Time-Varying Parameter VAR Model

with Stochastic Volatility:

An Overview of Methodology
and Empirical Applications
Jouchi Nakajima

This paper aims to provide a comprehensive overview of the estimation

methodology for the time-varying parameter structural vector auto-
regression (TVP-VAR) with stochastic volatility, in both methodology and
empirical applications. The TVP-VAR model, combined with stochastic
volatility, enables us to capture possible changes in underlying structure
of the economy in a flexible and robust manner. In this respect, as shown in
simulation exercises in the paper, the incorporation of stochastic volatility
into the TVP estimation significantly improves estimation performance.
The Markov chain Monte Carlo method is employed for the estimation of
the TVP-VAR models with stochastic volatility. As an example of empirical
application, the TVP-VAR model with stochastic volatility is estimated
using the Japanese data with significant structural changes in the dynamic
relationship between the macroeconomic variables.

Keywords: Bayesian inference; Markov chain Monte Carlo; Monetary

policy; State space model; Structural vector autoregression;
Stochastic volatility; Time-varying parameter
JEL Classification: C11, C15, E52

Economist, Institute for Monetary and Economic Studies, Bank of Japan. Currently in the Person-
nel and Corporate Affairs Department (studying at Duke University) (E-mail: jouchi.nakajima@
stat.duke.edu)

The author would like to thank Shigeru Iwata, Han Li, Toshiaki Watanabe, Tomoyoshi Yabu,
and the staff of the Institute for Monetary and Economic Studies (IMES), Bank of Japan (BOJ),
for their useful comments. Views expressed in this paper are those of the author and do not
necessarily reflect the official views of the BOJ.

MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

DO NOT REPRINT OR REPRODUCE WITHOUT PERMISSION 107
 I. Introduction

A vector autoregression (VAR) is a basic econometric tool in econometric analysis

with a wide range of applications. Among them, a time-varying parameter VAR (TVP-
VAR) model with stochastic volatility, proposed by Primiceri (2005), is broadly used,
especially in analyzing macroeconomic issues. The TVP-VAR model enables us to
capture the potential time-varying nature of the underlying structure in the economy
in a flexible and robust manner. All parameters in the VAR specification are assumed
to follow the first-order random walk process, thus allowing both a temporary and
permanent shift in the parameters.
Stochastic volatility plays an important role in the TVP-VAR model, although
the idea of stochastic volatility is originally proposed by Black (1976), followed by
numerous developments in financial econometrics (see, e.g., Ghysels, Harvey, and
Renault [2002] and Shephard [2005]). In recent years, stochastic volatility is also
more frequently incorporated into the empirical analysis in macroeconomics (e.g.,
Uhlig [1997], Cogley and Sargent [2005], and Primiceri [2005]). In many cases, a
data-generating process of economic variables seems to have drifting coefficients and
shocks of stochastic volatility. If that is the case, then application of a model with
time-varying coefficients but constant volatility raises the question of whether the es-
timated time-varying coefficients are likely to be biased because a possible variation
of the volatility in disturbances is ignored. To avoid this misspecification, stochastic
volatility is assumed in the TVP-VAR model. Although stochastic volatility makes the
estimation difficult because the likelihood function becomes intractable, the model can
be estimated using Markov chain Monte Carlo (MCMC) methods in the context of a
Bayesian inference.
To illustrate the estimation procedure of the TVP-VAR model, this paper begins by
reviewing an estimation algorithm for a TVP regression model with stochastic vola-
tility, which is a univariate case of the TVP-VAR model. Then the paper extends the
estimation algorithm to the multivariate case. The paper also provides simulation exer-
cises of the TVP regression model to examine its estimation performance against the
possibility of structural changes using simulated data. Such simulation exercises show
the important role of stochastic volatility in improving the estimation performance.1
Regarding the empirical application of the TVP-VAR model, this paper provides
empirical illustrations using Japanese macroeconomic data. The estimation results for
standard three-variable models reveal the time-varying structure of the Japanese econ-
omy and the Bank of Japan’s (BOJ’s) monetary policy from 1977 to 2007. During
the three decades of the sample period, the Japanese economy shows significantly dif-
ferent macroeconomic performance, thus implying the possibility of important struc-
tural changes in the economy over time. The time-varying impulse responses show
remarkable changes in the relations between the macroeconomic variables.

1. In this regard, the estimation performance of the TVP-VAR model differs significantly, depending on whether
the stochastic volatility is incorporated or not. Thus, we use the expression “TVP-VAR model with stochastic
volatility” if the inclusion of the stochastic volatility needs to be emphasized. Otherwise, we use just “TVP-VAR
model” for simplicity.

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Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

The paper is organized as follows. In Section II, the estimation methodology of the
TVP regression model is developed. Section III illustrates the simulation study of the
TVP regression model focusing on stochastic volatility. In Section IV, the model spec-
ification, the estimation scheme, and the literature survey of the TVP-VAR model are
provided. Section V presents the empirical results of the TVP-VAR model for Japanese
macroeconomic variables. Finally, Section VI concludes the paper.

II. TVP Regression Model with Stochastic Volatility

This section explains the basic estimation methodology of the TVP-VAR model
by reviewing an estimation algorithm for a univariate TVP regression model with
stochastic volatility.

A. Model
Consider the TVP regression model:
(Regression)

   z        (1)

(Time-varying coefficients)

          (2)

(Stochastic volatility)

   exp        

  
(3)

 
where  is a scalar of response;  and z are   and   vectors of covariates,

respectively; is a   vector of constant coefficients;  is a   vector 
of time-varying coefficients; and is stochastic volatility. We assume that  , 
   ,   , and . 


  

Equation (1) has two parts of covariates; one corresponds to the constant co-
efficients   and the other to the time-varying coefficients  . The effects of  on 
are assumed to be time-invariant, while the regression relations of z to  are assumed
to change over time.
The time-varying coefficients  are formulated to follow the first-order random
walk process in equation (2). It allows both temporary and permanent shifts in the
coefficients. The drifting coefficient is meant to capture a possible nonlinearity, such
as a gradual change or a structural break. In practice, this assumption implies a possi-
bility that the time-varying coefficients capture not only the true movement but also
some spurious movements, because the  can freely move under the random-walk
assumption. In other words, there is a risk that the time-varying coefficients overfit
the data if the relations of z and  are obscure. To avoid such a situation, it might
be better to assume a stationarity for the time-varying coefficients. For example, each
coefficient can be modeled to follow an AR process where the absolute value of

109
 the persistence parameter is less than one. However, in this formulation, a structural
change or a permanent shift of the coefficient would be difficult to estimate even if it
existed. After all, it is important to choose the model specification of the time-varying
coefficients that is considered to be suitable to data of interest, economic theories, and
the purpose of analysis (see, e.g., West and Harrison [1997]).
The disturbance of the regression, denoted by  , follows the normal distribution
with the time-varying variance  . The log-volatility, 
log   , is modeled to
follow the AR process in equation (3). Similar to the discussion on the assumption
of the time-varying coefficients above, the process of log-volatility can be modeled
following both stationary and non-stationary processes. For the following analysis in
this section, we assume that    and the initial condition is set based on the stationary
distribution as   
 
    . In the case of  , the log-volatility follows
the random walk process. The estimation algorithm for the random-walk case requires
only a slight modification for the algorithm developed below.2
We can consider reduced models in the class of the TVP regression model. If
the regression has only constant coefficients (i.e., z   ), the model reduces to a
standard (constant-parameter) linear regression model. If we assume that   , for 

  , the model forms the TVP regression model with the constant variance.

