The assumption that exogenous shocks evolve according to independent AR(1) is to some extent arbitrary.
Trying to generalize this assumption seems natural.
However, the more elaborate the exogenous propagation mechanism, the more difficult it becomes to disentangle endogenous from exogenous propagation.
This generates identification problems.
Technology growth shock \(\hat{z}_t\), government spending shock \index{government!spending shock} \(\hat{g}_t\) evolve:
\begin{align*} \left[ \begin{array}{c} \hat{z}_t \ \hat{g}_t \end{array} \right] &= \left[ \begin{array}{cc} \rho_z & \rho_{zg} \ \rho_{gz} & \rho_g \end{array} \right] \left[ \begin{array}{c} \hat{z}_{t-1} \ \hat{g}_{t-1} \end{array} \right]
This VAR process is combined with:
\begin{eqnarray*} \hat y_{t} &=& \mathbb{E}_t[\hat y_{t+1}] - \frac{1}{\tau} \bigg( \hat R_t - \mathbb{E}_t[\hat\pi_{t+1}] - \mathbb{E}_t[\hat{z}_{t+1}] \bigg)\\\ &&+ \hat{g}_t - \mathbb{E}_t[\hat{g}_{t+1}], \\\ \hat \pi_t &=& \beta \mathbb{E}_t[\hat \pi_{t+1}] + \kappa (\hat y_t- \hat g_t), \\\ \hat R_{t} &=& \rho_R \hat R_{t-1} + (1-\rho_R) \psi_1 \hat \pi_{t} + (1-\rho_R) \psi_2 \left( \hat y_{t} - \hat g_t \right)+ \epsilon_{R,t}. \end{eqnarray*}
We use agnostic priors:\( \quad \rho_g, \rho_z \sim U[0,1], \quad \rho_{gz}, \rho_{zg} \sim U[-1,1].\)
\begin{center} \includegraphics[width=4in]{static/dsge1_gen_shock_density.pdf} \end{center}
Notes: The two panels depict histograms of prior distributions (shaded area) and kernel density estimates of the posterior densities (solid lines).
\begin{center} \includegraphics[width=4in]{static/dsge1_gen_shock_irfs_exo.pdf} \end{center}
\begin{center} \includegraphics[width=3in]{static/dsge1_gen_shock_irfs_endo.pdf} \end{center}
\begin{table}[t!] \begin{center} \begin{tabular}{p{2in}p{2in}} \hline \hline RWMH-V & SMC \ \hline $N = 100,000$ & $N = 4,800$ \\\ $N_{burn} = 50,000 $ & $N_\phi = 500$ \\\ $N_{blocks} = 1$ & $N_{blocks} = 6$, $N_{MH}=1$ \\\ $c= 0.125 $ & $\lambda = 2$ \\\ Run Time: 00:28 (1 core) & Run Time: 05:52 (12 cores) \ \hline \hline\\\ \end{tabular} \end{center} \end{table}
Note: We run each algorithm \(N_{run}=50\) times. Run time is reported as mm:ss.
\begin{center} \begin{tabular}{cc} $\mathbb{P}_\pi\{\rho_{zg} > 0 \}$ & $\mathbb{P}_\pi\{\partial \hat{\pi}_t / \partial \hat{\epsilon}_{g,t} > 0 \}$ \\\ \includegraphics[width=0.46\textwidth]{static/dsge1_gen_shock_prob_rhozg.pdf} & \includegraphics[width=0.46\textwidth]{static/dsge1_gen_shock_prob_pi_g_irf.pdf}\\\ \end{tabular} \end{center}
Notes: Each symbol (50 in total) corresponds to one run of the SMC algorithm (dot) or the RWMH algorithm (triangle).
\begin{center} \includegraphics[width=3in]{static/dsge1_gen_shock_log_mdd.pdf} \end{center}
Notes: Each symbol (50 in total) corresponds to one run of the SMC algorithm (dot) or the RWMH algorithm (triangle). The SMC algorithm automatically generates an estimate of the MDD; for the RWMH algorithm we use Geweke’s modified harmonic mean estimator.
\begin{table}[t!] \begin{center} \begin{tabular}{lcc} \hline\hline Model & Mean($\ln \hat p(Y)$) & Std. Dev.($\ln \hat p(Y)$) \\\ \hline AR(1) Shocks & $-346.16$ & (0.07) \\\ VAR(1) Shocks & $-314.45$ & (0.05) \\\ \hline \hline\\\ \end{tabular} \end{center} \end{table}
Notes: Table shows mean and standard deviation of SMC-based estimate of the log marginal data density, computed over \(N_{run}=50\) runs of the SMC sampler.
Sort the posterior draws \(\{\theta_j^i\}_{i=1}^N\) and select
the \(\lfloor \tau N \rfloor\)’th element.
