| tvc {hdflex} | R Documentation |
Compute density forecasts based on univariate time-varying coefficient (TV-C) models in state-space form
Description
'tvc()' can be used to generate density forecasts based on univariate time-varying coefficient models. In each forecasting model, we include an intercept and one predictive signal. The predictive signal either represents the value of a 'simple' signal or the the value of an external point forecast. All models are estimated independently from each other and estimation and forecasting are carried out recursively.
Usage
tvc(y, X, Ext_F, lambda_grid, kappa_grid, init_length, n_cores)
Arguments
y |
A matrix of dimension 'T * 1' or numeric vector of length 'T' containing the observations of the target variable. |
X |
A matrix with 'T' rows containing the lagged 'simple' signals in each column. Use NULL if no 'simple' signal shall be included. |
Ext_F |
A matrix with 'T' rows containing point forecasts of y in each column. Use NULL if no point forecasts shall be included. |
lambda_grid |
A numeric vector denoting the discount factor(s) that control the dynamics of the coefficients. Each signal in combination with each value of lambda provides a separate candidate forecast. Constant coefficients are nested for the case 'lambda = 1'. |
kappa_grid |
A numeric vector to accommodate time-varying volatility. The observational variance is estimated via Exponentially Weighted Moving Average. Constant variance is nested for the case 'kappa = 1'. Each signal in combination with each value of kappa provides a separate forecast. |
init_length |
An integer that denotes the number of observations used to initialize the observational variance and the coefficients' variance. |
n_cores |
An integer that denotes the number of CPU-cores used for the computation. |
Value
A list that contains:
* (1) a matrix with the first moments (point forecasts) of the conditionally normal predictive distributions and
* (2) a matrix with the second moments (variance) of the conditionally normal predictive distributions.
Author(s)
Philipp Adämmer, Sven Lehmann, Rainer Schüssler
References
Beckmann, J., Koop, G., Korobilis, D., and Schüssler, R. A. (2020) "Exchange rate predictability and dynamic bayesian learning." Journal of Applied Econometrics, 35 (4): 410–421.
Dangl, T. and Halling, M. (2012) "Predictive regressions with time-varying coefficients." Journal of Financial Economics, 106 (1): 157–181.
Koop, G. and Korobilis, D. (2012) "Forecasting inflation using dynamic model averaging." International Economic Review, 53 (3): 867–886.
Koop, G. and Korobilis, D. (2023) "Bayesian dynamic variable selection in high dimensions." International Economic Review.
Raftery, A. E., Kárn'y, M., and Ettler, P. (2010) "Online prediction under model uncertainty via dynamic model averaging: Application to a cold rolling mill." Technometrics, 52 (1): 52–66.
West, M. and Harrison, J. (1997) "Bayesian forecasting and dynamic models" Springer, 2nd edn.
See Also
https://github.com/lehmasve/hdflex#readme
Examples
#########################################################
######### Forecasting quarterly U.S. inflation ##########
#### Please see Koop & Korobilis (2023) for further ####
#### details regarding the data & external forecasts ####
#########################################################
# Packages
library("hdflex")
########## Get Data ##########
# Load Data
inflation_data <- inflation_data
benchmark_ar2 <- benchmark_ar2
