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# Dynamic models and the Kalman filter

## 1.

State Space Representation ofDynamic Models and the

Kalman Filter

Joint Vienna Institute/ IMF ICD

Macro-econometric Forecasting and Analysis

JV16.12, L04, Vienna, Austria May 18, 2016

This training material is the property of the International Monetary Fund (IMF) and is intended

for use in IMF Institute courses. Any reuse requires the permission of the IMF Institute.

Presenter

Charis Christofides

## 2.

Introduction and Motivation• The dynamics of a time series can be influenced by

“unobservable” (sometimes called “latent”) variables.

• Examples include:

Potential output or the NAIRU

A common business-cycle

The equilibrium real interest rate

Yield curve factors: “level”, “slope”, “curvature”

• Classical regression analysis is not feasible when unobservable

variables are present:

• If the variables are estimated first and then used for estimation, the

estimates are typically biased and inconsistent.

## 3.

Introduction and Motivation (continued)• State space representation is a way to describe the law of

motion of these latent variables and their linkage with known

observations.

• The Kalman filter is a computational algorithm that uses

conditional means and expectations to obtain exact (from

a statistical point of view) finite sample linear predictions

of unobserved latent variables, given observed variables.

• Maximum Likelihood Estimation (MLE) and Bayesian methods

are often used to estimate such models and draw statistical

inferences.

## 4. Common Usage of These Techniques

• Macroeconomics, finance, time series models• Autopilot, radar tracking

• Orbit tracking, satellite navigation (historically important)

• Speech, picture enhancement

## 5. Another example

• Use nightlight data and the Kalman filter to adjust officialGDP growth statistics.

• The idea is that economic activity is closely related to

nightlight data.

• “Measuring Economic Growth from Outer Space” by

Henderson, Storeygard, and Weil AER(2012)

## 6. Measuring Long-Term Growth

## 7. Measuring Short-Term Growth

## 8. Measuring Short-Term Growth

## 9.

Content Outline: Lecture Segments• State Space Representation

• The Kalman Filter

• Maximum Likelihood Estimation and Kalman

Smoothing

## 10.

Content Outline: Workshops• Workshops

Estimation of equilibrium real interest rate, trend growth

rate, and potential output level: Laubach and Williams

(ReStat 2003);

Estimation of a term structure model of latent factors:

Diebold and Li (J. Econometrics 2006);

Estimation of output gap (various country examples).

## 11.

State Space Representation## 12.

Basic SetupLet yt be an (or a vector) observable variable(s) at time t.

E.g.,

• return on asset j

• nominal interest for period from t to t+j

• GDP growth

Let xt be a set of exogenous (pre-determined) variables.

E.g.,

• a constant and/or time trend

• the discount rate of the Central Bank

• demand from trading partners

Let st be one or a vector of (possibly) unobserved

variable/s: this is the so-called state variable

• Observable variables are assumed to depend on the

## 13.

Basic SetupThe state-space representation of the dynamics of yt is given by :

st st 1 ut

State equation

yt xt st t

Observation equation

We assume that:

• The two equations above represent the true data-generating

process for yt

• All parameters of the process are known

• Later we will relax this assumption when we discuss estimation

• The

unknown (unobserved) variables are

the last two representing error processes

st , ut , for

t

all t, with

## 14.

Basic SetupThe state-space representation of the dynamics of yt is given by :

st st 1 ut

State equation

yt xt st t

Observation equation

with

either a constant, or a matrix (if st is a vector)

either a constant, or a matrix (if xt is a vector)

either a constant, or a matrix (if st is a vector)

The coefficients in β are sometimes called the “loadings”.

