Package 'bpvars'

Forecasting with Bayesian Panel Vector Autoregressions

Description

Provides Bayesian estimation and forecasting of dynamic panel data using Bayesian Panel Vector Autoregressions with hierarchical prior distributions. The models include country-specific Vector Autoregressions (VARs) that share a global prior distribution that extend the model by Jarociński (2010) doi:10.1002/jae.1082. Under this prior expected value, each country's system follows a global VAR with country-invariant parameters. Further flexibility is provided by the hierarchical prior structure that retains the Minnesota prior interpretation for the global VAR and features estimated prior covariance matrices, shrinkage, and persistence levels. Bayesian forecasting is developed for models including exogenous variables, allowing conditional forecasts given the future trajectories of some variables and restricted forecasts assuring that rates are forecasted to stay positive and less than 100. The package implements the model specification, estimation, and forecasting routines, facilitating coherent workflows and reproducibility. It also includes automated pseudo-out-of-sample forecasting and computation of forecasting performance measures. Beautiful plots, informative summary functions, and extensive documentation complement all this. Extraordinary computational speed is achieved thanks to employing frontier econometric and numerical techniques and algorithms written in 'C++'. The 'bpvars' package is aligned regarding objects, workflows, and code structure with the R packages 'bsvars' by Woźniak (2024) doi:10.32614/CRAN.package.bsvars and 'bsvarSIGNs' by Wang & Woźniak (2025) doi:10.32614/CRAN.package.bsvarSIGNs, and they constitute an integrated toolset. Copyright: 2025 International Labour Organization. The International Labour Organization should not be held responsible for any issues arising from the use of the 'bpvars' package or from the results obtained with it.

Details

The package provides a set of functions for predictive analysis with the Bayesian Hierarchical Panel Vector Autoregression.

The Model. The model specification is initiated using function specify_bvarPANEL that creates an object of class BVARPANEL containing the prior specification, starting values for estimation, data matrices, and the setup of the Monte Carlo Markov Chain sampling algorithm.

The model features country-specific Vector Autoregressive (VAR) equation for N dependent variables with T_c observations for each country c. Its equation is given by

$\mathbf{Y}_c = \mathbf{A}_c\mathbf{X}_c + \mathbf{E}_c$

where $\mathbf{Y}_c$ is an T_c x N matrix of dependent variables for country c, $\mathbf{X}_c$ is a T_c x K matrix of explanatory variables, $\mathbf{E}_c$ is an T_c x N matrix of error terms, and $\mathbf{A}_c$ is an NxK matrix of country-specific autoregressive slope coefficients and parameters on deterministic terms in $\mathbf{X}_c$. The parameter matrix $\mathbf{A}_c$ includes autoregressive matrices capturing the effects of the lagged vectors of dependent variables at lags from 1 to p, a constant term and a set of exogenous variables.

The error terms for each of the periods have zero conditional mean and conditional covariance given by the NxN matrix $\mathbf{\Sigma}$. The errors are jointly normally distributed and serially uncorrelated. These assumptions are summarised using a matrix-variate normal distribution (see Woźniak, 2016):

$\mathbf{E}_c \sim MN(\mathbf{0}_{T_c x N}, \mathbf{\Sigma}, \mathbf{I}_{T_c})$

where the identity matrix $\mathbf{I}_{T_c}$ of order T_c and joint normality imply no serial autocorrelation. Matrix $\mathbf{0}_{T_c x N}$ denotes a T_c x N matrix of zeros.

Global Prior Distributions. The Hierarchical Panel VAR model features a sophisticated hierarchical prior structure that grants the model flexibility, interpretability, and improved forecasting performance.

The country-specific parameters follow a prior distribution that, at its mean value, represents a global VAR model with a global autoregressive parameter matrix $\mathbf{A}$ of dimension KxN and an NxN global covariance matrix $\mathbf{\Sigma}$:

$\mathbf{Y}_c = \mathbf{AX}_c + \mathbf{E}_c$

This global VAR model under the prior mean is represented by the parameters of the matrix-variate normal inverted Wishart distribution (see Woźniak, 2016) given by:

$\mathbf{A}_c, \mathbf{\Sigma}_c | \mathbf{A}, \mathbf{V}, \mathbf{\Sigma}, \nu \sim MNIW(\mathbf{A}, \mathbf{V}, (N - \nu - 1)\mathbf{\Sigma}, \nu)$

where $V$ is a KxK column-specific covariance matrix, $(N - \nu - 1)\mathbf{\Sigma}$ is the row-specific matrix, and $\nu > N+1$ is the degrees-of-freedom parameter.

All of the parameters of the prior distribution above feature their own prior distributions and are estimated. These prior distributions are given by:

$\mathbf{A} | \mathbf{V}, m, w, s \sim MN (m\underline{\mathbf{M}}, \mathbf{V}, s\underline{\mathbf{S}} )$

with the KxN mean matrix $m\underline{\mathbf{M}}$, the KxK column-specific covariance matrix $\mathbf{V}$, and the NxN matrix of row-specific covariance $s\underline{\mathbf{S}}$.

The global error term covariance matrix, $\mathbf{\Sigma}$, follows a Wishart distribution with NxN scale matrix $s\underline{\mathbf{S}}_\Sigma$ and shape parameter $\underline{\mu}_\Sigma$

$\mathbf{\Sigma} | s, \nu \sim W(s\underline{\mathbf{S}}_\Sigma,\underline{\mu}_\Sigma)$

Other Prior Distributions. The column-specific covariance $\mathbf{V}$ follows the inverse-Wishart distribution with scale $w\underline{\mathbf{W}}$ and shape $\underline{\eta}$:

$\mathbf{V} | w \sim IW(w\underline{\mathbf{W}}, \underline{\eta})$

The shape parameter $\nu$ follows an exponential distribution with mean $\underline\lambda$:

$\nu \sim\exp(\underline\lambda)$

Finally, the priors for the remaining scalar hyper-parameters are:

$m \sim N(\underline{\mu}_m, \underline{\sigma}_m^2)$

$w \sim G(\underline{s}_w, \underline{a}_w)$

$s \sim IG2 (\underline{s}_s, \underline{\nu}_s)$

The prior hyper-parameters in this note are grouped into those that are:

fixed

and denoted using underscore, such as e.g. $\underline{\mathbf{M}}$, $\underline{\mu}_\Sigma$, or $\underline{\nu}_s$. These hyper-parameters must be fixed and their default values are set by initiating the model specification using function specify_bvarPANEL. These values can be accesses from such generated object in its element prior and can be modified by the user.

estimated

not featuring the underscore in the notation, such as e.g. $\mathbf{A}$, $\mathbf{\Sigma}$, or $m$. These hyper-parameters are estimated and their posterior draws are available from an object generated after the estimation running the function estimate.

