[PPT] - State-space Title models and the Pawel Zabczyk Kalman filter PowerPoint Presentation

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Centre for Central Banking Studies

State-space models and the Kalman filter

Date Nairobi, April 26, 2016 Title Pawel Zabczyk pawel.zabczyk@bankofengland.co.uk

The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.

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Centre for Central Banking Studies Modelling and Forecasting 2

State space models are useful/crucial for applied research

Can be used to

measure how a relationship between variables changes
ver time
decompose a time-series into a cycle and trend component
determine if a common component is driving a group of

time series

calculate the likelihood function for many models (including

DSGE’s)

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Centre for Central Banking Studies Modelling and Forecasting 2

State space models are useful/crucial for applied research

Can be used to

measure how a relationship between variables changes
ver time
decompose a time-series into a cycle and trend component
determine if a common component is driving a group of

time series

calculate the likelihood function for many models (including

DSGE’s)

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Centre for Central Banking Studies Modelling and Forecasting 2

State space models are useful/crucial for applied research

Can be used to

measure how a relationship between variables changes
ver time
decompose a time-series into a cycle and trend component
determine if a common component is driving a group of

time series

calculate the likelihood function for many models (including

DSGE’s)

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Centre for Central Banking Studies Modelling and Forecasting 2

State space models are useful/crucial for applied research

Can be used to

measure how a relationship between variables changes
ver time
decompose a time-series into a cycle and trend component
determine if a common component is driving a group of

time series

calculate the likelihood function for many models (including

DSGE’s)

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 3

State space models

Observation equation

Data

yt = xt

Coefficient/Data State

βt + et Transition equation

βt = µt + Fβt−1 + vt

We assume

et ∼ i.i.d.N(0, R)
vt ∼ i.i.d.N(0, Q)
∀t, s E(etv′

s) = 0

where R and Q are variance-covariance matrices

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Centre for Central Banking Studies Modelling and Forecasting 4

Examples: A Time-varying parameter regression

Yt = ct + ZtBt + et, ct and Bt follow a random walk

How to write this in State Space form?

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Centre for Central Banking Studies Modelling and Forecasting 4

Examples: A Time-varying parameter regression

Yt = ct + ZtBt + et, ct and Bt follow a random walk

How to write this in State Space form?

Yt =

xt

1

Zt

βt

ct Bt

+ et, VAR(et) = R

βt

ct Bt

=

βt−1

ct−1 Bt−1

+

v1t v2t

, VAR(vt) = Q

Note: F = I and µ = 0

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Centre for Central Banking Studies Modelling and Forecasting 5

Examples: A Trend Cycle Model

Decomposing GDP into trend and cycle

Let Yt = Ct + τ t where
the cycle Ct follows an AR (2) process

Ct = c + ρ1Ct−1 + ρ2Ct−2 + v1t

the trend follows a random walk

τ t = τ t−1 + v2t

How to write this in state space form?

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Centre for Central Banking Studies Modelling and Forecasting 5

Examples: A Trend Cycle Model

Decomposing GDP into trend and cycle

Let Yt = Ct + τ t where
the cycle Ct follows an AR (2) process

Ct = c + ρ1Ct−1 + ρ2Ct−2 + v1t

the trend follows a random walk

τ t = τ t−1 + v2t

How to write this in state space form?

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Centre for Central Banking Studies Modelling and Forecasting 5

Examples: A Trend Cycle Model

Decomposing GDP into trend and cycle

Let Yt = Ct + τ t where
the cycle Ct follows an AR (2) process

Ct = c + ρ1Ct−1 + ρ2Ct−2 + v1t

the trend follows a random walk

τ t = τ t−1 + v2t

How to write this in state space form?