B. Estimation Methodology
1. State space model
Regarding  and as state variables, TVP regression forms the state space model.
The state space model has been well studied in many fields (see, e.g., Harvey [1993]
and Durbin and Koopman [2001] for econometric issues). To estimate the state space
model, several methods have been developed. For the TVP regression models, if the
variance of disturbance is assumed to be time-invariant (i.e., time-varying coefficient
and constant volatility), the parameters are easily estimated using the standard Kalman
filter for a linear Gaussian state space model (e.g., West and Harrison [1997]). However,
if it has stochastic volatility, the maximum likelihood estimation requires a heavy com-
putational burden to repeat the filtering many times to evaluate the likelihood function
for each set of parameters until we reach the maximum, because the model forms a
nonlinear state space model. Therefore, we alternatively take a Bayesian approach using
the MCMC method for a precise and efficient estimation of the TVP regression model.
This also has a great advantage when the model is extended to the TVP-VAR model,
as shown later.
2. Bayesian inference and MCMC sampling method
The MCMC method has become popular in econometrics. In recent years, a con-
siderable number of works on empirical macroeconomics have employed the MCMC
method. The MCMC method is considered in the context of Bayesian inference, and its
goal is to assess the joint posterior distribution of parameters of interest under a certain
prior probability density that the researchers set in advance. Given data, we repeat-
edly sample a Markov chain whose invariant (stationary) distribution is the posterior

2. The estimation algorithm in the case of   is provided in the appendix of Nakajima and Teranishi (2009).
See also Sekine (2006) and Sekine and Teranishi (2008) for investigation of the macroeconomic issues using the
TVP regression model with random-walk stochastic volatility.

110 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

distribution. There are many ways to construct the Markov chain with this property
(e.g., Chib and Greenberg [1996] and Chib [2001]).3
In the Bayesian inference, we specify the prior density, denoted by  , for a
vector of the unknown parameters . Let    denote the likelihood function for
data   
 . Inference is then based on the posterior distribution, denoted by
  , which is obtained by the Bayes’ theorem,


   
       
In principle, the prior information concerning is updated by observing the data  .
The quantity   
     is called the normalizing constant or marginal
distribution. In the case where the likelihood function or the normalizing constant is
intractable, the posterior distribution does not have a closed form. To overcome this
difficulty, many computational methods are developed for sampling from the posterior
distribution. Among them, the MCMC sampling methods are popular and powerful
algorithms that enable us to sample from the posterior distribution without comput-
ing the normalizing constant. The MCMC algorithm proceeds by sampling recursively
the conditional posterior distribution where the most recent values of the conditioning
parameters are used in the simulation.
The Gibbs sampler is one of the well-known MCMC methods. Consider a vector of
unknown parameters  
. The procedure is constructed as follows:
 

(1) Choose an arbitrary starting point  , and set  . 

 






(2) Given  

,    

(a) generate
 from the conditional posterior distribution  

 

 

 
 
,  

(b) generate
 

from 
   
,        

(c) generate
 from    
 
,
 

 

 

   

(d) generate
 
 
, in the same way.



  


(3) Set   , and go to (2).
 

These draws can be used as the basis for making inferences by appealing to suitable
ergodic theorems for Markov chains.
For the estimation of the TVP regression model, there are several reasons to use
the Bayesian inference and MCMC sampling method. First, the likelihood function is
intractable because the model includes the nonlinear state equations of stochastic vol-
atility, which precludes the maximum likelihood estimation method. Also, we cannot
assess the normalizing constant and therefore the posterior distribution analytically.
Second, using the MCMC method, since not only the parameters     
   


but also the state variables    and  are sampled simul- 

taneously, we can make the inference for the state variables with the uncertainty of the
parameters . Third, we can estimate the function of the parameters such as an impulse

3. Koop (2003) and Lancaster (2003) are helpful for understanding Bayesian econometrics as a primer.
Geweke (2005) and Gamerman and Lopes (2006) cover more comprehensive theories and practices of the
MCMC method.

111
 response function with the uncertainty of the parameters taken into consideration by
using the sample drawn through the MCMC procedure.

C. MCMC Algorithm for the TVP Regression Model

For the TVP regression model, specifying the prior density as  , we obtain the
posterior distribution,    .4 There are several ways to implement the MCMC
algorithm to explore this posterior distribution, though we develop the implementation
using the following algorithm:
(1) Initialize ,  , and .
(2) Sample    .
(3) Sample    .
(4) Sample   .
(5) Sample     . 

(6) Sample   . 

(7) Sample   .
 .


(8) Sample 
(9) Go to (2).
The details of the procedure are illustrated as follows.
1. Sample
We specify the prior for as   . We explore the conditional posterior
 

density of given by

   
 
exp          

        z  




exp            

   
where

            


 



   


 

and      z   , for   
. The conditional posterior density is propor-
tional to the kernel of the normal distribution whose mean and variance are and  ,  
respectively. Then, we draw a sample as      .  
2. Sample 
We consider how to sample  from its conditional posterior distribution. Regarding 
as the state variable, the model given by equations (1) and (2) forms the linear Gaussian
state space model. Given the parameters    , a primitive way to sample  is
to assess the conditional posterior density of  given      , where  is

4. Section A of the Appendix provides the functional form of the joint posterior distribution.

112 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

the  excluding  , i.e.,        . This manner of sampling

 

is often called a single-move sampler. The single-move sampler is quite simple, but
inefficient in the sense that the autocorrelation of the MCMC sample often goes ex-
tremely high. For instance, after the  is sampled given  (including   ), the  
is sampled given   (including the  , which has been just drawn). The recursive
 


chain depending on both sides of the sampled state variable yields an undesirable high
autocorrelation. If the MCMC sample has a high autocorrelation, the convergence of
the Markov chain is slow and an inference requires considerably many samples. To re-
duce the sample autocorrelation for  , we introduce the simulation smoother developed
by de Jong and Shephard (1995) and Durbin and Koopman (2002). This enables us to
sample  simultaneously from the conditional posterior distribution     ,
which can reduce the autocorrelation of the MCMC sample.
Following de Jong and Shephard (1995), we show the algorithm of the simulation
smoother on the state space model

  !   
 "   
  (4)

where   ,    , and     # . The simulation smoother draws  

     , where     , for   


 , and  denotes all the

parameters in the model. We initialize $  , %     , and recursively run the


   

Kalman filter:

     ! $ &  ! % !      " % ! & 

' "  ! $  " $   %   " % '    

for    . Then, letting   (  , and      , we run the simulation

smoother:

    (           )   ( '
   !  & '   )     (   !  & ! ' ( ' )    )

 

 

for     . For the initial state, we draw      ,     with

     (  . Once  is drawn, we can compute  using the state equation (4),
     

replacing   by  .
    