Estimation. The package implements Bayesian estimation using the Gibbs sampler. This algorithm provides a sample of random draws from the posterior distribution of the parameters of the model. The posterior distribution is defined by Bayes' Rule stating that the posterior distribution of the parameters given data and is proportional to the likelihood function and the prior distribution of the parameters:

$p(\mathbf{\theta} | \mathbf{Y}) \propto L(\mathbf{\theta}; \mathbf{Y}) p(\mathbf{\theta})$

where $\mathbf{\theta}$ collects all the parameters of the model to be estimated. At each of its iterations a single draw of all of the parameters of the model, including the estimated hyper-parameters, is obtained. This Bayesian procedure estimates jointly all the parameters of the model and is implemented in the estimate.BVARPANEL and estimate.PosteriorBVARPANEL functions.

Forecasting. The package implements Bayesian forecasting providing the a sample of draws from the joint predictive density defined as the joint density of the future unknown values to be predicted, $\mathbf{Y}_f$, given data, $\mathbf{Y}$ closely following Karlsson (2013):

$p(\mathbf{Y}_f | \mathbf{Y})$

The package offers the possibility of:

forecasting for models with exogenous variables

given the provided future values of the exogenous variables.

conditional predictions

given provided future projections for some of the variables.

trucated forecasts

for variables that represents rates from the interval $[0,100]$.

The forecasting is performed using function forecast.PosteriorBVARPANEL.

Note

This package is currently in active development.

Author(s)

Tomasz Woźniak [email protected]

References

Jarociński, M. (2010). Responses to Monetary Policy Shocks in the East and the West of Europe: a Comparison. Journal of Applied Econometrics, 25(5), 833-868, doi:10.1002/jae.1082.

Karlsson, S. (2013). Forecasting with Bayesian Vector Autoregression, in: Handbook of Economic Forecasting, Elsevier. volume 2, 791–897, doi:10.1016/B978-0-444-62731-5.00015-4.

Woźniak, T. (2016). Bayesian Vector Autoregressions, Australian Economic Review, 49, 365-380, doi:10.1111/1467-8462.12179.

See Also

specify_bvarPANEL, estimate.BVARPANEL, forecast.PosteriorBVARPANEL

Examples

# Basic estimation and forecasting example
############################################################
specification = specify_bvarPANEL$new(ilo_dynamic_panel[1:5]) # specify the model
burn_in       = estimate(specification, S = 5)          # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, S = 5)                # estimate the model; use say S = 10000
predictive    = forecast(posterior, 2)                  # forecast the future       

# workflow with the pipe |>
ilo_dynamic_panel[1:5] |>
  specify_bvarPANEL$new() |>
  estimate(S = 5) |> 
  estimate(S = 5) |> 
  forecast(horizon = 2) -> predictive

---

Computes forecasting performance measures for recursive pseudo-out-of-sample forecasts

Description

Computes forecasting performance measures selected from: log-predictive score "lps", root-mean-squared-forecast error "rmsfe", mean-absolute-forecast error "mafe" from the output of the recursive pseudo-out-of-sample forecastinge exercise performed using function forecast_poos_recursively.

Usage

compute_forecast_performance(forecasts, measures = c("pls", "rmsfe", "mafe"))

Arguments

forecasts	an object containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples generated using function forecast_poos_recursively.
measures	a character vector with any of the values "lps", "rmsfe", "mafe" indicating the forecasting performance measures to be computed.

Value

An object of class ForecastingPerformance

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast_poos_recursively, forecast_poos_recursively.BVARPANEL, forecast_poos_recursively.BVARGROUPPANEL

Examples

spec = specify_bvarPANEL$new(ilo_dynamic_panel[1:5])               # specify the model
poos = specify_poosf_exercise$new(spec, 2, 5, 1, 30)    # specify the forecasting  exercise
fore = forecast_poos_recursively(spec, poos)                  # perform the forecasting  exercise
fp   = compute_forecast_performance(fore, "pls")   # compute forecasting performance measures

---

Computes forecasting performance measures for recursive pseudo-out-of-sample forecasts

Description

Computes forecasting performance measures selected from: log-predictive score "lps", root-mean-squared-forecast error "rmsfe", mean-absolute-forecast error "mafe" from the output of the recursive pseudo-out-of-sample forecastinge exercise performed using function forecast_poos_recursively.

Usage

## S3 method for class 'ForecastsPANELpoos'
compute_forecast_performance(forecasts, measures = c("pls", "rmsfe", "mafe"))

Arguments

forecasts	an object of class ForecastsPANELpoos containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples generated using function forecast_poos_recursively.
measures	a character vector with any of the values "lps", "rmsfe", "mafe" indicating the forecasting performance measures to be computed.

Value

An object of class ForecastingPerformance

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast_poos_recursively, forecast_poos_recursively.BVARPANEL, forecast_poos_recursively.BVARGROUPPANEL

Examples

spec = specify_bvarPANEL$new(ilo_dynamic_panel[1:5])               # specify the model
poos = specify_poosf_exercise$new(spec, 2, 5, 1, 30)    # specify the forecasting  exercise
fore = forecast_poos_recursively(spec, poos)                  # perform the forecasting  exercise
fp   = compute_forecast_performance(fore, "pls")   # compute forecasting performance measures

---

Computes posterior draws of the forecast error variance decomposition

Description

For each country, each of the draws from the posterior estimation of the model is transformed into a draw from the posterior distribution of the forecast error variance decomposition.

Usage

## S3 method for class 'PosteriorBVARGROUPPANEL'
compute_variance_decompositions(posterior, horizon)

Arguments

posterior	posterior estimation outcome - an object of class PosteriorBVARGROUPPANEL obtained by running the estimate function.
horizon	a positive integer number denoting the forecast horizon for the forecast error variance decompositions.

Value

An object of class PosteriorFEVDPANEL, that is, a list with C elements containing NxNx(horizon+1)xS arrays of class PosteriorFEVD with S draws of country-specific forecast error variance decompositions.

Author(s)

Tomasz Woźniak [email protected]

References

Lütkepohl, H. (2017). Structural VAR Tools, Chapter 4, In: Structural vector autoregressive analysis. Cambridge University Press.

See Also

estimate.PosteriorBVARGROUPPANEL, summary.PosteriorFEVDPANEL, plot.PosteriorFEVDPANEL

Examples

# specify the model and set seed
specification  = specify_bvarGroupPANEL$new(            # specify the model
                  ilo_dynamic_panel[1:5],
                  group_allocation = country_grouping_region[1:5]
                )

# run the burn-in
burn_in        = estimate(specification, 5)

# estimate the model
posterior      = estimate(burn_in, 5)

# compute forecast error variance decomposition 4 years ahead
fevd           = compute_variance_decompositions(posterior, horizon = 4)

---

Computes posterior draws of the forecast error variance decomposition

Description

For each country, each of the draws from the posterior estimation of the model is transformed into a draw from the posterior distribution of the forecast error variance decomposition.

Usage

## S3 method for class 'PosteriorBVARPANEL'
compute_variance_decompositions(posterior, horizon)

Arguments

posterior	posterior estimation outcome - an object of class PosteriorBVARPANEL obtained by running the estimate function.
horizon	a positive integer number denoting the forecast horizon for the forecast error variance decompositions.