## 15.

Basic SetupThe error terms in the two equations are such that:

st st 1 ut

State equation

yt xt st t

Observation equation

E[ut ] 0 and E[ t ] 0 for every t

ìΩ for every t and j t

E[ut ut j ] í

î 0 for every t and j ¹ t

ì R for every t and j t

E [ t t j ] í

î 0 for every t and j ¹ t

E[ut t j ] 0 for every t and j

ìa var-cov matrix, if ut is a vector

Ω is í

î a variance if ut is one variable

ìa var-cov matrix, if t is a vector

R is í

î a variance if t is one variable

## 16. Basic Setup

The error terms in the two equations are such that:st st 1 ut

State equation

yt xt st t

Observation equation

• What if you know that uare

serially correlated:

t

, if t j

– ut ut 1 tand E[ t ] ,0 E[ t j ] ìí

î0, if t ¹ j

– Then E[ut ut 1 ] ¹ 0so one of the

assumptions is violated!

– What to do? Can you still apply the model?

## 17.

The State Space Representation: ExamplesExample #1: simple version of the CAPM

st

yt

Φ, α, and β

Ω and R

one variable, return on all invested wealth

one variable, return on an asset

constants

constants

st st 1 ut

State equation

yt st t

Observation equation

## 18.

The State Space Representation: ExamplesExample #2: growth and real business cycle (small open

economy with a large export sector)

• st

• Yt

sales

• xt

partner

• Φ, and Ω

s•t α, and

st β1 ut

• R

one variable, business cycle

vector, GDP growth, unemployment, retail

one variable, demand growth of trading

constants

vectors

matrix

y1,t 1,t

1,t

1,t

y2,t 2,t xt 2,t st 2,t

y

3, t 3, t

3 ,t

3, t

State equation

Observation equation

## 19.

The State Space Representation: ExamplesExample #3: interest rates on zero-coupon bonds of different

maturity

st

yt

xt

Φ, and Ω

α and β

R

one variable, latent variable

a vector with interest rates for diff. mat.

one variable, the Central Bank discount rate

constants

vectors of constants

matrix

st st 1 ut

y

1,t 1,t

1,t

1,t

y x s

2,t 2,t t 2,t t 2,t

y

n ,t n ,t

n ,t

n ,t

State equation

Observation

equation

## 20.

The State Space Representation: ExamplesExample #4: an AR(2) process

zt 1 zt 1 2 zt 2 vt , vt ~ N (0, v2 )

• Can we still apply the state space representation?

• Yes!

• Consider the following state equation:

zt 1 2 zt 1 1

z 1 0 z 0 vt

t 1 t 2

st

st 1

• And the observation equation:

zt 1 0 s t

yt

ut

## 21.

The State Space Representation: ExamplesExample #4: an AR(2) process

zt 1 zt 1 2 zt 2 vt , vt ~ N (0, v2 )

• The state equation:

zt 1 2 zt 1 1

z 1 0 z 0 vt

t 1 t 2

st

st 1

• And the observation equation:

ut

z 1 0 s

t t

yt

• What are matrices Ω (var-cov of ut) and R (var-cov of t) in this

case?

## 22.

The State Space Representation Is Not Unique!Consider the same AR(2) process

zt 1 zt 1 2 zt 2 vt , vt ~ N (0, v2 )

• Another possible state equation:

zt 1 1 zt 1 1

z 0 z 0 vt

t 1

2 2 t 2

2

st

st 1

ut

• And the corresponding observation equation:

z 1 0 s

t t

yt

• These two state space representations are equivalent!

• This example can be extended to AR(p) case

## 23.

The State Space Representation: ExamplesExample #5: an MA(2) process

zt vt vt 1 , vt ~ N (0, v2 )

• Consider the following state equation:

vt 0

v 1

t 1

st

• And the observation equation:

0 vt 1 1

vt

0 vt 2 0

st 1

ut

z 1 s

t t

yt

• What are matrices Ω (var-cov of ut) and R (var-cov of t) in this

case?