Quantile regression (Koenker and Basset, 1978)
\begin{eqnarray*} \hat{q}_\tau(\theta_j) &=& \mbox{argmin}_{q} ; \bigg[ (1-\tau) \frac{1}{N} \sum_{i: , \theta^i_j < q} (\theta_j^i - q) \&&+ \tau \frac{1}{N} \sum_{i: , \theta^i_j \ge q} (\theta_j^i - q) \bigg]. \nonumber \end{eqnarray*}
\begin{table}[t!] \begin{center} \begin{tabular}{p{2in}p{2in}} \hline \hline RWMH-V & SMC \ \hline $N = 10,000,000$ & $N = 12,000$ \\\ $N_{burn} = 5,000,000 $ & $N_\phi = 500$ \\\ $N_{blocks} = 1$ & $N_{blocks} = 6$, $N_{MH}=1$ \\\ $c= 0.08 $ & $\lambda = 2.1$ \\\ Run Time: 14:06 (1 core) & Run Time: 02:32 (24 cores) \ \hline \hline\\\ \end{tabular} \end{center} \end{table}
Note: We run each algorithm \(N_{run}=50\) times. Run time is reported as hh:mm.
\begin{center} \begin{tabular}{cc} \includegraphics[width=0.35\textwidth]{static/sw_diffuse_density_Neff_alp.pdf} & \includegraphics[width=0.35\textwidth]{static/sw_diffuse_density_Neff_rdely.pdf} \ \includegraphics[width=0.35\textwidth]{static/sw_diffuse_density_Neff_xiw.pdf} & \includegraphics[width=0.35\textwidth]{static/sw_diffuse_density_Neff_iotaw.pdf} \end{tabular} \end{center}
Notes: Each panel depicts a Kernel estimate of the posterior density (solid) and \(\ln (N_{eff}) = \ln (N/\mbox{InEff}_N)\) (light gray hatched bars correspond to RWMH and solid bars correspond to SMC) for various choices of \(\tau\) equal to \(0.05\), \(0.5\), and \(0.95\).
\begin{center} \includegraphics[width=0.38\textwidth]{static/sw_diffuse_scatter_Neffs_log.pdf} \end{center}
Notes: \(N_{eff}\) for the RWMH-V and SMC quantile approximations. Each dot corresponds to one parameter. The 45-degree line appears in solid.
The budget constraint of the households
\begin{align*} (1+\tau_t^c)c_t &+ i_t + b_t \\\ &= (1-\tau_t^l)w_tl_t + (1-\tau_t^k)R_t^ku_tk_{t-1} + R_{t-1}b_{t-1} + z_t. \nonumber \end{align*}
The budget constraint for the government, using capital letters to denote aggregate quantities \[ B_t + \tau_t^kR_t^ku_tK_{t-1} + \tau_t^l w_t L_t + \tau_t^cC_t = R_{t-1}B_{t-1} + G_t + Z_t. \]
The fiscal policy rules (\(\hat x_t\): log deviation from steady state of \(x_t\))
\begin{align*} \hat\tau_t^k &= \varphi_k\hat Y_t + \gamma_k \hat B_{t-1} + \phi_{kl}\hat u_t^l + \phi_{kc} \hat u_t^c + \hat u_t^k, \\\ \hat\tau_t^l &= \varphi_l\hat Y_t + \gamma_l \hat B_{t-1} + \phi_{lk}\hat u_t^k + \phi_{lc} \hat u_t^c + \hat u_t^l, \\\ \hat\tau_t^c &= \phi_{ck}\hat u_t^k + \phi_{cl} \hat u_t^l + \hat u_t^c. \end{align*}
The exogenous movements in taxes follow AR(1) processes
\begin{align*} \hat u_t^k &= \rho_k \hat u_{t-1}^k + \sigma_k \epsilon_t^k, \quad \epsilon_t^k \sim N(0,1), \\\ \hat u_t^l &= \rho_l \hat u_{t-1}^l + \sigma_l \epsilon_t^l, \quad \epsilon_t^l \sim N(0,1), \\\ \hat u_t^c &= \rho_c \hat u_{t-1}^c + \sigma_c \epsilon_t^c, \quad \epsilon_t^c \sim N(0,1). \end{align*}
The government spending rule is given by