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Centre for Central Banking Studies Modelling and Forecasting 5

Examples: A Trend Cycle Model

Decomposing GDP into trend and cycle

Let Yt = Ct + τ t where
the cycle Ct follows an AR (2) process

Ct = c + ρ1Ct−1 + ρ2Ct−2 + v1t

the trend follows a random walk

τ t = τ t−1 + v2t

How to write this in state space form?

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Centre for Central Banking Studies Modelling and Forecasting 5

Examples: A Trend Cycle Model

Decomposing GDP into trend and cycle

Let Yt = Ct + τ t where
the cycle Ct follows an AR (2) process

Ct = c + ρ1Ct−1 + ρ2Ct−2 + v1t

the trend follows a random walk

τ t = τ t−1 + v2t

How to write this in state space form?

Yt =

x

1

1

βt

  Ct τ t Ct−1  ; R ≡ 0

βt

  Ct τ t Ct−1   =

µ

  c   +

F

  ρ1 ρ2 1 1  

βt−1

  Ct−1 τ t−1 Ct−2   +   v1t v2t  

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Examples: A Dynamic factor model

A panel of series Yit has a common component Yit = BiFt + eit where Ft = c + ρ1Ft−1+ ρ2Ft−2 + vt

How to write this in state space form?

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Examples: A Dynamic factor model

A panel of series Yit has a common component Yit = BiFt + eit where Ft = c + ρ1Ft−1+ ρ2Ft−2 + vt

How to write this in state space form?

yt

    Y1t Y2t . YNt     =

x

    B1 B1 . . BN    

βt

Ft

Ft−1

+

    e1t e2t . eNt    

βt

Ft

Ft−1

=

µ

c

+

F

ρ1 ρ2 1

βt−1

Ft−1 Ft−2

+

v1t v2t

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Examples: Conditional Forecasting

Imagine you’ve just estimated the following VAR

GDPt it

βt

= c1 c2

µ

+ B1 B2 B3 B4

F

GDPt−1 it−1

βt−1

+ v1t v2t

vt

where it denotes the nominal interest rate. You could use the state-space toolkit to forecast GDP conditional on any path of interest rates icf,t+1, , . . . , icf,t+n

To do that take µ and F from the VAR and set-up
icf,t+s
ys

=

1
x

GDPt+s it+s

βs

GDPt+s it+s

βs

= c1 c2

µ

+ B1 B2 B3 B4

F

GDPt+s−1 it+s−1

βs−1

+ v1t+s v2t+s

vs

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Centre for Central Banking Studies Modelling and Forecasting 7

Examples: Conditional Forecasting

Imagine you’ve just estimated the following VAR

GDPt it

βt

= c1 c2

µ

+ B1 B2 B3 B4

F

GDPt−1 it−1

βt−1

+ v1t v2t

vt

where it denotes the nominal interest rate. You could use the state-space toolkit to forecast GDP conditional on any path of interest rates icf,t+1, , . . . , icf,t+n

To do that take µ and F from the VAR and set-up
icf,t+s
ys

=

1
x

GDPt+s it+s

βs

GDPt+s it+s

βs

= c1 c2

µ

+ B1 B2 B3 B4

F

GDPt+s−1 it+s−1

βs−1

+ v1t+s v2t+s

vs

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Examples: Interpolation of data

Suppose we have quarterly GDP and we want to estimate

a monthly series ˆ Yt using information in monthly data Zt. We can treat monthly GDP numbers as unobserved states!

Specifically, we could assume

ˆ Yt = θ ˆ Yt−1 + DZt + ut, ut = ρut−1 + gt, VAR(g) = σ2 and define y as           GDP3 GDP6 .           with x given by           1 1 1 1 1 1 . . . .           where the unobserved state βt =

ˆ

Yt, ˆ Yt−1, ˆ Yt−2, ut ′ .

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Examples: Interpolation of data

Suppose we have quarterly GDP and we want to estimate

a monthly series ˆ Yt using information in monthly data Zt. We can treat monthly GDP numbers as unobserved states!