In the case of the TVP regression model to sample  , the correspondence of the
variables is as follows:

   !  z      
 

"            
 

 


where  is a    zero vector, and  is a    identity matrix.

 

113
 3. Sample 
We derive the conditional posterior density of . If we specify the prior as 
IW   , where IW denotes the inverse-Wishart distribution, we obtain the
 


conditional posterior distribution for  as

  
 
 exp   tr  






 exp            



 
 


 

 exp   tr
  
(5)

where

           


    

   

Note that the posterior distribution for  depends on only  and (5) forms the kernel of
the inverse-Wishart distribution. Then, we draw the sample as   IW   .   

4. Sample 
Regarding stochastic volatility , the equations (1) and (3) form a nonlinear and non-
Gaussian state space model. We need more technical methods for sampling . A
simple way of sampling is to assess the conditional posterior distribution of given
       and other parameters. This method is called a single-move
sampler, similar to sampling  , and yields an undesirable high autocorrelation in
MCMC sample.
There are mainly two efficient methods for sampling stochastic volatility devel-
oped in the literature. One way to sample stochastic volatility is the approach of Kim,
Shephard, and Chib (1998), called the mixture sampler. The mixture sampler has been
widely used in financial and macroeconomics literature (Cogley and Sargent [2005]
and Primiceri [2005]). The other way is the multi-move sampler of Shephard and Pitt
(1997), modified by Watanabe and Omori (2004). The idea of the former method is to
approximate the nonlinear and non-Gaussian state space model by the normal mixture
distribution, converting the original model to the linear Gaussian state space form.
Though we draw samples from the posterior distribution based on the approximated
model, its approximation error is small enough to implement the original model, and
can be corrected by reweighting steps, as discussed by Kim, Shephard, and Chib (1998),
and Omori et al. (2007). On the other hand, the latter algorithm approaches to the model
by drawing samples from the exact posterior distribution of the original model. Both
methods are more efficient to draw samples of stochastic volatility than a single-move
sampler, while we use the latter one in this paper. The details of the multi-move sampler
are illustrated in Section B of the Appendix.

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Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

5. Sample 
We write the prior of  as , and assume that    Beta . This  

beta distribution is chosen to satisfy the restriction   . The conditional posterior

distribution of  is given by

  
  



       exp       

  
exp

  
 


       

  


 
exp
 



The conditional posterior density does not form any basic distribution from which we
can easily sample. If the term    is omitted, the rest of the term corresponds
to a kernel of the normal distribution. In this case, we use the Metropolis-Hasting (MH)
algorithm (e.g., Chib and Greenberg [1995]).
The idea of the MH algorithm is as follows. First, we draw samples (which we call
candidates) from a certain distribution (proposal distribution) that is close to the con-
ditional posterior distribution we want to sample from. We had better choose the pro-
posal distribution whose random sample can be easily generated. Next, we accept the
candidate as a new sample with a certain probability. When the candidate is rejected,
we use the old (current) sample we have just drawn in the previous iteration as the
new sample. Under certain conditions, the iterations of these steps produce the sample
from the target conditional posterior distribution (see, e.g., Chib and Greenberg [1995]).
There are many ways to choose the proposal density, which often depends on the target
conditional posterior distribution.
Specifically, let *    denote the probability density function of the proposal
 

given the current point , and   denote the acceptance rate from the current
 

point to the proposal  . The MH algorithm is written as the following algorithm:



, and set  . 


(1) Choose an arbitrary starting point
(2) Generate a candidate  from the proposal *   .  

(3) Accept  with the probability  , and set  . Otherwise, set
   

  
.  


(4) Set   , and go to (2).
The acceptance rate is given by

  min     *  

 *   


  

where    denotes the target posterior distribution.

115
 To sample  in our model, we first draw a candidate as   TN   ,
    , and
    

where TN refers to the truncated normal distribution on the domain

  

   
 
  


 
 

This proposal density is the one excluding the term    from the conditional
posterior distribution, considered to be close to our target conditional posterior distribu-
tion and truncated for the same domain of the target. Next, we calculate the probability
for acceptance. Let *  denote the probability density function of the proposal and  

denote the old sample (current point) drawn in the previous iteration. The acceptance
rate for the candidate   from the current point  , denoted by    , is given by 

   

     min
    
  



*  
*   

 min  
      
      

The acceptance rate is the ratio of the terms omitted from the conditional posterior
distribution. The acceptance step can be implemented by drawing a uniform random
number  (   to accept the candidate   when      . 

6. Sample 
We assume the prior of  as  IG  ) , where IG refers to the inverse
   

  
  
    exp

 ) 
gamma distribution. The conditional posterior distribution for  is obtained as




 

 


           


exp 
 
exp


 
 
 

  
    exp

)           

  

 

The conditional posterior distribution forms the kernel of the inverse gamma distribu-
tion. Thus, we draw samples as   IG  ) , where 
 

)  )


          

   

7. Sample 
Sampling  can be implemented in the same way as sampling  . We set the prior
IG  + . Then, the conditional posterior distribution for  is given by


as 

IG  + , where 
 

  +  +     z    


 

116 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

III. Simulation Study

This section carries out simulation exercises of the TVP regression model to examine
its estimation performance against the possibility of structural changes using simulated
data, with emphasis on the role of stochastic volatility.

A. Setup
The performance of the proposed estimation method for the TVP regression model
is illustrated using simulated data. In this simulation study, we investigate how the
parameters are estimated, and how the assumption of stochastic volatility affects the
estimates of other parameters.
Based on the TVP regression model of equations (1)–(3) with ,  , and  
    
 , we generate   and z  as  (   ,  ,, z (   ,  , for  
 -  , where     , z z z  , and ( $ . denotes the uniform
  
   

distribution on the domain $ . . Setting the true parameters as /  ,    

 

     
  ,  diag   ,   ,,   , and   , where diag  refers to
 


a diagonal matrix with the diagonal elements in the arguments, we generate  , , and
 recursively on the TVP regression model. The simulated state variables  and are
plotted in Figure 1. The volatility temporarily increases around  . 

Figure 1 Simulated State Variables  and  ( )

117
 B. Parameter Estimates
We estimate the TVP regression model using the simulated data by drawing 0 
  samples, after the initial 2,000 samples are discarded by assuming the following
prior distributions:5

      IW / /         
  Beta  ,  IG    IG  
 

Figure 2 shows the sample autocorrelation function, the sample paths, and the posterior
densities for the selected parameters. After discarding the samples in the burn-in period
(initial 2,000 samples), the sample paths look stable and the sample autocorrelations
drop stably, indicating that our sampling method efficiently produces the samples with
low autocorrelation.

Figure 2 Estimation Results of the TVP Regression Model (With Stochastic Volatility)
for the Simulated Data

Note: Sample autocorrelations (top), sample paths (middle), and posterior densities (bottom).

5. The computational results are generated using Ox version 4.02 (Doornik [2006]). All the codes for the algorithms
illustrated in this paper are available at http://sites.google.com/site/jnakajimaweb/program.