Value

An object of class PosteriorFEVDPANEL, that is, a list with C elements containing NxNx(horizon+1)xS arrays of class PosteriorFEVD with S draws of country-specific forecast error variance decompositions.

Author(s)

Tomasz Woźniak [email protected]

References

Lütkepohl, H. (2017). Structural VAR Tools, Chapter 4, In: Structural vector autoregressive analysis. Cambridge University Press.

See Also

estimate.PosteriorBVARPANEL, summary.PosteriorFEVDPANEL, plot.PosteriorFEVDPANEL

Examples

# specify the model and set seed
specification  = specify_bvarPANEL$new(ilo_dynamic_panel[1:5], p = 1)

# run the burn-in
burn_in        = estimate(specification, 5)

# estimate the model
posterior      = estimate(burn_in, 5)

# compute forecast error variance decomposition 4 years ahead
fevd           = compute_variance_decompositions(posterior, horizon = 4)

---

Computes posterior draws of the forecast error variance decomposition

Description

For each country, each of the draws from the posterior estimation of the model is transformed into a draw from the posterior distribution of the forecast error variance decomposition.

Usage

## S3 method for class 'PosteriorBVARs'
compute_variance_decompositions(posterior, horizon)

Arguments

posterior	posterior estimation outcome - an object of class PosteriorBVARs obtained by running the estimate function.
horizon	a positive integer number denoting the forecast horizon for the forecast error variance decompositions.

Value

An object of class PosteriorFEVDPANEL, that is, a list with C elements containing NxNx(horizon+1)xS arrays of class PosteriorFEVD with S draws of country-specific forecast error variance decompositions.

Author(s)

Tomasz Woźniak [email protected]

References

Lütkepohl, H. (2017). Structural VAR Tools, Chapter 4, In: Structural vector autoregressive analysis. Cambridge University Press.

See Also

estimate.PosteriorBVARs, summary.PosteriorFEVDPANEL, plot.PosteriorFEVDPANEL

Examples

# specify the model and set seed
specification  = specify_bvars$new(ilo_dynamic_panel[1:5]) # specify the model

# run the burn-in
burn_in        = estimate(specification, 5)

# estimate the model
posterior      = estimate(burn_in, 5)

# compute forecast error variance decomposition 4 years ahead
fevd           = compute_variance_decompositions(posterior, horizon = 4)

---

A vector with country grouping by income group for 189 countries

Description

Each of the country is classified into one of the 4 categories according to their geographical location. The categories are:

1

Low-income countries

2

Lower-middle-income countries

3

Upper-middle-income countries

4

High-income countries

Last data update was implemented on 2026-04-18.

Usage

data(country_grouping_incomegroup)

Format

A numeric vector with values from 1 to 4

Source

International Labour Organization. (2020). ILO modelled estimates database, ILOSTAT [database]. Available from https://ilostat.ilo.org/data/.

Examples

data(country_grouping_incomegroup)   # upload the data

# setup a fixed group allocation Panel VAR model
spec = specify_bvarGroupPANEL$new(
         ilo_dynamic_panel,
         group_allocation = country_grouping_incomegroup
)

---

A vector with country grouping by region for 189 countries

Description

Each of the country is classified into one of the 5 categories according to their geographical location. The categories are:

1

Asia and the Pacific

2

Africa

3

Europe and Central Asia

4

Arab States

5

Americas

Last data update was implemented on 2026-04-18.

Usage

data(country_grouping_region)

Format

A numeric vector with values from 1 to 5

Source

International Labour Organization. (2020). ILO modelled estimates database, ILOSTAT [database]. Available from https://ilostat.ilo.org/data/.

Examples

data(country_grouping_region)   # upload the data

# setup a fixed group allocation Panel VAR model
spec = specify_bvarGroupPANEL$new(
         ilo_dynamic_panel,
         group_allocation = country_grouping_region
)

---

A vector with country grouping by subregion for 189 countries

Description

Each of the country is classified into one of the 11 categories according to their geographical location. The categories are:

1

Southern Asia

2

Sub-Saharan Africa

3

Northern, Southern and Western Europe

4

Arab States

5

Latin America and the Caribbean

6

Central and Western Asia

7

South-Eastern Asia and the Pacific

8

Eastern Europe

9

Northern America

10

Eastern Asia

11

Northern Africa

Last data update was implemented on 2026-04-18.

Usage

data(country_grouping_subregionbroad)

Format

A numeric vector with values from 1 to 11

Source

International Labour Organization. (2020). ILO modelled estimates database, ILOSTAT [database]. Available from https://ilostat.ilo.org/data/.

Examples

data(country_grouping_subregionbroad)   # upload the data

# setup a fixed group allocation Panel VAR model
spec = specify_bvarGroupPANEL$new(
         ilo_dynamic_panel,
         group_allocation = country_grouping_subregionbroad
)

---

A vector with country grouping by detailed subregion for 189 countries

Description

Each of the country is classified into one of the 20 categories according to their geographical location. The categories are:

1

Southern Asia

2

Central Africa

3

Southern Europe

4

Arab States

5

South America

6

Western Asia

7

Pacific Islands

8

Western Europe

9

Eastern Africa

10

Western Africa

11

Eastern Europe

12

Caribbean

13

Central America

14

South-Eastern Asia

15

Southern Africa

16

Northern America

17

Northern Europe

18

Eastern Asia

19

Northern Africa

20

Central Asia

Last data update was implemented on 2026-04-18.

Usage

data(country_grouping_subregiondetailed)

Format

A numeric vector with values from 1 to 20

Source

International Labour Organization. (2020). ILO modelled estimates database, ILOSTAT [database]. Available from https://ilostat.ilo.org/data/.

Examples

data(country_grouping_subregiondetailed)   # upload the data

# setup a fixed group allocation Panel VAR model
spec = specify_bvarGroupPANEL$new(
         ilo_dynamic_panel,
         group_allocation = country_grouping_subregiondetailed
)

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression with fixed or estimated country grouping

Description

Estimates the Bayesian Hierarchical Panel VAR with fixed or estimated country grouping using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'BVARGROUPPANEL'
estimate(specification, S, thin = 1L, show_progress = TRUE)

Arguments

specification	an object of class BVARGROUPPANEL generated using the specify_bvarPANEL$new() function.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model with fixed or estimated country grouping is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}$

a KxN global autoregressive parameters matrix

$\mathbf{\Sigma}$

an NxN global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Parameters $\mathbf{A}_c$ and $\mathbf{\Sigma}_c$ are subject to country grouping which sets them to the group-specific values.