# Set Index for Target Variable
i <- 1 # (1 -> GDPCTPI; 2 -> PCECTPI; 3 -> CPIAUCSL; 4 -> CPILFESL)
# Subset Data (keep only data relevant for target variable i)
dataset <- inflation_data[, c(1+(i-1), # Target Variable
5+(i-1), # Lag 1
9+(i-1), # Lag 2
(13:16)[-i], # Remaining Price Series
17:452, # Exogenous Predictor Variables
seq(453+(i-1)*16,468+(i-1)*16))] # Ext. Point Forecasts
########## STSC ##########
### Part 1: TV-C Model ###
# Set Target Variable
y <- dataset[, 1, drop = FALSE]
# Set 'Simple' Signals
X <- dataset[, 2:442, drop = FALSE]
# Set External Point Forecasts (Koop & Korobilis 2023)
Ext_F <- dataset[, 443:458, drop = FALSE]
# Set TV-C-Parameter
sample_length <- 4 * 5
lambda_grid <- c(0.90, 0.95, 1)
kappa_grid <- 0.98
n_cores <- 1
# Apply TV-C-Function
results <- hdflex::tvc(y,
X,
Ext_F,
lambda_grid,
kappa_grid,
sample_length,
n_cores)
# Assign TV-C-Results
forecast_tvc <- results[[1]]
variance_tvc <- results[[2]]
# Define Burn-In Period
sample_period_idx <- 80:nrow(dataset)
sub_forecast_tvc <- forecast_tvc[sample_period_idx, , drop = FALSE]
sub_variance_tvc <- variance_tvc[sample_period_idx, , drop = FALSE]
sub_y <- y[sample_period_idx, , drop = FALSE]
sub_dates <- rownames(dataset)[sample_period_idx]
### Part 2: Dynamic Subset Combination ###
# Set DSC-Parameter
nr_mods <- ncol(sub_forecast_tvc)
gamma_grid <- c(0.40, 0.50, 0.60, 0.70, 0.80, 0.90,
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00)
psi_grid <- c(1:100)
delta <- 0.95
n_cores <- 1
# Apply DSC-Function
results <- hdflex::dsc(gamma_grid,
psi_grid,
sub_y,
sub_forecast_tvc,
sub_variance_tvc,
delta,
n_cores)
# Assign DSC-Results
sub_forecast_stsc <- results[[1]]
sub_variance_stsc <- results[[2]]
sub_chosen_gamma <- results[[3]]
sub_chosen_psi <- results[[4]]
sub_chosen_signals <- results[[5]]
# Define Evaluation Period
eval_date_start <- "1991-01-01"
eval_date_end <- "2021-12-31"
eval_period_idx <- which(sub_dates > eval_date_start & sub_dates <= eval_date_end)
# Trim Objects
oos_y <- sub_y[eval_period_idx, ]
oos_forecast_stsc <- sub_forecast_stsc[eval_period_idx]
oos_variance_stsc <- sub_variance_stsc[eval_period_idx]
oos_chosen_gamma <- sub_chosen_gamma[eval_period_idx]
oos_chosen_psi <- sub_chosen_psi[eval_period_idx]
oos_chosen_signals <- sub_chosen_signals[eval_period_idx, , drop = FALSE]
oos_dates <- sub_dates[eval_period_idx]
# Add Dates
names(oos_forecast_stsc) <- oos_dates
names(oos_variance_stsc) <- oos_dates
names(oos_chosen_gamma) <- oos_dates
names(oos_chosen_psi) <- oos_dates
rownames(oos_chosen_signals) <- oos_dates
### Part 3: Evaluation ###
# Apply Summary-Function
summary_results <- summary_stsc(oos_y,
benchmark_ar2[, i],
oos_forecast_stsc)
# Assign Summary-Results
cssed <- summary_results[[3]]
mse <- summary_results[[4]]
########## Results ##########
# Relative MSE
print(paste("Relative MSE:", round(mse[[1]] / mse[[2]], 4)))
# Plot CSSED
plot(x = as.Date(oos_dates),
y = cssed,
ylim = c(-0.0008, 0.0008),
main = "Cumulated squared error differences",
type = "l",
lwd = 1.5,
xlab = "Date",
ylab = "CSSED") + abline(h = 0, lty = 2, col = "darkgray")
# Plot Predictive Signals
vec <- seq_len(dim(oos_chosen_signals)[2])
mat <- oos_chosen_signals %*% diag(vec)
mat[mat == 0] <- NA
matplot(x = as.Date(oos_dates),
y = mat,
cex = 0.4,
pch = 20,
type = "p",
main = "Evolution of selected signal(s)",
xlab = "Date",
ylab = "Predictive Signal")
# Plot Psi
plot(x = as.Date(oos_dates),
y = oos_chosen_psi,
ylim = c(1, 100),
main = "Evolution of the subset size",
type = "p",
cex = 0.75,
pch = 20,
xlab = "Date",
ylab = "Psi")