This package contains the forecasting algorithm developed by Adämmer, Lehmann and Schüssler (2023). When using the package, please remember to cite the paper.
The package comprises three functions:
stsc() can be used to directly apply the âSignal-Transformed-Subset-Combinationâ forecasting algorithm described in Adämmer, Lehmann and Schüssler (2023).
tvc() can be used to transform predictive signals into univariate density forecasts via time-varying coefficient models, where each model generates a conditionally gaussian predictive density for each signal at each point in time (first part of the STSC algorithm).
dsc() can be used to dynamically select a subset of candidate forecast models for each period based on their past performance in terms of density forecast accuracy (second part of the STSC algorithm).
You can install the released version of hdflex from CRAN:
install.packages("hdflex")You can install hdflex from GitHub:
# install.packages("devtools")
devtools::install_github("https://github.com/lehmasve/hdflex")The package compiles some C++ source code for installation, which is why you need the appropriate compilers:
On Windows you need Rtools available from CRAN.
On macOS you need Xcode and a Fortran compiler - for more details see Compiler.
Example using the stsc() function:
#########################################################
######### Forecasting quarterly U.S. inflation ##########
#### Please see Koop & Korobilis (2023) for further ####
#### details regarding the data & external forecasts ####
#########################################################
# Load Package
library("hdflex")
library("ggplot2")
library("cowplot")
########## Get Data ##########
# Load Package Data
inflation_data <- inflation_data
# Set Target Variable
y <- inflation_data[, 1]
# Set 'P-Signals'
X <- inflation_data[, 2:442]
# Set 'F-Signals'
Ext_F <- inflation_data[, 443:462]
# Get Dates and Number of Observations
tdates <- rownames(inflation_data)
tlength <- length(tdates)
# First complete observation (no missing values)
first_complete <- which(complete.cases(inflation_data))[1]
########## Rolling AR2-Benchmark ##########
# Set up matrix for predictions
benchmark <- matrix(NA, nrow = tlength,
ncol = 1, dimnames = list(tdates, "AR2"))
# Set Window-Size (15 years of quarterly data)
window_size <- 15 * 4
# Time Sequence
t_seq <- seq(window_size, tlength - 1)
# Loop with rolling window
for (t in t_seq) {
# Split Data for Training Train Data
x_train <- cbind(int = 1, X[(t - window_size + 1):t, 1:2])
y_train <- y[(t - window_size + 1):t]
# Split Data for Prediction
x_pred <- cbind(int = 1, X[t + 1, 1:2, drop = FALSE])
# Fit AR-Model
model_ar <- .lm.fit(x_train, y_train)
# Predict and store in benchmark matrix
benchmark[t + 1, ] <- x_pred %*% model_ar$coefficients
}
########## STSC ##########
# Set TV-C-Parameter
init <- 5 * 4
lambda_grid <- c(0.90, 0.95, 1.00)
kappa_grid <- c(0.94, 0.96, 0.98)
bias <- TRUE
# Set DSC-Parameter
gamma_grid <- c(0.40, 0.50, 0.60, 0.70, 0.80, 0.90,
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00)
n_tvc <- (ncol(X) + ncol(Ext_F)) * length(lambda_grid) * length(kappa_grid)
psi_grid <- c(1:100, sapply(1:4, function(i) floor(i * n_tvc / 4)))
delta <- 0.95
burn_in <- first_complete + init / 2
burn_in_dsc <- 1
metric <- 5
equal_weight <- TRUE
incl <- NULL
parallel <- FALSE
n_threads <- NULL
# Apply STSC-Function
results <- hdflex::stsc(y,
X,
Ext_F,
init,
lambda_grid,
kappa_grid,
bias,
gamma_grid,
psi_grid,
delta,
burn_in,
burn_in_dsc,
metric,
equal_weight,
incl,
parallel,
n_threads,
NULL)
########## Evaluation ##########
# Define Evaluation Period (OOS-Period)
eval_period <- which(tdates >= "1991-04-01" & tdates <= "2021-12-01")
# Apply Evaluation Summary for STSC
eval_results <- summary(obj = results, eval_period = eval_period)
# Calculate (Mean-)Squared-Errors for AR2-Benchmark
se_ar2 <- (y[eval_period] - benchmark[eval_period, 1])^2
mse_ar2 <- mean(se_ar2)
# Create Cumulative Squared Error Differences (CSSED) Plot
cssed <- cumsum(se_ar2 - eval_results$MSE[[2]])
plot_cssed <- ggplot(
data.frame(eval_period, cssed),
aes(x = eval_period, y = cssed)
) +
geom_line() +
ylim(-0.0008, 0.0008) +
ggtitle("Cumulative Squared Error Differences") +
xlab("Time Index") +
ylab("CSSED") +
geom_hline(yintercept = 0, linetype = "dashed", color = "darkgray") +
theme_minimal(base_size = 15) +
theme(
panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
panel.border = element_rect(colour = "black", fill = NA),
axis.ticks = element_line(colour = "black"),
plot.title = element_text(hjust = 0.5)
)