118 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Table 1 Estimation Results of the TVP Regression Model for the Simulated Data with
(1) Stochastic Volatility and (2) Constant Volatility
[1] TVP Regression Model with Stochastic Volatility
Parameter True Mean Stdev. 95 percent interval CD Inefficiency
 4.0 4.0155 0.1166 [3.7837, 4.2441] 0.833 2.46
 −3.0 −2.8668 0.1371 [−3.1409, −2.6019] 0.909 4.37
 0.1 0.0440 0.0303 [0.0096, 0.1221] 0.144 38.02
 0.03 0.0201 0.0168 [0.0043, 0.0656] 0.217 57.05
 0.95 0.9735 0.0197 [0.9224, 0.9967] 0.895 52.39
 0.5 0.4508 0.1084 [0.2808, 0.7057] 0.506 33.55
 0.1 0.0445 0.0511 [0.0052, 0.1865] 0.908 116.44

[2] TVP Regression Model with Constant Volatility

Parameter True Mean Stdev. 95 percent interval CD Inefficiency
 4.0 4.2373 0.3118 [3.6256, 4.8447] 0.472 1.03
 −3.0 −2.7760 0.3369 [−3.4188, −2.1054] 0.398 1.52
 0.1 0.0173 0.0206 [0.0029, 0.0689] 0.533 68.50
 0.03 0.0123 0.0133 [0.0025, 0.0444] 0.136 70.39
 — 0.9451 0.0688 [0.8215, 1.0922] 0.456 1.87
Note: The true model is stochastic volatility.

Table 1 gives the estimates for posterior means, standard deviations, the 95 per-
cent credible intervals,6 the convergence diagnostics (CD) of Geweke (1992), and in-
efficiency factors, which are computed using the MCMC sample.7 In the estimated
result, the null hypothesis of the convergence to the posterior distribution is not rejected
for the parameters at the 5 percent significance level based on the CD statistics, and the
inefficiency factors are quite low except for  , which indicates an efficient sampling
for the parameters and state variables. Even for  , the inefficiency factor is about 100,

which implies that we obtain about 0   uncorrelated samples. It is considered
to be sufficient for the posterior inference. In addition, the estimated posterior mean is

6. In Bayesian inference, we use “credible intervals” to describe the uncertainty of the parameters, instead of
“confidence intervals” in the frequentist approach. In the MCMC analysis, we usually report the 2.5 percent and
97.5 percent quantiles of posterior draws, as taken here.
7. To check the convergence of the Markov chain, Geweke (1992) suggests the comparison between the first 
draws and the last  draws, dropping out the middle draws. The CD statistics are computed by

CD             


 

 


where       

   ,    is the  -th draw, and    is the standard error of  respectively

  

for    . If the sequence of the MCMC sampling is stationary, it converges in distribution to a standard
normal. We set   ,   ,   , and   . The  is computed using a Parzen win-
dow with bandwidth,   . The inefficiency factor is defined as  
 


 , where  is the sample
autocorrelation at lag , which is computed to measure how well the MCMC chain mixes (see, e.g., Chib [2001]).
It is the ratio of the numerical variance of the posterior sample mean to the variance of the sample mean from
uncorrelated draws. The inverse of the inefficiency factor is also known as relative numerical efficiency (Geweke
[1992]). When the inefficiency factor is equal to , we need to draw the MCMC sample  times as many as the
uncorrelated sample.

119
 close to the true value of the parameter, and the 95 percent credible intervals include it
for each parameter listed in Table 1 [1].

C. The Role of Stochastic Volatility

To assess the function of stochastic volatility in the TVP regression model, we estimate
the TVP regression model with constant volatility for the same simulated data. Because
the true specification is stochastic volatility, we investigate how the estimation result
changes with the misspecification. As mentioned in Section II.A, constant volatility is
specified by   , for    . If we assume the prior as  IG1  2 ,
  IG1  2 ,  
 

then the conditional posterior distribution of  is given by 

where 1 1  , and 2 2       z    . For the MCMC algorithm

 
for the TVP regression model, Steps 4–7 are replaced by the step of sampling  for
constant volatility.
In the simulation study, the prior  IG   is additionally assumed, and the
estimation procedure is the same as the TVP regression model with stochastic volatility
discussed above. Table 1 [2] reports the estimation results of the TVP regression model
with constant volatility for the simulated data. The standard deviations of   are 

evidently wider than the stochastic volatility model, and the posterior means are slightly
apart from the true value. The posterior means of    are estimated lower than


the stochastic volatility model.

We check how the time-varying coefficients are estimated. In addition to the above
two models, the constant coefficient and constant volatility model is estimated. The
posterior estimates of  are plotted in Figure 3. Figure 3 [1] clearly shows that the
constant coefficient model is unable to capture the time variation of the coefficients, and
the posterior mean is estimated around the averaged level of time-varying coefficients
over time. Figure 3 [2] plots the estimates based on the same time-invariant model with
structural breaks. To detect a possible break, the CUSUM of squares test proposed by
Brown, Durbin, and Evans (1975) is applied to divide the sample period into three parts

( – – –). Then, the constant coefficient and constant volatility model
is estimated for each subsample period.8 In the first and second subsample periods, the
posterior 95 percent credible intervals are wide, primarily due to the high volatility
of the disturbance. In the third subsample period, the posterior means seem to follow
the average level of the time-varying coefficient over each subsample period, and the
95 percent credible intervals are narrower. However, the true states are not traced well.
Figure 3 [3] exhibits the estimation results for the TVP regression with constant
volatility. The posterior means seem to follow the true states of the time-varying co-
efficients to some extent. However, for  , some true values do not drop in the 95 per-


cent credible intervals. On the other hand, for  , the intervals are too wide to capture
the movement of the true value. The constant volatility model neglects the behavior
of the volatility change and lacks the accuracy of estimates for  . The estimates 

of the TVP regression with stochastic volatility, which is the true model, are plotted

8. Modeling structural changes is one of the central issues of recent econometrics (see, e.g., Perron [2006]). As
well as the time-varying coefficients and stochastic volatility, structural changes can assess possible changes
in the underlying data generation process. Whether or not a true model has a structural break or time-varying
parameters such as the one in this paper, both models are intended to capture it by approximating its behavior
in each case.

120 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Figure 3 Estimation Results of  on the TVP Regression Model for the Simulated Data

[1] Constant Coefficient and Constant [2] Constant Coefficient and Constant
Volatility Volatility (With Break)

[3] Time-Varying Coefficient and [4] Time-Varying Coefficient and

Constant Volatility Stochastic Volatility

Note: True value (solid line), posterior mean (bold line), and 95 percent credible intervals
(dashed line). The true model is the time-varying coefficient and stochastic volatility [4].

in Figure 3 [4]. The posterior means trace the movement of the true values and the
95 percent credible intervals tend to be narrower overall than the constant volatility
model, and almost include the true values.
The simulation analysis here refers to a profound issue of identifying the source
of the shock. Focusing on the third case, the estimated constant variance  of the
disturbance is smaller in the first-half period and larger in the second half than the true
state of stochastic volatility, because the constant variance captures the average level of
volatility. For the first-half period, the 95 percent credible intervals are almost as wide
as the stochastic volatility model, although the posterior mean is less accurate with
respect to the distance between the estimated posterior means and true values, because
the shock to the disturbance is estimated to be smaller than the true state and the rest of
the shock is drawn up to the drifting  in a misspecified way. On the other hand, for


the second-half period, the posterior mean of the constant volatility model is relatively

121
 accurate compared to the first-half period, but the 95 percent credible intervals are wider
than the stochastic volatility model, because the constant volatility is over-estimated and
the vagueness remains in the drifting  . 