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARGROUPPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A, Sigma, V, nu, m, w, s.

last_draw

an object of class BVARGROUPPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarGroupPANEL, specify_posterior_bvarGroupPANEL, summary.PosteriorBVARGROUPPANEL, forecast.PosteriorBVARGROUPPANEL

Examples

# specify the model
specification = specify_bvarGroupPANEL$new(
     data = ilo_dynamic_panel[1:5],
     exogenous = ilo_exogenous_variables[1:5],
     group_allocation = country_grouping_region[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression with fixed or estimated country grouping for global priors

Description

Estimates the Bayesian Hierarchical Panel VAR with fixed or estimated country grouping for global prior parameters using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'BVARGROUPPRIORPANEL'
estimate(specification, S, thin = 1L, show_progress = TRUE)

Arguments

specification	an object of class BVARGROUPPRIORPANEL generated using the specify_bvarGroupPriorPANEL$new() function.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model with fixed or estimated country grouping for global prior parametrs is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}_g$

a KxNxG group-specific global autoregressive parameters matrix

$\mathbf{\Sigma}_g$

an NxN group-specific global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Parameters $\mathbf{A}_c$ and $\mathbf{\Sigma}_c$ have prior expected values determined by the group-specific prior parameters $\mathbf{A}_g$ and $\mathbf{\Sigma}_g$.

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARGROUPPRIORPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A_g, Sigma_g, V, nu, m, w, s.

last_draw

an object of class BVARGROUPPRIORPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarGroupPriorPANEL, specify_posterior_bvarGroupPriorPANEL, summary.PosteriorBVARGROUPPRIORPANEL, forecast.PosteriorBVARGROUPPRIORPANEL

Examples

# specify the model
specification = specify_bvarGroupPriorPANEL$new(
     data = ilo_dynamic_panel[1:5],
     exogenous = ilo_exogenous_variables[1:5],
     group_allocation = country_grouping_region[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression using Gibbs sampler

Description

Estimates the Bayesian Hierarchical Panel VAR using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'BVARPANEL'
estimate(specification, S, thin = 1L, show_progress = TRUE)

Arguments

specification	an object of class BVARPANEL generated using the specify_bvarPANEL$new() function.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model described in bpvars is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}$

a KxN global autoregressive parameters matrix

$\mathbf{\Sigma}$

an NxN global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A, Sigma, V, nu, m, w, s.

last_draw

an object of class BVARPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarPANEL, specify_posterior_bvarPANEL, summary.PosteriorBVARPANEL, forecast.PosteriorBVARPANEL

Examples

# specify the model
specification = specify_bvarPANEL$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Vector Autoregressions for cubic data using Gibbs sampler

Description

Estimates the Bayesian Hierarchical VARs for cubic data using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'BVARs'
estimate(specification, S, thin = 1L, show_progress = TRUE)

Arguments

specification	an object of class BVARs generated using the specify_bvars$new() function.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Vector Autoregressive models described in bpvars is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARs 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, nu, m, w, s.

last_draw

an object of class BVARs with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvars, specify_posterior_bvars, summary.PosteriorBVARs, forecast.PosteriorBVARs

Examples

# specify the model
specification = specify_bvars$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression with fixed or estimated country grouping

Description

Estimates the Bayesian Hierarchical Panel VAR with fixed or estimated country grouping using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'PosteriorBVARGROUPPANEL'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification	an object of class PosteriorBVARGROUPPANEL generated using the estimate.BVARGROUPPANEL() function. This setup facilitates the continuation of the MCMC sampling starting from the last draw of the previous run.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model with fixed or estimated country grouping is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}$

a KxN global autoregressive parameters matrix

$\mathbf{\Sigma}$

an NxN global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Parameters $\mathbf{A}_c$ and $\mathbf{\Sigma}_c$ are subject to country grouping which sets them to the group-specific values.

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARGROUPPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A, Sigma, V, nu, m, w, s.

last_draw

an object of class BVARGROUPPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarGroupPANEL, specify_posterior_bvarGroupPANEL, summary.PosteriorBVARGROUPPANEL, forecast.PosteriorBVARGROUPPANEL

Examples

# specify the model
specification = specify_bvarGroupPANEL$new(
     data = ilo_dynamic_panel[1:5],
     exogenous = ilo_exogenous_variables[1:5],
     group_allocation = country_grouping_region[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression with fixed or estimated country grouping for global priors

Description

Estimates the Bayesian Hierarchical Panel VAR with fixed or estimated country grouping for global prior parameters using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'PosteriorBVARGROUPPRIORPANEL'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification	an object of class PosteriorBVARGROUPPRIORPANEL generated using the estimate.BVARGROUPPRIORPANEL() function. This setup facilitates the continuation of the MCMC sampling starting from the last draw of the previous run.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model with fixed or estimated country grouping for global prior parametrs is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}_g$

a KxNxG group-specific global autoregressive parameters matrix

$\mathbf{\Sigma}_g$

an NxN group-specific global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Parameters $\mathbf{A}_c$ and $\mathbf{\Sigma}_c$ have prior expected values determined by the group-specific prior parameters $\mathbf{A}_g$ and $\mathbf{\Sigma}_g$.

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARGROUPPRIORPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A_g, Sigma_g, V, nu, m, w, s.

last_draw

an object of class BVARGROUPPRIORPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarGroupPriorPANEL, specify_posterior_bvarGroupPriorPANEL, summary.PosteriorBVARGROUPPRIORPANEL, forecast.PosteriorBVARGROUPPRIORPANEL

Examples

# specify the model
specification = specify_bvarGroupPriorPANEL$new(
     data = ilo_dynamic_panel[1:5],
     exogenous = ilo_exogenous_variables[1:5],
     group_allocation = country_grouping_region[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression using Gibbs sampler

Description

Estimates the Bayesian Hierarchical Panel VAR using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'PosteriorBVARPANEL'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification	an object of class PosteriorBVARPANEL generated using the estimate.BVARPANEL() function. This setup facilitates the continuation of the MCMC sampling starting from the last draw of the previous run.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Panel Vector Autoregressive model described in bpvars is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\mathbf{A}$

a KxN global autoregressive parameters matrix

$\mathbf{\Sigma}$

an NxN global covariance matrix

$\mathbf{V}$

a KxK covariance matrix of prior for global autoregressive parameters

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARPANEL 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A, Sigma, V, nu, m, w, s.

last_draw

an object of class BVARPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvarPANEL, specify_posterior_bvarPANEL, summary.PosteriorBVARPANEL, forecast.PosteriorBVARPANEL

Examples

# specify the model
specification = specify_bvarPANEL$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian estimation of a Bayesian Hierarchical Vector Autoregressions for cubic data using Gibbs sampler

Description

Estimates the Bayesian Hierarchical VARs for cubic data using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

## S3 method for class 'PosteriorBVARs'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification	an object of class PosteriorBVARs generated using the estimate.BVARs() function. This setup facilitates the continuation of the MCMC sampling starting from the last draw of the previous run.
S	a positive integer, the number of posterior draws to be generated
thin	a positive integer, specifying the frequency of MCMC output thinning
show_progress	a logical value, if TRUE the estimation progress bar is visible

Details

The Bayesian Hierarchical Vector Autoregressive models described in bpvars is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