# Show Plots
options(repr.plot.width = 15, repr.plot.height = 15)
plots_list <- eval_results$Plots
plots_list <- c(list(plot_cssed), plots_list)
cowplot::plot_grid(plotlist = plots_list, ncol = 2, nrow = 3, align = "hv")
# Relative MSE
print(paste("Relative MSE:", round(eval_results$MSE[[1]] / mse_ar2, 4)))Same example using the tvc() and dsc() functions:
#########################################################
######### Forecasting quarterly U.S. inflation ##########
#### Please see Koop & Korobilis (2023) for further ####
#### details regarding the data & external forecasts ####
#########################################################
# Load Package
library("hdflex")
library("ggplot2")
library("cowplot")
########## Get Data ##########
# Load Package Data
inflation_data <- inflation_data
# Set Target Variable
y <- inflation_data[, 1]
# Set 'P-Signals'
X <- inflation_data[, 2:442]
# Set 'F-Signals'
Ext_F <- inflation_data[, 443:462]
# Get Dates and Number of Observations
tdates <- rownames(inflation_data)
tlength <- length(tdates)
# First complete observation (no missing values)
first_complete <- which(complete.cases(inflation_data))[1]
########## Rolling AR2-Benchmark ##########
# Set up matrix for predictions
benchmark <- matrix(NA, nrow = tlength,
ncol = 1, dimnames = list(tdates, "AR2"))
# Set Window-Size (15 years of quarterly data)
window_size <- 15 * 4
# Time Sequence
t_seq <- seq(window_size, tlength - 1)
# Loop with rolling window
for (t in t_seq) {
# Split Data for Training Train Data
x_train <- cbind(int = 1, X[(t - window_size + 1):t, 1:2])
y_train <- y[(t - window_size + 1):t]
# Split Data for Prediction
x_pred <- cbind(int = 1, X[t + 1, 1:2, drop = FALSE])
# Fit AR-Model
model_ar <- .lm.fit(x_train, y_train)
# Predict and store in benchmark matrix
benchmark[t + 1, ] <- x_pred %*% model_ar$coefficients
}
########## STSC ##########
### Part 1: TVC-Function
# Set TV-C-Parameter
init <- 5 * 4
lambda_grid <- c(0.90, 0.95, 1.00)
kappa_grid <- c(0.94, 0.96, 0.98)
bias <- TRUE
# Apply TVC-Function
tvc_results <- hdflex::tvc(y,
X,
Ext_F,
init,
lambda_grid,
kappa_grid,
bias)
# Assign TVC-Results
forecast_tvc <- tvc_results$Forecasts$Point_Forecasts
variance_tvc <- tvc_results$Forecasts$Variance_Forecasts
# First complete forecast period (no missing values)
sub_period <- seq(which(complete.cases(forecast_tvc))[1], tlength)
### Part 2: DSC-Function
# Set DSC-Parameter
gamma_grid <- c(0.40, 0.50, 0.60, 0.70, 0.80, 0.90,
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00)
psi_grid <- c(1:100, sapply(1:4, function(i) floor(i * ncol(forecast_tvc) / 4)))
delta <- 0.95
burn_in <- (init / 2) + 1
burn_in_dsc <- 1
metric <- 5
equal_weight <- TRUE
incl <- NULL
# Apply DSC-Function
dsc_results <- hdflex::dsc(y[sub_period],
forecast_tvc[sub_period, , drop = FALSE],
variance_tvc[sub_period, , drop = FALSE],
gamma_grid,
psi_grid,
delta,
burn_in,
burn_in_dsc,
metric,
equal_weight,
incl,
NULL)
# Assign DSC-Results
pred_stsc <- dsc_results$Forecasts$Point_Forecasts
var_stsc <- dsc_results$Forecasts$Variance_Forecasts
########## Evaluation ##########
# Define Evaluation Period (OOS-Period)
eval_period <- which(tdates[sub_period] >= "1991-04-01" & tdates[sub_period] <= "2021-12-01")
# Get Evaluation Summary for STSC
eval_results <- summary(obj = dsc_results, eval_period = eval_period)
# Calculate (Mean-)Squared-Errors for AR2-Benchmark
oos_y <- y[sub_period][eval_period]
oos_benchmark <- benchmark[sub_period[eval_period], , drop = FALSE]
se_ar2 <- (oos_y - oos_benchmark)^2
mse_ar2 <- mean(se_ar2)
# Create Cumulative Squared Error Differences (CSSED) Plot
cssed <- cumsum(se_ar2 - eval_results$MSE[[2]])
plot_cssed <- ggplot(
data.frame(eval_period, cssed),
aes(x = eval_period, y = cssed)
) +
geom_line() +
ylim(-0.0008, 0.0008) +
ggtitle("Cumulative Squared Error Differences") +
xlab("Time Index") +
ylab("CSSED") +
geom_hline(yintercept = 0, linetype = "dashed", color = "darkgray") +
theme_minimal(base_size = 15) +
theme(
panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
panel.border = element_rect(colour = "black", fill = NA),
axis.ticks = element_line(colour = "black"),
plot.title = element_text(hjust = 0.5)
)
# Show Plots
options(repr.plot.width = 15, repr.plot.height = 15)
plots_list <- eval_results$Plots
plots_list <- c(list(plot_cssed), plots_list)
cowplot::plot_grid(plotlist = plots_list, ncol = 2, nrow = 3, align = "hv")
# Relative MSE
print(paste("Relative MSE:", round(eval_results$MSE[[1]] / mse_ar2, 4)))Philipp Adämmer, Sven Lehmann and Rainer Schüssler
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