B.1 Prior distributions

The parameter space is broken up into four blocks: (1) VAR coefficients Ψ, (2) structural coefficients B, (3) innovation equation coefficients μ, ρ and Q and (4) stochastic volatilities h_t.

VAR coefficients Ψ. I assume a normal distribution for the prior of Ψ. The mean and variance are calibrated based on a homoskedastic version of the model estimated with ordinary least squares using a training sample with data from 1947Q1–1959Q4.^[21] To account for the stochastic volatility in the mean, the estimation takes two steps. First, I estimate a homoskedastic VAR using the training sample and then I take the squared residuals as an initial guess of the stochastic volatility and re-estimate the VAR with the logarithm of this series^[22] (and its lags if necessary) as an additional explanatory variable. The OLS estimate of the VAR coefficients from this second step Ψ^OLS and four times the estimate of its variance P^OLSΨ=4V^(Ψ^OLS) are the mean and variance for the prior distribution, Ψ0∼N(Ψ^OLS,P^OLSΨ).

Structural coefficientsB. In order to apply convenient methods from Cogley and Sargent (2005) for simulating the posterior distribution, I follow Pereira and Lopes (2014) and modify the identification equations (4) to ensure that only exogenous variables appear on the right-hand-side of each regression. The modification involves the equation for output. Letting B=A~−1B~, the relationship between the reduced-form residuals and structural shocks is changed to

There is a one-to-one mapping between the coefficients in (4) and the ones in (23) and the structural shocks estimated using the two identification schemes are identical for the fiscal variables and up to a scale factor for output. The mapping of the coefficients between the identification in (4) and (23) is given by

so that,

The prior of the free parameters in B is also normal. I set the mean to the maximum likelihood estimates – since the identification scheme is non-triangular, a Cholesky decomposition cannot be applied – of the coefficients in B based on the training sample. Let α=[θg,ζτ,ζg]′ be the vector of free parameters in B. The prior is α0∼N(α^MLE,P^α), where, as in Benati and Mumtaz (2007) and Baumeister and Peersman (2013), P^α is a matrix with ten times the absolute value of the elements in α^MLE along the diagonal and zeros elsewhere. The scaling of the variance is to account for the relative magnitude of the coefficients, but otherwise arbitrary.

Innovation equation coefficients μ,ρ,Q. The priors for the intercepts and autoregressive coefficients in each of the stochastic volatilities are also normal. I set μ0i∼N(0,1) and ρ0i∼N(0.9,0.1), for i∈(τ,g,y). These values reflect the traditional modeling conventions that specify the volatility process as a random walk (e.g. see Cogley and Sargent, 2005; Primiceri, 2005). The priors for the diagonal elements of Q follow inverse Gamma distributions with scale parameter 0.5 and 1 degree of freedom, i.e. Q0i∼IG(0.52,12), where once again i∈(τ,g,y) and Q0i is the diagonal element in Q₀ corresponding to the volatility equation for variable i. The scale parameter is larger than typical choices in the literature – for example Jo (2014) and Baumeister and Peersman (2013) set it to 10⁻⁴ – and reflects my knowledge of estimates of similar parameters in Born and Pfeifer (2014) and Fernández-Villaverde et al. (2015).

Stochastic volatility h_t. The prior for the logarithm of volatility at time t = 0 is normal with the mean set to the OLS estimate of the structural shock variances based on the training sample and the variance set to the identity matrix, i.e. log⁡(h0)∼N(log⁡(h^0OLS),I3), where h^0OLS=[σ^τ2,σ^g2,σ^y2]′.

B.2 Initial values and estimation algorithm

I simulate the posterior distributions using the Gibbs sampler. I obtain the starting values for the algorithm by estimating a homoskedastic version of the model in the same way as for the prior distributions. I initialize the stochastic volatility using the squared residuals and set the VAR coefficients Ψ and structural coefficients α to their OLS and MLE estimates. I set the remaining coefficients equal to the means of their prior distributions. I adopt the common notation of using a superscript T to refer to the entire sample. For example, hT={ht}t=1T denotes the entire history of volatility states.

