A pair of simple modifications-in the forecast error and forecast error variance-to the Kalman filter recursions makes possible the filtering of models in which one or more state variables is truncated normal and latent. Such recursions are broadly applicable to macroeconometric models, such as vector autoregressions and estimated dynamic stochastic general equilibrium models, that have one or more probit-type equation.Macroeconomics - Econometric models

Michael Dueker

English

Research Papers in Economics

      Research Division 
          Federal Reserve Bank of St. Louis 
                   Working Paper Series 
 
 
 
 
Kalman Filtering with Truncated Normal State Variables 
for Bayesian Estimation of Macroeconomic Models 
 
 
 
 
Michael Dueker 
 
 
 
 
 
 
Working Paper 2005-057B 
http://research.stlouisfed.org/wp/2005/2005-057.pdf 
 
 
 
  July 2005 
Revised March 2006 
 
 
 
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cleared with the author or authors. Kalman ¯ltering with truncated normal
state variables for Bayesian estimation
of macroeconomic models
Michael Dueker¤
Federal Reserve Bank of St. Louis
Abstract
A pair of simple modi¯cations|in the forecast error and forecast error variance|
to the Kalman ¯lter recursions makes possible the ¯ltering of models in which one or
more state variables is truncated normal and latent. Such recursions are broadly applica-
ble to macroeconometric models, such as vector autoregressions and estimated dynamic
stochastic general equilibrium models, that have one or more probit-type equation.
JEL classi¯cations: C32, C35, E37
Key words: Kalman ¯lter, truncated normal, probit model, macroeconometric models
¤ The content is the responsibility of the authors and does not represent o±cial posi-
tions of the Federal Reserve Bank of St. Louis or the Federal Reserve System. Address:
P.O. Box 442, St. Louis, MO 63166; mdueker@stls.frb.org; ph. (314) 725-7902; fax (314)
444-8731
11 Introduction
Multivariate models that contain a probit-type equation are emerging in empirical macroe-
conomics, such as the Qual VAR of Dueker (2005)|a vector autoregression that includes
an equation that determines a qualitative variable. The probit-type equation could be
one that includes information on business cycle turning points, or it could be a monetary
policy equation that uses data on the central bank's interest rate target that changes by
discrete increments. Other macroeconomic phenomena also lead to qualitative variables
that evolve dynamically, such as currency pegs and central bank exchange-rate inter-
ventions. Dynamic probits aim to capture both the limited-dependent nature and serial
correlation inherent in qualitative time-series variables.1
Until now, however, multivariate macroeconometric models that contain a probit-type
equation have relied on single-move sampling techniques, even though multi-move sam-
pling dominates in terms of e±ciency and convergence to the joint posterior distribution.
In single-move Markov Chain Monte Carlo (MCMC) sampling of a vector, z, the density
used to draw an element t for iteration i depends on the previous iteration's draws:
¼(z
(i)
t j fz
(i)rgr<t;fz
(i¡1)
s gs>t;Data, parameters):
In multi-move sampling, in contrast, the entire vector of z is drawn without reference to
1The term \dynamic probit" is also used in some contexts to refer to a model in which the lagged
qualitative variable appears as a covariate; see Horowitz (1992). In this case, the presence of the lagged
qualitative variable adds persistence but not dynamics in the sense of time-varying expected durations to
