EC 831: Empirical Methods in Macroeconomics

Aeimit Lakdawala

Michigan State University

Linear State Space Models

Observation Equation

yt = Axt +Hξt + wt

yt ,wt are n x 1,ξt is r x 1 andxt is k x 1

State Equationξt+1 = F ξt + vt+1(

vtwt

)∼ iiN

([00

],

[Q 00 R

])Observed variables: yt ,xtUnobserved variables: ξt ,vt and wt

Consider an AR(1)yt = ρyt−1 + εt

Define ξt = yt , vt = εt and yt = yt , wt = 0, xt = 0 andF = ρ, H = 1 and A = 0

Observation Equation:

yt = Axt +Hξt + wt

yt = yt

State Equation

ξt+1 = F ξt + vt+1

yt+1 = ρyt + εt+1

Consider an AR(1)yt = ρyt−1 + εt

Define ξt = yt , vt = εt and yt = yt , wt = 0, xt = 0 andF = ρ, H = 1 and A = 0

Observation Equation:

yt = Axt +Hξt + wt

yt = yt

State Equation

ξt+1 = F ξt + vt+1

yt+1 = ρyt + εt+1

Consider a MA(1):

yt = µ + ut + θut−1

Define ξt = (ut , ut−1), yt = yt , xt = 1, wt = 0,vt = [ut , 0]′

Observation Equation:

yt = Axt +Hξt + wt

yt = µ + [ 1 θ ]

(utut−1

)State Equation

ξt+1 = F ξt + vt+1(ut+1

ut

)=

(0 01 0

)(utut−1

)+

(ut+1

0

)

Generally any ARMA(p,q) model can be put in state space form.

Solutions to linearized DSGE models can also be put in state spaceform.

Kalman Filter

The goal is to find the distribution of ξt given all informationavailable upto time t, Ωt = yt , yt−1, ..., y1, xt , xt−1, ..., x1

p(ξt |Ωt)

Assume ξ0 ∼ N(ξ0|0,P0|0)

• ξ0|0: Best guess of ξ0• P0|0: Uncertainty about this guess

With normal errors and linear structure we get

ξt |Ωt ∼ N(ξt|t ,Pt|t)

Goal is to find ξt|t and Pt|t

Kalman Filter

The goal is to find the distribution of ξt given all informationavailable upto time t, Ωt = yt , yt−1, ..., y1, xt , xt−1, ..., x1

p(ξt |Ωt)

Assume ξ0 ∼ N(ξ0|0,P0|0)

• ξ0|0: Best guess of ξ0• P0|0: Uncertainty about this guess

With normal errors and linear structure we get

ξt |Ωt ∼ N(ξt|t ,Pt|t)

Goal is to find ξt|t and Pt|t

Kalman Filter

The goal is to find the distribution of ξt given all informationavailable upto time t, Ωt = yt , yt−1, ..., y1, xt , xt−1, ..., x1

p(ξt |Ωt)

Assume ξ0 ∼ N(ξ0|0,P0|0)

• ξ0|0: Best guess of ξ0• P0|0: Uncertainty about this guess

With normal errors and linear structure we get

ξt |Ωt ∼ N(ξt|t ,Pt|t)

Goal is to find ξt|t and Pt|t

Kalman Filter

Two main steps:

1 Predict: Using all information upto time t − 1 we want toobtain an optimal forecast of ξt , let’s call it ξt|t−1.

2 Update: Once yt is realized, we update our forecast, call it ξt|t

Intuition for Kalman Filter

Assume z1 and z2 are jointly normal.[z1z2

]∼ N

([µ1

µ2

],

[Ω11 Ω12

Ω21 Ω22

])Suppose now you observe z1.Now what is the conditional distribution of z2?

z2|z1 ∼ N(mz2 ,Vz2)

mz2 = µ2 + Ω21Ω−111 [µ1 − z1]

Vz2 = Ω22 −Ω21Ω−111 Ω12

Intuition for Kalman Filter

Assume z1 and z2 are jointly normal.[z1z2

]∼ N

([µ1

µ2

],

[Ω11 Ω12

Ω21 Ω22

])Suppose now you observe z1.Now what is the conditional distribution of z2?

