--- title: "Random Walk stochastic volatility steady-state BVAR (Clark, 2011)" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Random Walk stochastic volatility steady-state BVAR (Clark, 2011)} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- Here we estimate the steady-state BVAR model with Random Walk stochastic volatility from Clark (2011), which is an extension of the original homoscedastic steady-state BVAR model (Villani, 2009). See `?bvar` for details. We will estimate the model on a quarterly US data set from Koop and Korobilis (2010) on the inflation rate $\Delta \pi_t$ (the annual percentage change in a chain-weighted GDP price index), the unemployment rate $u_t$ (seasonally adjusted civilian unemployment rate, all civilian workers aged 16 years or older) and the interest rate $r_t$ (yield on the three-month Treasury bill rate). The sample is 1953Q1-2006Q3 and we have the data vector $$ y_t = \begin{pmatrix} \Delta \pi_t \\ u_t \\ r_t \end{pmatrix} $$ First, let's load the package, then import and plot the data. ``` r library(SteadyStateBVAR) data("KoopKorobilis2010") yt <- KoopKorobilis2010 plot.ts(yt) ``` ![](figure/RW-1-1.png) Let's create the bvar object which we will use throughout here. ``` r bvar_obj <- bvar(data = yt) ``` We choose 2 lags and only a constant as the deterministic variable. ``` r bvar_obj <- setup(bvar_obj, p=2, deterministic = "constant") ``` We set the overall tightness to $\lambda_1 = 0.20$, cross-equation tightness to $\lambda_2 = 0.50$ and the lag decay rate to $\lambda_3 = 1.00$. For the prior means on the first own lags, we set them to $0.6$ for $\Delta \pi_t$ and $0.9$ for $u_t$ and $r_t$. Note that the prior mean on the first own lag of inflation is set to $0.6$ instead of $0$ to reflect some degree of persistence in the series (even though it is a growth rate variable). ``` r lambda_1 <- 0.20 lambda_2 <- 0.50 lambda_3 <- 1.00 fol_pm=c(0.6, # delta pi 0.9, #u 0.9) #R ``` Now, for the steady-state coefficients we use some toy values (let us pretend that they are expert based). Remember that we only have a constant now, so $q=1$ and therefore $\Psi$ only has one column $\psi_1=\Psi$. Since $d_t = 1 \ \forall \ t$, we have $\Psi d_t = \mu_t$ which simplifies to $\Psi = \mu$ and as such we can directly interpret $\Psi$ as the unconditional mean, i.e. the steady state. ``` r theta_Psi <- c( ppi(1.90, 2.10, interval=0.95)$mean, #Psi: delta pi ppi(3.80, 4.50, interval=0.95)$mean, #Psi: u ppi(2.60, 3.90, interval=0.95)$mean #Psi: r ) Omega_Psi <- diag( c( ppi(1.90, 2.10, interval=0.95)$var, #Psi: delta pi ppi(3.80, 4.50, interval=0.95)$var, #Psi: u ppi(2.60, 3.90, interval=0.95)$var #Psi: r ) ) ``` Now we need to specify our stochastic