The models used in this paper all begin with the TVP-VAR given in (1) and (2) in the body of the text. A key step in any of our MCMC algorithms will be to draw the states, = ( 01; ::; 0T)0. For known values of Ht; Qt and Rt, this can be done using any of the standard algorithms for state space models. We use the algorithm of Durbin and Koopman (2002) which (for the reasons given in that paper) is more e¢ cient than other popular alternatives.
State space algorithms such as this require a treatment of the initial condition
1. We do this by writing (1) as:
yt = Zt 0+ Zt t+ "t
and then initializing the algorithm for drawing states by setting 1 = 0.
Note that 0 can be interpreted as benchmark VAR coe¢ cients and the state equation as capturing deviations from this benchmark. The case where Rt = 0 for t = 1; ::; T then produces the standard VAR with time-invariant parameters.
Our MCMC algorithm involves drawing from the posterior of 0 condi-tional on the states and other model parameters. This is straightforward since we can re-arrange the previous equation as:
yt Zt t= Zt 0+ "t
and standard results for the multivariate Normal regression model (see, e.g., Koop, 2005, pages 140-141) can be used with yt Zt t as the dependent variable. In our models with stochastic volatility, we also use the Durbin and Koopman (2002) algorithm for the elements relating to the measurement error covariance matrix. In these cases, we treat initial conditions in the same manner.
Our MCMC algorithms involve cycling through the full posterior con-ditional distributions. For simplicity, we do not list all the conditioning arguments. But we stress that all of the posteriors noted below (which are labelled as being conditional on "Data") are the full conditionals required to set up a valid MCMC algorithm.
Model 1: A Time Varying VAR with Constant Error Covariance A simple benchmark model is the time-varying VAR with constant error covariance matrix. This is obtained by using (1) and (2) with Ht= H; Qt = Q
and Rt= Im. We follow the common practice of using Wishart priors for the error precision matrices in the measurement and state equations:
H 1 W H; H 1 (A.1)
and
Q 1 W Q; Q 1 (A.2)
The posterior for H 1 (conditional on the states) is Wishart:
H 1jData W H; H 1 (A.3)
where
H = T + H and
H 1 =
"
H + XT
t=1
(yt Zt t) (yt Zt t)0
# 1
: The posterior for Q 1 (conditional on the states) is Wishart:
Q 1jData W Q; Q 1 (A.4)
where
Q = T + Q and
Q 1 =
"
Q + XT
t=1
( t+1 t) ( t+1 t)0
# 1 :
A posterior simulator for this model involves drawing the states using the algorithm of Durbin and Koopman (2002) and drawing the other model parameters from (A.3) and (A.4).
Model 2: A Time Varying VAR with Stochastic Volatility Following the argument in the body of the paper, it is probably unrea-sonable to assume the error covariances are constant over time. We use a
triangular reduction of the measurement error covariance, Ht, given in (3) with evolution of the parameters given by (4) and (5).
To carry out posterior simulation of h = (h01; ::; h0T)0 (conditional on and the parameters of the model) we can, following Primiceri (2005), adapt an algorithm of Kim, Shephard and Chib (1998) as follows. Using (3) we can transform (1) as:
yt = At(yt Zt t) ;
where var (yt) = t 0t which is a diagonal matrix. Let yj;t for j = 1; ::; p denote the jth element of yt, yj;t = lnh
yj;t 2+ ci
and yt = y1;t; ::; yp;t 0. Note that c is referred to as an o¤set constant which has no e¤ect on the following theoretical derivations. Following standard practice we set c = 0:001.
We can now write our speci…cation for t as a state space model with measurement equation given by
yt = 2ht+ et (A.5)
and state equation (4).5 The only problem with using standard state space algorithms is that et is not Normally distributed. Note, however, that since yj;t and yi;t are independent of one another (for i 6= j), this independence property will carry over to et = (e1t; ::; ept)0. Thus, we can draw on the uni-variate results of Kim, Shephard and Chib (1998) as relating to ejt. Although ejt is not Normal, Kim, Shephard and Chib (1998) show how its distribution can be approximated to an extremely high degree of accuracy by a mix-ture of seven Normals with means and variances given in their Table 4. If Sjt 2 f1; 2; 3; ::; 7g denotes which of the seven Normals ejt is drawn from, we can construct Sj = (Sj1; ::; SjT)0 and S = S10; ::; Sp0 0 as component indica-tors for all elements of et. Conditional on S (and and other parameters), (A.5) and (4) is a Normal linear state space model and, hence, we can use the algorithm of Durbin and Koopman (2002) to draw ht.