## 24.

The State Space Representation: ExamplesExample #5: an MA(2) process

zt vt vt 1 , vt ~ N (0, v2 )

• Consider the following state equation:

yt 0

v 0

t

st

1 yt 1 1

vt

0 vt 1

• And the observation equation:

st 1

ut

z 1 0 s

t t

yt

• What are matrices Ω (var-cov of ut) and R (var-cov of t) in this

case?

## 25.

The State Space Representation: ExamplesExample #6: A random walk plus drift process

zt zt 1 vt , vt ~ N (0, v2 )

• State equation? Observation equation?

• What are the loadings ?

• What are matrices Ω (var-cov of ut ) and R (var-cov of )

tfor your state-space representation?

## 26.

The State Space Representation:System Stability

• In this course we will deal only with stable systems:

• Such systems that for any initial state s0 , the state variable

(vector) st converges to a unique s (the steady state)

• The necessary and sufficient condition for the state space

representation to be stable is that all eigenvalues of are less

than 1 in absolute value:

| i ( ) | 1 for all i

• Think of a simple univariate AR(1) process ( zt 1 zt 1 vt )

• It is stable as long as | 1 | 1

• Why? So that it is possible to be right at least in the “long-run”.

## 27.

The Kalman Filter## 28. Kalman Filter: Introduction

• State Space Representation [univariate case]:st st 1 ut

ut ~ i.i.d . N (0, )

2

u

yt xt st t t ~ i.i.d . N (0, )

2

( , , , u2 , 2 )

are known

• Notation:

– st|t 1 E ( st | y1 ,..., yt 1 ) is the best linear predictor of st

conditional on the information up to t-1.

– yt |t 1 E ( yt | y1 ,..., yt 1 ) is the best linear predictor of yt

conditional on the information up to t-1.

– st |t E ( st | y1 ,..., yt ) is the best linear predictor of st

conditional on the information up to t.

## 29. Kalman Filter: Main Idea Moving from t-1 to t

• Suppose we know st |t 1 and yt |t 1 at time t-1.• When arrive in period t we observe yt and xt

• Need to obtain st|t !

• If we know st|t ,

– using the state equation: st 1|t st|t

– using the observation equation: yt+1|t = xt+1 + st+1|t

• The key question: how to obtain st|t from yt ?

Why?

## 30. Kalman Filter: Main Idea How to update st|t ?

• Idea: use the observed prediction errorst |t

t,

yt ytot |t infer

the state at time

1

• It turns out it is optimal to update it using

st|t st |t 1 K t ( yt yt|t 1 )

• K t is called Kalman gain

– It measures how informative is the prediction error about the

underlying state vector

• How do you think it depends on the variance of the observation error?

– It is chosen so that the new prediction error is orthogonal to all of the

previous ones.

• Thus there is no (linear) predictable component in generated errors.

## 31. Kalman Filter: More Notations

Pt|t 1 E (( st st|t 1 ) 2 | y1 ,..., yt 1 )is the prediction error variance

of s given the history of observed variables up to t-1.

t

2

F

E

((

y

y

)

| y1 ,..., yt 1 )is the prediction error variance

t |t 1

t

t |t 1

of yt conditional on the information up to t-1.

2

P

E

((

s

s

)

| y1 ,..., yt ) is the prediction error variance of

t |t

t

t |t

st

conditional

on the information up to t.

• Intuitively the Kalman gain is chosen so that Pt|t is minimized.

– Will show this later.

## 32. Kalman Gain: Intuition

• Kalman gain is chosen so that Pt |t is minimized.• It can be shown that

K t Pt|t 1 ( 2 Pt|t 1 R) 1

• Intuition:

– If a big mistake is made forecasting s

(

is large), put

t |t 1 Pt |t 1

a lot weight on the new observation (K is large).

– If the new information is noisy (R is large), put less weight

on the new information (K is small).

