estimate_bsvar_sv {bsvars} | R Documentation |

Estimates the SVAR with Stochastic Volatility (SV) heteroskedasticity proposed by Lütkepohl, Shang, Uzeda, and Woźniak (2022).
Implements the Gibbs sampler proposed by Waggoner & Zha (2003)
for the structural matrix `B`

and the equation-by-equation sampler by Chan, Koop, & Yu (2021)
for the autoregressive slope parameters `A`

. Additionally, the parameter matrices `A`

and `B`

follow a Minnesota prior and generalised-normal prior distributions respectively with the matrix-specific
overall shrinkage parameters estimated thanks to a 3-level hierarchical prior distribution. The SV model
is estimated in a non-centred parameterisation using a range of techniques including:
simulation smoother, auxiliary mixture, ancillarity-sufficiency interweaving strategy,
and generalised inverse Gaussian distribution summarised by Kastner & Frühwirth-Schnatter (2014).
See section **Details** for the model equations.

```
estimate_bsvar_sv(S, specification, thin = 10, show_progress = TRUE)
```

`S` |
a positive integer, the number of posterior draws to be generated |

`specification` |
an object of class BSVAR-SV generated using the |

`thin` |
a positive integer, specifying the frequency of MCMC output thinning |

`show_progress` |
a logical value, if |

The heteroskedastic SVAR model is given by the reduced form equation:

`Y = AX + E`

where `Y`

is an `NxT`

matrix of dependent variables, `X`

is a `KxT`

matrix of explanatory variables,
`E`

is an `NxT`

matrix of reduced form error terms, and `A`

is an `NxK`

matrix of autoregressive slope coefficients and parameters on deterministic terms in `X`

.

The structural equation is given by

`BE = U`

where `U`

is an `NxT`

matrix of structural form error terms, and
`B`

is an `NxN`

matrix of contemporaneous relationships.

Finally, the structural shocks, `U`

, are temporally and contemporaneously independent and jointly normally distributed with zero mean.
The conditional variance of the `n`

th shock at time `t`

is given by:

`Var_{t-1}[u_{n.t}] = exp(w_n h_{n.t})`

where `w_n`

is the estimated conditional standard deviation of the log-conditional variance
and the log-volatility process `h_{n.t}`

follows an autoregressive process:

`h_{n.t} = g_n h_{n.t-1} + v_{n.t}`

where `h_{n.0}=0`

, `g_n`

is an autoregressive parameter and `v_{n.t}`

is a standard normal error term.

An object of class PosteriorBSVAR-SV containing the Bayesian estimation output and containing two elements:

`posterior`

a list with a collection of `S`

draws from the posterior distribution generated via Gibbs sampler containing:

- A
an

`NxKxS`

array with the posterior draws for matrix`A`

- B
an

`NxNxS`

array with the posterior draws for matrix`B`

- hyper
a

`5xS`

matrix with the posterior draws for the hyper-parameters of the hierarchical prior distribution- h
an

`NxTxS`

array with the posterior draws of the log-volatility processes- rho
an

`NxS`

matrix with the posterior draws of SV autoregressive parameters- omega
an

`NxS`

matrix with the posterior draws of SV process conditional standard deviations- S
an

`NxTxS`

array with the posterior draws of the auxiliary mixture component indicators- sigma2_omega
an

`NxS`

matrix with the posterior draws of the variances of the zero-mean normal prior for`omega`

- s_
an

`S`

-vector with the posterior draws of the scale of the gamma prior of the hierarchical prior for`sigma2_omega`

`last_draw`

an object of class BSVAR-SV with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using `bsvar_sv()`

.

Tomasz Woźniak wozniak.tom@pm.me

The model, prior distributions, and estimation algorithms were proposed by

Lütkepohl, H., Shang, F., Uzeda, L., and Woźniak, T. (2022) Partial Identification of Heteroskedastic Structural VARs: Theory and Bayesian Inference.

Sampling from the generalised-normal full conditional posterior distribution of matrix `B`

is implemented using the Gibbs sampler by:

Waggoner, D.F., and Zha, T., (2003) A Gibbs sampler for structural vector autoregressions. *Journal of Economic Dynamics and Control*, **28**, 349–366, doi:10.1016/S0165-1889(02)00168-9.

Sampling from the multivariate normal full conditional posterior distribution of each of the `A`

matrix row is implemented using the sampler by:

Chan, J.C.C., Koop, G, and Yu, X. (2021) Large Order-Invariant Bayesian VARs with Stochastic Volatility.

Many of the techniques employed for the estimation of the Stochastic Volatility model are summarised by:

Kastner, G. and Frühwirth-Schnatter, S. (2014) Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC
Estimation of Stochastic Volatility Models. *Computational Statistics & Data Analysis*, **76**, 408–423,
doi:10.1016/j.csda.2013.01.002.

`specify_bsvar_sv`

, `specify_posterior_bsvar_sv`

, `normalise_posterior`

```
# upload data
data(us_fiscal_lsuw)
# specify the model and set seed
specification = specify_bsvar_sv$new(us_fiscal_lsuw, p = 4)
set.seed(123)
# run the burn-in
burn_in = estimate_bsvar_sv(10, specification)
# estimate the model
posterior = estimate_bsvar_sv(50, burn_in$get_last_draw())
```

[Package *bsvars* version 1.0.0 Index]