The strategy above requires that we draw from the posterior of S condi-tional on the model parameters and states. Kim, Shephard and Chib (1998) derive the appropriate posterior conditional. Let qi; mi and v2i for i = 1; ::; 7
5We treat the initial conditions as in Primiceri (2005) by drawing from the training sample prior.
be the component probability, mean and variance of each of the components in the Normal mixture (obtained from their Table 4). Then
Pr (Sit= jjData; hi;t)/ qjfN yi;tj2hi;t+ mj 1:2704; v2j (A.6) for j = 1; ::; 7, i = 1; ::; p and t = 1; ::; T .
To complete the description of the MCMC algorithm relating to t, we need to work out the conditional posterior for W (where W is de…ned after equation 4). We use a Wishart prior for W 1:
W 1 W W; W 1 : (A.7)
The posterior for W 1 (conditional on the states) is then Wishart:
W 1jData W W; W 1 (A.8)
where
W = T + W and d
W 1 =
"
W + XT
t=1
(ht+1 ht) (ht+1 ht)0
# 1 :
Thus, to handle stochastic volatility in t, we add to the MCMC algo-rithm for Model 1 steps which draw h using the state space model (A.5) and (4), S using (A.6) and W using (A.8).
Next we describe an algorithm for drawing from At, the unrestricted
ele-ments of which we stack by rows into a p(p 1)2 vector as at= a21;t; a31;t; a32;t; ::; ap(p 1);t 0. These are allowed to evolve according to the state equation (5). We can
trans-form the original measurement equation so that the Durbin and Koopman (2002) algorithm can be used to draw the states. This can be done as follows.
De…ne
b
yt= yt Zt t and:
Atybt= t
where t is independent N (0; t t) (and independent of t) . We can use the structure of At to isolate ybt on the left hand side and write:
b
yt= Ctat+ t: (A.9)
Primiceri (2005), page 845 gives a general de…nition of Ct. For our empirical work we have p = 3 and, for this case,
Ct= 2 4
0 0 0
b
y1;t 0 0 0 by1;t yb2;t
3 5 ;
where ybi;t is the ith element of ybt. (A.9) and (5) is now in form of the state space model given in (1) and (2) and the algorithm of Durbin and Koopman (2002) can be used to draw at for t = 1; ::; T .
Recall that the error tin the state equation (5) has distribution N (0; C) : To complete the description of the MCMC algorithm relating to At, we need to work out the conditional posterior for C. We use a Wishart prior for C 1:
C 1 W C; C 1 : (A.10)
The posterior for C 1 (conditional on the states) is then Wishart:
C 1jData W C; C 1 (A.11)
where
C = T + C and
C 1 =
"
C + XT
t=1
(at+1 at) (at+1 at)0
# 1
:
To summarize, to handle the variation in At, we add to the MCMC algo-rithm, steps which draw at (for t = 1; ::; T ) using the state space model (A.9) and (5), and C using (A.11). To obtain draws of get the structural VAR (see equation 9), we can use the transformation t= At1 t:
Model 3: A Mixture Innovation Time-varying VAR with Sto-chastic Volatility
Our mixture innovation extension of the TVP-VAR with stochastic volatil-ity is given in (6) through (8). (8) de…nes a hierarchical prior which depends on the parameters pj for j = 1; 2; 3. We use a (conditionally) conjugate Beta prior for pj for j = 1; 2; 3: B
1j;
2j . With this choice, the conditional posterior for the breakpoint probabilities used in our MCMC algorithm is:
B 1j; 2j ; (A.12)
The MCMC algorithm for the time-varying parameter model (set out previously in this appendix) still, with one minor alteration, works (except now that the formulae set out above are additionally conditional on K). The alteration is that the degrees of freedom parameters, Q, W and C all have T in their formulae which should be changed to PT
t=1K1t, PT
t=1K2t and PT
t=1K3t, respectively.