$\mathbf{A}_c$

a KxN country-specific autoregressive parameters matrix for each of the countries $c = 1,\dots,C$

$\mathbf{\Sigma}_c$

an NxN country-specific covariance matrix for each of the countries $c = 1,\dots,C$

$\nu$

prior degrees of freedom parameter

$m$

prior average global persistence parameter

$w$

prior scaling parameter

$s$

prior scaling parameter

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters $\theta_1$ and $\theta_2$. In order to sample from the joint posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$ the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by $p(\theta_1|\theta_2,\mathbf{Y})$ and $p(\theta_2|\theta_1,\mathbf{Y})$. These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

1. Set the initial values of the parameters $\theta_2^{(0)}$

2. At each of the s iterations:

   1. Sample $\theta_1^{(s)}$ from $p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})$

   2. Sample $\theta_2^{(s)}$ from $p(\theta_2|\theta_1^{(s)},\mathbf{Y})$

3. Repeat step 2. S times. Return $\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}$ as a sample drawn from the posterior distribution $p(\theta_1,\theta_2|\mathbf{Y})$.

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Value

An object of class PosteriorBVARs 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. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, nu, m, w, s.

last_draw

an object of class BVARs with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Author(s)

Tomasz Woźniak [email protected]

See Also

bpvars, specify_bvars, specify_posterior_bvars, summary.PosteriorBVARs, forecast.PosteriorBVARs

Examples

# specify the model
specification = specify_bvars$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

---

Bayesian recursive pseudo-out-of-sample forecasting

Description

Performs the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.

Usage

forecast_poos_recursively(model_spec, poos_spec, show_progress = TRUE)

Arguments

model_spec	an object generated using one of the specify_* functions containing model specification.
poos_spec	an object of class POOSForecastSetup containing specification of the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.
show_progress	a logical value, if TRUE the estimation progress bar is visible

Value

An object of class ForecastsPOOS containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples. The object is a list with forecasting_sample elements, where forecasting_sample is equal to the sample size less the maximum of horizons and the training_sample plus one. Each element of the list is an object of class ForecastsPANEL containing the forecasts for each country, see forecast.PosteriorBVARPANEL.

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast.PosteriorBVARPANEL, specify_bvarPANEL, specify_poosf_exercise, estimate.BVARPANEL

Examples

spec = specify_bvarPANEL$new(ilo_dynamic_panel[1:5])   # specify the model
poos = specify_poosf_exercise$new(                # specify the forecasting exercise
         spec, 
         S = 5,                                   # use at least S = 5000
         S_burn = 2,                              # use at least S_burn = 1000
         horizons = 1,
         training_sample = 30
       )   
fore = forecast_poos_recursively(spec, poos)      # execute the forecasting exercise

---

Bayesian recursive pseudo-out-of-sample forecasting

Description

Performs the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.

Usage

## S3 method for class 'BVARGROUPPANEL'
forecast_poos_recursively(model_spec, poos_spec, show_progress = TRUE)

Arguments

model_spec	an object of class BVARGROUPPANEL generated using the specify_bvarGroupPANEL function and containing the Bayesian Panel VAR model specification.
poos_spec	an object of class POOSForecastSetup containing specification of the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.
show_progress	a logical value, if TRUE the estimation progress bar is visible

Value

An object of class ForecastsPOOS containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples. The object is a list with forecasting_sample elements, where forecasting_sample is equal to the sample size less the maximum of horizons and the training_sample plus one. Each element of the list is an object of class ForecastsPANEL containing the forecasts for each country, see forecast.PosteriorBVARPANEL.

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast.PosteriorBVARPANEL, specify_bvarPANEL, specify_poosf_exercise, estimate.BVARPANEL

Examples

spec = specify_bvarGroupPANEL$new(                # specify the model
         ilo_dynamic_panel[1:5], 
         group_allocation = country_grouping_region[1:5]
       )   
poos = specify_poosf_exercise$new(                # specify the forecasting exercise
         spec, 
         S = 5,                                  # use at least S = 5000
         S_burn = 2,                              # use at least S_burn = 1000
         horizons = 1,
         training_sample = 30
       )   
fore = forecast_poos_recursively(spec, poos)      # execute the forecasting exercise

---

Bayesian recursive pseudo-out-of-sample forecasting

Description

Performs the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.

Usage

## S3 method for class 'BVARGROUPPRIORPANEL'
forecast_poos_recursively(model_spec, poos_spec, show_progress = TRUE)

Arguments

model_spec	an object of class BVARGROUPPRIORPANEL generated using the specify_bvarGroupPANEL function and containing the Bayesian Panel VAR model specification with group-specific global parameters.
poos_spec	an object of class POOSForecastSetup containing specification of the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.
show_progress	a logical value, if TRUE the estimation progress bar is visible

Value

An object of class ForecastsPOOS containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples. The object is a list with forecasting_sample elements, where forecasting_sample is equal to the sample size less the maximum of horizons and the training_sample plus one. Each element of the list is an object of class ForecastsPANEL containing the forecasts for each country, see forecast.PosteriorBVARPANEL.

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast.PosteriorBVARPANEL, specify_bvarPANEL, specify_poosf_exercise, estimate.BVARPANEL

Examples

spec = specify_bvarGroupPriorPANEL$new(                # specify the model
         ilo_dynamic_panel[1:5], 
         group_allocation = country_grouping_region[1:5]
       )   
poos = specify_poosf_exercise$new(                # specify the forecasting exercise
         spec, 
         S = 5,                                  # use at least S = 5000
         S_burn = 5,                              # use at least S_burn = 1000
         horizons = 1,
         training_sample = 30
       )   
fore = forecast_poos_recursively(spec, poos)      # execute the forecasting exercise

---

Bayesian recursive pseudo-out-of-sample forecasting

Description

Performs the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.

Usage

## S3 method for class 'BVARPANEL'
forecast_poos_recursively(model_spec, poos_spec, show_progress = TRUE)

Arguments

model_spec	an object of class BVARPANEL generated using the specify_bvarPANEL function and containing the Bayesian Panel VAR model specification.
poos_spec	an object of class POOSForecastSetup containing specification of the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.
show_progress	a logical value, if TRUE the estimation progress bar is visible

Value

An object of class ForecastsPOOS containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples. The object is a list with forecasting_sample elements, where forecasting_sample is equal to the sample size less the maximum of horizons and the training_sample plus one. Each element of the list is an object of class ForecastsPANEL containing the forecasts for each country, see forecast.PosteriorBVARPANEL.

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast.PosteriorBVARPANEL, specify_bvarPANEL, specify_poosf_exercise, estimate.BVARPANEL

Examples

spec = specify_bvarPANEL$new(ilo_dynamic_panel[1:5])   # specify the model
poos = specify_poosf_exercise$new(                # specify the forecasting exercise
         spec, 
         S = 5,                                   # use at least S = 5000
         S_burn = 2,                              # use at least S_burn = 1000
         horizons = 1,
         training_sample = 30
       )   
fore = forecast_poos_recursively(spec, poos)      # execute the forecasting exercise

---

Bayesian recursive pseudo-out-of-sample forecasting

Description

Performs the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.