\begin{align*} \hat G_t &= -\varphi_g \hat Y_t - \gamma_g \hat B_{t-1} + \hat u_t^g, \\\ \hat u_t^g &= \rho_g \hat u_{t-1}^g + \sigma_g \epsilon_t^g, \quad \epsilon_t^g \sim N(0,1). \end{align*}
The transfer rule is given by
\begin{align*} \hat Z_t &= -\varphi_z \hat Y_t - \gamma_z \hat B_{t-1} + \hat u_t^z, \\\ \hat u_t^z &= \rho_z \hat u_{t-1}^z + \sigma_z \epsilon_t^z, \quad \epsilon_t^z \sim N(0,1). \end{align*}
\begin{table}[t!] \begin{center} \scalebox{0.93}{ \begin{tabular}{lcccccc} \hline\hline & \multicolumn{3}{c}{LPT Prior} & \multicolumn{3}{c}{Diffuse Prior} \\\ & Type & Para (1) & Para (2) & Type & Para (1) & Para (2)\\\ \hline \multicolumn{7}{c}{Debt Response Parameters} \\\ \hline $\gamma_{g}$ & G & 0.4 & 0.2 & U & 0 & 5 \\\ $\gamma_{tk}$ & G & 0.4 & 0.2 & U & 0 & 5 \\\ $\gamma_{tl}$ & G & 0.4 & 0.2 & U & 0 & 5 \\\ $\gamma_{z}$ & G & 0.4 & 0.2 & U & 0 & 5 \\\ \hline \multicolumn{7}{c}{Output Response Parameters} \\\ \hline $\varphi_{tk}$& G & 1.0 & 0.3 & N & 1.0 & 1 \\\ $\varphi_{tl}$& G & 0.5 & 0.25 & N & 0.5 & 1 \\\ $\varphi_{g}$ & G & 0.07 & 0.05 & N & 0.07 & 1 \\\ $\varphi_{z}$ & G & 0.2 & 0.1 & N & 0.2 & 1 \\\ \hline \multicolumn{7}{c}{Exogenous Tax Comovement Parameters} \\\ \hline $\phi_{kl}$ & N & 0.25 & 0.1 & N & 0.25 & 1 \\\ $\phi_{kc}$ & N & 0.05 & 0.1 & N & 0.05 & 1 \\\ $\phi_{lc}$ & N & 0.05 & 0.1 & N & 0.05 & 1 \\\ \hline \hline\\\ \end{tabular} } \end{center} /Notes:/ Para (1) and Para (2) correspond to the mean and standard deviation of the Beta (B), Gamma (G), and Normal (N) distributions and to the upper, lower bounds of the support for Uniform (U) distribution. \end{table}
\begin{table}[t!]
\begin{center}
\scalebox{0.93}{
\begin{tabular}{lccccccc} \hline\hline
& Type & Para (1) & Para (2) & & Type & Para (1) & Para (2)\\ \hline
\multicolumn{8}{c}{Endogenous Propagation Parameters} \\ \hline
$\gamma$ & G & 1.75 & 0.5 & $s''$ & G & 5 & 0.5 \\\\\\
$\kappa$ & G & 2.0 & 0.5 & $\delta\_2$ & G & 0.7 & 0.5 \\\\\\
$h$ & B & 0.5 & 0.2 & & & & \\\\\\
\hline
\multicolumn{8}{c}{Exogenous Process Parameters} \\\\\\
\hline
$\rho\_{a}$ & B & 0.7 & 0.2 & $\sigma\_{a}$ & IG & 1 & 4 \\\\\\
$\rho\_{b}$ & B & 0.7 & 0.2 & $\sigma\_{b}$ & IG & 1 & 4 \\\\\\
$\rho\_{l}$ & B & 0.7 & 0.2 & $\sigma\_{l}$ & IG & 1 & 4 \\\\\\
$\rho\_{i}$ & B & 0.7 & 0.2 & $\sigma\_{i}$ & IG & 1 & 4 \\\\\\
$\rho\_{g}$ & B & 0.7 & 0.2 & $\sigma\_{g}$ & IG & 1 & 4 \\\\\\
$\rho\_{tk}$ & B & 0.7 & 0.2 & $\sigma\_{tk}$ & IG & 1 & 4 \\\\\\
$\rho\_{tl}$ & B & 0.7 & 0.2 & $\sigma\_{tl}$ & IG & 1 & 4 \\\\\\
$\rho\_{tc}$ & B & 0.7 & 0.2 & $\sigma\_{tc}$ & IG & 1 & 4 \\\\\\
$\rho\_{z}$ & B & 0.7 & 0.2 & $\sigma\_{z}$ & IG & 1 & 4 \\\\\\
\hline \hline\\\\\\
\end{tabular}
}
\end{center}
/Notes:/ For the Inv. Gamma (IG) distribution, Para (1) and Para (2) refer to
$s$ and $\nu$, where $p(\sigma|\nu, s)\propto \sigma^{-\nu-1}e^{-\nu s^2/2\sigma^2}$.