D. Other Models
In addition, other interesting simulations in which the true model is not the TVP
regression form with time-varying coefficient and stochastic volatility are examined.
First, data are simulated from the TVP regression model with constant coefficient and
stochastic volatility. The true values are the same as the previous simulation study,
except   
 and  
, for all   
. The TVP regression model with
time-varying coefficient and stochastic volatility is estimated to examine how the
time-varying coefficient follows the time-invariant true state. The estimation results of
   are shown in Figure 4 [1]. Though the estimates of the posterior means are


not perfectly time-invariant, they are moving near the true states, and the 95 percent
credible intervals include the true value throughout the sample periods.
Second, data are simulated from the TVP regression model with stochastic vola-
tility, but with the time-varying coefficients    modeled to have the Markov-


switching structural change. Much of the literature considers the Markov-switching

type of time-varying parameters in macroeconomic issues. We assume that  and 
 
have two regimes      and      , respectively. The co- 

   

efficients    switch independently with the transition probabilities   

 

  
 

   
    , for    and -  . The TVP regression model with


 
 
 

 

time-varying coefficient (of the original form) and stochastic volatility is estimated
to examine how the time-varying coefficient follows the Markov-switching structural

Figure 4 Estimation Results of  on the TVP Regression Model for the Simulated Data

[1] Constant Coefficient and Stochastic [2] Markov-Switching Coefficient and

Volatility Stochastic Volatility

Note: True value (solid line), posterior mean (bold line), and 95 percent credible intervals
(dashed line). The true models are (1) constant coefficient and stochastic volatility, and
(2) Markov-switching coefficient and stochastic volatility. The TVP regression model with
time-varying coefficient and stochastic volatility is fitted.

122 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

change. Figure 4 [2] plots the estimation results of the coefficients. The true states
of  and  have two breaks and one break, respectively. For both coefficients, the


95 percent credible intervals include the true values. Around the structural breaks, the
posterior means of the coefficients follow the true states to some extent, although their
movements would not be so responsive, especially for  . The degree of adjustment
to the structural change depends on the size of the volatility of the disturbance in
regression. The posterior estimates tend to smooth the true states of the coefficients.
The simulations in this section are just one case of generated data for each setting.
However, the estimation results show the flexibility and the applicability of the TVP re-
gression models, which would help us to understand the importance of the time-varying
parameters in the regression models.

IV. Time-Varying Parameter VAR with Stochastic Volatility

This section extends the estimation algorithm for a univariate TVP estimation model to
a multivariate TVP-VAR model.

A. Model
To introduce the TVP-VAR model, we begin with a basic structural VAR model
defined as

  3    3      1 
    (6)

where  is the    vector of observed variables, and , 3 3 are    matrices

of coefficients. The disturbance  is a    structural shock and, we assume that
 

  , where
 

    




  
    

We specify the simultaneous relations of the structural shock by recursive identification,

 
assuming that is lower-triangular,

   

  $ 



$   $  
  

We rewrite model (6) as the following reduced form VAR model:

        
 
 

    

123
 where   3 , for   1. Stacking the elements in the rows of the  ’s to
 


form ( 1  vector), and defining       , where denotes
  
  

  

the Kronecker product, the model can be written as

   

(7)

Now, all parameters in equation (7) are time-invariant. We extend it to the TVP-VAR
model by allowing the parameters to change over time.
Consider the TVP-VAR model stochastic volatility specified by

  

 1  (8)

where the coefficients , and the parameters and  are all time varying.9 There
are many ways to model the process for these time-varying parameters.10 Follow-
ing Primiceri (2005), let  $ $ $ $  $   be a stacked vector of
and   with  
      

the lower-triangular elements in  log  , for


-  ,  1  
   

. We assume that the parameters in (8) follow a random

walk process as follows:

        
    
    

     # # # #

 # # 
   # #  # 

  # # #  

for   1  , where       ,          and  

   

    .
 

Several remarks are required for the specification of the TVP-VAR model. First,
the assumption of a lower-triangular matrix for is recursive identification for the
VAR system. This specification is simple and widely used, although an estimation of
structural models may require a more complicated identification to extract implications
for the economic structure, as pointed out by Christiano, Eichenbaum, and Evans (1999)
and other studies. In this paper, the estimation algorithm is explained in the model with
recursive identification for simplicity, although the estimation procedure is applicable
for the model with non-recursive identification by a slight modification of the variable
in the MCMC algorithm.
Second, the parameters are not assumed to follow a stationary process such as
AR, but the random walk process. As mentioned before, because the TVP-VAR
model has a number of parameters to estimate, we had better decrease the number of
parameters by assuming the random walk process for the innovation of parameters.
Most of studies that use the TVP-VAR model assume the random walk process for

 
9. Time-varying intercepts are incorporated in some literature on the TVP-VAR models. This case requires only
the modification of defining     Ý       Ý .
10. Hereafter, we use the “TVP-VAR model” to indicate that model with stochastic volatility for simplicity.

124 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

parameters. Note that the extension of the estimation algorithm to the case of stationary
process is straightforward.
Third, the variance and covariance structure for the innovations of the time-varying
parameters are governed by the parameters,  ,  , and  . Most of the articles assume
 

that  is a diagonal matrix. In this paper, we further assume that  is also a diagonal
 

matrix for simplicity. The experience of several estimations indicates that this diago-
nal assumption for  is not sensitive for the results, compared to the non-diagonal


assumption.
Fourth, when the TVP-VAR model is implemented in the Bayesian inference, the
priors should be carefully chosen because the TVP-VAR model has many state variables
and their process is modeled as a non-stationary random walk process. The TVP-VAR
model is so flexible that the state variables can capture both gradual and sudden changes
in the underlying economic structure. On the other hand, allowing time variation in
every parameter in the VAR model may cause an over-identification problem. As men-
tioned by Primiceri (2005), the tight prior for the covariance matrix of the disturbance in
the random walk process avoids the implausible behaviors of the time-varying param-
eters. The time-varying coefficient      requires a tighter prior than
the simultaneous relations       and the volatility      
 

   

of the structural shock for the variance of the disturbance in their time-varying pro-
cess. The structural shock we consider in the model unexpectedly hits the economic
system, and its size would fluctuate more widely over time than the possible change in
the autoregressive system of the economic variables specified by VAR coefficients. In
most of the related literature, a tighter prior is set for  and a rather diffuse prior for
 and  . A prior sensitivity analysis would be necessary to check the robustness of
 

the empirical result with respect to the prior tightness.