Specifically, we could assume

ˆ Yt = θ ˆ Yt−1 + DZt + ut, ut = ρut−1 + gt, VAR(g) = σ2 and define y as           GDP3 GDP6 .           with x given by           1 1 1 1 1 1 . . . .           where the unobserved state βt =

ˆ

Yt, ˆ Yt−1, ˆ Yt−2, ut ′ .

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Examples: Interpolation of data

The transition equation would then be given by      ˆ Yt ˆ Yt−1 ˆ Yt−2 ut     

βt

=     DZt    

µt

+     θ ρ 1 1 ρ    

F

     ˆ Yt−1 ˆ Yt−2 ˆ Yt−3 ut−1     

βt−1

+     gt gt     with Q =     σ2 σ2    

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Estimation Aims

Observation equation yt = xtβt + et, VAR(e) = R
Transition equation βt = µt + Fβt−1 + vt, VAR(v) = Q
Parameters of the state space and the state variables are

unknown

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Estimation Aims

Observation equation yt = xtβt + et, VAR(e) = R
Transition equation βt = µt + Fβt−1 + vt, VAR(v) = Q
Parameters of the state space and the state variables are

unknown

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Estimation Aims

Observation equation yt = xtβt + et, VAR(e) = R
Transition equation βt = µt + Fβt−1 + vt, VAR(v) = Q
Parameters of the state space and the state variables are

unknown

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Estimating the states via the Kalman filter

Consider the state space formulation yt = xtβt + et, et ∼ N(0, R) βt = µ + Fβt−1 + vt, vt ∼ N(0, Q) Notation: βt\t inference on βt given information up to time t Notation: Pt\t covariance of βt given information up to time t

Assume for the moment that R, Q, µ, F are known

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The Kalman filter (Overview)

Starting Values (time 0) Predict States (time 1...onwards) Calculate Prediction Error Update States

t\t1 yt x tt\t1 ft\t1 x tPt\t1x t

R

K Pt\t1x t

ft\t1 1

t\t t\t1 Kt\t1 Pt\t Pt\t1 Kx tPt\t1

t\

t 1 F t1 \t

1

P t\t 1 FP

t 1 \t 1F Q

0\0,P0\0

repeat for t = 1....T

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The Prediction equations of the Kalman filter

First equation βt\t−1 = µ + Fβt−1\t−1. This is simply the

expected value of the transition equation E

µ + Fβt−1 + vt
Second equation Pt\t−1 = FPt−1\t−1F ′ + Q. Can be

derived as VAR

µ + Fβt−1 + vt
Third equation of the Kalman filter just compares the
utcome to the prediction: ηt\t−1 = yt − xtβt\t−1
Fourth equation of the Kalman filter: ft\t−1 = xtPt\t−1x′

t + R

Can be derived as E([xtβt + vt] − xtβt\t−1)2 = E(xt

βt − βt\t−1

2 x′

t + vtv′ t )

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The Prediction equations of the Kalman filter

First equation βt\t−1 = µ + Fβt−1\t−1. This is simply the

expected value of the transition equation E

µ + Fβt−1 + vt
Second equation Pt\t−1 = FPt−1\t−1F ′ + Q. Can be

derived as VAR

µ + Fβt−1 + vt
Third equation of the Kalman filter just compares the
utcome to the prediction: ηt\t−1 = yt − xtβt\t−1
Fourth equation of the Kalman filter: ft\t−1 = xtPt\t−1x′

t + R

Can be derived as E([xtβt + vt] − xtβt\t−1)2 = E(xt

βt − βt\t−1

2 x′

t + vtv′ t )

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Centre for Central Banking Studies Modelling and Forecasting 13