volatility priors. See `?priors` for more information about the prior specification. I take some inspiration from Clark (2011) below. ``` r k <- bvar_obj$setup$k n_free_params_A <- bvar_obj$setup$n_free_params_A sigma2 <- diag(bvar_obj$setup$Sigma_AR) SV_priors_RW <- list( theta_A = rep(0, n_free_params_A), Omega_A = diag(10, n_free_params_A), mu_log_lambda_1 = log(sigma2), sigma2_log_lambda_1 = rep(4, k), alpha_phi = rep(2.5, k), beta_phi = rep(0.0875, k) ) ``` Here `sigma2` contains the residual variances from AR($p$) models (the same ones we used in the Minnesota prior). Let's put everything into the `priors()` function. ``` r bvar_obj <- priors(bvar_obj, lambda_1 = lambda_1, lambda_2 = lambda_2, lambda_3 = lambda_3, first_own_lag_prior_mean =fol_pm, theta_Psi = theta_Psi, Omega_Psi = Omega_Psi, SV = TRUE, SV_type = "RW", SV_priors = SV_priors_RW) ``` Now we can fit the model. Note that we can use arguments from `rstan::sampling()` such as `control` where we can tweak `max_treedepth` and `adapt_delta`. ``` r bvar_obj <- fit(bvar_obj, H = 40, d_pred = matrix(rep(1, 40)), iter = 4000, warmup = 1000, chains = 2, cores = 2, control = list(max_treedepth = 14, adapt_delta = 0.95)) ``` Now let's see the posterior means ``` r summary(bvar_obj, stat="mean", t = 215) #t = 215 for covariance matrix #> Posterior mean estimates #> ------------------------ #> #> #> beta #> -------------------------------------------------------------------------------- #> delta pi u r #> delta pi.l1 1.27 0.02 0.15 #> u.l1 -0.09 1.17 -0.16 #> r.l1 0.00 -0.01 1.04 #> delta pi.l2 -0.28 0.02 -0.10 #> u.l2 0.07 -0.23 0.17 #> r.l2 0.00 0.02 -0.11 #> -------------------------------------------------------------------------------- #> #> #> Psi #> -------------------------------------------------------------------------------- #> [,1] #> delta pi 2.00 #> u 4.29 #> r 3.50 #> -------------------------------------------------------------------------------- #> #> #> Sigma_u,t (t = 215) #> -------------------------------------------------------------------------------- #> delta pi u r #> delta pi 0.09 -0.01 0.02 #> u -0.01 0.02 -0.01 #> r 0.02 -0.01 0.16 #> -------------------------------------------------------------------------------- #> #> #> A #> -------------------------------------------------------------------------------- #> delta pi u r #> delta pi 1.00 0.00 0 #> u 0.12 1.00 0 #> r -0.22 0.51 1 #> -------------------------------------------------------------------------------- #> #> #> phi #> -------------------------------------------------------------------------------- #> delta pi u r #> 0.04 0.07 0.10 #> -------------------------------------------------------------------------------- ``` You can always look at