Finally, the prior of the initial state of the time-varying parameters is specified.
When the time-series model is a stationary process, we often assume the initial state
following a stationary distribution of the process (for instance,       

in the TVP regression model). However, our time-varying parameters are random
walks; thus, we specify a certain prior for  ,   , and   . We have two ways
     

to set the prior. First, following Primiceri (2005), we set a prior of normal distribution
whose mean and variance are chosen based on the estimates of a constant parameter
VAR model computed using the pre-sample period. It is reasonable to use the economic
structure estimated from the pre-sample period up to the initial period of the main
sample data. Second, we can set a reasonably flat prior for the initial state from the
standpoint that we have no information about the initial state a priori.11

B. Estimation Methodology
The estimation procedure for the TVP-VAR model is illustrated by extending several
parts of the algorithm for the TVP regression model. Let    , and    
   . We set the prior probability density as  for  . Given the data  ,


we draw samples from the posterior distribution,      , by the following

 

MCMC algorithm:

11. Koop and Korobilis (2010) provide a comprehensive discussion on the methodology for the TVP-VAR model,
including the issues about the prior specifications.

125
 (1) Initialize , , , and  .
(2) Sample     .
(3) Sample  .
(4) Sample    .
(5) Sample  .


(6) Sample    .


(7) Sample  .


(8) Go to (2).
The details of the procedure are illustrated as follows.
1. Sample
To sample from the conditional posterior distribution, the state space model with
respect to as the state variable is written as

  

 1 
     1 
where  
  , and     . We run the simulation smoother with the


correspondence of the variables to equation (4) as follows:

   !      #

 

"     #     #  
 

 


where  is the number of rows of .

2. Sample 
To sample  from the conditional posterior distribution, the expression of the state space
form with respect to  is a key to implementing the simulation smoother. Specifically,

        1 
    1
  

where    ,    ,     
 
    , and

 

   


    



 
  
  






  
       
 

126 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

for  1  . We run the simulation smoother to sample  with the

correspondences:

   !      #  

"     #     #  
 

  



where  is the number of rows of  .

3. Sample 


As for stochastic volatility , we make the inference for     separately for -

(   ), because we assume  and  are diagonal matrices. Let   denote the
  



 -th element of  . Then, we can write:

 

   exp    1 
  

    1 
       
   



  
 

where     , and  and   are the  -th diagonal elements of  and   ,

respectively, and  is the  -th element of  . We sample  
     

  using the     

multi-move sampler developed in Section B of the Appendix.

4. Sample 
Sampling  from its conditional posterior distribution is the same way to sample 
in the TVP regression model. Sampling the diagonal elements of  and  is also  

the same way to sample  in the TVP regression model. When the prior is the inverse


gamma distribution, so is the conditional posterior distribution.

C. Literature
The econometric analysis using the VAR model was originally developed by Sims
(1980). Numerous studies have been investigated in this context, and it has become
a standard econometric tool in macroeconomics literature (see, e.g., Leeper, Sims, and
Zha [1996] and Christiano, Eichenbaum, and Evans [1999] for a broader survey of
the literature).
Since the late 1990s, the time-varying components have been incorporated into the
VAR analysis. A salient analysis using the VAR model with time-varying coefficients
was developed by Cogley and Sargent (2001). They estimate a three-variable VAR
model (inflation, unemployment, and nominal short-term interest rates), focusing on
the persistence of inflation and the forecasts of inflation and unemployment for postwar
U.S. data. The dynamics of policy activism are also discussed based on their time-
varying VAR model. Among the discussions of their results, Sims (2001) and Stock
(2001) questioned the assumption of the constant variance ( and  in our notation)
for the VAR’s structural shock, and were concerned that the results for the drifting
coefficients of Cogley and Sargent (2001) might be exaggerated due to the neglect of

127
 a possible variation of the variance.12 Replying to them, Cogley and Sargent (2005)
incorporated stochastic volatility into the VAR model with time-varying coefficients.13
Primiceri (2005) proposes the TVP-VAR model that allows all parameters   
to vary over time, and estimate a three-variable VAR model (the same variables as
Cogley and Sargent [2001]) for the U.S. data.14 The empirical results reveal that the
responses of the policy interest rates to inflation and unemployment exhibit a trend
toward more aggressive behavior in recent decades, and this has a negligible effect on
the rest of the economy.
After Primiceri (2005)’s introduction of the TVP-VAR model, several papers have
analyzed the time-varying structure of the macroeconomy in specific ways. Benati and
Mumtaz (2005) estimate the TVP-VAR model for the U.K. data by imposing sign
restrictions on the impulse responses to assess the source of the “Great Stability” in
the United Kingdom as well as uncertainty for inflation forecasting (see also Benati
[2008]). Baumeister, Durinck, and Peersman (2008) estimate the TVP-VAR model for
the euro area data to assess the effects of excess liquidity shocks on macroeconomic
variables. D’Agostino, Gambetti, and Giannone (2010) examine the forecasting perfor-
mance of the TVP-VAR model over other standard VAR models. Nakajima, Kasuya,
and Watanabe (2009) and Nakajima, Shiratsuka, and Teranishi (2010) estimate the
TVP-VAR model for the Japanese macroeconomic data. An increasing number of
studies have examined the TVP-VAR models to provide empirical evidence of the
dynamic structure of the economy (see e.g., Benati and Surico [2008], Mumtaz and
Surico [2009], Baumeister and Benati [2010], and Clark and Terry [2010]). Given such
previous literature, we will show an empirical application of the TVP-VAR model to
Japanese data, with emphasis on the role of stochastic volatility in the estimation.

V. Empirical Results for the Japanese Economy

As mentioned above, this section applies the TVP-VAR model, developed so far, to
Japanese macroeconomic variables, with emphasis on the role of stochastic volatility in
the estimation.15

A. Data and Settings

A three-variable TVP-VAR model is estimated for quarterly data from the period
1977/Q1 to 2007/Q4, thereby examining the time-varying nature of macroeconomic
dynamics over the three decades of the sample period. To this end, two sets of variables

12. Cogley and Sargent (2005) state, “If the world were characterized by constant [coefficients of the VAR] and
drifting  [variance of the VAR], and we fit an approximating model with constant  and drifting , then it
seems likely that our estimates of would drift to compensate for misspecification of , thus exaggerating the
time variation in .”
13. Uhlig (1997) originally developed the VAR model with stochastic volatility.
14. In Cogley and Sargent (2005), it is assumed that the simultaneous relations, , of the structural shock remain
time-invariant.
15. Similar studies for Japanese macroeconomic data are analyzed by Nakajima, Kasuya, and Watanabe (2009)
and Nakajima, Shiratsuka, and Teranishi (2010). See the previous section for literature on the empirical studies
of the TVP-VAR models using other countries’ data.