## 33. Kalman Filter: Example

K t Pt|t 1 ( 2 Pt|t 1 R) 1• Kalman gain is

• Consider

2

s

u

,

u

~

N

(

0

,

t

t

t

u)

– State equation

– Observation equation yt st t , t ~ N (0, 2 )

– Additionally 2 2, where is a constant

u

• Assume that we picked P1|0 u (we don’t know anything

about s1 ).

• Can you calculate the Kalman gain in the 1 st period,K1?

• What is the interpretation?

2

## 34. Kalman Filter: The last step

• How do we get from Pt |t 1 to Pt 1|t using yt ?• Recall that for a bivariate normal distribution

1 12 12

z1

z 2 | z1 ~ N ( 2 12 ( 12 ) 1 ( z1 1 ), 22 12 ( 12 ) 1 12 )

~ N ,

2

2

z2

12

2

• Using this property and the fact that E[ st|t 1 | yt ] st|t

yt |t 1 Ft |t 1

yt

,

| ( y1 , , yt 1 ) ~ N

st|t 1 Pt|t 1

s

t

Pt|t 1

Pt|t 1

E ( st st|t 1 )( yt yt|t 1 ) | y1 , yt 1 E ( st st|t 1 )( ( st st|t 1 ) t ) | y1 , yt 1 Pt|t 1

• Thus, st|t = st|t-1+ Pt|t-1(Ft|t-1)-1(yt - yt|t-1) and

Pt|t = Pt|t-1 – Pt|t-1(Ft|t-1)-1 Pt|t-1

Kalman gain

## 35. Kalman Filter: Finally

• From the previous slidest|t = st|t-1+ Pt|t-1(Ft|t-1)-1(yt - yt|t-1)

Pt|t = Pt|t-1 – Pt|t-1(Ft|t-1)-1 Pt|t-1

• Need: from Pt |t 1 to Pt 1|t using yt

Ft |t 1 E ( yt yt|t 1 ) 2 | y1 , yt 1 E ( ( st st |t 1 ) t ) 2 | y1 , yt 1 2 Pt |t 1 R

• Thus, we get the expression for the Kalman gain:

• Similarly

K t Pt|t 1 ( 2 Pt|t 1 R) 1

Pt 1|t E ( st 1 st 1|t ) 2 | y1 , , yt E ( ( st st|t ) ut ) 2 | y1 , yt 2 Pt |t

• And we are done!

## 36. Kalman Filter: Review

• We start from st|t 1 and Pt |t 1 .yt|t-1 = xt + st|t-1

Ft|t 1 2 Pt|t 1 R

• Calculate Kalman gain

K t Pt|t 1 ( 2 Pt|t 1 R) 1

• Update using observed yt

st |t st|t 1 K t ( yt yt|t 1 )

Pt|t = Pt|t-1 – Pt|t-1(Ft|t-1)-1 Pt|t-1

• Construct forecasts for the next period

st 1|t st|t

• Repeat!

Pt 1|t 2 Pt|t

## 37. Kalman Filter: How to choose initial state

• If the sample size is large, the choice of the initial state is notvery important

• In short samples can have significant effect

• For stationary models

s1|0 s *

P1|0 P *

• Where

s* s*

P * P *

• Solution to the last equation is P * [ I ] 1 vec( )

• Why? Under some very general conditions

P P * as t

t |t 1

## 38. Kalman Filter as a Recursive Regression

• Consider a regular regression functionE[ s | y ] a by

where

a E[ s ] b E[ y ]

• Substituting

b Cov ( s, y ) (Var ( y )) 1

E[ s | y ] E[ s ] Cov ( s, y ) Var ( y ) 1 [ y E[ y ]]

• From one of the previous slides:

st|t = st|t-1+ Pt|t-1(Ft|t-1)-1(yt - yt|t-1)

## 39. Kalman Filter as a Recursive Regression

• Consider a regular regression functionE[ s | y ] a by

where

a E[ s ] b E[ y ]

b Cov ( s, y ) (Var ( y )) 1

• Substituting

E[ s | y ] E[ s ] Cov ( s, y ) Var ( y ) 1 [ y E[ y ]]