Usage

## S3 method for class 'BVARs'
forecast_poos_recursively(model_spec, poos_spec, show_progress = TRUE)

Arguments

model_spec	an object of class BVARs generated using the specify_bvars function and containing the Bayesian VAR models specification.
poos_spec	an object of class POOSForecastSetup containing specification of the recursive pseudo-out-of-sample forecasting exercise using expanding window samples.
show_progress	a logical value, if TRUE the estimation progress bar is visible

Value

An object of class ForecastsPOOS containing the outcome of Bayesian recursive pseudo-out-of-sample forecasting exercise using expanding window samples. The object is a list with forecasting_sample elements, where forecasting_sample is equal to the sample size less the maximum of horizons and the training_sample plus one. Each element of the list is an object of class ForecastsPANEL containing the forecasts for each country, see forecast.PosteriorBVARPANEL.

Author(s)

Tomasz Woźniak [email protected]

See Also

forecast.PosteriorBVARPANEL, specify_bvarPANEL, specify_poosf_exercise, estimate.BVARPANEL

Examples

spec = specify_bvars$new(ilo_dynamic_panel[1:5])  # specify the model
poos = specify_poosf_exercise$new(                # specify the forecasting exercise
         spec, 
         S = 5,                                  # use at least S = 5000
         S_burn = 5,                              # use at least S_burn = 1000
         horizons = 1,
         training_sample = 30
       )   
fore = forecast_poos_recursively(spec, poos)      # execute the forecasting exercise

---

Forecasting using Hierarchical Panel Vector Autoregressions

Description

Samples from the joint predictive density of the dependent variables for all countries at forecast horizons from 1 to horizon specified as an argument of the function. Also implements conditional forecasting based on the provided projections for some of the variables.

Usage

## S3 method for class 'PosteriorBVARGROUPPANEL'
forecast(
  object,
  horizon = 1,
  exogenous_forecast = NULL,
  conditional_forecast = NULL,
  ...
)

Arguments

object	posterior estimation outcome - an object of class PosteriorBVARGROUPPANEL obtained by running the estimate function.
horizon	a positive integer, specifying the forecasting horizon.
exogenous_forecast	not used here ATM; included for compatibility with generic forecast.
conditional_forecast	a list of length C containing horizon x N matrices with forecasted values for selected variables. These matrices should only contain numeric or NA values. The entries with NA values correspond to the values that are forecasted conditionally on the realisations provided as numeric values.
...	not used

Details

The package provides a range of options regarding the forecasting procedure. They are dependent on the model and forecast specifications and include Bayesian forecasting many periods ahead, conditional forecasting, and forecasting for models with exogenous variables.

One-period-ahead predictive density. The model assumptions provided in the documentation for bpvars determine the country-specific one-period ahead conditional predictive density for the unknown vector $\mathbf{y}_{c.t+1}$ given the data available at time $t$ and the parameters of the model. It is multivariate normal with the mean $\mathbf{A}_c' \mathbf{x}_{c.t+1}$ and the covariance matrix $\mathbf{\Sigma}_c$

$p(\mathbf{y}_{c.t+1} | \mathbf{x}_{c.t+1}, \mathbf{A}_c, \mathbf{\Sigma}_c) = N_N(\mathbf{A}_c' \mathbf{x}_{c.t+1}, \mathbf{\Sigma}_c)$

where $\mathbf{x}_{c.t+1}$ includes the lagged values of $\mathbf{y}_{c.t+1}$, the constant term, and, potentially, exogenous variables if they were specified by the user.

Bayesian predictive density. The one-period ahead predictive density is used to sample from the joint predictive density of the unknown future values. This predictive density is defined as a joint density of $\mathbf{y}_{c.t+h}$ at horizons $h = 1,\dots,H$, where $H$ corresponds to the value of argument horizon, given the data available at time $t$:

$p( \mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c) = \int p(\mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c, \mathbf{A}_c, \boldsymbol\Sigma_c) p( \mathbf{A}_c, \boldsymbol\Sigma_c | \mathbf{Y}_c, \mathbf{X}_c) d(\mathbf{A}_c, \boldsymbol\Sigma_c)$

Therefore, the Bayesian forecast does not depend on the parameter values as the parameters are integrated out with respect to their posterior distribution. Consequently, Bayesian forecasts incorporate the uncertainty with respect to estimation. Sampling from the density is facilitated using the draws from the posterior density and sequential sampling from the one-period ahead predictive density.

Conditional forecasting of some of the variables given the future values of the remaining variables is implemented following Waggoner and Zha (1999) and is based on the conditional normal density given the future projections of some of the variables created basing on the one-period ahead predictive density.

Exogenous variables. Forecasting with models for which specification argument exogenous_variables was specified required providing the future values of these exogenous variables in the argument exogenous_forecast of the forecast.PosteriorBVARPANEL function.

Truncated forecasts for variables of type 'rate'. The package provides the option to truncate the forecasts for variables of for which the corresponding element of argument type of the function specify_bvarPANEL$new() is set to "rate". The one-period-ahead predictive normal density for such variables is truncated to values from interval $[0,100]$.

Value

A list of class ForecastsPANEL with C elements containing the draws from the country-specific predictive density and data in a form of object class Forecasts that includes:

forecasts

an NxhorizonxS array with the draws from the country-specific predictive density

forecast_mean

an NxhorizonxS array with the mean of the country-specific predictive density

forecast_cov

an NxNxhorizonxS array with the covariance of the country-specific predictive density

Y

a T_cxN matrix with the country-specific data

Author(s)

Tomasz Woźniak [email protected]

References

Waggoner, D. F., & Zha, T. (1999) Conditional forecasts in dynamic multivariate models, Review of Economics and Statistics, 81(4), 639-651, doi:10.1162/003465399558508.

See Also

specify_bvarGroupPANEL, estimate.PosteriorBVARGROUPPANEL, summary.ForecastsPANEL, plot.ForecastsPANEL

Examples

# specify the model
specification = specify_bvarGroupPANEL$new(
                  ilo_dynamic_panel[1:5], 
                  exogenous = ilo_exogenous_variables[1:5],
                  group_allocation = country_grouping_incomegroup[1:5]
                )
burn_in       = estimate(specification, 5)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 5)                   # estimate the model; use say S = 10000

# forecast 3 years ahead
predictive    = forecast(
                  posterior, 
                  horizon = 3, 
                  exogenous_forecast = ilo_exogenous_forecasts[1:5]
                )

---

Forecasting using Hierarchical Panel Vector Autoregressions

Description

Samples from the joint predictive density of the dependent variables for all countries at forecast horizons from 1 to horizon specified as an argument of the function. Also implements conditional forecasting based on the provided projections for some of the variables.