128 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Table 2 Estimation Results of Selected Parameters in the TVP-VAR Model for the
Variable Set of   

Parameter Mean Stdev. 95 percent interval CD Inefficiency

  0.0531 0.0123 [0.0341, 0.0824] 0.165 3.97
  0.0567 0.0129 [0.0361, 0.0866] 0.253 10.25
  0.5575 0.4392 [0.1487, 1.7505] 0.511 45.58
  0.6148 0.5439 [0.1633, 1.9004] 0.383 60.34
  0.4453 0.2452 [0.1302, 1.0847] 0.382 33.64
  0.1300 0.0808 [0.0304, 0.3377] 0.526 43.37

Note: The estimates of  and  are multiplied by 100.

are examined:   .  and    , where  is the inflation rate;  is the output; . is

the medium-term interest rates; and  is the short-term interest rates.16
The number of the VAR lags is four,17 and we assume that  is a diagonal matrix in
this study for simplicity. Some experiences indicate that this assumption is not sensitive
for the results, compared to the non-diagonal assumption. The following priors are
assumed for the  -th diagonals of the covariance matrices:

  
Gamma/    
 
Gamma/  
   
Gamma/  
For the initial state of the time-varying parameter, rather flat priors are set;      
    
, and         . To compute the posterior estimates, we draw





0   samples after the initial 1,000 samples are discarded. Table 2 and Figure 5
  

report the estimation results for selected parameters of the TVP-VAR model for the
variable set   . . The results show that the MCMC algorithm produces posterior
draws efficiently.

B. Empirical Results
1. Estimation results for the first set of variables:   
First, the variable set of   .  is estimated. Figure 6 plots the posterior estimates of
stochastic volatility and the simultaneous relation. The time-series plots consist of the
posterior draws on each date. As for the simultaneous relation, which is specified by the
lower triangular matrix , the posterior estimates of the free elements in  , denoted 

16. The inflation rate is taken from the consumer price index (CPI, general excluding fresh food, log-difference,
the effects of the increase in the consumption tax removed, and seasonally adjusted). The output gap is a series
of deviations of GDP from its potential level, calculated by the BOJ. The medium-term bond interest rates are
a yield of five-year Japanese government bonds. Up to 1988/Q1, the five-year interest-bearing bank debenture,
and from 1988/Q2 a series of the generic index of Bloomberg, is used. The short-term interest rates are the
overnight call rate. Except for the output gap, the monthly data are arranged to a quarterly base by monthly
average. For both the interest rates, the (log-scale) difference of the original series from the trend of the HP
filter, that is, an interest rate gap from the trend, is computed for the variable of the estimation.
17. The marginal likelihood is estimated for different lag lengths (up to six) and the number of lags is determined
based on the highest marginal likelihood (see Nakajima, Kasuya, and Watanabe [2009] for the computation of
the marginal likelihood).

129
 Figure 5 Estimation Results of Selected Parameters in the TVP-VAR Model for the
Variable Set of   

Note: Sample autocorrelations (top), sample paths (middle), and posterior densities (bottom).
The estimates of  and  are multiplied by 100.

$
 , are plotted. This implies the size of the simultaneous effect of other variables to one
unit of the structural shock based on the recursive identification.
Stochastic volatility of inflation   exhibits a spike around 1980 due to the second
oil shock, and shows a general downward trend thereafter, with some cyclical ups and
downs around this downward trend. In particular, it remains low and stable during the
first half of the 2000s, when the Japanese economy experiences deflation. Stochastic
volatility of output   remains slightly high in the early 1980s and the late 1990s.
Nakajima, Kasuya, and Watanabe (2009) report that the estimated stochastic volatility
of the structural shock for industrial production becomes higher in the second half of
the 1990s and the beginning of the 2000s, compared to the 1980s. However, stochastic
volatility of the output gap in our analysis based on GDP shows relatively moderate
movements in the 1990s to 2000s. Stochastic volatility of the medium-term interest
rates .  declines significantly in the mid-1990s, when the BOJ reduces the overnight
interest rates close to zero. It declines further in the late 1990s, and remains very low
and stable in the late 1990s to mid-2000s, when the BOJ carries out the zero interest
rate policy from 1999 to 2000 and the quantitative easing policy from 2001 to 2006.
The time-varying simultaneous relation is one of the characteristics in the TVP-
VAR model. The simultaneous relation of the output to the inflation shock   

130 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Figure 6 Posterior Estimates for (1) Stochastic Volatility of the Structural Shock,  

exp  , and (2) Simultaneous Relation,  , for the Variable Set of   

[1] Stochastic Volatility [2] Simultaneous Relation

Note: Posterior mean (solid line) and 95 percent credible intervals (dotted line).

stays positive, and remains almost constant over the sample period. By contrast, the

simultaneous relations of the interest rates to the inflation shock  .  and the output
shock   . vary over time.
The impulse response is a basic tool to see the macroeconomic dynamics captured
by the estimated VAR system. For a standard VAR model whose parameters are all time-
invariant, the impulse responses are drawn for each set of two variables. By contrast, for
the TVP-VAR model, the impulse responses can be drawn in an additional dimension,
that is, the responses are computed at all points in time using the estimated time-varying
parameters. In this case, we have several ways to simulate the impulse response based
on the parameter estimates of the TVP-VAR model. Considering the comparability over
time, we propose to compute the impulse responses by fixing an initial shock size equal
to the time-series average of stochastic volatility over the sample period, and using the
simultaneous relations at each point in time. To compute the recursive innovation of the
variable, the estimated time-varying coefficients are used from the current date to future
periods. Around the end of the sample period, the coefficients are set constant in future
periods for convenience. A three-dimensional plot can be drawn for the time-varying
impulse responses, although a time series of impulse responses for selected horizons or
impulse responses for selected periods are often exhibited in the literature.

131
 Figure 7 shows the impulse responses of the constant VAR model and the time-
varying responses for the TVP-VAR model. The latter responses are drawn in a time-
series manner by showing the size of the impulses for one-quarter and one- to three-year
horizons over time. The time-varying nature of the macroeconomic dynamics between
the variables is shown in the impulse responses, and the shape of the impulse response
in the constant VAR model is associated with the average level of the response in the
TVP-VAR model to some extent.
The impulse responses of output to a positive inflation shock   
  are estimated
as being insignificantly different from zero using the constant-parameter VAR model,
although it is remarkable that the impulse responses vary significantly over time once
the TVP-VAR model is used: the impulse responses stay negative from the 1980s to the
early 1990s, and they turn positive in the mid-1990s. Basic economic theory tells us
that an inflation shock affects output negatively in the medium to long term, which is
consistent with the negative impulse responses observed in the first half of the sample
period. The positive impulse responses observed in the second half of the sample period
imply the possibility of a deflationary spiral, that is, mutual reinforcement between
deflation and recession. The impulse responses of inflation to a positive output shock
   decline rapidly in the early 1980s, and remain around zero thereafter. This
observation can be regarded as empirical evidence of the flattened Phillips curve. The
impulse responses of output to a positive interest rate shock   
  stay negative in
the 1980s, but approach very closely to zero in the mid-1990s, when nominal short-term
interest rates are close to zero, and have remained around zero since then.
2. Estimation results for the second set of variables:   
Next, the variable set of     is estimated. Figure 8 plots the results of stochastic
volatility and simultaneous relations. The stochastic volatilities of inflation and output
seem to be similar to the previous analysis, and stochastic volatility of short-term in-
terest rates   implies the changing variance of the monetary policy shock. Two major
hikes in the interest rate volatility are observed around 1981 and 1986, and the volatility
stays quite low from 1995 under virtually zero interest rate circumstances.

