple
physics application.

18. APPLICATION:
NOWCASTING GDP

19. Nowcasting
Key economic statistics like GDP growth rate delayed
“Higher-frequency” data released on GDP components
Production components like Industrial Production
Demand components like Personal Consumption
Challenges
Differing frequencies of releases

20. Observation Variables
Examples
Durable Goods Inventories Manufacturing monthly
Personal Consumption Retail & Consumption monthly
Disposable Personal Income Income monthly
Imports Trade monthly
Initial Jobless Claims Labor weekly

21. Nowcasting Modeling
Model a proxy for GDP growth rate, R, as the hidden state variable
Possible dynamics of R
Random walk
VAR (vector autoregression) model
Rt = a1Rt-1 + a2Rt-2+.. + 𝝐
Each observation has a measurement equation
Model must specify as, ws, 𝝐-
variances, and 𝝐i-covariances.

22. Estimating GDP growth rate
As observations flow in, the Kalman filter is used to update R
Hidden variable R is only a proxy for the GDP growth rate
Use regression to estimate the GDP growth rate
Average instantaneous growth rates to nowcast quarter growth
rate