\end{table}
\begin{table}[t!] \begin{center} \begin{tabular}{p{4cm}p{2.5cm}} \hline \hline $N = 6,000$ & $N_\phi = 500$\\\ $N_{blocks} = 3$ & $N_{MH}=1$ \\\ $\lambda = 4.0$ & \\\ \multicolumn{2}{l}{Run Time [mm:ss]: 48:00 (12 cores)} \ \hline \hline \end{tabular} \end{center} \end{table}
\begin{table}[t!]
\begin{center}
\footnotesize
\begin{tabular}{l@{\hspace\*{1cm}}cc@{\hspace\*{1cm}}cc} \hline\hline
& \multicolumn{2}{c}{Based on LPT Prior} & \multicolumn{2}{c}{Based on Diff. Prior} \\\\\\
& Mean & [5\%, 95\%] Int. & Mean & [5\%, 95\%] Int. \\ \hline
\multicolumn{5}{c}{Debt Response Parameters} \\ \hline
$\gamma\_{g}$ & 0.16 & [ 0.07, 0.27] & 0.10 & [ 0.01, 0.23] \\\\\\
$\gamma\_{tk}$ & 0.39 & [ 0.22, 0.60] & 0.38 & [ 0.16, 0.62] \\\\\\
$\gamma\_{tl}$ & 0.11 & [ 0.04, 0.21] & 0.04 & [ 0.00, 0.11] \\\\\\
$\gamma\_{z}$ & 0.32 & [ 0.17, 0.47] & 0.32 & [ 0.14, 0.49] \\\\\\
\hline \multicolumn{5}{c}{Output Response Parameters} \\ \hline
$\varphi\_{tk}$ & 1.67 & [ 1.18, 2.18] & 2.06 & [ 1.44, 2.69] \\\\\\
$\varphi\_{tl}$ & 0.29 & [ 0.11, 0.53] & 0.11 & [ -0.34, 0.58] \\\\\\
$\varphi\_{g}$ & 0.06 & [ 0.01, 0.13] & -0.43 & [ -0.87, 0.02] \\\\\\
$\varphi\_{z}$ & 0.17 & [ 0.06, 0.33] & -0.07 & [ -0.56, 0.41] \\\\\\
\hline \multicolumn{5}{c}{Exogenous Tax Comovement Parameters} \\ \hline
$\phi\_{kl}$ & 0.19 & [ 0.14, 0.24] & 1.57 & [ 1.29, 1.87] \\\\\\
$\phi\_{kc}$ & 0.03 & [ -0.03, 0.08] & -0.33 & [ -2.84, 2.73] \\\\\\
$\phi\_{lc}$ & -0.02 & [ -0.07, 0.04] & 0.20 & [ -1.23, 1.40] \\\\\\
\hline
\multicolumn{5}{c}{Innovations to Fiscal Rules} \\ \hline
$\sigma\_{g}$ & 3.03 & [ 2.79, 3.30] & 2.91 & [ 2.66, 3.19] \\\\\\
$\sigma\_{tk}$ & 4.36 & [ 4.01, 4.75] & 1.26 & [ 1.08, 1.46] \\\\\\
$\sigma\_{tl}$ & 2.95 & [ 2.71, 3.22] & 2.00 & [ 1.71, 2.33] \\\\\\
$\sigma\_{tc}$ & 3.99 & [ 3.67, 4.33] & 1.14 & [ 0.96, 1.35] \\\\\\
$\sigma\_{z}$ & 3.34 & [ 3.07, 3.63] & 3.34 & [ 3.07, 3.63] \\ \hline \hline
\end{tabular}
\footnotesize
\renewcommand{\baselinestretch}{2}
\normalsize
\end{center}
\end{table}
\begin{figure}[t!] \begin{center} \begin{tabular}{cc} \includegraphics[width=0.47\textwidth]{static/lpt_varphi_g.pdf} & \includegraphics[width=0.47\textwidth]{static/lpt_varphi_z.pdf} \end{tabular} \end{center} \end{figure}
Notes: The figure depicts posterior densities under the LPT prior (solid) and the diffuse prior (dashed).
\begin{center} \includegraphics[width=2.5in]{static/lpt_phi_scatter.pdf} \end{center}
Notes: The plots on the diagonal depict posterior densities under the LPT prior (solid) and the diffuse prior (dashed). The plots on the off-diagonals depict draws from the posterior distribution under the LPT prior (circles) and the diffuse prior (triangles).
\begin{center} \includegraphics[width=3in]{static/lpt_etn_irf.pdf} \end{center}
Notes: Figure depicts posterior mean impulse responses under LPT prior (solid); diffuse prior (dashed); diffuse prior with \(\phi_{lc} > 0\), \(\phi_{kl} < 0\) (dotted); and diffuse prior with \(\phi_{lc} < 0\), \(\phi_{kl} > 0\) (dots and short dashes). \(\hat{C}_t\), \(\hat{I}_t\) and \(\hat{L}_t\) are consumption, investment, and hours worked in deviation from steady state.