To complete our MCMC algorithm, we must specify a way of drawing K. The posterior for K conditional on the states takes a simple form. This motivated some early authors (e.g. McCulloch and Tsay, 1993) to draw from K conditional on the states. However, Gerlach, Carter and Kohn (2000) point out some limitations of such a strategy. Most importantly it can be extremely ine¢ cient since the states and K can be very highly correlated with one another. They develop an algorithm which integrates out the states analytically and draws from p KtjData; K( t) where K( t) denotes all the elements of K except for Kt. For state space models, Gerlach, Carter and Kohn (2000) use notation xs;t for all observations from s to t on any variable, x, and show that:
p KtjData; K( t) / p yt+1;Tjy1;t; K p ytjy1;t 1; K1;t p KtjK( t) : (A.13)
The term p KtjK( t) is simply the hierarchical prior and, thus, easy to draw from. Gerlach, Carter and Kohn (2000, pages 820-822) set out an e¢ cient algorithm for drawing from the other terms p yt+1;Tjy1;t; K and p (ytjy1;t 1; K1;t).
As discussed in Giordani and Kohn (2006), we can draw K1t,K2t and K3t separately from one another in the context of the three state space al-gorithms which make up the blocks of the MCMC algorithm for time vary-ing parameter model with stochastic volatility. Formally, this amounts to drawing from p K1tjData; K( t); K2t; K3t , p K2tjData; K( t); K1t; K3t and p K1tjData; K( t); K1t; K2t . That is, drawing t in the time varying para-meter model involves use of the algorithm of Durbin and Koopman (2002) conditional on all the model parameters including Ht (see our discussion of Model 1). K2t and K3t are used in the de…nition of Ht in Model 3. Thus, the algorithm of Gerlach, Carter and Kohn (2000) can be combined with Durbin and Koopman (2002) to draw from K1t and the VAR coe¢ cients (conditional on all other model parameters including K2t and K3t). Simi-larly, the algorithm of Gerlach, Carter and Kohn (2000) can be combined with Durbin and Koopman (2002) to draw from K3t and At (conditional on all other model parameters including K1t and K2t). Finally, the algorithm of Gerlach, Carter and Kohn (2000) can be combined with our extension of Kim, Shephard and Chib (1998) to draw from K2t and t (conditional on all other model parameters including K1t and K3t).
For the TVP-VAR the prior we use is the same as that used in Primiceri (2005). That is, we use a training sample prior with the …rst ten years of data to choose many of the key prior hyperparameters. To be precise, we use the training sample and a time-invariant VAR to produce OLS estimates of the VAR coe¢ cients, b0, and the error covariance matrix, b and decompose the latter as in (3) to produce ba0 and bh0 (where these are both vectors stacking the free elements as we did with At and t). We also obtain OLS estimates of the variance-covariance matrices of b0 and ba0 which we label bV and bVa. Using these, we construct the priors for the initial conditions in each of our state equations as:
0 N b0; 4 bV ; a0 N ba0; 4 bVa and
log (h0) N log bh0 ; I3 :
Next we describe the priors for the error variances in the state equations.
Note that we are choosing small degrees of freedom parameters (relative to sample size) and, thus, these prior contain a relatively small amount of information (relative to the data). For (A.2) we set Q = 40 and Q = 0:0001 bV For (A.7) we set W = 4 and W = 0:0001I3. For (A.10), we set
C = 3 and C = 0:01 bVa.
For the TVP-VAR this completes the speci…cation of the prior. For re-stricted versions of this model (e.g. the homoskedastic TVP-VAR or the stan-dard time-invariant VAR) we use the same prior for the parameters which are left unrestricted.