• From one of the previous slides

st|t = st|t-1+ Pt|t-1(Ft|t-1)-1(yt - yt|t-1)

Because

Cov ( st , yt | y1 , yt 1 ) Pt|t 1

Var ( yt | y1 ,..., yt 1 ) Ft|t 1

st|t 1 E ( st | y1 ,..., yt 1 )

yt|t 1 E ( yt | y1 ,..., yt 1 ) st|t E[ E ( st | y1 ,..., yt 1 ) | yt ]

## 40. Kalman Filter as a Recursive Regression

• Thus the Kalman filter can be interpreted as a recursiveregression of a type

t

st y vt

1

t

where vt st y is the forecasting error at time t

1

• The Kalman filter describes how to recursively estimate

t and thus obtain st |t E[ st | y1 ,... yt ]

## 41. Optimality of the Kalman Filter

• Using the property of OLS estimates that constructed residualsare uncorrelated with regressors

E[vt ] 0 E[vt yt ] 0 for all t

• Using the expression for

t

vt st y

1

and the state equation, it is easy to show that

for

E[vallyt and

] k=0..t-1

0

t

t k

• Thus the errors vt do not have any (linear) predictable

component!

## 42. Kalman Filter Some comments

Within the class of linear (in observables) predictors the Kalman filter

algorithm minimizes the mean squared prediction error (i.e., predictions

of the state variables based on the Kalman filter are best linear unbiased):

Min E[ s t ( st |t 1 K t ( yt yt |t 1 )) ]

2

Kt

Pt |t 1

Kt 2

Pt |t 1 R

If the model disturbances are normally distributed, predictions based on

the Kalman filter are optimal (its MSE is minimal) among all predictors:

E[ s t ( st|t 1 K t ( yt yt|t 1 )) ] Min E[ s t f ( y1 ,..., yt ) ]

2

2

f ( y1 ,..., yt )

In this sense, the Kalman filter delivers optimal predictions.

## 43.

Kalman Filter - Multivariate Case• The Kalman Filter algorithm can be easily generalized to the

generic multivariate state space representation, including

exogenous variables:

s t+1 =Φs t + u t+1 , E u tu 't Ω

y t = Axt +Βs t + ε t , E ε tε 't R

• Defining similarly as before:

s t|t-1 = E s t | y1 ,…, y t-1 , Pt|t-1

y t|t-1

'

E st st|t-1 st st|t-1 | y1 ,…, y t-1

'

= E y t | y1 ,…, y t-1 , Ft E y t - y t|t-1 y t - y t|t-1 | y 1 ,…, y t-1

• Now we have vectors and matrices

## 44. Kalman Filter Algorithm – Multivariate Case

Initialization: s1|0 , P1|01: y t|t-1 = Axt + Bs t|t-1

2 : Ft = BPt|t-1B' + R

3 : K t = Pt|t-1B' (Ft )-1

4 : s t|t = s t|t + K(y t - y t|t-1 )

5 : Pt|t = Pt|t-1 - Pt|t-1B' (Ft )-1 BPt|t-1

6 : s t+1|t =Φs t|t

7 : Pt+1|t =ΦP t|tΦ ' + Ω

Repeat 1,...,7 from t 1 to t T

## 45. Kalman Filter Algorithm – Multivariate Case (cont.)

How do we obtain these expressions?y t|t-1 = E Axt + Bs t +ε t |y 1,…, y t-1 Ax t + Bs t|t-1

Ft = E (y t y t|t-1 )(y t y t|t-1 )' | y 1 ,…, y t-1

'

E ( B (s t sεt|t-1 ) st )( Bs ( t εt|t-1 ) y ,…,

t ) | y1

BP

R

t-1 B +

t|t-1

'

Also:

E (y t y t|t-1 )(s t - s t|t-1 )' | y 1 ,…, y t-1 E (B(s t sεt|t-1 )s ts)( t yt|t-1,…,