the `stanfit` object `bvar_obj$fit$stan` directly if you want. Note that the `z`'s below are not parameters per se, they are simply used in a reparameterization trick to sample the log volatilities more efficiently. ``` r print(bvar_obj$fit$stan) #> Inference for Stan model: steady_state_bvar_RW_stochastic_volatility. #> 2 chains, each with iter=4000; warmup=1000; thin=1; #> post-warmup draws per chain=3000, total post-warmup draws=6000. #> #> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat #> beta[1,1] 1.27 0.00 0.06 1.16 1.23 1.27 1.31 1.38 10297 1 #> beta[1,2] 0.02 0.00 0.04 -0.06 -0.01 0.02 0.04 0.09 8494 1 #> beta[1,3] 0.15 0.00 0.08 -0.01 0.10 0.15 0.20 0.30 7773 1 #> beta[2,1] -0.09 0.00 0.03 -0.16 -0.12 -0.09 -0.07 -0.02 8907 1 #> beta[2,2] 1.17 0.00 0.06 1.06 1.13 1.17 1.21 1.28 8749 1 #> beta[2,3] -0.16 0.00 0.08 -0.32 -0.21 -0.16 -0.11 -0.01 8377 1 #> beta[3,1] 0.00 0.00 0.02 -0.03 -0.01 0.00 0.01 0.04 10036 1 #> beta[3,2] -0.01 0.00 0.02 -0.04 -0.02 -0.01 0.00 0.02 10459 1 #> beta[3,3] 1.04 0.00 0.06 0.92 1.00 1.04 1.08 1.15 10192 1 #> beta[4,1] -0.28 0.00 0.06 -0.39 -0.32 -0.28 -0.24 -0.17 10144 1 #> beta[4,2] 0.02 0.00 0.04 -0.06 -0.01 0.02 0.04 0.09 8920 1 #> beta[4,3] -0.10 0.00 0.08 -0.26 -0.16 -0.10 -0.05 0.05 7662 1 #> beta[5,1] 0.07 0.00 0.03 0.01 0.05 0.07 0.09 0.14 9346 1 #> beta[5,2] -0.23 0.00 0.05 -0.34 -0.27 -0.23 -0.20 -0.12 8545 1 #> beta[5,3] 0.17 0.00 0.07 0.03 0.12 0.17 0.22 0.32 9076 1 #> beta[6,1] 0.00 0.00 0.01 -0.03 -0.01 0.00 0.01 0.03 9794 1 #> beta[6,2] 0.02 0.00 0.02 -0.01 0.01 0.02 0.03 0.05 11413 1 #> beta[6,3] -0.11 0.00 0.06 -0.22 -0.15 -0.11 -0.07 0.01 9481 1 #> Psi[1,1] 2.00 0.00 0.05 1.90 1.96 2.00 2.03 2.10 15316 1 #> Psi[2,1] 4.29 0.00 0.18 3.93 4.17 4.30 4.41 4.65 15661 1 #> Psi[3,1] 3.50 0.00 0.32 2.87 3.28 3.50 3.72 4.11 13141 1 #> z[1,1] -0.05 0.00 0.25 -0.50 -0.22 -0.06 0.11 0.47 10574 1 #> z[1,2] 0.64 0.00 0.27 0.15 0.46 0.64 0.81 1.19 10220 1 #> z[1,3] -0.65 0.00 0.34 -1.29 -0.88 -0.66 -0.43 0.03 9569 1 #> z[2,1] -0.13 0.01 0.95 -1.97 -0.79 -0.13 0.52 1.75 15812 1 #> z[2,2] 0.18 0.01 0.99 -1.76 -0.49 0.18 0.86 2.11 13629 1 #> z[2,3] 0.00 0.01 1.00 -1.95 -0.69 0.00 0.68 1.96 11939 1 #> z[3,1] -0.05 0.01 0.99 -1.98 -0.71 -0.04 0.62 1.85 15000 1 #> z[3,2] 0.25 0.01 0.98 -1.70 -0.41 0.24 0.91 2.21 12218 1 #> z[3,3] -0.04 0.01 1.00 -2.00 -0.71 -0.04 0.65 1.90 15046 1 #> z[4,1] -0.09 0.01 0.98 -1.97 -0.75 -0.10 0.59 1.84 14747 1 #> z[4,2] -0.43 0.01 0.92 -2.19 -1.06 -0.43 0.19 1.36 16494 1 #> z[4,3] -0.26 0.01 0.95 -2.07 -0.89 -0.27 0.36 1.67 14793 1 #> z[5,1] 0.00 0.01 0.97 -1.90 -0.66 -0.01 0.66 1.92 14608 1 #> z[5,2] -0.46 0.01 0.95 -2.32 -1.11 -0.46 0.19 1.38 16332 1 #> z[5,3] -0.21 0.01 1.00 -2.13 -0.89 -0.22 0.46 1.72 15558 