Regarding the simultaneous relations, the effects of inflation on output    and

on interest rates    seem clearer than the previous specification. The simultaneous
effects of inflation on the short-term interest rate shock diminish from the mid-1980s.
At the same time, the simultaneous effects of output on interest rates     become
significantly positive temporarily in the mid-1990s, but decline to zero thereafter. These
observations suggest the possibility that monetary policy responses are constrained by
the zero lower bound (ZLB) of nominal interest rates from the mid-1990s.
Figure 9 shows the impulse responses of estimation results for the variables set of
   . The impulse responses between inflation   and output   are similar to the
previous specification. Regarding the response related to short-term interest rates, the
impulse responses of inflation to a positive short-term interest rate shock    differ
significantly from the previous specifications. The price puzzle in the 1980s becomes
less evident, but time-series movements of the impulse responses become more volatile,
especially from the mid-1980s.

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Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Figure 7 Impulse Responses of (1) Constant VAR and (2) TVP-VAR Models for the
Variable Set of   

[1] Constant VAR Model

[2] TVP-VAR Model (Time-Varying Impulse Responses)

Note: Posterior mean (solid line) and 95 percent intervals (dotted line) for the constant VAR
model. Time-varying responses for one-quarter (dotted line), one-year (dashed line),
two-year (solid line), and three-year (bold line) horizons for the TVP-VAR model.

133
 Figure 8 Posterior Estimates for (1) Stochastic Volatility of the Structural Shock,  

exp  , and (2) Simultaneous Relation,  , for the Variable Set of   

[1] Stochastic Volatility [2] Simultaneous Relation

Note: Posterior mean (solid line) and 95 percent credible intervals (dotted line).

VI. Concluding Remarks

This paper provided an overview of the empirical methodology of the TVP-VAR model
with stochastic volatility as well as its application to the Japanese data. The simula-
tion exercises of the TVP regression model revealed the importance of incorporating
stochastic volatility into the TVP regression models. The empirical applications using
the Japanese data showed the time-varying nature of the dynamic relationships between
macroeconomic variables.
Some words of caution are in order regarding the empirical application of the TVP-
VAR model to data including an extremely low level of interest rates due to the ZLB of
nominal interest rates. Nominal interest rates cannot become negative in the real world,
although the ZLB of nominal interest rates is not assumed explicitly in the standard
specification of the TVP-VAR model, as developed in this paper. Under the ZLB of
nominal interest rates, structural shocks should not be observed in the VAR system. It
is natural that stochastic volatility of the short-term interest rates is estimated to be very
low in the related periods and that the time-varying impulse response of interest rates to
some shocks of economic variables is equal to zero. However, other impulse responses

134 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Figure 9 Impulse Responses of (1) Constant VAR and (2) TVP-VAR Models for the
Variable Set of   

[1] Constant VAR Model

[2] TVP-VAR Model (Time-Varying Impulse Responses)

Note: Posterior mean (solid line) and 95 percent intervals (dotted line) for the constant VAR
model. Time-varying responses for one-quarter (dotted line), one-year (dashed line),
two-year (solid line), and three-year (bold line) horizons for the TVP-VAR model.

135
 related to the interest rates in Figure 9 are not zero but fluctuating for the involved
periods in which the short-term interest rates never change. To solve this problem,
Nakajima (2011) proposes a TVP-VAR model with the ZLB of nominal interest rates
and presents empirical findings using Japanese economic data.
The technique of the TVP-VAR model has been recently extended to the factor-
augmented VAR (FAVAR, originally proposed by Bernanke, Boivin, and Eliasz [2005])
models. The MCMC algorithm illustrated in this paper can be straightforwardly ap-
plied to the estimation of the TVP-FAVAR model. Several studies show the empirical
evidence of the TVP-FAVAR models (e.g., Korobilis [2009] and Baumeister, Liu, and
Mumtaz [2010]). The TVP-VAR model has great potential as a very flexible toolkit to
analyze the evolving structure of the modern economy.

APPENDIX
A. Joint Posterior Distribution for the TVP Regression Model
Given data  , we obtain the joint posterior distribution of    as

  

  
  exp
 
     z  
 
  




             



exp


 
 

    exp      



  
   



 
 


exp    
  
 exp     

 

    

B. Multi-Move Sampler for the TVP Regression Model

In this paper, the multi-move sampler is applied to draw samples from the conditional
posterior density of stochastic volatility in the TVP regression model. This appendix
shows the algorithm of the multi-move sampler following Shephard and Pitt (1997) and
Watanabe and Omori (2004). We rewrite the model as

   exp   
 
    

136 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

       
Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications


    



where         z     ,  , and       . For sampling
(note that   ,   ,
  

a typical block    from its joint posterior density

   ), we consider the draw of
  

        
          
         


 
   

 
 
 
exp
 


 
 

(A.1)

where
    
   exp 
(if   )
   
   


 exp (if   )
     


    
(if  

    
exp )
   
 


 (if   )


and           . The posterior draw of 
       can be    

obtained by running the state equation with the draw of       given  .       

We sample       from the density (A.1) using the acceptance-

    

rejection MH (AR-MH) algorithm (see, e.g., Tierney [1994] and Chib and Greenberg
[1995]) with the following proposal distribution. Our construction of the proposal
density begins with the second-order Taylor expansion of

      

around a certain point  , which is discussed later, namely,

                   

             
    

137
 We have

              
  
 

We use the proposal density formed as

*     

 

exp

    


   

 

 
    

  

where

            (A.2)
 
for       , and     (when    ). For     (when
   ),
  

 
 
       
(A.3)

                 
                
 (A.4)

The choice of this proposal density is derived from its correspondence to the state
space model
     
       

    
 (A.5)



      


with     , when  , and  


    . Given  , we draw  

a candidate point of       for the AR-MH algorithm by running the
    

simulation smoother over the state-space representation (A.5).

Now we find   

 , for which it is desirable to be near the mode of the
 

posterior density for an efficient sampling. We loop the following steps several times
enough to reach near the mode:
 
 ,      by (A.2) and (A.4).
(1) Initialize  .
(2) Compute  
  

(3) Run the moment smoother using the current  

 
 ,      on
   

  
     

(A.5) and obtain 4  for     .

(4) Replace    by  
 
 .   
 
  

(5) Go to (2).

138 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2011

Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications

Note that the 4   is the product in the simulation smoother as  

with   . We divide   into   blocks, say,      for
  
    with   and   , and sample each block recursively. One
   

remark should be made about the determination of   . The method called

  

stochastic knots (Shephard and Pitt [1997]) proposes  

int  (   , for
 


   , where ( is a random sample from the uniform distribution (  . We
 

randomly choose    for every iteration of MCMC sampling for a flexible



 

draw of   .

139
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