The Prediction equations of the Kalman filter

First equation βt\t−1 = µ + Fβt−1\t−1. This is simply the

expected value of the transition equation E

µ + Fβt−1 + vt
Second equation Pt\t−1 = FPt−1\t−1F ′ + Q. Can be

derived as VAR

µ + Fβt−1 + vt
Third equation of the Kalman filter just compares the
utcome to the prediction: ηt\t−1 = yt − xtβt\t−1
Fourth equation of the Kalman filter: ft\t−1 = xtPt\t−1x′

t + R

Can be derived as E([xtβt + vt] − xtβt\t−1)2 = E(xt

βt − βt\t−1

2 x′

t + vtv′ t )

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Centre for Central Banking Studies Modelling and Forecasting 13

The Prediction equations of the Kalman filter

First equation βt\t−1 = µ + Fβt−1\t−1. This is simply the

expected value of the transition equation E

µ + Fβt−1 + vt
Second equation Pt\t−1 = FPt−1\t−1F ′ + Q. Can be

derived as VAR

µ + Fβt−1 + vt
Third equation of the Kalman filter just compares the
utcome to the prediction: ηt\t−1 = yt − xtβt\t−1
Fourth equation of the Kalman filter: ft\t−1 = xtPt\t−1x′

t + R

Can be derived as E([xtβt + vt] − xtβt\t−1)2 = E(xt

βt − βt\t−1

2 x′

t + vtv′ t )

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The Updating equations of the Kalman filter

Fifth equation: βt\t = βt\t−1 + Kηt\t−1 where

K = Pt\t−1x′

tf −1 t\t−1

Updates old estimate βt\t−1 with information contained in

ηt\t−1

The Kalman gain K is weight given to new information
The prediction error contains information that is new

relative to the past data

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The Updating equations of the Kalman filter

Fifth equation: βt\t = βt\t−1 + Kηt\t−1 where

K = Pt\t−1x′

tf −1 t\t−1

Updates old estimate βt\t−1 with information contained in

ηt\t−1

The Kalman gain K is weight given to new information
The prediction error contains information that is new

relative to the past data

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The Updating equations of the Kalman filter

Fifth equation: βt\t = βt\t−1 + Kηt\t−1 where

K = Pt\t−1x′

tf −1 t\t−1

Updates old estimate βt\t−1 with information contained in

ηt\t−1

The Kalman gain K is weight given to new information
The prediction error contains information that is new

relative to the past data

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The Updating equations of the Kalman filter

Fifth equation: βt\t = βt\t−1 + Kηt\t−1 where

K = Pt\t−1x′

tf −1 t\t−1

Updates old estimate βt\t−1 with information contained in

ηt\t−1

The Kalman gain K is weight given to new information
The prediction error contains information that is new

relative to the past data

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Centre for Central Banking Studies Modelling and Forecasting 15

The Updating equations of the Kalman filter

The Kalman gain in a univariate setting is given by

K = 1 xt Pt\t−1x2

t [due to uncertainty about state]

Pt\t−1x2

t + R [due to uncertainty about shock]

Note that K increases as Pt\t−1x2

t rises and more weight is

placed on new information in ηt\t−1

K falls as R increases and the shock is less informative

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Centre for Central Banking Studies Modelling and Forecasting 15

The Updating equations of the Kalman filter

The Kalman gain in a univariate setting is given by

K = 1 xt Pt\t−1x2

t [due to uncertainty about state]

Pt\t−1x2

t + R [due to uncertainty about shock]

Note that K increases as Pt\t−1x2

t rises and more weight is

placed on new information in ηt\t−1

K falls as R increases and the shock is less informative

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Centre for Central Banking Studies Modelling and Forecasting 15

The Updating equations of the Kalman filter

The Kalman gain in a univariate setting is given by

K = 1 xt Pt\t−1x2

t [due to uncertainty about state]

Pt\t−1x2

t + R [due to uncertainty about shock]

Note that K increases as Pt\t−1x2

t rises and more weight is

placed on new information in ηt\t−1

K falls as R increases and the shock is less informative

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The initial values

For a stationary transition equation

β0\0 = (I − F)−1 µ P0\0 = (I − F ⊗ F)−1 vec (Q)