the qualitative regimes. We refer here to dynamic probits as those for which the latent variable governing
the qualitative variable is autoregressive and, thereby, displays dynamic behavior. Eichengreen et al.
(1985) is the ¯rst example of an autoregressive dynamic probit.
1the last iteration's draws:
¼(fz
(i)tg
T
t=1 j Data, parameters):
One hallmark feature of MCMC estimation of probit-type models is the sampling of
truncated normal latent variables; see Albert and Chib (1993). At the same time, the
Kalman ¯lter has seen great use in Bayesian estimation as a tool to derive multi-move
sampling distributions, as pioneered by Carter and Kohn (1994), Fruhwirth-Schnatter
(1994), Shephard (1994) and De Jong and Shephard (1995). This paper demonstrates
that multi-move sampling of truncated normal variables is possible if one modi¯es the
Kalman ¯lter recursions for the case where a state variable is truncated normal. To the
author's knowledge, this speci¯c modi¯cation of the Kalman ¯lter recursions for truncated
state variables has not appeared in the literature to date. With such a modi¯ed Kalman
¯lter, it is possible to extend multi-move MCMC sampling to the Qual VAR model of
Dueker (2005) and also to extend estimation of dynamic stochastic general equilibrium
(DSGE) models to include probit-type features, such as the central bank's discrete target
interest rate. Dueker and Nelson (2006) use a Qual VAR to conduct counterfactual
business cycle ¯ltering of macroeconomic data. Because many linearized DSGE models
take the form of a vector autoregression, subject to coe±cient restrictions, the the form
of the model we present here covers a wide range of macroeconometric models.
2 State-space model and truncation
Thus, consider a macroeconometric model written as a p-order vector autoregression,
where z is the latent variable that lies behind the qualitative data, zq, and observed
macroeconomic time seris, X:
2Yt = cY +
p X
i=1
©
(i)Yt¡i + ²t; (1)
where Yt = (Xt;zt)0 is k £ 1 and
©
(i) =
Ã
©
(i)
XX ©
(i)
Xz
©
(i)
zX ©(i)
zz
!
:
A standard state-space representation of the VAR from eq. (1) has the following state
equation2 :
0
B B
B B
@
Yt
Yt¡1
. . .
Yt¡p+1
1
C C
C C
A
=
0
B B
B B
@
cY
0
. . .
0
1
C C
C C
A
+
0
B B
B B
@
©(1) ©(2) ¢¢¢ ©(p)
I 0 ¢¢¢ 0
0 ... . . .
0 ¢¢¢ I 0
1
C C
C C
A
0
B B
B B
@
Yt¡1
Yt¡2
. . .
Yt¡p
1
C C
C C
A
+
0
B B
B B
@
²Y;t
0
. . .
0
1
C C
C C
A
The measurement equation for the Qual VAR is simply
Xt =
³
Ik¡1 0 ¢¢¢ 0
´
0
B B
B B
@
Yt
Yt¡1
. . .
Yt¡p+1
1
C C
C C
A
(2)
To achieve multi-move sampling, it is necessary to derive the conditional distribution
of the entire vector of the latent variable conditional on the parameter vector, £, the
observed qualitative data, zq, and the other macroeconomic data, Xt:
¼(fztg
T
t=1 j £;fz
q
tg
T
t=1;fXtg
T
t=1):
We can summarize the above state-space model (suppressing the constants) as:
2Macroeconometric models have made great use of the state-space form to take account of latent state
variables, e.g., Kuttner (1994).
3Xt = HSt
St = FSt¡1 + ²t (3)
The number of rows in the state vector is k and Fk denotes row k of F.
Without loss of generality, assume that z is the last element in the state vector S.
Because we want to allow for the possiblity that the covariance matrix of ²t has non-zero
o®-diagonal elements, let
²t = W´t;
where W is an upper-triangular Cholesky decomposition of the covariance matrix Q of ²t.
The truncation from having z
q
t in category j implies that zt lies in the range (dj;dj+1);
and this truncation of z will change the mean and variance of the last element of ´ from
