An **R** package for Bayesian Estimation of Structural Vector Autoregressions Identified by Sign, Zero, and Narrative Restrictions

Implements state-of-the-art algorithms for the Bayesian analysis of Structural Vector Autoregressions identified by sign, zero, and narrative restrictions. The core model is based on a flexible Vector Autoregression with estimated hyper-parameters of the Minnesota prior and the dummy observation priors as in Giannone, Lenza, Primiceri (2015). The sign restrictions are implemented employing the methods proposed by Rubio-Ramírez, Waggoner & Zha (2010), while identification through sign and zero restrictions follows the approach developed by Arias, Rubio-Ramírez, & Waggoner (2018). Furthermore, our tool provides algorithms for identification via sign and narrative restrictions, in line with the methods introduced by Antolín-Díaz and Rubio-Ramírez (2018). Users can also estimate a model with sign, zero, and narrative restrictions imposed at once. The package facilitates predictive and structural analyses using impulse responses, forecast error variance and historical decompositions, forecasting and conditional forecasting, as well as analyses of structural shocks and fitted values. All this is complemented by colourful plots, user-friendly summary functions, and comprehensive documentation. The **bsvarSIGNs** package is aligned regarding objects, workflows, and code structure with the **R** package **bsvars** by Woźniak (2024), and they constitute an integrated toolset.

## Features

#### Structural Vector Autoregressions

- All the models in the
**bsvarSIGNs**package consist of the Vector Autoregressive equation, with autoregressive parameters`A`

and error terms`E`

, and the structural equation with a structural matrix`B`

and shocks`U`

- All the models feature a hierarchical Minnesota prior following the specification proposed by Giannone, Lenza, Primiceri (2015)
- The identification of the SVAR model is achieved by imposing:
- sign restrictions on the structural matrix $\mathbf{B}$,
- sign and zero restrictions on the zero-horizon impulse responses $\mathbf{\Theta}_0 = \mathbf{B}^{-1}$,
- sign restrictions on the impulse responses at other horizons $\mathbf{\Theta}_i$ for $i = 1, 2, \ldots$,
- sign restrictions on selected structural shocks $\mathbf{u}_t$,
- two types of sign restrictions on the historical decompositions.

#### Simple workflows

- Specify the models using
`specify_bsvarSIGN$new()`

function - Estimate the models using the
`estimate()`

method - Predict the future using the
`forecast()`

method - Provide structural analyses using
**impulse responses**, forecast error variance decompositions, historical decompositions, and structural shocks using functions`compute_impulse_responses()`

,`compute_variance_decompositions()`

,`compute_historical_decompositions()`

, and`compute_structural_shocks()`

respectively - Analyse the fitted values, time-varying volatility using functions
`compute_fitted_values()`

and`compute_conditional_sd()`

respectively - Use
`plot()`

and`summary()`

methods to gain the insights into the core of the empirical problem.

#### Fast and efficient computations

- Blazingly fast computations are obtained by combining
- the application of frontier econometric and numerical techniques, and
- the implementation using compiled code written in
**cpp**

- It combines the best of two worlds: the ease of data analysis with
**R**and fast**cpp**algorithms - The algorithms used here are very fast. But still, Bayesian estimation might take a little time. Look at our beautiful
**progress bar**in the meantime:

```
**************************************************|
bsvarSIGNs: Bayesian Structural VAR with sign, |
zero and narrative restrictions |
**************************************************|
Progress of simulation for 1000 independent draws
Press Esc to interrupt the computations
**************************************************|
0% 10 20 30 40 50 60 70 80 90 100%
[----|----|----|----|----|----|----|----|----|----|
***********************************
```

## Examples

Replication of the results by Arias, Rubio-Ramírez, & Waggoner (2018).

```
# investigate the effects of the optimism shock
data(optimism)
# specify identifying restrictions:
# + no effect on productivity (zero restriction)
# + positive effect on stock prices (positive sign restriction)
sign_irf = matrix(c(0, 1, rep(NA, 23)), 5, 5)
# specify the model
specification = specify_bsvarSIGN$new(optimism * 100,
p = 4,
sign_irf = sign_irf)
# estimate the model
posterior = estimate(specification, S = 100)
# compute and plot impulse responses
irf = compute_impulse_responses(posterior, horizon = 40)
plot(irf, probability = 0.68)
```

Replication of the results by Antolín-Díaz and Rubio-Ramírez (2018).

```
# investigate the effects of the contractionary monetary policy shock
data(monetary)
# specify identifying restrictions:
# + sign restrictions on the impulse responses at horizons from 0 to 5
sign_irf = matrix(NA, 6, 6)
sign_irf[, 1] = c(NA, -1, -1, NA, -1, 1)
sign_irf = array(sign_irf, dim = c(6, 6, 6))
# + narrative sign restriction: the shock is positive in October 1979
sign_narrative = list(
specify_narrative(start = 166, periods = 1, type = "S", sign = 1, shock = 1),
specify_narrative(start = 166, periods = 1, type = "B", sign = 1, shock = 1, var = 6)
)
# specify the model
specification = specify_bsvarSIGN$new(monetary * 100,
p = 12,
sign_irf = sign_irf,
sign_narrative = sign_narrative)
# estimate the model
posterior = estimate(specification, S = 100)
# compute and plot impulse responses
irf = compute_impulse_responses(posterior, horizon = 60)
plot(irf, probability = 0.68)
```

## Installation

#### The first time you install the package

You must have a **cpp** compiler. Follow the instructions from Section 1.3. by Eddelbuettel & François (2023). In short, for **Windows:** install RTools, for **macOS:** install Xcode Command Line Tools, and for **Linux:** install the standard development packages.

## Development

The package is under intensive development. Your help is most welcome! Please, have a look at the roadmap and report a bug. Thank you!

## About the authors

**Xiaolei** is a PhD candidate at the University of Melbourne. He is interested in Bayesian econometrics and time series analysis and is the author and maintainer of the **bsvarSIGNs** package.

**Tomasz** is a Bayesian econometrician and a Senior Lecturer at the University of Melbourne. He develops methodology for empirical macroeconomic analyses and programs in **R** and **cpp** using **Rcpp**.