)' | y1

BPt|t-1

Thus:

y t|t-1 Ft

BPt|t-1

yt

,

| y 1 ,…, y t-1 ~ N

(BPt|t-1 ) ' Pt|t-1

st

s t|t-1

t-1

## 46. Kalman Filter Algorithm – Multivariate Case (cont.)

How we obtained these expression? (cont.)y t|t-1 Ft

BPt|t-1

yt

,

| y 1 ,…, y t-1 ~ N

P

B'

P

t|t-1

st

s t|t-1 t|t-1

Using the property for a multivariate normal distribution to get

conditional disttribution:

s t|t ~ N s t|t-1 + Pt|t-1B'(Ft )-1 y t - y t|t-1 + , Pt|t-1 Pt|t-1B'(Ft )-1 BPt|t-1

1 42 43

1 4 4 44 2 4 4 4 43

Kt

Pt|t

1

4

4

4

4

4

2

4

4

4

4

43

st|t

Also:

sΦs

+E

u

t+1|t

yt ,…,

t+1y|

1

PΦ

) s

t+1|t s Es ( +( ut t|tΦ

Φs

t

t+1s)( ( t

t|t

'

)

+

u

)

t|t

t+1 | y 1 ,…, yΦP

t Φ

'

Ω

t|t

## 47.

ML Estimation and Kalman Smoothing## 48.

Maximum Likelihood Estimation• The algorithm in the previous section assumes knowledge of

the parameters. If these are not known, estimates are needed.

• Consider the univariate case:

st 1 st ut 1 , E ut2 u2

yt xt st t , E t2 2

and using that st is normally distributed (ut is normal) then

( yt | y1 ,..., yt 1 ) ~ N ( yt |t 1 , Ft|t 1 )

• Thus we can do maximum likelihood estimation

1

1

1

2

log l ( , , , u , ) log 2 log | Ft|t 1 |

( yt yt|t 1 )

2

2 Ft |t 1

t 1

2

T

• Similarly with the multivariate case:

(y t | x t , y 1 ,…, y t-1 ) ~ N (y t|t-1 , Ft )

## 49.

Maximum Likelihood EstimationTo estimate model parameters through maximizing loglikelihood:

Step 1: For every set of the underlying parameters, θ

Step 2: run the Kalman filter to obtain estimates for the

sequence

yt|t 1 , Ft|t 1

Step 3: Construct the likelihood function as a function of θ

Step 4: Maximize with respect to the parameters.

## 50.

Kalman Smoothing• For each period t, the Kalman filter uses only information

available up to time t:

E[s t | y 1 ,…, y t-1 ] s t|t-1

• Is it possible to use all the information available so as to obtain

an even better estimate of st: E[st | y 1 ,…, y T ] ?

• This is called smoothed inference of the state and denoted by

s t|T

• In general, we can obtain the smoothed inference

s t|τ , > t

## 51.

Kalman SmoothingUsing the same principles for normal conditional distribution, it is

possible to show that there is a recursive algorithm to compute

s t|Tstarting from

s T|T

:

Step 1: use Kalman filter to estimate s1|1 , …, sT|T

Step 2: use recursive method to obtain, s t|T , the smoothed

estimate of st:

s t|T = s t|t + J t (s t+1|T - s t+1|t )

where

'

J t = PΦ

P(

t|t

1

)

t+1|t

## 52.

Conclusion• Many models require estimations of unobserved variables,

either because these are of economic interest, or because one

needs them to estimate the model parameters (example,

ARMA).

• The Kalman filter is a recursive algorithm that:

• provides efficient estimates of unobserved variables, and

their MSE;

• can be used for forecasting given estimates of MSE;

• is used to initialize maximum likelihood estimation of models

(for example, of ARMA models) by first producing good

estimates of un-observed variables;

• can also be used to smooth series for unobserved variables.