Step 1: drawing structural coefficients α. Conditional on Ψ and h^T, the reduced-form residuals are observable and related to the structural innovations by the following set of regression equations,

Due to the identity in (27) and since θ_y is known, (28) can be rewritten as

which is a regression equation with standard normal innovations. The posterior for θ_g is also normal. Letting R be the left-hand-side variable and M be the right-hand-side variable, the posterior, conditional on the data and other parameters in the model, is θg∼N(α1,P1), where P1=((P^iα)−1+M′M)−1, α1=P1((P^iα)−1α^MLEi+M′R) and the index i on the prior mean and variance selects the value corresponding to the revenue equation.

Once θ_g is drawn, the structural innovations εtτ are identified and the same procedure is applied to (29), which is rewritten as

The coefficients, conditional on the data and other parameters are drawn from [ζτ,ζg]∼N(α1,P1), where α₁ and P₁ are defined in the same way as above but with values corresponding to the output equation.

Step 2: drawing VAR coefficients Ψ. Conditional on h^T and α, (19) is a linear regression with a known form of heteroskedasticity. This equation can be transformed into a state-space model,

where (30) is the observation equation and (31) is the transition equation. The posterior distribution for the VAR coefficients Ψ is normal with mean ΨT|T=E(ΨT|hT,XT,α) and variance PT|T=Cov(ΨT|hT,XT,α). I use the Carter and Kohn (1994) algorithm and obtain Ψ_T|T and P_T|T from the final iteration of a Kalman filter applied to (30) and (31).

Step 3: drawing innovation equation coefficients μ,ρ,Q. Conditional on h^T, the coefficients μ, ρ, Q are drawn using the standard methods for linear regressions. For μ and ρ, the posterior distributions are normal with means and variances combining information from the likelihood and prior in the same way as for the structural coefficients α described above. The posterior for the variance of the error terms in each of the innovation equations has an inverse Gamma distribution, Qi∼IG(∑i=1Tν^i,t2+0.52,T+12), where ν^i,t are the residuals for the stochastic volatility equation corresponding to variable i based on the posterior draws for μ_i and ρ_i. To prevent drawing explosive processes, I use rejection sampling to impose |ρi|≤1.

Step 4: drawing volatility states h^T. Conditional on the VAR coefficients Ψ, the structural coefficients α and the innovation equation coefficients μ, ρ, Q, (20) and (21) form a nonlinear state-space model. Following Cogley and Sargent (2005), I apply the date-by-date independence Metropolis-Hastings algorithm from Jacquier, Polson, and Rossi (1994). This is a univariate algorithm that I can apply to each equation separately due to the diagonality of Q, which makes the equations independent. For details on the sampling distributions and acceptance probability, see B.2.5 in Cogley and Sargent (2005).

Selecting the number of draws and monitoring convergence. Iteratively drawing from the conditional distributions from the four steps described above converges to a sample of coefficients drawn from their joint posterior distribution. To select the number of draws to discard and the number of draws to keep for inference I look at a few convergence diagnostics. A convenient method for determining the number of draws to discard is based on trace plots. These plots trace out consecutive draws for a given parameter over the simulation period. The draws often settle around a stationary mean after starting at some distant point in the parameter space. In my model, this convergence occurs within the first few thousand draws, which I discard.

The number of draws to keep for inference is a less obvious choice because it depends on the persistence of the Markov Chain. If it takes the Gibbs sampler many draws to move from one area of the posterior to another (slow mixing), then the chain needs to run longer to obtain a sufficient amount of independent draws for inference relative to the situation in which the algorithm moves around the posterior quickly (fast mixing). The mixing speed for a given parameter can be determined by the autocorrelation function of its draws. A common practice to alleviate the problems associated with slow-mixing is thinning: keeping only every nth draw where, for example, n can be set to ten. However, as argued by Link and Eaton (2012), thinning is inefficient and – given advancements in computing (particularly for storage) – outdated. Therefore, I choose the number of draws by looking at the autocorrelation functions of a thinned sample and then use the entire sample for inference. A common criterion (e.g. see Primiceri, 2005; Pereira and Lopes, 2014) is to ensure that the 20th autocorrelation for each parameter is close to zero. For instance, if I want to have 1000 draws for inference and thinning the sample by 15 achieves this criteria, then I set the number of draws for inference to 15,000.