z2|z1 ∼ N(mz2 ,Vz2)

mz2 = µ2 + Ω21Ω−111 [µ1 − z1]

Vz2 = Ω22 −Ω21Ω−111 Ω12

1st Predict step

ξ1|Ω0 ∼ N(ξ1|0,P1|0)

ξ1|0 = F ξ0|0

P1|0 = FP0|0F′ +Q

1st Update step

[y1|x1, Ω0

ξ1|x1, Ω0

]∼ N

([µ1

µ2

],

[Ω11 Ω12

Ω21 Ω22

])where

µ1 = Ax1 +Hξ1|0µ2 = ξ1|0

Ω11 = HP1|0H′ + R

Ω21 = P1|0H′

Ω22 = P1|0

1st Update step

ξ1|y1, x1, Ω0 = ξ1|Ω1 ∼ N(ξ1|1,P1|1)

ξ1|1 = ξ1|0 +Kη1|0P1|1 = P1|0 −KHP1|0

where

K = P1|0H′(HP1|0H

′ + R)−1

η1|0 = y1 − Ax1 −Hξ1|0

Kalman Filter Recursions

Predict:

ξt|t−1 = F ξt−1|t−1

Pt|t−1 = FPt−1|t−1F′ +Q

Update:

ξt|t = ξt|t−1 +Kηt|t−1Pt|t = Pt|t−1 −KHPt|t−1

where

ηt|t−1 = yt − Axt −Hξt|t−1

ft|t−1 = HPt|t−1H′ + R

K = Pt|t−1H′f −1t|t−1

How do we specify ξ0|0 and P0|0?

ξt+1 = F ξt + vt

If this is stationary, i.e. eigenvalues of F are inside the unit circle,then

ξ0|0 = E (ξ0) = 0

P0|0 = E (ξ0ξ ′0)

vec(P0|0) = (I − F ⊗ F )−1vec(Q)

If not, set diagonal elements of P0|0 to large numbers

Bayesian Perspective

ξ0 ∼ N(ξ0|0,P0|0) is the prior.

ξt|t is the posterior Bayesian expectation of ξt |Ωt for given valuesof F,Q,A,H,R

So how do we estimate the parameters?

Parameter Estimation from classical perspective

What is the distribution of yt |Ωt−1, xt , θ where θ contains all theparameters.

yt |Ωt−1, xt , θ ∼ N(yt|t−1, ft|t−1)

where

yt|t−1 = Axt +Hξt|t−1

ft|t−1 = HPt|t−1H′ + R

Likelihood (based on ”Prediction Error Decomposition”)

lnL = −1

2

T

∑t=1

ln(2πft|t−1)−1

2

T

∑t=1

η′t|t−1f−1t|t−1ηt|t−1

Parameter Estimation from classical perspective

lnL = −1

2

T

∑t=1

ln(2πft|t−1)−1

2

T

∑t=1

η′t|t−1f−1t|t−1ηt|t−1

Note ft|t−1 and ηt|t−1 are functions of θ

Maximize the log-likelihood w.r.t θ.

Note ξt|t is our best guess given all data upto time t.

But as econometricians we observe T ≥ t

• Can we improve our best guess using all available data?

Yes!

We can use ”smoothed” inference.

ξt |ΩT ∼ N(ξt|T ,Pt|T )

Same idea used as Kalman filter derivations.

Note ξt|t is our best guess given all data upto time t.

But as econometricians we observe T ≥ t

• Can we improve our best guess using all available data?

Yes!

We can use ”smoothed” inference.

ξt |ΩT ∼ N(ξt|T ,Pt|T )

Same idea used as Kalman filter derivations.

Smoothing

ξt|T = ξt|t + Jt(ξt+1|T − ξt+1|t

)Pt|T = Pt|t + Jt(Pt+1|T − Pt+1|t)J

′t

Jt = Pt|tF′P−1

t+1|t

For smoothed inference.

1 Filter forward to get ξt|t and Pt|t2 Smooth backward using above formulae to get ξt|T and Pt|T

What if Pt+1|t is singular?

See Durbin & Koopman (2002)Time Series Analysis by State Space Methods

• Alternative formulation of Kalman filter recursion in Ch 4.

• Does not require inversion of Pt+1|t for smoothing

• Their method is also computationally more efficient