Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 427

9.7.1 The augmented Kalman filter

The Kalman filter (KF) is a fundamental algorithm for the statistical treatment of
a state-space model. Under the Gaussian assumption it produces the minimum
mean square estimator of the state vector along with its mean square error matrix,
conditional on past information; this is used to build the one-step-ahead predictor
ofytand its mean square error matrix. Due to the independence of the one-step-
ahead prediction errors, the likelihood can be evaluated via the prediction error
decomposition.
The case whenδis a fixed vector (fixed initial conditions) has been considered
by Rosenberg (1973). He showed thatδcan be concentrated out of the likelihood
function and that its generalized least square estimate is obtained from the output
of an augmented KF. The diffuse case has been dealt with by de Jong (1989).
DefiningA 1 | 0 =−W 1 ,q 0 =0,s 0 = 0 ,S 0 = 0 , the augmented KF is given by the
following recursive formulae and definitions fort=1,...,n:

vt∗=yt−Ztα ̃∗t|t− 1 , Vt=−ZtAt|t− 1 ,

F∗t=ZtP∗t|t− 1 Zt′+GtG′t, Kt=Tt+ 1 P∗t|t− 1 Z′tF∗−t^1 ,

qt=qt− 1 +v∗

′

tF

∗− 1

t v

∗

t, st=st− 1 +V

′

tF

∗− 1

t v

∗

t, St=St− 1 +V

′

tF

∗− 1

t Vt,

α ̃∗t+ 1 |t=Tt+ 1 α ̃∗t|t− 1 +Ktv∗t, At+ 1 |t=Tt+ 1 At|t− 1 +KtVt

P∗t+ 1 |t=Tt+ 1 Pt∗|t− 1 T′t+ 1 +Ht+ 1 H′t+ 1 −KtF∗tK′t.

(9.32)

The diffuse likelihood is defined as follows (de Jong, 1991):

(y 1 ,...,yn;)=−

1

2

⎛

⎝

∑

t

ln|F∗t|+ln|Sn|+qn−s′nS−n^1 sn

⎞

⎠. (9.33)

DenotingYt={y 1 ,y 2 ,...,yt}, the innovations,vt=yt−E(yt|Yt− 1 ), the con-
ditional covariance matrixFt=Var(yt|Yt− 1 ), the one-step-ahead prediction of
the state vectorα ̃t|t− 1 =E(αt|Yt− 1 ), and the corresponding covariance matrices,
Var(αt|Yt− 1 )=Pt|t− 1 , are given by:

vt = v∗t−VtS−t−^11 st− 1 , Ft = F∗t+VtS−t−^11 V′t,

α ̃t|t− 1 = α ̃∗t|t− 1 −At|t− 1 S−t−^11 st− 1 , Pt|t− 1 = P∗t|t− 1 +At|t− 1 S−t−^11 A′t|t− 1.

(9.34)

9.7.2 Real-time (updated) estimates

Theupdated(orreal-time,filtered) estimates of the state vector,α ̃t|t=E(αt|Yt), and
the covariance matrix of the real-time estimation error are, respectively:

α ̃t|t=α ̃∗t|t− 1 −At|t− 1 S−t^1 st+P∗t|t− 1 Z′tF−t^1 (v∗t−VtS−t^1 st),

Pt|t=P∗t|t− 1 −P∗t|t− 1 Z′tFt∗−^1 ZtP∗t|t− 1 +(At|t− 1 +P∗t|t− 1 Z′tF∗−t^1 Vt)S−t^1 (At|t− 1

+P∗t|t− 1 Z′tF∗−t^1 Vt)′.