Forecasting using Hierarchical Panel Vector Autoregressions

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(
  posterior,
  horizon = 1,
  exogenous_forecast = NULL,
  conditional_forecast = NULL
)

Arguments

posterior: 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.

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 horizonxNxS array with the draws from the country-specific predictive density
Y: a T_cxN matrix with the country-specific data

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 bvarPANELs 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 $[0,100]$.

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 .

Author

Tomasz Woźniak wozniak.tom@pm.me

Examples

data(ilo_dynamic_panel)                                 # load the data
data(ilo_exogenous_variables)                           # load the exogenous variables
data(ilo_exogenous_forecasts)                           # load the exogenous forecast
set.seed(123)

# specify the model
specification = specify_bvarPANEL$new(ilo_dynamic_panel, exogenous = ilo_exogenous_variables)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|

# forecast 6 years ahead
predictive    = forecast(posterior, 6, exogenous_forecast = ilo_exogenous_forecasts)
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of sampling 10 draws from
#>     the predictive density for 189 countries
#>     Press Esc to interrupt the computations
#> **************************************************|

# workflow with the pipe |>
############################################################
set.seed(123)
ilo_dynamic_panel |>
  specify_bvarPANEL$new() |>
  estimate(S = 10) |> 
  estimate(S = 20) |> 
  forecast(horizon = 2) -> predictive
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 20 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of sampling 20 draws from
#>     the predictive density for 189 countries
#>     Press Esc to interrupt the computations
#> **************************************************|

# conditional forecasting 6 years ahead conditioning on 
#  provided future values for the Gross Domestic Product 
#  and truncated forecasts for the rates
############################################################
data(ilo_conditional_forecasts)                        # load the conditional forecasts of dgdp
specification = specify_bvarPANEL$new(
                  ilo_dynamic_panel,
                  type = c("real", rep("rate", 3))
                )   # specify the model
burn_in       = estimate(specification, 10)            # run the burn-in; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
posterior     = estimate(burn_in, 10)                  # estimate the model; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
# forecast 6 years ahead
predictive    = forecast(posterior, 6, conditional_forecast = ilo_conditional_forecasts)
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of sampling 10 draws from
#>     the predictive density for 189 countries
#>     Press Esc to interrupt the computations
#> **************************************************|

# workflow with the pipe |>
############################################################
set.seed(123)
ilo_dynamic_panel |>
  specify_bvarPANEL$new(type = c("real", rep("rate", 3))) |>
  estimate(S = 10) |> 
  estimate(S = 20) |> 
  forecast(
    horizon = 6, 
    conditional_forecast = ilo_conditional_forecasts
  ) -> predictive
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 20 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of sampling 20 draws from
#>     the predictive density for 189 countries
#>     Press Esc to interrupt the computations
#> **************************************************|