The preceding prior choices were the same as Primiceri (2005) and were calibrated with the TVP-VAR with stochastic volatility in mind. With our mixture innovation extension of the TVP-VAR, we have to additionally elicit the prior hyperparameters 1j and 2j. These are discussed in the empirical section. With regards to the remaining parameters, we make one alteration on Primiceri’s prior. The latter was a prior calculated for a TVP-VAR with stochastic volatility which assumed a structural break occurred in every time period (a “many small breaks” model). We want our prior for the mixture innovation extension to allow for this, but also to allow for fewer breaks, potentially of a larger magnitude. Accordingly, we allow the mean of the error covariance matrices for the state equation to depend on our prior about the number of breaks which occur. Note that, the Beta prior in (A.12) implies that
E (pj) = 1j
1j +
2j
:
If we let T0j = E (pj) T, we modify our previous prior hyperparameters as Q = 0:0001 bV TT
01, W = 0:0001I3 T
T02 and C = 0:01 bVaTT
03. Thus, if we set E (pj) = 1we get Primiceri’s prior, but if we use a prior for pj which implies fewer breaks, then our prior for the state equation error variances allows for large shifts in the parameters to occur.
The Gerlach, Carter and Kohn (2000) algorithm allows us to calculate the marginal likelihood and the expected value of the likelihood in a straight-forward manner. Let Y stack all the data on the dependent variables and
denote all the parameters in the model except for K1; K2 and K3. Equa-tion (3) and Lemmas 3 and 4 of Gerlach, Carter and Kohn (2000) describe how we can calculate p (Y jK1) for the model studied in that paper. Our algorithm uses the Gerlach, Carter and Kohn (2000) three times (i.e. for K1; K2 and K3). But we can use the Gerlach, Carter and Kohn (2005) result as holding for p (Y jK1; K2; K3; ). By averaging over MCMC draws of all of these parameters (i.e. K1; K2; K3; ), we can obtain the expected value of the log-likelihood function. To calculate the marginal likelihood, we use these draws of the likelihood function in the approach to marginal likelihood calculation of Gelfand and Dey (1994). Note that this approach involves integrating out the states before calculating p (Y jK) (and, hence, is much more computationally e¢ cient than using the Kalman …lter to evaluate the likelihood function). Finally, note that some of the models set elements of K to particular values and, for these, we simply condition on these values.
For instance, for the Benchmark model with t being constant, we calculate p (YjK11= K12= :: = K1T = 0).
The use of the expected log-likelihood can be motivated as in Section 6.5.1 of Carlin and Louis (2000). Note that Carlin and Louis’s penalized likelihood criteria are closely related to conventional information criteria such as the Schwarz criteria, but (instead of evaluating them at the maximum likelihood estimate) use the posterior and are based on the expected value of the log of the likelihood function. Like information criteria, such features do not involve the prior (except insofar as the prior enters the posterior and, thus, the MCMC algorithm) and, thus, will be less sensitive to prior choice (and can be considered as approximations to the log of the marginal likelihood).
Finally, we turn to the calculation of impulse responses. In linear (time-invariant) VARs, impulse responses can be taken directly from the Vector Moving Average (VMA) representation implied by the VAR. However, with a TVP-VAR the implied VMA is changing over time. Suppose the VMA representation of a standard VAR is given by:
yt= X1
i=0
iut i;
then the usual result is that an impulse response h periods in the future is the appropriate element of h. With a TVP-VAR the implied VMA will, of course, have time varying coe¢ cients:
yt= X1
i=0
t i;iut i:
This raises two issues when calculating impulse responses. The …rst is that the impulse responses will be changing over time. Hence, we have to either plot impulse responses for every time period or choose a few time periods for detailed study. We adopt both these strategies in our empirical work. A second and more subtle issue arises due to the treatment of shocks other than the one being perturbed. To explain this issue, suppose we are interested in the e¤ect of a shock of size one (to the structural errors in the measurement equation) which occurs at time on the variables at time +h.
Strictly speaking, an impulse response is usually interpreted as a di¤erence in conditional expectations such as:
E (y +hjI ; u = 1) E (y +hjI ) ;
where I denotes information through time . In any nonlinear time series model, these expectations can be calculated using simulation methods (as in Koop, 1996). However, this can be computationally demanding, so it is much easier to simply take the structural VAR coe¢ cients at time (i.e.
and ) and calculate a conventional impulse response function. In linear models, these two strategies are identical, but with nonlinear models they can be slightly di¤erent. Nevertheless, in this paper we adopt this second simpler strategy. Formally, it can be interpreted as an impulse response function calculated assuming all shocks to the model (including the shocks to the state equations) between time and + h are simply set to their expected values of zero.