For the non-stationary case

β0\0 = arbitrary P0\0 = large number

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The initial values

For a stationary transition equation

β0\0 = (I − F)−1 µ P0\0 = (I − F ⊗ F)−1 vec (Q)

For the non-stationary case

β0\0 = arbitrary P0\0 = large number

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Kalman Smoothing

The Kalman filter provides estimates of the state given

current information

One may be interested in estimating the state vector at

date t based on information contained in the entire data set

Kalman smoothing updates βt\t for this information, it

calculates E

βt\βt+1, yt
for each t

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Centre for Central Banking Studies Modelling and Forecasting 17

Kalman Smoothing

The Kalman filter provides estimates of the state given

current information

One may be interested in estimating the state vector at

date t based on information contained in the entire data set

Kalman smoothing updates βt\t for this information, it

calculates E

βt\βt+1, yt
for each t

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Centre for Central Banking Studies Modelling and Forecasting 17

Kalman Smoothing

The Kalman filter provides estimates of the state given

current information

One may be interested in estimating the state vector at

date t based on information contained in the entire data set

Kalman smoothing updates βt\t for this information, it

calculates E

βt\βt+1, yt
for each t

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Kalman Smoothing

First run the Kalman filter and save βt+1\t , βt\t , Pt+1/t, Pt/t

Starting values (time T) For T=t‐1,...1 Update states

T/T,PT/T t\T  t\t  Pt\tF′Pt1\t

−1 t1\T − Ft\t − 

Pt\T  Pt\t  Pt\tF′Pt1\t

−1 Pt1\T − Pt1\tPt1\t −1′ FPt\t ′

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Kalman Smoothing

Starting values (time T) For T=t‐1,...1 Update states

T/T,PT/T t\T  t\t  Pt\tF′Pt1\t

−1 t1\T − Ft\t − 

Pt\T  Pt\t  Pt\tF′Pt1\t

−1 Pt1\T − Pt1\tPt1\t −1′ FPt\t ′

Kalman filtered state gain Prediction error

Knowledge of yt+j, xt+j redundant as this is already

incorporated in βt+1

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Example of filtering and smoothing

Consider the following Time-varying regression for UK inflation πt = ct + btπt−1 + εt, VAR (εt) = R = 0.6 ct bt

βt

=

µ

+ 1 1

F

ct−1 bt−1

βt−1

+ v1t v2t

,

VAR v1t v2t

=

Q = 0.006 0.001

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Example of filtering and smoothing

0.2 0.4 0.6 0.8 1.0 1970 1975 1980 1985 1990 1995 2000 2005

AR Coefficient Filtered

0.0 0.4 0.8 1.2 1.6 2.0 1970 1975 1980 1985 1990 1995 2000 2005

Constant Filtered

.2 .3 .4 .5 .6 .7 1970 1975 1980 1985 1990 1995 2000 2005

AR coefficient Smoothed

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1970 1975 1980 1985 1990 1995 2000 2005

Constant Smoothed

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Maximum Likelihood estimation

R, Q, µ and F are generally not known but need to be

estimated

The Kalman filter provides us with an estimate of the

likelihood function of the model

Recall

yt = xβt + et, VAR(e) = R βt = µ + Fβt−1 + vt, VAR(v) = Q

Assuming normal error terms →

yt\xt ∼ N

xβt, xPt\t−1x′ + R
The likelihood function for each observation is

(2π)−n/2 xPt\t−1x′ + R

−0.5

exp

−0.5 (yt − xβt)′

xPt\t−1x′ + R −1 (yt − xβt)

r (2π)−n/2

ft\t−1

−0.5 exp
−0.5η′

t\t−1

ft\t−1

−1 ηt\t−1

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Maximum Likelihood estimation