C.1 Prior distributions

The parameter space is broken up into the same four blocks as in Appendix B and I use all of the same priors described therein. The time-varying version of the model adds only two new parameters, which are the covariance matrices for the stochastic innovations in (13) and (14). As in Primiceri (2005), I assume that the prior distributions for these matrices are inverse Wishart. The covariance matrix S is block diagonal and consists of two blocks: S₁ corresponds to the revenue equation and S₂ corresponds to the output equation. I assume the following prior distributions W∼IW(kW2×T0×P^OLSΨ,T0) , S1∼IW(kS2×P^1α,10), S2∼IW(kS2×P^2α,10), where T₀ is the number of observations in the training sample and, as before,

where the coefficient estimates come from an estimation of a homoskedastic VAR on the training sample as in Appendix B. The parameters k_W and k_S reflect the prior belief about the amount of time-variation in the coefficients. Following Baumeister and Peersman (2013) and Benati and Mumtaz (2007), I set kS2=kW2=10−4.

As discussed by Primiceri (2005), another factor in determining the prior for the amount of time-variation in the model coefficients arises from the fact that it has a strong influence on the estimation. One the one hand, I want the data to be able to reflect changes in coefficients arising from underlying changes in the transmission mechanism. On the other hand, if the time-variation is not restricted in some way then the coefficients will adjust at each time period to reduce the residuals to as close to zero as possible. Therefore, the choice of kS2 and kW2 reflects a balance between these forces. I find that when these values are loosened the model misbehaves.^[23]

The same issue concerns the choice of the degrees of freedom. The minimum degrees of freedom required to have a proper prior distribution is equal to the number of coefficients plus one. For W this value is 3(3p+3(pλ+1)+c)+1, for S₁ it is 2 and for S₂ it is 3. I set the degrees of freedom to values higher than these minimums to avert implausible behaviors of the time-varying coefficients. Primiceri (2005) also sets the degrees of freedom for the prior of W to the number of observations in the training sample for the same reasons.

C.2 Initial values and estimation algorithm

I initialize the Gibbs sampler in the same way as described in Appendix B.2 except that I use an estimate of h^T from the SVAR-SV-M model to initialize the stochastic volatility rather than using the squared residuals from the homoskedastic regression. The estimation proceeds in the same steps as described in Appendix B.2 but with modifications to the procedure for drawing the VAR coefficients and structural equation coefficients to account for time-variation.

Drawing VAR coefficients Ψ^T and W. Conditional on h^T, α^T and W, the VAR coefficients have a state-space representation where (10) is the observation equation and (13) is the transition equation. As in Cogley and Sargent (2005), the joint distribution for the entire history of the VAR coefficients Ψ^T is given by

where all the densities on the right-hand-side are conditionally Gaussian and can be obtained by backward recursion from the terminal state of the forward Kalman filter.

I use the Carter and Kohn (1994) algorithm, set the initial values to the prior mean and covariance and iterate on the following equations

These equations are identical to the ones for the fixed-parameter version of the model except for a key difference in (32), where I augment the variance to account for the stochastic evolution of the coefficients. The final iteration of the forward recursion delivers the mean and variance for the posterior distribution of Ψ_T, i.e. ΨT∼N(ΨT|T,PT|T). The remaining history of Ψ^T is drawn period-by-period through backward recursion where the conditional means and variances are updated to include information about Ψ_{t + 1} for drawing Ψ_t. For t = 1, …, T − 1, the posterior distribution of Ψ_t is

The algorithm generates smoothed draws for Ψ^T that account for information in the entire sample. Conditional on Ψ^T, the covariance for the innovation equation is drawn from an inverse Wishart distribution,

where ω^t=Ψt−Ψt−1 and ω^T is the entire history of these residuals.

Drawing structural equation coefficients α^T and S. Drawing the structural equation coefficients follows the same procedure as described above for the VAR coefficients. Conditional on Ψ^T, h^T and S, the reduced-form residuals and structural shocks form the following set of state-space models

with (33) specifying two measurement equations and (34) their corresponding transition equations.

With utg=htgεtg, the structural equation for tax revenue forms the following state-space model

I use the Carter and Kohn (1994) algorithm to run the Kalman filter forward and then take draws for θgT using backward recursion form the terminal state. The draw for θgT makes εtτ observable in the output equation, which forms a second state-space model given by

Once again, I obtain draws of the coefficients ζτT and ζgT from their posterior distributions using the Kalman filter with backward recursion.

Conditional on α^T, the covariance matrices for each of the innovation equations are drawn from inverse Wishart distributions,

where

and the entire histories of these residuals are given by e^1T and e^2T, respectively.

Drawing μ, ρ, Q and h^T follows exactly the same steps as in Appendix B.1. I also monitor convergence and determine the number of draws using the methods described in that section.