| stsc {hdflex} | R Documentation |
Signal-Transform-Subset-Combination (STSC)
Description
'stsc()' is a time series forecasting method designed to handle vast sets of predictive signals, many of which are irrelevant or short-lived. The method transforms heterogeneous scalar-valued signals into candidate density forecasts via time-varying coefficient models (TV-C), and subsequently, combines them into a final density forecast via dynamic subset combination (DSC).
Usage
stsc(
y,
X,
Ext_F,
sample_length,
lambda_grid,
kappa_grid,
burn_in_tvc,
gamma_grid,
psi_grid,
delta,
burn_in_dsc,
method,
equal_weight,
risk_aversion = NULL,
min_weight = NULL,
max_weight = NULL
)
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. |
sample_length |
An integer that denotes the number of observations used to initialize the observational variance and the coefficients' variance in the TV-C models. |
lambda_grid |
A numeric vector with values between 0 and 1 denoting the discount factor(s) that control the dynamics of the time-varying 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 between 0 and 1 to accommodate time-varying volatility in the TV-C models. 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. |
burn_in_tvc |
An integer value '>= 1' that denotes the number of observations used to 'initialize' the TV-C models. After 'burn_in_tvc' observations the generated sum of discounted predictive log-likelihoods (DPLLs) of each Candidate Model (TV-C model) and Subset Combination (combination of gamma and psi) is resetted. 'burn_in_tvc = 1' means no burn-in period is applied. |
gamma_grid |
A numerical vector that contains discount factors between 0 and 1 to exponentially down-weight the past predictive performance of the candidate forecasts. |
psi_grid |
An integer vector that controls the (possible) sizes of the active subsets. |
delta |
A numeric value between 0 and 1 denoting the discount factor used to down-weight the past predictive performance of the subset combinations. |
burn_in_dsc |
An integer value '>= 1' that denotes the number of observations used to 'initialize' the Dynamic Subset Combinations. After 'burn_in_dsc' observations the generated sum of discounted predictive log-likelihoods (DPLLs) of each Subset Combination (combination of gamma and psi) is resetted. 'burn_in_dsc = 1' means no burn-in period is applied. |
method |
An integer of the set '1, 2, 3, 4' that denotes the method used to rank the Candidate Models (TV-C models) and Subset Combinations according to their performance. Default is 'method = 1' which ranks according to their generated sum of discounted predictive log-likelihoods (DPLLs). 'method = 2' uses Squared-Error (SE) instead of DPLLs. 'method = 3' uses Absolute-Error (AE) and 'method = 4' uses Compounded-Returns (in this case the target variable y has to be a time series of financial returns). |
equal_weight |
A boolean that denotes whether equal weights are used to combine the candidate forecasts within a subset. If 'FALSE', the weights are calculated using the softmax-function on the predictive log-scores of the candidate models. The method proposed in Adaemmer et al (2023) uses equal weights to combine the candidate forecasts. |
risk_aversion |
A double '>= 0' that denotes the risk aversion of an investor. A higher value indicates a risk avoiding behaviour. |
min_weight |
A double that denotes the lower bound for the weight placed on the market. A non-negative value rules out short sales. |
max_weight |
A double that denotes the upper bound for the weight placed on the market. A value of e.g. 2 allows for a maximum leverage ratio of two. |
Value
A list that contains: * (1) a vector with the first moments (point forecasts) of the STSC-Model, * (2) a vector with the second moments (variance) of the STSC-Model, * (3) a vector that contains the selected values for gamma, * (4) a vector that contains the selected values for psi and * (5) a matrix that indicates the selected signals for every point in time.
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.
Del Negro, M., Hasegawa, R. B., and Schorfheide, F. (2016) "Dynamic prediction pools: An investigation of financial frictions and forecasting performance." Journal of Econometrics, 192 (2): 391–405.
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 ##########
# 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 Dates
dates <- rownames(dataset)
# Set TV-C-Parameter
sample_length <- 4 * 5
lambda_grid <- c(0.90, 0.95, 1)
kappa_grid <- 0.98
# 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)
delta <- 0.95
# Apply STSC-Function
results <- hdflex::stsc(y,
X,
Ext_F,
sample_length,
lambda_grid,
kappa_grid,
burn_in_tvc = 79,
gamma_grid,
psi_grid,
delta,
burn_in_dsc = 1,
method = 1,
equal_weight = TRUE,
risk_aversion = NULL,
min_weight = NULL,
max_weight = NULL)
# Assign DSC-Results
forecast_stsc <- results[[1]]
variance_stsc <- results[[2]]
chosen_gamma <- results[[3]]
chosen_psi <- results[[4]]
chosen_signals <- results[[5]]
# Define Evaluation Period
eval_date_start <- "1991-01-01"
eval_date_end <- "2021-12-31"
eval_period_idx <- which(dates > eval_date_start & dates <= eval_date_end)
# Trim Objects
oos_y <- y[eval_period_idx, ]
oos_forecast_stsc <- forecast_stsc[eval_period_idx]
oos_variance_stsc <- variance_stsc[eval_period_idx]
oos_chosen_gamma <- chosen_gamma[eval_period_idx]
oos_chosen_psi <- chosen_psi[eval_period_idx]
oos_chosen_signals <- chosen_signals[eval_period_idx, , drop = FALSE]
oos_dates <- 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 2: 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")