1 #> z[6,1] -0.01 0.01 0.99 -2.00 -0.69 -0.01 0.66 1.96 13342 1 #> z[6,2] -0.34 0.01 0.97 -2.26 -1.00 -0.33 0.33 1.60 15816 1 #> z[6,3] -0.13 0.01 0.97 -2.02 -0.80 -0.13 0.53 1.73 14987 1 #> z[7,1] 0.10 0.01 0.95 -1.78 -0.54 0.10 0.76 1.94 20300 1 #> z[7,2] -0.21 0.01 0.98 -2.13 -0.86 -0.22 0.46 1.74 14686 1 #> z[7,3] -0.04 0.01 0.95 -1.92 -0.67 -0.03 0.61 1.85 13893 1 #> z[8,1] 0.19 0.01 0.99 -1.79 -0.47 0.17 0.86 2.12 19128 1 #> z[8,2] -0.25 0.01 0.97 -2.18 -0.90 -0.26 0.39 1.61 15872 1 #> z[8,3] 0.06 0.01 0.92 -1.75 -0.57 0.05 0.69 1.88 12407 1 #> z[9,1] 0.16 0.01 0.97 -1.74 -0.49 0.16 0.84 2.07 15461 1 #> z[9,2] -0.15 0.01 0.96 -2.02 -0.79 -0.15 0.51 1.71 16838 1 #> z[9,3] 0.19 0.01 0.97 -1.72 -0.48 0.19 0.85 2.03 15960 1 #> z[10,1] -0.18 0.01 0.97 -2.08 -0.85 -0.18 0.48 1.66 15155 1 #> z[10,2] -0.12 0.01 0.97 -2.01 -0.78 -0.11 0.55 1.82 15984 1 #> z[10,3] -0.04 0.01 0.97 -1.98 -0.69 -0.03 0.59 1.88 18843 1 #> z[11,1] -0.17 0.01 0.94 -2.04 -0.81 -0.17 0.47 1.71 13422 1 #> z[11,2] -0.14 0.01 0.95 -2.00 -0.79 -0.14 0.50 1.74 14516 1 #> z[11,3] -0.17 0.01 0.97 -2.03 -0.82 -0.18 0.49 1.71 15496 1 #> z[12,1] -0.32 0.01 0.96 -2.21 -0.97 -0.32 0.31 1.53 13949 1 #> z[12,2] -0.04 0.01 0.96 -1.91 -0.68 -0.04 0.60 1.86 15602 1 #> z[12,3] -0.09 0.01 0.94 -1.92 -0.73 -0.09 0.55 1.74 14597 1 #> z[13,1] -0.29 0.01 0.98 -2.22 -0.95 -0.29 0.39 1.64 16269 1 #> z[13,2] 0.09 0.01 0.97 -1.80 -0.55 0.08 0.73 2.00 14781 1 #> z[13,3] 0.04 0.01 0.96 -1.81 -0.61 0.04 0.68 1.92 14488 1 #> z[14,1] -0.32 0.01 0.97 -2.16 -0.98 -0.32 0.33 1.57 15088 1 #> z[14,2] 0.13 0.01 0.95 -1.74 -0.51 0.14 0.78 1.96 15283 1 #> z[14,3] 0.03 0.01 0.96 -1.84 -0.62 0.02 0.65 1.95 16390 1 #> z[15,1] -0.25 0.01 0.99 -2.19 -0.90 -0.27 0.42 1.70 14537 1 #> z[15,2] 0.06 0.01 0.96 -1.87 -0.58 0.07 0.71 1.90 14451 1 #> z[15,3] 0.11 0.01 0.97 -1.77 -0.55 0.12 0.78 1.97 14430 1 #> z[16,1] -0.21 0.01 0.96 -2.06 -0.87 -0.22 0.44 1.68 15286 1 #> z[16,2] 0.12 0.01 0.98 -1.82 -0.53 0.13 0.78 2.06 15161 1 #> z[16,3] 0.26 0.01 0.96 -1.64 -0.39 0.26 0.90 2.15 13452 1 #> z[17,1] -0.21 0.01 0.97 -2.10 -0.87 -0.22 0.45 1.71 14900 1 #> z[17,2] 0.18 0.01 0.95 -1.67 -0.46 0.19 0.83 2.04 13575 1 #> z[17,3] 0.30 0.01 0.96 -1.57 -0.35 0.30 0.94 2.17 12809 1 #> z[18,1] -0.28 0.01 0.96 -2.18 -0.94 -0.28 0.37 1.57 16645 1 #> z[18,2] 0.26 0.01 0.95 -1.60 -0.39 0.26 0.90 2.13 16719 1 #> z[18,3] 0.43 0.01 0.95 -1.42 -0.22 0.44 1.08 2.27 12369 1 #> z[19,1] -0.20 0.01 0.97 -2.09 -0.88 -0.22 0.47 1.71 15822 1 #> z[19,2] 0.38 0.01 0.95 -1.50 -0.25 0.37 1.01 2.24 15054 1 #> z[19,3] 0.37 0.01 0.97 -1.50 -0.29 0.37 1.02 2.27 13218 1 #> z[20,1] -0.19 0.01 0.98 -2.08 -0.85 -0.19 0.48 1.71 16124 1 #> z[20,2] 0.04 0.01 0.93 -1.79 -0.58 0.04 0.67 1.79 14503 1 #> z[20,3] 0.31 0.01 0.99 -1.61 -0.37 0.32 1.00 2.26 17200 1 #> z[21,1] -0.12 0.01 0.96 -2.04 -0.74 -0.11 0.51 1.79 14224 1 #> z[21,2] -0.43 0.01 0.97 -2.30 -1.09 -0.44 0.20 1.51 15180 1 #> z[21,3] 0.38 0.01 0.94 -1.43 -0.27 0.38 1.02 2.21 14878 1 #> z[22,1] -0.11 0.01 0.97 -1.97 -0.77 -0.12 0.55 1.79 14179 1 #> z[22,2] -0.32 0.01 0.94 -2.17 -0.95 -0.34 0.30 1.54 14549 1 #> z[22,3] 0.50 0.01 0.94 -1.34 -0.14 0.52 1.14 2.35 16356 1 #> z[23,1] -0.06 0.01 0.98 -2.01 -0.72 -0.05 0.59 1.83 15152 1 #> z[23,2] -0.36 0.01 0.98 -2.28 -1.03 -0.36 0.31 1.57 13117 1 #> z[23,3] -0.17 0.01 0.95 -2.07 -0.81 -0.17 0.46 1.67 16037 1 #> z[24,1] -0.19 0.01 0.94 -2.02 -0.82 -0.19 0.44 1.67 15235 1 #> z[24,2] -0.29 0.01 0.95 -2.15 -0.93 -0.29 0.36 1.52 13414 1 #> z[24,3] -0.07 0.01 0.94 -1.92 -0.70 -0.08 0.57 1.78 15798 1 #> z[25,1] -0.17 0.01 0.97 -2.10 -0.81 -0.18 0.47 1.76 18826 1 #> z[25,2] -0.29 0.01 0.95 -2.15 -0.92 -0.30 0.34 1.57 13711 1 #> z[25,3] 0.06 0.01 0.95 -1.81 -0.59 0.07 0.71 1.88 13770 1 #> z[26,1] -0.05 0.01 0.98 -2.02 -0.71 -0.04 0.61 1.87 16092 1 #> z[26,2] -0.19 0.01 0.95 -2.04 -0.86 -0.17 0.45 1.66 13991 1 #> z[26,3] 0.20 0.01 0.96 -1.70 -0.45 0.21 0.84 2.07 14431 1 #> z[27,1] -0.10 0.01 0.92 -1.94 -0.69 -0.09 0.52 1.76 14331 1 #> [ reached 'max' / getOption("max.print") -- omitted 3759 rows ] #> #> Samples were drawn using NUTS(diag_e) at Tue Jul 28 03:46:18 2026. #> For each parameter, n_eff is a crude measure of effective sample size, #> and Rhat is the potential scale reduction factor on split chains (at #> convergence, Rhat=1). ``` We can forecast ``` r forecast(bvar_obj, pi = 0.68, show_all = TRUE) ``` ![](figure/RW-2-1.png)![](figure/RW-2-2.png)![](figure/RW-2-3.png) Let us plot the log volatility estimates and predictions ``` r stochastic_volatility_plot(bvar_obj, ci = 0.95, vol = "log_lambda") ``` ![](figure/RW-3-1.png)![](figure/RW-3-2.png)![](figure/RW-3-3.png) Let us plot the estimates and predictions of the implied innovation standard deviations ``` r stochastic_volatility_plot(bvar_obj, vol = "sd") ``` ![](figure/RW-4-1.png)![](figure/RW-4-2.png)![](figure/RW-4-3.png) We can also produce orthogonalized IRFs ``` r IRF(bvar_obj, method = "OIRF", t=215, ci=0.68) #latest t ``` ![](figure/RW-5-1.png) ## References Clark, T. E. (2011). Real-time density forecasts from Bayesian vector autoregressions with stochastic volatility. *Journal of Business \& Economic Statistics*, 29(3), pp. 327–341. Koop, G. and Korobilis, D. (2010). Bayesian multivariate time series methods for empirical macroeconomics. *Foundations and Trends in Econometrics*, 3(4), pp. 267–358. Villani, M. (2009). Steady-state priors for vector autoregressions. *Journal of Applied Econometrics*, 24(4), pp. 630–650.