R, Q, µ and F are generally not known but need to be

estimated

The Kalman filter provides us with an estimate of the

likelihood function of the model

Recall

yt = xβt + et, VAR(e) = R βt = µ + Fβt−1 + vt, VAR(v) = Q

Assuming normal error terms →

yt\xt ∼ N

xβt, xPt\t−1x′ + R
The likelihood function for each observation is

(2π)−n/2 xPt\t−1x′ + R

−0.5

exp

−0.5 (yt − xβt)′

xPt\t−1x′ + R −1 (yt − xβt)

r (2π)−n/2

ft\t−1

−0.5 exp
−0.5η′

t\t−1

ft\t−1

−1 ηt\t−1

SLIDE 53

Centre for Central Banking Studies Modelling and Forecasting 22

Maximum Likelihood estimation

R, Q, µ and F are generally not known but need to be

estimated

The Kalman filter provides us with an estimate of the

likelihood function of the model

Recall

yt = xβt + et, VAR(e) = R βt = µ + Fβt−1 + vt, VAR(v) = Q

Assuming normal error terms →

yt\xt ∼ N

xβt, xPt\t−1x′ + R
The likelihood function for each observation is

(2π)−n/2 xPt\t−1x′ + R

−0.5

exp

−0.5 (yt − xβt)′

xPt\t−1x′ + R −1 (yt − xβt)

r (2π)−n/2

ft\t−1

−0.5 exp
−0.5η′

t\t−1

ft\t−1

−1 ηt\t−1

SLIDE 54

Centre for Central Banking Studies Modelling and Forecasting 22

Maximum Likelihood estimation

R, Q, µ and F are generally not known but need to be

estimated

The Kalman filter provides us with an estimate of the

likelihood function of the model

Recall

yt = xβt + et, VAR(e) = R βt = µ + Fβt−1 + vt, VAR(v) = Q

Assuming normal error terms →

yt\xt ∼ N

xβt, xPt\t−1x′ + R
The likelihood function for each observation is

(2π)−n/2 xPt\t−1x′ + R

−0.5

exp

−0.5 (yt − xβt)′

xPt\t−1x′ + R −1 (yt − xβt)

r (2π)−n/2

ft\t−1

−0.5 exp
−0.5η′

t\t−1

ft\t−1

−1 ηt\t−1

SLIDE 55

Centre for Central Banking Studies Modelling and Forecasting 22

Maximum Likelihood estimation

R, Q, µ and F are generally not known but need to be

estimated

The Kalman filter provides us with an estimate of the

likelihood function of the model

Recall

yt = xβt + et, VAR(e) = R βt = µ + Fβt−1 + vt, VAR(v) = Q

Assuming normal error terms →

yt\xt ∼ N

xβt, xPt\t−1x′ + R
The likelihood function for each observation is

(2π)−n/2 xPt\t−1x′ + R

−0.5

exp

−0.5 (yt − xβt)′

xPt\t−1x′ + R −1 (yt − xβt)

r (2π)−n/2

ft\t−1

−0.5 exp
−0.5η′

t\t−1

ft\t−1

−1 ηt\t−1

SLIDE 56

Centre for Central Banking Studies Modelling and Forecasting 23

Maximum Likelihood estimation

Starting Values (time 0) Predict States (time 1...onwards) Calculate Prediction Error Update States Calculate Likelihood

t\t1 yt x tt\t1 ft\t1 x tPt\t1x t

R

K Pt\t1x t

ft\t1 1

t\t t\t1 Kt\t1 Pt\t Pt\t1 Kx tPt\t1 Lt 2n/2|ft\t1 |0.5 exp 0.5t\t1

ft\t1 1t\t1
t\t

1 F

t

1 \t 1

P t\t 1 FP

t 1 \ t 1 F Q

0\0,P0\0

maximise T

t=1 Lt wrt R, Q, µ, F, A

SLIDE 57

Centre for Central Banking Studies Modelling and Forecasting 24