Usage

## S3 method for class 'PosteriorBVARGROUPPRIORPANEL'
forecast(
  object,
  horizon = 1,
  exogenous_forecast = NULL,
  conditional_forecast = NULL,
  ...
)

Arguments

object	posterior estimation outcome - an object of class PosteriorBVARGROUPPRIORPANEL obtained by running the estimate function.
horizon	a positive integer, specifying the forecasting horizon.
exogenous_forecast	not used here ATM; included for compatibility with generic forecast.
conditional_forecast	a list of length C containing horizon x N matrices with forecasted values for selected variables. These matrices should only contain numeric or NA values. The entries with NA values correspond to the values that are forecasted conditionally on the realisations provided as numeric values.
...	not used

Details

The package provides a range of options regarding the forecasting procedure. They are dependent on the model and forecast specifications and include Bayesian forecasting many periods ahead, conditional forecasting, and forecasting for models with exogenous variables.

One-period-ahead predictive density. The model assumptions provided in the documentation for bpvars determine the country-specific one-period ahead conditional predictive density for the unknown vector $\mathbf{y}_{c.t+1}$ given the data available at time $t$ and the parameters of the model. It is multivariate normal with the mean $\mathbf{A}_c' \mathbf{x}_{c.t+1}$ and the covariance matrix $\mathbf{\Sigma}_c$

$p(\mathbf{y}_{c.t+1} | \mathbf{x}_{c.t+1}, \mathbf{A}_c, \mathbf{\Sigma}_c) = N_N(\mathbf{A}_c' \mathbf{x}_{c.t+1}, \mathbf{\Sigma}_c)$

where $\mathbf{x}_{c.t+1}$ includes the lagged values of $\mathbf{y}_{c.t+1}$, the constant term, and, potentially, exogenous variables if they were specified by the user.

Bayesian predictive density. The one-period ahead predictive density is used to sample from the joint predictive density of the unknown future values. This predictive density is defined as a joint density of $\mathbf{y}_{c.t+h}$ at horizons $h = 1,\dots,H$, where $H$ corresponds to the value of argument horizon, given the data available at time $t$:

$p( \mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c) = \int p(\mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c, \mathbf{A}_c, \boldsymbol\Sigma_c) p( \mathbf{A}_c, \boldsymbol\Sigma_c | \mathbf{Y}_c, \mathbf{X}_c) d(\mathbf{A}_c, \boldsymbol\Sigma_c)$

Therefore, the Bayesian forecast does not depend on the parameter values as the parameters are integrated out with respect to their posterior distribution. Consequently, Bayesian forecasts incorporate the uncertainty with respect to estimation. Sampling from the density is facilitated using the draws from the posterior density and sequential sampling from the one-period ahead predictive density.

Conditional forecasting of some of the variables given the future values of the remaining variables is implemented following Waggoner and Zha (1999) and is based on the conditional normal density given the future projections of some of the variables created basing on the one-period ahead predictive density.

Exogenous variables. Forecasting with models for which specification argument exogenous_variables was specified required providing the future values of these exogenous variables in the argument exogenous_forecast of the forecast.PosteriorBVARPANEL function.

Truncated forecasts for variables of type 'rate'. The package provides the option to truncate the forecasts for variables of for which the corresponding element of argument type of the function specify_bvarPANEL$new() is set to "rate". The one-period-ahead predictive normal density for such variables is truncated to values from interval $[0,100]$.

Value

A list of class ForecastsPANEL with C elements containing the draws from the country-specific predictive density and data in a form of object class Forecasts that includes:

forecasts

an NxhorizonxS array with the draws from the country-specific predictive density

forecast_mean

an NxhorizonxS array with the mean of the country-specific predictive density

forecast_cov

an NxNxhorizonxS array with the covariance of the country-specific predictive density

Y

a T_cxN matrix with the country-specific data

Author(s)

Tomasz Woźniak [email protected]

References

Waggoner, D. F., & Zha, T. (1999) Conditional forecasts in dynamic multivariate models, Review of Economics and Statistics, 81(4), 639-651, doi:10.1162/003465399558508.

See Also

specify_bvarGroupPriorPANEL, estimate.PosteriorBVARGROUPPRIORPANEL, summary.ForecastsPANEL, plot.ForecastsPANEL

Examples

# specify the model
specification = specify_bvarGroupPriorPANEL$new(
                  ilo_dynamic_panel[1:5],
                  group_allocation = country_grouping_incomegroup[1:5]
                )
burn_in       = estimate(specification, 5)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 5)                   # estimate the model; use say S = 10000

# forecast 3 years ahead
predictive    = forecast(posterior, horizon = 3)

---

Forecasting using Hierarchical Panel Vector Autoregressions

Description

Samples from the joint predictive density of the dependent variables for all countries at forecast horizons from 1 to horizon specified as an argument of the function. Also implements conditional forecasting based on the provided projections for some of the variables.

Usage

## S3 method for class 'PosteriorBVARPANEL'
forecast(
  object,
  horizon = 1,
  exogenous_forecast = NULL,
  conditional_forecast = NULL,
  ...
)

Arguments

object	posterior estimation outcome - an object of class PosteriorBVARPANEL obtained by running the estimate function.
horizon	a positive integer, specifying the forecasting horizon.
exogenous_forecast	not used here ATM; included for compatibility with generic forecast.
conditional_forecast	a list of length C containing horizon x N matrices with forecasted values for selected variables. These matrices should only contain numeric or NA values. The entries with NA values correspond to the values that are forecasted conditionally on the realisations provided as numeric values.
...	not used

Details

The package provides a range of options regarding the forecasting procedure. They are dependent on the model and forecast specifications and include Bayesian forecasting many periods ahead, conditional forecasting, and forecasting for models with exogenous variables.

One-period-ahead predictive density. The model assumptions provided in the documentation for bpvars determine the country-specific one-period ahead conditional predictive density for the unknown vector $\mathbf{y}_{c.t+1}$ given the data available at time $t$ and the parameters of the model. It is multivariate normal with the mean $\mathbf{A}_c' \mathbf{x}_{c.t+1}$ and the covariance matrix $\mathbf{\Sigma}_c$

$p(\mathbf{y}_{c.t+1} | \mathbf{x}_{c.t+1}, \mathbf{A}_c, \mathbf{\Sigma}_c) = N_N(\mathbf{A}_c' \mathbf{x}_{c.t+1}, \mathbf{\Sigma}_c)$

where $\mathbf{x}_{c.t+1}$ includes the lagged values of $\mathbf{y}_{c.t+1}$, the constant term, and, potentially, exogenous variables if they were specified by the user.

Bayesian predictive density. The one-period ahead predictive density is used to sample from the joint predictive density of the unknown future values. This predictive density is defined as a joint density of $\mathbf{y}_{c.t+h}$ at horizons $h = 1,\dots,H$, where $H$ corresponds to the value of argument horizon, given the data available at time $t$:

$p( \mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c) = \int p(\mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c, \mathbf{A}_c, \boldsymbol\Sigma_c) p( \mathbf{A}_c, \boldsymbol\Sigma_c | \mathbf{Y}_c, \mathbf{X}_c) d(\mathbf{A}_c, \boldsymbol\Sigma_c)$

Therefore, the Bayesian forecast does not depend on the parameter values as the parameters are integrated out with respect to their posterior distribution. Consequently, Bayesian forecasts incorporate the uncertainty with respect to estimation. Sampling from the density is facilitated using the draws from the posterior density and sequential sampling from the one-period ahead predictive density.