the unconditional values of zero and one, respectively.
The mean of ², conditional on trunction, becomes
E[²t+1 j z
q
t+1] = W
0
B B
B
B
@
0
. . .
0
a
1
C C
C
C
A
;
where we denote ®j+1 = W ¡1[zt+1 ¡FkStjt +dj+1], ®j = W ¡1[zt+1 ¡FkStjt +dj] such that
a =
Á(®j+1) ¡ Á(®j)
©(®j+1) ¡ ©(®j)
:
and the variance of ², conditional on truncation, becomes
~ Q = W
0
B
B B
B
@
1 0 ¢¢¢ 0
0 ... 0 0
0 ¢¢¢ 1 0
0 ¢¢¢ 0 v
1
C
C C
C
A
W
0;
4where
v = 1 ¡ a
2 ¡
®j+1Á(®j+1) ¡ ®jÁ(®j)
©(®j+1) ¡ ©(®j)
:
3 Kalman ¯lter recursions
The Kalman ¯lter with the truncated state variable proceeeds as follows:
1. Data forecast error (DFE) conditional on zq:
yt+1 ¡ yt+1jt = yt+1 ¡ HFStjt ¡ HE[²t+1 j z
q
t+1]
2. State forecast variance:
Pt+1jt = FPtjtF
0 + ~ Q
3. Data forecast error variance:
Vart(DFEt+1) = HPt+1jtH
0
The equations to update the state and the precision matrix are based on
St+1jt+1 = St+1jt + E[(St+1 ¡ St+1jt)DFEt+1] £ [HPt+1H
0]
¡1DFEt+1:
Nevertheless, because
E[(St+1 ¡ St+1jt)DFEt+1] = Pt+1jtH
0;
the Kalman ¯lter update equations take the usual form, although the truncation infor-
mation alters the data forecast error (DFE) and forecast variance inputs:
St+1jt+1 = St+1jt + Pt+1jtH
0[HPt+1jtH
0]
¡1DFEt+1 (4)
Pt+1jt+1 = Pt+1jt ¡ Pt+1jtH
0[HPt+1jtH
0]
¡1HPt+1jt (5)
5Thus, these modi¯cations to the Kalman ¯lter recursions show that the Kalman ¯lter
remains a useful tool to calculate conditional densities in the case where one or more of
the state variables is truncated normal. The same logic would apply if an observation
equation variable were truncated normal.
After the modi¯ed Kalman ¯ltering, the usual (no modi¯cation necessary) Kalman
smoothing equations can be applied in order to draw values of z backwards from the end of
the sample, with the net result being a draw from the density ¼(fzg j fzqg;fXg;parameters).
See Fruwirth-Schnatter (1994) for a discussion of backwards sampling from Kalman ¯l-
tered and smoothed conditional densities.
4 Conclusion
A few simple modi¯cations of the Kalman ¯lter recursions make the approach applicable
to the case where one or more state variables is truncated normal. No approximation
of the Kalman ¯lter is required, in contrast to the extended Kalman ¯lter and its ¯rst-
order Taylor approximations. This extension of the Kalman ¯lter makes it possible to
do multi-move sampling for Bayesian estimation of vector autoregressions and dynamic
stochastic general equilibrium macroeconomic models that involve a probit-type equation
that makes use of qualitative or discrete-valued data.
6References
Albert, James H. and Siddhartha Chib (1993), \Bayesian Analysis of Binary and Poly-
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Carter, C.K. and P. Kohn (1994), \On Gibbs Sampling for State-Space Models," Biometrika
81, 541-53.
Chib, Siddhartha and Edward Greenberg (1996), \Markov Chain Monte Carlo Simulation
Methods in Econometrics," Econometric Theory 12, 409-31.
De Jong, Piet and Neil Shepard (1995), \The Simulation Smoother for Time Series Mod-
els," Biometrika 82, 339-50.
Dueker, Michael (2005), \Dynamic Forecasts of Qualitative Variables: A Qual VAR of
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Dueker, Michael and Charles R. Nelson (2006), \Business Cycle Filtering of Macroeco-
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7

Kalman filtering with truncated normal state variables for Bayesian estimation of macroeconomic models

http://research.stlouisfed.org/wp/2005/2005-057.pdf

Kalman filtering with truncated normal state variables for Bayesian estimation of macroeconomic models

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