Conditional forecasting of some of the variables given the future values of the remaining variables is implemented following Waggoner and Zha (1999) and is based on the conditional normal density given the future projections of some of the variables created basing on the one-period ahead predictive density.

Exogenous variables. Forecasting with models for which specification argument exogenous_variables was specified required providing the future values of these exogenous variables in the argument exogenous_forecast of the forecast.PosteriorBVARPANEL function.

Truncated forecasts for variables of type 'rate'. The package provides the option to truncate the forecasts for variables of for which the corresponding element of argument type of the function specify_bvarPANEL$new() is set to "rate". The one-period-ahead predictive normal density for such variables is truncated to values from interval $[0,100]$.

Value

A list of class ForecastsPANEL with C elements containing the draws from the country-specific predictive density and data in a form of object class Forecasts that includes:

forecasts

an NxhorizonxS array with the draws from the country-specific predictive density

forecast_mean

an NxhorizonxS array with the mean of the country-specific predictive density

forecast_cov

an NxNxhorizonxS array with the covariance of the country-specific predictive density

Y

a T_cxN matrix with the country-specific data

Author(s)

Tomasz Woźniak [email protected]

References

Waggoner, D. F., & Zha, T. (1999) Conditional forecasts in dynamic multivariate models, Review of Economics and Statistics, 81(4), 639-651, doi:10.1162/003465399558508.

See Also

specify_bvarPANEL, estimate.PosteriorBVARPANEL, summary.ForecastsPANEL, plot.ForecastsPANEL

Examples

# specify the model
specification = specify_bvarPANEL$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 5)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 5)                   # estimate the model; use say S = 10000

# forecast 3 years ahead
predictive    = forecast(posterior, 3, exogenous_forecast = ilo_exogenous_forecasts[1:5])

---

Forecasting using Hierarchical Vector Autoregressions for Dynamic Panel Data

Description

Samples from the joint predictive density of the dependent variables for all countries at forecast horizons from 1 to horizon specified as an argument of the function. Also implements conditional forecasting based on the provided projections for some of the variables.

Usage

## S3 method for class 'PosteriorBVARs'
forecast(
  object,
  horizon = 1,
  exogenous_forecast = NULL,
  conditional_forecast = NULL,
  ...
)

Arguments

object	posterior estimation outcome - an object of class PosteriorBVARs obtained by running the estimate function.
horizon	a positive integer, specifying the forecasting horizon.
exogenous_forecast	not used here ATM; included for compatibility with generic forecast.
conditional_forecast	a list of length C containing horizon x N matrices with forecasted values for selected variables. These matrices should only contain numeric or NA values. The entries with NA values correspond to the values that are forecasted conditionally on the realisations provided as numeric values.
...	not used

Details

The package provides a range of options regarding the forecasting procedure. They are dependent on the model and forecast specifications and include Bayesian forecasting many periods ahead, conditional forecasting, and forecasting for models with exogenous variables.

One-period-ahead predictive density. The model assumptions provided in the documentation for bpvars determine the country-specific one-period ahead conditional predictive density for the unknown vector $\mathbf{y}_{c.t+1}$ given the data available at time $t$ and the parameters of the model. It is multivariate normal with the mean $\mathbf{A}_c' \mathbf{x}_{c.t+1}$ and the covariance matrix $\mathbf{\Sigma}_c$

$p(\mathbf{y}_{c.t+1} | \mathbf{x}_{c.t+1}, \mathbf{A}_c, \mathbf{\Sigma}_c) = N_N(\mathbf{A}_c' \mathbf{x}_{c.t+1}, \mathbf{\Sigma}_c)$

where $\mathbf{x}_{c.t+1}$ includes the lagged values of $\mathbf{y}_{c.t+1}$, the constant term, and, potentially, exogenous variables if they were specified by the user.

Bayesian predictive density. The one-period ahead predictive density is used to sample from the joint predictive density of the unknown future values. This predictive density is defined as a joint density of $\mathbf{y}_{c.t+h}$ at horizons $h = 1,\dots,H$, where $H$ corresponds to the value of argument horizon, given the data available at time $t$:

$p( \mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c) = \int p(\mathbf{y}_{c.T_c + H}, \dots, \mathbf{y}_{c.T_c + 1} | \mathbf{Y}_c, \mathbf{X}_c, \mathbf{A}_c, \boldsymbol\Sigma_c) p( \mathbf{A}_c, \boldsymbol\Sigma_c | \mathbf{Y}_c, \mathbf{X}_c) d(\mathbf{A}_c, \boldsymbol\Sigma_c)$

Therefore, the Bayesian forecast does not depend on the parameter values as the parameters are integrated out with respect to their posterior distribution. Consequently, Bayesian forecasts incorporate the uncertainty with respect to estimation. Sampling from the density is facilitated using the draws from the posterior density and sequential sampling from the one-period ahead predictive density.

Conditional forecasting of some of the variables given the future values of the remaining variables is implemented following Waggoner and Zha (1999) and is based on the conditional normal density given the future projections of some of the variables created basing on the one-period ahead predictive density.

Exogenous variables. Forecasting with models for which specification argument exogenous_variables was specified required providing the future values of these exogenous variables in the argument exogenous_forecast of the forecast.PosteriorBVARPANEL function.

Truncated forecasts for variables of type 'rate'. The package provides the option to truncate the forecasts for variables of for which the corresponding element of argument type of the function specify_bvarPANEL$new() is set to "rate". The one-period-ahead predictive normal density for such variables is truncated to values from interval $[0,100]$.

Value

A list of class ForecastsPANEL with C elements containing the draws from the country-specific predictive density and data in a form of object class Forecasts that includes:

forecasts

an NxhorizonxS array with the draws from the country-specific predictive density

forecast_mean

an NxhorizonxS array with the mean of the country-specific predictive density

forecast_cov

an NxNxhorizonxS array with the covariance of the country-specific predictive density

Y

a T_cxN matrix with the country-specific data

Author(s)

Tomasz Woźniak [email protected]

References

Waggoner, D. F., & Zha, T. (1999) Conditional forecasts in dynamic multivariate models, Review of Economics and Statistics, 81(4), 639-651, doi:10.1162/003465399558508.

See Also

specify_bvars, estimate.PosteriorBVARs, summary.ForecastsPANEL, plot.ForecastsPANEL

Examples

# specify the model
specification = specify_bvars$new( 
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 5)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 5)                   # estimate the model; use say S = 10000

# forecast 3 years ahead
predictive    = forecast(posterior, 3, exogenous_forecast = ilo_exogenous_forecasts[1:5])

---